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annotate texts/XML/archimedes/el/eucli_eleme_105_el.xml @ 6:22d6a63640c6
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| date | Fri, 07 Dec 2012 17:05:22 +0100 |
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1 <?xml version="1.0"?> |
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2 <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <info> |
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3 <author>Euclides</author> |
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4 <title>Elementa</title> |
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5 <date>1969-1916</date> |
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6 <place>Leipzig</place> |
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7 <translator></translator> |
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8 <lang>el</lang> |
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9 <cvs_file></cvs_file> |
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10 <cvs_version></cvs_version> |
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11 <locator>105</locator> |
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12 </info> <text> <front></front><body> |
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13 <pb></pb> |
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14 |
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15 <chap><p type="main"><s>Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.</s> <lb n="1799,001,,,,1;HOR,1,1"></lb> |
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16 <s> |
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17 Γραμμὴ δὲ μῆκος ἀπλατές.</s> <lb n="1799,001,,,,1;HOR,2,1"></lb> |
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18 <s>Γραμμῆς δὲ πέρατα σημεῖα.</s> <lb n="1799,001,,,,1;HOR,3,1"></lb> |
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19 <s>Εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ' ἑαυτῆς <lb n="1799,001,,,,1;HOR,4,1"></lb> |
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20 σημείοις κεῖται.</s> <lb n="1799,001,,,,1;HOR,4,2"></lb> |
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21 <s>Ἐπιφάνεια δέ ἐστιν, ὃ μῆκος καὶ πλάτος μόνον ἔχει.</s> <lb n="1799,001,,,,1;HOR,5,1"></lb> |
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22 <s>Ἐπιφανείας δὲ πέρατα γραμμαί.</s> <lb n="1799,001,,,,1;HOR,6,1"></lb> |
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23 <s>Ἐπίπεδος ἐπιφάνειά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐφ' <lb n="1799,001,,,,1;HOR,7,1"></lb> |
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24 ἑαυτῆς εὐθείαις κεῖται.</s> <lb n="1799,001,,,,1;HOR,7,2"></lb> |
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25 <s>Ἐπίπεδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν <lb n="1799,001,,,,1;HOR,8,1"></lb> |
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26 ἁπτομένων ἀλλήλων καὶ μὴ ἐπ' εὐθείας κειμένων πρὸς <lb n="1799,001,,,,1;HOR,8,2"></lb> |
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27 ἀλλήλας τῶν γραμμῶν κλίσις.</s> <lb n="1799,001,,,,1;HOR,8,3"></lb> |
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28 <s>Ὅταν δὲ αἱ περιέχουσαι τὴν γωνίαν γραμμαὶ εὐθεῖαι <lb n="1799,001,,,,1;HOR,9,1"></lb> |
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29 ὦσιν, εὐθύγραμμος καλεῖται ἡ γωνία.</s> <s><pb></pb> <lb n="1799,001,,,,1;HOR,9,2"></lb> |
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30 Ὅταν δὲ εὐθεῖα ἐπ' εὐθεῖαν σταθεῖσα τὰς ἐφεξῆς <lb n="1799,001,,,,1;HOR,10,1"></lb> |
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31 γωνίας ἴσας ἀλλήλαις ποιῇ, ὀρθὴ ἑκατέρα τῶν ἴσων γω-<lb n="1799,001,,,,1;HOR,10,2"></lb> |
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32 νιῶν ἐστι, καὶ ἡ ἐφεστηκυῖα εὐθεῖα κάθετος καλεῖται, <lb n="1799,001,,,,1;HOR,10,3"></lb> |
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33 ἐφ' ἣν ἐφέστηκεν.</s> <lb n="1799,001,,,,1;HOR,10,4"></lb> |
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34 <s>Ἀμβλεῖα γωνία ἐστὶν ἡ μείζων ὀρθῆς.</s> <lb n="1799,001,,,,1;HOR,11,1"></lb> |
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35 <s>Ὀξεῖα δὲ ἡ ἐλάσσων ὀρθῆς.</s> <lb n="1799,001,,,,1;HOR,12,1"></lb> |
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36 <s>Ὅρος ἐστίν, ὅ τινός ἐστι πέρας.</s> <lb n="1799,001,,,,1;HOR,13,1"></lb> |
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37 <s>Σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον.</s> <lb n="1799,001,,,,1;HOR,14,1"></lb> |
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38 <s>Κύκλος ἐστὶ σχῆμα ἐπίπεδον ὑπὸ μιᾶς γραμμῆς <lb n="1799,001,,,,1;HOR,15,1"></lb> |
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39 περιεχόμενον [ἣ καλεῖται περιφέρεια], πρὸς ἣν ἀφ' ἑνὸς <lb n="1799,001,,,,1;HOR,15,2"></lb> |
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40 σημείου τῶν ἐντὸς τοῦ σχήματος κειμένων πᾶσαι αἱ <lb n="1799,001,,,,1;HOR,15,3"></lb> |
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41 προσπίπτουσαι εὐθεῖαι [πρὸς τὴν τοῦ κύκλου περι-<lb n="1799,001,,,,1;HOR,15,4"></lb> |
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42 φέρειαν] ἴσαι ἀλλήλαις εἰσίν.</s> <lb n="1799,001,,,,1;HOR,15,5"></lb> |
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43 <s>Κέντρον δὲ τοῦ κύκλου τὸ σημεῖον καλεῖται.</s> <s><pb></pb> <lb n="1799,001,,,,1;HOR,16,1"></lb> |
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44 Διάμετρος δὲ τοῦ κύκλου ἐστὶν εὐθεῖά τις διὰ τοῦ <lb n="1799,001,,,,1;HOR,17,1"></lb> |
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45 κέντρου ἠγμένη καὶ περατουμένη ἐφ' ἑκάτερα τὰ μέρη <lb n="1799,001,,,,1;HOR,17,2"></lb> |
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46 ὑπὸ τῆς τοῦ κύκλου περιφερείας, ἥτις καὶ δίχα τέμνει τὸν <lb n="1799,001,,,,1;HOR,17,3"></lb> |
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47 κύκλον.</s> <lb n="1799,001,,,,1;HOR,17,4"></lb> |
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48 <s>Ἡμικύκλιον δέ ἐστι τὸ περιεχόμενον σχῆμα ὑπό τε <lb n="1799,001,,,,1;HOR,18,1"></lb> |
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49 τῆς διαμέτρου καὶ τῆς ἀπολαμβανομένης ὑπ' αὐτῆς περι-<lb n="1799,001,,,,1;HOR,18,2"></lb> |
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50 φερείας.</s> <s>κέντρον δὲ τοῦ ἡμικυκλίου τὸ αὐτό, ὃ καὶ τοῦ <lb n="1799,001,,,,1;HOR,18,3"></lb> |
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51 κύκλου ἐστίν.</s> <lb n="1799,001,,,,1;HOR,18,4"></lb> |
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52 <s>Σχήματα εὐθύγραμμά ἐστι τὰ ὑπὸ εὐθειῶν περιεχό-<lb n="1799,001,,,,1;HOR,19,1"></lb> |
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53 μενα, τρίπλευρα μὲν τὰ ὑπὸ τριῶν, τετράπλευρα δὲ τὰ <lb n="1799,001,,,,1;HOR,19,2"></lb> |
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54 ὑπὸ τεσσάρων, πολύπλευρα δὲ τὰ ὑπὸ πλειόνων ἢ τεσσά-<lb n="1799,001,,,,1;HOR,19,3"></lb> |
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55 ρων εὐθειῶν περιεχόμενα.</s> <lb n="1799,001,,,,1;HOR,19,4"></lb> |
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56 <s>Τῶν δὲ τριπλεύρων σχημάτων ἰσόπλευρον μὲν τρί-<lb n="1799,001,,,,1;HOR,20,1"></lb> |
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57 γωνόν ἐστι τὸ τὰς τρεῖς ἴσας ἔχον πλευράς, ἰσοσκελὲς δὲ <lb n="1799,001,,,,1;HOR,20,2"></lb> |
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58 τὸ τὰς δύο μόνας ἴσας ἔχον πλευράς, σκαληνὸν δὲ τὸ τὰς <lb n="1799,001,,,,1;HOR,20,3"></lb> |
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59 τρεῖς ἀνίσους ἔχον πλευράς.</s> <lb n="1799,001,,,,1;HOR,20,4"></lb> |
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60 <s>Ἔτι δὲ τῶν τριπλεύρων σχημάτων ὀρθογώνιον μὲν <lb n="1799,001,,,,1;HOR,21,1"></lb> |
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61 τρίγωνόν ἐστι τὸ ἔχον ὀρθὴν γωνίαν, ἀμβλυγώνιον δὲ τὸ <pb></pb> <lb n="1799,001,,,,1;HOR,21,2"></lb> |
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62 ἔχον ἀμβλεῖαν γωνίαν, ὀξυγώνιον δὲ τὸ τὰς τρεῖς ὀξείας <lb n="1799,001,,,,1;HOR,21,3"></lb> |
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63 ἔχον γωνίας.</s> <lb n="1799,001,,,,1;HOR,21,4"></lb> |
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64 <s>Τῶν δὲ τετραπλεύρων σχημάτων τετράγωνον μέν <lb n="1799,001,,,,1;HOR,22,1"></lb> |
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65 ἐστιν, ὃ ἰσόπλευρόν τέ ἐστι καὶ ὀρθογώνιον, ἑτερόμηκες <lb n="1799,001,,,,1;HOR,22,2"></lb> |
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66 δέ, ὃ ὀρθογώνιον μέν, οὐκ ἰσόπλευρον δέ, ῥόμβος δέ, ὃ <lb n="1799,001,,,,1;HOR,22,3"></lb> |
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67 ἰσόπλευρον μέν, οὐκ ὀρθογώνιον δέ, ῥομβοειδὲς δὲ τὸ τὰς <lb n="1799,001,,,,1;HOR,22,4"></lb> |
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68 ἀπεναντίον πλευράς τε καὶ γωνίας ἴσας ἀλλήλαις ἔχον, <lb n="1799,001,,,,1;HOR,22,5"></lb> |
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69 ὃ οὔτε ἰσόπλευρόν ἐστιν οὔτε ὀρθογώνιον· τὰ δὲ παρὰ <lb n="1799,001,,,,1;HOR,22,6"></lb> |
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70 ταῦτα τετράπλευρα τραπέζια καλείσθω.</s> <lb n="1799,001,,,,1;HOR,22,7"></lb> |
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71 <s>Παράλληλοί εἰσιν εὐθεῖαι, αἵτινες ἐν τῷ αὐτῷ ἐπι-<lb n="1799,001,,,,1;HOR,23,1"></lb> |
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72 πέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ' ἑκάτερα τὰ <lb n="1799,001,,,,1;HOR,23,2"></lb> |
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73 μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις.</s> <lb n="1799,001,,,,1;HOR,23,3"></lb> |
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74 <s></s></p> |
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75 </chap> |
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76 <chap><p type="main"><s>Ἠιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον <lb n="1799,001,,,,1;AI,1,1"></lb> |
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77 |
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78 εὐθεῖαν γραμμὴν ἀγαγεῖν.</s> <lb n="1799,001,,,,1;AI,1,2"></lb> |
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79 <s>Καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ' εὐ-<lb n="1799,001,,,,1;AI,2,1"></lb> |
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80 θείας ἐκβαλεῖν.</s> <s><pb></pb> <lb n="1799,001,,,,1;AI,2,2"></lb> |
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81 Καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γρά-<lb n="1799,001,,,,1;AI,3,1"></lb> |
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82 φεσθαι.</s> <lb n="1799,001,,,,1;AI,3,2"></lb> |
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83 <s>Καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις εἶναι.</s> <lb n="1799,001,,,,1;AI,4,1"></lb> |
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84 <s>Καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς <lb n="1799,001,,,,1;AI,5,1"></lb> |
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85 ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας <lb n="1799,001,,,,1;AI,5,2"></lb> |
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86 ποιῇ, ἐκβαλλομένας τὰς δύο εὐθείας ἐπ' ἄπειρον συμπί-<lb n="1799,001,,,,1;AI,5,3"></lb> |
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87 πτειν, ἐφ' ἃ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες.</s> <lb n="1799,001,,,,1;AI,5,4"></lb> |
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88 <s></s></p> |
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89 </chap> |
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90 <chap><p type="main"><s>Τὰ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα.</s> <lb n="1799,001,,,,1;KE,1,1"></lb> |
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91 <s> |
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92 Καὶ ἐὰν ἴσοις ἴσα προστεθῇ, τὰ ὅλα ἐστὶν ἴσα.</s> <lb n="1799,001,,,,1;KE,2,1"></lb> |
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93 <s>Καὶ ἐὰν ἀπὸ ἴσων ἴσα ἀφαιρεθῇ, τὰ καταλειπόμενά <lb n="1799,001,,,,1;KE,3,1"></lb> |
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94 ἐστιν ἴσα.</s> <lb n="1799,001,,,,1;KE,3,2"></lb> |
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95 <s>[Καὶ ἐὰν ἀνίσοις ἴσα προστεθῇ, τὰ ὅλα ἐστὶν ἄνισα.</s> <s><pb></pb> <lb n="1799,001,,,,1;KE,4,1"></lb> |
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96 Καὶ τὰ τοῦ αὐτοῦ διπλάσια ἴσα ἀλλήλοις ἐστίν.</s> <lb n="1799,001,,,,1;KE,5,1"></lb> |
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97 <s>Καὶ τὰ τοῦ αὐτοῦ ἡμίση ἴσα ἀλλήλοις ἐστίν.] <lb n="1799,001,,,,1;KE,6,1"></lb> |
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98 Καὶ τὰ ἐφαρμόζοντα ἐπ' ἄλληλα ἴσα ἀλλήλοις ἐστίν.</s> <lb n="1799,001,,,,1;KE,7,1"></lb> |
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99 <s>Καὶ τὸ ὅλον τοῦ μέρους μεῖζον [ἐστιν].</s> <lb n="1799,001,,,,1;KE,8,1"></lb> |
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100 <s>Καὶ δύο εὐθεῖαι χωρίον οὐ περιέχουσιν.</s> <s><pb></pb> <lb n="1799,001,,,,1;KE,9,1"></lb> |
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101 </s></p> |
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102 </chap> |
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103 <chap><p type="main"><s>Ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τρίγωνον ἰσό-<lb n="1799,001,,,,1,1,1"></lb> |
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104 |
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105 πλευρον συστήσασθαι.</s> <lb n="1799,001,,,,1,1,2"></lb> |
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106 <s>Ἔστω ἡ δοθεῖσα εὐθεῖα πεπερασμένη ἡ ΑΒ.</s> <lb n="1799,001,,,,1,1,3"></lb> |
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107 <s>Δεῖ δὴ ἐπὶ τῆς ΑΒ εὐθείας τρίγωνον ἰσόπλευρον συστή-<lb n="1799,001,,,,1,1,4"></lb> |
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108 σασθαι.</s> <lb n="1799,001,,,,1,1,5"></lb> |
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109 <s>Κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγρά-<lb n="1799,001,,,,1,1,6"></lb> |
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110 φθω ὁ ΒΓΔ, καὶ πάλιν κέντρῳ μὲν τῷ Β διαστήματι δὲ <lb n="1799,001,,,,1,1,7"></lb> |
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111 τῷ ΒΑ κύκλος γεγράφθω ὁ ΑΓΕ, καὶ ἀπὸ τοῦ Γ ση-<lb n="1799,001,,,,1,1,8"></lb> |
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112 μείου, καθ' ὃ τέμνουσιν ἀλλή- <lb n="1799,001,,,,1,1,9"></lb> |
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113 λους οἱ κύκλοι, ἐπὶ τὰ Α, Β <lb n="1799,001,,,,1,1,10"></lb> |
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114 σημεῖα ἐπεζεύχθωσαν εὐθεῖαι <lb n="1799,001,,,,1,1,11"></lb> |
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115 αἱ ΓΑ, ΓΒ.</s> <lb n="1799,001,,,,1,1,12"></lb> |
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116 <s>Καὶ ἐπεὶ τὸ Α σημεῖον κέν-<lb n="1799,001,,,,1,1,13"></lb> |
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117 τρον ἐστὶ τοῦ ΓΔΒ κύκλου, <lb n="1799,001,,,,1,1,14"></lb> |
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118 ἴση ἐστὶν ἡ ΑΓ τῇ ΑΒ· πάλιν, <lb n="1799,001,,,,1,1,15"></lb> |
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119 ἐπεὶ τὸ Β σημεῖον κέντρον <pb></pb> <lb n="1799,001,,,,1,1,16"></lb> |
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120 ἐστὶ τοῦ ΓΑΕ κύκλου, ἴση ἐστὶν ἡ ΒΓ τῇ ΒΑ.</s> <s>ἐδείχθη δὲ <lb n="1799,001,,,,1,1,17"></lb> |
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121 καὶ ἡ ΓΑ τῇ ΑΒ ἴση· ἑκατέρα ἄρα τῶν ΓΑ, ΓΒ τῇ ΑΒ <lb n="1799,001,,,,1,1,18"></lb> |
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122 ἐστὶν ἴση.</s> <s>τὰ δὲ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα· καὶ ἡ <lb n="1799,001,,,,1,1,19"></lb> |
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123 ΓΑ ἄρα τῇ ΓΒ ἐστὶν ἴση· αἱ τρεῖς ἄρα αἱ ΓΑ, ΑΒ, ΒΓ <lb n="1799,001,,,,1,1,20"></lb> |
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124 ἴσαι ἀλλήλαις εἰσίν.</s> <lb n="1799,001,,,,1,1,21"></lb> |
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125 <s>Ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον, καὶ συνέσταται <lb n="1799,001,,,,1,1,22"></lb> |
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126 ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τῆς ΑΒ.</s> <lb n="1799,001,,,,1,1,23"></lb> |
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127 <s>[Ἐπὶ τῆς δοθείσης ἄρα εὐθείας πεπερασμένης τρίγωνον <lb n="1799,001,,,,1,1,24"></lb> |
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128 ἰσόπλευρον συνέσταται]· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,1,25"></lb> |
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129 <s>Πρὸς τῷ δοθέντι σημείῳ τῇ δοθείσῃ εὐθείᾳ ἴσην εὐθεῖαν <lb n="1799,001,,,,1,2,1"></lb> |
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130 θέσθαι.</s> <lb n="1799,001,,,,1,2,2"></lb> |
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131 <s>Ἔστω τὸ μὲν δοθὲν σημεῖον τὸ Α, ἡ δὲ δοθεῖσα εὐθεῖα <lb n="1799,001,,,,1,2,3"></lb> |
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132 ἡ ΒΓ· δεῖ δὴ πρὸς τῷ Α σημείῳ τῇ δοθείσῃ εὐθείᾳ τῇ ΒΓ <lb n="1799,001,,,,1,2,4"></lb> |
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133 ἴσην εὐθεῖαν θέσθαι.</s> <lb n="1799,001,,,,1,2,5"></lb> |
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134 <s>Ἐπεζεύχθω γὰρ ἀπὸ τοῦ Α σημείου ἐπὶ τὸ Β σημεῖον <lb n="1799,001,,,,1,2,6"></lb> |
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135 εὐθεῖα ἡ ΑΒ, καὶ συνεστάτω ἐπ' αὐτῆς τρίγωνον ἰσόπλευ-<lb n="1799,001,,,,1,2,7"></lb> |
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136 ρον τὸ ΔΑΒ, καὶ ἐκβεβλήσθωσαν ἐπ' εὐθείας ταῖς ΔΑ, <lb n="1799,001,,,,1,2,8"></lb> |
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137 ΔΒ εὐθεῖαι αἱ ΑΕ, ΒΖ, καὶ κέντρῳ μὲν τῷ Β διαστήματι <lb n="1799,001,,,,1,2,9"></lb> |
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138 δὲ τῷ ΒΓ κύκλος γεγράφθω ὁ ΓΗΘ, καὶ πάλιν κέντρῳ <lb n="1799,001,,,,1,2,10"></lb> |
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139 τῷ Δ καὶ διαστήματι τῷ ΔΗ κύκλος γεγράφθω ὁ ΗΚΛ.</s> <s><pb></pb> <lb n="1799,001,,,,1,2,11"></lb> |
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140 Ἐπεὶ οὖν τὸ Β σημεῖον κέν- <lb n="1799,001,,,,1,2,12"></lb> |
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141 τρον ἐστὶ τοῦ ΓΗΘ κύκλου, ἴση <lb n="1799,001,,,,1,2,13"></lb> |
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142 ἐστὶν ἡ ΒΓ τῇ ΒΗ.</s> <s>πάλιν, ἐπεὶ <lb n="1799,001,,,,1,2,14"></lb> |
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143 τὸ Δ σημεῖον κέντρον ἐστὶ <lb n="1799,001,,,,1,2,15"></lb> |
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144 τοῦ ΚΛΗ κύκλου, ἴση ἐστὶν <lb n="1799,001,,,,1,2,16"></lb> |
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145 ἡ ΔΛ τῇ ΔΗ, ὧν ἡ ΔΑ τῇ <lb n="1799,001,,,,1,2,17"></lb> |
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146 ΔΒ ἴση ἐστίν.</s> <s>λοιπὴ ἄρα ἡ <lb n="1799,001,,,,1,2,18"></lb> |
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147 ΑΛ λοιπῇ τῇ ΒΗ ἐστὶν ἴση.</s> <lb n="1799,001,,,,1,2,19"></lb> |
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148 <s>ἐδείχθη δὲ καὶ ἡ ΒΓ τῇ ΒΗ <lb n="1799,001,,,,1,2,20"></lb> |
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149 ἴση· ἑκατέρα ἄρα τῶν ΑΛ, <lb n="1799,001,,,,1,2,21"></lb> |
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150 ΒΓ τῇ ΒΗ ἐστὶν ἴση.</s> <s>τὰ δὲ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις <lb n="1799,001,,,,1,2,22"></lb> |
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151 ἐστὶν ἴσα· καὶ ἡ ΑΛ ἄρα τῇ ΒΓ ἐστὶν ἴση.</s> <lb n="1799,001,,,,1,2,23"></lb> |
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152 <s>Πρὸς ἄρα τῷ δοθέντι σημείῳ τῷ Α τῇ δοθείσῃ εὐθείᾳ <lb n="1799,001,,,,1,2,24"></lb> |
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153 τῇ ΒΓ ἴση εὐθεῖα κεῖται ἡ ΑΛ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,2,25"></lb> |
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154 <s>Δύο δοθεισῶν εὐθειῶν ἀνίσων ἀπὸ τῆς μείζονος τῇ <lb n="1799,001,,,,1,3,1"></lb> |
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155 ἐλάσσονι ἴσην εὐθεῖαν ἀφελεῖν.</s> <lb n="1799,001,,,,1,3,2"></lb> |
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156 <s>Ἔστωσαν αἱ δοθεῖσαι δύο εὐθεῖαι <lb n="1799,001,,,,1,3,3"></lb> |
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157 ἄνισοι αἱ ΑΒ, Γ, ὧν μείζων ἔστω ἡ <lb n="1799,001,,,,1,3,4"></lb> |
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158 ΑΒ· δεῖ δὴ ἀπὸ τῆς μείζονος τῆς ΑΒ <lb n="1799,001,,,,1,3,5"></lb> |
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159 τῇ ἐλάσσονι τῇ Γ ἴσην εὐθεῖαν ἀφελεῖν.</s> <lb n="1799,001,,,,1,3,6"></lb> |
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160 <s>Κείσθω πρὸς τῷ Α σημείῳ τῇ <lb n="1799,001,,,,1,3,7"></lb> |
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161 Γ εὐθείᾳ ἴση ἡ ΑΔ· καὶ κέντρῳ <pb></pb> <lb n="1799,001,,,,1,3,8"></lb> |
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162 μὲν τῷ Α διαστήματι δὲ τῷ ΑΔ κύκλος γεγράφθω <lb n="1799,001,,,,1,3,9"></lb> |
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163 ὁ ΔΕΖ.</s> <lb n="1799,001,,,,1,3,10"></lb> |
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164 <s>Καὶ ἐπεὶ τὸ Α σημεῖον κέντρον ἐστὶ τοῦ ΔΕΖ κύκλου, <lb n="1799,001,,,,1,3,11"></lb> |
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165 ἴση ἐστὶν ἡ ΑΕ τῇ ΑΔ· ἀλλὰ καὶ ἡ Γ τῇ ΑΔ ἐστιν ἴση.</s> <lb n="1799,001,,,,1,3,12"></lb> |
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166 <s>ἑκατέρα ἄρα τῶν ΑΕ, Γ τῇ ΑΔ ἐστιν ἴση· ὥστε καὶ ἡ <lb n="1799,001,,,,1,3,13"></lb> |
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167 ΑΕ τῇ Γ ἐστιν ἴση.</s> <lb n="1799,001,,,,1,3,14"></lb> |
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168 <s>Δύο ἄρα δοθεισῶν εὐθειῶν ἀνίσων τῶν ΑΒ, Γ ἀπὸ τῆς <lb n="1799,001,,,,1,3,15"></lb> |
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169 μείζονος τῆς ΑΒ τῇ ἐλάσσονι τῇ Γ ἴση ἀφῄρηται ἡ ΑΕ· <lb n="1799,001,,,,1,3,16"></lb> |
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170 ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,3,17"></lb> |
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171 <s>Ἐὰν δύο τρίγωνα τὰς δύο πλευρὰς [ταῖσ] δυσὶ πλευραῖς <lb n="1799,001,,,,1,4,1"></lb> |
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172 ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ καὶ τὴν γωνίαν τῇ γωνίᾳ ἴσην <lb n="1799,001,,,,1,4,2"></lb> |
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173 ἔχῃ τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην, καὶ τὴν βάσιν <lb n="1799,001,,,,1,4,3"></lb> |
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174 τῇ βάσει ἴσην ἕξει, καὶ τὸ τρίγωνον τῷ τριγώνῳ ἴσον <lb n="1799,001,,,,1,4,4"></lb> |
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175 ἔσται, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι ἔσον-<lb n="1799,001,,,,1,4,5"></lb> |
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176 ται ἑκατέρα ἑκατέρᾳ, ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑποτείνουσιν.</s> <lb n="1799,001,,,,1,4,6"></lb> |
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177 <s>Ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ <lb n="1799,001,,,,1,4,7"></lb> |
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178 τὰς δύο πλευρὰς τὰς ΑΒ, ΑΓ ταῖς δυσὶ <lb n="1799,001,,,,1,4,8"></lb> |
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179 πλευραῖς ταῖς ΔΕ, ΔΖ ἴσας ἔχοντα <lb n="1799,001,,,,1,4,9"></lb> |
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180 ἑκατέραν ἑκατέρᾳ τὴν μὲν ΑΒ τῇ ΔΕ <lb n="1799,001,,,,1,4,10"></lb> |
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181 τὴν δὲ ΑΓ τῇ ΔΖ καὶ γωνίαν τὴν ὑπὸ <lb n="1799,001,,,,1,4,11"></lb> |
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182 ΒΑΓ γωνίᾳ τῇ ὑπὸ ΕΔΖ ἴσην.</s> <s>λέγω, <lb n="1799,001,,,,1,4,12"></lb> |
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183 ὅτι καὶ βάσις ἡ ΒΓ βάσει τῇ ΕΖ ἴση <lb n="1799,001,,,,1,4,13"></lb> |
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184 ἐστίν, καὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ <lb n="1799,001,,,,1,4,14"></lb> |
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185 τριγώνῳ ἴσον ἔσται, καὶ αἱ λοιπαὶ <lb n="1799,001,,,,1,4,15"></lb> |
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186 γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι ἔσον- <pb></pb><lb n="1799,001,,,,1,4,16"></lb> |
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187 ται ἑκατέρα ἑκατέρᾳ, ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑποτείνου-<lb n="1799,001,,,,1,4,17"></lb> |
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188 σιν, ἡ μὲν ὑπὸ ΑΒΓ τῇ ὑπὸ ΔΕΖ, ἡ δὲ ὑπὸ Α*γΒ τῇ <lb n="1799,001,,,,1,4,18"></lb> |
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189 ὑπὸ ΔΖΕ.</s> <lb n="1799,001,,,,1,4,19"></lb> |
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190 <s>Ἐφαρμοζομένου γὰρ τοῦ ΑΒΓ τριγώνου ἐπὶ τὸ ΔΕΖ <lb n="1799,001,,,,1,4,20"></lb> |
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191 τρίγωνον καὶ τιθεμένου τοῦ μὲν Α σημείου ἐπὶ τὸ Δ ση-<lb n="1799,001,,,,1,4,21"></lb> |
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192 μεῖον τῆς δὲ ΑΒ εὐθείας ἐπὶ τὴν ΔΕ, ἐφαρμόσει καὶ τὸ <lb n="1799,001,,,,1,4,22"></lb> |
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193 Β σημεῖον ἐπὶ τὸ Ε διὰ τὸ ἴσην εἶναι τὴν ΑΒ τῇ ΔΕ· ἐφαρ-<lb n="1799,001,,,,1,4,23"></lb> |
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194 μοσάσης δὴ τῆς ΑΒ ἐπὶ τὴν ΔΕ ἐφαρμόσει καὶ ἡ ΑΓ <lb n="1799,001,,,,1,4,24"></lb> |
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195 εὐθεῖα ἐπὶ τὴν ΔΖ διὰ τὸ ἴσην εἶναι τὴν ὑπὸ ΒΑΓ γω-<lb n="1799,001,,,,1,4,25"></lb> |
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196 νίαν τῇ ὑπὸ ΕΔΖ· ὥστε καὶ τὸ Γ σημεῖον ἐπὶ τὸ Ζ ση-<lb n="1799,001,,,,1,4,26"></lb> |
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197 μεῖον ἐφαρμόσει διὰ τὸ ἴσην πάλιν εἶναι τὴν ΑΓ τῇ ΔΖ.</s> <lb n="1799,001,,,,1,4,27"></lb> |
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198 <s>ἀλλὰ μὴν καὶ τὸ Β ἐπὶ τὸ Ε ἐφηρμόκει· ὥστε βάσις ἡ ΒΓ <lb n="1799,001,,,,1,4,28"></lb> |
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199 ἐπὶ βάσιν τὴν ΕΖ ἐφαρμόσει.</s> <s>εἰ γὰρ τοῦ μὲν Β ἐπὶ τὸ Ε <lb n="1799,001,,,,1,4,29"></lb> |
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200 ἐφαρμόσαντος τοῦ δὲ Γ ἐπὶ τὸ Ζ ἡ ΒΓ βάσις ἐπὶ τὴν ΕΖ <lb n="1799,001,,,,1,4,30"></lb> |
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201 οὐκ ἐφαρμόσει, δύο εὐθεῖαι χωρίον περιέξουσιν· ὅπερ <lb n="1799,001,,,,1,4,31"></lb> |
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202 ἐστὶν ἀδύνατον.</s> <s>ἐφαρμόσει ἄρα ἡ ΒΓ βάσις ἐπὶ τὴν ΕΖ <lb n="1799,001,,,,1,4,32"></lb> |
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203 καὶ ἴση αὐτῇ ἔσται· ὥστε καὶ ὅλον τὸ ΑΒΓ τρίγωνον ἐπὶ <lb n="1799,001,,,,1,4,33"></lb> |
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204 ὅλον τὸ ΔΕΖ τρίγωνον ἐφαρμόσει καὶ ἴσον αὐτῷ ἔσται, <lb n="1799,001,,,,1,4,34"></lb> |
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205 καὶ αἱ λοιπαὶ γωνίαι ἐπὶ τὰς λοιπὰς γωνίας ἐφαρμόσουσι <lb n="1799,001,,,,1,4,35"></lb> |
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206 καὶ ἴσαι αὐταῖς ἔσονται, ἡ μὲν ὑπὸ ΑΒΓ τῇ ὑπὸ ΔΕΖ ἡ <lb n="1799,001,,,,1,4,36"></lb> |
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207 δὲ ὑπὸ ΑΓΒ τῇ ὑπὸ ΔΖΕ.</s> <lb n="1799,001,,,,1,4,37"></lb> |
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208 <s>Ἐὰν ἄρα δύο τρίγωνα τὰς δύο πλευρὰς [ταῖσ] δύο πλευ-<lb n="1799,001,,,,1,4,38"></lb> |
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209 ραῖς ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ καὶ τὴν γωνίαν τῇ γωνίᾳ <lb n="1799,001,,,,1,4,39"></lb> |
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210 ἴσην ἔχῃ τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην, καὶ τὴν <lb n="1799,001,,,,1,4,40"></lb> |
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211 βάσιν τῇ βάσει ἴσην ἕξει, καὶ τὸ τρίγωνον τῷ τριγώνῳ <pb></pb> <lb n="1799,001,,,,1,4,41"></lb> |
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212 ἴσον ἔσται, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι <lb n="1799,001,,,,1,4,42"></lb> |
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213 ἔσονται ἑκατέρα ἑκατέρᾳ, ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑποτεί-<lb n="1799,001,,,,1,4,43"></lb> |
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214 νουσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,4,44"></lb> |
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215 <s>Τῶν ἰσοσκελῶν τριγώνων αἱ πρὸς τῇ βάσει γωνίαι ἴσαι <lb n="1799,001,,,,1,5,1"></lb> |
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216 ἀλλήλαις εἰσίν, καὶ προσεκβληθεισῶν τῶν ἴσων εὐθειῶν <lb n="1799,001,,,,1,5,2"></lb> |
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217 αἱ ὑπὸ τὴν βάσιν γωνίαι ἴσαι ἀλλήλαις ἔσονται.</s> <lb n="1799,001,,,,1,5,3"></lb> |
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218 <s>Ἔστω τρίγωνον ἰσοσκελὲς τὸ ΑΒΓ ἴσην ἔχον τὴν ΑΒ <lb n="1799,001,,,,1,5,4"></lb> |
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219 πλευρὰν τῇ ΑΓ πλευρᾷ, καὶ προσεκβεβλήσθωσαν ἐπ' <lb n="1799,001,,,,1,5,5"></lb> |
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220 εὐθείας ταῖς ΑΒ, ΑΓ εὐθεῖαι αἱ ΒΔ, <lb n="1799,001,,,,1,5,6"></lb> |
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221 ΓΕ· λέγω, ὅτι ἡ μὲν ὑπὸ ΑΒΓ γωνία <lb n="1799,001,,,,1,5,7"></lb> |
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222 τῇ ὑπὸ ΑΓΒ ἴση ἐστίν, ἡ δὲ ὑπὸ ΓΒΔ <lb n="1799,001,,,,1,5,8"></lb> |
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223 τῇ ὑπὸ ΒΓΕ.</s> <lb n="1799,001,,,,1,5,9"></lb> |
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224 <s>Εἰλήφθω γὰρ ἐπὶ τῆς ΒΔ τυχὸν ση-<lb n="1799,001,,,,1,5,10"></lb> |
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225 μεῖον τὸ Ζ, καὶ ἀφῃρήσθω ἀπὸ τῆς μεί-<lb n="1799,001,,,,1,5,11"></lb> |
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226 ζονος τῆς ΑΕ τῇ ἐλάσσονι τῇ ΑΖ ἴση <lb n="1799,001,,,,1,5,12"></lb> |
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227 ἡ ΑΗ, καὶ ἐπεζεύχθωσαν αἱ ΖΓ, ΗΒ <lb n="1799,001,,,,1,5,13"></lb> |
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228 εὐθεῖαι.</s> <lb n="1799,001,,,,1,5,14"></lb> |
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229 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ μὲν ΑΖ τῇ ΑΗ ἡ δὲ ΑΒ τῇ ΑΓ, <lb n="1799,001,,,,1,5,15"></lb> |
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230 δύο δὴ αἱ ΖΑ, ΑΓ δυσὶ ταῖς ΗΑ, ΑΒ ἴσαι εἰσὶν ἑκατέρα <lb n="1799,001,,,,1,5,16"></lb> |
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231 ἑκατέρᾳ· καὶ γωνίαν κοινὴν περιέχουσι τὴν ὑπὸ ΖΑΗ· <lb n="1799,001,,,,1,5,17"></lb> |
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232 βάσις ἄρα ἡ ΖΓ βάσει τῇ ΗΒ ἴση ἐστίν, καὶ τὸ ΑΖΓ <lb n="1799,001,,,,1,5,18"></lb> |
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233 τρίγωνον τῷ ΑΗΒ τριγώνῳ ἴσον ἔσται, καὶ αἱ λοιπαὶ γω-<lb n="1799,001,,,,1,5,19"></lb> |
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234 νίαι ταῖς λοιπαῖς γωνίαις ἴσαι ἔσονται ἑκατέρα ἑκατέρᾳ, <pb></pb> <lb n="1799,001,,,,1,5,20"></lb> |
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235 ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑποτείνουσιν, ἡ μὲν ὑπὸ ΑΓΖ <lb n="1799,001,,,,1,5,21"></lb> |
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236 τῇ ὑπὸ ΑΒΗ, ἡ δὲ ὑπὸ ΑΖΓ τῇ ὑπὸ ΑΗΒ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,1,5,22"></lb> |
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237 ὅλη ἡ ΑΖ ὅλῃ τῇ ΑΗ ἐστιν ἴση, ὧν ἡ ΑΒ τῇ ΑΓ ἐστιν <lb n="1799,001,,,,1,5,23"></lb> |
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238 ἴση, λοιπὴ ἄρα ἡ ΒΖ λοιπῇ τῇ ΓΗ ἐστιν ἴση.</s> <s>ἐδείχθη <lb n="1799,001,,,,1,5,24"></lb> |
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239 δὲ καὶ ἡ ΖΓ τῇ ΗΒ ἴση· δύο δὴ αἱ ΒΖ, ΖΓ δυσὶ ταῖς <lb n="1799,001,,,,1,5,25"></lb> |
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240 ΓΗ, ΗΒ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ <lb n="1799,001,,,,1,5,26"></lb> |
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241 ΒΖΓ γωνίᾳ τῇ ὑπὸ ΓΗΒ ἴση, καὶ βάσις αὐτῶν κοινὴ ἡ <lb n="1799,001,,,,1,5,27"></lb> |
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242 ΒΓ· καὶ τὸ ΒΖΓ ἄρα τρίγωνον τῷ ΓΗΒ τριγώνῳ ἴσον <lb n="1799,001,,,,1,5,28"></lb> |
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243 ἔσται, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι ἔσον-<lb n="1799,001,,,,1,5,29"></lb> |
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244 ται ἑκατέρα ἑκατέρᾳ, ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑποτείνουσιν· <lb n="1799,001,,,,1,5,30"></lb> |
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245 ἴση ἄρα ἐστὶν ἡ μὲν ὑπὸ ΖΒΓ τῇ ὑπὸ ΗΓΒ ἡ δὲ ὑπὸ <lb n="1799,001,,,,1,5,31"></lb> |
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246 ΒΓΖ τῇ ὑπὸ ΓΒΗ.</s> <s>ἐπεὶ οὖν ὅλη ἡ ὑπὸ ΑΒΗ γωνία <lb n="1799,001,,,,1,5,32"></lb> |
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247 ὅλῃ τῇ ὑπὸ ΑΓΖ γωνίᾳ ἐδείχθη ἴση, ὧν ἡ ὑπὸ ΓΒΗ τῇ <lb n="1799,001,,,,1,5,33"></lb> |
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248 ὑπὸ ΒΓΖ ἴση, λοιπὴ ἄρα ἡ ὑπὸ ΑΒΓ λοιπῇ τῇ ὑπὸ ΑΓΒ <lb n="1799,001,,,,1,5,34"></lb> |
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249 ἐστιν ἴση· καί εἰσι πρὸς τῇ βάσει τοῦ ΑΒΓ τριγώνου.</s> <lb n="1799,001,,,,1,5,35"></lb> |
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250 <s>ἐδείχθη δὲ καὶ ἡ ὑπὸ ΖΒΓ τῇ ὑπὸ ΗΓΒ ἴση· καί εἰσιν <lb n="1799,001,,,,1,5,36"></lb> |
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251 ὑπὸ τὴν βάσιν.</s> <lb n="1799,001,,,,1,5,37"></lb> |
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252 <s>Τῶν ἄρα ἰσοσκελῶν τριγώνων αἱ πρὸς τῇ βάσει γωνίαι <lb n="1799,001,,,,1,5,38"></lb> |
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253 ἴσαι ἀλλήλαις εἰσίν, καὶ προσεκβληθεισῶν τῶν ἴσων <lb n="1799,001,,,,1,5,39"></lb> |
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254 εὐθειῶν αἱ ὑπὸ τὴν βάσιν γωνίαι ἴσαι ἀλλήλαις ἔσονται· <lb n="1799,001,,,,1,5,40"></lb> |
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255 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,5,41"></lb> |
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256 <s>Ἐὰν τριγώνου αἱ δύο γωνίαι ἴσαι ἀλλήλαις ὦσιν, καὶ αἱ <lb n="1799,001,,,,1,6,1"></lb> |
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257 ὑπὸ τὰς ἴσας γωνίαις ὑποτείνουσαι πλευραὶ ἴσαι ἀλλήλαις <lb n="1799,001,,,,1,6,2"></lb> |
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258 ἔσονται.</s> <s><pb></pb> <lb n="1799,001,,,,1,6,3"></lb> |
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259 Ἔστω τρίγωνον τὸ ΑΒΓ ἴσην ἔχον τὴν ὑπὸ ΑΒΓ γω-<lb n="1799,001,,,,1,6,4"></lb> |
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260 νίαν τῇ ὑπὸ ΑΓΒ γωνίᾳ· λέγω, ὅτι καὶ πλευρὰ ἡ ΑΒ <lb n="1799,001,,,,1,6,5"></lb> |
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261 πλευρᾷ τῇ ΑΓ ἐστιν ἴση.</s> <lb n="1799,001,,,,1,6,6"></lb> |
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262 <s>Εἰ γὰρ ἄνισός ἐστιν ἡ ΑΒ τῇ ΑΓ, ἡ ἑτέρα <lb n="1799,001,,,,1,6,7"></lb> |
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263 αὐτῶν μείζων ἐστίν.</s> <s>ἔστω μείζων ἡ ΑΒ, <lb n="1799,001,,,,1,6,8"></lb> |
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264 καὶ ἀφῃρήσθω ἀπὸ τῆς μείζονος τῆς ΑΒ <lb n="1799,001,,,,1,6,9"></lb> |
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265 τῇ ἐλάττονι τῇ ΑΓ ἴση ἡ ΔΒ, καὶ ἐπεζεύ-<lb n="1799,001,,,,1,6,10"></lb> |
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266 χθω ἡ ΔΓ.</s> <lb n="1799,001,,,,1,6,11"></lb> |
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267 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΒ τῇ ΑΓ κοινὴ <lb n="1799,001,,,,1,6,12"></lb> |
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268 δὲ ἡ ΒΓ, δύο δὴ αἱ ΔΒ, ΒΓ δύο ταῖς ΑΓ, <lb n="1799,001,,,,1,6,13"></lb> |
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269 ΓΒ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ, καὶ γωνία ἡ ὑπὸ ΔΒΓ <lb n="1799,001,,,,1,6,14"></lb> |
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270 γωνίᾳ τῇ ὑπὸ ΑΓΒ ἐστιν ἴση· βάσις ἄρα ἡ ΔΓ βάσει τῇ <lb n="1799,001,,,,1,6,15"></lb> |
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271 ΑΒ ἴση ἐστίν, καὶ τὸ ΔΒΓ τρίγωνον τῷ ΑΓΒ τριγώνῳ <lb n="1799,001,,,,1,6,16"></lb> |
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272 ἴσον ἔσται, τὸ ἔλασσον τῷ μείζονι· ὅπερ ἄτοπον· οὐκ ἄρα <lb n="1799,001,,,,1,6,17"></lb> |
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273 ἄνισός ἐστιν ἡ ΑΒ τῇ ΑΓ· ἴση ἄρα.</s> <lb n="1799,001,,,,1,6,18"></lb> |
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274 <s>Ἐὰν ἄρα τριγώνου αἱ δύο γωνίαι ἴσαι ἀλλήλαις ὦσιν, <lb n="1799,001,,,,1,6,19"></lb> |
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275 καὶ αἱ ὑπὸ τὰς ἴσας γωνίας ὑποτείνουσαι πλευραὶ ἴσαι <lb n="1799,001,,,,1,6,20"></lb> |
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276 ἀλλήλαις ἔσονται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,6,21"></lb> |
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277 <s>Ἐπὶ τῆς αὐτῆς εὐθείας δύο ταῖς αὐταῖς εὐθείαις ἄλλαι <lb n="1799,001,,,,1,7,1"></lb> |
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278 δύο εὐθεῖαι ἴσαι ἑκατέρα ἑκατέρᾳ οὐ συσταθήσονται πρὸς <lb n="1799,001,,,,1,7,2"></lb> |
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279 ἄλλῳ καὶ ἄλλῳ σημείῳ ἐπὶ τὰ αὐτὰ μέρη τὰ αὐτὰ πέρατα <lb n="1799,001,,,,1,7,3"></lb> |
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280 ἔχουσαι ταῖς ἐξ ἀρχῆς εὐθείαις.</s> <s><pb></pb> <lb n="1799,001,,,,1,7,4"></lb> |
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281 Εἰ γὰρ δυνατόν, ἐπὶ τῆς αὐτῆς εὐθείας τῆς ΑΒ δύο <lb n="1799,001,,,,1,7,5"></lb> |
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282 ταῖς αὐταῖς εὐθείαις ταῖς ΑΓ, ΓΒ ἄλλαι δύο εὐθεῖαι αἱ <lb n="1799,001,,,,1,7,6"></lb> |
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283 ΑΔ, ΔΒ ἴσαι ἑκατέρα ἑκατέρᾳ συνεστάτωσαν πρὸς ἄλλῳ <lb n="1799,001,,,,1,7,7"></lb> |
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284 καὶ ἄλλῳ σημείῳ τῷ τε Γ καὶ Δ ἐπὶ <lb n="1799,001,,,,1,7,8"></lb> |
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285 τὰ αὐτὰ μέρη τὰ αὐτὰ πέρατα ἔχου-<lb n="1799,001,,,,1,7,9"></lb> |
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286 σαι, ὥστε ἴσην εἶναι τὴν μὲν ΓΑ τῇ <lb n="1799,001,,,,1,7,10"></lb> |
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287 ΔΑ τὸ αὐτὸ πέρας ἔχουσαν αὐτῇ τὸ Α, <lb n="1799,001,,,,1,7,11"></lb> |
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288 τὴν δὲ ΓΒ τῇ ΔΒ τὸ αὐτὸ πέρας <lb n="1799,001,,,,1,7,12"></lb> |
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289 ἔχουσαν αὐτῇ τὸ Β, καὶ ἐπεζεύχθω <lb n="1799,001,,,,1,7,13"></lb> |
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290 ἡ ΓΔ.</s> <lb n="1799,001,,,,1,7,14"></lb> |
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291 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΑΓ τῇ ΑΔ, ἴση ἐστὶ καὶ γωνία <lb n="1799,001,,,,1,7,15"></lb> |
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292 ἡ ὑπὸ ΑΓΔ τῇ ὑπὸ ΑΔΓ· μείζων ἄρα ἡ ὑπὸ ΑΔΓ τῆς <lb n="1799,001,,,,1,7,16"></lb> |
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293 ὑπὸ ΔΓΒ· πολλῷ ἄρα ἡ ὑπὸ ΓΔΒ μείζων ἐστὶ τῆς ὑπὸ <lb n="1799,001,,,,1,7,17"></lb> |
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294 ΔΓΒ.</s> <s>πάλιν ἐπεὶ ἴση ἐστὶν ἡ ΓΒ τῇ ΔΒ, ἴση ἐστὶ καὶ <lb n="1799,001,,,,1,7,18"></lb> |
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295 γωνία ἡ ὑπὸ ΓΔΒ γωνίᾳ τῇ ὑπὸ ΔΓΒ.</s> <s>ἐδείχθη δὲ αὐ-<lb n="1799,001,,,,1,7,19"></lb> |
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296 τῆς καὶ πολλῷ μείζων· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,1,7,20"></lb> |
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297 <s>Οὐκ ἄρα ἐπὶ τῆς αὐτῆς εὐθείας δύο ταῖς αὐταῖς εὐ-<lb n="1799,001,,,,1,7,21"></lb> |
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298 θείαις ἄλλαι δύο εὐθεῖαι ἴσαι ἑκατέρα ἑκατέρᾳ συστα-<lb n="1799,001,,,,1,7,22"></lb> |
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299 θήσονται πρὸς ἄλλῳ καὶ ἄλλῳ σημείῳ ἐπὶ τὰ αὐτὰ μέρη <lb n="1799,001,,,,1,7,23"></lb> |
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300 τὰ αὐτὰ πέρατα ἔχουσαι ταῖς ἐξ ἀρχῆς εὐθείαις· ὅπερ <lb n="1799,001,,,,1,7,24"></lb> |
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301 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,7,25"></lb> |
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302 <s>Ἐὰν δύο τρίγωνα τὰς δύο πλευρὰς [ταῖσ] δύο πλευραῖς <lb n="1799,001,,,,1,8,1"></lb> |
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303 ἴσας ἔχῃ ἑκατέραν ἑκατέρα, ἔχῃ δὲ καὶ τὴν βάσιν τῇ <pb></pb> <lb n="1799,001,,,,1,8,2"></lb> |
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304 βάσει ἴσην, καὶ τὴν γωνίαν τῇ γωνίᾳ ἴσην ἕξει τὴν ὑπὸ <lb n="1799,001,,,,1,8,3"></lb> |
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305 τῶν ἴσων εὐθειῶν περιεχομένην.</s> <lb n="1799,001,,,,1,8,4"></lb> |
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306 <s>Ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ τὰς δύο πλευρὰς <lb n="1799,001,,,,1,8,5"></lb> |
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307 τὰς ΑΒ, ΑΓ ταῖς δύο πλευραῖς ταῖς ΔΕ, ΔΖ ἴσας ἔχον-<lb n="1799,001,,,,1,8,6"></lb> |
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308 τα ἑκατέραν ἑκατέρᾳ, τὴν μὲν <lb n="1799,001,,,,1,8,7"></lb> |
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309 ΑΒ τῇ ΔΕ τὴν δὲ ΑΓ τῇ ΔΖ· <lb n="1799,001,,,,1,8,8"></lb> |
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310 ἐχέτω δὲ καὶ βάσιν τὴν ΒΓ βάσει <lb n="1799,001,,,,1,8,9"></lb> |
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311 τῇ ΕΖ ἴσην· λέγω, ὅτι καὶ γωνία <lb n="1799,001,,,,1,8,10"></lb> |
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312 ἡ ὑπὸ ΒΑΓ γωνίᾳ τῇ ὑπὸ ΕΔΖ <lb n="1799,001,,,,1,8,11"></lb> |
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313 ἐστιν ἴση.</s> <lb n="1799,001,,,,1,8,12"></lb> |
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314 <s>Ἐφαρμοζομένου γὰρ τοῦ ΑΒΓ <lb n="1799,001,,,,1,8,13"></lb> |
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315 τριγώνου ἐπὶ τὸ ΔΕΖ τρίγωνον <lb n="1799,001,,,,1,8,14"></lb> |
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316 καὶ τιθεμένου τοῦ μὲν Β σημείου ἐπὶ τὸ Ε σημεῖον <lb n="1799,001,,,,1,8,15"></lb> |
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317 τῆς δὲ ΒΓ εὐθείας ἐπὶ τὴν ΕΖ ἐφαρμόσει καὶ τὸ Γ <lb n="1799,001,,,,1,8,16"></lb> |
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318 σημεῖον ἐπὶ τὸ Ζ διὰ τὸ ἴσην εἶναι τὴν ΒΓ τῇ ΕΖ· <lb n="1799,001,,,,1,8,17"></lb> |
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319 ἐφαρμοσάσης δὴ τῆς ΒΓ ἐπὶ τὴν ΕΖ ἐφαρμόσουσι καὶ <lb n="1799,001,,,,1,8,18"></lb> |
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320 αἱ ΒΑ, ΓΑ ἐπὶ τὰς ΕΔ, ΔΖ.</s> <s>εἰ γὰρ βάσις μὲν ἡ ΒΓ <lb n="1799,001,,,,1,8,19"></lb> |
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321 ἐπὶ βάσιν τὴν ΕΖ ἐφαρμόσει, αἱ δὲ ΒΑ, ΑΓ πλευραὶ <lb n="1799,001,,,,1,8,20"></lb> |
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322 ἐπὶ τὰς ΕΔ, ΔΖ οὐκ ἐφαρμόσουσιν ἀλλὰ παραλλάξου-<lb n="1799,001,,,,1,8,21"></lb> |
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323 σιν ὡς αἱ ΕΗ, ΗΖ, συσταθήσονται ἐπὶ τῆς αὐτῆς <lb n="1799,001,,,,1,8,22"></lb> |
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324 εὐθείας δύο ταῖς αὐταῖς εὐθείαις ἄλλαι δύο εὐθεῖαι <lb n="1799,001,,,,1,8,23"></lb> |
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325 ἴσαι ἑκατέρα ἑκατέρᾳ πρὸς ἄλλῳ καὶ ἄλλῳ σημείῳ ἐπὶ <lb n="1799,001,,,,1,8,24"></lb> |
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326 τὰ αὐτὰ μέρη τὰ αὐτὰ πέρατα ἔχουσαι.</s> <s>οὐ συνίστανται <lb n="1799,001,,,,1,8,25"></lb> |
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327 δέ· οὐκ ἄρα ἐφαρμοζομένης τῆς ΒΓ βάσεως ἐπὶ τὴν ΕΖ <lb n="1799,001,,,,1,8,26"></lb> |
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328 βάσιν οὐκ ἐφαρμόσουσι καὶ αἱ ΒΑ, ΑΓ πλευραὶ ἐπὶ <lb n="1799,001,,,,1,8,27"></lb> |
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329 τὰς ΕΔ, ΔΖ.</s> <s>ἐφαρμόσουσιν ἄρα· ὥστε καὶ γωνία ἡ ὑπὸ <lb n="1799,001,,,,1,8,28"></lb> |
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330 ΒΑΓ ἐπὶ γωνίαν τὴν ὑπὸ ΕΔΖ ἐφαρμόσει καὶ ἴση αὐτῇ <lb n="1799,001,,,,1,8,29"></lb> |
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331 ἔσται.</s> <s><pb></pb> <lb n="1799,001,,,,1,8,30"></lb> |
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332 Ἐὰν ἄρα δύο τρίγωνα τὰς δύο πλευρὰς [ταῖσ] δύο πλευ-<lb n="1799,001,,,,1,8,31"></lb> |
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333 ραῖς ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ καὶ τὴν βάσιν τῇ βάσει <lb n="1799,001,,,,1,8,32"></lb> |
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334 ἴσην ἔχῃ, καὶ τὴν γωνίαν τῇ γωνίᾳ ἴσην ἕξει τὴν ὑπὸ <lb n="1799,001,,,,1,8,33"></lb> |
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335 τῶν ἴσων εὐθειῶν περιεχομένην· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,8,34"></lb> |
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336 <s>Τὴν δοθεῖσαν γωνίαν εὐθύγραμμον δίχα τεμεῖν.</s> <lb n="1799,001,,,,1,9,1"></lb> |
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337 <s>Ἔστω ἡ δοθεῖσα γωνία εὐθύγραμμος ἡ ὑπὸ ΒΑΓ.</s> <s>δεῖ <lb n="1799,001,,,,1,9,2"></lb> |
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338 δὴ αὐτὴν δίχα τεμεῖν.</s> <lb n="1799,001,,,,1,9,3"></lb> |
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339 <s>Εἰλήφθω ἐπὶ τῆς ΑΒ τυχὸν σημεῖον τὸ Δ, καὶ ἀφῃρή-<lb n="1799,001,,,,1,9,4"></lb> |
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340 σθω ἀπὸ τῆς ΑΓ τῇ ΑΔ ἴση ἡ ΑΕ, καὶ ἐπεζεύχθω <lb n="1799,001,,,,1,9,5"></lb> |
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341 ἡ ΔΕ, καὶ συνεστάτω ἐπὶ τῆς ΔΕ τρίγωνον ἰσόπλευρον <lb n="1799,001,,,,1,9,6"></lb> |
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342 τὸ ΔΕΖ, καὶ ἐπεζεύχθω ἡ ΑΖ· λέγω, ὅτι ἡ ὑπὸ ΒΑΓ <lb n="1799,001,,,,1,9,7"></lb> |
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343 γωνία δίχα τέτμηται ὑπὸ τῆς ΑΖ εὐθείας.</s> <lb n="1799,001,,,,1,9,8"></lb> |
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344 <s>Ἐπεὶ γὰρ ἴση ἐστὶν ἡ ΑΔ τῇ ΑΕ, κοινὴ δὲ ἡ ΑΖ, <lb n="1799,001,,,,1,9,9"></lb> |
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345 δύο δὴ αἱ ΔΑ, ΑΖ δυσὶ ταῖς ΕΑ, ΑΖ <lb n="1799,001,,,,1,9,10"></lb> |
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346 ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ.</s> <s>καὶ βάσις ἡ <lb n="1799,001,,,,1,9,11"></lb> |
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347 ΔΖ βάσει τῇ ΕΖ ἴση ἐστίν· γωνία ἄρα <lb n="1799,001,,,,1,9,12"></lb> |
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348 ἡ ὑπὸ ΔΑΖ γωνίᾳ τῇ ὑπὸ ΕΑΖ ἴση <lb n="1799,001,,,,1,9,13"></lb> |
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349 ἐστίν.</s> <lb n="1799,001,,,,1,9,14"></lb> |
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350 <s>Ἡ ἄρα δοθεῖσα γωνία εὐθύγραμμος ἡ <lb n="1799,001,,,,1,9,15"></lb> |
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351 ὑπὸ ΒΑΓ δίχα τέτμηται ὑπὸ τῆς ΑΖ <lb n="1799,001,,,,1,9,16"></lb> |
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352 εὐθείας· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,1,9,17"></lb> |
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353 Τὴν δοθεῖσαν εὐθεῖαν πεπερασμένην δίχα τεμεῖν.</s> <lb n="1799,001,,,,1,10,1"></lb> |
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354 <s>Ἔστω ἡ δοθεῖσα εὐθεῖα πεπερασμένη ἡ ΑΒ· δεῖ δὴ <lb n="1799,001,,,,1,10,2"></lb> |
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355 τὴν ΑΒ εὐθεῖαν πεπερασμένην δίχα τεμεῖν.</s> <lb n="1799,001,,,,1,10,3"></lb> |
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356 <s>Συνεστάτω ἐπ' αὐτῆς τρίγωνον ἰσό-<lb n="1799,001,,,,1,10,4"></lb> |
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357 πλευρον τὸ ΑΒΓ, καὶ τετμήσθω ἡ ὑπὸ <lb n="1799,001,,,,1,10,5"></lb> |
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358 ΑΓΒ γωνία δίχα τῇ ΓΔ εὐθείᾳ· λέγω, <lb n="1799,001,,,,1,10,6"></lb> |
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359 ὅτι ἡ ΑΒ εὐθεῖα δίχα τέτμηται κατὰ <lb n="1799,001,,,,1,10,7"></lb> |
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360 τὸ Δ σημεῖον.</s> <lb n="1799,001,,,,1,10,8"></lb> |
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361 <s>Ἐπεὶ γὰρ ἴση ἐστὶν ἡ ΑΓ τῇ ΓΒ, <lb n="1799,001,,,,1,10,9"></lb> |
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362 κοινὴ δὲ ἡ ΓΔ, δύο δὴ αἱ ΑΓ, ΓΔ δύο <lb n="1799,001,,,,1,10,10"></lb> |
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363 ταῖς ΒΓ, ΓΔ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία ἡ <lb n="1799,001,,,,1,10,11"></lb> |
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364 ὑπὸ ΑΓΔ γωνίᾳ τῇ ὑπὸ ΒΓΔ ἴση ἐστίν· βάσις ἄρα ἡ <lb n="1799,001,,,,1,10,12"></lb> |
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365 ΑΔ βάσει τῇ ΒΔ ἴση ἐστίν.</s> <lb n="1799,001,,,,1,10,13"></lb> |
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366 <s>Ἡ ἄρα δοθεῖσα εὐθεῖα πεπερασμένη ἡ ΑΒ δίχα τέ-<lb n="1799,001,,,,1,10,14"></lb> |
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367 τμηται κατὰ τὸ Δ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,10,15"></lb> |
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368 <s>Τῇ δοθείσῃ εὐθείᾳ ἀπὸ τοῦ πρὸς αὐτῇ δοθέντος ση-<lb n="1799,001,,,,1,11,1"></lb> |
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369 μείου πρὸς ὀρθὰς γωνίας εὐθεῖαν γραμμὴν ἀγαγεῖν.</s> <lb n="1799,001,,,,1,11,2"></lb> |
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370 <s>Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ τὸ δὲ δοθὲν σημεῖον <lb n="1799,001,,,,1,11,3"></lb> |
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371 ἐπ' αὐτῆς τὸ Γ· δεῖ δὴ ἀπὸ τοῦ Γ σημείου τῇ ΑΒ εὐ-<lb n="1799,001,,,,1,11,4"></lb> |
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372 θείᾳ πρὸς ὀρθὰς γωνίας εὐθεῖαν γραμμὴν ἀγαγεῖν.</s> <s><pb></pb> <lb n="1799,001,,,,1,11,5"></lb> |
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373 Εἰλήφθω ἐπὶ τῆς ΑΓ τυχὸν σημεῖον τὸ Δ, καὶ κείσθω <lb n="1799,001,,,,1,11,6"></lb> |
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374 τῇ ΓΔ ἴση ἡ ΓΕ, καὶ συνεστάτω ἐπὶ τῆς ΔΕ τρίγωνον <lb n="1799,001,,,,1,11,7"></lb> |
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375 ἰσόπλευρον τὸ ΖΔΕ, καὶ ἐπεζεύχθω ἡ ΖΓ· λέγω, ὅτι τῇ <lb n="1799,001,,,,1,11,8"></lb> |
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376 δοθείσῃ εὐθείᾳ τῇ ΑΒ ἀπὸ <lb n="1799,001,,,,1,11,9"></lb> |
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377 τοῦ πρὸς αὐτῇ δοθέντος ση-<lb n="1799,001,,,,1,11,10"></lb> |
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378 μείου τοῦ Γ πρὸς ὀρθὰς γω-<lb n="1799,001,,,,1,11,11"></lb> |
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379 νίας εὐθεῖα γραμμὴ ἦκται ἡ <lb n="1799,001,,,,1,11,12"></lb> |
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380 ΖΓ.</s> <lb n="1799,001,,,,1,11,13"></lb> |
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381 <s>Ἐπεὶ γὰρ ἴση ἐστὶν ἡ ΔΓ <lb n="1799,001,,,,1,11,14"></lb> |
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382 τῇ ΓΕ, κοινὴ δὲ ἡ ΓΖ, δύο <lb n="1799,001,,,,1,11,15"></lb> |
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383 δὴ αἱ ΔΓ, ΓΖ δυσὶ ταῖς ΕΓ, ΓΖ ἴσαι εἰσὶν ἑκατέρα <lb n="1799,001,,,,1,11,16"></lb> |
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384 ἑκατέρᾳ· καὶ βάσις ἡ ΔΖ βάσει τῇ ΖΕ ἴση ἐστίν· γωνία <lb n="1799,001,,,,1,11,17"></lb> |
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385 ἄρα ἡ ὑπὸ ΔΓΖ γωνίᾳ τῇ ὑπὸ ΕΓΖ ἴση ἐστίν· καί <lb n="1799,001,,,,1,11,18"></lb> |
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386 εἰσιν ἐφεξῆς.</s> <s>ὅταν δὲ εὐθεῖα ἐπ' εὐθεῖαν σταθεῖσα τὰς <lb n="1799,001,,,,1,11,19"></lb> |
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387 ἐφεξῆς γωνίας ἴσας ἀλλήλαις ποιῇ, ὀρθὴ ἑκατέρα τῶν <lb n="1799,001,,,,1,11,20"></lb> |
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388 ἴσων γωνιῶν ἐστιν· ὀρθὴ ἄρα ἐστὶν ἑκατέρα τῶν ὑπὸ <lb n="1799,001,,,,1,11,21"></lb> |
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389 ΔΓΖ, ΖΓΕ.</s> <lb n="1799,001,,,,1,11,22"></lb> |
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390 <s>Τῇ ἄρα δοθείσῃ εὐθείᾳ τῇ ΑΒ ἀπὸ τοῦ πρὸς αὐτῇ δο-<lb n="1799,001,,,,1,11,23"></lb> |
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391 θέντος σημείου τοῦ Γ πρὸς ὀρθὰς γωνίας εὐθεῖα γραμμὴ <lb n="1799,001,,,,1,11,24"></lb> |
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392 ἦκται ἡ ΓΖ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,11,25"></lb> |
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393 <s>Ἐπὶ τὴν δοθεῖσαν εὐθεῖαν ἄπειρον ἀπὸ τοῦ δοθέντος <lb n="1799,001,,,,1,12,1"></lb> |
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394 σημείου, ὃ μή ἐστιν ἐπ' αὐτῆς, κάθετον εὐθεῖαν γραμ-<lb n="1799,001,,,,1,12,2"></lb> |
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395 μὴν ἀγαγεῖν.</s> <s><pb></pb> <lb n="1799,001,,,,1,12,3"></lb> |
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396 Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἄπειρος ἡ ΑΒ τὸ δὲ δο-<lb n="1799,001,,,,1,12,4"></lb> |
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397 θὲν σημεῖον, ὃ μή ἐστιν ἐπ' αὐτῆς, τὸ Γ· δεῖ δὴ ἐπὶ τὴν <lb n="1799,001,,,,1,12,5"></lb> |
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398 δοθεῖσαν εὐθεῖαν ἄπειρον τὴν ΑΒ ἀπὸ τοῦ δοθέντος <lb n="1799,001,,,,1,12,6"></lb> |
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399 σημείου τοῦ Γ, ὃ μή ἐστιν ἐπ' αὐτῆς, κάθετον εὐθεῖαν <lb n="1799,001,,,,1,12,7"></lb> |
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400 γραμμὴν ἀγαγεῖν.</s> <lb n="1799,001,,,,1,12,8"></lb> |
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401 <s>Εἰλήφθω γὰρ ἐπὶ τὰ ἕτερα μέρη τῆς ΑΒ εὐθείας τυ-<lb n="1799,001,,,,1,12,9"></lb> |
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402 χὸν σημεῖον τὸ Δ, καὶ κέντρῳ μὲν τῷ Γ διαστήματι δὲ <lb n="1799,001,,,,1,12,10"></lb> |
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403 τῷ ΓΔ κύκλος γεγράφθω ὁ ΕΖΗ, καὶ τετμήσθω ἡ ΕΗ <lb n="1799,001,,,,1,12,11"></lb> |
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404 εὐθεῖα δίχα κατὰ τὸ Θ, καὶ <lb n="1799,001,,,,1,12,12"></lb> |
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405 ἐπεζεύχθωσαν αἱ ΓΗ, ΓΘ, <lb n="1799,001,,,,1,12,13"></lb> |
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406 ΓΕ εὐθεῖαι· λέγω, ὅτι ἐπὶ <lb n="1799,001,,,,1,12,14"></lb> |
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407 τὴν δοθεῖσαν εὐθεῖαν ἄπειρον <lb n="1799,001,,,,1,12,15"></lb> |
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408 τὴν ΑΒ ἀπὸ τοῦ δοθέντος <lb n="1799,001,,,,1,12,16"></lb> |
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409 σημείου τοῦ Γ, ὃ μή ἐστιν <lb n="1799,001,,,,1,12,17"></lb> |
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410 ἐπ' αὐτῆς, κάθετος ἦκται ἡ <lb n="1799,001,,,,1,12,18"></lb> |
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411 ΓΘ.</s> <lb n="1799,001,,,,1,12,19"></lb> |
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412 <s>Ἐπεὶ γὰρ ἴση ἐστὶν ἡ ΗΘ τῇ ΘΕ, κοινὴ δὲ ἡ ΘΓ, <lb n="1799,001,,,,1,12,20"></lb> |
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413 δύο δὴ αἱ ΗΘ, ΘΓ δύο ταῖς ΕΘ, ΘΓ ἴσαι εἰσὶν ἑκα-<lb n="1799,001,,,,1,12,21"></lb> |
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414 τέρα ἑκατέρᾳ· καὶ βάσις ἡ ΓΗ βάσει τῇ ΓΕ ἐστιν ἴση· <lb n="1799,001,,,,1,12,22"></lb> |
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415 γωνία ἄρα ἡ ὑπὸ ΓΘΗ γωνίᾳ τῇ ὑπὸ ΕΘΓ ἐστιν ἴση.</s> <lb n="1799,001,,,,1,12,23"></lb> |
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416 <s>καί εἰσιν ἐφεξῆς.</s> <s>ὅταν δὲ εὐθεῖα ἐπ' εὐθεῖαν σταθεῖσα <lb n="1799,001,,,,1,12,24"></lb> |
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417 τὰς ἐφεξῆς γωνίας ἴσας ἀλλήλαις ποιῇ, ὀρθὴ ἑκατέρα <lb n="1799,001,,,,1,12,25"></lb> |
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418 τῶν ἴσων γωνιῶν ἐστιν, καὶ ἡ ἐφεστηκυῖα εὐθεῖα κάθ-<lb n="1799,001,,,,1,12,26"></lb> |
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419 ετος καλεῖται ἐφ' ἣν ἐφέστηκεν.</s> <lb n="1799,001,,,,1,12,27"></lb> |
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420 <s>Ἐπὶ τὴν δοθεῖσαν ἄρα εὐθεῖαν ἄπειρον τὴν ΑΒ ἀπὸ <lb n="1799,001,,,,1,12,28"></lb> |
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421 τοῦ δοθέντος σημείου τοῦ Γ, ὃ μή ἐστιν ἐπ' αὐτῆς, <lb n="1799,001,,,,1,12,29"></lb> |
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422 κάθετος ἦκται ἡ ΓΘ· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,1,12,30"></lb> |
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423 Ἐὰν εὐθεῖα ἐπ' εὐθεῖαν σταθεῖσα γωνίας ποιῇ, ἤτοι <lb n="1799,001,,,,1,13,1"></lb> |
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424 δύο ὀρθὰς ἢ δυσὶν ὀρθαῖς ἴσας ποιήσει.</s> <lb n="1799,001,,,,1,13,2"></lb> |
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425 <s>Εὐθεῖα γάρ τις ἡ ΑΒ ἐπ' εὐθεῖαν τὴν ΓΔ σταθεῖσα <lb n="1799,001,,,,1,13,3"></lb> |
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426 γωνίας ποιείτω τὰς ὑπὸ ΓΒΑ, ΑΒΔ· λέγω, ὅτι αἱ ὑπὸ <lb n="1799,001,,,,1,13,4"></lb> |
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427 ΓΒΑ, ΑΒΔ γωνίαι ἤτοι δύο ὀρθαί <lb n="1799,001,,,,1,13,5"></lb> |
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428 εἰσιν ἢ δυσὶν ὀρθαῖς ἴσαι.</s> <lb n="1799,001,,,,1,13,6"></lb> |
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429 <s>Εἰ μὲν οὖν ἴση ἐστὶν ἡ ὑπὸ ΓΒΑ <lb n="1799,001,,,,1,13,7"></lb> |
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430 τῇ ὑπὸ ΑΒΔ, δύο ὀρθαί εἰσιν.</s> <s>εἰ δὲ <lb n="1799,001,,,,1,13,8"></lb> |
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431 οὔ, ἤχθω ἀπὸ τοῦ Β σημείου τῇ ΓΔ <lb n="1799,001,,,,1,13,9"></lb> |
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432 [εὐθείᾳ] πρὸς ὀρθὰς ἡ ΒΕ· αἱ ἄρα ὑπὸ <lb n="1799,001,,,,1,13,10"></lb> |
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433 ΓΒΕ, ΕΒΔ δύο ὀρθαί εἰσιν· καὶ ἐπεὶ <lb n="1799,001,,,,1,13,11"></lb> |
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434 ἡ ὑπὸ ΓΒΕ δυσὶ ταῖς ὑπὸ ΓΒΑ, <lb n="1799,001,,,,1,13,12"></lb> |
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435 ΑΒΕ ἴση ἐστίν, κοινὴ προσκείσθω ἡ ὑπὸ ΕΒΔ· αἱ ἄρα <lb n="1799,001,,,,1,13,13"></lb> |
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436 ὑπὸ ΓΒΕ, ΕΒΔ τρισὶ ταῖς ὑπὸ ΓΒΑ, ΑΒΕ, ΕΒΔ <lb n="1799,001,,,,1,13,14"></lb> |
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437 ἴσαι εἰσίν.</s> <s>πάλιν, ἐπεὶ ἡ ὑπὸ ΔΒΑ δυσὶ ταῖς ὑπὸ ΔΒΕ, <lb n="1799,001,,,,1,13,15"></lb> |
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438 ΕΒΑ ἴση ἐστίν, κοινὴ προσκείσθω ἡ ὑπὸ ΑΒΓ· αἱ ἄρα <lb n="1799,001,,,,1,13,16"></lb> |
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439 ὑπὸ ΔΒΑ, ΑΒΓ τρισὶ ταῖς ὑπὸ ΔΒΕ, ΕΒΑ, ΑΒΓ ἴσαι <lb n="1799,001,,,,1,13,17"></lb> |
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440 εἰσίν.</s> <s>ἐδείχθησαν δὲ καὶ αἱ ὑπὸ ΓΒΕ, ΕΒΔ τρισὶ ταῖς <lb n="1799,001,,,,1,13,18"></lb> |
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441 αὐταῖς ἴσαι· τὰ δὲ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα· <lb n="1799,001,,,,1,13,19"></lb> |
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442 καὶ αἱ ὑπὸ ΓΒΕ, ΕΒΔ ἄρα ταῖς ὑπὸ ΔΒΑ, ΑΒΓ ἴσαι <pb></pb> <lb n="1799,001,,,,1,13,20"></lb> |
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443 εἰσίν· ἀλλὰ αἱ ὑπὸ ΓΒΕ, ΕΒΔ δύο ὀρθαί εἰσιν· καὶ αἱ <lb n="1799,001,,,,1,13,21"></lb> |
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444 ὑπὸ ΔΒΑ, ΑΒΓ ἄρα δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <lb n="1799,001,,,,1,13,22"></lb> |
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445 <s>Ἐὰν ἄρα εὐθεῖα ἐπ' εὐθεῖαν σταθεῖσα γωνίας ποιῇ, <lb n="1799,001,,,,1,13,23"></lb> |
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446 ἤτοι δύο ὀρθὰς ἢ δυσὶν ὀρθαῖς ἴσας ποιήσει· ὅπερ ἔδει <lb n="1799,001,,,,1,13,24"></lb> |
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447 δεῖξαι.</s> <lb n="1799,001,,,,1,13,25"></lb> |
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448 <s>Ἐὰν πρός τινι εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ δύο <lb n="1799,001,,,,1,14,1"></lb> |
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449 εὐθεῖαι μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς γωνίας <lb n="1799,001,,,,1,14,2"></lb> |
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450 δυσὶν ὀρθαῖς ἴσας ποιῶσιν, ἐπ' εὐθείας ἔσονται ἀλλήλαις <lb n="1799,001,,,,1,14,3"></lb> |
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451 αἱ εὐθεῖαι.</s> <lb n="1799,001,,,,1,14,4"></lb> |
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452 <s>Πρὸς γάρ τινι εὐθείᾳ τῇ ΑΒ καὶ τῷ πρὸς αὐτῇ ση-<lb n="1799,001,,,,1,14,5"></lb> |
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453 μείῳ τῷ Β δύο εὐθεῖαι αἱ ΒΓ, ΒΔ μὴ ἐπὶ τὰ αὐτὰ μέρη <lb n="1799,001,,,,1,14,6"></lb> |
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454 κείμεναι τὰς ἐφεξῆς γωνίας τὰς ὑπὸ <lb n="1799,001,,,,1,14,7"></lb> |
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455 ΑΒΓ, ΑΒΔ δύο ὀρθαῖς ἴσας ποιείτω-<lb n="1799,001,,,,1,14,8"></lb> |
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456 σαν· λέγω, ὅτι ἐπ' εὐθείας ἐστὶ τῇ <lb n="1799,001,,,,1,14,9"></lb> |
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457 ΓΒ ἡ ΒΔ.</s> <lb n="1799,001,,,,1,14,10"></lb> |
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458 <s>Εἰ γὰρ μή ἐστι τῇ ΒΓ ἐπ' εὐθείας ἡ <lb n="1799,001,,,,1,14,11"></lb> |
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459 ΒΔ, ἔστω τῇ ΓΒ ἐπ' εὐθείας ἡ ΒΕ.</s> <lb n="1799,001,,,,1,14,12"></lb> |
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460 <s>Ἐπεὶ οὖν εὐθεῖα ἡ ΑΒ ἐπ' εὐθεῖαν τὴν ΓΒΕ ἐφέστη-<lb n="1799,001,,,,1,14,13"></lb> |
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461 κεν, αἱ ἄρα ὑπὸ ΑΒΓ, ΑΒΕ γωνίαι δύο ὀρθαῖς ἴσαι <lb n="1799,001,,,,1,14,14"></lb> |
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462 εἰσίν· εἰσὶ δὲ καὶ αἱ ὑπὸ ΑΒΓ, ΑΒΔ δύο ὀρθαῖς ἴσαι· <lb n="1799,001,,,,1,14,15"></lb> |
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463 αἱ ἄρα ὑπὸ ΓΒΑ, ΑΒΕ ταῖς ὑπὸ ΓΒΑ, ΑΒΔ ἴσαι <pb></pb> <lb n="1799,001,,,,1,14,16"></lb> |
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464 εἰσίν.</s> <s>κοινὴ ἀφῃρήσθω ἡ ὑπὸ ΓΒΑ· λοιπὴ ἄρα ἡ ὑπὸ <lb n="1799,001,,,,1,14,17"></lb> |
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465 ΑΒΕ λοιπῇ τῇ ὑπὸ ΑΒΔ ἐστιν ἴση, ἡ ἐλάσσων τῇ μεί-<lb n="1799,001,,,,1,14,18"></lb> |
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466 ζονι· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ἐπ' εὐθείας ἐστὶν ἡ <lb n="1799,001,,,,1,14,19"></lb> |
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467 ΒΕ τῇ ΓΒ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι οὐδὲ ἄλλη τις πλὴν <lb n="1799,001,,,,1,14,20"></lb> |
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468 τῆς ΒΔ· ἐπ' εὐθείας ἄρα ἐστὶν ἡ ΓΒ τῇ ΒΔ.</s> <lb n="1799,001,,,,1,14,21"></lb> |
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469 <s>Ἐὰν ἄρα πρός τινι εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ <lb n="1799,001,,,,1,14,22"></lb> |
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470 δύο εὐθεῖαι μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς <lb n="1799,001,,,,1,14,23"></lb> |
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471 γωνίας δυσὶν ὀρθαῖς ἴσας ποιῶσιν, ἐπ' εὐθείας ἔσονται <lb n="1799,001,,,,1,14,24"></lb> |
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472 ἀλλήλαις αἱ εὐθεῖαι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,14,25"></lb> |
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473 <s>Ἐὰν δύο εὐθεῖαι τέμνωσιν ἀλλήλας, τὰς κατὰ κορυφὴν <lb n="1799,001,,,,1,15,1"></lb> |
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474 γωνίας ἴσας ἀλλήλαις ποιοῦσιν.</s> <lb n="1799,001,,,,1,15,2"></lb> |
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475 <s>Δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΓΔ τεμνέτωσαν ἀλλήλας κατὰ <lb n="1799,001,,,,1,15,3"></lb> |
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476 τὸ Ε σημεῖον· λέγω, ὅτι ἴση ἐστὶν ἡ μὲν ὑπὸ ΑΕΓ γω-<lb n="1799,001,,,,1,15,4"></lb> |
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477 νία τῇ ὑπὸ ΔΕΒ, ἡ δὲ ὑπὸ ΓΕΒ τῇ ὑπὸ ΑΕΔ.</s> <lb n="1799,001,,,,1,15,5"></lb> |
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478 <s>Ἐπεὶ γὰρ εὐθεῖα ἡ ΑΕ ἐπ' εὐθεῖαν <lb n="1799,001,,,,1,15,6"></lb> |
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479 τὴν ΓΔ ἐφέστηκε γωνίας ποιοῦσα <lb n="1799,001,,,,1,15,7"></lb> |
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480 τὰς ὑπὸ ΓΕΑ, ΑΕΔ, αἱ ἄρα ὑπὸ <lb n="1799,001,,,,1,15,8"></lb> |
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481 ΓΕΑ, ΑΕΔ γωνίαι δυσὶν ὀρθαῖς <lb n="1799,001,,,,1,15,9"></lb> |
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482 ἴσαι εἰσίν.</s> <s>πάλιν, ἐπεὶ εὐθεῖα ἡ ΔΕ <lb n="1799,001,,,,1,15,10"></lb> |
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483 ἐπ' εὐθεῖαν τὴν ΑΒ ἐφέστηκε γω- <pb></pb><lb n="1799,001,,,,1,15,11"></lb> |
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484 νίας ποιοῦσα τὰς ὑπὸ ΑΕΔ, ΔΕΒ, αἱ ἄρα ὑπὸ ΑΕΔ, <lb n="1799,001,,,,1,15,12"></lb> |
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485 ΔΕΒ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <s>ἐδείχθησαν δὲ καὶ αἱ <lb n="1799,001,,,,1,15,13"></lb> |
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486 ὑπὸ ΓΕΑ, ΑΕΔ δυσὶν ὀρθαῖς ἴσαι· αἱ ἄρα ὑπὸ ΓΕΑ, <lb n="1799,001,,,,1,15,14"></lb> |
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487 ΑΕΔ ταῖς ὑπὸ ΑΕΔ, ΔΕΒ ἴσαι εἰσίν.</s> <s>κοινὴ ἀφῃρήσθω ἡ <lb n="1799,001,,,,1,15,15"></lb> |
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488 ὑπὸ ΑΕΔ· λοιπὴ ἄρα ἡ ὑπὸ ΓΕΑ λοιπῇ τῇ ὑπὸ ΒΕΔ ἴση <lb n="1799,001,,,,1,15,16"></lb> |
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489 ἐστίν· ὁμοίως δὴ δειχθήσεται, ὅτι καὶ αἱ ὑπὸ ΓΕΒ, <lb n="1799,001,,,,1,15,17"></lb> |
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490 ΔΕΑ ἴσαι εἰσίν.</s> <lb n="1799,001,,,,1,15,18"></lb> |
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491 <s>Ἐὰν ἄρα δύο εὐθεῖαι τέμνωσιν ἀλλήλας, τὰς κατὰ κο-<lb n="1799,001,,,,1,15,19"></lb> |
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492 ρυφὴν γωνίας ἴσας ἀλλήλαις ποιοῦσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,15,20"></lb> |
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493 <s>[Πόρισμα <lb n="1799,001,,,,1,15,21"></lb> |
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494 Ἐκ δὴ τούτου φανερὸν ὅτι, ἐὰν δύο εὐθεῖαι τέμνωσιν <lb n="1799,001,,,,1,15,22"></lb> |
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495 ἀλλήλας, τὰς πρὸς τῇ τομῇ γωνίας τέτρασιν ὀρθαῖς ἴσας <lb n="1799,001,,,,1,15,23"></lb> |
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496 ποιήσουσιν.] <lb n="1799,001,,,,1,15,24"></lb> |
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497 Παντὸς τριγώνου μιᾶς τῶν πλευρῶν προσεκβληθείσης <lb n="1799,001,,,,1,16,1"></lb> |
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498 ἡ ἐκτὸς γωνία ἑκατέρας τῶν ἐντὸς καὶ ἀπεναντίον γω-<lb n="1799,001,,,,1,16,2"></lb> |
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499 νιῶν μείζων ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,1,16,3"></lb> |
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500 Ἔστω τρίγωνον τὸ ΑΒΓ, καὶ προσεκβεβλήσθω αὐτοῦ <lb n="1799,001,,,,1,16,4"></lb> |
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501 μία πλευρὰ ἡ ΒΓ ἐπὶ τὸ Δ· λέγω, ὅτι ἡ ἐκτὸς γωνία <lb n="1799,001,,,,1,16,5"></lb> |
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502 ἡ ὑπὸ ΑΓΔ μείζων ἐστὶν ἑκατέρας τῶν <lb n="1799,001,,,,1,16,6"></lb> |
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503 ἐντὸς καὶ ἀπεναντίον τῶν ὑπὸ ΓΒΑ, <lb n="1799,001,,,,1,16,7"></lb> |
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504 ΒΑΓ γωνιῶν.</s> <lb n="1799,001,,,,1,16,8"></lb> |
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505 <s>Τετμήσθω ἡ ΑΓ δίχα κατὰ τὸ Ε, καὶ <lb n="1799,001,,,,1,16,9"></lb> |
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506 ἐπιζευχθεῖσα ἡ ΒΕ ἐκβεβλήσθω ἐπ' εὐ-<lb n="1799,001,,,,1,16,10"></lb> |
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507 θείας ἐπὶ τὸ Ζ, καὶ κείσθω τῇ ΒΕ ἴση ἡ <lb n="1799,001,,,,1,16,11"></lb> |
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508 ΕΖ, καὶ ἐπεζεύχθω ἡ ΖΓ, καὶ διήχθω ἡ <lb n="1799,001,,,,1,16,12"></lb> |
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509 ΑΓ ἐπὶ τὸ Η.</s> <lb n="1799,001,,,,1,16,13"></lb> |
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510 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ μὲν ΑΕ τῇ ΕΓ, ἡ δὲ ΒΕ τῇ ΕΖ, <lb n="1799,001,,,,1,16,14"></lb> |
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511 δύο δὴ αἱ ΑΕ, ΕΒ δυσὶ ταῖς ΓΕ, ΕΖ ἴσαι εἰσὶν ἑκατέρα <lb n="1799,001,,,,1,16,15"></lb> |
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512 ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ ΑΕΒ γωνίᾳ τῇ ὑπὸ ΖΕΓ ἴση <lb n="1799,001,,,,1,16,16"></lb> |
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513 ἐστίν· κατὰ κορυφὴν γάρ· βάσις ἄρα ἡ ΑΒ βάσει τῇ ΖΓ <lb n="1799,001,,,,1,16,17"></lb> |
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514 ἴση ἐστίν, καὶ τὸ ΑΒΕ τρίγωνον τῷ ΖΕΓ τριγώνῳ <lb n="1799,001,,,,1,16,18"></lb> |
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515 ἐστὶν ἴσον, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις <lb n="1799,001,,,,1,16,19"></lb> |
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516 ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ, ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑπο-<lb n="1799,001,,,,1,16,20"></lb> |
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517 τείνουσιν· ἴση ἄρα ἐστὶν ἡ ὑπὸ ΒΑΕ τῇ ὑπὸ ΕΓΖ.</s> <s>μεί-<lb n="1799,001,,,,1,16,21"></lb> |
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518 ζων δέ ἐστιν ἡ ὑπὸ ΕΓΔ τῆς ὑπὸ ΕΓΖ· μείζων ἄρα ἡ <lb n="1799,001,,,,1,16,22"></lb> |
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519 ὑπὸ ΑΓΔ τῆς ὑπὸ ΒΑΕ.</s> <s>ὁμοίως δὴ τῆς ΒΓ τετμη-<lb n="1799,001,,,,1,16,23"></lb> |
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520 μένης δίχα δειχθήσεται καὶ ἡ ὑπὸ ΒΓΗ, τουτέστιν ἡ <lb n="1799,001,,,,1,16,24"></lb> |
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521 ὑπὸ ΑΓΔ, μείζων καὶ τῆς ὑπὸ ΑΒΓ.</s> <lb n="1799,001,,,,1,16,25"></lb> |
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522 <s>Παντὸς ἄρα τριγώνου μιᾶς τῶν πλευρῶν προσεκβλη-<lb n="1799,001,,,,1,16,26"></lb> |
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523 θείσης ἡ ἐκτὸς γωνία ἑκατέρας τῶν ἐντὸς καὶ ἀπεναντίον <lb n="1799,001,,,,1,16,27"></lb> |
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524 γωνιῶν μείζων ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,1,16,28"></lb> |
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525 Παντὸς τριγώνου αἱ δύο γωνίαι δύο ὀρθῶν ἐλάσσονές <lb n="1799,001,,,,1,17,1"></lb> |
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526 εἰσι πάντῃ μεταλαμβανόμεναι.</s> <lb n="1799,001,,,,1,17,2"></lb> |
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527 <s>Ἔστω τρίγωνον τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ τριγώ-<lb n="1799,001,,,,1,17,3"></lb> |
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528 νου αἱ δύο γωνίαι δύο ὀρθῶν ἐλάττο-<lb n="1799,001,,,,1,17,4"></lb> |
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529 νές εἰσι πάντῃ μεταλαμβανόμεναι.</s> <lb n="1799,001,,,,1,17,5"></lb> |
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530 <s>Ἐκβεβλήσθω γὰρ ἡ ΒΓ ἐπὶ τὸ Δ.</s> <lb n="1799,001,,,,1,17,6"></lb> |
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531 <s>Καὶ ἐπεὶ τριγώνου τοῦ ΑΒΓ ἐκτός <lb n="1799,001,,,,1,17,7"></lb> |
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532 ἐστι γωνία ἡ ὑπὸ ΑΓΔ, μείζων ἐστὶ <lb n="1799,001,,,,1,17,8"></lb> |
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533 τῆς ἐντὸς καὶ ἀπεναντίον τῆς ὑπὸ <lb n="1799,001,,,,1,17,9"></lb> |
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534 ΑΒΓ.</s> <s>κοινὴ προσκείσθω ἡ ὑπὸ ΑΓΒ· αἱ ἄρα ὑπὸ ΑΓΔ, <lb n="1799,001,,,,1,17,10"></lb> |
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535 ΑΓΒ τῶν ὑπὸ ΑΒΓ, ΒΓΑ μείζονές εἰσιν.</s> <s>ἀλλ' αἱ ὑπὸ <lb n="1799,001,,,,1,17,11"></lb> |
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536 ΑΓΔ, ΑΓΒ δύο ὀρθαῖς ἴσαι εἰσίν· αἱ ἄρα ὑπὸ ΑΒΓ, <lb n="1799,001,,,,1,17,12"></lb> |
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537 ΒΓΑ δύο ὀρθῶν ἐλάσσονές εἰσιν.</s> <s>ὁμοίως δὴ δείξομεν, <lb n="1799,001,,,,1,17,13"></lb> |
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538 ὅτι καὶ αἱ ὑπὸ ΒΑΓ, ΑΓΒ δύο ὀρθῶν ἐλάσσονές εἰσι <lb n="1799,001,,,,1,17,14"></lb> |
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539 καὶ ἔτι αἱ ὑπὸ ΓΑΒ, ΑΒΓ.</s> <lb n="1799,001,,,,1,17,15"></lb> |
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540 <s>Παντὸς ἄρα τριγώνου αἱ δύο γωνίαι δύο ὀρθῶν ἐλάσ-<lb n="1799,001,,,,1,17,16"></lb> |
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541 σονές εἰσι πάντῃ μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,17,17"></lb> |
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542 <s>Παντὸς τριγώνου ἡ μείζων πλευρὰ τὴν μείζονα γω-<lb n="1799,001,,,,1,18,1"></lb> |
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543 νίαν ὑποτείνει.</s> <lb n="1799,001,,,,1,18,2"></lb> |
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544 <s>Ἔστω γὰρ τρίγωνον τὸ ΑΒΓ μείζονα ἔχον τὴν ΑΓ <lb n="1799,001,,,,1,18,3"></lb> |
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545 πλευρὰν τῆς ΑΒ· λέγω, ὅτι καὶ γωνία ἡ ὑπὸ ΑΒΓ μεί-<lb n="1799,001,,,,1,18,4"></lb> |
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546 ζων ἐστὶ τῆς ὑπὸ ΒΓΑ.</s> <s><pb></pb> <lb n="1799,001,,,,1,18,5"></lb> |
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547 Ἐπεὶ γὰρ μείζων ἐστὶν ἡ ΑΓ τῆς ΑΒ, κείσθω τῇ <lb n="1799,001,,,,1,18,6"></lb> |
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548 ΑΒ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΒΔ.</s> <lb n="1799,001,,,,1,18,7"></lb> |
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549 <s>Καὶ ἐπεὶ τριγώνου τοῦ ΒΓΔ ἐκτός ἐστι γωνία ἡ ὑπὸ <lb n="1799,001,,,,1,18,8"></lb> |
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550 ΑΔΒ, μείζων ἐστὶ τῆς ἐντὸς καὶ <lb n="1799,001,,,,1,18,9"></lb> |
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551 ἀπεναντίον τῆς ὑπὸ ΔΓΒ· ἴση <lb n="1799,001,,,,1,18,10"></lb> |
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552 δὲ ἡ ὑπὸ ΑΔΒ τῇ ὑπὸ ΑΒΔ, ἐπεὶ <lb n="1799,001,,,,1,18,11"></lb> |
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553 καὶ πλευρὰ ἡ ΑΒ τῇ ΑΔ ἐστιν <lb n="1799,001,,,,1,18,12"></lb> |
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554 ἴση· μείζων ἄρα καὶ ἡ ὑπὸ ΑΒΔ <lb n="1799,001,,,,1,18,13"></lb> |
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555 τῆς ὑπὸ ΑΓΒ· πολλῷ ἄρα ἡ ὑπὸ <lb n="1799,001,,,,1,18,14"></lb> |
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556 ΑΒΓ μείζων ἐστὶ τῆς ὑπὸ ΑΓΒ.</s> <lb n="1799,001,,,,1,18,15"></lb> |
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557 <s>Παντὸς ἄρα τριγώνου ἡ μείζων πλευρὰ τὴν μείζονα <lb n="1799,001,,,,1,18,16"></lb> |
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558 γωνίαν ὑποτείνει· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,18,17"></lb> |
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559 <s>Παντὸς τριγώνου ὑπὸ τὴν μείζονα γωνίαν ἡ μείζων <lb n="1799,001,,,,1,19,1"></lb> |
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560 πλευρὰ ὑποτείνει.</s> <lb n="1799,001,,,,1,19,2"></lb> |
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561 <s>Ἔστω τρίγωνον τὸ ΑΒΓ μείζονα ἔχον <lb n="1799,001,,,,1,19,3"></lb> |
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562 τὴν ὑπὸ ΑΒΓ γωνίαν τῆς ὑπὸ ΒΓΑ· λέγω, <lb n="1799,001,,,,1,19,4"></lb> |
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563 ὅτι καὶ πλευρὰ ἡ ΑΓ πλευρᾶς τῆς ΑΒ <lb n="1799,001,,,,1,19,5"></lb> |
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564 μείζων ἐστίν.</s> <lb n="1799,001,,,,1,19,6"></lb> |
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565 <s>Εἰ γὰρ μή, ἤτοι ἴση ἐστὶν ἡ ΑΓ τῇ ΑΒ <lb n="1799,001,,,,1,19,7"></lb> |
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566 ἢ ἐλάσσων· ἴση μὲν οὖν οὐκ ἔστιν ἡ ΑΓ τῇ <lb n="1799,001,,,,1,19,8"></lb> |
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567 ΑΒ· ἴση γὰρ ἂν ἦν καὶ γωνία ἡ ὑπὸ ΑΒΓ <lb n="1799,001,,,,1,19,9"></lb> |
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568 τῇ ὑπὸ ΑΓΒ· οὐκ ἔστι δέ· οὐκ ἄρα ἴση <pb></pb> <lb n="1799,001,,,,1,19,10"></lb> |
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569 ἐστὶν ἡ ΑΓ τῇ ΑΒ.</s> <s>οὐδὲ μὴν ἐλάσσων ἐστὶν ἡ ΑΓ <lb n="1799,001,,,,1,19,11"></lb> |
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570 τῆς ΑΒ· ἐλάσσων γὰρ ἂν ἦν καὶ γωνία ἡ ὑπὸ ΑΒΓ τῆς <lb n="1799,001,,,,1,19,12"></lb> |
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571 ὑπὸ ΑΓΒ· οὐκ ἔστι δέ· οὐκ ἄρα ἐλάσσων ἐστὶν ἡ ΑΓ <lb n="1799,001,,,,1,19,13"></lb> |
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572 τῆς ΑΒ.</s> <s>ἐδείχθη δέ, ὅτι οὐδὲ ἴση ἐστίν.</s> <s>μείζων ἄρα ἐστὶν <lb n="1799,001,,,,1,19,14"></lb> |
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573 ἡ ΑΓ τῆς ΑΒ.</s> <lb n="1799,001,,,,1,19,15"></lb> |
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574 <s>Παντὸς ἄρα τριγώνου ὑπὸ τὴν μείζονα γωνίαν ἡ μεί-<lb n="1799,001,,,,1,19,16"></lb> |
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575 ζων πλευρὰ ὑποτείνει· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,19,17"></lb> |
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576 <s>Παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές <lb n="1799,001,,,,1,20,1"></lb> |
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577 εἰσι πάντῃ μεταλαμβανόμεναι.</s> <lb n="1799,001,,,,1,20,2"></lb> |
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578 <s>Ἔστω γὰρ τρίγωνον τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ <lb n="1799,001,,,,1,20,3"></lb> |
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579 τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ <lb n="1799,001,,,,1,20,4"></lb> |
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580 μεταλαμβανόμεναι, αἱ μὲν ΒΑ, ΑΓ τῆς ΒΓ, αἱ δὲ ΑΒ, <lb n="1799,001,,,,1,20,5"></lb> |
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581 ΒΓ τῆς ΑΓ, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.</s> <lb n="1799,001,,,,1,20,6"></lb> |
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582 <s>Διήχθω γὰρ ἡ ΒΑ ἐπὶ τὸ Δ σημεῖον, <lb n="1799,001,,,,1,20,7"></lb> |
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583 καὶ κείσθω τῇ ΓΑ ἴση ἡ ΑΔ, καὶ ἐπ-<lb n="1799,001,,,,1,20,8"></lb> |
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584 εζεύχθω ἡ ΔΓ.</s> <lb n="1799,001,,,,1,20,9"></lb> |
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585 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΑ τῇ ΑΓ, <lb n="1799,001,,,,1,20,10"></lb> |
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586 ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΑΔΓ τῇ <lb n="1799,001,,,,1,20,11"></lb> |
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587 ὑπὸ ΑΓΔ· μείζων ἄρα ἡ ὑπὸ ΒΓΔ <lb n="1799,001,,,,1,20,12"></lb> |
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588 τῆς ὑπὸ ΑΔΓ· καὶ ἐπεὶ τρίγωνόν ἐστι <lb n="1799,001,,,,1,20,13"></lb> |
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589 τὸ ΔΓΒ μείζονα ἔχον τὴν ὑπὸ ΒΓΔ <pb></pb> <lb n="1799,001,,,,1,20,14"></lb> |
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590 γωνίαν τῆς ὑπὸ ΒΔΓ, ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μεί-<lb n="1799,001,,,,1,20,15"></lb> |
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591 ζων πλευρὰ ὑποτείνει, ἡ ΔΒ ἄρα τῆς ΒΓ ἐστι μείζων.</s> <lb n="1799,001,,,,1,20,16"></lb> |
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592 <s>ἴση δὲ ἡ ΔΑ τῇ ΑΓ· μείζονες ἄρα αἱ ΒΑ, ΑΓ τῆς ΒΓ· <lb n="1799,001,,,,1,20,17"></lb> |
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593 ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ μὲν ΑΒ, ΒΓ τῆς ΓΑ <lb n="1799,001,,,,1,20,18"></lb> |
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594 μείζονές εἰσιν, αἱ δὲ ΒΓ, ΓΑ τῆς ΑΒ.</s> <lb n="1799,001,,,,1,20,19"></lb> |
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595 <s>Παντὸς ἄρα τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μεί-<lb n="1799,001,,,,1,20,20"></lb> |
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596 ζονές εἰσι πάντῃ μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,20,21"></lb> |
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597 <s>Ἐὰν τριγώνου ἐπὶ μιᾶς τῶν πλευρῶν ἀπὸ τῶν περάτων <lb n="1799,001,,,,1,21,1"></lb> |
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598 δύο εὐθεῖαι ἐντὸς συσταθῶσιν, αἱ συσταθεῖσαι τῶν λοι-<lb n="1799,001,,,,1,21,2"></lb> |
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599 πῶν τοῦ τριγώνου δύο πλευρῶν ἐλάττονες μὲν ἔσονται, <lb n="1799,001,,,,1,21,3"></lb> |
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600 μείζονα δὲ γωνίαν περιέξουσιν.</s> <lb n="1799,001,,,,1,21,4"></lb> |
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601 <s>Τριγώνου γὰρ τοῦ ΑΒΓ ἐπὶ μιᾶς τῶν πλευρῶν τῆς <lb n="1799,001,,,,1,21,5"></lb> |
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602 ΒΓ ἀπὸ τῶν περάτων τῶν Β, Γ δύο εὐθεῖαι ἐντὸς συν-<lb n="1799,001,,,,1,21,6"></lb> |
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603 εστάτωσαν αἱ ΒΔ, ΔΓ· λέγω, ὅτι αἱ ΒΔ, ΔΓ τῶν λοι-<lb n="1799,001,,,,1,21,7"></lb> |
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604 πῶν τοῦ τριγώνου δύο πλευρῶν τῶν ΒΑ, ΑΓ ἐλάσσονες <lb n="1799,001,,,,1,21,8"></lb> |
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605 μέν εἰσιν, μείζονα δὲ γωνίαν περιέχουσι τὴν ὑπὸ ΒΔΓ <lb n="1799,001,,,,1,21,9"></lb> |
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606 τῆς ὑπὸ ΒΑΓ.</s> <lb n="1799,001,,,,1,21,10"></lb> |
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607 <s>Διήχθω γὰρ ἡ ΒΔ ἐπὶ τὸ Ε.</s> <s>καὶ ἐπεὶ παντὸς τριγώνου <lb n="1799,001,,,,1,21,11"></lb> |
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608 αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσιν, τοῦ ΑΒΕ ἄρα <lb n="1799,001,,,,1,21,12"></lb> |
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609 τριγώνου αἱ δύο πλευραὶ αἱ ΑΒ, ΑΕ τῆς ΒΕ μείζονές <pb></pb> <lb n="1799,001,,,,1,21,13"></lb> |
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610 εἰσιν· κοινὴ προσκείσθω ἡ ΕΓ· αἱ ἄρα ΒΑ, ΑΓ τῶν <lb n="1799,001,,,,1,21,14"></lb> |
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611 ΒΕ, ΕΓ μείζονές εἰσιν.</s> <s>πάλιν, ἐπεὶ τοῦ ΓΕΔ τριγώνου <lb n="1799,001,,,,1,21,15"></lb> |
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612 αἱ δύο πλευραὶ αἱ ΓΕ, ΕΔ τῆς <lb n="1799,001,,,,1,21,16"></lb> |
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613 ΓΔ μείζονές εἰσιν, κοινὴ προσ-<lb n="1799,001,,,,1,21,17"></lb> |
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614 κείσθω ἡ ΔΒ· αἱ ΓΕ, ΕΒ ἄρα <lb n="1799,001,,,,1,21,18"></lb> |
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615 τῶν ΓΔ, ΔΒ μείζονές εἰσιν.</s> <lb n="1799,001,,,,1,21,19"></lb> |
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616 <s>ἀλλὰ τῶν ΒΕ, ΕΓ μείζονες <lb n="1799,001,,,,1,21,20"></lb> |
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617 ἐδείχθησαν αἱ ΒΑ, ΑΓ· πολλῷ <lb n="1799,001,,,,1,21,21"></lb> |
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618 ἄρα αἱ ΒΑ, ΑΓ τῶν ΒΔ, ΔΓ <lb n="1799,001,,,,1,21,22"></lb> |
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619 μείζονές εἰσιν.</s> <lb n="1799,001,,,,1,21,23"></lb> |
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620 <s>Πάλιν, ἐπεὶ παντὸς τριγώνου ἡ ἐκτὸς γωνία τῆς ἐντὸς <lb n="1799,001,,,,1,21,24"></lb> |
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621 καὶ ἀπεναντίον μείζων ἐστίν, τοῦ ΓΔΕ ἄρα τριγώνου ἡ <lb n="1799,001,,,,1,21,25"></lb> |
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622 ἐκτὸς γωνία ἡ ὑπὸ ΒΔΓ μείζων ἐστὶ τῆς ὑπὸ ΓΕΔ.</s> <lb n="1799,001,,,,1,21,26"></lb> |
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623 <s>διὰ ταὐτὰ τοίνυν καὶ τοῦ ΑΒΕ τριγώνου ἡ ἐκτὸς γωνία <lb n="1799,001,,,,1,21,27"></lb> |
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624 ἡ ὑπὸ ΓΕΒ μείζων ἐστὶ τῆς ὑπὸ ΒΑΓ.</s> <s>ἀλλὰ τῆς ὑπὸ <lb n="1799,001,,,,1,21,28"></lb> |
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625 ΓΕΒ μείζων ἐδείχθη ἡ ὑπὸ ΒΔΓ· πολλῷ ἄρα ἡ ὑπὸ <lb n="1799,001,,,,1,21,29"></lb> |
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626 ΒΔΓ μείζων ἐστὶ τῆς ὑπὸ ΒΑΓ.</s> <lb n="1799,001,,,,1,21,30"></lb> |
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627 <s>Ἐὰν ἄρα τριγώνου ἐπὶ μιᾶς τῶν πλευρῶν ἀπὸ τῶν πε-<lb n="1799,001,,,,1,21,31"></lb> |
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628 ράτων δύο εὐθεῖαι ἐντὸς συσταθῶσιν, αἱ συσταθεῖσαι <lb n="1799,001,,,,1,21,32"></lb> |
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629 τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν ἐλάττονες μέν <lb n="1799,001,,,,1,21,33"></lb> |
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630 εἰσιν, μείζονα δὲ γωνίαν περιέχουσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,21,34"></lb> |
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631 <s>Ἐκ τριῶν εὐθειῶν, αἵ εἰσιν ἴσαι τρισὶ ταῖς δοθείσαις <lb n="1799,001,,,,1,22,1"></lb> |
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632 [εὐθείαισ], τρίγωνον συστήσασθαι· δεῖ δὲ τὰς δύο τῆς <pb></pb> <lb n="1799,001,,,,1,22,2"></lb> |
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633 λοιπῆς μείζονας εἶναι πάντῃ μεταλαμβανομένας [διὰ τὸ <lb n="1799,001,,,,1,22,3"></lb> |
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634 καὶ παντὸς τριγώνου τὰς δύο πλευρὰς τῆς λοιπῆς μείζο-<lb n="1799,001,,,,1,22,4"></lb> |
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635 νας εἶναι πάντῃ μεταλαμβανομένασ].</s> <lb n="1799,001,,,,1,22,5"></lb> |
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636 <s>Ἔστωσαν αἱ δοθεῖσαι τρεῖς εὐθεῖαι αἱ Α, Β, Γ, ὧν αἱ <lb n="1799,001,,,,1,22,6"></lb> |
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637 δύο τῆς λοιπῆς μείζονες ἔστωσαν πάντῃ μεταλαμβανό-<lb n="1799,001,,,,1,22,7"></lb> |
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638 μεναι, αἱ μὲν Α, Β τῆς Γ, αἱ δὲ Α, Γ τῆς Β, καὶ ἔτι αἱ Β, <lb n="1799,001,,,,1,22,8"></lb> |
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639 Γ τῆς Α· δεῖ δὴ ἐκ τῶν ἴσων ταῖς Α, Β, Γ τρίγωνον <lb n="1799,001,,,,1,22,9"></lb> |
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640 συστήσασθαι.</s> <lb n="1799,001,,,,1,22,10"></lb> |
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641 <s>Ἐκκείσθω τις εὐθεῖα ἡ ΔΕ πεπερασμένη μὲν κατὰ τὸ <lb n="1799,001,,,,1,22,11"></lb> |
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642 Δ ἄπειρος δὲ κατὰ τὸ Ε, καὶ κείσθω τῇ μὲν Α ἴση ἡ <lb n="1799,001,,,,1,22,12"></lb> |
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643 ΔΖ, τῇ δὲ Β ἴση ἡ ΖΗ, τῇ δὲ Γ ἴση ἡ ΗΘ· καὶ κέντρῳ <lb n="1799,001,,,,1,22,13"></lb> |
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644 μὲν τῷ Ζ, διαστήματι δὲ τῷ ΖΔ κύκλος γεγράφθω ὁ <lb n="1799,001,,,,1,22,14"></lb> |
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645 ΔΚΛ· πάλιν κέντρῳ μὲν τῷ Η, διαστήματι δὲ τῷ ΗΘ <lb n="1799,001,,,,1,22,15"></lb> |
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646 κύκλος γεγράφθω ὁ ΚΛΘ, καὶ ἐπεζεύχθωσαν αἱ ΚΖ, <lb n="1799,001,,,,1,22,16"></lb> |
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647 ΚΗ· λέγω, ὅτι ἐκ τριῶν εὐθειῶν τῶν ἴσων ταῖς Α, Β, Γ <lb n="1799,001,,,,1,22,17"></lb> |
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648 τρίγωνον συνέσταται τὸ ΚΖΗ. <pb></pb> <lb n="1799,001,,,,1,22,18"></lb> |
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649 Ἐπεὶ γὰρ τὸ Ζ σημεῖον κέντρον ἐστὶ τοῦ ΔΚΛ κύ-<lb n="1799,001,,,,1,22,19"></lb> |
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650 κλου, ἴση ἐστὶν ἡ ΖΔ τῇ ΖΚ· ἀλλὰ ἡ ΖΔ τῇ Α ἐστιν ἴση.</s> <lb n="1799,001,,,,1,22,20"></lb> |
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651 <s>καὶ ἡ ΚΖ ἄρα τῇ Α ἐστιν ἴση.</s> <s>πάλιν, ἐπεὶ τὸ Η σημεῖον <lb n="1799,001,,,,1,22,21"></lb> |
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652 κέντρον ἐστὶ τοῦ ΛΚΘ κύκλου, ἴση ἐστὶν ἡ ΗΘ τῇ ΗΚ· <lb n="1799,001,,,,1,22,22"></lb> |
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653 ἀλλὰ ἡ ΗΘ τῇ Γ ἐστιν ἴση· καὶ ἡ ΚΗ ἄρα τῇ Γ ἐστιν <lb n="1799,001,,,,1,22,23"></lb> |
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654 ἴση.</s> <s>ἐστὶ δὲ καὶ ἡ ΖΗ τῇ Β ἴση· αἱ τρεῖς ἄρα εὐθεῖαι αἱ <lb n="1799,001,,,,1,22,24"></lb> |
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655 ΚΖ, ΖΗ, ΗΚ τρισὶ ταῖς Α, Β, Γ ἴσαι εἰσίν.</s> <lb n="1799,001,,,,1,22,25"></lb> |
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656 <s>Ἐκ τριῶν ἄρα εὐθειῶν τῶν ΚΖ, ΖΗ, ΗΚ, αἵ εἰσιν <lb n="1799,001,,,,1,22,26"></lb> |
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657 ἴσαι τρισὶ ταῖς δοθείσαις εὐθείαις ταῖς Α, Β, Γ, τρίγω-<lb n="1799,001,,,,1,22,27"></lb> |
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658 νον συνέσταται τὸ ΚΖΗ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,22,28"></lb> |
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659 <s>Πρὸς τῇ δοθείσῃ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῇ <lb n="1799,001,,,,1,23,1"></lb> |
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660 δοθείσῃ γωνίᾳ εὐθυγράμμῳ ἴσην γωνίαν εὐθύγραμμον <lb n="1799,001,,,,1,23,2"></lb> |
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661 συστήσασθαι.</s> <lb n="1799,001,,,,1,23,3"></lb> |
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662 <s>Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ πρὸς αὐτῇ <lb n="1799,001,,,,1,23,4"></lb> |
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663 σημεῖον τὸ Α, ἡ δὲ δοθεῖσα γωνία εὐθύγραμμος ἡ ὑπὸ <lb n="1799,001,,,,1,23,5"></lb> |
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664 ΔΓΕ· δεῖ δὴ πρὸς τῇ δοθείσῃ εὐθείᾳ <lb n="1799,001,,,,1,23,6"></lb> |
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665 τῇ ΑΒ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ <lb n="1799,001,,,,1,23,7"></lb> |
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666 Α τῇ δοθείσῃ γωνίᾳ εὐθυγράμμῳ τῇ <lb n="1799,001,,,,1,23,8"></lb> |
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667 ὑπὸ ΔΓΕ ἴσην γωνίαν εὐθύγραμμον <lb n="1799,001,,,,1,23,9"></lb> |
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668 συστήσασθαι.</s> <lb n="1799,001,,,,1,23,10"></lb> |
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669 <s>Εἰλήφθω ἐφ' ἑκατέρας τῶν ΓΔ, ΓΕ <lb n="1799,001,,,,1,23,11"></lb> |
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670 τυχόντα σημεῖα τὰ Δ, Ε, καὶ ἐπεζεύ-<lb n="1799,001,,,,1,23,12"></lb> |
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671 χθω ἡ ΔΕ· καὶ ἐκ τριῶν εὐθειῶν, <pb></pb> <lb n="1799,001,,,,1,23,13"></lb> |
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672 αἵ εἰσιν ἴσαι τρισὶ ταῖς ΓΔ, ΔΕ, ΓΕ, τρίγωνον συν-<lb n="1799,001,,,,1,23,14"></lb> |
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673 εστάτω τὸ ΑΖΗ, ὥστε ἴσην εἶναι τὴν μὲν ΓΔ τῇ ΑΖ, <lb n="1799,001,,,,1,23,15"></lb> |
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674 τὴν δὲ ΓΕ τῇ ΑΗ, καὶ ἔτι τὴν ΔΕ τῇ ΖΗ.</s> <lb n="1799,001,,,,1,23,16"></lb> |
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675 <s>Ἐπεὶ οὖν δύο αἱ ΔΓ, ΓΕ δύο ταῖς ΖΑ, ΑΗ ἴσαι <lb n="1799,001,,,,1,23,17"></lb> |
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676 εἰσὶν ἑκατέρα ἑκατέρᾳ, καὶ βάσις ἡ ΔΕ βάσει τῇ ΖΗ ἴση, <lb n="1799,001,,,,1,23,18"></lb> |
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677 γωνία ἄρα ἡ ὑπὸ ΔΓΕ γωνίᾳ τῇ ὑπὸ ΖΑΗ ἐστιν ἴση.</s> <lb n="1799,001,,,,1,23,19"></lb> |
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678 <s>Πρὸς ἄρα τῇ δοθείσῃ εὐθείᾳ τῇ ΑΒ καὶ τῷ πρὸς αὐτῇ <lb n="1799,001,,,,1,23,20"></lb> |
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679 σημείῳ τῷ Α τῇ δοθείσῃ γωνίᾳ εὐθυγράμμῳ τῇ ὑπὸ <lb n="1799,001,,,,1,23,21"></lb> |
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680 ΔΓΕ ἴση γωνία εὐθύγραμμος συνέσταται ἡ ὑπὸ ΖΑΗ· <lb n="1799,001,,,,1,23,22"></lb> |
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681 ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,23,23"></lb> |
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682 <s>Ἐὰν δύο τρίγωνα τὰς δύο πλευρὰς [ταῖσ] δύο πλευραῖς <lb n="1799,001,,,,1,24,1"></lb> |
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683 ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ, τὴν δὲ γωνίαν τῆς γωνίας <lb n="1799,001,,,,1,24,2"></lb> |
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684 μείζονα ἔχῃ τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην, <lb n="1799,001,,,,1,24,3"></lb> |
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685 καὶ τὴν βάσιν τῆς βάσεως μείζονα ἕξει.</s> <lb n="1799,001,,,,1,24,4"></lb> |
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686 <s>Ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ τὰς δύο πλευρὰς τὰς <lb n="1799,001,,,,1,24,5"></lb> |
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687 ΑΒ, ΑΓ ταῖς δύο πλευραῖς ταῖς ΔΕ, ΔΖ ἴσας ἔχοντα <lb n="1799,001,,,,1,24,6"></lb> |
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688 ἑκατέραν ἑκατέρᾳ, τὴν μὲν ΑΒ τῇ ΔΕ τὴν δὲ ΑΓ τῇ <lb n="1799,001,,,,1,24,7"></lb> |
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689 ΔΖ, ἡ δὲ πρὸς τῷ Α γωνία τῆς πρὸς τῷ Δ γωνίας μεί-<lb n="1799,001,,,,1,24,8"></lb> |
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690 ζων ἔστω· λέγω, ὅτι καὶ βάσις ἡ ΒΓ βάσεως τῆς ΕΖ <lb n="1799,001,,,,1,24,9"></lb> |
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691 μείζων ἐστίν.</s> <lb n="1799,001,,,,1,24,10"></lb> |
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692 <s>Ἐπεὶ γὰρ μείζων ἡ ὑπὸ ΒΑΓ γωνία τῆς ὑπὸ ΕΔΖ <pb></pb> <lb n="1799,001,,,,1,24,11"></lb> |
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693 γωνίας, συνεστάτω πρὸς τῇ ΔΕ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ <lb n="1799,001,,,,1,24,12"></lb> |
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694 σημείῳ τῷ Δ τῇ ὑπὸ ΒΑΓ γωνίᾳ ἴση ἡ ὑπὸ ΕΔΗ, καὶ <lb n="1799,001,,,,1,24,13"></lb> |
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695 κείσθω ὁποτέρᾳ τῶν ΑΓ, ΔΖ ἴση ἡ ΔΗ, καὶ ἐπεζεύ-<lb n="1799,001,,,,1,24,14"></lb> |
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696 χθωσαν αἱ ΕΗ, ΖΗ.</s> <lb n="1799,001,,,,1,24,15"></lb> |
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697 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ μὲν ΑΒ τῇ ΔΕ, ἡ δὲ ΑΓ τῇ ΔΗ, <lb n="1799,001,,,,1,24,16"></lb> |
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698 δύο δὴ αἱ ΒΑ, ΑΓ δυσὶ ταῖς ΕΔ, ΔΗ ἴσαι εἰσὶν ἑκα-<lb n="1799,001,,,,1,24,17"></lb> |
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699 τέρα ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ <lb n="1799,001,,,,1,24,18"></lb> |
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700 ΒΑΓ γωνίᾳ τῇ ὑπὸ ΕΔΗ ἴση· <lb n="1799,001,,,,1,24,19"></lb> |
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701 βάσις ἄρα ἡ ΒΓ βάσει τῇ ΕΗ <lb n="1799,001,,,,1,24,20"></lb> |
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702 ἐστιν ἴση.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν <lb n="1799,001,,,,1,24,21"></lb> |
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703 ἡ ΔΖ τῇ ΔΗ, ἴση ἐστὶ καὶ ἡ <lb n="1799,001,,,,1,24,22"></lb> |
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704 ὑπὸ ΔΗΖ γωνία τῇ ὑπὸ ΔΖΗ· <lb n="1799,001,,,,1,24,23"></lb> |
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705 μείζων ἄρα ἡ ὑπὸ ΔΖΗ τῆς <lb n="1799,001,,,,1,24,24"></lb> |
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706 ὑπὸ ΕΗΖ· πολλῷ ἄρα μείζων ἐστὶν ἡ ὑπὸ ΕΖΗ τῆς <lb n="1799,001,,,,1,24,25"></lb> |
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707 ὑπὸ ΕΗΖ.</s> <s>καὶ ἐπεὶ τρίγωνόν ἐστι τὸ ΕΖΗ μείζονα <lb n="1799,001,,,,1,24,26"></lb> |
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708 ἔχον τὴν ὑπὸ ΕΖΗ γωνίαν τῆς ὑπὸ ΕΗΖ, ὑπὸ δὲ τὴν <lb n="1799,001,,,,1,24,27"></lb> |
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709 μείζονα γωνίαν ἡ μείζων πλευρὰ ὑποτείνει, μείζων ἄρα <lb n="1799,001,,,,1,24,28"></lb> |
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710 καὶ πλευρὰ ἡ ΕΗ τῆς ΕΖ.</s> <s>ἴση δὲ ἡ ΕΗ τῇ ΒΓ· μείζων <lb n="1799,001,,,,1,24,29"></lb> |
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711 ἄρα καὶ ἡ ΒΓ τῆς ΕΖ.</s> <lb n="1799,001,,,,1,24,30"></lb> |
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712 <s>Ἐὰν ἄρα δύο τρίγωνα τὰς δύο πλευρὰς δυσὶ πλευραῖς <lb n="1799,001,,,,1,24,31"></lb> |
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713 ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ, τὴν δὲ γωνίαν τῆς γωνίας <lb n="1799,001,,,,1,24,32"></lb> |
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714 μείζονα ἔχῃ τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην, <lb n="1799,001,,,,1,24,33"></lb> |
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715 καὶ τὴν βάσιν τῆς βάσεως μείζονα ἕξει· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,1,24,34"></lb> |
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716 Ἐὰν δύο τρίγωνα τὰς δύο πλευρὰς δυσὶ πλευραῖς ἴσας <lb n="1799,001,,,,1,25,1"></lb> |
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717 ἔχῃ ἑκατέραν ἑκατέρᾳ, τὴν δὲ βάσιν τῆς βάσεως μείζονα <lb n="1799,001,,,,1,25,2"></lb> |
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718 ἔχῃ, καὶ τὴν γωνίαν τῆς γωνίας μείζονα ἕξει τὴν ὑπὸ <lb n="1799,001,,,,1,25,3"></lb> |
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719 τῶν ἴσων εὐθειῶν περιεχομένην.</s> <lb n="1799,001,,,,1,25,4"></lb> |
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720 <s>Ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ τὰς δύο πλευρὰς <lb n="1799,001,,,,1,25,5"></lb> |
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721 τὰς ΑΒ, ΑΓ ταῖς δύο πλευραῖς ταῖς ΔΕ, ΔΖ ἴσας ἔχοντα <lb n="1799,001,,,,1,25,6"></lb> |
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722 ἑκατέραν ἑκατέρᾳ, τὴν μὲν ΑΒ τῇ <lb n="1799,001,,,,1,25,7"></lb> |
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723 ΔΕ, τὴν δὲ ΑΓ τῇ ΔΖ· βάσις δὲ <lb n="1799,001,,,,1,25,8"></lb> |
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724 ἡ ΒΓ βάσεως τῆς ΕΖ μείζων ἔστω· <lb n="1799,001,,,,1,25,9"></lb> |
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725 λέγω, ὅτι καὶ γωνία ἡ ὑπὸ ΒΑΓ <lb n="1799,001,,,,1,25,10"></lb> |
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726 γωνίας τῆς ὑπὸ ΕΔΖ μείζων ἐστίν· <lb n="1799,001,,,,1,25,11"></lb> |
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727 Εἰ γὰρ μή, ἤτοι ἴση ἐστὶν αὐτῇ <lb n="1799,001,,,,1,25,12"></lb> |
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728 ἢ ἐλάσσων· ἴση μὲν οὖν οὐκ ἔστιν <lb n="1799,001,,,,1,25,13"></lb> |
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729 ἡ ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ· ἴση γὰρ ἂν ἦν καὶ βάσις <lb n="1799,001,,,,1,25,14"></lb> |
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730 ἡ ΒΓ βάσει τῇ ΕΖ· οὐκ ἔστι δέ.</s> <s>οὐκ ἄρα ἴση ἐστὶ γωνία <lb n="1799,001,,,,1,25,15"></lb> |
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731 ἡ ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ· οὐδὲ μὴν ἐλάσσων ἐστὶν ἡ ὑπὸ <lb n="1799,001,,,,1,25,16"></lb> |
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732 ΒΑΓ τῆς ὑπὸ ΕΔΖ· ἐλάσσων γὰρ ἂν ἦν καὶ βάσις ἡ ΒΓ <lb n="1799,001,,,,1,25,17"></lb> |
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733 βάσεως τῆς ΕΖ· οὐκ ἔστι δέ· οὐκ ἄρα ἐλάσσων ἐστὶν <lb n="1799,001,,,,1,25,18"></lb> |
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734 ἡ ὑπὸ ΒΑΓ γωνία τῆς ὑπὸ ΕΔΖ.</s> <s>ἐδείχθη δὲ ὅτι οὐδὲ <lb n="1799,001,,,,1,25,19"></lb> |
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735 ἴση· μείζων ἄρα ἐστὶν ἡ ὑπὸ ΒΑΓ τῆς ὑπὸ ΕΔΖ.</s> <lb n="1799,001,,,,1,25,20"></lb> |
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736 <s>Ἐὰν ἄρα δύο τρίγωνα τὰς δύο πλευρὰς δυσὶ πλευραῖς <lb n="1799,001,,,,1,25,21"></lb> |
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737 ἴσας ἔχῃ ἑκατέραν ἑκάτερᾳ, τὴν δὲ βάσιν τῆς βάσεως <pb></pb> <lb n="1799,001,,,,1,25,22"></lb> |
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738 μείζονα ἔχῃ, καὶ τὴν γωνίαν τῆς γωνίας μείζονα ἕξει τὴν <lb n="1799,001,,,,1,25,23"></lb> |
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739 ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,25,24"></lb> |
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740 <s>Ἐὰν δύο τρίγωνα τὰς δύο γωνίας δυσὶ γωνίαις ἴσας ἔχῃ <lb n="1799,001,,,,1,26,1"></lb> |
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741 ἑκατέραν ἑκατέρᾳ καὶ μίαν πλευρὰν μιᾷ πλευρᾷ ἴσην ἤτοι <lb n="1799,001,,,,1,26,2"></lb> |
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742 τὴν πρὸς ταῖς ἴσαις γωνίαις ἢ τὴν ὑποτείνουσαν ὑπὸ μίαν <lb n="1799,001,,,,1,26,3"></lb> |
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743 τῶν ἴσων γωνιῶν, καὶ τὰς λοιπὰς πλευρὰς ταῖς λοιπαῖς <lb n="1799,001,,,,1,26,4"></lb> |
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744 πλευραῖς ἴσας ἕξει [ἑκατέραν ἑκατέρᾳ] καὶ τὴν λοιπὴν <lb n="1799,001,,,,1,26,5"></lb> |
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745 γωνίαν τῇ λοιπῇ γωνίᾳ.</s> <lb n="1799,001,,,,1,26,6"></lb> |
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746 <s>Ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ τὰς δύο γωνίας τὰς <lb n="1799,001,,,,1,26,7"></lb> |
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747 ὑπὸ ΑΒΓ, ΒΓΑ δυσὶ ταῖς ὑπὸ ΔΕΖ, ΕΖΔ ἴσας ἔχοντα <lb n="1799,001,,,,1,26,8"></lb> |
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748 ἑκατέραν ἑκατέρᾳ, τὴν μὲν ὑπὸ <lb n="1799,001,,,,1,26,9"></lb> |
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749 ΑΒΓ τῇ ὑπὸ ΔΕΖ, τὴν δὲ ὑπὸ <lb n="1799,001,,,,1,26,10"></lb> |
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750 ΒΓΑ τῇ ὑπὸ ΕΖΔ· ἐχέτω δὲ καὶ <lb n="1799,001,,,,1,26,11"></lb> |
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751 μίαν πλευρὰν μιᾷ πλευρᾷ ἴσην, <lb n="1799,001,,,,1,26,12"></lb> |
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752 πρότερον τὴν πρὸς ταῖς ἴσαις <lb n="1799,001,,,,1,26,13"></lb> |
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753 γωνίαις τὴν ΒΓ τῇ ΕΖ· λέγω, <lb n="1799,001,,,,1,26,14"></lb> |
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754 ὅτι καὶ τὰς λοιπὰς πλευρὰς ταῖς <lb n="1799,001,,,,1,26,15"></lb> |
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755 λοιπαῖς πλευραῖς ἴσας ἕξει ἑκατέραν ἑκατέρᾳ, τὴν μὲν ΑΒ <lb n="1799,001,,,,1,26,16"></lb> |
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756 τῇ ΔΕ τὴν δὲ ΑΓ τῇ ΔΖ, καὶ τὴν λοιπὴν γωνίαν τῇ λοιπῇ <lb n="1799,001,,,,1,26,17"></lb> |
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757 γωνίᾳ, τὴν ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ.</s> <lb n="1799,001,,,,1,26,18"></lb> |
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758 <s>Εἰ γὰρ ἄνισός ἐστιν ἡ ΑΒ τῇ ΔΕ, μία αὐτῶν μείζων <lb n="1799,001,,,,1,26,19"></lb> |
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759 ἐστίν.</s> <s>ἔστω μείζων ἡ ΑΒ, καὶ κείσθω τῇ ΔΕ ἴση ἡ ΒΗ, <lb n="1799,001,,,,1,26,20"></lb> |
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760 καὶ ἐπεζεύχθω ἡ ΗΓ.</s> <s><pb></pb> <lb n="1799,001,,,,1,26,21"></lb> |
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761 Ἐπεὶ οὖν ἴση ἐστὶν ἡ μὲν ΒΗ τῇ ΔΕ, ἡ δὲ ΒΓ τῇ ΕΖ, <lb n="1799,001,,,,1,26,22"></lb> |
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762 δύο δὴ αἱ ΒΗ, ΒΓ δυσὶ ταῖς ΔΕ, ΕΖ ἴσαι εἰσὶν ἑκατέρα <lb n="1799,001,,,,1,26,23"></lb> |
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763 ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ ΗΒΓ γωνίᾳ τῇ ὑπὸ ΔΕΖ ἴση <lb n="1799,001,,,,1,26,24"></lb> |
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764 ἐστίν· βάσις ἄρα ἡ ΗΓ βάσει τῇ ΔΖ ἴση ἐστίν, καὶ τὸ <lb n="1799,001,,,,1,26,25"></lb> |
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765 ΗΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ ἴσον ἐστίν, καὶ αἱ λοι-<lb n="1799,001,,,,1,26,26"></lb> |
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766 παὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι ἔσονται, ὑφ' ἃς αἱ <lb n="1799,001,,,,1,26,27"></lb> |
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767 ἴσαι πλευραὶ ὑποτείνουσιν· ἴση ἄρα ἡ ὑπὸ ΗΓΒ γωνία τῇ <lb n="1799,001,,,,1,26,28"></lb> |
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768 ὑπὸ ΔΖΕ.</s> <s>ἀλλὰ ἡ ὑπὸ ΔΖΕ τῇ ὑπὸ ΒΓΑ ὑπόκειται ἴση· <lb n="1799,001,,,,1,26,29"></lb> |
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769 καὶ ἡ ὑπὸ ΒΓΗ ἄρα τῇ ὑπὸ ΒΓΑ ἴση ἐστίν, ἡ ἐλάσσων <lb n="1799,001,,,,1,26,30"></lb> |
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770 τῇ μείζονι· ὅπερ ἀδύνατον.</s> <s>οὐκ ἄρα ἄνισός ἐστιν ἡ ΑΒ τῇ <lb n="1799,001,,,,1,26,31"></lb> |
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771 ΔΕ.</s> <s>ἴση ἄρα.</s> <s>ἔστι δὲ καὶ ἡ ΒΓ τῇ ΕΖ ἴση· δύο δὴ αἱ ΑΒ, <lb n="1799,001,,,,1,26,32"></lb> |
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772 ΒΓ δυσὶ ταῖς ΔΕ, ΕΖ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ <lb n="1799,001,,,,1,26,33"></lb> |
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773 γωνία ἡ ὑπὸ ΑΒΓ γωνίᾳ τῇ ὑπὸ ΔΕΖ ἐστιν ἴση· βάσις <lb n="1799,001,,,,1,26,34"></lb> |
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774 ἄρα ἡ ΑΓ βάσει τῇ ΔΖ ἴση ἐστίν, καὶ λοιπὴ γωνία ἡ ὑπὸ <lb n="1799,001,,,,1,26,35"></lb> |
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775 ΒΑΓ τῇ λοιπῇ γωνίᾳ τῇ ὑπὸ ΕΔΖ ἴση ἐστίν.</s> <lb n="1799,001,,,,1,26,36"></lb> |
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776 <s>Ἀλλὰ δὴ πάλιν ἔστωσαν αἱ ὑπὸ τὰς ἴσας γωνίας πλευ-<lb n="1799,001,,,,1,26,37"></lb> |
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777 ραὶ ὑποτείνουσαι ἴσαι, ὡς ἡ ΑΒ τῇ ΔΕ· λέγω πάλιν, ὅτι <lb n="1799,001,,,,1,26,38"></lb> |
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778 καὶ αἱ λοιπαὶ πλευραὶ ταῖς λοιπαῖς πλευραῖς ἴσαι ἔσονται, <lb n="1799,001,,,,1,26,39"></lb> |
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779 ἡ μὲν ΑΓ τῇ ΔΖ, ἡ δὲ ΒΓ τῇ ΕΖ καὶ ἔτι ἡ λοιπὴ γωνία ἡ <lb n="1799,001,,,,1,26,40"></lb> |
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780 ὑπὸ ΒΑΓ τῇ λοιπῇ γωνίᾳ τῇ ὑπὸ ΕΔΖ ἴση ἐστίν.</s> <lb n="1799,001,,,,1,26,41"></lb> |
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781 <s>Εἰ γὰρ ἄνισός ἐστιν ἡ ΒΓ τῇ ΕΖ, μία αὐτῶν μείζων <lb n="1799,001,,,,1,26,42"></lb> |
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782 ἐστίν.</s> <s>ἔστω μείζων, εἰ δυνατόν, ἡ ΒΓ, καὶ κείσθω τῇ ΕΖ <lb n="1799,001,,,,1,26,43"></lb> |
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783 ἴση ἡ ΒΘ, καὶ ἐπεζεύχθω ἡ ΑΘ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ μὲν <lb n="1799,001,,,,1,26,44"></lb> |
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784 ΒΘ τῇ ΕΖ ἡ δὲ ΑΒ τῇ ΔΕ, δύο δὴ αἱ ΑΒ, ΒΘ δυσὶ ταῖς <lb n="1799,001,,,,1,26,45"></lb> |
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785 ΔΕ, ΕΖ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνίας ἴσας <pb></pb> <lb n="1799,001,,,,1,26,46"></lb> |
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786 περιέχουσιν· βάσις ἄρα ἡ ΑΘ βάσει τῇ ΔΖ ἴση ἐστίν, καὶ <lb n="1799,001,,,,1,26,47"></lb> |
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787 τὸ ΑΒΘ τρίγωνον τῷ ΔΕΖ τριγώνῳ ἴσον ἐστίν, καὶ αἱ <lb n="1799,001,,,,1,26,48"></lb> |
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788 λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι ἔσονται, ὑφ' ἃς αἱ <lb n="1799,001,,,,1,26,49"></lb> |
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789 ἴσαι πλευραὶ ὑποτείνουσιν· ἴση ἄρα ἐστὶν ἡ ὑπὸ ΒΘΑ <lb n="1799,001,,,,1,26,50"></lb> |
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790 γωνία τῇ ὑπὸ ΕΖΔ.</s> <s>ἀλλὰ ἡ ὑπὸ ΕΖΔ τῇ ὑπὸ ΒΓΑ ἐστιν <lb n="1799,001,,,,1,26,51"></lb> |
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791 ἴση· τριγώνου δὴ τοῦ ΑΘΓ ἡ ἐκτὸς γωνία ἡ ὑπὸ ΒΘΑ <lb n="1799,001,,,,1,26,52"></lb> |
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792 ἴση ἐστὶ τῇ ἐντὸς καὶ ἀπεναντίον τῇ ὑπὸ ΒΓΑ· ὅπερ <lb n="1799,001,,,,1,26,53"></lb> |
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793 ἀδύνατον.</s> <s>οὐκ ἄρα ἄνισός ἐστιν ἡ ΒΓ τῇ ΕΖ· ἴση ἄρα.</s> <lb n="1799,001,,,,1,26,54"></lb> |
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794 <s>ἐστὶ δὲ καὶ ἡ ΑΒ τῇ ΔΕ ἴση.</s> <s>δύο δὴ αἱ ΑΒ, ΒΓ δύο ταῖς <lb n="1799,001,,,,1,26,55"></lb> |
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795 ΔΕ, ΕΖ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνίας ἴσας <lb n="1799,001,,,,1,26,56"></lb> |
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796 περιέχουσι· βάσις ἄρα ἡ ΑΓ βάσει τῇ ΔΖ ἴση ἐστίν, καὶ <lb n="1799,001,,,,1,26,57"></lb> |
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797 τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ ἴσον καὶ λοιπὴ <lb n="1799,001,,,,1,26,58"></lb> |
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798 γωνία ἡ ὑπὸ ΒΑΓ τῇ λοιπῇ γωνίᾳ τῇ ὑπὸ ΕΔΖ ἴση.</s> <lb n="1799,001,,,,1,26,59"></lb> |
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799 <s>Ἐὰν ἄρα δύο τρίγωνα τὰς δύο γωνίας δυσὶ γωνίαις ἴσας <lb n="1799,001,,,,1,26,60"></lb> |
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800 ἔχῃ ἑκατέραν ἑκατέρᾳ καὶ μίαν πλευρὰν μιᾷ πλευρᾷ ἴσην <lb n="1799,001,,,,1,26,61"></lb> |
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801 ἤτοι τὴν πρὸς ταῖς ἴσαις γωνίαις, ἢ τὴν ὑποτείνουσαν <lb n="1799,001,,,,1,26,62"></lb> |
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802 ὑπὸ μίαν τῶν ἴσων γωνιῶν, καὶ τὰς λοιπὰς πλευρὰς ταῖς <lb n="1799,001,,,,1,26,63"></lb> |
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803 λοιπαῖς πλευραῖς ἴσας ἕξει καὶ τὴν λοιπὴν γωνίαν τῇ <lb n="1799,001,,,,1,26,64"></lb> |
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804 λοιπῇ γωνίᾳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,26,65"></lb> |
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805 <s>Ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐναλλὰξ <lb n="1799,001,,,,1,27,1"></lb> |
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806 γωνίας ἴσας ἀλλήλαις ποιῇ, παράλληλοι ἔσονται ἀλλήλαις <lb n="1799,001,,,,1,27,2"></lb> |
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807 αἱ εὐθεῖαι.</s> <s><pb></pb> <lb n="1799,001,,,,1,27,3"></lb> |
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808 Εἰς γὰρ δύο εὐθείας τὰς ΑΒ, ΓΔ εὐθεῖα ἐμπίπτουσα ἡ <lb n="1799,001,,,,1,27,4"></lb> |
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809 ΕΖ τὰς ἐναλλὰξ γωνίας τὰς ὑπὸ ΑΕΖ, ΕΖΔ ἴσας ἀλλή-<lb n="1799,001,,,,1,27,5"></lb> |
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810 λαις ποιείτω· λέγω, ὅτι παράλληλός ἐστιν ἡ ΑΒ τῇ ΓΔ.</s> <lb n="1799,001,,,,1,27,6"></lb> |
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811 <s>Εἰ γὰρ μή, ἐκβαλλόμεναι αἱ ΑΒ, ΓΔ συμπεσοῦνται <lb n="1799,001,,,,1,27,7"></lb> |
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812 ἤτοι ἐπὶ τὰ Β, Δ μέρη ἢ ἐπὶ τὰ Α, Γ.</s> <s>ἐκβεβλήσθωσαν καὶ <lb n="1799,001,,,,1,27,8"></lb> |
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813 συμπιπτέτωσαν ἐπὶ τὰ Β, Δ μέρη κατὰ τὸ Η.</s> <s>τριγώνου <lb n="1799,001,,,,1,27,9"></lb> |
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814 δὴ τοῦ ΗΕΖ ἡ ἐκτὸς γωνία ἡ ὑπὸ ΑΕΖ ἴση ἐστὶ τῇ ἐντὸς <lb n="1799,001,,,,1,27,10"></lb> |
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815 καὶ ἀπεναντίον τῇ ὑπὸ ΕΖΗ· ὅπερ ἐστὶν ἀδύνατον· οὐκ <lb n="1799,001,,,,1,27,11"></lb> |
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816 ἄρα αἱ ΑΒ, ΓΔ ἐκβαλλόμεναι συμπεσοῦνται ἐπὶ τὰ Β, <lb n="1799,001,,,,1,27,12"></lb> |
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817 Δ μέρη.</s> <s>ὁμοίως δὴ δειχθήσεται, ὅτι οὐδὲ ἐπὶ τὰ Α, Γ· αἱ <lb n="1799,001,,,,1,27,13"></lb> |
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818 δὲ ἐπὶ μηδέτερα τὰ μέρη συμπίπτουσαι παράλληλοί εἰσιν· <lb n="1799,001,,,,1,27,14"></lb> |
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819 παράλληλος ἄρα ἐστὶν ἡ ΑΒ τῇ ΓΔ.</s> <lb n="1799,001,,,,1,27,15"></lb> |
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820 <s>Ἐὰν ἄρα εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐναλλὰξ <lb n="1799,001,,,,1,27,16"></lb> |
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821 γωνίας ἴσας ἀλλήλαις ποιῇ, παράλληλοι ἔσονται αἱ <lb n="1799,001,,,,1,27,17"></lb> |
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822 εὐθεῖαι· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,1,27,18"></lb> |
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823 Ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὴν ἐκτὸς γω-<lb n="1799,001,,,,1,28,1"></lb> |
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824 νίαν τῇ ἐντὸς καὶ ἀπεναντίον καὶ ἐπὶ τὰ αὐτὰ μέρη ἴσην <lb n="1799,001,,,,1,28,2"></lb> |
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825 ποιῇ ἢ τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη δυσὶν ὀρθαῖς ἴσας, <lb n="1799,001,,,,1,28,3"></lb> |
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826 παράλληλοι ἔσονται ἀλλήλαις αἱ εὐθεῖαι.</s> <lb n="1799,001,,,,1,28,4"></lb> |
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827 <s>Εἰς γὰρ δύο εὐθείας τὰς ΑΒ, ΓΔ εὐθεῖα ἐμπίπτουσα ἡ <lb n="1799,001,,,,1,28,5"></lb> |
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828 ΕΖ τὴν ἐκτὸς γωνίαν τὴν ὑπὸ ΕΗΒ τῇ ἐντὸς καὶ ἀπεναν-<lb n="1799,001,,,,1,28,6"></lb> |
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829 τίον γωνίᾳ τῇ ὑπὸ ΗΘΔ ἴσην ποιείτω ἢ τὰς ἐντὸς καὶ ἐπὶ <lb n="1799,001,,,,1,28,7"></lb> |
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830 τὰ αὐτὰ μέρη τὰς ὑπὸ ΒΗΘ, ΗΘΔ δυσὶν ὀρθαῖς ἴσας· <lb n="1799,001,,,,1,28,8"></lb> |
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831 λέγω, ὅτι παράλληλός ἐστιν ἡ ΑΒ τῇ ΓΔ.</s> <lb n="1799,001,,,,1,28,9"></lb> |
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832 <s>Ἐπεὶ γὰρ ἴση ἐστὶν ἡ ὑπὸ ΕΗΒ τῇ ὑπὸ ΗΘΔ, ἀλλὰ ἡ <lb n="1799,001,,,,1,28,10"></lb> |
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833 ὑπὸ ΕΗΒ τῇ ὑπὸ ΑΗΘ ἐστιν ἴση, καὶ ἡ ὑπὸ ΑΗΘ ἄρα <lb n="1799,001,,,,1,28,11"></lb> |
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834 τῇ ὑπὸ ΗΘΔ ἐστιν ἴση· καί εἰσιν ἐναλλάξ· παράλληλος <lb n="1799,001,,,,1,28,12"></lb> |
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835 ἄρα ἐστὶν ἡ ΑΒ τῇ ΓΔ.</s> <lb n="1799,001,,,,1,28,13"></lb> |
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836 <s>Πάλιν, ἐπεὶ αἱ ὑπὸ ΒΗΘ, ΗΘΔ δύο ὀρθαῖς ἴσαι εἰσίν, <lb n="1799,001,,,,1,28,14"></lb> |
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837 εἰσὶ δὲ καὶ αἱ ὑπὸ ΑΗΘ, ΒΗΘ δυ-<lb n="1799,001,,,,1,28,15"></lb> |
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838 σὶν ὀρθαῖς ἴσαι, αἱ ἄρα ὑπὸ ΑΗΘ, <lb n="1799,001,,,,1,28,16"></lb> |
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839 ΒΗΘ ταῖς ὑπὸ ΒΗΘ, ΗΘΔ ἴσαι <lb n="1799,001,,,,1,28,17"></lb> |
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840 εἰσίν· κοινὴ ἀφῃρήσθω ἡ ὑπὸ ΒΗΘ· <lb n="1799,001,,,,1,28,18"></lb> |
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841 λοιπὴ ἄρα ἡ ὑπὸ ΑΗΘ λοιπῇ τῇ ὑπὸ <lb n="1799,001,,,,1,28,19"></lb> |
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842 ΗΘΔ ἐστιν ἴση· καί εἰσιν ἐναλλάξ· <lb n="1799,001,,,,1,28,20"></lb> |
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843 παράλληλος ἄρα ἐστὶν ἡ ΑΒ τῇ ΓΔ.</s> <s><pb></pb> <lb n="1799,001,,,,1,28,21"></lb> |
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844 Ἐὰν ἄρα εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὴν ἐκτὸς <lb n="1799,001,,,,1,28,22"></lb> |
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845 γωνίαν τῇ ἐντὸς καὶ ἀπεναντίον καὶ ἐπὶ τὰ αὐτὰ μέρη ἴσην <lb n="1799,001,,,,1,28,23"></lb> |
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846 ποιῇ ἢ τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη δυσὶν ὀρθαῖς ἴσας, <lb n="1799,001,,,,1,28,24"></lb> |
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847 παράλληλοι ἔσονται αἱ εὐθεῖαι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,28,25"></lb> |
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848 <s>Ἡ εἰς τὰς παραλλήλους εὐθείας εὐθεῖα ἐμπίπτουσα <lb n="1799,001,,,,1,29,1"></lb> |
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849 τάς τε ἐναλλὰξ γωνίας ἴσας ἀλλήλαις ποιεῖ καὶ τὴν ἐκτὸς <lb n="1799,001,,,,1,29,2"></lb> |
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850 τῇ ἐντὸς καὶ ἀπεναντίον ἴσην καὶ τὰς ἐντὸς καὶ ἐπὶ τὰ <lb n="1799,001,,,,1,29,3"></lb> |
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851 αὐτὰ μέρη δυσὶν ὀρθαῖς ἴσας.</s> <lb n="1799,001,,,,1,29,4"></lb> |
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852 <s>Εἰς γὰρ παραλλήλους εὐθείας τὰς ΑΒ, ΓΔ εὐθεῖα <lb n="1799,001,,,,1,29,5"></lb> |
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853 ἐμπιπτέτω ἡ ΕΖ· λέγω, ὅτι τὰς ἐναλλὰξ γωνίας τὰς <lb n="1799,001,,,,1,29,6"></lb> |
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854 ὑπὸ ΑΗΘ, ΗΘΔ ἴσας ποιεῖ καὶ τὴν ἐκτὸς γωνίαν τὴν ὑπὸ <lb n="1799,001,,,,1,29,7"></lb> |
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855 ΕΗΒ τῇ ἐντὸς καὶ ἀπεναντίον τῇ ὑπὸ ΗΘΔ ἴσην καὶ τὰς <lb n="1799,001,,,,1,29,8"></lb> |
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856 ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη τὰς ὑπὸ <lb n="1799,001,,,,1,29,9"></lb> |
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857 ΒΗΘ, ΗΘΔ δυσὶν ὀρθαῖς ἴσας.</s> <lb n="1799,001,,,,1,29,10"></lb> |
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858 <s>Εἰ γὰρ ἄνισός ἐστιν ἡ ὑπὸ ΑΗΘ <lb n="1799,001,,,,1,29,11"></lb> |
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859 τῇ ὑπὸ ΗΘΔ, μία αὐτῶν μείζων <lb n="1799,001,,,,1,29,12"></lb> |
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860 ἐστίν.</s> <s>ἔστω μείζων ἡ ὑπὸ ΑΗΘ· <lb n="1799,001,,,,1,29,13"></lb> |
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861 κοινὴ προσκείσθω ἡ ὑπὸ ΒΗΘ· αἱ <lb n="1799,001,,,,1,29,14"></lb> |
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862 ἄρα ὑπὸ ΑΗΘ, ΒΗΘ τῶν ὑπὸ <lb n="1799,001,,,,1,29,15"></lb> |
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863 ΒΗΘ, ΗΘΔ μείζονές εἰσιν.</s> <s>ἀλλὰ αἱ ὑπὸ ΑΗΘ, ΒΗΘ <lb n="1799,001,,,,1,29,16"></lb> |
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864 δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <s>[καὶ] αἱ ἄρα ὑπὸ ΒΗΘ, <pb></pb> <lb n="1799,001,,,,1,29,17"></lb> |
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865 ΗΘΔ δύο ὀρθῶν ἐλάσσονές εἰσιν.</s> <s>αἱ δὲ ἀπ' ἐλασσόνων <lb n="1799,001,,,,1,29,18"></lb> |
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866 ἢ δύο ὀρθῶν ἐκβαλλόμεναι εἰς ἄπειρον συμπίπτουσιν· αἱ <lb n="1799,001,,,,1,29,19"></lb> |
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867 ἄρα ΑΒ, ΓΔ ἐκβαλλόμεναι εἰς ἄπειρον συμπεσοῦνται· <lb n="1799,001,,,,1,29,20"></lb> |
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868 οὐ συμπίπτουσι δὲ διὰ τὸ παραλλήλους αὐτὰς ὑποκεῖσθαι· <lb n="1799,001,,,,1,29,21"></lb> |
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869 οὐκ ἄρα ἄνισός ἐστιν ἡ ὑπὸ ΑΗΘ τῇ ὑπὸ ΗΘΔ· ἴση ἄρα.</s> <lb n="1799,001,,,,1,29,22"></lb> |
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870 <s>ἀλλὰ ἡ ὑπὸ ΑΗΘ τῇ ὑπὸ ΕΗΒ ἐστιν ἴση· καὶ ἡ ὑπὸ <lb n="1799,001,,,,1,29,23"></lb> |
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871 ΕΗΒ ἄρα τῇ ὑπὸ ΗΘΔ ἐστιν ἴση.</s> <s>κοινὴ προσκείσθω ἡ <lb n="1799,001,,,,1,29,24"></lb> |
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872 ὑπὸ ΒΗΘ· αἱ ἄρα ὑπὸ ΕΗΒ, ΒΗΘ ταῖς ὑπὸ ΒΗΘ, <lb n="1799,001,,,,1,29,25"></lb> |
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873 ΗΘΔ ἴσαι εἰσίν.</s> <s>ἀλλὰ αἱ ὑπὸ ΕΗΒ, ΒΗΘ δύο ὀρθαῖς <lb n="1799,001,,,,1,29,26"></lb> |
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874 ἴσαι εἰσίν· καὶ αἱ ὑπὸ ΒΗΘ, ΗΘΔ ἄρα δύο ὀρθαῖς ἴσαι <lb n="1799,001,,,,1,29,27"></lb> |
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875 εἰσίν.</s> <lb n="1799,001,,,,1,29,28"></lb> |
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876 <s>Ἡ ἄρα εἰς τὰς παραλλήλους εὐθείας εὐθεῖα ἐμπίπτουσα <lb n="1799,001,,,,1,29,29"></lb> |
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877 τάς τε ἐναλλὰξ γωνίας ἴσας ἀλλήλαις ποιεῖ καὶ τὴν ἐκτὸς <lb n="1799,001,,,,1,29,30"></lb> |
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878 τῇ ἐντὸς καὶ ἀπεναντίον ἴσην καὶ τὰς ἐντὸς καὶ ἐπὶ τὰ <lb n="1799,001,,,,1,29,31"></lb> |
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879 αὐτὰ μέρη δυσὶν ὀρθαῖς ἴσας· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,29,32"></lb> |
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880 <s>Αἱ τῇ αὐτῇ εὐθείᾳ παράλληλοι καὶ ἀλλήλαις εἰσὶ παρ-<lb n="1799,001,,,,1,30,1"></lb> |
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881 άλληλοι.</s> <lb n="1799,001,,,,1,30,2"></lb> |
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882 <s>Ἔστω ἑκατέρα τῶν ΑΒ, ΓΔ τῇ ΕΖ παράλληλος· λέγω, <lb n="1799,001,,,,1,30,3"></lb> |
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883 ὅτι καὶ ἡ ΑΒ τῇ ΓΔ ἐστι παράλληλος.</s> <lb n="1799,001,,,,1,30,4"></lb> |
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884 <s>Ἐμπιπτέτω γὰρ εἰς αὐτὰς εὐθεῖα ἡ ΗΚ.</s> <lb n="1799,001,,,,1,30,5"></lb> |
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885 <s>Καὶ ἐπεὶ εἰς παραλλήλους εὐθείας τὰς ΑΒ, ΕΖ εὐθεῖα <lb n="1799,001,,,,1,30,6"></lb> |
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886 ἐμπέπτωκεν ἡ ΗΚ, ἴση ἄρα ἡ ὑπὸ ΑΗΚ τῇ ὑπὸ ΗΘΖ.</s> <s><pb></pb> <lb n="1799,001,,,,1,30,7"></lb> |
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887 πάλιν, ἐπεὶ εἰς παραλλήλους εὐθείας τὰς ΕΖ, ΓΔ εὐθεῖα <lb n="1799,001,,,,1,30,8"></lb> |
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888 ἐμπέπτωκεν ἡ ΗΚ, ἴση ἐστὶν ἡ ὑπὸ ΗΘΖ τῇ ὑπὸ ΗΚΔ.</s> <lb n="1799,001,,,,1,30,9"></lb> |
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889 <s>ἐδείχθη δὲ καὶ ἡ ὑπὸ ΑΗΚ τῇ ὑπὸ ΗΘΖ ἴση.</s> <s>καὶ ἡ ὑπὸ <lb n="1799,001,,,,1,30,10"></lb> |
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890 ΑΗΚ ἄρα τῇ ὑπὸ ΗΚΔ ἐστιν ἴση· <lb n="1799,001,,,,1,30,11"></lb> |
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891 καί εἰσιν ἐναλλάξ.</s> <s>παράλληλος ἄρα <lb n="1799,001,,,,1,30,12"></lb> |
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892 ἐστὶν ἡ ΑΒ τῇ ΓΔ.</s> <lb n="1799,001,,,,1,30,13"></lb> |
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893 <s>[Αἱ ἄρα τῇ αὐτῇ εὐθείᾳ παράλληλοι <lb n="1799,001,,,,1,30,14"></lb> |
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894 καὶ ἀλλήλαις εἰσὶ παράλληλοι·] ὅπερ <lb n="1799,001,,,,1,30,15"></lb> |
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895 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,30,16"></lb> |
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896 <s>Διὰ τοῦ δοθέντος σημείου τῇ δοθείσῃ εὐθείᾳ παράλλη-<lb n="1799,001,,,,1,31,1"></lb> |
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897 λον εὐθεῖαν γραμμὴν ἀγαγεῖν.</s> <lb n="1799,001,,,,1,31,2"></lb> |
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898 <s>Ἔστω τὸ μὲν δοθὲν σημεῖον τὸ Α, ἡ δὲ δοθεῖσα εὐθεῖα <lb n="1799,001,,,,1,31,3"></lb> |
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899 ἡ ΒΓ· δεῖ δὴ διὰ τοῦ Α σημείου τῇ <lb n="1799,001,,,,1,31,4"></lb> |
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900 ΒΓ εὐθείᾳ παράλληλον εὐθεῖαν γραμ-<lb n="1799,001,,,,1,31,5"></lb> |
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901 μὴν ἀγαγεῖν.</s> <lb n="1799,001,,,,1,31,6"></lb> |
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902 <s>Εἰλήφθω ἐπὶ τῆς ΒΓ τυχὸν σημεῖον <lb n="1799,001,,,,1,31,7"></lb> |
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903 τὸ Δ, καὶ ἐπεζεύχθω ἡ ΑΔ· καὶ <lb n="1799,001,,,,1,31,8"></lb> |
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904 συνεστάτω πρὸς τῇ ΔΑ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ ση-<lb n="1799,001,,,,1,31,9"></lb> |
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905 μείῳ τῷ Α τῇ ὑπὸ ΑΔΓ γωνίᾳ ἴση ἡ ὑπὸ ΔΑΕ· καὶ <lb n="1799,001,,,,1,31,10"></lb> |
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906 ἐκβεβλήσθω ἐπ' εὐθείας τῇ ΕΑ εὐθεῖα ἡ ΑΖ.</s> <lb n="1799,001,,,,1,31,11"></lb> |
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907 <s>Καὶ ἐπεὶ εἰς δύο εὐθείας τὰς ΒΓ, ΕΖ εὐθεῖα ἐμπίπτουσα <pb></pb> <lb n="1799,001,,,,1,31,12"></lb> |
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908 ἡ ΑΔ τὰς ἐναλλὰξ γωνίας τὰς ὑπὸ ΕΑΔ, ΑΔΓ ἴσας ἀλλή-<lb n="1799,001,,,,1,31,13"></lb> |
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909 λαις πεποίηκεν, παράλληλος ἄρα ἐστὶν ἡ ΕΑΖ τῇ ΒΓ.</s> <lb n="1799,001,,,,1,31,14"></lb> |
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910 <s>Διὰ τοῦ δοθέντος ἄρα σημείου τοῦ Α τῇ δοθείσῃ εὐθείᾳ <lb n="1799,001,,,,1,31,15"></lb> |
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911 τῇ ΒΓ παράλληλος εὐθεῖα γραμμὴ ἦκται ἡ ΕΑΖ· ὅπερ <lb n="1799,001,,,,1,31,16"></lb> |
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912 ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,31,17"></lb> |
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913 <s>Παντὸς τριγώνου μιᾶς τῶν πλευρῶν προσεκβληθείσης <lb n="1799,001,,,,1,32,1"></lb> |
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914 ἡ ἐκτὸς γωνία δυσὶ ταῖς ἐντὸς καὶ ἀπεναντίον ἴση ἐστίν, <lb n="1799,001,,,,1,32,2"></lb> |
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915 καὶ αἱ ἐντὸς τοῦ τριγώνου τρεῖς γωνίαι δυσὶν ὀρθαῖς ἴσαι <lb n="1799,001,,,,1,32,3"></lb> |
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916 εἰσίν.</s> <lb n="1799,001,,,,1,32,4"></lb> |
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917 <s>Ἔστω τρίγωνον τὸ ΑΒΓ, καὶ προσεκβεβλήσθω αὐτοῦ <lb n="1799,001,,,,1,32,5"></lb> |
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918 μία πλευρὰ ἡ ΒΓ ἐπὶ τὸ Δ· λέγω, ὅτι ἡ ἐκτὸς γωνία ἡ <lb n="1799,001,,,,1,32,6"></lb> |
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919 ὑπὸ ΑΓΔ ἴση ἐστὶ δυσὶ ταῖς ἐντὸς καὶ ἀπ-<lb n="1799,001,,,,1,32,7"></lb> |
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920 εναντίον ταῖς ὑπὸ ΓΑΒ, ΑΒΓ, καὶ αἱ ἐν-<lb n="1799,001,,,,1,32,8"></lb> |
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921 τὸς τοῦ τριγώνου τρεῖς γωνίαι αἱ ὑπὸ ΑΒΓ, <lb n="1799,001,,,,1,32,9"></lb> |
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922 ΒΓΑ, ΓΑΒ δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <lb n="1799,001,,,,1,32,10"></lb> |
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923 <s>Ἤχθω γὰρ διὰ τοῦ Γ σημείου τῇ ΑΒ <lb n="1799,001,,,,1,32,11"></lb> |
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924 εὐθείᾳ παράλληλος ἡ ΓΕ.</s> <lb n="1799,001,,,,1,32,12"></lb> |
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925 <s>Καὶ ἐπεὶ παράλληλός ἐστιν ἡ ΑΒ τῇ ΓΕ, καὶ εἰς αὐτὰς <lb n="1799,001,,,,1,32,13"></lb> |
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926 ἐμπέπτωκεν ἡ ΑΓ, αἱ ἐναλλὰξ γωνίαι αἱ ὑπὸ ΒΑΓ, <lb n="1799,001,,,,1,32,14"></lb> |
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927 ΑΓΕ ἴσαι ἀλλήλαις εἰσίν.</s> <s>πάλιν, ἐπεὶ παράλληλός <pb></pb> <lb n="1799,001,,,,1,32,15"></lb> |
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928 ἐστιν ἡ ΑΒ τῇ ΓΕ, καὶ εἰς αὐτὰς ἐμπέπτωκεν εὐθεῖα <lb n="1799,001,,,,1,32,16"></lb> |
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929 ἡ ΒΔ, ἡ ἐκτὸς γωνία ἡ ὑπὸ ΕΓΔ ἴση ἐστὶ τῇ ἐντὸς καὶ <lb n="1799,001,,,,1,32,17"></lb> |
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930 ἀπεναντίον τῇ ὑπὸ ΑΒΓ.</s> <s>ἐδείχθη δὲ καὶ ἡ ὑπὸ ΑΓΕ τῇ <lb n="1799,001,,,,1,32,18"></lb> |
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931 ὑπὸ ΒΑΓ ἴση· ὅλη ἄρα ἡ ὑπὸ ΑΓΔ γωνία ἴση ἐστὶ δυσὶ <lb n="1799,001,,,,1,32,19"></lb> |
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932 ταῖς ἐντὸς καὶ ἀπεναντίον ταῖς ὑπὸ ΒΑΓ, ΑΒΓ.</s> <lb n="1799,001,,,,1,32,20"></lb> |
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933 <s>Κοινὴ προσκείσθω ἡ ὑπὸ ΑΓΒ· αἱ ἄρα ὑπὸ ΑΓΔ, ΑΓΒ <lb n="1799,001,,,,1,32,21"></lb> |
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934 τρισὶ ταῖς ὑπὸ ΑΒΓ, ΒΓΑ, ΓΑΒ ἴσαι εἰσίν.</s> <s>ἀλλ' αἱ ὑπὸ <lb n="1799,001,,,,1,32,22"></lb> |
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935 ΑΓΔ, ΑΓΒ δυσὶν ὀρθαῖς ἴσαι εἰσίν· καὶ αἱ ὑπὸ ΑΓΒ, <lb n="1799,001,,,,1,32,23"></lb> |
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936 ΓΒΑ, ΓΑΒ ἄρα δυσίν ὀρθαῖς ἴσαι εἰσίν.</s> <lb n="1799,001,,,,1,32,24"></lb> |
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937 <s>Παντὸς ἄρα τριγώνου μιᾶς τῶν πλευρῶν προσεκβληθεί-<lb n="1799,001,,,,1,32,25"></lb> |
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938 σης ἡ ἐκτὸς γωνία δυσὶ ταῖς ἐντὸς καὶ ἀπεναντίον ἴση <lb n="1799,001,,,,1,32,26"></lb> |
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939 ἐστίν, καὶ αἱ ἐντὸς τοῦ τριγώνου τρεῖς γωνίαι δυσὶν ὀρθαῖς <lb n="1799,001,,,,1,32,27"></lb> |
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940 ἴσαι εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,32,28"></lb> |
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941 <s>Αἱ τὰς ἴσας τε καὶ παραλλήλους ἐπὶ τὰ αὐτὰ μέρη ἐπι-<lb n="1799,001,,,,1,33,1"></lb> |
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942 ζευγνύουσαι εὐθεῖαι καὶ αὐταὶ ἴσαι τε καὶ παράλληλοί <lb n="1799,001,,,,1,33,2"></lb> |
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943 εἰσιν.</s> <lb n="1799,001,,,,1,33,3"></lb> |
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944 <s>Ἔστωσαν ἴσαι τε καὶ παράλληλοι <lb n="1799,001,,,,1,33,4"></lb> |
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945 αἱ ΑΒ, ΓΔ, καὶ ἐπιζευγνύτωσαν <lb n="1799,001,,,,1,33,5"></lb> |
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946 αὐτὰς ἐπὶ τὰ αὐτὰ μέρη εὐθεῖαι <pb></pb> <lb n="1799,001,,,,1,33,6"></lb> |
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947 αἱ ΑΓ, ΒΔ· λέγω, ὅτι καὶ αἱ ΑΓ, ΒΔ ἴσαι τε καὶ <lb n="1799,001,,,,1,33,7"></lb> |
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948 παράλληλοί εἰσιν.</s> <lb n="1799,001,,,,1,33,8"></lb> |
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949 <s>Ἐπεζεύχθω ἡ ΒΓ.</s> <s>καὶ ἐπεὶ παράλληλός ἐστιν ἡ ΑΒ <lb n="1799,001,,,,1,33,9"></lb> |
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950 τῇ ΓΔ, καὶ εἰς αὐτὰς ἐμπέπτωκεν ἡ ΒΓ, αἱ ἐναλλὰξ <lb n="1799,001,,,,1,33,10"></lb> |
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951 γωνίαι αἱ ὑπὸ ΑΒΓ, ΒΓΔ ἴσαι ἀλλήλαις εἰσίν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,1,33,11"></lb> |
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952 ἴση ἐστὶν ἡ ΑΒ τῇ ΓΔ κοινὴ δὲ ἡ ΒΓ, δύο δὴ αἱ ΑΒ, ΒΓ <lb n="1799,001,,,,1,33,12"></lb> |
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953 δύο ταῖς ΒΓ, ΓΔ ἴσαι εἰσίν· καὶ γωνία ἡ ὑπὸ ΑΒΓ <lb n="1799,001,,,,1,33,13"></lb> |
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954 γωνίᾳ τῇ ὑπὸ ΒΓΔ ἴση· βάσις ἄρα ἡ ΑΓ βάσει τῇ ΒΔ <lb n="1799,001,,,,1,33,14"></lb> |
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955 ἐστιν ἴση, καὶ τὸ ΑΒΓ τρίγωνον τῷ ΒΓΔ τριγώνῳ ἴσον <lb n="1799,001,,,,1,33,15"></lb> |
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956 ἐστίν, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι <lb n="1799,001,,,,1,33,16"></lb> |
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957 ἔσονται ἑκατέρα ἑκατέρᾳ, ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑποτεί-<lb n="1799,001,,,,1,33,17"></lb> |
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958 νουσιν· ἴση ἄρα ἡ ὑπὸ ΑΓΒ γωνία τῇ ὑπὸ ΓΒΔ.</s> <s>καὶ <lb n="1799,001,,,,1,33,18"></lb> |
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959 ἐπεὶ εἰς δύο εὐθείας τὰς ΑΓ, ΒΔ εὐθεῖα ἐμπίπτουσα ἡ <lb n="1799,001,,,,1,33,19"></lb> |
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960 ΒΓ τὰς ἐναλλὰξ γωνίας ἴσας ἀλλήλαις πεποίηκεν, παράλ-<lb n="1799,001,,,,1,33,20"></lb> |
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961 ληλος ἄρα ἐστὶν ἡ ΑΓ τῇ ΒΔ.</s> <s>ἐδείχθη δὲ αὐτῇ καὶ ἴση.</s> <lb n="1799,001,,,,1,33,21"></lb> |
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962 <s>Αἱ ἄρα τὰς ἴσας τε καὶ παραλλήλους ἐπὶ τὰ αὐτὰ μέρη <lb n="1799,001,,,,1,33,22"></lb> |
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963 ἐπιζευγνύουσαι εὐθεῖαι καὶ αὐταὶ ἴσαι τε καὶ παράλληλοί <lb n="1799,001,,,,1,33,23"></lb> |
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964 εἰσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,33,24"></lb> |
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965 <s>Τῶν παραλληλογράμμων χωρίων αἱ ἀπεναντίον πλευ-<lb n="1799,001,,,,1,34,1"></lb> |
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966 ραί τε καὶ γωνίαι ἴσαι ἀλλήλαις εἰσίν, καὶ ἡ διάμετρος <lb n="1799,001,,,,1,34,2"></lb> |
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967 αὐτὰ δίχα τέμνει.</s> <s><pb></pb> <lb n="1799,001,,,,1,34,3"></lb> |
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968 Ἔστω παραλληλόγραμμον χωρίον τὸ ΑΓΔΒ, διά-<lb n="1799,001,,,,1,34,4"></lb> |
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969 μετρος δὲ αὐτοῦ ἡ ΒΓ· λέγω, ὅτι τοῦ ΑΓΔΒ παραλλη-<lb n="1799,001,,,,1,34,5"></lb> |
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970 λογράμμου αἱ ἀπεναντίον πλευραί τε καὶ γωνίαι ἴσαι <lb n="1799,001,,,,1,34,6"></lb> |
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971 ἀλλήλαις εἰσίν, καὶ ἡ ΒΓ διάμετρος <lb n="1799,001,,,,1,34,7"></lb> |
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972 αὐτὸ δίχα τέμνει.</s> <lb n="1799,001,,,,1,34,8"></lb> |
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973 <s>Ἐπεὶ γὰρ παράλληλός ἐστιν ἡ ΑΒ <lb n="1799,001,,,,1,34,9"></lb> |
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974 τῇ ΓΔ, καὶ εἰς αὐτὰς ἐμπέπτωκεν εὐ-<lb n="1799,001,,,,1,34,10"></lb> |
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975 θεῖα ἡ ΒΓ, αἱ ἐναλλὰξ γωνίαι αἱ ὑπὸ <lb n="1799,001,,,,1,34,11"></lb> |
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976 ΑΒΓ, ΒΓΔ ἴσαι ἀλλήλαις εἰσίν.</s> <s>πάλιν, ἐπεὶ παράλληλός <lb n="1799,001,,,,1,34,12"></lb> |
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977 ἐστιν ἡ ΑΓ τῇ ΒΔ, καὶ εἰς αὐτὰς ἐμπέπτωκεν ἡ ΒΓ, αἱ <lb n="1799,001,,,,1,34,13"></lb> |
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978 ἐναλλὰξ γωνίαι αἱ ὑπὸ ΑΓΒ, ΓΒΔ ἴσαι ἀλλήλαις εἰσίν.</s> <lb n="1799,001,,,,1,34,14"></lb> |
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979 <s>δύο δὴ τρίγωνά ἐστι τὰ ΑΒΓ, ΒΓΔ τὰς δύο γωνίας τὰς <lb n="1799,001,,,,1,34,15"></lb> |
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980 ὑπὸ ΑΒΓ, ΒΓΑ δυσὶ ταῖς ὑπὸ ΒΓΔ, ΓΒΔ ἴσας ἔχοντα <lb n="1799,001,,,,1,34,16"></lb> |
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981 ἑκατέραν ἑκατέρᾳ καὶ μίαν πλευρὰν μιᾷ πλευρᾷ ἴσην τὴν <lb n="1799,001,,,,1,34,17"></lb> |
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982 πρὸς ταῖς ἴσαις γωνίαις κοινὴν αὐτῶν τὴν ΒΓ· καὶ τὰς <lb n="1799,001,,,,1,34,18"></lb> |
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983 λοιπὰς ἄρα πλευρὰς ταῖς λοιπαῖς ἴσας ἕξει ἑκατέραν ἑκα-<lb n="1799,001,,,,1,34,19"></lb> |
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984 τέρᾳ καὶ τὴν λοιπὴν γωνίαν τῇ λοιπῇ γωνίᾳ· ἴση ἄρα <lb n="1799,001,,,,1,34,20"></lb> |
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985 ἡ μὲν ΑΒ πλευρὰ τῇ ΓΔ, ἡ δὲ ΑΓ τῇ ΒΔ, καὶ ἔτι ἴση <lb n="1799,001,,,,1,34,21"></lb> |
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986 ἐστὶν ἡ ὑπὸ ΒΑΓ γωνία τῇ ὑπὸ ΓΔΒ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν <lb n="1799,001,,,,1,34,22"></lb> |
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987 ἡ μὲν ὑπὸ ΑΒΓ γωνία τῇ ὑπὸ ΒΓΔ, ἡ δὲ ὑπὸ ΓΒΔ τῇ <lb n="1799,001,,,,1,34,23"></lb> |
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988 ὑπὸ ΑΓΒ, ὅλη ἄρα ἡ ὑπὸ ΑΒΔ ὅλῃ τῇ ὑπὸ ΑΓΔ ἐστιν <lb n="1799,001,,,,1,34,24"></lb> |
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989 ἴση.</s> <s>ἐδείχθη δὲ καὶ ἡ ὑπὸ ΒΑΓ τῇ ὑπὸ ΓΔΒ ἴση.</s> <lb n="1799,001,,,,1,34,25"></lb> |
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990 <s>Τῶν ἄρα παραλληλογράμμων χωρίων αἱ ἀπεναντίον <lb n="1799,001,,,,1,34,26"></lb> |
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991 πλευραί τε καὶ γωνίαι ἴσαι ἀλλήλαις εἰσίν.</s> <s><pb></pb> <lb n="1799,001,,,,1,34,27"></lb> |
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992 Λέγω δή, ὅτι καὶ ἡ διάμετρος αὐτὰ δίχα τέμνει.</s> <s>ἐπεὶ <lb n="1799,001,,,,1,34,28"></lb> |
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993 γὰρ ἴση ἐστὶν ἡ ΑΒ τῇ ΓΔ, κοινὴ δὲ ἡ ΒΓ, δύο δὴ αἱ ΑΒ, <lb n="1799,001,,,,1,34,29"></lb> |
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994 ΒΓ δυσὶ ταῖς ΓΔ, ΒΓ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ <lb n="1799,001,,,,1,34,30"></lb> |
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995 γωνία ἡ ὑπὸ ΑΒΓ γωνίᾳ τῇ ὑπὸ ΒΓΔ ἴση.</s> <s>καὶ βάσις ἄρα <lb n="1799,001,,,,1,34,31"></lb> |
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996 ἡ ΑΓ τῇ ΔΒ ἴση.</s> <s>καὶ τὸ ΑΒΓ [ἄρα] τρίγωνον τῷ ΒΓΔ <lb n="1799,001,,,,1,34,32"></lb> |
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997 τριγώνῳ ἴσον ἐστίν.</s> <lb n="1799,001,,,,1,34,33"></lb> |
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998 <s>Ἡ ἄρα ΒΓ διάμετρος δίχα τέμνει τὸ ΑΒΓΔ παραλλη-<lb n="1799,001,,,,1,34,34"></lb> |
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999 λόγραμμον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,34,35"></lb> |
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1000 <s>Τὰ παραλληλόγραμμα τὰ ἐπὶ τῆς αὐτῆς βάσεως ὄντα <lb n="1799,001,,,,1,35,1"></lb> |
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1001 καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἴσα ἀλλήλοις ἐστίν.</s> <lb n="1799,001,,,,1,35,2"></lb> |
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1002 <s>Ἔστω παραλληλόγραμμα τὰ ΑΒΓΔ, <lb n="1799,001,,,,1,35,3"></lb> |
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1003 ΕΒΓΖ ἐπὶ τῆς αὐτῆς βάσεως τῆς ΒΓ <lb n="1799,001,,,,1,35,4"></lb> |
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1004 καὶ ἐν ταῖς αὐταῖς παραλλήλοις ταῖς <lb n="1799,001,,,,1,35,5"></lb> |
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1005 ΑΖ, ΒΓ· λέγω, ὅτι ἴσον ἐστὶ τὸ ΑΒΓΔ <lb n="1799,001,,,,1,35,6"></lb> |
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1006 τῷ ΕΒΓΖ παραλληλογράμμῳ.</s> <lb n="1799,001,,,,1,35,7"></lb> |
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1007 <s>Ἐπεὶ γὰρ παραλληλόγραμμόν ἐστι τὸ ΑΒΓΔ, ἴση ἐστὶν <lb n="1799,001,,,,1,35,8"></lb> |
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1008 ἡ ΑΔ τῇ ΒΓ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΕΖ τῇ ΒΓ ἐστιν <lb n="1799,001,,,,1,35,9"></lb> |
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1009 ἴση· ὥστε καὶ ἡ ΑΔ τῇ ΕΖ ἐστιν ἴση· καὶ κοινὴ ἡ ΔΕ· <lb n="1799,001,,,,1,35,10"></lb> |
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1010 ὅλη ἄρα ἡ ΑΕ ὅλῃ τῇ ΔΖ ἐστιν ἴση.</s> <s>ἔστι δὲ καὶ ἡ ΑΒ <pb></pb> <lb n="1799,001,,,,1,35,11"></lb> |
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1011 τῇ ΔΓ ἴση· δύο δὴ αἱ ΕΑ, ΑΒ δύο ταῖς ΖΔ, ΔΓ ἴσαι <lb n="1799,001,,,,1,35,12"></lb> |
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1012 εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ ΖΔΓ γωνίᾳ τῇ <lb n="1799,001,,,,1,35,13"></lb> |
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1013 ὑπὸ ΕΑΒ ἐστιν ἴση ἡ ἐκτὸς τῇ ἐντός· βάσις ἄρα ἡ ΕΒ <lb n="1799,001,,,,1,35,14"></lb> |
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1014 βάσει τῇ ΖΓ ἴση ἐστίν, καὶ τὸ ΕΑΒ τρίγωνον τῷ ΔΖΓ <lb n="1799,001,,,,1,35,15"></lb> |
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1015 τριγώνῳ ἴσον ἔσται· κοινὸν ἀφῃρήσθω τὸ ΔΗΕ· λοιπὸν <lb n="1799,001,,,,1,35,16"></lb> |
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1016 ἄρα τὸ ΑΒΗΔ τραπέζιον λοιπῷ τῷ ΕΗΓΖ τραπεζίῳ <lb n="1799,001,,,,1,35,17"></lb> |
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1017 ἐστὶν ἴσον· κοινὸν προσκείσθω τὸ ΗΒΓ τρίγωνον· ὅλον <lb n="1799,001,,,,1,35,18"></lb> |
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1018 ἄρα τὸ ΑΒΓΔ παραλληλόγραμμον ὅλῳ τῷ ΕΒΓΖ παραλ-<lb n="1799,001,,,,1,35,19"></lb> |
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1019 ληλογράμμῳ ἴσον ἐστίν.</s> <lb n="1799,001,,,,1,35,20"></lb> |
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1020 <s>Τὰ ἄρα παραλληλόγραμμα τὰ ἐπὶ τῆς αὐτῆς βάσεως <lb n="1799,001,,,,1,35,21"></lb> |
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1021 ὄντα καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἴσα ἀλλήλοις ἐστίν· <lb n="1799,001,,,,1,35,22"></lb> |
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1022 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,35,23"></lb> |
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1023 <s>Τὰ παραλληλόγραμμα τὰ ἐπὶ ἴσων βάσεων ὄντα καὶ <lb n="1799,001,,,,1,36,1"></lb> |
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1024 ἐν ταῖς αὐταῖς παραλλήλοις ἴσα ἀλλήλοις ἐστίν.</s> <lb n="1799,001,,,,1,36,2"></lb> |
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1025 <s>Ἔστω παραλληλόγραμμα τὰ ΑΒΓΔ, ΕΖΗΘ ἐπὶ ἴσων <lb n="1799,001,,,,1,36,3"></lb> |
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1026 βάσεων ὄντα τῶν ΒΓ, ΖΗ <lb n="1799,001,,,,1,36,4"></lb> |
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1027 καὶ ἐν ταῖς αὐταῖς παραλλή-<lb n="1799,001,,,,1,36,5"></lb> |
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1028 λοις ταῖς ΑΘ, ΒΗ· λέγω, ὅτι <lb n="1799,001,,,,1,36,6"></lb> |
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1029 ἴσον ἐστὶ τὸ ΑΒΓΔ παραλ-<lb n="1799,001,,,,1,36,7"></lb> |
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1030 ληλόγραμμον τῷ ΕΖΗΘ.</s> <s><pb></pb> <lb n="1799,001,,,,1,36,8"></lb> |
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1031 Ἐπεζεύχθωσαν γὰρ αἱ ΒΕ, ΓΘ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ <lb n="1799,001,,,,1,36,9"></lb> |
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1032 ΒΓ τῇ ΖΗ, ἀλλὰ ἡ ΖΗ τῇ ΕΘ ἐστιν ἴση, καὶ ἡ ΒΓ ἄρα <lb n="1799,001,,,,1,36,10"></lb> |
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1033 τῇ ΕΘ ἐστιν ἴση.</s> <s>εἰσὶ δὲ καὶ παράλληλοι.</s> <s>καὶ ἐπιζευγνύ-<lb n="1799,001,,,,1,36,11"></lb> |
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1034 ουσιν αὐτὰς αἱ ΕΒ, ΘΓ· αἱ δὲ τὰς ἴσας τε καὶ παραλλή-<lb n="1799,001,,,,1,36,12"></lb> |
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1035 λους ἐπὶ τὰ αὐτὰ μέρη ἐπιζευγνύουσαι ἴσαι τε καὶ παράλ-<lb n="1799,001,,,,1,36,13"></lb> |
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1036 ληλοί εἰσι· [καὶ αἱ ΕΒ, ΘΓ ἄρα ἴσαι τέ εἰσι καὶ παράλ-<lb n="1799,001,,,,1,36,14"></lb> |
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1037 ληλοι].</s> <s>παραλληλόγραμμον ἄρα ἐστὶ τὸ ΕΒΓΘ.</s> <s>καί ἐστιν <lb n="1799,001,,,,1,36,15"></lb> |
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1038 ἴσον τῷ ΑΒΓΔ· βάσιν τε γὰρ αὐτῷ τὴν αὐτὴν ἔχει τὴν <lb n="1799,001,,,,1,36,16"></lb> |
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1039 ΒΓ, καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστὶν αὐτῷ ταῖς ΒΓ, <lb n="1799,001,,,,1,36,17"></lb> |
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1040 ΑΘ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ ΕΖΗΘ τῷ αὐτῷ τῷ ΕΒΓΘ <lb n="1799,001,,,,1,36,18"></lb> |
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1041 ἐστιν ἴσον· ὥστε καὶ τὸ ΑΒΓΔ παραλληλόγραμμον τῷ <lb n="1799,001,,,,1,36,19"></lb> |
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1042 ΕΖΗΘ ἐστιν ἴσον.</s> <lb n="1799,001,,,,1,36,20"></lb> |
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1043 <s>Τὰ ἄρα παραλληλόγραμμα τὰ ἐπὶ ἴσων βάσεων ὄντα καὶ <lb n="1799,001,,,,1,36,21"></lb> |
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1044 ἐν ταῖς αὐταῖς παραλλήλοις ἴσα ἀλλήλοις ἐστίν· ὅπερ ἔδει <lb n="1799,001,,,,1,36,22"></lb> |
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1045 δεῖξαι.</s> <lb n="1799,001,,,,1,36,23"></lb> |
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1046 <s>Τὰ τρίγωνα τὰ ἐπὶ τῆς αὐτῆς βάσεως ὄντα καὶ ἐν ταῖς <lb n="1799,001,,,,1,37,1"></lb> |
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1047 αὐταῖς παραλλήλοις ἴσα ἀλλήλοις ἐστίν.</s> <lb n="1799,001,,,,1,37,2"></lb> |
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1048 <s>Ἔστω τρίγωνα τὰ ΑΒΓ, ΔΒΓ ἐπὶ τῆς αὐτῆς βάσεως <lb n="1799,001,,,,1,37,3"></lb> |
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1049 τῆς ΒΓ καὶ ἐν ταῖς αὐταῖς <lb n="1799,001,,,,1,37,4"></lb> |
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1050 παραλλήλοις ταῖς ΑΔ, ΒΓ· <lb n="1799,001,,,,1,37,5"></lb> |
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1051 λέγω, ὅτι ἴσον ἐστὶ τὸ ΑΒΓ <lb n="1799,001,,,,1,37,6"></lb> |
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1052 τρίγωνον τῷ ΔΒΓ τριγώνῳ.</s> <s><pb></pb> <lb n="1799,001,,,,1,37,7"></lb> |
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1053 Ἐκβεβλήσθω ἡ ΑΔ ἐφ' ἑκάτερα τὰ μέρη ἐπὶ τὰ Ε, Ζ, <lb n="1799,001,,,,1,37,8"></lb> |
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1054 καὶ διὰ μὲν τοῦ Β τῇ ΓΑ παράλληλος ἤχθω ἡ ΒΕ, διὰ <lb n="1799,001,,,,1,37,9"></lb> |
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1055 δὲ τοῦ Γ τῇ ΒΔ παράλληλος ἤχθω ἡ ΓΖ.</s> <s>παραλληλό-<lb n="1799,001,,,,1,37,10"></lb> |
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1056 γραμμον ἄρα ἐστὶν ἑκάτερον τῶν ΕΒΓΑ, ΔΒΓΖ· καί <lb n="1799,001,,,,1,37,11"></lb> |
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1057 εἰσιν ἴσα· ἐπί τε γὰρ τῆς αὐτῆς βάσεώς εἰσι τῆς ΒΓ καὶ <lb n="1799,001,,,,1,37,12"></lb> |
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1058 ἐν ταῖς αὐταῖς παραλλήλοις ταῖς ΒΓ, ΕΖ· καί ἐστι τοῦ <lb n="1799,001,,,,1,37,13"></lb> |
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1059 μὲν ΕΒΓΑ παραλληλογράμμου ἥμισυ τὸ ΑΒΓ τρίγω-<lb n="1799,001,,,,1,37,14"></lb> |
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1060 νον· ἡ γὰρ ΑΒ διάμετρος αὐτὸ δίχα τέμνει· τοῦ δὲ ΔΒΓΖ <lb n="1799,001,,,,1,37,15"></lb> |
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1061 παραλληλογράμμου ἥμισυ τὸ ΔΒΓ τρίγωνον· ἡ γὰρ ΔΓ <lb n="1799,001,,,,1,37,16"></lb> |
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1062 διάμετρος αὐτὸ δίχα τέμνει.</s> <s>[τὰ δὲ τῶν ἴσων ἡμίση ἴσα <lb n="1799,001,,,,1,37,17"></lb> |
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1063 ἀλλήλοις ἐστίν].</s> <s>ἴσον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΒΓ <lb n="1799,001,,,,1,37,18"></lb> |
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1064 τριγώνῳ.</s> <lb n="1799,001,,,,1,37,19"></lb> |
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1065 <s>Τὰ ἄρα τρίγωνα τὰ ἐπὶ τῆς αὐτῆς βάσεως ὄντα καὶ ἐν <lb n="1799,001,,,,1,37,20"></lb> |
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1066 ταῖς αὐταῖς παραλλήλοις ἴσα ἀλλήλοις ἐστίν· ὅπερ ἔδει <lb n="1799,001,,,,1,37,21"></lb> |
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1067 δεῖξαι.</s> <lb n="1799,001,,,,1,37,22"></lb> |
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1068 <s>Τὰ τρίγωνα τὰ ἐπὶ ἴσων βάσεων ὄντα καὶ ἐν ταῖς <lb n="1799,001,,,,1,38,1"></lb> |
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1069 αὐταῖς παραλλήλοις ἴσα ἀλλήλοις ἐστίν.</s> <lb n="1799,001,,,,1,38,2"></lb> |
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1070 <s>Ἔστω τρίγωνα τὰ ΑΒΓ, <lb n="1799,001,,,,1,38,3"></lb> |
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1071 ΔΕΖ ἐπὶ ἴσων βάσεων τῶν <lb n="1799,001,,,,1,38,4"></lb> |
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1072 ΒΓ, ΕΖ καὶ ἐν ταῖς αὐταῖς <lb n="1799,001,,,,1,38,5"></lb> |
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1073 παραλλήλοις ταῖς ΒΖ, ΑΔ· <pb></pb> <lb n="1799,001,,,,1,38,6"></lb> |
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1074 λέγω, ὅτι ἴσον ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ <lb n="1799,001,,,,1,38,7"></lb> |
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1075 τριγώνῳ.</s> <lb n="1799,001,,,,1,38,8"></lb> |
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1076 <s>Ἐκβεβλήσθω γὰρ ἡ ΑΔ ἐφ' ἑκάτερα τὰ μέρη ἐπὶ τὰ Η, <lb n="1799,001,,,,1,38,9"></lb> |
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1077 Θ, καὶ διὰ μὲν τοῦ Β τῇ ΓΑ παράλληλος ἤχθω ἡ ΒΗ, <lb n="1799,001,,,,1,38,10"></lb> |
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1078 διὰ δὲ τοῦ Ζ τῇ ΔΕ παράλληλος ἤχθω ἡ ΖΘ.</s> <s>παραλλη-<lb n="1799,001,,,,1,38,11"></lb> |
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1079 λόγραμμον ἄρα ἐστὶν ἑκάτερον τῶν ΗΒΓΑ, ΔΕΖΘ· <lb n="1799,001,,,,1,38,12"></lb> |
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1080 καὶ ἴσον τὸ ΗΒΓΑ τῷ ΔΕΖΘ· ἐπί τε γὰρ ἴσων βάσεών <lb n="1799,001,,,,1,38,13"></lb> |
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1081 εἰσι τῶν ΒΓ, ΕΖ καὶ ἐν ταῖς αὐταῖς παραλλήλοις ταῖς <lb n="1799,001,,,,1,38,14"></lb> |
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1082 ΒΖ, ΗΘ· καί ἐστι τοῦ μὲν ΗΒΓΑ παραλληλογράμμου <lb n="1799,001,,,,1,38,15"></lb> |
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1083 ἥμισυ τὸ ΑΒΓ τρίγωνον.</s> <s>ἡ γὰρ ΑΒ διάμετρος αὐτὸ δίχα <lb n="1799,001,,,,1,38,16"></lb> |
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1084 τέμνει· τοῦ δὲ ΔΕΖΘ παραλληλογράμμου ἥμισυ τὸ <lb n="1799,001,,,,1,38,17"></lb> |
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1085 ΖΕΔ τρίγωνον· ἡ γὰρ ΔΖ διάμετρος αὐτὸ δίχα τέμνει· <lb n="1799,001,,,,1,38,18"></lb> |
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1086 [τὰ δὲ τῶν ἴσων ἡμίση ἴσα ἀλλήλοις ἐστίν].</s> <s>ἴσον ἄρα ἐστὶ <lb n="1799,001,,,,1,38,19"></lb> |
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1087 τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ.</s> <lb n="1799,001,,,,1,38,20"></lb> |
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1088 <s>Τὰ ἄρα τρίγωνα τὰ ἐπὶ ἴσων βάσεων ὄντα καὶ ἐν ταῖς <lb n="1799,001,,,,1,38,21"></lb> |
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1089 αὐταῖς παραλλήλοις ἴσα ἀλλήλοις ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,38,22"></lb> |
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1090 <s>Τὰ ἴσα τρίγωνα τὰ ἐπὶ τῆς αὐτῆς βάσεως ὄντα καὶ ἐπὶ <lb n="1799,001,,,,1,39,1"></lb> |
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1091 τὰ αὐτὰ μέρη καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,1,39,2"></lb> |
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1092 Ἔστω ἴσα τρίγωνα τὰ ΑΒΓ, ΔΒΓ ἐπὶ τῆς αὐτῆς <lb n="1799,001,,,,1,39,3"></lb> |
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1093 βάσεως ὄντα καὶ ἐπὶ τὰ αὐτὰ μέρη τῆς ΒΓ· λέγω, ὅτι <lb n="1799,001,,,,1,39,4"></lb> |
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1094 καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστίν.</s> <lb n="1799,001,,,,1,39,5"></lb> |
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1095 <s>Ἐπεζεύχθω γὰρ ἡ ΑΔ· λέγω, ὅτι παράλληλός ἐστιν <lb n="1799,001,,,,1,39,6"></lb> |
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1096 ἡ ΑΔ τῇ ΒΓ.</s> <lb n="1799,001,,,,1,39,7"></lb> |
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1097 <s>Εἰ γὰρ μή, ἤχθω διὰ τοῦ Α σημείου τῇ ΒΓ εὐθείᾳ <lb n="1799,001,,,,1,39,8"></lb> |
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1098 παράλληλος ἡ ΑΕ, καὶ ἐπεζεύχθω ἡ ΕΓ.</s> <s>ἴσον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,1,39,9"></lb> |
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1099 ΑΒΓ τρίγωνον τῷ ΕΒΓ τριγώνῳ· ἐπί <lb n="1799,001,,,,1,39,10"></lb> |
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1100 τε γὰρ τῆς αὐτῆς βάσεώς ἐστιν αὐτῷ τῆς <lb n="1799,001,,,,1,39,11"></lb> |
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1101 ΒΓ καὶ ἐν ταῖς αὐταῖς παραλλήλοις.</s> <s>ἀλλὰ <lb n="1799,001,,,,1,39,12"></lb> |
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1102 τὸ ΑΒΓ τῷ ΔΒΓ ἐστιν ἴσον· καὶ τὸ ΔΒΓ <lb n="1799,001,,,,1,39,13"></lb> |
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1103 ἄρα τῷ ΕΒΓ ἴσον ἐστὶ τὸ μεῖζον τῷ <lb n="1799,001,,,,1,39,14"></lb> |
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1104 ἐλάσσονι· ὅπερ ἐστὶν ἀδύνατον· οὐκ ἄρα παράλληλός <lb n="1799,001,,,,1,39,15"></lb> |
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1105 ἐστιν ἡ ΑΕ τῇ ΒΓ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι οὐδ' ἄλλη τις <lb n="1799,001,,,,1,39,16"></lb> |
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1106 πλὴν τῆς ΑΔ· ἡ ΑΔ ἄρα τῇ ΒΓ ἐστι παράλληλος.</s> <lb n="1799,001,,,,1,39,17"></lb> |
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1107 <s>Τὰ ἄρα ἴσα τρίγωνα τὰ ἐπὶ τῆς αὐτῆς βάσεως ὄντα καὶ <lb n="1799,001,,,,1,39,18"></lb> |
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1108 ἐπὶ τὰ αὐτὰ μέρη καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστίν· <lb n="1799,001,,,,1,39,19"></lb> |
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1109 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,39,20"></lb> |
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1110 <s>Τὰ ἴσα τρίγωνα τὰ ἐπὶ ἴσων βάσεων ὄντα καὶ ἐπὶ τὰ <lb n="1799,001,,,,1,40,1"></lb> |
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1111 αὐτὰ μέρη καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,1,40,2"></lb> |
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1112 Ἔστω ἴσα τρίγωνα τὰ ΑΒΓ, ΓΔΕ <lb n="1799,001,,,,1,40,3"></lb> |
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1113 ἐπὶ ἴσων βάσεων τῶν ΒΓ, ΓΕ καὶ ἐπὶ <lb n="1799,001,,,,1,40,4"></lb> |
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1114 τὰ αὐτὰ μέρη.</s> <s>λέγω, ὅτι καὶ ἐν ταῖς αὐ-<lb n="1799,001,,,,1,40,5"></lb> |
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1115 ταῖς παραλλήλοις ἐστίν.</s> <lb n="1799,001,,,,1,40,6"></lb> |
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1116 <s>Ἐπεζεύχθω γὰρ ἡ ΑΔ· λέγω, ὅτι παράλληλός <lb n="1799,001,,,,1,40,7"></lb> |
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1117 ἐστιν ἡ ΑΔ τῇ ΒΕ.</s> <lb n="1799,001,,,,1,40,8"></lb> |
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1118 <s>Εἰ γὰρ μή, ἤχθω διὰ τοῦ Α τῇ ΒΕ παράλληλος ἡ ΑΖ, <lb n="1799,001,,,,1,40,9"></lb> |
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1119 καὶ ἐπεζεύχθω ἡ ΖΕ.</s> <s>ἴσον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον <lb n="1799,001,,,,1,40,10"></lb> |
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1120 τῷ ΖΓΕ τριγώνῳ· ἐπί τε γὰρ ἴσων βάσεών εἰσι τῶν ΒΓ, <lb n="1799,001,,,,1,40,11"></lb> |
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1121 ΓΕ καὶ ἐν ταῖς αὐταῖς παραλλήλοις ταῖς ΒΕ, ΑΖ.</s> <s>ἀλλὰ <lb n="1799,001,,,,1,40,12"></lb> |
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1122 τὸ ΑΒΓ τρίγωνον ἴσον ἐστὶ τῷ ΔΓΕ [τριγώνῳ]· καὶ τὸ <lb n="1799,001,,,,1,40,13"></lb> |
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1123 ΔΓΕ ἄρα [τρίγωνον] ἴσον ἐστὶ τῷ ΖΓΕ τριγώνῳ τὸ μεῖ-<lb n="1799,001,,,,1,40,14"></lb> |
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1124 ζον τῷ ἐλάσσονι· ὅπερ ἐστὶν ἀδύνατον· οὐκ ἄρα παράλλη-<lb n="1799,001,,,,1,40,15"></lb> |
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1125 λος ἡ ΑΖ τῇ ΒΕ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι οὐδ' ἄλλη τις <lb n="1799,001,,,,1,40,16"></lb> |
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1126 πλὴν τῆς ΑΔ· ἡ ΑΔ ἄρα τῇ ΒΕ ἐστι παράλληλος.</s> <lb n="1799,001,,,,1,40,17"></lb> |
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1127 <s>Τὰ ἄρα ἴσα τρίγωνα τὰ ἐπὶ ἴσων βάσεων ὄντα καὶ <lb n="1799,001,,,,1,40,18"></lb> |
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1128 ἐπὶ τὰ αὐτὰ μέρη καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστίν· <lb n="1799,001,,,,1,40,19"></lb> |
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1129 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,40,20"></lb> |
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1130 <s>Ἐὰν παραλληλόγραμμον τριγώνῳ βάσιν τε ἔχῃ τὴν αὐτὴν <lb n="1799,001,,,,1,41,1"></lb> |
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1131 καὶ ἐν ταῖς αὐταῖς παραλλήλοις ᾖ, διπλάσιόν ἐστι τὸ παρ-<lb n="1799,001,,,,1,41,2"></lb> |
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1132 αλληλόγραμμον τοῦ τριγώνου.</s> <s><pb></pb> <lb n="1799,001,,,,1,41,3"></lb> |
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1133 Παραλληλόγραμμον γὰρ τὸ ΑΒΓΔ τριγώνῳ τῷ ΕΒΓ <lb n="1799,001,,,,1,41,4"></lb> |
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1134 βάσιν τε ἐχέτω τὴν αὐτὴν τὴν ΒΓ καὶ ἐν ταῖς αὐταῖς <lb n="1799,001,,,,1,41,5"></lb> |
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1135 παραλλήλοις ἔστω ταῖς ΒΓ, ΑΕ· λέγω, ὅτι διπλάσιόν <lb n="1799,001,,,,1,41,6"></lb> |
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1136 ἐστι τὸ ΑΒΓΔ παραλληλόγραμμον τοῦ <lb n="1799,001,,,,1,41,7"></lb> |
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1137 ΒΕΓ τριγώνου.</s> <lb n="1799,001,,,,1,41,8"></lb> |
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1138 <s>Ἐπεζεύχθω γὰρ ἡ ΑΓ.</s> <s>ἴσον δή ἐστι <lb n="1799,001,,,,1,41,9"></lb> |
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1139 τὸ ΑΒΓ τρίγωνον τῷ ΕΒΓ τριγώνῳ· <lb n="1799,001,,,,1,41,10"></lb> |
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1140 ἐπί τε γὰρ τῆς αὐτῆς βάσεώς ἐστιν αὐτῷ <lb n="1799,001,,,,1,41,11"></lb> |
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1141 τῆς ΒΓ καὶ ἐν ταῖς αὐταῖς παραλλήλοις <lb n="1799,001,,,,1,41,12"></lb> |
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1142 ταῖς ΒΓ, ΑΕ.</s> <s>ἀλλὰ τὸ ΑΒΓΔ παραλληλόγραμμον διπλά-<lb n="1799,001,,,,1,41,13"></lb> |
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1143 σιόν ἐστι τοῦ ΑΒΓ τριγώνου· ἡ γὰρ ΑΓ διάμετρος αὐτὸ <lb n="1799,001,,,,1,41,14"></lb> |
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1144 δίχα τέμνει· ὥστε τὸ ΑΒΓΔ παραλληλόγραμμον καὶ τοῦ <lb n="1799,001,,,,1,41,15"></lb> |
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1145 ΕΒΓ τριγώνου ἐστὶ διπλάσιον.</s> <lb n="1799,001,,,,1,41,16"></lb> |
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1146 <s>Ἐὰν ἄρα παραλληλόγραμμον τριγώνῳ βάσιν τε ἔχῃ τὴν <lb n="1799,001,,,,1,41,17"></lb> |
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1147 αὐτὴν καὶ ἐν ταῖς αὐταῖς παραλλήλοις ᾖ, διπλάσιόν ἐστι <lb n="1799,001,,,,1,41,18"></lb> |
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1148 τὸ παραλληλόγραμμον τοῦ τριγώνου· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,41,19"></lb> |
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1149 <s>Τῷ δοθέντι τριγώνῳ ἴσον παραλληλόγραμμον συστή-<lb n="1799,001,,,,1,42,1"></lb> |
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1150 σασθαι ἐν τῇ δοθείσῃ γωνίᾳ εὐθυγράμμῳ.</s> <s><pb></pb> <lb n="1799,001,,,,1,42,2"></lb> |
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1151 Ἔστω τὸ μὲν δοθὲν τρίγωνον τὸ ΑΒΓ, ἡ δὲ δοθεῖσα <lb n="1799,001,,,,1,42,3"></lb> |
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1152 γωνία εὐθύγραμμος ἡ Δ· δεῖ δὴ τῷ ΑΒΓ τριγώνῳ ἴσον <lb n="1799,001,,,,1,42,4"></lb> |
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1153 παραλληλόγραμμον συστήσασθαι ἐν τῇ Δ γωνίᾳ εὐθυ-<lb n="1799,001,,,,1,42,5"></lb> |
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1154 γράμμῳ.</s> <lb n="1799,001,,,,1,42,6"></lb> |
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1155 <s>Τετμήσθω ἡ ΒΓ δίχα κατὰ τὸ Ε, καὶ ἐπεζεύχθω ἡ <lb n="1799,001,,,,1,42,7"></lb> |
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1156 ΑΕ, καὶ συνεστάτω πρὸς τῇ ΕΓ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ <lb n="1799,001,,,,1,42,8"></lb> |
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1157 σημείῳ τῷ Ε τῇ Δ γωνίᾳ ἴση ἡ ὑπὸ <lb n="1799,001,,,,1,42,9"></lb> |
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1158 ΓΕΖ, καὶ διὰ μὲν τοῦ Α τῇ ΕΓ <lb n="1799,001,,,,1,42,10"></lb> |
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1159 παράλληλος ἤχθω ἡ ΑΗ, διὰ δὲ <lb n="1799,001,,,,1,42,11"></lb> |
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1160 τοῦ Γ τῇ ΕΖ παράλληλος ἤχθω ἡ <lb n="1799,001,,,,1,42,12"></lb> |
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1161 ΓΗ· παραλληλόγραμμον ἄρα ἐστὶ <lb n="1799,001,,,,1,42,13"></lb> |
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1162 τὸ ΖΕΓΗ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΕ <lb n="1799,001,,,,1,42,14"></lb> |
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1163 τῇ ΕΓ, ἴσον ἐστὶ καὶ τὸ ΑΒΕ τρίγωνον τῷ ΑΕΓ τριγώνῳ· <lb n="1799,001,,,,1,42,15"></lb> |
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1164 ἐπί τε γὰρ ἴσων βάσεών εἰσι τῶν ΒΕ, ΕΓ καὶ ἐν ταῖς αὐ-<lb n="1799,001,,,,1,42,16"></lb> |
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1165 ταῖς παραλλήλοις ταῖς ΒΓ, ΑΗ· διπλάσιον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,1,42,17"></lb> |
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1166 ΑΒΓ τρίγωνον τοῦ ΑΕΓ τριγώνου.</s> <s>ἔστι δὲ καὶ τὸ ΖΕΓΗ <lb n="1799,001,,,,1,42,18"></lb> |
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1167 παραλληλόγραμμον διπλάσιον τοῦ ΑΕΓ τριγώνου· βάσιν τε <lb n="1799,001,,,,1,42,19"></lb> |
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1168 γὰρ αὐτῷ τὴν αὐτὴν ἔχει καὶ ἐν ταῖς αὐταῖς ἐστιν αὐτῷ <lb n="1799,001,,,,1,42,20"></lb> |
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1169 παραλλήλοις· ἴσον ἄρα ἐστὶ τὸ ΖΕΓΗ παραλληλόγραμ-<lb n="1799,001,,,,1,42,21"></lb> |
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1170 μον τῷ ΑΒΓ τριγώνῳ.</s> <s>καὶ ἔχει τὴν ὑπὸ ΓΕΖ γωνίαν <lb n="1799,001,,,,1,42,22"></lb> |
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1171 ἴσην τῇ δοθείσῃ τῇ Δ.</s> <lb n="1799,001,,,,1,42,23"></lb> |
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1172 <s>Τῷ ἄρα δοθέντι τριγώνῳ τῷ ΑΒΓ ἴσον παραλληλό- <pb></pb><lb n="1799,001,,,,1,42,24"></lb> |
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1173 γραμμον συνέσταται τὸ ΖΕΓΗ ἐν γωνίᾳ τῇ ὑπὸ ΓΕΖ, <lb n="1799,001,,,,1,42,25"></lb> |
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1174 ἥτις ἐστὶν ἴση τῇ Δ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,42,26"></lb> |
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1175 <s>Παντὸς παραλληλογράμμου τῶν περὶ τὴν διάμετρον <lb n="1799,001,,,,1,43,1"></lb> |
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1176 παραλληλογράμμων τὰ παραπληρώματα ἴσα ἀλλήλοις <lb n="1799,001,,,,1,43,2"></lb> |
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1177 ἐστίν.</s> <lb n="1799,001,,,,1,43,3"></lb> |
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1178 <s>Ἔστω παραλληλόγραμμον τὸ ΑΒΓΔ, διάμετρος δὲ αὐ-<lb n="1799,001,,,,1,43,4"></lb> |
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1179 τοῦ ἡ ΑΓ, περὶ δὲ τὴν ΑΓ παραλληλόγραμμα μὲν ἔστω τὰ <lb n="1799,001,,,,1,43,5"></lb> |
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1180 ΖΘ, ΖΗ, τὰ δὲ λεγόμενα παραπληρώ- <lb n="1799,001,,,,1,43,6"></lb> |
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1181 ματα τὰ ΒΚ, ΚΔ· λέγω, ὅτι ἴσον ἐστὶ <lb n="1799,001,,,,1,43,7"></lb> |
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1182 τὸ ΒΚ παραπλήρωμα τῷ ΚΔ παρα-<lb n="1799,001,,,,1,43,8"></lb> |
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1183 πληρώματι.</s> <lb n="1799,001,,,,1,43,9"></lb> |
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1184 <s>Ἐπεὶ γὰρ παραλληλόγραμμόν ἐστι τὸ <lb n="1799,001,,,,1,43,10"></lb> |
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1185 ΑΒΓΔ, διάμετρος δὲ αὐτοῦ ἡ ΑΓ, ἴσον <lb n="1799,001,,,,1,43,11"></lb> |
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1186 ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΑΓΔ τριγώνῳ.</s> <s>πάλιν, ἐπεὶ <lb n="1799,001,,,,1,43,12"></lb> |
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1187 παραλληλόγραμμόν ἐστι τὸ ΕΘ, διάμετρος δὲ αὐτοῦ ἐστιν <lb n="1799,001,,,,1,43,13"></lb> |
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1188 ἡ ΑΚ, ἴσον ἐστὶ τὸ ΑΕΚ τρίγωνον τῷ ΑΘΚ τριγώνῳ.</s> <lb n="1799,001,,,,1,43,14"></lb> |
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1189 <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ ΚΖΓ τρίγωνον τῷ ΚΗΓ ἐστιν ἴσον.</s> <lb n="1799,001,,,,1,43,15"></lb> |
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1190 <s>ἐπεὶ οὖν τὸ μὲν ΑΕΚ τρίγωνον τῷ ΑΘΚ τριγώνῳ ἐστὶν <pb></pb> <lb n="1799,001,,,,1,43,16"></lb> |
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1191 ἴσον, τὸ δὲ ΚΖΓ τῷ ΚΗΓ, τὸ ΑΕΚ τρίγωνον μετὰ τοῦ <lb n="1799,001,,,,1,43,17"></lb> |
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1192 ΚΗΓ ἴσον ἐστὶ τῷ ΑΘΚ τριγώνῳ μετὰ τοῦ ΚΖΓ· ἔστι <lb n="1799,001,,,,1,43,18"></lb> |
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1193 δὲ καὶ ὅλον τὸ ΑΒΓ τρίγωνον ὅλῳ τῷ ΑΔΓ ἴσον· λοιπὸν <lb n="1799,001,,,,1,43,19"></lb> |
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1194 ἄρα τὸ ΒΚ παραπλήρωμα λοιπῷ τῷ ΚΔ παραπληρώματί <lb n="1799,001,,,,1,43,20"></lb> |
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1195 ἐστιν ἴσον.</s> <lb n="1799,001,,,,1,43,21"></lb> |
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1196 <s>Παντὸς ἄρα παραλληλογράμμου χωρίου τῶν περὶ τὴν <lb n="1799,001,,,,1,43,22"></lb> |
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1197 διάμετρον παραλληλογράμμων τὰ παραπληρώματα ἴσα <lb n="1799,001,,,,1,43,23"></lb> |
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1198 ἀλλήλοις ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,43,24"></lb> |
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1199 <s>Παρὰ τὴν δοθεῖσαν εὐθεῖαν τῷ δοθέντι τριγώνῳ ἴσον <lb n="1799,001,,,,1,44,1"></lb> |
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1200 παραλληλόγραμμον παραβαλεῖν ἐν τῇ δοθείσῃ γωνίᾳ <lb n="1799,001,,,,1,44,2"></lb> |
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1201 εὐθυγράμμῳ.</s> <lb n="1799,001,,,,1,44,3"></lb> |
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1202 <s>Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ δοθὲν τρίγω-<lb n="1799,001,,,,1,44,4"></lb> |
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1203 νον τὸ Γ, ἡ δὲ δοθεῖσα γωνία εὐθύ-<lb n="1799,001,,,,1,44,5"></lb> |
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1204 γραμμος ἡ Δ· δεῖ δὴ παρὰ τὴν δοθεῖσαν <lb n="1799,001,,,,1,44,6"></lb> |
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1205 εὐθεῖαν τὴν ΑΒ τῷ δοθέντι τριγώνῳ <lb n="1799,001,,,,1,44,7"></lb> |
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1206 τῷ Γ ἴσον παραλληλόγραμμον παραβα-<lb n="1799,001,,,,1,44,8"></lb> |
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1207 λεῖν ἐν ἴσῃ τῇ Δ γωνίᾳ.</s> <lb n="1799,001,,,,1,44,9"></lb> |
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1208 <s>Συνεστάτω τῷ Γ τριγώνῳ ἴσον παραλ-<lb n="1799,001,,,,1,44,10"></lb> |
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1209 ληλόγραμμον τὸ ΒΕΖΗ ἐν γωνίᾳ τῇ <lb n="1799,001,,,,1,44,11"></lb> |
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1210 ὑπὸ ΕΒΗ, ἥ ἐστιν ἴση τῇ Δ· καὶ <lb n="1799,001,,,,1,44,12"></lb> |
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1211 κείσθω ὥστε ἐπ' εὐθείας εἶναι τὴν ΒΕ <lb n="1799,001,,,,1,44,13"></lb> |
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1212 τῇ ΑΒ, καὶ διήχθω ἡ ΖΗ ἐπὶ τὸ Θ, <pb></pb> <lb n="1799,001,,,,1,44,14"></lb> |
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1213 καὶ διὰ τοῦ Α ὁποτέρᾳ τῶν ΒΗ, ΕΖ παράλληλος ἤχθω <lb n="1799,001,,,,1,44,15"></lb> |
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1214 ἡ ΑΘ, καὶ ἐπεζεύχθω ἡ ΘΒ.</s> <s>καὶ ἐπεὶ εἰς παραλλήλους <lb n="1799,001,,,,1,44,16"></lb> |
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1215 τὰς ΑΘ, ΕΖ εὐθεῖα ἐνέπεσεν ἡ ΘΖ, αἱ ἄρα ὑπὸ ΑΘΖ, <lb n="1799,001,,,,1,44,17"></lb> |
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1216 ΘΖΕ γωνίαι δυσὶν ὀρθαῖς εἰσιν ἴσαι.</s> <s>αἱ ἄρα ὑπὸ ΒΘΗ, <lb n="1799,001,,,,1,44,18"></lb> |
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1217 ΗΖΕ δύο ὀρθῶν ἐλάσσονές εἰσιν· αἱ δὲ ἀπὸ ἐλασσόνων <lb n="1799,001,,,,1,44,19"></lb> |
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1218 ἢ δύο ὀρθῶν εἰς ἄπειρον ἐκβαλλόμεναι συμπίπτουσιν· <lb n="1799,001,,,,1,44,20"></lb> |
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1219 αἱ ΘΒ, ΖΕ ἄρα ἐκβαλλόμεναι συμπεσοῦνται.</s> <s>ἐκβεβλή-<lb n="1799,001,,,,1,44,21"></lb> |
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1220 σθωσαν καὶ συμπιπτέτωσαν κατὰ τὸ Κ, καὶ διὰ τοῦ <lb n="1799,001,,,,1,44,22"></lb> |
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1221 Κ σημείου ὁποτέρᾳ τῶν ΕΑ, ΖΘ παράλληλος ἤχθω ἡ <lb n="1799,001,,,,1,44,23"></lb> |
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1222 ΚΛ, καὶ ἐκβεβλήσθωσαν αἱ ΘΑ, ΗΒ ἐπὶ τὰ Λ, Μ ση-<lb n="1799,001,,,,1,44,24"></lb> |
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1223 μεῖα.</s> <s>παραλληλόγραμμον ἄρα ἐστὶ τὸ ΘΛΚΖ, διάμετρος <lb n="1799,001,,,,1,44,25"></lb> |
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1224 δὲ αὐτοῦ ἡ ΘΚ, περὶ δὲ τὴν ΘΚ παραλληλόγραμμα <lb n="1799,001,,,,1,44,26"></lb> |
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1225 μὲν τὰ ΑΗ, ΜΕ, τὰ δὲ λεγόμενα παραπληρώματα τὰ <lb n="1799,001,,,,1,44,27"></lb> |
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1226 ΛΒ, ΒΖ· ἴσον ἄρα ἐστὶ τὸ ΛΒ τῷ ΒΖ.</s> <s>ἀλλὰ τὸ ΒΖ τῷ <lb n="1799,001,,,,1,44,28"></lb> |
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1227 Γ τριγώνῳ ἐστὶν ἴσον· καὶ τὸ ΛΒ ἄρα τῷ Γ ἐστιν ἴσον.</s> <lb n="1799,001,,,,1,44,29"></lb> |
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1228 <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ὑπὸ ΗΒΕ γωνία τῇ ὑπὸ ΑΒΜ, ἀλλὰ <lb n="1799,001,,,,1,44,30"></lb> |
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1229 ἡ ὑπὸ ΗΒΕ τῇ Δ ἐστιν ἴση, καὶ ἡ ὑπὸ ΑΒΜ ἄρα τῇ Δ <lb n="1799,001,,,,1,44,31"></lb> |
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1230 γωνίᾳ ἐστὶν ἴση.</s> <lb n="1799,001,,,,1,44,32"></lb> |
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1231 <s>Παρὰ τὴν δοθεῖσαν ἄρα εὐθεῖαν τὴν ΑΒ τῷ δοθέντι τριγώ-<lb n="1799,001,,,,1,44,33"></lb> |
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1232 νῳ τῷ Γ ἴσον παραλληλόγραμμον παραβέβληται τὸ ΛΒ ἐν <lb n="1799,001,,,,1,44,34"></lb> |
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1233 γωνίᾳ τῇ ὑπὸ ΑΒΜ, ἥ ἐστιν ἴση τῇ Δ· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,1,44,35"></lb> |
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1234 Τῷ δοθέντι εὐθυγράμμῳ ἴσον παραλληλόγραμμον <lb n="1799,001,,,,1,45,1"></lb> |
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1235 συστήσασθαι ἐν τῇ δοθείσῃ γωνίᾳ εὐθυγράμμῳ.</s> <lb n="1799,001,,,,1,45,2"></lb> |
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1236 <s>Ἔστω τὸ μὲν δοθὲν εὐθύγραμμον τὸ <lb n="1799,001,,,,1,45,3"></lb> |
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1237 ΑΒΓΔ, ἡ δὲ δοθεῖσα γωνία εὐθύγραμμος <lb n="1799,001,,,,1,45,4"></lb> |
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1238 ἡ Ε· δεῖ δὴ τῷ ΑΒΓΔ εὐθυγράμμῳ ἴσον <lb n="1799,001,,,,1,45,5"></lb> |
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1239 παραλληλόγραμμον συστήσασθαι ἐν τῇ δο-<lb n="1799,001,,,,1,45,6"></lb> |
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1240 θείσῃ γωνίᾳ τῇ Ε.</s> <lb n="1799,001,,,,1,45,7"></lb> |
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1241 <s>Ἐπεζεύχθω ἡ ΔΒ, καὶ συνεστάτω τῷ <lb n="1799,001,,,,1,45,8"></lb> |
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1242 ΑΒΔ τριγώνῳ ἴσον παραλληλόγραμμον τὸ <lb n="1799,001,,,,1,45,9"></lb> |
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1243 ΖΘ ἐν τῇ ὑπὸ ΘΚΖ γωνίᾳ, ἥ ἐστιν ἴση <lb n="1799,001,,,,1,45,10"></lb> |
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1244 τῇ Ε· καὶ παραβεβλήσθω παρὰ τὴν ΗΘ <lb n="1799,001,,,,1,45,11"></lb> |
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1245 εὐθεῖαν τῷ ΔΒΓ τριγώνῳ ἴσον παραλληλό-<lb n="1799,001,,,,1,45,12"></lb> |
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1246 γραμμον τὸ ΗΜ ἐν τῇ ὑπὸ ΗΘΜ γωνίᾳ, ἥ ἐστιν ἴση τῇ Ε.</s> <lb n="1799,001,,,,1,45,13"></lb> |
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1247 <s>καὶ ἐπεὶ ἡ Ε γωνία ἑκατέρᾳ τῶν ὑπὸ ΘΚΖ, ΗΘΜ ἐστιν ἴση, <lb n="1799,001,,,,1,45,14"></lb> |
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1248 καὶ ἡ ὑπὸ ΘΚΖ ἄρα τῇ ὑπὸ ΗΘΜ ἐστιν ἴση.</s> <s>κοινὴ προσκεί-<lb n="1799,001,,,,1,45,15"></lb> |
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1249 σθω ἡ ὑπὸ ΚΘΗ· αἱ ἄρα ὑπὸ ΖΚΘ, ΚΘΗ ταῖς ὑπὸ ΚΘΗ, <lb n="1799,001,,,,1,45,16"></lb> |
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1250 ΗΘΜ ἴσαι εἰσίν.</s> <s>ἀλλ' αἱ ὑπὸ ΖΚΘ, ΚΘΗ δυσὶν ὀρθαῖς <lb n="1799,001,,,,1,45,17"></lb> |
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1251 ἴσαι εἰσίν· καὶ αἱ ὑπὸ ΚΘΗ, ΗΘΜ ἄρα δύο ὀρθαῖς ἴσαι <lb n="1799,001,,,,1,45,18"></lb> |
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1252 εἰσίν.</s> <s>πρὸς δή τινι εὐθείᾳ τῇ ΗΘ καὶ τῷ πρὸς αὐτῇ σημείῳ <lb n="1799,001,,,,1,45,19"></lb> |
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1253 τῷ Θ δύο εὐθεῖαι αἱ ΚΘ, ΘΜ μὴ ἐπὶ τὰ αὐτὰ μέρη κείμε-<lb n="1799,001,,,,1,45,20"></lb> |
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1254 ναι τὰς ἐφεξῆς γωνίας δύο ὀρθαῖς ἴσας ποιοῦσιν· ἐπ' <pb></pb> <lb n="1799,001,,,,1,45,21"></lb> |
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1255 εὐθείας ἄρα ἐστὶν ἡ ΚΘ τῇ ΘΜ· καὶ ἐπεὶ εἰς παραλλήλους <lb n="1799,001,,,,1,45,22"></lb> |
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1256 τὰς ΚΜ, ΖΗ εὐθεῖα ἐνέπεσεν ἡ ΘΗ, αἱ ἐναλλὰξ γωνίαι <lb n="1799,001,,,,1,45,23"></lb> |
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1257 αἱ ὑπὸ ΜΘΗ, ΘΗΖ ἴσαι ἀλλήλαις εἰσίν.</s> <s>κοινὴ προσκεί-<lb n="1799,001,,,,1,45,24"></lb> |
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1258 σθω ἡ ὑπὸ ΘΗΛ· αἱ ἄρα ὑπὸ ΜΘΗ, ΘΗΛ ταῖς ὑπὸ <lb n="1799,001,,,,1,45,25"></lb> |
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1259 ΘΗΖ, ΘΗΛ ἴσαι εἰσίν.</s> <s>ἀλλ' αἱ ὑπὸ ΜΘΗ, ΘΗΛ δύο <lb n="1799,001,,,,1,45,26"></lb> |
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1260 ὀρθαῖς ἴσαι εἰσίν· καὶ αἱ ὑπὸ ΘΗΖ, ΘΗΛ ἄρα δύο ὀρθαῖς <lb n="1799,001,,,,1,45,27"></lb> |
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1261 ἴσαι εἰσίν· ἐπ' εὐθείας ἄρα ἐστὶν ἡ ΖΗ τῇ ΗΛ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,1,45,28"></lb> |
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1262 ἡ ΖΚ τῇ ΘΗ ἴση τε καὶ παράλληλός ἐστιν, ἀλλὰ καὶ ἡ <lb n="1799,001,,,,1,45,29"></lb> |
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1263 ΘΗ τῇ ΜΛ, καὶ ἡ ΚΖ ἄρα τῇ ΜΛ ἴση τε καὶ παράλλη-<lb n="1799,001,,,,1,45,30"></lb> |
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1264 λός ἐστιν· καὶ ἐπιζευγνύουσιν αὐτὰς εὐθεῖαι αἱ ΚΜ, ΖΛ· <lb n="1799,001,,,,1,45,31"></lb> |
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1265 καὶ αἱ ΚΜ, ΖΛ ἄρα ἴσαι τε καὶ παράλληλοί εἰσιν· παραλ-<lb n="1799,001,,,,1,45,32"></lb> |
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1266 ληλόγραμμον ἄρα ἐστὶ τὸ ΚΖΛΜ.</s> <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ <lb n="1799,001,,,,1,45,33"></lb> |
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1267 μὲν ΑΒΔ τρίγωνον τῷ ΖΘ παραλληλογράμμῳ, τὸ δὲ <lb n="1799,001,,,,1,45,34"></lb> |
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1268 ΔΒΓ τῷ ΗΜ, ὅλον ἄρα τὸ ΑΒΓΔ εὐθύγραμμον ὅλῳ <lb n="1799,001,,,,1,45,35"></lb> |
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1269 τῷ ΚΖΛΜ παραλληλογράμμῳ ἐστὶν ἴσον.</s> <lb n="1799,001,,,,1,45,36"></lb> |
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1270 <s>Τῷ ἄρα δοθέντι εὐθυγράμμῳ τῷ ΑΒΓΔ ἴσον παραλλη-<lb n="1799,001,,,,1,45,37"></lb> |
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1271 λόγραμμον συνέσταται τὸ ΚΖΛΜ ἐν γωνίᾳ τῇ ὑπὸ ΖΚΜ, <lb n="1799,001,,,,1,45,38"></lb> |
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1272 ἥ ἐστιν ἴση τῇ δοθείσῃ τῇ Ε· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,1,45,39"></lb> |
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1273 Ἀπὸ τῆς δοθείσης εὐθείας τετράγωνον ἀναγράψαι.</s> <lb n="1799,001,,,,1,46,1"></lb> |
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1274 <s>Ἔστω ἡ δοθεῖσα εὐθεῖα ἡ ΑΒ· δεῖ δὴ ἀπὸ τῆς ΑΒ εὐ-<lb n="1799,001,,,,1,46,2"></lb> |
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1275 θείας τετράγωνον ἀναγράψαι.</s> <lb n="1799,001,,,,1,46,3"></lb> |
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1276 <s>Ἤχθω τῇ ΑΒ εὐθείᾳ ἀπὸ τοῦ πρὸς αὐτῇ σημείου τοῦ <lb n="1799,001,,,,1,46,4"></lb> |
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1277 Α πρὸς ὀρθὰς ἡ ΑΓ, καὶ κείσθω τῇ ΑΒ ἴση ἡ ΑΔ· καὶ <lb n="1799,001,,,,1,46,5"></lb> |
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1278 διὰ μὲν τοῦ Δ σημείου τῇ ΑΒ παράλληλος <lb n="1799,001,,,,1,46,6"></lb> |
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1279 ἤχθω ἡ ΔΕ, διὰ δὲ τοῦ Β σημείου τῇ ΑΔ <lb n="1799,001,,,,1,46,7"></lb> |
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1280 παράλληλος ἤχθω ἡ ΒΕ.</s> <s>Παραλληλόγραμμον <lb n="1799,001,,,,1,46,8"></lb> |
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1281 ἄρα ἐστὶ τὸ ΑΔΕΒ· ἴση ἄρα ἐστὶν ἡ μὲν ΑΒ <lb n="1799,001,,,,1,46,9"></lb> |
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1282 τῇ ΔΕ, ἡ δὲ ΑΔ τῇ ΒΕ.</s> <s>ἀλλὰ ἡ ΑΒ τῇ ΑΔ <lb n="1799,001,,,,1,46,10"></lb> |
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1283 ἐστιν ἴση· αἱ τέσσαρες ἄρα αἱ ΒΑ, ΑΔ, ΔΕ, <lb n="1799,001,,,,1,46,11"></lb> |
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1284 ΕΒ ἴσαι ἀλλήλαις εἰσίν· ἰσόπλευρον ἄρα ἐστὶ <lb n="1799,001,,,,1,46,12"></lb> |
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1285 τὸ ΑΔΕΒ παραλληλόγραμμον.</s> <s>λέγω δή, ὅτι <lb n="1799,001,,,,1,46,13"></lb> |
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1286 καὶ ὀρθογώνιον.</s> <s>ἐπεὶ γὰρ εἰς παραλλήλους τὰς <lb n="1799,001,,,,1,46,14"></lb> |
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1287 ΑΒ, ΔΕ εὐθεῖα ἐνέπεσεν ἡ ΑΔ, αἱ ἄρα ὑπὸ ΒΑΔ, ΑΔΕ <lb n="1799,001,,,,1,46,15"></lb> |
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1288 γωνίαι δύο ὀρθαῖς ἴσαι εἰσίν.</s> <s>ὀρθὴ δὲ ἡ ὑπὸ ΒΑΔ· ὀρθὴ <lb n="1799,001,,,,1,46,16"></lb> |
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1289 ἄρα καὶ ἡ ὑπὸ ΑΔΕ.</s> <s>τῶν δὲ παραλληλογράμμων χωρίων <lb n="1799,001,,,,1,46,17"></lb> |
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1290 αἱ ἀπεναντίον πλευραί τε καὶ γωνίαι ἴσαι ἀλλήλαις εἰσίν· <lb n="1799,001,,,,1,46,18"></lb> |
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1291 ὀρθὴ ἄρα καὶ ἑκατέρα τῶν ἀπεναντίον τῶν ὑπὸ ΑΒΕ, <lb n="1799,001,,,,1,46,19"></lb> |
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1292 ΒΕΔ γωνιῶν· ὀρθογώνιον ἄρα ἐστὶ τὸ ΑΔΕΒ.</s> <s>ἐδείχθη <lb n="1799,001,,,,1,46,20"></lb> |
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1293 δὲ καὶ ἰσόπλευρον.</s> <s><pb></pb> <lb n="1799,001,,,,1,46,21"></lb> |
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1294 Τετράγωνον ἄρα ἐστίν· καί ἐστιν ἀπὸ τῆς ΑΒ εὐθείας <lb n="1799,001,,,,1,46,22"></lb> |
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1295 ἀναγεγραμμένον· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,1,46,23"></lb> |
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1296 <s>Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν <lb n="1799,001,,,,1,47,1"></lb> |
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1297 γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς <lb n="1799,001,,,,1,47,2"></lb> |
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1298 ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετρα-<lb n="1799,001,,,,1,47,3"></lb> |
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1299 γώνοις.</s> <lb n="1799,001,,,,1,47,4"></lb> |
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1300 <s>Ἔστω τρίγωνον ὀρθογώνιον τὸ ΑΒΓ ὀρθὴν ἔχον τὴν <lb n="1799,001,,,,1,47,5"></lb> |
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1301 ὑπὸ ΒΑΓ γωνίαν· λέγω, ὅτι τὸ ἀπὸ τῆς ΒΓ τετράγωνον <lb n="1799,001,,,,1,47,6"></lb> |
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1302 ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ τετραγώνοις.</s> <lb n="1799,001,,,,1,47,7"></lb> |
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1303 <s>Ἀναγεγράφθω γὰρ ἀπὸ μὲν τῆς ΒΓ τετράγωνον τὸ <lb n="1799,001,,,,1,47,8"></lb> |
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1304 ΒΔΕΓ, ἀπὸ δὲ τῶν ΒΑ, ΑΓ τὰ ΗΒ, ΘΓ, καὶ διὰ τοῦ <lb n="1799,001,,,,1,47,9"></lb> |
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1305 Α ὁποτέρᾳ τῶν ΒΔ, ΓΕ παράλληλος ἤχθω ἡ ΑΛ· καὶ <lb n="1799,001,,,,1,47,10"></lb> |
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1306 ἐπεζεύχθωσαν αἱ ΑΔ, ΖΓ.</s> <s>καὶ ἐπεὶ ὀρθή ἐστιν ἑκατέρα <lb n="1799,001,,,,1,47,11"></lb> |
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1307 τῶν ὑπὸ ΒΑΓ, ΒΑΗ γωνιῶν, πρὸς δή τινι εὐθείᾳ τῇ ΒΑ <lb n="1799,001,,,,1,47,12"></lb> |
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1308 καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Α δύο εὐθεῖαι αἱ ΑΓ, ΑΗ <lb n="1799,001,,,,1,47,13"></lb> |
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1309 μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς γωνίας δυσὶν <lb n="1799,001,,,,1,47,14"></lb> |
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1310 ὀρθαῖς ἴσας ποιοῦσιν· ἐπ' εὐθείας ἄρα ἐστὶν ἡ ΓΑ τῇ <lb n="1799,001,,,,1,47,15"></lb> |
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1311 ΑΗ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΒΑ τῇ ΑΘ ἐστιν ἐπ' εὐθείας.</s> <lb n="1799,001,,,,1,47,16"></lb> |
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1312 <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ὑπὸ ΔΒΓ γωνία τῇ ὑπὸ ΖΒΑ· ὀρθὴ <lb n="1799,001,,,,1,47,17"></lb> |
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1313 γὰρ ἑκατέρα· κοινὴ προσκείσθω ἡ ὑπὸ ΑΒΓ· ὅλη ἄρα <pb></pb> <lb n="1799,001,,,,1,47,18"></lb> |
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1314 ἡ ὑπὸ ΔΒΑ ὅλῃ τῇ ὑπὸ ΖΒΓ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ ἴση <lb n="1799,001,,,,1,47,19"></lb> |
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1315 ἐστὶν ἡ μὲν ΔΒ τῇ ΒΓ, ἡ δὲ ΖΒ τῇ ΒΑ, δύο δὴ αἱ ΔΒ, <lb n="1799,001,,,,1,47,20"></lb> |
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1316 ΒΑ δύο ταῖς ΖΒ, ΒΓ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ <lb n="1799,001,,,,1,47,21"></lb> |
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1317 γωνία ἡ ὑπὸ ΔΒΑ γωνίᾳ τῇ ὑπὸ <lb n="1799,001,,,,1,47,22"></lb> |
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1318 ΖΒΓ ἴση· βάσις ἄρα ἡ ΑΔ βάσει <lb n="1799,001,,,,1,47,23"></lb> |
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1319 τῇ ΖΓ [ἐστιν] ἴση, καὶ τὸ ΑΒΔ τρί-<lb n="1799,001,,,,1,47,24"></lb> |
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1320 γωνον τῷ ΖΒΓ τριγώνῳ ἐστὶν ἴσον· <lb n="1799,001,,,,1,47,25"></lb> |
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1321 καὶ [ἐστὶ] τοῦ μὲν ΑΒΔ τριγώνου <lb n="1799,001,,,,1,47,26"></lb> |
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1322 διπλάσιον τὸ ΒΛ παραλληλόγραμ-<lb n="1799,001,,,,1,47,27"></lb> |
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1323 μον· βάσιν τε γὰρ τὴν αὐτὴν ἔχουσι <lb n="1799,001,,,,1,47,28"></lb> |
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1324 τὴν ΒΔ καὶ ἐν ταῖς αὐταῖς εἰσι <lb n="1799,001,,,,1,47,29"></lb> |
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1325 παραλλήλοις ταῖς ΒΔ, ΑΛ· τοῦ <lb n="1799,001,,,,1,47,30"></lb> |
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1326 δὲ ΖΒΓ τριγώνου διπλάσιον τὸ <lb n="1799,001,,,,1,47,31"></lb> |
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1327 ΗΒ τετράγωνον· βάσιν τε γὰρ πάλιν τὴν αὐτὴν ἔχουσι <lb n="1799,001,,,,1,47,32"></lb> |
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1328 τὴν ΖΒ καὶ ἐν ταῖς αὐταῖς εἰσι παραλλήλοις ταῖς ΖΒ, ΗΓ.</s> <lb n="1799,001,,,,1,47,33"></lb> |
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1329 <s>[τὰ δὲ τῶν ἴσων διπλάσια ἴσα ἀλλήλοις ἐστίν·] ἴσον ἄρα <lb n="1799,001,,,,1,47,34"></lb> |
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1330 ἐστὶ καὶ τὸ ΒΛ παραλληλόγραμμον τῷ ΗΒ τετραγώνῳ.</s> <lb n="1799,001,,,,1,47,35"></lb> |
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1331 <s>ὁμοίως δὴ ἐπιζευγνυμένων τῶν ΑΕ, ΒΚ δειχθήσεται καὶ <lb n="1799,001,,,,1,47,36"></lb> |
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1332 τὸ ΓΛ παραλληλόγραμμον ἴσον τῷ ΘΓ τετραγώνῳ· <pb></pb> <lb n="1799,001,,,,1,47,37"></lb> |
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1333 ὅλον ἄρα τὸ ΒΔΕΓ τετράγωνον δυσὶ τοῖς ΗΒ, ΘΓ τετρα-<lb n="1799,001,,,,1,47,38"></lb> |
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1334 γώνοις ἴσον ἐστίν.</s> <s>καί ἐστι τὸ μὲν ΒΔΕΓ τετράγωνον <lb n="1799,001,,,,1,47,39"></lb> |
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1335 ἀπὸ τῆς ΒΓ ἀναγραφέν, τὰ δὲ ΗΒ, ΘΓ ἀπὸ τῶν ΒΑ, <lb n="1799,001,,,,1,47,40"></lb> |
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1336 ΑΓ.</s> <s>τὸ ἄρα ἀπὸ τῆς ΒΓ πλευρᾶς τετράγωνον ἴσον ἐστὶ <lb n="1799,001,,,,1,47,41"></lb> |
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1337 τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν τετραγώνοις.</s> <lb n="1799,001,,,,1,47,42"></lb> |
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1338 <s>Ἐν ἄρα τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν <lb n="1799,001,,,,1,47,43"></lb> |
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1339 γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς <lb n="1799,001,,,,1,47,44"></lb> |
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1340 ἀπὸ τῶν τὴν ὀρθὴν [γωνίαν] περιεχουσῶν πλευρῶν τετρα-<lb n="1799,001,,,,1,47,45"></lb> |
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1341 γώνοις· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,1,47,46"></lb> |
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1342 <s>Ἐὰν τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον <lb n="1799,001,,,,1,48,1"></lb> |
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1343 ἴσον ᾖ τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν <lb n="1799,001,,,,1,48,2"></lb> |
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1344 τετραγώνοις, ἡ περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ <lb n="1799,001,,,,1,48,3"></lb> |
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1345 τριγώνου δύο πλευρῶν ὀρθή ἐστιν.</s> <lb n="1799,001,,,,1,48,4"></lb> |
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1346 <s>Τριγώνου γὰρ τοῦ ΑΒΓ τὸ ἀπὸ μιᾶς τῆς ΒΓ πλευρᾶς <lb n="1799,001,,,,1,48,5"></lb> |
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1347 τετράγωνον ἴσον ἔστω τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν <lb n="1799,001,,,,1,48,6"></lb> |
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1348 τετραγώνοις· λέγω, ὅτι ὀρθή ἐστιν ἡ ὑπὸ ΒΑΓ γωνία.</s> <lb n="1799,001,,,,1,48,7"></lb> |
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1349 <s>Ἤχθω γὰρ ἀπὸ τοῦ Α σημείου τῇ ΑΓ εὐθείᾳ πρὸς <lb n="1799,001,,,,1,48,8"></lb> |
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1350 ὀρθὰς ἡ ΑΔ καὶ κείσθω τῇ ΒΑ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω <lb n="1799,001,,,,1,48,9"></lb> |
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1351 ἡ ΔΓ.</s> <s>ἐπεὶ ἴση ἐστὶν ἡ ΔΑ τῇ ΑΒ, ἴσον ἐστὶ καὶ τὸ ἀπὸ <pb></pb> <lb n="1799,001,,,,1,48,10"></lb> |
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1352 τῆς ΔΑ τετράγωνον τῷ ἀπὸ τῆς ΑΒ τετραγώνῳ.</s> <s>κοινὸν <lb n="1799,001,,,,1,48,11"></lb> |
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1353 προσκείσθω τὸ ἀπὸ τῆς ΑΓ τετράγωνον· τὰ ἄρα ἀπὸ τῶν <lb n="1799,001,,,,1,48,12"></lb> |
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1354 ΔΑ, ΑΓ τετράγωνα ἴσα ἐστὶ τοῖς ἀπὸ τῶν <lb n="1799,001,,,,1,48,13"></lb> |
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1355 ΒΑ, ΑΓ τετραγώνοις.</s> <s>ἀλλὰ τοῖς μὲν ἀπὸ <lb n="1799,001,,,,1,48,14"></lb> |
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1356 τῶν ΔΑ, ΑΓ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΔΓ· ὀρθὴ <lb n="1799,001,,,,1,48,15"></lb> |
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1357 γάρ ἐστιν ἡ ὑπὸ ΔΑΓ γωνία· τοῖς δὲ ἀπὸ <lb n="1799,001,,,,1,48,16"></lb> |
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1358 τῶν ΒΑ, ΑΓ ἴσον ἐστὶ τὸ ἀπὸ ΒΓ· ὑπόκειται <lb n="1799,001,,,,1,48,17"></lb> |
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1359 γάρ· τὸ ἄρα ἀπὸ τῆς ΔΓ τετράγωνον ἴσον <lb n="1799,001,,,,1,48,18"></lb> |
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1360 ἐστὶ τῷ ἀπὸ τῆς ΒΓ τετραγώνῳ· ὥστε καὶ <lb n="1799,001,,,,1,48,19"></lb> |
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1361 πλευρὰ ἡ ΔΓ τῇ ΒΓ ἐστιν ἴση· καὶ ἐπεὶ ἴση <lb n="1799,001,,,,1,48,20"></lb> |
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1362 ἐστὶν ἡ ΔΑ τῇ ΑΒ, κοινὴ δὲ ἡ ΑΓ, δύο δὴ αἱ <lb n="1799,001,,,,1,48,21"></lb> |
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1363 ΔΑ, ΑΓ δύο ταῖς ΒΑ, ΑΓ ἴσαι εἰσίν· καὶ βάσις ἡ ΔΓ <lb n="1799,001,,,,1,48,22"></lb> |
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1364 βάσει τῇ ΒΓ ἴση· γωνία ἄρα ἡ ὑπὸ ΔΑΓ γωνίᾳ τῇ ὑπὸ <lb n="1799,001,,,,1,48,23"></lb> |
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1365 ΒΑΓ [ἐστιν] ἴση.</s> <s>ὀρθὴ δὲ ἡ ὑπὸ ΔΑΓ· ὀρθὴ ἄρα καὶ ἡ <lb n="1799,001,,,,1,48,24"></lb> |
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1366 ὑπὸ ΒΑΓ.</s> <lb n="1799,001,,,,1,48,25"></lb> |
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1367 <s>Ἐὰν ἄρα τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον <lb n="1799,001,,,,1,48,26"></lb> |
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1368 ἴσον ᾖ τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν <lb n="1799,001,,,,1,48,27"></lb> |
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1369 τετραγώνοις, ἡ περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ <lb n="1799,001,,,,1,48,28"></lb> |
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1370 τριγώνου δύο πλευρῶν ὀρθή ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,1,48,29"></lb> |
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1371 </s></p> |
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1372 </chap> |
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1373 <chap><p type="main"><s>Πᾶν παραλληλόγραμμον ὀρθογώνιον περιέχεσθαι <lb n="1799,001,,,,2;HOR,1,1"></lb> |
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1374 |
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1375 λέγεται ὑπὸ δύο τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν <lb n="1799,001,,,,2;HOR,1,2"></lb> |
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1376 εὐθειῶν.</s> <lb n="1799,001,,,,2;HOR,1,3"></lb> |
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1377 <s>Παντὸς δὲ παραλληλογράμμου χωρίου τῶν περὶ <lb n="1799,001,,,,2;HOR,2,1"></lb> |
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1378 τὴν διάμετρον αὐτοῦ παραλληλογράμμων ἓν ὁποιονοῦν <lb n="1799,001,,,,2;HOR,2,2"></lb> |
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1379 σὺν τοῖς δυσὶ παραπληρώμασι γνώμων καλείσθω.</s> <lb n="1799,001,,,,2;HOR,2,3"></lb> |
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1380 <s></s></p> |
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1381 </chap> |
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1382 <chap><p type="main"><s>Ἐὰν ὦσι δύο εὐθεῖαι, τμηθῇ δὲ ἡ ἑτέρα αὐτῶν εἰς ὁσα-<lb n="1799,001,,,,2,1,1"></lb> |
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1383 |
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1384 δηποτοῦν τμήματα, τὸ περιεχόμενον ὀρθογώνιον ὑπὸ τῶν <lb n="1799,001,,,,2,1,2"></lb> |
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1385 δύο εὐθειῶν ἴσον ἐστὶ τοῖς ὑπό τε τῆς ἀτμήτου καὶ ἑκά-<lb n="1799,001,,,,2,1,3"></lb> |
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1386 στου τῶν τμημάτων περιεχομένοις ὀρθογωνίοις.</s> <lb n="1799,001,,,,2,1,4"></lb> |
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1387 <s>Ἔστωσαν δύο εὐθεῖαι αἱ Α, ΒΓ, καὶ τετμήσθω ἡ ΒΓ, <lb n="1799,001,,,,2,1,5"></lb> |
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1388 ὡς ἔτυχεν, κατὰ τὰ Δ, Ε σημεῖα· λέγω, ὅτι τὸ ὑπὸ τῶν Α, <lb n="1799,001,,,,2,1,6"></lb> |
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1389 ΒΓ περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ τε ὑπὸ τῶν Α, <pb></pb> <lb n="1799,001,,,,2,1,7"></lb> |
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1390 ΒΔ περιεχομένῳ ὀρθογωνίῳ καὶ τῷ ὑπὸ τῶν Α, ΔΕ καὶ <lb n="1799,001,,,,2,1,8"></lb> |
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1391 ἔτι τῷ ὑπὸ τῶν Α, ΕΓ.</s> <lb n="1799,001,,,,2,1,9"></lb> |
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1392 <s>Ἤχθω γὰρ ἀπὸ τοῦ Β τῇ ΒΓ πρὸς ὀρθὰς ἡ ΒΖ, καὶ <lb n="1799,001,,,,2,1,10"></lb> |
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1393 κείσθω τῇ Α ἴση ἡ ΒΗ, καὶ διὰ μὲν τοῦ Η τῇ ΒΓ παράλ-<lb n="1799,001,,,,2,1,11"></lb> |
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1394 ληλος ἤχθω ἡ ΗΘ, διὰ δὲ τῶν Δ, Ε, Γ <lb n="1799,001,,,,2,1,12"></lb> |
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1395 τῇ ΒΗ παράλληλοι ἤχθωσαν αἱ ΔΚ, <lb n="1799,001,,,,2,1,13"></lb> |
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1396 ΕΛ, ΓΘ.</s> <lb n="1799,001,,,,2,1,14"></lb> |
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1397 <s>Ἴσον δή ἐστι τὸ ΒΘ τοῖς ΒΚ, ΔΛ, <lb n="1799,001,,,,2,1,15"></lb> |
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1398 ΕΘ.</s> <s>καί ἐστι τὸ μὲν ΒΘ τὸ ὑπὸ τῶν <lb n="1799,001,,,,2,1,16"></lb> |
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1399 Α, ΒΓ· περιέχεται μὲν γὰρ ὑπὸ τῶν <lb n="1799,001,,,,2,1,17"></lb> |
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1400 ΗΒ, ΒΓ, ἴση δὲ ἡ ΒΗ τῇ Α· τὸ δὲ <lb n="1799,001,,,,2,1,18"></lb> |
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1401 ΒΚ τὸ ὑπὸ τῶν Α, ΒΔ· περιέχεται <lb n="1799,001,,,,2,1,19"></lb> |
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1402 μὲν γὰρ ὑπὸ τῶν ΗΒ, ΒΔ, ἴση δὲ ἡ ΒΗ τῇ Α.</s> <s>τὸ δὲ <lb n="1799,001,,,,2,1,20"></lb> |
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1403 ΔΛ τὸ ὑπὸ τῶν Α, ΔΕ· ἴση γὰρ ἡ ΔΚ, τουτέστιν ἡ <lb n="1799,001,,,,2,1,21"></lb> |
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1404 ΒΗ, τῇ Α.</s> <s>καὶ ἔτι ὁμοίως τὸ ΕΘ τὸ ὑπὸ τῶν Α, ΕΓ· <lb n="1799,001,,,,2,1,22"></lb> |
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1405 τὸ ἄρα ὑπὸ τῶν Α, ΒΓ ἴσον ἐστὶ τῷ τε ὑπὸ Α, ΒΔ <lb n="1799,001,,,,2,1,23"></lb> |
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1406 καὶ τῷ ὑπὸ Α, ΔΕ καὶ ἔτι τῷ ὑπὸ Α, ΕΓ.</s> <lb n="1799,001,,,,2,1,24"></lb> |
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1407 <s>Ἐὰν ἄρα ὦσι δύο εὐθεῖαι, τμηθῇ δὲ ἡ ἑτέρα αὐτῶν εἰς <lb n="1799,001,,,,2,1,25"></lb> |
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1408 ὁσαδηποτοῦν τμήματα, τὸ περιεχόμενον ὀρθογώνιον ὑπὸ <lb n="1799,001,,,,2,1,26"></lb> |
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1409 τῶν δύο εὐθειῶν ἴσον ἐστὶ τοῖς ὑπό τε τῆς ἀτμήτου καὶ <lb n="1799,001,,,,2,1,27"></lb> |
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1410 ἑκάστου τῶν τμημάτων περιεχομένοις ὀρθογωνίοις· ὅπερ <lb n="1799,001,,,,2,1,28"></lb> |
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1411 ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,2,1,29"></lb> |
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1412 Ἐὰν εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ὑπὸ τῆς ὅλης <lb n="1799,001,,,,2,2,1"></lb> |
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1413 καὶ ἑκατέρου τῶν τμημάτων περιεχόμενον ὀρθογώνιον <lb n="1799,001,,,,2,2,2"></lb> |
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1414 ἴσον ἐστὶ τῷ ἀπὸ τῆς ὅλης τετραγώνῳ.</s> <lb n="1799,001,,,,2,2,3"></lb> |
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1415 <s>Εὐθεῖα γὰρ ἡ ΑΒ τετμήσθω, ὡς ἔτυχεν, κατὰ τὸ Γ <lb n="1799,001,,,,2,2,4"></lb> |
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1416 σημεῖον· λέγω, ὅτι τὸ ὑπὸ τῶν ΑΒ, ΒΓ περιεχόμενον <lb n="1799,001,,,,2,2,5"></lb> |
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1417 ὀρθογώνιον μετὰ τοῦ ὑπὸ ΒΑ, ΑΓ περιεχομένου ὀρθογω-<lb n="1799,001,,,,2,2,6"></lb> |
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1418 νίου ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ τετραγώνῳ.</s> <lb n="1799,001,,,,2,2,7"></lb> |
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1419 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔΕΒ, <lb n="1799,001,,,,2,2,8"></lb> |
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1420 καὶ ἤχθω διὰ τοῦ Γ ὁποτέρᾳ τῶν ΑΔ, ΒΕ παράλληλος ἡ ΓΖ.</s> <lb n="1799,001,,,,2,2,9"></lb> |
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1421 <s>Ἴσον δή ἐστι τὸ ΑΕ τοῖς ΑΖ, ΓΕ.</s> <s>καί <lb n="1799,001,,,,2,2,10"></lb> |
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1422 ἐστι τὸ μὲν ΑΕ τὸ ἀπὸ τῆς ΑΒ τετρά-<lb n="1799,001,,,,2,2,11"></lb> |
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1423 γωνον, τὸ δὲ ΑΖ τὸ ὑπὸ τῶν ΒΑ, ΑΓ <lb n="1799,001,,,,2,2,12"></lb> |
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1424 περιεχόμενον ὀρθογώνιον· περιέχεται μὲν <lb n="1799,001,,,,2,2,13"></lb> |
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1425 γὰρ ὑπὸ τῶν ΔΑ, ΑΓ, ἴση δὲ ἡ ΑΔ τῇ <lb n="1799,001,,,,2,2,14"></lb> |
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1426 ΑΒ· τὸ δὲ ΓΕ τὸ ὑπὸ τῶν ΑΒ, ΒΓ· ἴση <lb n="1799,001,,,,2,2,15"></lb> |
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1427 γὰρ ἡ ΒΕ τῇ ΑΒ.</s> <s>τὸ ἄρα ὑπὸ τῶν ΒΑ, ΑΓ μετὰ τοῦ <lb n="1799,001,,,,2,2,16"></lb> |
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1428 ὑπὸ τῶν ΑΒ, ΒΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ τετραγώνῳ.</s> <lb n="1799,001,,,,2,2,17"></lb> |
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1429 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ὑπὸ τῆς <lb n="1799,001,,,,2,2,18"></lb> |
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1430 ὅλης καὶ ἑκατέρου τῶν τμημάτων περιεχόμενον ὀρθογώ-<lb n="1799,001,,,,2,2,19"></lb> |
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1431 νιον ἴσον ἐστὶ τῷ ἀπὸ τῆς ὅλης τετραγώνῳ· ὅπερ ἔδει <lb n="1799,001,,,,2,2,20"></lb> |
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1432 δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,2,2,21"></lb> |
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1433 Ἐὰν εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ὑπὸ τῆς <lb n="1799,001,,,,2,3,1"></lb> |
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1434 ὅλης καὶ ἑνὸς τῶν τμημάτων περιεχόμενον ὀρθογώνιον <lb n="1799,001,,,,2,3,2"></lb> |
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1435 ἴσον ἐστὶ τῷ τε ὑπὸ τῶν τμημάτων περιεχομένῳ ὀρθογω-<lb n="1799,001,,,,2,3,3"></lb> |
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1436 νίῳ καὶ τῷ ἀπὸ τοῦ προειρημένου τμήματος τετραγώνῳ.</s> <lb n="1799,001,,,,2,3,4"></lb> |
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1437 <s>Εὐθεῖα γὰρ ἡ ΑΒ τετμήσθω, ὡς ἔτυχεν, κατὰ τὸ Γ· <lb n="1799,001,,,,2,3,5"></lb> |
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1438 λέγω, ὅτι τὸ ὑπὸ τῶν ΑΒ, ΒΓ περιεχόμενον ὀρθογώνιον <lb n="1799,001,,,,2,3,6"></lb> |
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1439 ἴσον ἐστὶ τῷ τε ὑπὸ τῶν ΑΓ, ΓΒ περιεχομένῳ ὀρθογωνίῳ <lb n="1799,001,,,,2,3,7"></lb> |
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1440 μετὰ τοῦ ἀπὸ τῆς ΒΓ τετραγώνου.</s> <lb n="1799,001,,,,2,3,8"></lb> |
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1441 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΓΒ τετράγωνον τὸ ΓΔΕΒ, <lb n="1799,001,,,,2,3,9"></lb> |
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1442 καὶ διήχθω ἡ ΕΔ ἐπὶ τὸ Ζ, καὶ διὰ τοῦ Α ὁποτέρᾳ τῶν <lb n="1799,001,,,,2,3,10"></lb> |
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1443 ΓΔ, ΒΕ παράλληλος ἤχθω ἡ ΑΖ.</s> <s>ἴσον <lb n="1799,001,,,,2,3,11"></lb> |
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1444 δή ἐστι τὸ ΑΕ τοῖς ΑΔ, ΓΕ· καί ἐστι <lb n="1799,001,,,,2,3,12"></lb> |
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1445 τὸ μὲν ΑΕ τὸ ὑπὸ τῶν ΑΒ, ΒΓ περιεχό-<lb n="1799,001,,,,2,3,13"></lb> |
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1446 μενον ὀρθογώνιον· περιέχεται μὲν γὰρ <lb n="1799,001,,,,2,3,14"></lb> |
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1447 ὑπὸ τῶν ΑΒ, ΒΕ, ἴση δὲ ἡ ΒΕ τῇ ΒΓ· <lb n="1799,001,,,,2,3,15"></lb> |
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1448 τὸ δὲ ΑΔ τὸ ὑπὸ τῶν ΑΓ, ΓΒ· ἴση γὰρ <lb n="1799,001,,,,2,3,16"></lb> |
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1449 ἡ ΔΓ τῇ ΓΒ· τὸ δὲ ΔΒ τὸ ἀπὸ τῆς ΓΒ τετράγωνον· τὸ <lb n="1799,001,,,,2,3,17"></lb> |
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1450 ἄρα ὑπὸ τῶν ΑΒ, ΒΓ περιεχόμενον ὀρθογώνιον ἴσον <lb n="1799,001,,,,2,3,18"></lb> |
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1451 ἐστὶ τῷ ὑπὸ τῶν ΑΓ, ΓΒ περιεχομένῳ ὀρθογωνίῳ μετὰ <lb n="1799,001,,,,2,3,19"></lb> |
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1452 τοῦ ἀπὸ τῆς ΒΓ τετραγώνου.</s> <lb n="1799,001,,,,2,3,20"></lb> |
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1453 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ὑπὸ τῆς <lb n="1799,001,,,,2,3,21"></lb> |
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1454 ὅλης καὶ ἑνὸς τῶν τμημάτων περιεχόμενον ὀρθογώνιον <pb></pb> <lb n="1799,001,,,,2,3,22"></lb> |
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1455 ἴσον ἐστὶ τῷ τε ὑπὸ τῶν τμημάτων περιεχομένῳ ὀρθογω-<lb n="1799,001,,,,2,3,23"></lb> |
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1456 νίῳ καὶ τῷ ἀπὸ τοῦ προειρημένου τμήματος τετραγώνῳ· <lb n="1799,001,,,,2,3,24"></lb> |
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1457 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,2,3,25"></lb> |
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1458 <s>Ἐὰν εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ἀπὸ τῆς ὅλης <lb n="1799,001,,,,2,4,1"></lb> |
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1459 τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν τμημάτων τετρα-<lb n="1799,001,,,,2,4,2"></lb> |
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1460 γώνοις καὶ τῷ δὶς ὑπὸ τῶν τμημάτων περιεχομένῳ <lb n="1799,001,,,,2,4,3"></lb> |
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1461 ὀρθογωνίῳ.</s> <lb n="1799,001,,,,2,4,4"></lb> |
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1462 <s>Εὐθεῖα γὰρ γραμμὴ ἡ ΑΒ τετμήσθω, ὡς ἔτυχεν, κατὰ <lb n="1799,001,,,,2,4,5"></lb> |
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1463 τὸ Γ.</s> <s>λέγω, ὅτι τὸ ἀπὸ τῆς ΑΒ τετράγωνον ἴσον ἐστὶ τοῖς <lb n="1799,001,,,,2,4,6"></lb> |
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1464 τε ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν <lb n="1799,001,,,,2,4,7"></lb> |
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1465 ΑΓ, ΓΒ περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,2,4,8"></lb> |
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1466 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔΕΒ, <lb n="1799,001,,,,2,4,9"></lb> |
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1467 καὶ ἐπεζεύχθω ἡ ΒΔ, καὶ διὰ μὲν τοῦ Γ ὁποτέρᾳ τῶν ΑΔ, <lb n="1799,001,,,,2,4,10"></lb> |
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1468 ΕΒ παράλληλος ἤχθω ἡ ΓΖ, διὰ δὲ τοῦ Η ὁποτέρᾳ τῶν <lb n="1799,001,,,,2,4,11"></lb> |
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1469 ΑΒ, ΔΕ παράλληλος ἤχθω ἡ ΘΚ.</s> <s>καὶ ἐπεὶ παράλληλός <lb n="1799,001,,,,2,4,12"></lb> |
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1470 ἐστιν ἡ ΓΖ τῇ ΑΔ, καὶ εἰς αὐτὰς ἐμπέπτωκεν ἡ ΒΔ, ἡ <lb n="1799,001,,,,2,4,13"></lb> |
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1471 ἐκτὸς γωνία ἡ ὑπὸ ΓΗΒ ἴση ἐστὶ τῇ ἐντὸς καὶ <lb n="1799,001,,,,2,4,14"></lb> |
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1472 ἀπεναντίον τῇ ὑπὸ ΑΔΒ.</s> <s>ἀλλ' ἡ ὑπὸ ΑΔΒ τῇ <lb n="1799,001,,,,2,4,15"></lb> |
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1473 ὑπὸ ΑΒΔ ἐστιν ἴση, ἐπεὶ καὶ πλευρὰ ἡ ΒΑ τῇ <lb n="1799,001,,,,2,4,16"></lb> |
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1474 ΑΔ ἐστιν ἴση· καὶ ἡ ὑπὸ ΓΗΒ ἄρα γωνία τῇ <lb n="1799,001,,,,2,4,17"></lb> |
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1475 ὑπὸ ΗΒΓ ἐστιν ἴση· ὥστε καὶ πλευρὰ ἡ ΒΓ <lb n="1799,001,,,,2,4,18"></lb> |
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1476 πλευρᾷ τῇ ΓΗ ἐστιν ἴση· ἀλλ' ἡ μὲν ΓΒ τῇ <pb></pb> <lb n="1799,001,,,,2,4,19"></lb> |
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1477 ΗΚ ἐστιν ἴση, ἡ δὲ ΓΗ τῇ ΚΒ· καὶ ἡ ΗΚ ἄρα τῇ ΚΒ ἐστιν <lb n="1799,001,,,,2,4,20"></lb> |
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1478 ἴση· ἰσόπλευρον ἄρα ἐστὶ τὸ ΓΗΚΒ.</s> <s>λέγω δή, ὅτι καὶ <lb n="1799,001,,,,2,4,21"></lb> |
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1479 ὀρθογώνιον.</s> <s>ἐπεὶ γὰρ παράλληλός ἐστιν ἡ ΓΗ τῇ ΒΚ <lb n="1799,001,,,,2,4,22"></lb> |
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1480 [καὶ εἰς αὐτὰς ἐμπέπτωκεν εὐθεῖα ἡ ΓΒ], αἱ ἄρα ὑπὸ <lb n="1799,001,,,,2,4,23"></lb> |
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1481 ΚΒΓ, ΗΓΒ γωνίαι δύο ὀρθαῖς εἰσιν ἴσαι.</s> <s>ὀρθὴ δὲ ἡ ὑπὸ <lb n="1799,001,,,,2,4,24"></lb> |
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1482 ΚΒΓ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΒΓΗ· ὥστε καὶ αἱ ἀπεναν-<lb n="1799,001,,,,2,4,25"></lb> |
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1483 τίον αἱ ὑπὸ ΓΗΚ, ΗΚΒ ὀρθαί εἰσιν.</s> <s>ὀρθογώνιον ἄρα <lb n="1799,001,,,,2,4,26"></lb> |
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1484 ἐστὶ τὸ ΓΗΚΒ· ἐδείχθη δὲ καὶ ἰσόπλευρον· τετράγωνον <lb n="1799,001,,,,2,4,27"></lb> |
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1485 ἄρα ἐστίν· καί ἐστιν ἀπὸ τῆς ΓΒ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ <lb n="1799,001,,,,2,4,28"></lb> |
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1486 ΘΖ τετράγωνόν ἐστιν· καί ἐστιν ἀπὸ τῆς ΘΗ, τουτέστιν <lb n="1799,001,,,,2,4,29"></lb> |
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1487 [ἀπὸ] τῆς ΑΓ· τὰ ἄρα ΘΖ, Κ*γ τετράγωνα ἀπὸ τῶν ΑΓ, <lb n="1799,001,,,,2,4,30"></lb> |
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1488 ΓΒ εἰσιν.</s> <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΑΗ τῷ ΗΕ, καί ἐστι τὸ <lb n="1799,001,,,,2,4,31"></lb> |
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1489 ΑΗ τὸ ὑπὸ τῶν ΑΓ, ΓΒ· ἴση γὰρ ἡ ΗΓ τῇ ΓΒ· καὶ τὸ <lb n="1799,001,,,,2,4,32"></lb> |
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1490 ΗΕ ἄρα ἴσον ἐστὶ τῷ ὑπὸ ΑΓ, ΓΒ· τὰ ἄρα ΑΗ, ΗΕ <lb n="1799,001,,,,2,4,33"></lb> |
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1491 ἴσα ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ.</s> <s>ἔστι δὲ καὶ τὰ ΘΖ, ΓΚ <lb n="1799,001,,,,2,4,34"></lb> |
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1492 τετράγωνα ἀπὸ τῶν ΑΓ, ΓΒ· τὰ ἄρα τέσσαρα τὰ ΘΖ, <lb n="1799,001,,,,2,4,35"></lb> |
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1493 ΓΚ, ΑΗ, ΗΕ ἴσα ἐστὶ τοῖς τε ἀπὸ τῶν ΑΓ, ΓΒ τετρα-<lb n="1799,001,,,,2,4,36"></lb> |
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1494 γώνοις καὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ περιεχομένῳ ὀρθογω-<lb n="1799,001,,,,2,4,37"></lb> |
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1495 νίῳ.</s> <s>ἀλλὰ τὰ ΘΖ, ΓΚ, ΑΗ, ΗΕ ὅλον ἐστὶ τὸ ΑΔΕΒ, ὅ <lb n="1799,001,,,,2,4,38"></lb> |
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1496 ἐστιν ἀπὸ τῆς ΑΒ τετράγωνον· τὸ ἄρα ἀπὸ τῆς ΑΒ τετρά- <pb></pb><lb n="1799,001,,,,2,4,39"></lb> |
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1497 γωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνοις καὶ <lb n="1799,001,,,,2,4,40"></lb> |
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1498 τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,2,4,41"></lb> |
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1499 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ἀπὸ τῆς <lb n="1799,001,,,,2,4,42"></lb> |
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1500 ὅλης τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν τμημάτων <lb n="1799,001,,,,2,4,43"></lb> |
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1501 τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν τμημάτων περιεχομένῳ <lb n="1799,001,,,,2,4,44"></lb> |
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1502 ὀρθογωνίῳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,2,4,45"></lb> |
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1503 <s>[Πόρισμα <lb n="1799,001,,,,2,4,46"></lb> |
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1504 Ἐκ δὴ τούτου φανερόν, ὅτι ἐν τοῖς τετραγώνοις χωρίοις <lb n="1799,001,,,,2,4,47"></lb> |
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1505 τὰ περὶ τὴν διάμετρον παραλληλόγραμμα τετράγωνά <lb n="1799,001,,,,2,4,48"></lb> |
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1506 ἐστιν].</s> <lb n="1799,001,,,,2,4,49"></lb> |
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1507 <s>Ἐὰν εὐθεῖα γραμμὴ τμηθῇ εἰς ἴσα καὶ ἄνισα, τὸ ὑπὸ <lb n="1799,001,,,,2,5,1"></lb> |
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1508 τῶν ἀνίσων τῆς ὅλης τμημάτων περιεχόμενον ὀρθογώ-<lb n="1799,001,,,,2,5,2"></lb> |
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1509 νιον μετὰ τοῦ ἀπὸ τῆς μεταξὺ τῶν τομῶν τετραγώνου <lb n="1799,001,,,,2,5,3"></lb> |
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1510 ἴσον ἐστὶ τῷ ἀπὸ τῆς ἡμισείας τετραγώνῳ.</s> <lb n="1799,001,,,,2,5,4"></lb> |
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1511 <s>Εὐθεῖα γάρ τις ἡ ΑΒ τετμήσθω εἰς μὲν ἴσα κατὰ τὸ Γ, <lb n="1799,001,,,,2,5,5"></lb> |
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1512 εἰς δὲ ἄνισα κατὰ τὸ Δ· λέγω, ὅτι τὸ ὑπὸ τῶν ΑΔ, ΔΒ <lb n="1799,001,,,,2,5,6"></lb> |
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1513 περιεχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΓΔ τετρα-<lb n="1799,001,,,,2,5,7"></lb> |
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1514 γώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΒ τετραγώνῳ.</s> <s><pb></pb> <lb n="1799,001,,,,2,5,8"></lb> |
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1515 Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΓΒ τετράγωνον τὸ ΓΕΖΒ, <lb n="1799,001,,,,2,5,9"></lb> |
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1516 καὶ ἐπεζεύχθω ἡ ΒΕ, καὶ διὰ μὲν τοῦ Δ ὁποτέρᾳ τῶν ΓΕ, <lb n="1799,001,,,,2,5,10"></lb> |
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1517 ΒΖ παράλληλος ἤχθω ἡ ΔΗ, διὰ δὲ τοῦ Θ ὁποτέρᾳ τῶν <lb n="1799,001,,,,2,5,11"></lb> |
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1518 ΑΒ, ΕΖ παράλληλος πάλιν ἤχθω ἡ ΚΜ, καὶ πάλιν διὰ <lb n="1799,001,,,,2,5,12"></lb> |
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1519 τοῦ Α ὁποτέρᾳ τῶν ΓΛ, ΒΜ <lb n="1799,001,,,,2,5,13"></lb> |
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1520 παράλληλος ἤχθω ἡ ΑΚ.</s> <s>καὶ <lb n="1799,001,,,,2,5,14"></lb> |
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1521 ἐπεὶ ἴσον ἐστὶ τὸ ΓΘ παρα-<lb n="1799,001,,,,2,5,15"></lb> |
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1522 πλήρωμα τῷ ΘΖ παραπλη-<lb n="1799,001,,,,2,5,16"></lb> |
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1523 ρώματι, κοινὸν προσκείσθω <lb n="1799,001,,,,2,5,17"></lb> |
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1524 τὸ ΔΜ· ὅλον ἄρα τὸ ΓΜ <lb n="1799,001,,,,2,5,18"></lb> |
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1525 ὅλῳ τῷ ΔΖ ἴσον ἐστίν.</s> <s>ἀλλὰ <lb n="1799,001,,,,2,5,19"></lb> |
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1526 τὸ ΓΜ τῷ ΑΛ ἴσον ἐστίν, <lb n="1799,001,,,,2,5,20"></lb> |
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1527 ἐπεὶ καὶ ἡ ΑΓ τῇ ΓΒ ἐστιν ἴση· καὶ τὸ ΑΛ ἄρα τῷ ΔΖ <lb n="1799,001,,,,2,5,21"></lb> |
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1528 ἴσον ἐστίν.</s> <s>κοινὸν προσκείσθω τὸ ΓΘ· ὅλον ἄρα τὸ ΑΘ <lb n="1799,001,,,,2,5,22"></lb> |
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1529 τῷ ΜΝΞ γνώμονι ἴσον ἐστίν.</s> <s>ἀλλὰ τὸ ΑΘ τὸ ὑπὸ τῶν <lb n="1799,001,,,,2,5,23"></lb> |
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1530 ΑΔ, ΔΒ ἐστιν· ἴση γὰρ ἡ ΔΘ τῇ ΔΒ· καὶ ὁ ΜΝΞ ἄρα <lb n="1799,001,,,,2,5,24"></lb> |
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1531 γνώμων ἴσος ἐστὶ τῷ ὑπὸ ΑΔ, ΔΒ.</s> <s>κοινὸν προσκείσθω <lb n="1799,001,,,,2,5,25"></lb> |
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1532 τὸ ΛΗ, ὅ ἐστιν ἴσον τῷ ἀπὸ τῆς ΓΔ· ὁ ἄρα ΜΝΞ γνώ-<lb n="1799,001,,,,2,5,26"></lb> |
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1533 μων καὶ τὸ ΛΗ ἴσα ἐστὶ τῷ ὑπὸ τῶν ΑΔ, ΔΒ περιεχο-<lb n="1799,001,,,,2,5,27"></lb> |
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1534 μένῳ ὀρθογωνίῳ καὶ τῷ ἀπὸ τῆς ΓΔ τετραγώνῳ.</s> <s>ἀλλὰ ὁ <lb n="1799,001,,,,2,5,28"></lb> |
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1535 ΜΝΞ γνώμων καὶ τὸ ΛΗ ὅλον ἐστὶ τὸ ΓΕΖΒ τετρά-<lb n="1799,001,,,,2,5,29"></lb> |
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1536 γωνον, ὅ ἐστιν ἀπὸ τῆς ΓΒ· τὸ ἄρα ὑπὸ τῶν ΑΔ, ΔΒ <pb></pb> <lb n="1799,001,,,,2,5,30"></lb> |
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1537 περιεχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΓΔ τετρα-<lb n="1799,001,,,,2,5,31"></lb> |
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1538 γώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΒ τετραγώνῳ.</s> <lb n="1799,001,,,,2,5,32"></lb> |
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1539 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ εἰς ἴσα καὶ ἄνισα, τὸ <lb n="1799,001,,,,2,5,33"></lb> |
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1540 ὑπὸ τῶν ἀνίσων τῆς ὅλης τμημάτων περιεχόμενον ὀρθο-<lb n="1799,001,,,,2,5,34"></lb> |
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1541 γώνιον μετὰ τοῦ ἀπὸ τῆς μεταξὺ τῶν τομῶν τετραγώνου <lb n="1799,001,,,,2,5,35"></lb> |
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1542 ἴσον ἐστὶ τῷ ἀπὸ τῆς ἡμισείας τετραγώνῳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,2,5,36"></lb> |
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1543 <s>Ἐὰν εὐθεῖα γραμμὴ τμηθῇ δίχα, προστεθῇ δέ τις αὐτῇ <lb n="1799,001,,,,2,6,1"></lb> |
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1544 εὐθεῖα ἐπ' εὐθείας, τὸ ὑπὸ τῆς ὅλης σὺν τῇ προσκειμένῃ <lb n="1799,001,,,,2,6,2"></lb> |
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1545 καὶ τῆς προσκειμένης περιεχόμενον ὀρθογώνιον μετὰ τοῦ <lb n="1799,001,,,,2,6,3"></lb> |
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1546 ἀπὸ τῆς ἡμισείας τετραγώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς συγκει-<lb n="1799,001,,,,2,6,4"></lb> |
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1547 μένης ἔκ τε τῆς ἡμισείας καὶ τῆς προσκειμένης τετραγώνῳ.</s> <lb n="1799,001,,,,2,6,5"></lb> |
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1548 <s>Εὐθεῖα γάρ τις ἡ ΑΒ τετμήσθω δίχα <lb n="1799,001,,,,2,6,6"></lb> |
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1549 κατὰ τὸ Γ σημεῖον, προσκείσθω δέ τις <lb n="1799,001,,,,2,6,7"></lb> |
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1550 αὐτῇ εὐθεῖα ἐπ' εὐθείας ἡ ΒΔ· λέγω, <lb n="1799,001,,,,2,6,8"></lb> |
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1551 ὅτι τὸ ὑπὸ τῶν ΑΔ, ΔΒ περιεχόμενον <lb n="1799,001,,,,2,6,9"></lb> |
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1552 ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΓΒ τετρα-<lb n="1799,001,,,,2,6,10"></lb> |
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1553 γώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΔ τετρα-<lb n="1799,001,,,,2,6,11"></lb> |
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1554 γώνῳ.</s> <lb n="1799,001,,,,2,6,12"></lb> |
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1555 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΓΔ τετράγωνον τὸ ΓΕΖΔ, <lb n="1799,001,,,,2,6,13"></lb> |
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1556 καὶ ἐπεζεύχθω ἡ ΔΕ, καὶ διὰ μὲν τοῦ Β σημείου ὁποτέρᾳ <lb n="1799,001,,,,2,6,14"></lb> |
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1557 τῶν ΕΓ, ΔΖ παράλληλος ἤχθω ἡ ΒΗ, διὰ δὲ τοῦ Θ <lb n="1799,001,,,,2,6,15"></lb> |
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1558 σημείου ὁποτέρᾳ τῶν ΑΒ, ΕΖ παράλληλος ἤχθω ἡ ΚΜ, <pb></pb> <lb n="1799,001,,,,2,6,16"></lb> |
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1559 καὶ ἔτι διὰ τοῦ Α ὁποτέρᾳ τῶν ΓΛ, ΔΜ παράλληλος ἤχθω <lb n="1799,001,,,,2,6,17"></lb> |
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1560 ἡ ΑΚ.</s> <lb n="1799,001,,,,2,6,18"></lb> |
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1561 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΑΓ τῇ ΓΒ, ἴσον ἐστὶ καὶ τὸ ΑΛ <lb n="1799,001,,,,2,6,19"></lb> |
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1562 τῷ ΓΘ.</s> <s>ἀλλὰ τὸ ΓΘ τῷ ΘΖ ἴσον ἐστίν.</s> <s>καὶ τὸ ΑΛ ἄρα <lb n="1799,001,,,,2,6,20"></lb> |
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1563 τῷ ΘΖ ἐστιν ἴσον.</s> <s>κοινὸν προσκείσθω τὸ ΓΜ· ὅλον ἄρα <lb n="1799,001,,,,2,6,21"></lb> |
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1564 τὸ ΑΜ τῷ ΝΞΟ γνώμονί ἐστιν ἴσον.</s> <s>ἀλλὰ τὸ ΑΜ ἐστι <lb n="1799,001,,,,2,6,22"></lb> |
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1565 τὸ ὑπὸ τῶν ΑΔ, ΔΒ· ἴση γάρ ἐστιν ἡ ΔΜ τῇ ΔΒ· καὶ ὁ <lb n="1799,001,,,,2,6,23"></lb> |
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1566 ΝΞΟ ἄρα γνώμων ἴσος ἐστὶ τῷ ὑπὸ τῶν ΑΔ, ΔΒ [περι-<lb n="1799,001,,,,2,6,24"></lb> |
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1567 εχομένῳ ὀρθογωνίῳ].</s> <s>κοινὸν προσκείσθω τὸ ΛΗ, ὅ ἐστιν <lb n="1799,001,,,,2,6,25"></lb> |
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1568 ἴσον τῷ ἀπὸ τῆς ΒΓ τετραγώνῳ· τὸ ἄρα ὑπὸ τῶν ΑΔ, <lb n="1799,001,,,,2,6,26"></lb> |
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1569 ΔΒ περιεχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΓΒ τε-<lb n="1799,001,,,,2,6,27"></lb> |
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1570 τραγώνου ἴσον ἐστὶ τῷ ΝΞΟ γνώμονι καὶ τῷ ΛΗ.</s> <s>ἀλλὰ <lb n="1799,001,,,,2,6,28"></lb> |
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1571 ὁ ΝΞΟ γνώμων καὶ τὸ ΛΗ ὅλον ἐστὶ τὸ ΓΕΖΔ τετράγω-<lb n="1799,001,,,,2,6,29"></lb> |
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1572 νον, ὅ ἐστιν ἀπὸ τῆς ΓΔ· τὸ ἄρα ὑπὸ τῶν ΑΔ, ΔΒ περι-<lb n="1799,001,,,,2,6,30"></lb> |
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1573 εχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΓΒ τετραγώνου <lb n="1799,001,,,,2,6,31"></lb> |
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1574 ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΔ τετραγώνῳ.</s> <lb n="1799,001,,,,2,6,32"></lb> |
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1575 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ δίχα, προστεθῇ δέ τις <lb n="1799,001,,,,2,6,33"></lb> |
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1576 αὐτῇ εὐθεῖα ἐπ' εὐθείας, τὸ ὑπὸ τῆς ὅλης σὺν τῇ προσκει-<lb n="1799,001,,,,2,6,34"></lb> |
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1577 μένῃ καὶ τῆς προσκειμένης περιεχόμενον ὀρθογώνιον μετὰ <lb n="1799,001,,,,2,6,35"></lb> |
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1578 τοῦ ἀπὸ τῆς ἡμισείας τετραγώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,2,6,36"></lb> |
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1579 συγκειμένης ἔκ τε τῆς ἡμισείας καὶ τῆς προσκειμένης <lb n="1799,001,,,,2,6,37"></lb> |
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1580 τετραγώνῳ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,2,6,38"></lb> |
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1581 Ἐὰν εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ἀπὸ τῆς ὅλης <lb n="1799,001,,,,2,7,1"></lb> |
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1582 καὶ τὸ ἀφ' ἑνὸς τῶν τμημάτων τὰ συναμφότερα τετρά-<lb n="1799,001,,,,2,7,2"></lb> |
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1583 γωνα ἴσα ἐστὶ τῷ τε δὶς ὑπὸ τῆς ὅλης καὶ τοῦ εἰρημένου <lb n="1799,001,,,,2,7,3"></lb> |
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1584 τμήματος περιεχομένῳ ὀρθογωνίῳ καὶ τῷ ἀπὸ τοῦ λοιποῦ <lb n="1799,001,,,,2,7,4"></lb> |
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1585 τμήματος τετραγώνῳ.</s> <lb n="1799,001,,,,2,7,5"></lb> |
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1586 <s>Εὐθεῖα γάρ τις ἡ ΑΒ τετμήσθω, ὡς ἔτυχεν, κατὰ <lb n="1799,001,,,,2,7,6"></lb> |
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1587 τὸ Γ σημεῖον· λέγω, ὅτι τὰ ἀπὸ τῶν ΑΒ, ΒΓ τετράγωνα <lb n="1799,001,,,,2,7,7"></lb> |
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1588 ἴσα ἐστὶ τῷ τε δὶς ὑπὸ τῶν ΑΒ, ΒΓ περιεχομένῳ ὀρθογω-<lb n="1799,001,,,,2,7,8"></lb> |
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1589 νίῳ καὶ τῷ ἀπὸ τῆς ΓΑ τετραγώνῳ.</s> <lb n="1799,001,,,,2,7,9"></lb> |
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1590 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔΕΒ· <lb n="1799,001,,,,2,7,10"></lb> |
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1591 καὶ καταγεγράφθω τὸ σχῆμα.</s> <lb n="1799,001,,,,2,7,11"></lb> |
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1592 <s>Ἐπεὶ οὖν ἴσον ἐστὶ τὸ ΑΗ τῷ ΗΕ, κοινὸν προσκείσθω <lb n="1799,001,,,,2,7,12"></lb> |
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1593 τὸ ΓΖ· ὅλον ἄρα τὸ ΑΖ ὅλῳ τῷ ΓΕ ἴσον ἐστίν· τὰ ἄρα <lb n="1799,001,,,,2,7,13"></lb> |
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1594 ΑΖ, ΓΕ διπλάσιά ἐστι τοῦ ΑΖ.</s> <s>ἀλλὰ τὰ ΑΖ, ΓΕ ὁ ΚΛΜ <lb n="1799,001,,,,2,7,14"></lb> |
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1595 ἐστι γνώμων καὶ τὸ ΓΖ τετράγωνον· ὁ ΚΛΜ ἄρα γνώμων <lb n="1799,001,,,,2,7,15"></lb> |
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1596 καὶ τὸ ΓΖ διπλάσιά ἐστι τοῦ ΑΖ.</s> <s>ἔστι <lb n="1799,001,,,,2,7,16"></lb> |
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1597 δὲ τοῦ ΑΖ διπλάσιον καὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,2,7,17"></lb> |
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1598 ΑΒ, ΒΓ· ἴση γὰρ ἡ ΒΖ τῇ ΒΓ· ὁ ἄρα <lb n="1799,001,,,,2,7,18"></lb> |
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1599 ΚΛΜ γνώμων καὶ τὸ ΓΖ τετράγωνον <lb n="1799,001,,,,2,7,19"></lb> |
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1600 ἴσον ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ.</s> <lb n="1799,001,,,,2,7,20"></lb> |
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1601 <s>κοινὸν προσκείσθω τὸ ΔΗ, ὅ ἐστιν ἀπὸ <pb></pb> <lb n="1799,001,,,,2,7,21"></lb> |
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1602 τῆς ΑΓ τετράγωνον· ὁ ἄρα ΚΛΜ γνώμων καὶ τὰ ΒΗ, <lb n="1799,001,,,,2,7,22"></lb> |
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1603 ΗΔ τετράγωνα ἴσα ἐστὶ τῷ τε δὶς ὑπὸ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,2,7,23"></lb> |
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1604 περιεχομένῳ ὀρθογωνίῳ καὶ τῷ ἀπὸ τῆς ΑΓ τετραγώνῳ.</s> <lb n="1799,001,,,,2,7,24"></lb> |
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1605 <s>ἀλλὰ ὁ ΚΛΜ γνώμων καὶ τὰ ΒΗ, ΗΔ τετράγωνα ὅλον <lb n="1799,001,,,,2,7,25"></lb> |
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1606 ἐστὶ τὸ ΑΔΕΒ καὶ τὸ ΓΖ, ἅ ἐστιν ἀπὸ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,2,7,26"></lb> |
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1607 τετράγωνα· τὰ ἄρα ἀπὸ τῶν ΑΒ, ΒΓ τετράγωνα ἴσα <lb n="1799,001,,,,2,7,27"></lb> |
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1608 ἐστὶ τῷ [τε] δὶς ὑπὸ τῶν ΑΒ, ΒΓ περιεχομένῳ ὀρθογω-<lb n="1799,001,,,,2,7,28"></lb> |
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1609 νίῳ μετὰ τοῦ ἀπὸ τῆς ΑΓ τετραγώνου.</s> <lb n="1799,001,,,,2,7,29"></lb> |
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1610 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ἀπὸ τῆς <lb n="1799,001,,,,2,7,30"></lb> |
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1611 ὅλης καὶ τὸ ἀφ' ἑνὸς τῶν τμημάτων τὰ συναμφότερα <lb n="1799,001,,,,2,7,31"></lb> |
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1612 τετράγωνα ἴσα ἐστὶ τῷ τε δὶς ὑπὸ τῆς ὅλης καὶ τοῦ εἰρημέ-<lb n="1799,001,,,,2,7,32"></lb> |
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1613 νου τμήματος περιεχομένῳ ὀρθογωνίῳ καὶ τῷ ἀπὸ τοῦ <lb n="1799,001,,,,2,7,33"></lb> |
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1614 λοιποῦ τμήματος τετραγώνῳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,2,7,34"></lb> |
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1615 <s>Ἐὰν εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ τετράκις ὑπὸ <lb n="1799,001,,,,2,8,1"></lb> |
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1616 τῆς ὅλης καὶ ἑνὸς τῶν τμημάτων περιεχόμενον ὀρθογώ-<lb n="1799,001,,,,2,8,2"></lb> |
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1617 νιον μετὰ τοῦ ἀπὸ τοῦ λοιποῦ τμήματος τετραγώνου ἴσον <lb n="1799,001,,,,2,8,3"></lb> |
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1618 ἐστὶ τῷ ἀπό τε τῆς ὅλης καὶ τοῦ εἰρημένου τμήματος ὡς <lb n="1799,001,,,,2,8,4"></lb> |
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1619 ἀπὸ μιᾶς ἀναγραφέντι τετραγώνῳ.</s> <lb n="1799,001,,,,2,8,5"></lb> |
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1620 <s>Εὐθεῖα γάρ τις ἡ ΑΒ τετμήσθω, ὡς ἔτυχεν, κατὰ τὸ <lb n="1799,001,,,,2,8,6"></lb> |
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1621 Γ σημεῖον· λέγω, ὅτι τὸ τετράκις ὑπὸ τῶν ΑΒ, ΒΓ περι-<lb n="1799,001,,,,2,8,7"></lb> |
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1622 εχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΑΓ τετραγώνου <lb n="1799,001,,,,2,8,8"></lb> |
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1623 ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ, ΒΓ ὡς ἀπὸ μιᾶς ἀναγραφέντι <lb n="1799,001,,,,2,8,9"></lb> |
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1624 τετραγώνῳ.</s> <s><pb></pb> <lb n="1799,001,,,,2,8,10"></lb> |
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1625 Ἐκβεβλήσθω γὰρ ἐπ' εὐθείας [τῇ ΑΒ εὐθεῖα] ἡ ΒΔ, <lb n="1799,001,,,,2,8,11"></lb> |
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1626 καὶ κείσθω τῇ ΓΒ ἴση ἡ ΒΔ, καὶ ἀναγεγράφθω ἀπὸ τῆς <lb n="1799,001,,,,2,8,12"></lb> |
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1627 ΑΔ τετράγωνον τὸ ΑΕΖΔ, καὶ κατα- <lb n="1799,001,,,,2,8,13"></lb> |
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1628 γεγράφθω διπλοῦν τὸ σχῆμα.</s> <lb n="1799,001,,,,2,8,14"></lb> |
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1629 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΓΒ τῇ ΒΔ, ἀλλὰ <lb n="1799,001,,,,2,8,15"></lb> |
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1630 ἡ μὲν ΓΒ τῇ ΗΚ ἐστιν ἴση, ἡ δὲ ΒΔ τῇ <lb n="1799,001,,,,2,8,16"></lb> |
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1631 ΚΝ, καὶ ἡ ΗΚ ἄρα τῇ ΚΝ ἐστιν ἴση.</s> <lb n="1799,001,,,,2,8,17"></lb> |
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1632 <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΠΡ τῇ ΡΟ ἐστιν <lb n="1799,001,,,,2,8,18"></lb> |
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1633 ἴση.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΓ τῇ ΒΔ, ἡ <lb n="1799,001,,,,2,8,19"></lb> |
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1634 δὲ ΗΚ τῇ ΚΝ, ἴσον ἄρα ἐστὶ καὶ τὸ μὲν <lb n="1799,001,,,,2,8,20"></lb> |
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1635 ΓΚ τῷ ΚΔ, τὸ δὲ ΗΡ τῷ ΡΝ.</s> <s>ἀλλὰ τὸ <lb n="1799,001,,,,2,8,21"></lb> |
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1636 ΓΚ τῷ ΡΝ ἐστιν ἴσον· παραπληρώματα γὰρ τοῦ ΓΟ παραλ-<lb n="1799,001,,,,2,8,22"></lb> |
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1637 ληλογράμμου· καὶ τὸ ΚΔ ἄρα τῷ ΗΡ ἴσον ἐστίν· τὰ τέσ-<lb n="1799,001,,,,2,8,23"></lb> |
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1638 σαρα ἄρα τὰ ΔΚ, ΓΚ, ΗΡ, ΡΝ ἴσα ἀλλήλοις ἐστίν.</s> <s>τὰ <lb n="1799,001,,,,2,8,24"></lb> |
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1639 τέσσαρα ἄρα τετραπλάσιά ἐστι τοῦ ΓΚ.</s> <s>πάλιν, ἐπεὶ ἴση <lb n="1799,001,,,,2,8,25"></lb> |
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1640 ἐστὶν ἡ ΓΒ τῇ ΒΔ, ἀλλὰ ἡ μὲν ΒΔ τῇ ΒΚ, τουτέστι τῇ <lb n="1799,001,,,,2,8,26"></lb> |
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1641 ΓΗ ἴση, ἡ δὲ ΓΒ τῇ ΗΚ, τουτέστι τῇ ΗΠ, ἐστιν ἴση, <lb n="1799,001,,,,2,8,27"></lb> |
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1642 καὶ ἡ ΓΗ ἄρα τῇ ΗΠ ἴση ἐστίν.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ μὲν <lb n="1799,001,,,,2,8,28"></lb> |
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1643 ΓΗ τῇ ΗΠ, ἡ δὲ ΠΡ τῇ ΡΟ, ἴσον ἐστὶ καὶ τὸ μὲν ΑΗ <pb></pb> <lb n="1799,001,,,,2,8,29"></lb> |
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1644 τῷ ΜΠ, τὸ δὲ ΠΛ τῷ ΡΖ.</s> <s>ἀλλὰ τὸ ΜΠ τῷ ΠΛ ἐστιν <lb n="1799,001,,,,2,8,30"></lb> |
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1645 ἴσον· παραπληρώματα γὰρ τοῦ ΜΛ παραλληλογράμμου· <lb n="1799,001,,,,2,8,31"></lb> |
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1646 καὶ τὸ ΑΗ ἄρα τῷ ΡΖ ἴσον ἐστίν· τὰ τέσσαρα ἄρα τὰ ΑΗ, <lb n="1799,001,,,,2,8,32"></lb> |
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1647 ΜΠ, ΠΛ, ΡΖ ἴσα ἀλλήλοις ἐστίν· τὰ τέσσαρα ἄρα τοῦ <lb n="1799,001,,,,2,8,33"></lb> |
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1648 ΑΗ ἐστι τετραπλάσια.</s> <s>ἐδείχθη δὲ καὶ τὰ τέσσαρα τὰ <lb n="1799,001,,,,2,8,34"></lb> |
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1649 ΓΚ, ΚΔ, ΗΡ, ΡΝ τοῦ ΓΚ τετραπλάσια· τὰ ἄρα ὀκτώ, <lb n="1799,001,,,,2,8,35"></lb> |
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1650 ἃ περιέχει τὸν ΣΤΥ γνώμονα, τετραπλάσιά ἐστι τοῦ ΑΚ.</s> <lb n="1799,001,,,,2,8,36"></lb> |
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1651 <s>καὶ ἐπεὶ τὸ ΑΚ τὸ ὑπὸ τῶν ΑΒ, ΒΔ ἐστιν· ἴση γὰρ ἡ ΒΚ <lb n="1799,001,,,,2,8,37"></lb> |
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1652 τῇ ΒΔ· τὸ ἄρα τετράκις ὑπὸ τῶν ΑΒ, ΒΔ τετραπλάσιόν <lb n="1799,001,,,,2,8,38"></lb> |
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1653 ἐστι τοῦ ΑΚ.</s> <s>ἐδείχθη δὲ τοῦ ΑΚ τετραπλάσιος καὶ ὁ <lb n="1799,001,,,,2,8,39"></lb> |
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1654 ΣΤΥ γνώμων· τὸ ἄρα τετράκις ὑπὸ τῶν ΑΒ, ΒΔ ἴσον <lb n="1799,001,,,,2,8,40"></lb> |
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1655 ἐστὶ τῷ ΣΤΥ γνώμονι.</s> <s>κοινὸν προσκείσθω τὸ ΞΘ, ὅ <lb n="1799,001,,,,2,8,41"></lb> |
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1656 ἐστιν ἴσον τῷ ἀπὸ τῆς ΑΓ τετραγώνῳ· τὸ ἄρα τετράκις <lb n="1799,001,,,,2,8,42"></lb> |
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1657 ὑπὸ τῶν ΑΒ, ΒΔ περιεχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ <lb n="1799,001,,,,2,8,43"></lb> |
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1658 ΑΓ τετραγώνου ἴσον ἐστὶ τῷ ΣΤΥ γνώμονι καὶ τῷ ΞΘ.</s> <lb n="1799,001,,,,2,8,44"></lb> |
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1659 <s>ἀλλὰ ὁ ΣΤΥ γνώμων καὶ τὸ ΞΘ ὅλον ἐστὶ τὸ ΑΕΖΔ <lb n="1799,001,,,,2,8,45"></lb> |
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1660 τετράγωνον, ὅ ἐστιν ἀπὸ τῆς ΑΔ· τὸ ἄρα τετράκις ὑπὸ <lb n="1799,001,,,,2,8,46"></lb> |
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1661 τῶν ΑΒ, ΒΔ μετὰ τοῦ ἀπὸ ΑΓ ἴσον ἐστὶ τῷ ἀπὸ ΑΔ τετρα-<lb n="1799,001,,,,2,8,47"></lb> |
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1662 γώνῳ· ἴση δὲ ἡ ΒΔ τῇ ΒΓ.</s> <s>τὸ ἄρα τετράκις ὑπὸ τῶν ΑΒ, <lb n="1799,001,,,,2,8,48"></lb> |
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1663 ΒΓ περιεχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ ΑΓ τετρα-<lb n="1799,001,,,,2,8,49"></lb> |
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1664 γώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΔ, τουτέστι τῷ ἀπὸ τῆς ΑΒ <lb n="1799,001,,,,2,8,50"></lb> |
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1665 καὶ ΒΓ ὡς ἀπὸ μιᾶς ἀναγραφέντι τετραγώνῳ.</s> <lb n="1799,001,,,,2,8,51"></lb> |
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1666 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ τετράκις <lb n="1799,001,,,,2,8,52"></lb> |
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1667 ὑπὸ τῆς ὅλης καὶ ἑνὸς τῶν τμημάτων περιεχόμενον ὀρθο-<lb n="1799,001,,,,2,8,53"></lb> |
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1668 γώνιον μετὰ τοῦ ἀπὸ τοῦ λοιποῦ τμήματος τετραγώνου <pb></pb> <lb n="1799,001,,,,2,8,54"></lb> |
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1669 ἴσον ἐστὶ τῷ ἀπό τε τῆς ὅλης καὶ τοῦ εἰρημένου τμήματος <lb n="1799,001,,,,2,8,55"></lb> |
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1670 ὡς ἀπὸ μιᾶς ἀναγραφέντι τετραγώνῳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,2,8,56"></lb> |
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1671 <s>Ἐὰν εὐθεῖα γραμμὴ τμηθῇ εἰς ἴσα καὶ ἄνισα, τὰ ἀπὸ <lb n="1799,001,,,,2,9,1"></lb> |
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1672 τῶν ἀνίσων τῆς ὅλης τμημάτων τετράγωνα διπλάσιά ἐστι <lb n="1799,001,,,,2,9,2"></lb> |
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1673 τοῦ τε ἀπὸ τῆς ἡμισείας καὶ τοῦ ἀπὸ τῆς μεταξὺ τῶν <lb n="1799,001,,,,2,9,3"></lb> |
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1674 τομῶν τετραγώνου.</s> <lb n="1799,001,,,,2,9,4"></lb> |
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1675 <s>Εὐθεῖα γάρ τις ἡ ΑΒ τετμήσθω εἰς μὲν ἴσα κατὰ τὸ Γ, <lb n="1799,001,,,,2,9,5"></lb> |
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1676 εἰς δὲ ἄνισα κατὰ τὸ Δ· λέγω, ὅτι τὰ ἀπὸ τῶν ΑΔ, ΔΒ <lb n="1799,001,,,,2,9,6"></lb> |
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1677 τετράγωνα διπλάσιά ἐστι τῶν ἀπὸ τῶν <lb n="1799,001,,,,2,9,7"></lb> |
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1678 ΑΓ, ΓΔ τετραγώνων.</s> <lb n="1799,001,,,,2,9,8"></lb> |
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1679 <s>Ἤχθω γὰρ ἀπὸ τοῦ Γ τῇ ΑΒ πρὸς <lb n="1799,001,,,,2,9,9"></lb> |
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1680 ὀρθὰς ἡ ΓΕ, καὶ κείσθω ἴση ἑκατέρᾳ τῶν <lb n="1799,001,,,,2,9,10"></lb> |
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1681 ΑΓ, ΓΒ, καὶ ἐπεζεύχθωσαν αἱ ΕΑ, ΕΒ, <lb n="1799,001,,,,2,9,11"></lb> |
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1682 καὶ διὰ μὲν τοῦ Δ τῇ ΕΓ παράλληλος ἤχθω <lb n="1799,001,,,,2,9,12"></lb> |
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1683 ἡ ΔΖ, διὰ δὲ τοῦ Ζ τῇ ΑΒ ἡ ΖΗ, καὶ ἐπεζεύχθω ἡ ΑΖ.</s> <lb n="1799,001,,,,2,9,13"></lb> |
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1684 <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΓ τῇ ΓΕ, ἴση ἐστὶ καὶ ἡ ὑπὸ ΕΑΓ <lb n="1799,001,,,,2,9,14"></lb> |
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1685 γωνία τῇ ὑπὸ ΑΕΓ.</s> <s>καὶ ἐπεὶ ὀρθή ἐστιν ἡ πρὸς τῷ Γ, <lb n="1799,001,,,,2,9,15"></lb> |
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1686 λοιπαὶ ἄρα αἱ ὑπὸ ΕΑΓ, ΑΕΓ μιᾷ ὀρθῇ ἴσαι εἰσίν· καί <lb n="1799,001,,,,2,9,16"></lb> |
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1687 εἰσιν ἴσαι· ἡμίσεια ἄρα ὀρθῆς ἐστιν ἑκατέρα τῶν ὑπὸ <pb></pb> <lb n="1799,001,,,,2,9,17"></lb> |
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1688 ΓΕΑ, ΓΑΕ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἑκατέρα τῶν ὑπὸ ΓΕΒ, <lb n="1799,001,,,,2,9,18"></lb> |
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1689 ΕΒΓ ἡμίσειά ἐστιν ὀρθῆς· ὅλη ἄρα ἡ ὑπὸ ΑΕΒ ὀρθή <lb n="1799,001,,,,2,9,19"></lb> |
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1690 ἐστιν.</s> <s>καὶ ἐπεὶ ἡ ὑπὸ ΗΕΖ ἡμίσειά ἐστιν ὀρθῆς, ὀρθὴ <lb n="1799,001,,,,2,9,20"></lb> |
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1691 δὲ ἡ ὑπὸ ΕΗΖ· ἴση γάρ ἐστι τῇ ἐντὸς καὶ ἀπεναντίον τῇ <lb n="1799,001,,,,2,9,21"></lb> |
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1692 ὑπὸ ΕΓΒ· λοιπὴ ἄρα ἡ ὑπὸ ΕΖΗ ἡμίσειά ἐστιν ὀρθῆς· <lb n="1799,001,,,,2,9,22"></lb> |
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1693 ἴση ἄρα [ἐστὶν] ἡ ὑπὸ ΗΕΖ γωνία τῇ ὑπὸ ΕΖΗ· ὥστε <lb n="1799,001,,,,2,9,23"></lb> |
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1694 καὶ πλευρὰ ἡ ΕΗ τῇ ΗΖ ἐστιν ἴση.</s> <s>πάλιν ἐπεὶ ἡ πρὸς τῷ <lb n="1799,001,,,,2,9,24"></lb> |
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1695 Β γωνία ἡμίσειά ἐστιν ὀρθῆς, ὀρθὴ δὲ ἡ ὑπὸ ΖΔΒ· ἴση <lb n="1799,001,,,,2,9,25"></lb> |
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1696 γὰρ πάλιν ἐστὶ τῇ ἐντὸς καὶ ἀπεναντίον τῇ ὑπὸ ΕΓΒ· λοι-<lb n="1799,001,,,,2,9,26"></lb> |
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1697 πὴ ἄρα ἡ ὑπὸ ΒΖΔ ἡμίσειά ἐστιν ὀρθῆς· ἴση ἄρα ἡ πρὸς <lb n="1799,001,,,,2,9,27"></lb> |
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1698 τῷ Β γωνία τῇ ὑπὸ ΔΖΒ· ὥστε καὶ πλευρὰ ἡ ΖΔ πλευ-<lb n="1799,001,,,,2,9,28"></lb> |
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1699 ρᾷ τῇ ΔΒ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΓ τῇ ΓΕ, ἴσον <lb n="1799,001,,,,2,9,29"></lb> |
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1700 ἐστὶ καὶ τὸ ἀπὸ ΑΓ τῷ ἀπὸ ΓΕ· τὰ ἄρα ἀπὸ τῶν ΑΓ, <lb n="1799,001,,,,2,9,30"></lb> |
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1701 ΓΕ τετράγωνα διπλάσιά ἐστι τοῦ ἀπὸ ΑΓ.</s> <s>τοῖς δὲ ἀπὸ <lb n="1799,001,,,,2,9,31"></lb> |
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1702 τῶν ΑΓ, ΓΕ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΕΑ τετράγωνον· ὀρθὴ <lb n="1799,001,,,,2,9,32"></lb> |
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1703 γὰρ ἡ ὑπὸ ΑΓΕ γωνία· τὸ ἄρα ἀπὸ τῆς ΕΑ διπλάσιόν <lb n="1799,001,,,,2,9,33"></lb> |
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1704 ἐστι τοῦ ἀπὸ τῆς ΑΓ.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΕΗ τῇ ΗΖ, <lb n="1799,001,,,,2,9,34"></lb> |
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1705 ἴσον καὶ τὸ ἀπὸ τῆς ΕΗ τῷ ἀπὸ τῆς ΗΖ· τὰ ἄρα ἀπὸ <lb n="1799,001,,,,2,9,35"></lb> |
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1706 τῶν ΕΗ, ΗΖ τετράγωνα διπλάσιά ἐστι τοῦ ἀπὸ τῆς ΗΖ <lb n="1799,001,,,,2,9,36"></lb> |
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1707 τετραγώνου.</s> <s>τοῖς δὲ ἀπὸ τῶν ΕΗ, ΗΖ τετραγώνοις ἴσον <lb n="1799,001,,,,2,9,37"></lb> |
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1708 ἐστὶ τὸ ἀπὸ τῆς ΕΖ τετράγωνον· τὸ ἄρα ἀπὸ τῆς ΕΖ <pb></pb> <lb n="1799,001,,,,2,9,38"></lb> |
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1709 τετράγωνον διπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΗΖ.</s> <s>ἴση δὲ ἡ <lb n="1799,001,,,,2,9,39"></lb> |
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1710 ΗΖ τῇ ΓΔ· τὸ ἄρα ἀπὸ τῆς ΕΖ διπλάσιόν ἐστι τοῦ <lb n="1799,001,,,,2,9,40"></lb> |
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1711 ἀπὸ τῆς ΓΔ.</s> <s>ἔστι δὲ καὶ τὸ ἀπὸ τῆς ΕΑ διπλάσιον τοῦ ἀπὸ <lb n="1799,001,,,,2,9,41"></lb> |
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1712 τῆς ΑΓ· τὰ ἄρα ἀπὸ τῶν ΑΕ, ΕΖ τετράγωνα διπλάσιά <lb n="1799,001,,,,2,9,42"></lb> |
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1713 ἐστι τῶν ἀπὸ τῶν ΑΓ, ΓΔ τετραγώνων.</s> <s>τοῖς δὲ ἀπὸ τῶν <lb n="1799,001,,,,2,9,43"></lb> |
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1714 ΑΕ, ΕΖ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΑΖ τετράγωνον· ὀρθὴ γάρ <lb n="1799,001,,,,2,9,44"></lb> |
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1715 ἐστιν ἡ ὑπὸ ΑΕΖ γωνία· τὸ ἄρα ἀπὸ τῆς ΑΖ τετρά-<lb n="1799,001,,,,2,9,45"></lb> |
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1716 γωνον διπλάσιόν ἐστι τῶν ἀπὸ τῶν ΑΓ, ΓΔ.</s> <s>τῷ δὲ ἀπὸ <lb n="1799,001,,,,2,9,46"></lb> |
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1717 τῆς ΑΖ ἴσα τὰ ἀπὸ τῶν ΑΔ, ΔΖ· ὀρθὴ γὰρ ἡ πρὸς τῷ Δ <lb n="1799,001,,,,2,9,47"></lb> |
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1718 γωνία· τὰ ἄρα ἀπὸ τῶν ΑΔ, ΔΖ διπλάσιά ἐστι τῶν ἀπὸ <lb n="1799,001,,,,2,9,48"></lb> |
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1719 τῶν ΑΓ, ΓΔ τετραγώνων.</s> <s>ἴση δὲ ἡ ΔΖ τῇ ΔΒ· τὰ ἄρα <lb n="1799,001,,,,2,9,49"></lb> |
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1720 ἀπὸ τῶν ΑΔ, ΔΒ τετράγωνα διπλάσιά ἐστι τῶν ἀπὸ τῶν <lb n="1799,001,,,,2,9,50"></lb> |
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1721 ΑΓ, ΓΔ τετραγώνων.</s> <lb n="1799,001,,,,2,9,51"></lb> |
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1722 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ εἰς ἴσα καὶ ἄνισα, τὰ <lb n="1799,001,,,,2,9,52"></lb> |
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1723 ἀπὸ τῶν ἀνίσων τῆς ὅλης τμημάτων τετράγωνα διπλά-<lb n="1799,001,,,,2,9,53"></lb> |
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1724 σιά ἐστι τοῦ τε ἀπὸ τῆς ἡμισείας καὶ τοῦ ἀπὸ τῆς μεταξὺ <lb n="1799,001,,,,2,9,54"></lb> |
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1725 τῶν τομῶν τετραγώνου· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,2,9,55"></lb> |
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1726 <s>Ἐὰν εὐθεῖα γραμμὴ τμηθῇ δίχα, προστεθῇ δέ τις αὐτῇ <lb n="1799,001,,,,2,10,1"></lb> |
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1727 εὐθεῖα ἐπ' εὐθείας, τὸ ἀπὸ τῆς ὅλης σὺν τῇ προσκειμένῃ <lb n="1799,001,,,,2,10,2"></lb> |
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1728 καὶ τὸ ἀπὸ τῆς προσκειμένης τὰ συναμφότερα τετράγωνα <lb n="1799,001,,,,2,10,3"></lb> |
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1729 διπλάσιά ἐστι τοῦ τε ἀπὸ τῆς ἡμισείας καὶ τοῦ ἀπὸ τῆς <pb></pb> <lb n="1799,001,,,,2,10,4"></lb> |
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1730 συγκειμένης ἔκ τε τῆς ἡμισείας καὶ τῆς προσκειμένης <lb n="1799,001,,,,2,10,5"></lb> |
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1731 ὡς ἀπὸ μιᾶς ἀναγραφέντος τετραγώνου.</s> <lb n="1799,001,,,,2,10,6"></lb> |
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1732 <s>Εὐθεῖα γάρ τις ἡ ΑΒ τετμήσθω δίχα κατὰ τὸ Γ, προσ-<lb n="1799,001,,,,2,10,7"></lb> |
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1733 κείσθω δέ τις αὐτῇ εὐθεῖα ἐπ' εὐθείας ἡ ΒΔ· λέγω, ὅτι <lb n="1799,001,,,,2,10,8"></lb> |
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1734 τὰ ἀπὸ τῶν ΑΔ, ΔΒ τετράγωνα δι-<lb n="1799,001,,,,2,10,9"></lb> |
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1735 πλάσιά ἐστι τῶν ἀπὸ τῶν ΑΓ, ΓΔ <lb n="1799,001,,,,2,10,10"></lb> |
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1736 τετραγώνων.</s> <lb n="1799,001,,,,2,10,11"></lb> |
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1737 <s>Ἤχθω γὰρ ἀπὸ τοῦ Γ σημείου τῇ <lb n="1799,001,,,,2,10,12"></lb> |
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1738 ΑΒ πρὸς ὀρθὰς ἡ ΓΕ, καὶ κείσθω <lb n="1799,001,,,,2,10,13"></lb> |
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1739 ἴση ἑκατέρᾳ, τῶν ΑΓ, ΓΒ, καὶ ἐπ-<lb n="1799,001,,,,2,10,14"></lb> |
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1740 εζεύχθωσαν αἱ ΕΑ, ΕΒ· καὶ διὰ μὲν τοῦ Ε τῇ ΑΔ παράλ-<lb n="1799,001,,,,2,10,15"></lb> |
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1741 ληλος ἤχθω ἡ ΕΖ, διὰ δὲ τοῦ Δ τῇ ΓΕ παράλληλος ἤχθω <lb n="1799,001,,,,2,10,16"></lb> |
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1742 ἡ ΖΔ.</s> <s>καὶ ἐπεὶ εἰς παραλλήλους εὐθείας τὰς ΕΓ, ΖΔ εὐ-<lb n="1799,001,,,,2,10,17"></lb> |
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1743 θεῖά τις ἐνέπεσεν ἡ ΕΖ, αἱ ὑπὸ ΓΕΖ, ΕΖΔ ἄρα δυσὶν <lb n="1799,001,,,,2,10,18"></lb> |
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1744 ὀρθαῖς ἴσαι εἰσίν· αἱ ἄρα ὑπὸ ΖΕΒ, ΕΖΔ δύο ὀρθῶν <lb n="1799,001,,,,2,10,19"></lb> |
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1745 ἐλάσσονές εἰσιν· αἱ δὲ ἀπ' ἐλασσόνων ἢ δύο ὀρθῶν ἐκβαλ-<lb n="1799,001,,,,2,10,20"></lb> |
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1746 λόμεναι συμπίπτουσιν· αἱ ἄρα ΕΒ, ΖΔ ἐκβαλλόμεναι <lb n="1799,001,,,,2,10,21"></lb> |
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1747 ἐπὶ τὰ Β, Δ μέρη συμπεσοῦνται.</s> <s>ἐκβεβλήσθωσαν καὶ συμ-<lb n="1799,001,,,,2,10,22"></lb> |
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1748 πιπτέτωσαν κατὰ τὸ Η, καὶ ἐπεζεύχθω ἡ ΑΗ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,2,10,23"></lb> |
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1749 ἴση ἐστὶν ἡ ΑΓ τῇ ΓΕ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΕΑΓ <lb n="1799,001,,,,2,10,24"></lb> |
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1750 τῇ ὑπὸ ΑΕΓ· καὶ ὀρθὴ ἡ πρὸς τῷ Γ· ἡμίσεια ἄρα ὀρθῆς <lb n="1799,001,,,,2,10,25"></lb> |
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1751 [ἐστιν] ἑκατέρα τῶν ὑπὸ ΕΑΓ, ΑΕΓ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,2,10,26"></lb> |
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1752 ἑκατέρα τῶν ὑπὸ ΓΕΒ, ΕΒΓ ἡμίσειά ἐστιν ὀρθῆς· ὀρθὴ <lb n="1799,001,,,,2,10,27"></lb> |
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1753 ἄρα ἐστὶν ἡ ὑπὸ ΑΕΒ.</s> <s>καὶ ἐπεὶ ἡμίσεια ὀρθῆς ἐστιν ἡ <lb n="1799,001,,,,2,10,28"></lb> |
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1754 ὑπὸ Ε*βΓ, ἡμίσεια ἄρα ὀρθῆς καὶ ἡ ὑπὸ ΔΒΗ.</s> <s>ἔστι δὲ <pb></pb> <lb n="1799,001,,,,2,10,29"></lb> |
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1755 καὶ ἡ ὑπὸ ΒΔΗ ὀρθή· ἴση γάρ ἐστι τῇ ὑπὸ ΔΓΕ· ἐναλλὰξ <lb n="1799,001,,,,2,10,30"></lb> |
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1756 γάρ· λοιπὴ ἄρα ἡ ὑπὸ ΔΗΒ ἡμίσειά ἐστιν ὀρθῆς· ἡ ἄρα <lb n="1799,001,,,,2,10,31"></lb> |
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1757 ὑπὸ ΔΗΒ τῇ ὑπὸ ΔΒΗ ἐστιν ἴση· ὥστε καὶ πλευρὰ ἡ ΒΔ <lb n="1799,001,,,,2,10,32"></lb> |
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1758 πλευρᾷ τῇ ΗΔ ἐστιν ἴση.</s> <s>πάλιν, ἐπεὶ ἡ ὑπὸ ΕΗΖ ἡμίσειά <lb n="1799,001,,,,2,10,33"></lb> |
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1759 ἐστιν ὀρθῆς, ὀρθὴ δὲ ἡ πρὸς τῷ Ζ· ἴση γάρ ἐστι τῇ ἀπεναν-<lb n="1799,001,,,,2,10,34"></lb> |
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1760 τίον τῇ πρὸς τῷ Γ· λοιπὴ ἄρα ἡ ὑπὸ ΖΕΗ ἡμίσειά ἐστιν <lb n="1799,001,,,,2,10,35"></lb> |
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1761 ὀρθῆς· ἴση ἄρα ἡ ὑπὸ ΕΗΖ γωνία τῇ ὑπὸ ΖΕΗ· ὥστε <lb n="1799,001,,,,2,10,36"></lb> |
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1762 καὶ πλευρὰ ἡ ΗΖ πλευρᾷ τῇ ΕΖ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ [ἴση <lb n="1799,001,,,,2,10,37"></lb> |
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1763 ἐστὶν ἡ ΕΓ τῇ ΓΑ,] ἴσον ἐστὶ [καὶ] τὸ ἀπὸ τῆς ΕΓ τετρά-<lb n="1799,001,,,,2,10,38"></lb> |
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1764 γωνον τῷ ἀπὸ τῆς ΓΑ τετραγώνῳ· τὰ ἄρα ἀπὸ τῶν ΕΓ, <lb n="1799,001,,,,2,10,39"></lb> |
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1765 ΓΑ τετράγωνα διπλάσιά ἐστι τοῦ ἀπὸ τῆς ΓΑ τετραγώ-<lb n="1799,001,,,,2,10,40"></lb> |
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1766 νου.</s> <s>τοῖς δὲ ἀπὸ τῶν ΕΓ, ΓΑ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΕΑ· <lb n="1799,001,,,,2,10,41"></lb> |
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1767 τὸ ἄρα ἀπὸ τῆς ΕΑ τετράγωνον διπλάσιόν ἐστι τοῦ ἀπὸ <lb n="1799,001,,,,2,10,42"></lb> |
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1768 τῆς ΑΓ τετραγώνου.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΖΗ τῇ ΕΖ, <lb n="1799,001,,,,2,10,43"></lb> |
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1769 ἴσον ἐστὶ καὶ τὸ ἀπὸ τῆς ΖΗ τῷ ἀπὸ τῆς ΖΕ· τὰ ἄρα <lb n="1799,001,,,,2,10,44"></lb> |
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1770 ἀπὸ τῶν ΗΖ, ΖΕ διπλάσιά ἐστι τοῦ ἀπὸ τῆς ΕΖ.</s> <s>τοῖς <lb n="1799,001,,,,2,10,45"></lb> |
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1771 δὲ ἀπὸ τῶν ΗΖ, ΖΕ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΕΗ· τὸ ἄρα <lb n="1799,001,,,,2,10,46"></lb> |
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1772 ἀπὸ τῆς ΕΗ διπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΕΖ.</s> <s>ἴση <pb></pb> <lb n="1799,001,,,,2,10,47"></lb> |
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1773 δὲ ἡ ΕΖ τῇ ΓΔ· τὸ ἄρα ἀπὸ τῆς ΕΗ τετράγωνον <lb n="1799,001,,,,2,10,48"></lb> |
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1774 διπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΓΔ.</s> <s>ἐδείχθη δὲ καὶ τὸ <lb n="1799,001,,,,2,10,49"></lb> |
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1775 ἀπὸ τῆς ΕΑ διπλάσιον τοῦ ἀπὸ τῆς ΑΓ· τὰ ἄρα ἀπὸ τῶν <lb n="1799,001,,,,2,10,50"></lb> |
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1776 ΑΕ, ΕΗ τετράγωνα διπλάσιά ἐστι τῶν ἀπὸ τῶν ΑΓ, ΓΔ <lb n="1799,001,,,,2,10,51"></lb> |
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1777 τετραγώνων.</s> <s>τοῖς δὲ ἀπὸ τῶν ΑΕ, ΕΗ τετραγώνοις ἴσον <lb n="1799,001,,,,2,10,52"></lb> |
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1778 ἐστὶ τὸ ἀπὸ τῆς ΑΗ τετράγωνον· τὸ ἄρα ἀπὸ τῆς ΑΗ δι-<lb n="1799,001,,,,2,10,53"></lb> |
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1779 πλάσιόν ἐστι τῶν ἀπὸ τῶν ΑΓ, ΓΔ.</s> <s>τῷ δὲ ἀπὸ τῆς ΑΗ <lb n="1799,001,,,,2,10,54"></lb> |
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1780 ἴσα ἐστὶ τὰ ἀπὸ τῶν ΑΔ, ΔΗ· τὰ ἄρα ἀπὸ τῶν ΑΔ, ΔΗ <lb n="1799,001,,,,2,10,55"></lb> |
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1781 [τετράγωνα] διπλάσιά ἐστι τῶν ἀπὸ τῶν ΑΓ, ΓΔ [τετρα-<lb n="1799,001,,,,2,10,56"></lb> |
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1782 γώνων].</s> <s>ἴση δὲ ἡ ΔΗ τῇ ΔΒ· τὰ ἄρα ἀπὸ τῶν ΑΔ, ΔΒ <lb n="1799,001,,,,2,10,57"></lb> |
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1783 [τετράγωνα] διπλάσιά ἐστι τῶν ἀπὸ τῶν ΑΓ, ΓΔ τετρα-<lb n="1799,001,,,,2,10,58"></lb> |
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1784 γώνων.</s> <lb n="1799,001,,,,2,10,59"></lb> |
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1785 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ δίχα, προστεθῇ δέ τις <lb n="1799,001,,,,2,10,60"></lb> |
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1786 αὐτῇ εὐθεῖα ἐπ' εὐθείας, τὸ ἀπὸ τῆς ὅλης σὺν τῇ προσ-<lb n="1799,001,,,,2,10,61"></lb> |
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1787 κειμένῃ καὶ τὸ ἀπὸ τῆς προσκειμένης τὰ συναμφότερα <lb n="1799,001,,,,2,10,62"></lb> |
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1788 τετράγωνα διπλάσιά ἐστι τοῦ τε ἀπὸ τῆς ἡμισείας καὶ τοῦ <lb n="1799,001,,,,2,10,63"></lb> |
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1789 ἀπὸ τῆς συγκειμένης ἔκ τε τῆς ἡμισείας καὶ τῆς προσκει-<lb n="1799,001,,,,2,10,64"></lb> |
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1790 μένης ὡς ἀπὸ μιᾶς ἀναγραφέντος τετραγώνου· ὅπερ ἔδει <lb n="1799,001,,,,2,10,65"></lb> |
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1791 δεῖξαι.</s> <lb n="1799,001,,,,2,10,66"></lb> |
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1792 <s>Τὴν δοθεῖσαν εὐθεῖαν τεμεῖν ὥστε τὸ ὑπὸ τῆς ὅλης καὶ <lb n="1799,001,,,,2,11,1"></lb> |
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1793 τοῦ ἑτέρου τῶν τμημάτων περιεχόμενον ὀρθογώνιον ἴσον <lb n="1799,001,,,,2,11,2"></lb> |
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1794 εἶναι τῷ ἀπὸ τοῦ λοιποῦ τμήματος τετραγώνῳ.</s> <s><pb></pb> <lb n="1799,001,,,,2,11,3"></lb> |
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1795 Ἔστω ἡ δοθεῖσα εὐθεῖα ἡ ΑΒ· δεῖ δὴ τὴν ΑΒ τεμεῖν <lb n="1799,001,,,,2,11,4"></lb> |
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1796 ὥστε τὸ ὑπὸ τῆς ὅλης καὶ τοῦ ἑτέρου τῶν τμημάτων <lb n="1799,001,,,,2,11,5"></lb> |
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1797 περιεχόμενον ὀρθογώνιον ἴσον εἶναι τῷ ἀπὸ τοῦ λοιποῦ <lb n="1799,001,,,,2,11,6"></lb> |
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1798 τμήματος τετραγώνῳ.</s> <lb n="1799,001,,,,2,11,7"></lb> |
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1799 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΒΔΓ, <lb n="1799,001,,,,2,11,8"></lb> |
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1800 καὶ τετμήσθω ἡ ΑΓ δίχα κατὰ τὸ Ε σημεῖον, καὶ ἐπεζεύ-<lb n="1799,001,,,,2,11,9"></lb> |
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1801 χθω ἡ ΒΕ, καὶ διήχθω ἡ ΓΑ ἐπὶ τὸ Ζ, καὶ <lb n="1799,001,,,,2,11,10"></lb> |
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1802 κείσθω τῇ ΒΕ ἴση ἡ ΕΖ, καὶ ἀναγεγράφθω <lb n="1799,001,,,,2,11,11"></lb> |
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1803 ἀπὸ τῆς ΑΖ τετράγωνον τὸ ΖΘ, καὶ διήχθω <lb n="1799,001,,,,2,11,12"></lb> |
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1804 ἡ ΗΘ ἐπὶ τὸ Κ· λέγω, ὅτι ἡ ΑΒ τέτμηται <lb n="1799,001,,,,2,11,13"></lb> |
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1805 κατὰ τὸ Θ, ὥστε τὸ ὑπὸ τῶν ΑΒ, ΒΘ <lb n="1799,001,,,,2,11,14"></lb> |
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1806 περιεχόμενον ὀρθογώνιον ἴσον ποιεῖν τῷ ἀπὸ <lb n="1799,001,,,,2,11,15"></lb> |
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1807 τῆς ΑΘ τετραγώνῳ.</s> <lb n="1799,001,,,,2,11,16"></lb> |
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1808 <s>Ἐπεὶ γὰρ εὐθεῖα ἡ ΑΓ τέτμηται δίχα κατὰ τὸ Ε, πρόσ-<lb n="1799,001,,,,2,11,17"></lb> |
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1809 κειται δὲ αὐτῇ ἡ ΖΑ, τὸ ἄρα ὑπὸ τῶν ΓΖ, ΖΑ περιεχό-<lb n="1799,001,,,,2,11,18"></lb> |
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1810 μενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΑΕ τετραγώνου <lb n="1799,001,,,,2,11,19"></lb> |
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1811 ἴσον ἐστὶ τῷ ἀπὸ τῆς ΕΖ τετραγώνῳ.</s> <s>ἴση δὲ ἡ ΕΖ τῇ <lb n="1799,001,,,,2,11,20"></lb> |
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1812 ΕΒ· τὸ ἄρα ὑπὸ τῶν ΓΖ, ΖΑ μετὰ τοῦ ἀπὸ τῆς ΑΕ ἴσον <lb n="1799,001,,,,2,11,21"></lb> |
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1813 ἐστὶ τῷ ἀπὸ ΕΒ.</s> <s>ἀλλὰ τῷ ἀπὸ ΕΒ ἴσα ἐστὶ τὰ ἀπὸ τῶν <lb n="1799,001,,,,2,11,22"></lb> |
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1814 ΒΑ, ΑΕ· ὀρθὴ γὰρ ἡ πρὸς τῷ Α γωνία· τὸ ἄρα ὑπὸ τῶν <lb n="1799,001,,,,2,11,23"></lb> |
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1815 ΓΖ, ΖΑ μετὰ τοῦ ἀπὸ τῆς ΑΕ ἴσον ἐστὶ τοῖς ἀπὸ τῶν <lb n="1799,001,,,,2,11,24"></lb> |
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1816 ΒΑ, ΑΕ.</s> <s>κοινὸν ἀφῃρήσθω τὸ ἀπὸ τῆς ΑΕ· λοιπὸν ἄρα <lb n="1799,001,,,,2,11,25"></lb> |
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1817 τὸ ὑπὸ τῶν ΓΖ, ΖΑ περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ <pb></pb> <lb n="1799,001,,,,2,11,26"></lb> |
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1818 τῷ ἀπὸ τῆς ΑΒ τετραγώνῳ.</s> <s>καί ἐστι τὸ μὲν ὑπὸ τῶν <lb n="1799,001,,,,2,11,27"></lb> |
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1819 ΓΖ, ΖΑ τὸ ΖΚ· ἴση γὰρ ἡ ΑΖ τῇ ΖΗ· τὸ δὲ ἀπὸ τῆς <lb n="1799,001,,,,2,11,28"></lb> |
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1820 ΑΒ τὸ ΑΔ· τὸ ἄρα ΖΚ ἴσον ἐστὶ τῷ ΑΔ.</s> <s>κοινὸν ἀφῃρήσθω <lb n="1799,001,,,,2,11,29"></lb> |
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1821 τὸ ΑΚ· λοιπὸν ἄρα τὸ ΖΘ τῷ ΘΔ ἴσον ἐστίν.</s> <s>καί ἐστι τὸ <lb n="1799,001,,,,2,11,30"></lb> |
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1822 μὲν ΘΔ τὸ ὑπὸ τῶν ΑΒ, ΒΘ· ἴση γὰρ ἡ ΑΒ τῇ ΒΔ· τὸ <lb n="1799,001,,,,2,11,31"></lb> |
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1823 δὲ ΖΘ τὸ ἀπὸ τῆς ΑΘ· τὸ ἄρα ὑπὸ τῶν ΑΒ, ΒΘ περι-<lb n="1799,001,,,,2,11,32"></lb> |
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1824 εχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ἀπὸ ΘΑ τετραγώνῳ.</s> <lb n="1799,001,,,,2,11,33"></lb> |
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1825 <s>Ἡ ἄρα δοθεῖσα εὐθεῖα ἡ ΑΒ τέτμηται κατὰ τὸ Θ ὥστε <lb n="1799,001,,,,2,11,34"></lb> |
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1826 τὸ ὑπὸ τῶν ΑΒ, ΒΘ περιεχόμενον ὀρθογώνιον ἴσον ποιεῖν <lb n="1799,001,,,,2,11,35"></lb> |
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1827 τῷ ἀπὸ τῆς ΘΑ τετραγώνῳ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,2,11,36"></lb> |
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1828 <s>Ἐν τοῖς ἀμβλυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ἀμβλεῖ-<lb n="1799,001,,,,2,12,1"></lb> |
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1829 αν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον μεῖζόν ἐστι <lb n="1799,001,,,,2,12,2"></lb> |
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1830 τῶν ἀπὸ τῶν τὴν ἀμβλεῖαν γωνίαν περιεχουσῶν πλευρῶν <lb n="1799,001,,,,2,12,3"></lb> |
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1831 τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε μιᾶς τῶν περὶ τὴν <lb n="1799,001,,,,2,12,4"></lb> |
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1832 ἀμβλεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος πίπτει, καὶ τῆς ἀπο-<lb n="1799,001,,,,2,12,5"></lb> |
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1833 λαμβανομένης ἐκτὸς ὑπὸ τῆς καθέτου πρὸς τῇ ἀμβλείᾳ <lb n="1799,001,,,,2,12,6"></lb> |
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1834 γωνίᾳ.</s> <lb n="1799,001,,,,2,12,7"></lb> |
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1835 <s>Ἔστω ἀμβλυγώνιον τρίγωνον τὸ ΑΒΓ ἀμβλεῖαν ἔχον <lb n="1799,001,,,,2,12,8"></lb> |
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1836 τὴν ὑπὸ ΒΑΓ, καὶ ἤχθω ἀπὸ τοῦ Β σημείου ἐπὶ τὴν ΓΑ <lb n="1799,001,,,,2,12,9"></lb> |
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1837 ἐκβληθεῖσαν κάθετος ἡ ΒΔ.</s> <s>λέγω, ὅτι τὸ ἀπὸ τῆς ΒΓ <lb n="1799,001,,,,2,12,10"></lb> |
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1838 τετράγωνον μεῖζόν ἐστι τῶν ἀπὸ τῶν ΒΑ, ΑΓ τετρα-<lb n="1799,001,,,,2,12,11"></lb> |
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1839 γώνων τῷ δὶς ὑπὸ τῶν ΓΑ, ΑΔ περιεχομένῳ ὀρθογωνίῳ.</s> <s><pb></pb> <lb n="1799,001,,,,2,12,12"></lb> |
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1840 Ἐπεὶ γὰρ εὐθεῖα ἡ ΓΑ τέτμηται, ὡς ἔτυχεν, κατὰ <lb n="1799,001,,,,2,12,13"></lb> |
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1841 τὸ Α σημεῖον, τὸ ἄρα ἀπὸ τῆς ΔΓ ἴσον ἐστὶ τοῖς ἀπὸ <lb n="1799,001,,,,2,12,14"></lb> |
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1842 τῶν ΓΑ, ΑΔ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΓΑ, ΑΔ <lb n="1799,001,,,,2,12,15"></lb> |
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1843 περιεχομένῳ ὀρθογωνίῳ.</s> <s>κοινὸν προσκείσθω τὸ ἀπὸ τῆς <lb n="1799,001,,,,2,12,16"></lb> |
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1844 ΔΒ· τὰ ἄρα ἀπὸ τῶν ΓΔ, ΔΒ ἴσα ἐστὶ τοῖς τε <lb n="1799,001,,,,2,12,17"></lb> |
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1845 ἀπὸ τῶν ΓΑ, ΑΔ, ΔΒ τετραγώνοις καὶ τῷ δὶς <lb n="1799,001,,,,2,12,18"></lb> |
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1846 ὑπὸ τῶν ΓΑ, ΑΔ [περιεχομένῳ ὀρθογωνίῳ].</s> <lb n="1799,001,,,,2,12,19"></lb> |
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1847 <s>ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΓΔ, ΔΒ ἴσον ἐστὶ τὸ <lb n="1799,001,,,,2,12,20"></lb> |
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1848 ἀπὸ τῆς ΓΒ· ὀρθὴ γὰρ ἡ πρὸς τῷ Δ γωνία· <lb n="1799,001,,,,2,12,21"></lb> |
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1849 τοῖς δὲ ἀπὸ τῶν ΑΔ, ΔΒ ἴσον τὸ ἀπὸ τῆς <lb n="1799,001,,,,2,12,22"></lb> |
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1850 ΑΒ· τὸ ἄρα ἀπὸ τῆς ΓΒ τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ <lb n="1799,001,,,,2,12,23"></lb> |
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1851 τῶν ΓΑ, ΑΒ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΓΑ, ΑΔ <lb n="1799,001,,,,2,12,24"></lb> |
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1852 περιεχομένῳ ὀρθογωνίῳ· ὥστε τὸ ἀπὸ τῆς ΓΒ τετράγω-<lb n="1799,001,,,,2,12,25"></lb> |
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1853 νον τῶν ἀπὸ τῶν ΓΑ, ΑΒ τετραγώνων μεῖζόν ἐστι τῷ δὶς <lb n="1799,001,,,,2,12,26"></lb> |
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1854 ὑπὸ τῶν ΓΑ, ΑΔ περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,2,12,27"></lb> |
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1855 <s>Ἐν ἄρα τοῖς ἀμβλυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν <lb n="1799,001,,,,2,12,28"></lb> |
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1856 ἀμβλεῖαν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον μεῖ-<lb n="1799,001,,,,2,12,29"></lb> |
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1857 ζόν ἐστι τῶν ἀπὸ τῶν τὴν ἀμβλεῖαν γωνίαν περιεχου-<lb n="1799,001,,,,2,12,30"></lb> |
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1858 σῶν πλευρῶν τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε <lb n="1799,001,,,,2,12,31"></lb> |
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1859 μιᾶς τῶν περὶ τὴν ἀμβλεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος <lb n="1799,001,,,,2,12,32"></lb> |
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1860 πίπτει, καὶ τῆς ἀπολαμβανομένης ἐκτὸς ὑπὸ τῆς καθέτου <lb n="1799,001,,,,2,12,33"></lb> |
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1861 πρὸς τῇ ἀμβλείᾳ γωνίᾳ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,2,12,34"></lb> |
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1862 Ἐν τοῖς ὀξυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀξεῖαν <lb n="1799,001,,,,2,13,1"></lb> |
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1863 γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἔλαττόν ἐστι <lb n="1799,001,,,,2,13,2"></lb> |
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1864 τῶν ἀπὸ τῶν τὴν ὀξεῖαν γωνίαν περιεχουσῶν πλευρῶν <lb n="1799,001,,,,2,13,3"></lb> |
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1865 τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε μιᾶς τῶν περὶ <lb n="1799,001,,,,2,13,4"></lb> |
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1866 τὴν ὀξεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος πίπτει, καὶ τῆς ἀπο-<lb n="1799,001,,,,2,13,5"></lb> |
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1867 λαμβανομένης ἐντὸς ὑπὸ τῆς καθέτου πρὸς τῇ ὀξείᾳ γωνίᾳ.</s> <lb n="1799,001,,,,2,13,6"></lb> |
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1868 <s>Ἔστω ὀξυγώνιον τρίγωνον τὸ ΑΒΓ ὀξεῖαν <lb n="1799,001,,,,2,13,7"></lb> |
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1869 ἔχον τὴν πρὸς τῷ Β γωνίαν, καὶ ἤχθω ἀπὸ <lb n="1799,001,,,,2,13,8"></lb> |
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1870 τοῦ Α σημείου ἐπὶ τὴν ΒΓ κάθετος ἡ ΑΔ· <lb n="1799,001,,,,2,13,9"></lb> |
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1871 λέγω, ὅτι τὸ ἀπὸ τῆς ΑΓ τετράγωνον ἔλαττόν <lb n="1799,001,,,,2,13,10"></lb> |
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1872 ἐστι τῶν ἀπὸ τῶν ΓΒ, ΒΑ τετραγώνων τῷ δὶς <lb n="1799,001,,,,2,13,11"></lb> |
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1873 ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,2,13,12"></lb> |
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1874 <s>Ἐπεὶ γὰρ εὐθεῖα ἡ ΓΒ τέτμηται, ὡς ἔτυχεν, κατὰ τὸ <lb n="1799,001,,,,2,13,13"></lb> |
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1875 Δ, τὰ ἄρα ἀπὸ τῶν ΓΒ, ΒΔ τετράγωνα ἴσα ἐστὶ τῷ <lb n="1799,001,,,,2,13,14"></lb> |
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1876 τε δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ καὶ τῷ <lb n="1799,001,,,,2,13,15"></lb> |
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1877 ἀπὸ τῆς ΔΓ τετραγώνῳ.</s> <s>κοινὸν προσκείσθω τὸ ἀπὸ τῆς <lb n="1799,001,,,,2,13,16"></lb> |
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1878 ΔΑ τετράγωνον· τὰ ἄρα ἀπὸ τῶν ΓΒ, ΒΔ, ΔΑ τετρά-<lb n="1799,001,,,,2,13,17"></lb> |
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1879 γωνα ἴσα ἐστὶ τῷ τε δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ <lb n="1799,001,,,,2,13,18"></lb> |
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1880 ὀρθογωνίῳ καὶ τοῖς ἀπὸ τῶν ΑΔ, ΔΓ τετραγώνοις.</s> <s>ἀλλὰ <lb n="1799,001,,,,2,13,19"></lb> |
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1881 τοῖς μὲν ἀπὸ τῶν ΒΔ, ΔΑ ἴσον τὸ ἀπὸ τῆς ΑΒ· ὀρθὴ γὰρ <lb n="1799,001,,,,2,13,20"></lb> |
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1882 ἡ πρὸς τῷ Δ γωνίᾳ· τοῖς δὲ ἀπὸ τῶν ΑΔ, ΔΓ ἴσον τὸ ἀπὸ <lb n="1799,001,,,,2,13,21"></lb> |
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1883 τῆς ΑΓ· τὰ ἄρα ἀπὸ τῶν ΓΒ, ΒΑ ἴσα ἐστὶ τῷ τε ἀπὸ τῆς <lb n="1799,001,,,,2,13,22"></lb> |
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1884 ΑΓ καὶ τῷ δὶς ὑπὸ τῶν ΓΒ, ΒΔ· ὥστε μόνον τὸ ἀπὸ τῆς <pb></pb> <lb n="1799,001,,,,2,13,23"></lb> |
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1885 ΑΓ ἔλαττόν ἐστι τῶν ἀπὸ τῶν ΓΒ, ΒΑ τετραγώνων τῷ <lb n="1799,001,,,,2,13,24"></lb> |
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1886 δὶς ὑπὸ τῶν ΓΒ, ΒΔ περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,2,13,25"></lb> |
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1887 <s>Ἐν ἄρα τοῖς ὀξυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀξεῖαν <lb n="1799,001,,,,2,13,26"></lb> |
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1888 γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἔλαττόν ἐστι <lb n="1799,001,,,,2,13,27"></lb> |
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1889 τῶν ἀπὸ τῶν τὴν ὀξεῖαν γωνίαν περιεχουσῶν πλευρῶν <lb n="1799,001,,,,2,13,28"></lb> |
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1890 τετραγώνων τῷ περιεχομένῳ δὶς ὑπό τε μιᾶς τῶν περὶ <lb n="1799,001,,,,2,13,29"></lb> |
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1891 τὴν ὀξεῖαν γωνίαν, ἐφ' ἣν ἡ κάθετος πίπτει, καὶ τῆς <lb n="1799,001,,,,2,13,30"></lb> |
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1892 ἀπολαμβανομένης ἐντὸς ὑπὸ τῆς καθέτου πρὸς τῇ ὀξείᾳ <lb n="1799,001,,,,2,13,31"></lb> |
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1893 γωνίᾳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,2,13,32"></lb> |
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1894 <s>Τῷ δοθέντι εὐθυγράμμῳ ἴσον τετράγωνον συστήσασθαι.</s> <lb n="1799,001,,,,2,14,1"></lb> |
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1895 <s>Ἔστω τὸ δοθὲν εὐθύγραμμον τὸ Α· δεῖ δὴ τῷ Α εὐθυ-<lb n="1799,001,,,,2,14,2"></lb> |
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1896 γράμμῳ ἴσον τετράγωνον συστήσασθαι.</s> <lb n="1799,001,,,,2,14,3"></lb> |
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1897 <s>Συνεστάτω γὰρ τῷ Α εὐθυγράμμῳ ἴσον παραλληλό-<lb n="1799,001,,,,2,14,4"></lb> |
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1898 γραμμον ὀρθογώνιον τὸ ΒΔ· εἰ μὲν οὖν ἴση ἐστὶν ἡ ΒΕ <lb n="1799,001,,,,2,14,5"></lb> |
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1899 τῇ ΕΔ, γεγονὸς ἂν εἴη τὸ ἐπιταχθέν.</s> <s>συνέσταται γὰρ τῷ <lb n="1799,001,,,,2,14,6"></lb> |
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1900 Α εὐθυγράμμῳ ἴσον τετράγωνον τὸ <lb n="1799,001,,,,2,14,7"></lb> |
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1901 ΒΔ· εἰ δὲ οὔ, μία τῶν ΒΕ, ΕΔ <lb n="1799,001,,,,2,14,8"></lb> |
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1902 μείζων ἐστίν.</s> <s>ἔστω μείζων ἡ ΒΕ, <lb n="1799,001,,,,2,14,9"></lb> |
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1903 καὶ ἐκβεβλήσθω ἐπὶ τὸ Ζ, καὶ κείσθω <lb n="1799,001,,,,2,14,10"></lb> |
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1904 τῇ ΕΔ ἴση ἡ ΕΖ, καὶ τετμήσθω ἡ ΒΖ <lb n="1799,001,,,,2,14,11"></lb> |
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1905 δίχα κατὰ τὸ Η, καὶ κέντρῳ τῷ Η, <lb n="1799,001,,,,2,14,12"></lb> |
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1906 διαστήματι δὲ ἑνὶ τῶν ΗΒ, ΗΖ <pb></pb> <lb n="1799,001,,,,2,14,13"></lb> |
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1907 ἡμικύκλιον γεγράφθω τὸ ΒΘΖ, καὶ ἐκβεβλήσθω ἡ ΔΕ <lb n="1799,001,,,,2,14,14"></lb> |
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1908 ἐπὶ τὸ Θ, καὶ ἐπεζεύχθω ἡ ΗΘ.</s> <lb n="1799,001,,,,2,14,15"></lb> |
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1909 <s>Ἐπεὶ οὖν εὐθεῖα ἡ ΒΖ τέτμηται εἰς μὲν ἴσα κατὰ τὸ Η, <lb n="1799,001,,,,2,14,16"></lb> |
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1910 εἰς δὲ ἄνισα κατὰ τὸ Ε, τὸ ἄρα ὑπὸ τῶν ΒΕ, ΕΖ περι-<lb n="1799,001,,,,2,14,17"></lb> |
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1911 εχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΕΗ τετραγώνου <lb n="1799,001,,,,2,14,18"></lb> |
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1912 ἴσον ἐστὶ τῷ ἀπὸ τῆς ΗΖ τετραγώνῳ.</s> <s>ἴση δὲ ἡ ΗΖ τῇ <lb n="1799,001,,,,2,14,19"></lb> |
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1913 ΗΘ· τὸ ἄρα ὑπὸ τῶν ΒΕ, ΕΖ μετὰ τοῦ ἀπὸ τῆς ΗΕ <lb n="1799,001,,,,2,14,20"></lb> |
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1914 ἴσον ἐστὶ τῷ ἀπὸ τῆς ΗΘ.</s> <s>τῷ δὲ ἀπὸ τῆς ΗΘ ἴσα ἐστὶ τὰ <lb n="1799,001,,,,2,14,21"></lb> |
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1915 ἀπὸ τῶν ΘΕ, ΕΗ τετράγωνα· τὸ ἄρα ὑπὸ τῶν ΒΕ, ΕΖ <lb n="1799,001,,,,2,14,22"></lb> |
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1916 μετὰ τοῦ ἀπὸ ΗΕ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΘΕ, ΕΗ.</s> <s>κοι-<lb n="1799,001,,,,2,14,23"></lb> |
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1917 νὸν ἀφῃρήσθω τὸ ἀπὸ τῆς ΗΕ τετράγωνον· λοιπὸν ἄρα <lb n="1799,001,,,,2,14,24"></lb> |
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1918 τὸ ὑπὸ τῶν ΒΕ, ΕΖ περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ <lb n="1799,001,,,,2,14,25"></lb> |
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1919 τῷ ἀπὸ τῆς ΕΘ τετραγώνῳ.</s> <s>ἀλλὰ τὸ ὑπὸ τῶν ΒΕ, ΕΖ τὸ <lb n="1799,001,,,,2,14,26"></lb> |
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1920 ΒΔ ἐστιν· ἴση γὰρ ἡ ΕΖ τῇ ΕΔ· τὸ ἄρα ΒΔ παραλληλό-<lb n="1799,001,,,,2,14,27"></lb> |
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1921 γραμμον ἴσον ἐστὶ τῷ ἀπὸ τῆς ΘΕ τετραγώνῳ.</s> <s>ἴσον δὲ τὸ <lb n="1799,001,,,,2,14,28"></lb> |
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1922 ΒΔ τῷ Α εὐθυγράμμῳ.</s> <s>καὶ τὸ Α ἄρα εὐθύγραμμον ἴσον <lb n="1799,001,,,,2,14,29"></lb> |
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1923 ἐστὶ τῷ ἀπὸ τῆς ΕΘ ἀναγραφησομένῳ τετραγώνῳ.</s> <lb n="1799,001,,,,2,14,30"></lb> |
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1924 <s>Τῷ ἄρα δοθέντι εὐθυγράμμῳ τῷ Α ἴσον τετράγωνον συν-<lb n="1799,001,,,,2,14,31"></lb> |
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1925 έσταται τὸ ἀπὸ τῆς ΕΘ ἀναγραφησόμενον· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,2,14,32"></lb> |
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1926 </s></p> |
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1927 </chap> |
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1928 <chap><p type="main"><s>Ἴσοι κύκλοι εἰσίν, ὧν αἱ διάμετροι ἴσαι εἰσίν, ἢ ὧν <lb n="1799,001,,,,3;HOR,1,1"></lb> |
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1929 |
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1930 αἱ ἐκ τῶν κέντρων ἴσαι εἰσίν.</s> <lb n="1799,001,,,,3;HOR,1,2"></lb> |
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1931 <s>Εὐθεῖα κύκλου ἐφάπτεσθαι λέγεται, ἥτις ἁπτομένη <lb n="1799,001,,,,3;HOR,2,1"></lb> |
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1932 τοῦ κύκλου καὶ ἐκβαλλομένη οὐ τέμνει τὸν κύκλον.</s> <lb n="1799,001,,,,3;HOR,2,2"></lb> |
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1933 <s>Κύκλοι ἐφάπτεσθαι ἀλλήλων λέγονται οἵτινες ἁπτό-<lb n="1799,001,,,,3;HOR,3,1"></lb> |
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1934 μενοι ἀλλήλων οὐ τέμνουσιν ἀλλήλους.</s> <lb n="1799,001,,,,3;HOR,3,2"></lb> |
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1935 <s>Ἐν κύκλῳ ἴσον ἀπέχειν ἀπὸ τοῦ κέντρου εὐθεῖαι <lb n="1799,001,,,,3;HOR,4,1"></lb> |
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1936 λέγονται, ὅταν αἱ ἀπὸ τοῦ κέντρου ἐπ' αὐτὰς κάθετοι <lb n="1799,001,,,,3;HOR,4,2"></lb> |
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1937 ἀγόμεναι ἴσαι ὦσιν.</s> <lb n="1799,001,,,,3;HOR,4,3"></lb> |
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1938 <s>Μεῖζον δὲ ἀπέχειν λέγεται, ἐφ' ἣν ἡ μείζων κάθετος <lb n="1799,001,,,,3;HOR,5,1"></lb> |
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1939 πίπτει.</s> <lb n="1799,001,,,,3;HOR,5,2"></lb> |
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1940 <s>Τμῆμα κύκλου ἐστὶ τὸ περιεχόμενον σχῆμα ὑπό τε <lb n="1799,001,,,,3;HOR,6,1"></lb> |
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1941 εὐθείας καὶ κύκλου περιφερείας.</s> <lb n="1799,001,,,,3;HOR,6,2"></lb> |
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1942 <s>Τμήματος δὲ γωνία ἐστὶν ἡ περιεχομένη ὑπό τε <lb n="1799,001,,,,3;HOR,7,1"></lb> |
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1943 εὐθείας καὶ κύκλου περιφερείας.</s> <lb n="1799,001,,,,3;HOR,7,2"></lb> |
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1944 <s>Ἐν τμήματι δὲ γωνία ἐστίν, ὅταν ἐπὶ τῆς περιφερείας <lb n="1799,001,,,,3;HOR,8,1"></lb> |
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1945 τοῦ τμήματος ληφθῇ τι σημεῖον καὶ ἀπ' αὐτοῦ ἐπὶ τὰ <pb></pb> <lb n="1799,001,,,,3;HOR,8,2"></lb> |
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1946 πέρατα τῆς εὐθείας, ἥ ἐστι βάσις τοῦ τμήματος, ἐπιζευ-<lb n="1799,001,,,,3;HOR,8,3"></lb> |
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1947 χθῶσιν εὐθεῖαι, ἡ περιεχομένη γωνία ὑπὸ τῶν ἐπιζευ-<lb n="1799,001,,,,3;HOR,8,4"></lb> |
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1948 χθεισῶν εὐθειῶν.</s> <lb n="1799,001,,,,3;HOR,8,5"></lb> |
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1949 <s>Ὅταν δὲ αἱ περιέχουσαι τὴν γωνίαν εὐθεῖαι ἀπο-<lb n="1799,001,,,,3;HOR,9,1"></lb> |
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1950 λαμβάνωσί τινα περιφέρειαν, ἐπ' ἐκείνης λέγεται βεβη-<lb n="1799,001,,,,3;HOR,9,2"></lb> |
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1951 κέναι ἡ γωνία.</s> <lb n="1799,001,,,,3;HOR,9,3"></lb> |
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1952 <s>Τομεὺς δὲ κύκλου ἐστίν, ὅταν πρὸς τῷ κέντρῳ τοῦ <lb n="1799,001,,,,3;HOR,10,1"></lb> |
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1953 κύκλου συσταθῇ γωνία, τὸ περιεχόμενον σχῆμα ὑπό τε <lb n="1799,001,,,,3;HOR,10,2"></lb> |
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1954 τῶν τὴν γωνίαν περιεχουσῶν εὐθειῶν καὶ τῆς ἀπολαμ-<lb n="1799,001,,,,3;HOR,10,3"></lb> |
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1955 βανομένης ὑπ' αὐτῶν περιφερείας.</s> <lb n="1799,001,,,,3;HOR,10,4"></lb> |
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1956 <s>Ὅμοια τμήματα κύκλων ἐστὶ τὰ δεχόμενα γωνίας <lb n="1799,001,,,,3;HOR,11,1"></lb> |
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1957 ἴσας, ἢ ἐν οἷς αἱ γωνίαι ἴσαι ἀλλήλαις εἰσίν.</s> <lb n="1799,001,,,,3;HOR,11,2"></lb> |
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1958 <s></s></p> |
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1959 </chap> |
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1960 <chap><p type="main"><s>Τοῦ δοθέντος κύκλου τὸ κέντρον εὑρεῖν.</s> <lb n="1799,001,,,,3,1,1"></lb> |
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1961 <s> |
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1962 Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓ· δεῖ δὴ τοῦ ΑΒΓ κύκλου <lb n="1799,001,,,,3,1,2"></lb> |
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1963 τὸ κέντρον εὑρεῖν.</s> <lb n="1799,001,,,,3,1,3"></lb> |
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1964 <s>Διήχθω τις εἰς αὐτόν, ὡς ἔτυχεν, εὐθεῖα ἡ ΑΒ, καὶ <lb n="1799,001,,,,3,1,4"></lb> |
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1965 τετμήσθω δίχα κατὰ τὸ Δ σημεῖον, καὶ ἀπὸ τοῦ Δ τῇ ΑΒ <lb n="1799,001,,,,3,1,5"></lb> |
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1966 πρὸς ὀρθὰς ἤχθω ἡ ΔΓ καὶ διήχθω ἐπὶ τὸ <lb n="1799,001,,,,3,1,6"></lb> |
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1967 Ε, καὶ τετμήσθω ἡ ΓΕ δίχα κατὰ τὸ Ζ· <lb n="1799,001,,,,3,1,7"></lb> |
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1968 λέγω, ὅτι τὸ Ζ κέντρον ἐστὶ τοῦ ΑΒΓ <lb n="1799,001,,,,3,1,8"></lb> |
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1969 [κύκλου].</s> <lb n="1799,001,,,,3,1,9"></lb> |
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1970 <s>Μὴ γάρ, ἀλλ' εἰ δυνατόν, ἔστω τὸ Η, <lb n="1799,001,,,,3,1,10"></lb> |
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1971 καὶ ἐπεζεύχθωσαν αἱ ΗΑ, ΗΔ, ΗΒ.</s> <s>καὶ <pb></pb> <lb n="1799,001,,,,3,1,11"></lb> |
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1972 ἐπεὶ ἴση ἐστὶν ἡ ΑΔ τῇ ΔΒ, κοινὴ δὲ ἡ ΔΗ, δύο δὴ αἱ <lb n="1799,001,,,,3,1,12"></lb> |
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1973 ΑΔ, ΔΗ δύο ταῖς ΗΔ, ΔΒ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· <lb n="1799,001,,,,3,1,13"></lb> |
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1974 καὶ βάσις ἡ ΗΑ βάσει τῇ ΗΒ ἐστιν ἴση· ἐκ κέντρου <lb n="1799,001,,,,3,1,14"></lb> |
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1975 γάρ· γωνία ἄρα ἡ ὑπὸ ΑΔΗ γωνίᾳ τῇ ὑπὸ ΗΔΒ ἴση <lb n="1799,001,,,,3,1,15"></lb> |
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1976 ἐστίν.</s> <s>ὅταν δὲ εὐθεῖα ἐπ' εὐθεῖαν σταθεῖσα τὰς ἐφεξῆς <lb n="1799,001,,,,3,1,16"></lb> |
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1977 γωνίας ἴσας ἀλλήλαις ποιῇ, ὀρθὴ ἑκατέρα τῶν ἴσων γω-<lb n="1799,001,,,,3,1,17"></lb> |
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1978 νιῶν ἐστιν· ὀρθὴ ἄρα ἐστὶν ἡ ὑπὸ ΗΔΒ.</s> <s>ἐστὶ δὲ καὶ ἡ <lb n="1799,001,,,,3,1,18"></lb> |
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1979 ὑπὸ ΖΔΒ ὀρθή· ἴση ἄρα ἡ ὑπὸ ΖΔΒ τῇ ὑπὸ ΗΔΒ, ἡ <lb n="1799,001,,,,3,1,19"></lb> |
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1980 μείζων τῇ ἐλάττονι· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τὸ Η <lb n="1799,001,,,,3,1,20"></lb> |
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1981 κέντρον ἐστὶ τοῦ ΑΒΓ κύκλου.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι <lb n="1799,001,,,,3,1,21"></lb> |
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1982 οὐδ' ἄλλο τι πλὴν τοῦ Ζ.</s> <lb n="1799,001,,,,3,1,22"></lb> |
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1983 <s>Τὸ Ζ ἄρα σημεῖον κέντρον ἐστὶ τοῦ ΑΒΓ [κύκλου].</s> <lb n="1799,001,,,,3,1,23"></lb> |
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1984 <s>Πόρισμα <lb n="1799,001,,,,3,1,24"></lb> |
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1985 Ἐκ δὴ τούτου φανερόν, ὅτι ἐὰν ἐν κύκλῳ εὐθεῖά τις <lb n="1799,001,,,,3,1,25"></lb> |
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1986 εὐθεῖάν τινα δίχα καὶ πρὸς ὀρθὰς τέμνῃ, ἐπὶ τῆς τεμνού-<lb n="1799,001,,,,3,1,26"></lb> |
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1987 σης ἐστὶ τὸ κέντρον τοῦ κύκλου· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,3,1,27"></lb> |
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1988 <s>Ἐὰν κύκλου ἐπὶ τῆς περιφερείας ληφθῇ δύο τυχόντα <lb n="1799,001,,,,3,2,1"></lb> |
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1989 σημεῖα, ἡ ἐπὶ τὰ σημεῖα ἐπιζευγνυμένη εὐθεῖα ἐντὸς <lb n="1799,001,,,,3,2,2"></lb> |
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1990 πεσεῖται τοῦ κύκλου.</s> <s><pb></pb> <lb n="1799,001,,,,3,2,3"></lb> |
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1991 Ἔστω κύκλος ὁ ΑΒΓ, καὶ ἐπὶ τῆς περιφερείας αὐτοῦ <lb n="1799,001,,,,3,2,4"></lb> |
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1992 εἰλήφθω δύο τυχόντα σημεῖα τὰ Α, Β· λέγω, ὅτι ἡ ἀπὸ <lb n="1799,001,,,,3,2,5"></lb> |
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1993 τοῦ Α ἐπὶ τὸ Β ἐπιζευγνυμένη εὐθεῖα ἐντὸς πεσεῖται τοῦ <lb n="1799,001,,,,3,2,6"></lb> |
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1994 κύκλου.</s> <lb n="1799,001,,,,3,2,7"></lb> |
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1995 <s>Μὴ γάρ, ἀλλ' εἰ δυνατόν, πιπτέτω ἐκτὸς ὡς ἡ ΑΕΒ, <lb n="1799,001,,,,3,2,8"></lb> |
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1996 καὶ εἰλήφθω τὸ κέντρον τοῦ ΑΒΓ κύκλου, καὶ ἔστω τὸ <lb n="1799,001,,,,3,2,9"></lb> |
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1997 Δ, καὶ ἐπεζεύχθωσαν αἱ ΔΑ, ΔΒ, καὶ διήχθω ἡ ΔΖΕ.</s> <lb n="1799,001,,,,3,2,10"></lb> |
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1998 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΑ τῇ ΔΒ, ἴση <lb n="1799,001,,,,3,2,11"></lb> |
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1999 ἄρα καὶ γωνία ἡ ὑπὸ ΔΑΕ τῇ ὑπὸ <lb n="1799,001,,,,3,2,12"></lb> |
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2000 ΔΒΕ· καὶ ἐπεὶ τριγώνου τοῦ ΔΑΕ <lb n="1799,001,,,,3,2,13"></lb> |
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2001 μία πλευρὰ προσεκβέβληται ἡ ΑΕΒ, <lb n="1799,001,,,,3,2,14"></lb> |
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2002 μείζων ἄρα ἡ ὑπὸ ΔΕΒ γωνία τῆς ὑπὸ <lb n="1799,001,,,,3,2,15"></lb> |
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2003 ΔΑΕ.</s> <s>ἴση δὲ ἡ ὑπὸ ΔΑΕ τῇ ὑπὸ ΔΒΕ· <lb n="1799,001,,,,3,2,16"></lb> |
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2004 μείζων ἄρα ἡ ὑπὸ ΔΕΒ τῆς ὑπὸ ΔΒΕ.</s> <lb n="1799,001,,,,3,2,17"></lb> |
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2005 <s>ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων <lb n="1799,001,,,,3,2,18"></lb> |
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2006 πλευρὰ ὑποτείνει· μείζων ἄρα ἡ ΔΒ τῆς ΔΕ.</s> <s>ἴση δὲ ἡ <lb n="1799,001,,,,3,2,19"></lb> |
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2007 ΔΒ τῇ ΔΖ.</s> <s>μείζων ἄρα ἡ ΔΖ τῆς ΔΕ ἡ ἐλάττων τῆς <lb n="1799,001,,,,3,2,20"></lb> |
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2008 μείζονος· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ἡ ἀπὸ τοῦ Α ἐπὶ <lb n="1799,001,,,,3,2,21"></lb> |
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2009 τὸ Β ἐπιζευγνυμένη εὐθεῖα ἐκτὸς πεσεῖται τοῦ κύκλου.</s> <lb n="1799,001,,,,3,2,22"></lb> |
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2010 <s>ὁμοίως δὴ δείξομεν, ὅτι οὐδὲ ἐπ' αὐτῆς τῆς περιφερείας· <lb n="1799,001,,,,3,2,23"></lb> |
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2011 ἐντὸς ἄρα.</s> <lb n="1799,001,,,,3,2,24"></lb> |
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2012 <s>Ἐὰν ἄρα κύκλου ἐπὶ τῆς περιφερείας ληφθῇ δύο τυχόντα <lb n="1799,001,,,,3,2,25"></lb> |
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2013 σημεῖα, ἡ ἐπὶ τὰ σημεῖα ἐπιζευγνυμένη εὐθεῖα ἐντὸς πεσεῖ-<lb n="1799,001,,,,3,2,26"></lb> |
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2014 ται τοῦ κύκλου· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,3,2,27"></lb> |
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2015 Ἐὰν ἐν κύκλῳ εὐθεῖά τις διὰ τοῦ κέντρου εὐθεῖάν τινα <lb n="1799,001,,,,3,3,1"></lb> |
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2016 μὴ διὰ τοῦ κέντρου δίχα τέμνῃ, καὶ πρὸς ὀρθὰς αὐτὴν <lb n="1799,001,,,,3,3,2"></lb> |
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2017 τέμνει· καὶ ἐὰν πρὸς ὀρθὰς αὐτὴν τέμνῃ, καὶ δίχα αὐτὴν <lb n="1799,001,,,,3,3,3"></lb> |
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2018 τέμνει.</s> <lb n="1799,001,,,,3,3,4"></lb> |
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2019 <s>Ἔστω κύκλος ὁ ΑΒΓ, καὶ ἐν αὐτῷ εὐθεῖά τις διὰ τοῦ <lb n="1799,001,,,,3,3,5"></lb> |
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2020 κέντρου ἡ ΓΔ εὐθεῖάν τινα μὴ διὰ τοῦ κέντρου τὴν ΑΒ <lb n="1799,001,,,,3,3,6"></lb> |
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2021 δίχα τεμνέτω κατὰ τὸ Ζ σημεῖον· λέγω, ὅτι καὶ πρὸς <lb n="1799,001,,,,3,3,7"></lb> |
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2022 ὀρθὰς αὐτὴν τέμνει.</s> <lb n="1799,001,,,,3,3,8"></lb> |
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2023 <s>Εἰλήφθω γὰρ τὸ κέντρον τοῦ ΑΒΓ κύκλου, καὶ ἔστω <lb n="1799,001,,,,3,3,9"></lb> |
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2024 τὸ Ε, καὶ ἐπεζεύχθωσαν αἱ ΕΑ, ΕΒ.</s> <lb n="1799,001,,,,3,3,10"></lb> |
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2025 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΖ τῇ ΖΒ, κοινὴ δὲ ἡ ΖΕ, δύο <lb n="1799,001,,,,3,3,11"></lb> |
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2026 δυσὶν ἴσαι [εἰσίν].</s> <s>καὶ βάσις ἡ ΕΑ βάσει τῇ ΕΒ ἴση· <lb n="1799,001,,,,3,3,12"></lb> |
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2027 γωνία ἄρα ἡ ὑπὸ ΑΖΕ γωνίᾳ τῇ ὑπὸ ΒΖΕ ἴση ἐστίν.</s> <lb n="1799,001,,,,3,3,13"></lb> |
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2028 <s>ὅταν δὲ εὐθεῖα ἐπ' εὐθεῖαν σταθεῖσα τὰς ἐφεξῆς γωνίας <lb n="1799,001,,,,3,3,14"></lb> |
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2029 ἴσας ἀλλήλαις ποιῇ, ὀρθὴ ἑκατέρα τῶν ἴσων γωνιῶν ἐστιν· <lb n="1799,001,,,,3,3,15"></lb> |
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2030 ἑκατέρα ἄρα τῶν ὑπὸ ΑΖΕ, ΒΖΕ ὀρθή ἐστιν.</s> <s>ἡ ΓΔ ἄρα <lb n="1799,001,,,,3,3,16"></lb> |
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2031 διὰ τοῦ κέντρου οὖσα τὴν ΑΒ μὴ διὰ τοῦ κέντρου <lb n="1799,001,,,,3,3,17"></lb> |
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2032 οὖσαν δίχα τέμνουσα καὶ πρὸς ὀρθὰς <lb n="1799,001,,,,3,3,18"></lb> |
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2033 τέμνει.</s> <lb n="1799,001,,,,3,3,19"></lb> |
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2034 <s>Ἀλλὰ δὴ ἡ ΓΔ τὴν ΑΒ πρὸς ὀρθὰς <lb n="1799,001,,,,3,3,20"></lb> |
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2035 τεμνέτω· λέγω, ὅτι καὶ δίχα αὐτὴν τέμνει, <lb n="1799,001,,,,3,3,21"></lb> |
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2036 τουτέστιν, ὅτι ἴση ἐστὶν ἡ ΑΖ τῇ ΖΒ.</s> <lb n="1799,001,,,,3,3,22"></lb> |
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2037 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεὶ <lb n="1799,001,,,,3,3,23"></lb> |
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2038 ἴση ἐστὶν ἡ ΕΑ τῇ ΕΒ, ἴση ἐστὶ καὶ γωνία <pb></pb> <lb n="1799,001,,,,3,3,24"></lb> |
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2039 ἡ ὑπὸ ΕΑΖ τῇ ὑπὸ ΕΒΖ.</s> <s>ἐστὶ δὲ καὶ ὀρθὴ ἡ ὑπὸ ΑΖΕ <lb n="1799,001,,,,3,3,25"></lb> |
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2040 ὀρθῇ τῇ ὑπὸ ΒΖΕ ἴση· δύο ἄρα τρίγωνά ἐστι τὰ ΕΑΖ, <lb n="1799,001,,,,3,3,26"></lb> |
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2041 ΕΖΒ τὰς δύο γωνίας δυσὶ γωνίαις ἴσας ἔχοντα καὶ μίαν <lb n="1799,001,,,,3,3,27"></lb> |
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2042 πλευρὰν μιᾷ πλευρᾷ ἴσην κοινὴν αὐτῶν τὴν ΕΖ ὑποτείνου-<lb n="1799,001,,,,3,3,28"></lb> |
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2043 σαν ὑπὸ μίαν τῶν ἴσων γωνιῶν· καὶ τὰς λοιπὰς ἄρα πλευρὰς <lb n="1799,001,,,,3,3,29"></lb> |
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2044 ταῖς λοιπαῖς πλευραῖς ἴσας ἕξει· ἴση ἄρα ἡ ΑΖ τῇ ΖΒ.</s> <lb n="1799,001,,,,3,3,30"></lb> |
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2045 <s>Ἐὰν ἄρα ἐν κύκλῳ εὐθεῖά τις διὰ τοῦ κέντρου εὐθεῖάν <lb n="1799,001,,,,3,3,31"></lb> |
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2046 τινα μὴ διὰ τοῦ κέντρου δίχα τέμνῃ, καὶ πρὸς ὀρθὰς αὐ-<lb n="1799,001,,,,3,3,32"></lb> |
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2047 τὴν τέμνει· καὶ ἐὰν πρὸς ὀρθὰς αὐτὴν τέμνῃ, καὶ δίχα <lb n="1799,001,,,,3,3,33"></lb> |
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2048 αὐτὴν τέμνει· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,3,34"></lb> |
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2049 <s>Ἐὰν ἐν κύκλῳ δύο εὐθεῖαι τέμνωσιν ἀλλήλας μὴ διὰ τοῦ <lb n="1799,001,,,,3,4,1"></lb> |
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2050 κέντρου οὖσαι, οὐ τέμνουσιν ἀλλήλας δίχα.</s> <lb n="1799,001,,,,3,4,2"></lb> |
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2051 <s>Ἔστω κύκλος ὁ ΑΒΓΔ, καὶ ἐν αὐτῷ <lb n="1799,001,,,,3,4,3"></lb> |
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2052 δύο εὐθεῖαι αἱ ΑΓ, ΒΔ τεμνέτωσαν ἀλλή-<lb n="1799,001,,,,3,4,4"></lb> |
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2053 λας κατὰ τὸ Ε μὴ διὰ τοῦ κέντρου οὖσαι· <lb n="1799,001,,,,3,4,5"></lb> |
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2054 λέγω, ὅτι οὐ τέμνουσιν ἀλλήλας δίχα.</s> <lb n="1799,001,,,,3,4,6"></lb> |
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2055 <s>Εἰ γὰρ δυνατόν, τεμνέτωσαν ἀλλήλας <lb n="1799,001,,,,3,4,7"></lb> |
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2056 δίχα ὥστε ἴσην εἶναι τὴν μὲν ΑΕ τῇ ΕΓ, <lb n="1799,001,,,,3,4,8"></lb> |
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2057 τὴν δὲ ΒΕ τῇ ΕΔ· καὶ εἰλήφθω τὸ κέντρον τοῦ ΑΒΓΔ <lb n="1799,001,,,,3,4,9"></lb> |
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2058 κύκλου, καὶ ἔστω τὸ Ζ, καὶ ἐπεζεύχθω ἡ ΖΕ.</s> <lb n="1799,001,,,,3,4,10"></lb> |
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2059 <s>Ἐπεὶ οὖν εὐθεῖά τις διὰ τοῦ κέντρου ἡ ΖΕ εὐθεῖάν τινα <lb n="1799,001,,,,3,4,11"></lb> |
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2060 μὴ διὰ τοῦ κέντρου τὴν ΑΓ δίχα τέμνει, καὶ πρὸς ὀρθὰς <pb></pb> <lb n="1799,001,,,,3,4,12"></lb> |
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2061 αὐτὴν τέμνει· ὀρθὴ ἄρα ἐστὶν ἡ ὑπὸ ΖΕΑ· πάλιν, ἐπεὶ <lb n="1799,001,,,,3,4,13"></lb> |
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2062 εὐθεῖά τις ἡ ΖΕ εὐθεῖάν τινα τὴν ΒΔ δίχα τέμνει, καὶ <lb n="1799,001,,,,3,4,14"></lb> |
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2063 πρὸς ὀρθὰς αὐτὴν τέμνει· ὀρθὴ ἄρα ἡ ὑπὸ ΖΕΒ.</s> <s>ἐδείχθη <lb n="1799,001,,,,3,4,15"></lb> |
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2064 δὲ καὶ ἡ ὑπὸ ΖΕΑ ὀρθή· ἴση ἄρα ἡ ὑπὸ ΖΕΑ τῇ ὑπὸ <lb n="1799,001,,,,3,4,16"></lb> |
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2065 ΖΕΒ ἡ ἐλάττων τῇ μείζονι· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ <lb n="1799,001,,,,3,4,17"></lb> |
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2066 ἄρα αἱ ΑΓ, ΒΔ τέμνουσιν ἀλλήλας δίχα.</s> <lb n="1799,001,,,,3,4,18"></lb> |
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2067 <s>Ἐὰν ἄρα ἐν κύκλῳ δύο εὐθεῖαι τέμνωσιν ἀλλήλας μὴ διὰ <lb n="1799,001,,,,3,4,19"></lb> |
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2068 τοῦ κέντρου οὖσαι, οὐ τέμνουσιν ἀλλήλας δίχα· ὅπερ ἔδει <lb n="1799,001,,,,3,4,20"></lb> |
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2069 δεῖξαι.</s> <lb n="1799,001,,,,3,4,21"></lb> |
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2070 <s>Ἐὰν δύο κύκλοι τέμνωσιν ἀλλήλους, οὐκ ἔσται αὐτῶν <lb n="1799,001,,,,3,5,1"></lb> |
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2071 τὸ αὐτὸ κέντρον.</s> <lb n="1799,001,,,,3,5,2"></lb> |
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2072 <s>Δύο γὰρ κύκλοι οἱ ΑΒΓ, ΓΔΗ τεμνέτωσαν ἀλλήλους <lb n="1799,001,,,,3,5,3"></lb> |
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2073 κατὰ τὰ Β, Γ σημεῖα.</s> <s>λέγω, ὅτι οὐκ ἔσται αὐτῶν τὸ αὐτὸ <lb n="1799,001,,,,3,5,4"></lb> |
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2074 κέντρον.</s> <lb n="1799,001,,,,3,5,5"></lb> |
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2075 <s>Εἰ γὰρ δυνατόν, ἔστω τὸ Ε, καὶ ἐπεζεύχθω ἡ ΕΓ, καὶ <lb n="1799,001,,,,3,5,6"></lb> |
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2076 διήχθω ἡ ΕΖΗ, ὡς ἔτυχεν.</s> <s>καὶ ἐπεὶ τὸ Ε σημεῖον κέν-<lb n="1799,001,,,,3,5,7"></lb> |
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2077 τρον ἐστὶ τοῦ ΑΒΓ κύκλου, ἴση ἐστὶν ἡ ΕΓ τῇ ΕΖ.</s> <s>πάλιν, <lb n="1799,001,,,,3,5,8"></lb> |
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2078 ἐπεὶ τὸ Ε σημεῖον κέντρον ἐστὶ τοῦ ΓΔΗ <lb n="1799,001,,,,3,5,9"></lb> |
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2079 κύκλου, ἴση ἐστὶν ἡ ΕΓ τῇ ΕΗ· ἐδείχθη <lb n="1799,001,,,,3,5,10"></lb> |
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2080 δὲ ἡ ΕΓ καὶ τῇ ΕΖ ἴση· καὶ ἡ ΕΖ ἄρα τῇ <lb n="1799,001,,,,3,5,11"></lb> |
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2081 ΕΗ ἐστιν ἴση ἡ ἐλάσσων τῇ μείζονι· ὅπερ <lb n="1799,001,,,,3,5,12"></lb> |
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2082 ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τὸ Ε σημεῖον <lb n="1799,001,,,,3,5,13"></lb> |
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2083 κέντρον ἐστὶ τῶν ΑΒΓ, ΓΔΗ κύκλων.</s> <s><pb></pb> <lb n="1799,001,,,,3,5,14"></lb> |
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2084 Ἐὰν ἄρα δύο κύκλοι τέμνωσιν ἀλλήλους, οὐκ ἔστιν <lb n="1799,001,,,,3,5,15"></lb> |
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2085 αὐτῶν τὸ αὐτὸ κέντρον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,5,16"></lb> |
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2086 <s>Ἐὰν δύο κύκλοι ἐφάπτωνται ἀλλήλων, οὐκ ἔσται αὐτῶν <lb n="1799,001,,,,3,6,1"></lb> |
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2087 τὸ αὐτὸ κέντρον.</s> <lb n="1799,001,,,,3,6,2"></lb> |
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2088 <s>Δύο γὰρ κύκλοι οἱ ΑΒΓ, ΓΔΕ ἐφαπτέσθωσαν ἀλλήλων <lb n="1799,001,,,,3,6,3"></lb> |
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2089 κατὰ τὸ Γ σημεῖον· λέγω, ὅτι οὐκ ἔσται αὐτῶν τὸ αὐτὸ <lb n="1799,001,,,,3,6,4"></lb> |
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2090 κέντρον.</s> <lb n="1799,001,,,,3,6,5"></lb> |
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2091 <s>Εἰ γὰρ δυνατόν, ἔστω τὸ Ζ, καὶ ἐπεζεύχθω ἡ ΖΓ, <lb n="1799,001,,,,3,6,6"></lb> |
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2092 καὶ διήχθω, ὡς ἔτυχεν, ἡ ΖΕΒ.</s> <lb n="1799,001,,,,3,6,7"></lb> |
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2093 <s>Ἐπεὶ οὖν τὸ Ζ σημεῖον κέντρον <lb n="1799,001,,,,3,6,8"></lb> |
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2094 ἐστὶ τοῦ ΑΒΓ κύκλου, ἴση ἐστὶν ἡ ΖΓ <lb n="1799,001,,,,3,6,9"></lb> |
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2095 τῇ ΖΒ.</s> <s>πάλιν, ἐπεὶ τὸ Ζ σημεῖον κέν-<lb n="1799,001,,,,3,6,10"></lb> |
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2096 τρον ἐστὶ τοῦ ΓΔΕ κύκλου, ἴση ἐστὶν <lb n="1799,001,,,,3,6,11"></lb> |
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2097 ἡ ΖΓ τῇ ΖΕ.</s> <s>ἐδείχθη δὲ ἡ ΖΓ τῇ ΖΒ <lb n="1799,001,,,,3,6,12"></lb> |
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2098 ἴση· καὶ ἡ ΖΕ ἄρα τῇ ΖΒ ἐστιν ἴση, ἡ <lb n="1799,001,,,,3,6,13"></lb> |
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2099 ἐλάττων τῇ μείζονι· ὅπερ ἐστὶν ἀδύνα-<lb n="1799,001,,,,3,6,14"></lb> |
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2100 τον.</s> <s>οὐκ ἄρα τὸ Ζ σημεῖον κέντρον ἐστὶ <lb n="1799,001,,,,3,6,15"></lb> |
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2101 τῶν ΑΒΓ, ΓΔΕ κύκλων.</s> <lb n="1799,001,,,,3,6,16"></lb> |
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2102 <s>Ἐὰν ἄρα δύο κύκλοι ἐφάπτωνται ἀλλήλων, οὐκ ἔσται <lb n="1799,001,,,,3,6,17"></lb> |
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2103 αὐτῶν τὸ αὐτὸ κέντρον· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,3,6,18"></lb> |
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2104 Ἐὰν κύκλου ἐπὶ τῆς διαμέτρου ληφθῇ τι σημεῖον, ὃ μή <lb n="1799,001,,,,3,7,1"></lb> |
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2105 ἐστι κέντρον τοῦ κύκλου, ἀπὸ δὲ τοῦ σημείου πρὸς τὸν <lb n="1799,001,,,,3,7,2"></lb> |
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2106 κύκλον προσπίπτωσιν εὐθεῖαί τινες, μεγίστη μὲν ἔσται, <lb n="1799,001,,,,3,7,3"></lb> |
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2107 ἐφ' ἧς τὸ κέντρον, ἐλαχίστη δὲ ἡ λοιπή, τῶν δὲ ἄλλων <lb n="1799,001,,,,3,7,4"></lb> |
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2108 ἀεὶ ἡ ἔγγιον τῆς διὰ τοῦ κέντρου τῆς ἀπώτερον μείζων <lb n="1799,001,,,,3,7,5"></lb> |
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2109 ἐστίν, δύο δὲ μόνον ἴσαι ἀπὸ τοῦ σημείου προσπεσοῦνται <lb n="1799,001,,,,3,7,6"></lb> |
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2110 πρὸς τὸν κύκλον ἐφ' ἑκάτερα τῆς ἐλαχίστης.</s> <lb n="1799,001,,,,3,7,7"></lb> |
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2111 <s>Ἔστω κύκλος ὁ ΑΒΓΔ, διάμετρος δὲ αὐτοῦ ἔστω ἡ <lb n="1799,001,,,,3,7,8"></lb> |
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2112 ΑΔ, καὶ ἐπὶ τῆς ΑΔ εἰλήφθω τι σημεῖον τὸ Ζ, ὃ μή ἐστι <lb n="1799,001,,,,3,7,9"></lb> |
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2113 κέντρον τοῦ κύκλου, κέντρον δὲ τοῦ κύκλου ἔστω τὸ Ε, <lb n="1799,001,,,,3,7,10"></lb> |
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2114 καὶ ἀπὸ τοῦ Ζ πρὸς τὸν ΑΒΓΔ κύκλον προσπιπτέτωσαν <lb n="1799,001,,,,3,7,11"></lb> |
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2115 εὐθεῖαί τινες αἱ ΖΒ, ΖΓ, ΖΗ· λέγω, ὅτι μεγίστη μέν <lb n="1799,001,,,,3,7,12"></lb> |
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2116 ἐστιν ἡ ΖΑ, ἐλαχίστη δὲ ἡ ΖΔ, τῶν δὲ <lb n="1799,001,,,,3,7,13"></lb> |
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2117 ἄλλων ἡ μὲν ΖΒ τῆς ΖΓ μείζων, ἡ δὲ <lb n="1799,001,,,,3,7,14"></lb> |
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2118 ΖΓ τῆς ΖΗ.</s> <lb n="1799,001,,,,3,7,15"></lb> |
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2119 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΒΕ, ΓΕ, ΗΕ.</s> <lb n="1799,001,,,,3,7,16"></lb> |
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2120 <s>καὶ ἐπεὶ παντὸς τριγώνου αἱ δύο πλευραὶ <lb n="1799,001,,,,3,7,17"></lb> |
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2121 τῆς λοιπῆς μείζονές εἰσιν, αἱ ἄρα ΕΒ, <lb n="1799,001,,,,3,7,18"></lb> |
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2122 ΕΖ τῆς ΒΖ μείζονές εἰσιν.</s> <s>ἴση δὲ ἡ ΑΕ <lb n="1799,001,,,,3,7,19"></lb> |
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2123 τῇ ΒΕ [αἱ ἄρα ΒΕ, ΕΖ ἴσαι εἰσὶ τῇ <lb n="1799,001,,,,3,7,20"></lb> |
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2124 ΑΖ]· μείζων ἄρα ἡ ΑΖ τῆς ΒΖ.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν ἡ <lb n="1799,001,,,,3,7,21"></lb> |
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2125 ΒΕ τῇ ΓΕ, κοινὴ δὲ ἡ ΖΕ, δύο δὴ αἱ ΒΕ, ΕΖ δυσὶ ταῖς <lb n="1799,001,,,,3,7,22"></lb> |
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2126 ΓΕ, ΕΖ ἴσαι εἰσίν.</s> <s>ἀλλὰ καὶ γωνία ἡ ὑπὸ ΒΕΖ γωνίας <pb></pb> <lb n="1799,001,,,,3,7,23"></lb> |
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2127 τῆς ὑπὸ ΓΕΖ μείζων.</s> <s>βάσις ἄρα ἡ ΒΖ βάσεως τῆς <lb n="1799,001,,,,3,7,24"></lb> |
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2128 ΓΖ μείζων ἐστίν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΓΖ τῆς ΖΗ <lb n="1799,001,,,,3,7,25"></lb> |
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2129 μείζων ἐστίν.</s> <lb n="1799,001,,,,3,7,26"></lb> |
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2130 <s>Πάλιν, ἐπεὶ αἱ ΗΖ, ΖΕ τῆς ΕΗ μείζονές εἰσιν, ἴση δὲ <lb n="1799,001,,,,3,7,27"></lb> |
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2131 ἡ ΕΗ τῇ ΕΔ, αἱ ἄρα ΗΖ, ΖΕ τῆς ΕΔ μείζονές εἰσιν.</s> <lb n="1799,001,,,,3,7,28"></lb> |
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2132 <s>κοινὴ ἀφῃρήσθω ἡ ΕΖ· λοιπὴ ἄρα ἡ ΗΖ λοιπῆς τῆς ΖΔ <lb n="1799,001,,,,3,7,29"></lb> |
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2133 μείζων ἐστίν.</s> <s>μεγίστη μὲν ἄρα ἡ ΖΑ, ἐλαχίστη δὲ ἡ ΖΔ, <lb n="1799,001,,,,3,7,30"></lb> |
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2134 μείζων δὲ ἡ μὲν ΖΒ τῆς ΖΓ, ἡ δὲ ΖΓ τῆς ΖΗ.</s> <lb n="1799,001,,,,3,7,31"></lb> |
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2135 <s>Λέγω, ὅτι καὶ ἀπὸ τοῦ Ζ σημείου δύο μόνον ἴσαι προσ-<lb n="1799,001,,,,3,7,32"></lb> |
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2136 πεσοῦνται πρὸς τὸν ΑΒΓΔ κύκλον ἐφ' ἑκάτερα τῆς ΖΔ <lb n="1799,001,,,,3,7,33"></lb> |
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2137 ἐλαχίστης.</s> <s>συνεστάτω γὰρ πρὸς τῇ ΕΖ εὐθείᾳ καὶ τῷ <lb n="1799,001,,,,3,7,34"></lb> |
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2138 πρὸς αὐτῇ σημείῳ τῷ Ε τῇ ὑπὸ ΗΕΖ γωνίᾳ ἴση ἡ ὑπὸ <lb n="1799,001,,,,3,7,35"></lb> |
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2139 ΖΕΘ, καὶ ἐπεζεύχθω ἡ ΖΘ.</s> <s>ἐπεὶ οὖν ἴση ἐστὶν ἡ ΗΕ τῇ <lb n="1799,001,,,,3,7,36"></lb> |
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2140 ΕΘ, κοινὴ δὲ ἡ ΕΖ, δύο δὴ αἱ ΗΕ, ΕΖ δυσὶ ταῖς ΘΕ, <lb n="1799,001,,,,3,7,37"></lb> |
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2141 ΕΖ ἴσαι εἰσίν· καὶ γωνία ἡ ὑπὸ ΗΕΖ γωνίᾳ τῇ ὑπὸ ΘΕΖ <lb n="1799,001,,,,3,7,38"></lb> |
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2142 ἴση· βάσις ἄρα ἡ ΖΗ βάσει τῇ ΖΘ ἴση ἐστίν.</s> <s>λέγω δή, ὅτι <lb n="1799,001,,,,3,7,39"></lb> |
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2143 τῇ ΖΗ ἄλλη ἴση οὐ προσπεσεῖται πρὸς τὸν κύκλον ἀπὸ <lb n="1799,001,,,,3,7,40"></lb> |
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2144 τοῦ Ζ σημείου.</s> <s>εἰ γὰρ δυνατόν, προσπιπτέτω ἡ ΖΚ.</s> <s>καὶ <lb n="1799,001,,,,3,7,41"></lb> |
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2145 ἐπεὶ ἡ ΖΚ τῇ ΖΗ ἴση ἐστίν, ἀλλὰ ἡ ΖΘ τῇ ΖΗ [ἴση ἐστίν], <lb n="1799,001,,,,3,7,42"></lb> |
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2146 καὶ ἡ ΖΚ ἄρα τῇ ΖΘ ἐστιν ἴση, ἡ ἔγγιον τῆς διὰ τοῦ κέν-<lb n="1799,001,,,,3,7,43"></lb> |
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2147 τρου τῇ ἀπώτερον ἴση· ὅπερ ἀδύνατον.</s> <s>οὐκ ἄρα ἀπὸ τοῦ <pb></pb> <lb n="1799,001,,,,3,7,44"></lb> |
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2148 Ζ σημείου ἑτέρα τις προσπεσεῖται πρὸς τὸν κύκλον ἴση <lb n="1799,001,,,,3,7,45"></lb> |
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2149 τῇ ΗΖ· μία ἄρα μόνη.</s> <lb n="1799,001,,,,3,7,46"></lb> |
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2150 <s>Ἐὰν ἄρα κύκλου ἐπὶ τῆς διαμέτρου ληφθῇ τι σημεῖον, <lb n="1799,001,,,,3,7,47"></lb> |
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2151 ὃ μή ἐστι κέντρον τοῦ κύκλου, ἀπὸ δὲ τοῦ σημείου πρὸς <lb n="1799,001,,,,3,7,48"></lb> |
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2152 τὸν κύκλον προσπίπτωσιν εὐθεῖαί τινες, μεγίστη μὲν <lb n="1799,001,,,,3,7,49"></lb> |
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2153 ἔσται, ἐφ' ἧς τὸ κέντρον, ἐλαχίστη δὲ ἡ λοιπή, τῶν δὲ <lb n="1799,001,,,,3,7,50"></lb> |
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2154 ἄλλων ἀεὶ ἡ ἔγγιον τῆς διὰ τοῦ κέντρου τῆς ἀπώτερον <lb n="1799,001,,,,3,7,51"></lb> |
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2155 μείζων ἐστίν, δύο δὲ μόνον ἴσαι ἀπὸ τοῦ αὐτοῦ σημείου <lb n="1799,001,,,,3,7,52"></lb> |
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2156 προσπεσοῦνται πρὸς τὸν κύκλον ἐφ' ἑκάτερα τῆς ἐλαχί-<lb n="1799,001,,,,3,7,53"></lb> |
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2157 στης· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,7,54"></lb> |
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2158 <s>Ἐὰν κύκλου ληφθῇ τι σημεῖον ἐκτός, ἀπὸ δὲ τοῦ ση-<lb n="1799,001,,,,3,8,1"></lb> |
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2159 μείου πρὸς τὸν κύκλον διαχθῶσιν εὐθεῖαί τινες, ὧν μία <lb n="1799,001,,,,3,8,2"></lb> |
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2160 μὲν διὰ τοῦ κέντρου, αἱ δὲ λοιπαί, ὡς ἔτυχεν, τῶν μὲν <lb n="1799,001,,,,3,8,3"></lb> |
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2161 πρὸς τὴν κοίλην περιφέρειαν προσπιπτουσῶν εὐθειῶν <lb n="1799,001,,,,3,8,4"></lb> |
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2162 μεγίστη μέν ἐστιν ἡ διὰ τοῦ κέντρου, τῶν δὲ ἄλλων ἀεὶ ἡ <lb n="1799,001,,,,3,8,5"></lb> |
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2163 ἔγγιον τῆς διὰ τοῦ κέντρου τῆς ἀπώτερον μείζων ἐστίν, <lb n="1799,001,,,,3,8,6"></lb> |
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2164 τῶν δὲ πρὸς τὴν κυρτὴν περιφέρειαν προσπιπτουσῶν <lb n="1799,001,,,,3,8,7"></lb> |
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2165 εὐθειῶν ἐλαχίστη μέν ἐστιν ἡ μεταξὺ τοῦ τε σημείου καὶ <lb n="1799,001,,,,3,8,8"></lb> |
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2166 τῆς διαμέτρου, τῶν δὲ ἄλλων ἀεὶ ἡ ἔγγιον τῆς ἐλαχίστης <pb></pb> <lb n="1799,001,,,,3,8,9"></lb> |
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2167 τῆς ἀπώτερόν ἐστιν ἐλάττων, δύο δὲ <lb n="1799,001,,,,3,8,10"></lb> |
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2168 μόνον ἴσαι ἀπὸ τοῦ σημείου προσπεσοῦν-<lb n="1799,001,,,,3,8,11"></lb> |
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2169 ται πρὸς τὸν κύκλον ἐφ' ἑκάτερα τῆς <lb n="1799,001,,,,3,8,12"></lb> |
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2170 ἐλαχίστης.</s> <lb n="1799,001,,,,3,8,13"></lb> |
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2171 <s>Ἔστω κύκλος ὁ ΑΒΓ, καὶ τοῦ ΑΒΓ <lb n="1799,001,,,,3,8,14"></lb> |
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2172 εἰλήφθω τι σημεῖον ἐκτὸς τὸ Δ, καὶ <lb n="1799,001,,,,3,8,15"></lb> |
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2173 ἀπ' αὐτοῦ διήχθωσαν εὐθεῖαί τινες αἱ <lb n="1799,001,,,,3,8,16"></lb> |
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2174 ΔΑ, ΔΕ, ΔΖ, ΔΓ, ἔστω δὲ ἡ ΔΑ διὰ <lb n="1799,001,,,,3,8,17"></lb> |
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2175 τοῦ κέντρου.</s> <s>λέγω, ὅτι τῶν μὲν πρὸς τὴν <lb n="1799,001,,,,3,8,18"></lb> |
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2176 ΑΕΖΓ κοίλην περιφέρειαν προσπιπτουσῶν εὐθειῶν με-<lb n="1799,001,,,,3,8,19"></lb> |
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2177 γίστη μέν ἐστιν ἡ διὰ τοῦ κέντρου ἡ ΔΑ, μείζων δὲ ἡ μὲν <lb n="1799,001,,,,3,8,20"></lb> |
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2178 ΔΕ τῆς ΔΖ ἡ δὲ ΔΖ τῆς ΔΓ, τῶν δὲ πρὸς τὴν ΘΛΚΗ <lb n="1799,001,,,,3,8,21"></lb> |
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2179 κυρτὴν περιφέρειαν προσπιπτουσῶν εὐθειῶν ἐλαχίστη μέν <lb n="1799,001,,,,3,8,22"></lb> |
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2180 ἐστιν ἡ ΔΗ ἡ μεταξὺ τοῦ σημείου καὶ τῆς διαμέτρου τῆς <lb n="1799,001,,,,3,8,23"></lb> |
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2181 ΑΗ, ἀεὶ δὲ ἡ ἔγγιον τῆς ΔΗ ἐλαχίστης ἐλάττων ἐστὶ <lb n="1799,001,,,,3,8,24"></lb> |
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2182 τῆς ἀπώτερον, ἡ μὲν ΔΚ τῆς ΔΛ, ἡ δὲ ΔΛ τῆς ΔΘ.</s> <s><pb></pb> <lb n="1799,001,,,,3,8,25"></lb> |
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2183 Εἰλήφθω γὰρ τὸ κέντρον τοῦ ΑΒΓ κύκλου καὶ ἔστω <lb n="1799,001,,,,3,8,26"></lb> |
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2184 τὸ Μ· καὶ ἐπεζεύχθωσαν αἱ ΜΕ, ΜΖ, ΜΓ, ΜΚ, ΜΛ, <lb n="1799,001,,,,3,8,27"></lb> |
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2185 ΜΘ.</s> <lb n="1799,001,,,,3,8,28"></lb> |
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2186 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΜ τῇ ΕΜ, κοινὴ προσκείσθω ἡ <lb n="1799,001,,,,3,8,29"></lb> |
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2187 ΜΔ· ἡ ἄρα ΑΔ ἴση ἐστὶ ταῖς ΕΜ, ΜΔ.</s> <s>ἀλλ' αἱ ΕΜ, <lb n="1799,001,,,,3,8,30"></lb> |
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2188 ΜΔ τῆς ΕΔ μείζονές εἰσιν· καὶ ἡ ΑΔ ἄρα τῆς ΕΔ μεί-<lb n="1799,001,,,,3,8,31"></lb> |
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2189 ζων ἐστίν.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΜΕ τῇ Μ*ζ, κοινὴ δὲ <lb n="1799,001,,,,3,8,32"></lb> |
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2190 ἡ ΜΔ, αἱ ΕΜ, ΜΔ ἄρα ταῖς ΖΜ, ΜΔ ἴσαι εἰσίν· καὶ <lb n="1799,001,,,,3,8,33"></lb> |
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2191 γωνία ἡ ὑπὸ ΕΜΔ γωνίας τῆς ὑπὸ ΖΜΔ μείζων ἐστίν.</s> <lb n="1799,001,,,,3,8,34"></lb> |
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2192 <s>βάσις ἄρα ἡ ΕΔ βάσεως τῆς ΖΔ μείζων ἐστίν.</s> <s>ὁμοίως δὴ <lb n="1799,001,,,,3,8,35"></lb> |
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2193 δείξομεν, ὅτι καὶ ἡ ΖΔ τῆς ΓΔ μείζων ἐστίν· μεγίστη μὲν <lb n="1799,001,,,,3,8,36"></lb> |
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2194 ἄρα ἡ ΔΑ, μείζων δὲ ἡ μὲν ΔΕ τῆς ΔΖ, ἡ δὲ ΔΖ τῆς ΔΓ.</s> <lb n="1799,001,,,,3,8,37"></lb> |
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2195 <s>Καὶ ἐπεὶ αἱ ΜΚ, ΚΔ τῆς ΜΔ μείζονές εἰσιν, ἴση δὲ ἡ <lb n="1799,001,,,,3,8,38"></lb> |
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2196 ΜΗ τῇ ΜΚ, λοιπὴ ἄρα ἡ ΚΔ λοιπῆς τῆς ΗΔ μείζων <lb n="1799,001,,,,3,8,39"></lb> |
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2197 ἐστίν· ὥστε ἡ ΗΔ τῆς ΚΔ ἐλάττων ἐστίν· καὶ ἐπεὶ τρι-<lb n="1799,001,,,,3,8,40"></lb> |
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2198 γώνου τοῦ ΜΛΔ ἐπὶ μιᾶς τῶν πλευρῶν τῆς ΜΔ δύο <lb n="1799,001,,,,3,8,41"></lb> |
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2199 εὐθεῖαι ἐντὸς συνεστάθησαν αἱ ΜΚ, ΚΔ, αἱ ἄρα ΜΚ, <lb n="1799,001,,,,3,8,42"></lb> |
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2200 ΚΔ τῶν ΜΛ, ΛΔ ἐλάττονές εἰσιν· ἴση δὲ ἡ ΜΚ τῇ ΜΛ· <pb></pb> <lb n="1799,001,,,,3,8,43"></lb> |
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2201 λοιπὴ ἄρα ἡ ΔΚ λοιπῆς τῆς ΔΛ ἐλάττων ἐστίν.</s> <s>ὁμοίως <lb n="1799,001,,,,3,8,44"></lb> |
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2202 δὴ δείξομεν, ὅτι καὶ ἡ ΔΛ τῆς ΔΘ ἐλάττων ἐστίν· ἐλαχί-<lb n="1799,001,,,,3,8,45"></lb> |
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2203 στη μὲν ἄρα ἡ ΔΗ, ἐλάττων δὲ ἡ μὲν ΔΚ τῆς ΔΛ ἡ δὲ <lb n="1799,001,,,,3,8,46"></lb> |
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2204 ΔΛ τῆς ΔΘ.</s> <lb n="1799,001,,,,3,8,47"></lb> |
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2205 <s>Λέγω, ὅτι καὶ δύο μόνον ἴσαι ἀπὸ τοῦ Δ σημείου προσ-<lb n="1799,001,,,,3,8,48"></lb> |
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2206 πεσοῦνται πρὸς τὸν κύκλον ἐφ' ἑκάτερα τῆς ΔΗ ἐλαχί-<lb n="1799,001,,,,3,8,49"></lb> |
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2207 στης· συνεστάτω πρὸς τῇ ΜΔ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ <lb n="1799,001,,,,3,8,50"></lb> |
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2208 σημείῳ τῷ Μ τῇ ὑπὸ ΚΜΔ γωνίᾳ ἴση γωνία ἡ ὑπὸ ΔΜΒ <lb n="1799,001,,,,3,8,51"></lb> |
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2209 καὶ ἐπεζεύχθω ἡ ΔΒ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΜΚ τῇ ΜΒ, <lb n="1799,001,,,,3,8,52"></lb> |
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2210 κοινὴ δὲ ἡ ΜΔ, δύο δὴ αἱ ΚΜ, ΜΔ δύο ταῖς ΒΜ, ΜΔ <lb n="1799,001,,,,3,8,53"></lb> |
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2211 ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ ΚΜΔ γωνίᾳ <lb n="1799,001,,,,3,8,54"></lb> |
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2212 τῇ ὑπὸ ΒΜΔ ἴση· βάσις ἄρα ἡ ΔΚ βάσει τῇ ΔΒ ἴση <lb n="1799,001,,,,3,8,55"></lb> |
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2213 ἐστίν.</s> <s>λέγω [δή], ὅτι τῇ ΔΚ εὐθείᾳ ἄλλη ἴση οὐ προσπεσεῖ-<lb n="1799,001,,,,3,8,56"></lb> |
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2214 ται πρὸς τὸν κύκλον ἀπὸ τοῦ Δ σημείου.</s> <s>εἰ γὰρ δυνατόν, <lb n="1799,001,,,,3,8,57"></lb> |
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2215 προσπιπτέτω καὶ ἔστω ἡ ΔΝ.</s> <s>ἐπεὶ οὖν ἡ ΔΚ τῇ ΔΝ <lb n="1799,001,,,,3,8,58"></lb> |
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2216 ἐστιν ἴση, ἀλλ' ἡ ΔΚ τῇ ΔΒ ἐστιν ἴση, καὶ ἡ ΔΒ ἄρα τῇ <lb n="1799,001,,,,3,8,59"></lb> |
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2217 ΔΝ ἐστιν ἴση, ἡ ἔγγιον τῆς ΔΗ ἐλαχίστης τῇ ἀπώτερον <lb n="1799,001,,,,3,8,60"></lb> |
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2218 [ἐστιν] ἴση· ὅπερ ἀδύνατον ἐδείχθη.</s> <s>οὐκ ἄρα πλείους ἢ <lb n="1799,001,,,,3,8,61"></lb> |
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2219 δύο ἴσαι πρὸς τὸν ΑΒΓ κύκλον ἀπὸ τοῦ Δ σημείου ἐφ' <lb n="1799,001,,,,3,8,62"></lb> |
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2220 ἑκάτερα τῆς ΔΗ ἐλαχίστης προσπεσοῦνται.</s> <s><pb></pb> <lb n="1799,001,,,,3,8,63"></lb> |
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2221 Ἐὰν ἄρα κύκλου ληφθῇ τι σημεῖον ἐκτός, ἀπὸ δὲ τοῦ <lb n="1799,001,,,,3,8,64"></lb> |
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2222 σημείου πρὸς τὸν κύκλον διαχθῶσιν εὐθεῖαί τινες, ὧν <lb n="1799,001,,,,3,8,65"></lb> |
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2223 μία μὲν διὰ τοῦ κέντρου αἱ δὲ λοιπαί, ὡς ἔτυχεν, τῶν μὲν <lb n="1799,001,,,,3,8,66"></lb> |
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2224 πρὸς τὴν κοίλην περιφέρειαν προσπιπτουσῶν εὐθειῶν <lb n="1799,001,,,,3,8,67"></lb> |
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2225 μεγίστη μέν ἐστιν ἡ διὰ τοῦ κέντρου, τῶν δὲ ἄλλων ἀεὶ <lb n="1799,001,,,,3,8,68"></lb> |
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2226 ἡ ἔγγιον τῆς διὰ τοῦ κέντρου τῆς ἀπώτερον μείζων ἐστίν, <lb n="1799,001,,,,3,8,69"></lb> |
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2227 τῶν δὲ πρὸς τὴν κυρτὴν περιφέρειαν προσπιπτουσῶν <lb n="1799,001,,,,3,8,70"></lb> |
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2228 εὐθειῶν ἐλαχίστη μέν ἐστιν ἡ μεταξὺ τοῦ τε σημείου καὶ <lb n="1799,001,,,,3,8,71"></lb> |
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2229 τῆς διαμέτρου, τῶν δὲ ἄλλων ἀεὶ ἡ ἔγγιον τῆς ἐλαχίστης <lb n="1799,001,,,,3,8,72"></lb> |
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2230 τῆς ἀπώτερόν ἐστιν ἐλάττων, δύο δὲ μόνον ἴσαι ἀπὸ τοῦ <lb n="1799,001,,,,3,8,73"></lb> |
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2231 σημείου προσπεσοῦνται πρὸς τὸν κύκλον ἐφ' ἑκάτερα τῆς <lb n="1799,001,,,,3,8,74"></lb> |
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2232 ἐλαχίστης· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,8,75"></lb> |
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2233 <s>Ἐὰν κύκλου ληφθῇ τι σημεῖον ἐντός, ἀπὸ δὲ τοῦ σημείου <lb n="1799,001,,,,3,9,1"></lb> |
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2234 πρὸς τὸν κύκλον προσπίπτωσι πλείους ἢ δύο ἴσαι εὐθεῖαι, <lb n="1799,001,,,,3,9,2"></lb> |
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2235 τὸ ληφθὲν σημεῖον κέντρον ἐστὶ τοῦ κύκλου.</s> <lb n="1799,001,,,,3,9,3"></lb> |
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2236 <s>Ἔστω κύκλος ὁ ΑΒΓ, ἐντὸς δὲ αὐτοῦ σημεῖον τὸ Δ, <lb n="1799,001,,,,3,9,4"></lb> |
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2237 καὶ ἀπὸ τοῦ Δ πρὸς τὸν ΑΒΓ κύκλον προσπιπτέτωσαν <lb n="1799,001,,,,3,9,5"></lb> |
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2238 πλείους ἢ δύο ἴσαι εὐθεῖαι αἱ ΔΑ, <lb n="1799,001,,,,3,9,6"></lb> |
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2239 ΔΒ, ΔΓ· λέγω, ὅτι τὸ Δ σημεῖον <lb n="1799,001,,,,3,9,7"></lb> |
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2240 κέντρον ἐστὶ τοῦ ΑΒΓ κύκλου.</s> <lb n="1799,001,,,,3,9,8"></lb> |
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2241 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΑΒ, ΒΓ <lb n="1799,001,,,,3,9,9"></lb> |
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2242 καὶ τετμήσθωσαν δίχα κατὰ τὰ <lb n="1799,001,,,,3,9,10"></lb> |
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2243 Ε, Ζ σημεῖα, καὶ ἐπιζευχθεῖσαι <lb n="1799,001,,,,3,9,11"></lb> |
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2244 αἱ ΕΔ, ΖΔ διήχθωσαν ἐπὶ τὰ <lb n="1799,001,,,,3,9,12"></lb> |
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2245 Η, Κ, Θ, Λ σημεῖα.</s> <lb n="1799,001,,,,3,9,13"></lb> |
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2246 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΑΕ <lb n="1799,001,,,,3,9,14"></lb> |
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2247 τῇ ΕΒ, κοινὴ δὲ ἡ ΕΔ, δύο <pb></pb> <lb n="1799,001,,,,3,9,15"></lb> |
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2248 δὴ αἱ ΑΕ, ΕΔ δύο ταῖς ΒΕ, ΕΔ ἴσαι εἰσίν· καὶ βάσις ἡ <lb n="1799,001,,,,3,9,16"></lb> |
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2249 ΔΑ βάσει τῇ ΔΒ ἴση· γωνία ἄρα ἡ ὑπὸ ΑΕΔ γωνίᾳ τῇ <lb n="1799,001,,,,3,9,17"></lb> |
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2250 ὑπὸ ΒΕΔ ἴση ἐστίν· ὀρθὴ ἄρα ἑκατέρα τῶν ὑπὸ ΑΕΔ, <lb n="1799,001,,,,3,9,18"></lb> |
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2251 ΒΕΔ γωνιῶν· ἡ ΗΚ ἄρα τὴν ΑΒ τέμνει δίχα καὶ πρὸς <lb n="1799,001,,,,3,9,19"></lb> |
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2252 ὀρθάς.</s> <s>καὶ ἐπεί, ἐὰν ἐν κύκλῳ εὐθεῖά τις εὐθεῖάν τινα δίχα <lb n="1799,001,,,,3,9,20"></lb> |
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2253 τε καὶ πρὸς ὀρθὰς τέμνῃ, ἐπὶ τῆς τεμνούσης ἐστὶ τὸ <lb n="1799,001,,,,3,9,21"></lb> |
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2254 κέντρον τοῦ κύκλου, ἐπὶ τῆς ΗΚ ἄρα ἐστὶ τὸ κέντρον τοῦ <lb n="1799,001,,,,3,9,22"></lb> |
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2255 κύκλου.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἐπὶ τῆς ΘΛ ἐστι τὸ κέντρον <lb n="1799,001,,,,3,9,23"></lb> |
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2256 τοῦ ΑΒΓ κύκλου.</s> <s>καὶ οὐδὲν ἕτερον κοινὸν ἔχουσιν αἱ ΗΚ, <lb n="1799,001,,,,3,9,24"></lb> |
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2257 ΘΛ εὐθεῖαι ἢ τὸ Δ σημεῖον· τὸ Δ ἄρα σημεῖον κέντρον <lb n="1799,001,,,,3,9,25"></lb> |
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2258 ἐστὶ τοῦ ΑΒΓ κύκλου.</s> <lb n="1799,001,,,,3,9,26"></lb> |
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2259 <s>Ἐὰν ἄρα κύκλου ληφθῇ τι σημεῖον ἐντός, ἀπὸ δὲ τοῦ <lb n="1799,001,,,,3,9,27"></lb> |
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2260 σημείου πρὸς τὸν κύκλον προσπίπτωσι πλείους ἢ δύο <lb n="1799,001,,,,3,9,28"></lb> |
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2261 ἴσαι εὐθεῖαι, τὸ ληφθὲν σημεῖον κέντρον ἐστὶ τοῦ κύκλου· <lb n="1799,001,,,,3,9,29"></lb> |
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2262 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,9,30"></lb> |
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2263 <s>Κύκλος κύκλον οὐ τέμνει κατὰ πλείονα σημεῖα ἢ δύο.</s> <lb n="1799,001,,,,3,10,1"></lb> |
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2264 <s>Εἰ γὰρ δυνατόν, κύκλος ὁ ΑΒΓ κύκλον τὸν ΔΕΖ <lb n="1799,001,,,,3,10,2"></lb> |
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2265 τεμνέτω κατὰ πλείονα σημεῖα ἢ δύο τὰ Β, Η, Ζ, Θ, καὶ <lb n="1799,001,,,,3,10,3"></lb> |
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2266 ἐπιζευχθεῖσαι αἱ ΒΘ, ΒΗ δίχα τεμνέσθωσαν κατὰ τὰ <pb></pb> <lb n="1799,001,,,,3,10,4"></lb> |
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2267 Κ, Λ σημεῖα· καὶ ἀπὸ τῶν Κ, Λ ταῖς ΒΘ, ΒΗ πρὸς ὀρθὰς <lb n="1799,001,,,,3,10,5"></lb> |
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2268 ἀχθεῖσαι αἱ ΚΓ, ΛΜ διήχθωσαν ἐπὶ τὰ Α, Ε σημεῖα.</s> <lb n="1799,001,,,,3,10,6"></lb> |
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2269 <s>Ἐπεὶ οὖν ἐν κύκλῳ τῷ <lb n="1799,001,,,,3,10,7"></lb> |
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2270 ΑΒΓ εὐθεῖά τις ἡ ΑΓ εὐ-<lb n="1799,001,,,,3,10,8"></lb> |
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2271 θεῖάν τινα τὴν ΒΘ δίχα <lb n="1799,001,,,,3,10,9"></lb> |
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2272 καὶ πρὸς ὀρθὰς τέμνει, <lb n="1799,001,,,,3,10,10"></lb> |
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2273 ἐπὶ τῆς ΑΓ ἄρα ἐστὶ τὸ <lb n="1799,001,,,,3,10,11"></lb> |
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2274 κέντρον τοῦ ΑΒΓ κύκλου.</s> <lb n="1799,001,,,,3,10,12"></lb> |
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2275 <s>πάλιν, ἐπεὶ ἐν κύκλῳ τῷ <lb n="1799,001,,,,3,10,13"></lb> |
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2276 αὐτῷ τῷ ΑΒΓ εὐθεῖά τις <lb n="1799,001,,,,3,10,14"></lb> |
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2277 ἡ ΝΞ εὐθεῖάν τινα τὴν <lb n="1799,001,,,,3,10,15"></lb> |
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2278 ΒΗ δίχα καὶ πρὸς ὀρθὰς <lb n="1799,001,,,,3,10,16"></lb> |
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2279 τέμνει, ἐπὶ τῆς ΝΞ ἄρα <lb n="1799,001,,,,3,10,17"></lb> |
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2280 ἐστὶ τὸ κέντρον τοῦ ΑΒΓ κύκλου.</s> <s>ἐδείχθη δὲ καὶ ἐπὶ τῆς <lb n="1799,001,,,,3,10,18"></lb> |
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2281 ΑΓ, καὶ κατ' οὐδὲν συμβάλλουσιν αἱ ΑΓ, ΝΞ εὐθεῖαι ἢ <lb n="1799,001,,,,3,10,19"></lb> |
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2282 κατὰ τὸ Ο· τὸ Ο ἄρα σημεῖον κέντρον ἐστὶ τοῦ ΑΒΓ <lb n="1799,001,,,,3,10,20"></lb> |
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2283 κύκλου.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ τοῦ ΔΕΖ κύκλου <lb n="1799,001,,,,3,10,21"></lb> |
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2284 κέντρον ἐστὶ τὸ Ο· δύο ἄρα κύκλων τεμνόντων ἀλλήλους <lb n="1799,001,,,,3,10,22"></lb> |
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2285 τῶν ΑΒΓ, ΔΕΖ τὸ αὐτό ἐστι κέντρον τὸ Ο· ὅπερ ἐστὶν <lb n="1799,001,,,,3,10,23"></lb> |
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2286 ἀδύνατον.</s> <lb n="1799,001,,,,3,10,24"></lb> |
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2287 <s>Οὐκ ἄρα κύκλος κύκλον τέμνει κατὰ πλείονα σημεῖα ἢ <lb n="1799,001,,,,3,10,25"></lb> |
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2288 δύο· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,10,26"></lb> |
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2289 <s>Ἐὰν δύο κύκλοι ἐφάπτωνται ἀλλήλων ἐντός, καὶ ληφθῇ <lb n="1799,001,,,,3,11,1"></lb> |
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2290 αὐτῶν τὰ κέντρα, ἡ ἐπὶ τὰ κέντρα αὐτῶν ἐπιζευγνυμένη <pb></pb> <lb n="1799,001,,,,3,11,2"></lb> |
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2291 εὐθεῖα καὶ ἐκβαλλομένη ἐπὶ τὴν συναφὴν πεσεῖται τῶν <lb n="1799,001,,,,3,11,3"></lb> |
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2292 κύκλων.</s> <lb n="1799,001,,,,3,11,4"></lb> |
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2293 <s>Δύο γὰρ κύκλοι οἱ ΑΒΓ, ΑΔΕ ἐφαπτέσθωσαν ἀλλή-<lb n="1799,001,,,,3,11,5"></lb> |
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2294 λων ἐντὸς κατὰ τὸ Α σημεῖον, καὶ εἰλήφθω τοῦ μὲν ΑΒΓ <lb n="1799,001,,,,3,11,6"></lb> |
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2295 κύκλου κέντρον τὸ Ζ, τοῦ δὲ ΑΔΕ τὸ Η· λέγω, ὅτι ἡ ἀπὸ <lb n="1799,001,,,,3,11,7"></lb> |
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2296 τοῦ Η ἐπὶ τὸ Ζ ἐπιζευγνυμένη εὐθεῖα ἐκβαλλομένη ἐπὶ τὸ <lb n="1799,001,,,,3,11,8"></lb> |
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2297 Α πεσεῖται.</s> <lb n="1799,001,,,,3,11,9"></lb> |
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2298 <s>Μὴ γάρ, ἀλλ' εἰ δυνατόν, πιπτέτω ὡς ἡ ΖΗΘ, καὶ <lb n="1799,001,,,,3,11,10"></lb> |
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2299 ἐπεζεύχθωσαν αἱ ΑΖ, ΑΗ.</s> <lb n="1799,001,,,,3,11,11"></lb> |
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2300 <s>Ἐπεὶ οὖν αἱ ΑΗ, ΗΖ τῆς ΖΑ, τουτέστι τῆς ΖΘ, μεί-<lb n="1799,001,,,,3,11,12"></lb> |
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2301 ζονές εἰσιν, κοινὴ ἀφῃρήσθω ἡ ΖΗ· λοιπὴ ἄρα ἡ ΑΗ <lb n="1799,001,,,,3,11,13"></lb> |
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2302 λοιπῆς τῆς ΗΘ μείζων ἐστίν.</s> <s>ἴση δὲ ἡ ΑΗ τῇ ΗΔ· καὶ <lb n="1799,001,,,,3,11,14"></lb> |
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2303 ἡ ΗΔ ἄρα τῆς ΗΘ μείζων ἐστὶν ἡ ἐλάτ-<lb n="1799,001,,,,3,11,15"></lb> |
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2304 των τῆς μείζονος· ὅπερ ἐστὶν ἀδύνατον· <lb n="1799,001,,,,3,11,16"></lb> |
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2305 οὐκ ἄρα ἡ ἀπὸ τοῦ Ζ ἐπὶ τὸ Η ἐπιζευ-<lb n="1799,001,,,,3,11,17"></lb> |
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2306 γνυμένη εὐθεῖα ἐκτὸς πεσεῖται· κατὰ <lb n="1799,001,,,,3,11,18"></lb> |
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2307 τὸ Α ἄρα ἐπὶ τῆς συναφῆς πεσεῖται.</s> <lb n="1799,001,,,,3,11,19"></lb> |
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2308 <s>Ἐὰν ἄρα δύο κύκλοι ἐφάπτωνται ἀλλή-<lb n="1799,001,,,,3,11,20"></lb> |
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2309 λων ἐντός, [καὶ ληφθῇ αὐτῶν τὰ κέντρα], <pb></pb> <lb n="1799,001,,,,3,11,21"></lb> |
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2310 ἡ ἐπὶ τὰ κέντρα αὐτῶν ἐπιζευγνυμένη εὐθεῖα [καὶ ἐκ-<lb n="1799,001,,,,3,11,22"></lb> |
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2311 βαλλομένη] ἐπὶ τὴν συναφὴν πεσεῖται τῶν κύκλων· ὅπερ <lb n="1799,001,,,,3,11,23"></lb> |
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2312 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,11,24"></lb> |
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2313 <s>Ἐὰν δύο κύκλοι ἐφάπτωνται ἀλλήλων ἐκτός, ἡ ἐπὶ τὰ <lb n="1799,001,,,,3,12,1"></lb> |
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2314 κέντρα αὐτῶν ἐπιζευγνυμένη διὰ τῆς ἐπαφῆς ἐλεύσεται.</s> <lb n="1799,001,,,,3,12,2"></lb> |
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2315 <s>Δύο γὰρ κύκλοι οἱ ΑΒΓ, ΑΔΕ ἐφαπτέσθωσαν ἀλλή-<lb n="1799,001,,,,3,12,3"></lb> |
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2316 λων ἐκτὸς κατὰ τὸ Α σημεῖον, καὶ εἰλήφθω τοῦ μὲν ΑΒΓ <lb n="1799,001,,,,3,12,4"></lb> |
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2317 κέντρον τὸ Ζ, τοῦ δὲ ΑΔΕ τὸ Η· λέγω, ὅτι ἡ ἀπὸ τοῦ Ζ <lb n="1799,001,,,,3,12,5"></lb> |
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2318 ἐπὶ τὸ Η ἐπιζευγνυμένη εὐθεῖα διὰ τῆς κατὰ τὸ Α ἐπαφῆς <lb n="1799,001,,,,3,12,6"></lb> |
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2319 ἐλεύσεται.</s> <lb n="1799,001,,,,3,12,7"></lb> |
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2320 <s>Μὴ γάρ, ἀλλ' εἰ δυνατόν, ἐρχέσθω ὡς ἡ ΖΓΔΗ, καὶ <lb n="1799,001,,,,3,12,8"></lb> |
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2321 ἐπεζεύχθωσαν αἱ ΑΖ, ΑΗ.</s> <lb n="1799,001,,,,3,12,9"></lb> |
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2322 <s>Ἐπεὶ οὖν τὸ Ζ σημεῖον κέντρον ἐστὶ τοῦ ΑΒΓ κύκλου, <lb n="1799,001,,,,3,12,10"></lb> |
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2323 ἴση ἐστὶν ἡ ΖΑ τῇ ΖΓ.</s> <s>πάλιν, ἐπεὶ τὸ Η σημεῖον κέντρον <lb n="1799,001,,,,3,12,11"></lb> |
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2324 ἐστὶ τοῦ ΑΔΕ κύκλου, ἴση ἐστὶν ἡ ΗΑ <lb n="1799,001,,,,3,12,12"></lb> |
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2325 τῇ ΗΔ.</s> <s>ἐδείχθη δὲ καὶ ἡ ΖΑ τῇ ΖΓ <lb n="1799,001,,,,3,12,13"></lb> |
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2326 ἴση· αἱ ἄρα ΖΑ, ΑΗ ταῖς ΖΓ, ΗΔ <lb n="1799,001,,,,3,12,14"></lb> |
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2327 ἴσαι εἰσίν· ὥστε ὅλη ἡ ΖΗ τῶν ΖΑ, <lb n="1799,001,,,,3,12,15"></lb> |
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2328 ΑΗ μείζων ἐστίν· ἀλλὰ καὶ ἐλάττων· <lb n="1799,001,,,,3,12,16"></lb> |
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2329 ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ἡ ἀπὸ <lb n="1799,001,,,,3,12,17"></lb> |
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2330 τοῦ Ζ ἐπὶ τὸ Η ἐπιζευγνυμένη εὐθεῖα <lb n="1799,001,,,,3,12,18"></lb> |
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2331 διὰ τῆς κατὰ τὸ Α ἐπαφῆς οὐκ ἐλεύσε-<lb n="1799,001,,,,3,12,19"></lb> |
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2332 ται· δι' αὐτῆς ἄρα.</s> <s><pb></pb> <lb n="1799,001,,,,3,12,20"></lb> |
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2333 Ἐὰν ἄρα δύο κύκλοι ἐφάπτωνται ἀλλήλων ἐκτός, ἡ ἐπὶ <lb n="1799,001,,,,3,12,21"></lb> |
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2334 τὰ κέντρα αὐτῶν ἐπιζευγνυμένη [εὐθεῖα] διὰ τῆς ἐπ-<lb n="1799,001,,,,3,12,22"></lb> |
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2335 αφῆς ἐλεύσεται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,12,23"></lb> |
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2336 <s>Κύκλος κύκλου οὐκ ἐφάπτεται κατὰ πλείονα σημεῖα ἢ <lb n="1799,001,,,,3,13,1"></lb> |
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2337 καθ' ἕν, ἐάν τε ἐντὸς ἐάν τε ἐκτὸς ἐφάπτηται.</s> <lb n="1799,001,,,,3,13,2"></lb> |
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2338 <s>Εἰ γὰρ δυνατόν, κύκλος ὁ ΑΒΓΔ κύκλου τοῦ ΕΒΖΔ <lb n="1799,001,,,,3,13,3"></lb> |
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2339 ἐφαπτέσθω πρότερον ἐντὸς κατὰ πλείονα σημεῖα ἢ ἓν τὰ <lb n="1799,001,,,,3,13,4"></lb> |
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2340 Δ, Β.</s> <lb n="1799,001,,,,3,13,5"></lb> |
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2341 <s>Καὶ εἰλήφθω τοῦ μὲν ΑΒΓΔ κύκλου κέντρον τὸ Η, <lb n="1799,001,,,,3,13,6"></lb> |
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2342 τοῦ δὲ ΕΒΖΔ τὸ Θ.</s> <lb n="1799,001,,,,3,13,7"></lb> |
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2343 <s>Ἡ ἄρα ἀπὸ τοῦ Η ἐπὶ τὸ Θ ἐπιζευγνυμένη ἐπὶ τὰ Β, Δ <lb n="1799,001,,,,3,13,8"></lb> |
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2344 πεσεῖται.</s> <s>πιπτέτω ὡς ἡ ΒΗΘΔ.</s> <s>καὶ ἐπεὶ τὸ Η σημεῖον <lb n="1799,001,,,,3,13,9"></lb> |
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2345 κέντρον ἐστὶ τοῦ ΑΒΓΔ κύκλου, ἴση ἐστὶν ἡ ΒΗ τῇ ΗΔ· <lb n="1799,001,,,,3,13,10"></lb> |
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2346 μείζων ἄρα ἡ ΒΗ τῆς ΘΔ· πολλῷ <lb n="1799,001,,,,3,13,11"></lb> |
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2347 ἄρα μείζων ἡ ΒΘ τῆς ΘΔ.</s> <s>πάλιν, <lb n="1799,001,,,,3,13,12"></lb> |
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2348 ἐπεὶ τὸ Θ σημεῖον κέντρον ἐστὶ τοῦ <lb n="1799,001,,,,3,13,13"></lb> |
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2349 ΕΒΖΔ κύκλου, ἴση ἐστὶν ἡ ΒΘ τῇ <lb n="1799,001,,,,3,13,14"></lb> |
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2350 ΘΔ· ἐδείχθη δὲ αὐτῆς καὶ πολλῷ <lb n="1799,001,,,,3,13,15"></lb> |
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2351 μείζων· ὅπερ ἀδύνατον· οὐκ ἄρα <lb n="1799,001,,,,3,13,16"></lb> |
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2352 κύκλος κύκλου ἐφάπτεται ἐντὸς κατὰ <lb n="1799,001,,,,3,13,17"></lb> |
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2353 πλείονα σημεῖα ἢ ἕν.</s> <s><pb></pb> <lb n="1799,001,,,,3,13,18"></lb> |
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2354 Λέγω δή, ὅτι οὐδὲ ἐκτός.</s> <lb n="1799,001,,,,3,13,19"></lb> |
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2355 <s>Εἰ γὰρ δυνατόν, κύκλος ὁ ΑΓΚ κύκλου τοῦ ΑΒΓΔ <lb n="1799,001,,,,3,13,20"></lb> |
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2356 ἐφαπτέσθω ἐκτὸς κατὰ πλείονα σημεῖα ἢ ἓν τὰ Α, Γ, καὶ <lb n="1799,001,,,,3,13,21"></lb> |
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2357 ἐπεζεύχθω ἡ ΑΓ.</s> <lb n="1799,001,,,,3,13,22"></lb> |
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2358 <s>Ἐπεὶ οὖν κύκλων τῶν ΑΒΓΔ, ΑΓΚ εἴληπται ἐπὶ τῆς <lb n="1799,001,,,,3,13,23"></lb> |
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2359 περιφερείας ἑκατέρου δύο τυχόντα σημεῖα τὰ Α, Γ, ἡ ἐπὶ <lb n="1799,001,,,,3,13,24"></lb> |
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2360 τὰ σημεῖα ἐπιζευγνυμένη εὐθεῖα ἐντὸς ἑκατέρου πεσεῖται· <lb n="1799,001,,,,3,13,25"></lb> |
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2361 ἀλλὰ τοῦ μὲν ΑΒΓΔ ἐντὸς ἔπεσεν, τοῦ δὲ ΑΓΚ ἐκτός· <lb n="1799,001,,,,3,13,26"></lb> |
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2362 ὅπερ ἄτοπον· οὐκ ἄρα κύκλος κύκλου ἐφάπτεται ἐκτὸς <lb n="1799,001,,,,3,13,27"></lb> |
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2363 κατὰ πλείονα σημεῖα ἢ ἕν.</s> <s>ἐδείχθη δέ, ὅτι οὐδὲ ἐντός.</s> <lb n="1799,001,,,,3,13,28"></lb> |
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2364 <s>Κύκλος ἄρα κύκλου οὐκ ἐφάπτεται κατὰ πλείονα σημεῖα <lb n="1799,001,,,,3,13,29"></lb> |
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2365 ἢ [καθ'] ἕν, ἐάν τε ἐντὸς ἐάν τε ἐκτὸς ἐφάπτηται· ὅπερ <lb n="1799,001,,,,3,13,30"></lb> |
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2366 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,13,31"></lb> |
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2367 <s>Ἐν κύκλῳ αἱ ἴσαι εὐθεῖαι ἴσον ἀπέχουσιν ἀπὸ τοῦ <lb n="1799,001,,,,3,14,1"></lb> |
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2368 κέντρου, καὶ αἱ ἴσον ἀπέχουσαι ἀπὸ τοῦ κέντρου ἴσαι ἀλλή-<lb n="1799,001,,,,3,14,2"></lb> |
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2369 λαις εἰσίν.</s> <lb n="1799,001,,,,3,14,3"></lb> |
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2370 <s>Ἔστω κύκλος ὁ ΑΒΓΔ, καὶ ἐν αὐτῷ ἴσαι εὐθεῖαι ἔστω-<lb n="1799,001,,,,3,14,4"></lb> |
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2371 σαν αἱ ΑΒ, ΓΔ· λέγω, ὅτι αἱ ΑΒ, ΓΔ ἴσον ἀπέχουσιν <lb n="1799,001,,,,3,14,5"></lb> |
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2372 ἀπὸ τοῦ κέντρου.</s> <lb n="1799,001,,,,3,14,6"></lb> |
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2373 <s>Εἰλήφθω γὰρ τὸ κέντρον τοῦ ΑΒΓΔ κύκλου καὶ ἔστω <lb n="1799,001,,,,3,14,7"></lb> |
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2374 τὸ Ε, καὶ ἀπὸ τοῦ Ε ἐπὶ τὰς ΑΒ, ΓΔ κάθετοι ἤχθωσαν <lb n="1799,001,,,,3,14,8"></lb> |
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2375 αἱ ΕΖ, ΕΗ, καὶ ἐπεζεύχθωσαν αἱ ΑΕ, ΕΓ.</s> <s><pb></pb> <lb n="1799,001,,,,3,14,9"></lb> |
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2376 Ἐπεὶ οὖν εὐθεῖά τις διὰ τοῦ κέντρου ἡ ΕΖ εὐθεῖάν τινα <lb n="1799,001,,,,3,14,10"></lb> |
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2377 μὴ διὰ τοῦ κέντρου τὴν ΑΒ πρὸς ὀρθὰς τέμνει, καὶ δίχα <lb n="1799,001,,,,3,14,11"></lb> |
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2378 αὐτὴν τέμνει.</s> <s>ἴση ἄρα ἡ ΑΖ τῇ ΖΒ· <lb n="1799,001,,,,3,14,12"></lb> |
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2379 διπλῆ ἄρα ἡ ΑΒ τῆς ΑΖ.</s> <s>διὰ τὰ αὐτὰ <lb n="1799,001,,,,3,14,13"></lb> |
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2380 δὴ καὶ ἡ ΓΔ τῆς ΓΗ ἐστι διπλῆ· καί <lb n="1799,001,,,,3,14,14"></lb> |
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2381 ἐστιν ἴση ἡ ΑΒ τῇ ΓΔ· ἴση ἄρα καὶ ἡ <lb n="1799,001,,,,3,14,15"></lb> |
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2382 ΑΖ τῇ ΓΗ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΕ <lb n="1799,001,,,,3,14,16"></lb> |
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2383 τῇ ΕΓ, ἴσον καὶ τὸ ἀπὸ τῆς ΑΕ τῷ <lb n="1799,001,,,,3,14,17"></lb> |
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2384 ἀπὸ τῆς ΕΓ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΕ <lb n="1799,001,,,,3,14,18"></lb> |
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2385 ἴσα τὰ ἀπὸ τῶν ΑΖ, ΕΖ· ὀρθὴ γὰρ ἡ <lb n="1799,001,,,,3,14,19"></lb> |
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2386 πρὸς τῷ Ζ γωνία· τῷ δὲ ἀπὸ τῆς ΕΓ ἴσα τὰ ἀπὸ τῶν <lb n="1799,001,,,,3,14,20"></lb> |
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2387 ΕΗ, ΗΓ· ὀρθὴ γὰρ ἡ πρὸς τῷ Η γωνία· τὰ ἄρα ἀπὸ τῶν <lb n="1799,001,,,,3,14,21"></lb> |
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2388 ΑΖ, ΖΕ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΓΗ, ΗΕ, ὧν τὸ ἀπὸ τῆς <lb n="1799,001,,,,3,14,22"></lb> |
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2389 ΑΖ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΗ· ἴση γάρ ἐστιν ἡ ΑΖ τῇ <lb n="1799,001,,,,3,14,23"></lb> |
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2390 ΓΗ· λοιπὸν ἄρα τὸ ἀπὸ τῆς ΖΕ τῷ ἀπὸ τῆς ΕΗ ἴσον <lb n="1799,001,,,,3,14,24"></lb> |
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2391 ἐστίν· ἴση ἄρα ἡ ΕΖ τῇ ΕΗ.</s> <s>ἐν δὲ κύκλῳ ἴσον ἀπέχειν <lb n="1799,001,,,,3,14,25"></lb> |
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2392 ἀπὸ τοῦ κέντρου εὐθεῖαι λέγονται, ὅταν αἱ ἀπὸ τοῦ κέν-<lb n="1799,001,,,,3,14,26"></lb> |
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2393 τρου ἐπ' αὐτὰς κάθετοι ἀγόμεναι ἴσαι ὦσιν· αἱ ἄρα ΑΒ, <lb n="1799,001,,,,3,14,27"></lb> |
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2394 ΓΔ ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου.</s> <lb n="1799,001,,,,3,14,28"></lb> |
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2395 <s>Ἀλλὰ δὴ αἱ ΑΒ, ΓΔ εὐθεῖαι ἴσον ἀπεχέτωσαν ἀπὸ τοῦ <lb n="1799,001,,,,3,14,29"></lb> |
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2396 κέντρου, τουτέστιν ἴση ἔστω ἡ ΕΖ τῇ ΕΗ.</s> <s>λέγω, ὅτι ἴση <lb n="1799,001,,,,3,14,30"></lb> |
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2397 ἐστὶ καὶ ἡ ΑΒ τῇ ΓΔ.</s> <lb n="1799,001,,,,3,14,31"></lb> |
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2398 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, <lb n="1799,001,,,,3,14,32"></lb> |
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2399 ὅτι διπλῆ ἐστιν ἡ μὲν ΑΒ τῆς ΑΖ, ἡ δὲ ΓΔ τῆς ΓΗ· καὶ <pb></pb> <lb n="1799,001,,,,3,14,33"></lb> |
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2400 ἐπεὶ ἴση ἐστὶν ἡ ΑΕ τῇ ΓΕ, ἴσον ἐστὶ τὸ ἀπὸ τῆς ΑΕ τῷ <lb n="1799,001,,,,3,14,34"></lb> |
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2401 ἀπὸ τῆς ΓΕ· ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΕ ἴσα ἐστὶ τὰ ἀπὸ <lb n="1799,001,,,,3,14,35"></lb> |
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2402 τῶν ΕΖ, ΖΑ, τῷ δὲ ἀπὸ τῆς ΓΕ ἴσα τὰ ἀπὸ τῶν ΕΗ, <lb n="1799,001,,,,3,14,36"></lb> |
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2403 ΗΓ.</s> <s>τὰ ἄρα ἀπὸ τῶν ΕΖ, ΖΑ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΕΗ, <lb n="1799,001,,,,3,14,37"></lb> |
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2404 ΗΓ· ὧν τὸ ἀπὸ τῆς ΕΖ τῷ ἀπὸ τῆς ΕΗ ἐστιν ἴσον· ἴση <lb n="1799,001,,,,3,14,38"></lb> |
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2405 γὰρ ἡ ΕΖ τῇ ΕΗ· λοιπὸν ἄρα τὸ ἀπὸ τῆς ΑΖ ἴσον ἐστὶ τῷ <lb n="1799,001,,,,3,14,39"></lb> |
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2406 ἀπὸ τῆς ΓΗ· ἴση ἄρα ἡ ΑΖ τῇ ΓΗ· καί ἐστι τῆς μὲν <lb n="1799,001,,,,3,14,40"></lb> |
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2407 ΑΖ διπλῆ ἡ ΑΒ, τῆς δὲ ΓΗ διπλῆ ἡ ΓΔ· ἴση ἄρα ἡ ΑΒ <lb n="1799,001,,,,3,14,41"></lb> |
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2408 τῇ ΓΔ.</s> <lb n="1799,001,,,,3,14,42"></lb> |
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2409 <s>Ἐν κύκλῳ ἄρα αἱ ἴσαι εὐθεῖαι ἴσον ἀπέχουσιν ἀπὸ τοῦ <lb n="1799,001,,,,3,14,43"></lb> |
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2410 κέντρου, καὶ αἱ ἴσον ἀπέχουσαι ἀπὸ τοῦ κέντρου ἴσαι <lb n="1799,001,,,,3,14,44"></lb> |
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2411 ἀλλήλαις εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,14,45"></lb> |
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2412 <s>Ἐν κύκλῳ μεγίστη μὲν ἡ διάμετρος τῶν δὲ ἄλλων ἀεὶ <lb n="1799,001,,,,3,15,1"></lb> |
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2413 ἡ ἔγγιον τοῦ κέντρου τῆς ἀπώτερον μείζων ἐστίν.</s> <lb n="1799,001,,,,3,15,2"></lb> |
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2414 <s>Ἔστω κύκλος ὁ ΑΒΓΔ, διάμετρος δὲ αὐτοῦ ἔστω ἡ <lb n="1799,001,,,,3,15,3"></lb> |
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2415 ΑΔ, κέντρον δὲ τὸ Ε, καὶ ἔγγιον μὲν τῆς ΑΔ διαμέτρου <lb n="1799,001,,,,3,15,4"></lb> |
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2416 ἔστω ἡ ΒΓ, ἀπώτερον δὲ ἡ ΖΗ· λέγω, ὅτι μεγίστη μέν <lb n="1799,001,,,,3,15,5"></lb> |
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2417 ἐστιν ἡ ΑΔ, μείζων δὲ ἡ ΒΓ τῆς ΖΗ.</s> <lb n="1799,001,,,,3,15,6"></lb> |
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2418 <s>Ἤχθωσαν γὰρ ἀπὸ τοῦ Ε κέντρου ἐπὶ τὰς ΒΓ, ΖΗ <lb n="1799,001,,,,3,15,7"></lb> |
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2419 κάθετοι αἱ ΕΘ, ΕΚ.</s> <s>καὶ ἐπεὶ ἔγγιον μὲν τοῦ κέντρου <pb></pb> <lb n="1799,001,,,,3,15,8"></lb> |
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2420 ἐστὶν ἡ ΒΓ, ἀπώτερον δὲ ἡ ΖΗ, μείζων ἄρα ἡ ΕΚ τῆς <lb n="1799,001,,,,3,15,9"></lb> |
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2421 ΕΘ.</s> <s>κείσθω τῇ ΕΘ ἴση ἡ ΕΛ, καὶ διὰ τοῦ Λ τῇ ΕΚ <lb n="1799,001,,,,3,15,10"></lb> |
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2422 πρὸς ὀρθὰς ἀχθεῖσα ἡ ΛΜ διήχθω ἐπὶ τὸ Ν, καὶ ἐπεζεύ-<lb n="1799,001,,,,3,15,11"></lb> |
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2423 χθωσαν αἱ ΜΕ, ΕΝ, ΖΕ, ΕΗ.</s> <lb n="1799,001,,,,3,15,12"></lb> |
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2424 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΕΘ τῇ ΕΛ, ἴση ἐστὶ καὶ ἡ ΒΓ <lb n="1799,001,,,,3,15,13"></lb> |
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2425 τῇ ΜΝ.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν ἡ μὲν ΑΕ τῇ ΕΜ, ἡ δὲ ΕΔ <lb n="1799,001,,,,3,15,14"></lb> |
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2426 τῇ ΕΝ, ἡ ἄρα ΑΔ ταῖς ΜΕ, ΕΝ ἴση ἐστίν.</s> <lb n="1799,001,,,,3,15,15"></lb> |
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2427 <s>ἀλλ' αἱ μὲν ΜΕ, ΕΝ τῆς ΜΝ μείζονές <lb n="1799,001,,,,3,15,16"></lb> |
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2428 εἰσιν [καὶ ἡ ΑΔ τῆς ΜΝ μείζων ἐστίν, ἴση <lb n="1799,001,,,,3,15,17"></lb> |
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2429 δὲ ἡ ΜΝ τῇ ΒΓ· ἡ ΑΔ ἄρα τῆς ΒΓ μεί-<lb n="1799,001,,,,3,15,18"></lb> |
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2430 ζων ἐστίν.</s> <s>καὶ ἐπεὶ δύο αἱ ΜΕ, ΕΝ δύο <lb n="1799,001,,,,3,15,19"></lb> |
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2431 ταῖς ΖΕ, ΕΗ ἴσαι εἰσίν, καὶ γωνία ἡ ὑπὸ <lb n="1799,001,,,,3,15,20"></lb> |
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2432 ΜΕΝ γωνίας τῆς ὑπὸ ΖΕΗ μείζων [ἐστίν], <lb n="1799,001,,,,3,15,21"></lb> |
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2433 βάσις ἄρα ἡ ΜΝ βάσεως τῆς ΖΗ μείζων <lb n="1799,001,,,,3,15,22"></lb> |
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2434 ἐστίν.</s> <s>ἀλλὰ ἡ ΜΝ τῇ ΒΓ ἐδείχθη ἴση [καὶ ἡ ΒΓ τῆς <lb n="1799,001,,,,3,15,23"></lb> |
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2435 ΖΗ μείζων ἐστίν].</s> <s>μεγίστη μὲν ἄρα ἡ ΑΔ διάμετρος, <lb n="1799,001,,,,3,15,24"></lb> |
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2436 μείζων δὲ ἡ ΒΓ τῆς ΖΗ.</s> <lb n="1799,001,,,,3,15,25"></lb> |
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2437 <s>Ἐν κύκλῳ ἄρα μεγίστη μέν ἐστιν ἡ διάμετρος, τῶν δὲ <lb n="1799,001,,,,3,15,26"></lb> |
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2438 ἄλλων ἀεὶ ἡ ἔγγιον τοῦ κέντρου τῆς ἀπώτερον μείζων <lb n="1799,001,,,,3,15,27"></lb> |
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2439 ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,3,15,28"></lb> |
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2440 Ἡ τῇ διαμέτρῳ τοῦ κύκλου πρὸς ὀρθὰς ἀπ' ἄκρας <lb n="1799,001,,,,3,16,1"></lb> |
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2441 ἀγομένη ἐκτὸς πεσεῖται τοῦ κύκλου, καὶ εἰς τὸν μεταξὺ <lb n="1799,001,,,,3,16,2"></lb> |
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2442 τόπον τῆς τε εὐθείας καὶ τῆς περιφερείας ἑτέρα εὐθεῖα <lb n="1799,001,,,,3,16,3"></lb> |
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2443 οὐ παρεμπεσεῖται, καὶ ἡ μὲν τοῦ ἡμικυκλίου γωνία ἁπά-<lb n="1799,001,,,,3,16,4"></lb> |
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2444 σης γωνίας ὀξείας εὐθυγράμμου μείζων ἐστίν, ἡ δὲ λοιπὴ <lb n="1799,001,,,,3,16,5"></lb> |
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2445 ἐλάττων.</s> <lb n="1799,001,,,,3,16,6"></lb> |
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2446 <s>Ἔστω κύκλος ὁ ΑΒΓ περὶ κέντρον τὸ Δ καὶ διάμετρον <lb n="1799,001,,,,3,16,7"></lb> |
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2447 τὴν ΑΒ· λέγω, ὅτι ἡ ἀπὸ τοῦ Α τῇ ΑΒ πρὸς ὀρθὰς <lb n="1799,001,,,,3,16,8"></lb> |
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2448 ἀπ' ἄκρας ἀγομένη ἐκτὸς πεσεῖται τοῦ <lb n="1799,001,,,,3,16,9"></lb> |
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2449 κύκλου.</s> <lb n="1799,001,,,,3,16,10"></lb> |
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2450 <s>Μὴ γάρ, ἀλλ' εἰ δυνατόν, πιπτέτω <lb n="1799,001,,,,3,16,11"></lb> |
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2451 ἐντὸς ὡς ἡ ΓΑ, καὶ ἐπεζεύχθω ἡ ΔΓ.</s> <lb n="1799,001,,,,3,16,12"></lb> |
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2452 <s>Ἐπεὶ ἴση ἐστὶν ἡ ΔΑ τῇ ΔΓ, ἴση ἐστὶ <lb n="1799,001,,,,3,16,13"></lb> |
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2453 καὶ γωνία ἡ ὑπὸ ΔΑΓ γωνίᾳ τῇ ὑπὸ <lb n="1799,001,,,,3,16,14"></lb> |
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2454 ΑΓΔ.</s> <s>ὀρθὴ δὲ ἡ ὑπὸ ΔΑΓ· ὀρθὴ <lb n="1799,001,,,,3,16,15"></lb> |
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2455 ἄρα καὶ ἡ ὑπὸ ΑΓΔ· τριγώνου δὴ τοῦ ΑΓΔ αἱ δύο γωνίαι <lb n="1799,001,,,,3,16,16"></lb> |
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2456 αἱ ὑπὸ ΔΑΓ, ΑΓΔ δύο ὀρθαῖς ἴσαι εἰσίν· ὅπερ ἐστὶν ἀδύ-<lb n="1799,001,,,,3,16,17"></lb> |
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2457 νατον.</s> <s>οὐκ ἄρα ἡ ἀπὸ τοῦ Α σημείου τῇ ΒΑ πρὸς ὀρθὰς <lb n="1799,001,,,,3,16,18"></lb> |
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2458 ἀγομένη ἐντὸς πεσεῖται τοῦ κύκλου.</s> <s>ὁμοίως δὴ δείξομεν, <lb n="1799,001,,,,3,16,19"></lb> |
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2459 ὅτι οὐδ' ἐπὶ τῆς περιφερείας· ἐκτὸς ἄρα.</s> <lb n="1799,001,,,,3,16,20"></lb> |
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2460 <s>Πιπτέτω ὡς ἡ ΑΕ· λέγω δή, ὅτι εἰς τὸν μεταξὺ τόπον <pb></pb> <lb n="1799,001,,,,3,16,21"></lb> |
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2461 τῆς τε ΑΕ εὐθείας καὶ τῆς ΓΘΑ περιφερείας ἑτέρα εὐθεῖα <lb n="1799,001,,,,3,16,22"></lb> |
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2462 οὐ παρεμπεσεῖται.</s> <lb n="1799,001,,,,3,16,23"></lb> |
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2463 <s>Εἰ γὰρ δυνατόν, παρεμπιπτέτω ὡς ἡ ΖΑ, καὶ ἤχθω <lb n="1799,001,,,,3,16,24"></lb> |
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2464 ἀπὸ τοῦ Δ σημείου ἐπὶ τὴν ΖΑ κάθετος ἡ ΔΗ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,3,16,25"></lb> |
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2465 ὀρθή ἐστιν ἡ ὑπὸ ΑΗΔ, ἐλάττων δὲ ὀρθῆς ἡ ὑπὸ ΔΑΗ, <lb n="1799,001,,,,3,16,26"></lb> |
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2466 μείζων ἄρα ἡ ΑΔ τῆς ΔΗ.</s> <s>ἴση δὲ ἡ ΔΑ τῇ ΔΘ· μείζων <lb n="1799,001,,,,3,16,27"></lb> |
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2467 ἄρα ἡ ΔΘ τῆς ΔΗ, ἡ ἐλάττων τῆς μείζονος· ὅπερ ἐστὶν <lb n="1799,001,,,,3,16,28"></lb> |
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2468 ἀδύνατον.</s> <s>οὐκ ἄρα εἰς τὸν μεταξὺ τόπον τῆς τε εὐθείας <lb n="1799,001,,,,3,16,29"></lb> |
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2469 καὶ τῆς περιφερείας ἑτέρα εὐθεῖα παρεμπεσεῖται.</s> <lb n="1799,001,,,,3,16,30"></lb> |
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2470 <s>Λέγω, ὅτι καὶ ἡ μὲν τοῦ ἡμικυκλίου γωνία ἡ περιεχο-<lb n="1799,001,,,,3,16,31"></lb> |
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2471 μένη ὑπό τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περιφερείας <lb n="1799,001,,,,3,16,32"></lb> |
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2472 ἁπάσης γωνίας ὀξείας εὐθυγράμμου μείζων ἐστίν, ἡ δὲ <lb n="1799,001,,,,3,16,33"></lb> |
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2473 λοιπὴ ἡ περιεχομένη ὑπό τε τῆς ΓΘΑ περιφερείας καὶ <lb n="1799,001,,,,3,16,34"></lb> |
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2474 τῆς ΑΕ εὐθείας ἁπάσης γωνίας ὀξείας εὐθυγράμμου <lb n="1799,001,,,,3,16,35"></lb> |
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2475 ἐλάττων ἐστίν.</s> <lb n="1799,001,,,,3,16,36"></lb> |
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2476 <s>Εἰ γὰρ ἐστί τις γωνία εὐθύγραμμος μείζων μὲν τῆς <lb n="1799,001,,,,3,16,37"></lb> |
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2477 περιεχομένης ὑπό τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περι-<lb n="1799,001,,,,3,16,38"></lb> |
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2478 φερείας, ἐλάττων δὲ τῆς περιεχομένης ὑπό τε τῆς ΓΘΑ <lb n="1799,001,,,,3,16,39"></lb> |
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2479 περιφερείας καὶ τῆς ΑΕ εὐθείας, εἰς τὸν μεταξὺ τόπον <lb n="1799,001,,,,3,16,40"></lb> |
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2480 τῆς τε ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας εὐθεῖα <lb n="1799,001,,,,3,16,41"></lb> |
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2481 περεμπεσεῖται, ἥτις ποιήσει μείζονα μὲν τῆς περιεχο-<lb n="1799,001,,,,3,16,42"></lb> |
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2482 μένης ὑπό τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περιφερείας <lb n="1799,001,,,,3,16,43"></lb> |
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2483 ὑπὸ εὐθειῶν περιεχομένην, ἐλάττονα δὲ τῆς περιεχομένης <lb n="1799,001,,,,3,16,44"></lb> |
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2484 ὑπό τε τῆς ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας.</s> <s>οὐ <pb></pb> <lb n="1799,001,,,,3,16,45"></lb> |
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2485 παρεμπίπτει δέ· οὐκ ἄρα τῆς περιεχομένης γωνίας ὑπό <lb n="1799,001,,,,3,16,46"></lb> |
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2486 τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περιφερείας ἔσται μεί-<lb n="1799,001,,,,3,16,47"></lb> |
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2487 ζων ὀξεῖα ὑπὸ εὐθειῶν περιεχομένη, οὐδὲ μὴν ἐλάττων <lb n="1799,001,,,,3,16,48"></lb> |
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2488 τῆς περιεχομένης ὑπό τε τῆς ΓΘΑ περιφερείας καὶ τῆς <lb n="1799,001,,,,3,16,49"></lb> |
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2489 ΑΕ εὐθείας.</s> <lb n="1799,001,,,,3,16,50"></lb> |
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2490 <s>Πόρισμα <lb n="1799,001,,,,3,16,51"></lb> |
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2491 Ἐκ δὴ τούτου φανερόν, ὅτι ἡ τῇ διαμέτρῳ τοῦ κύκλου <lb n="1799,001,,,,3,16,52"></lb> |
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2492 πρὸς ὀρθὰς ἀπ' ἄκρας ἀγομένη ἐφάπτεται τοῦ κύκλου <lb n="1799,001,,,,3,16,53"></lb> |
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2493 [καὶ ὅτι εὐθεῖα κύκλου καθ' ἓν μόνον ἐφάπτεται σημεῖον, <lb n="1799,001,,,,3,16,54"></lb> |
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2494 ἐπειδήπερ καὶ ἡ κατὰ δύο αὐτῷ συμβάλλουσα ἐντὸς <lb n="1799,001,,,,3,16,55"></lb> |
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2495 αὐτοῦ πίπτουσα ἐδείχθη].</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,16,56"></lb> |
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2496 <s>Ἀπὸ τοῦ δοθέντος σημείου τοῦ δοθέντος κύκλου ἐφ-<lb n="1799,001,,,,3,17,1"></lb> |
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2497 απτομένην εὐθεῖαν γραμμὴν ἀγαγεῖν.</s> <lb n="1799,001,,,,3,17,2"></lb> |
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2498 <s>Ἔστω τὸ μὲν δοθὲν σημεῖον τὸ Α, ὁ δὲ δοθεὶς κύκλος <lb n="1799,001,,,,3,17,3"></lb> |
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2499 ὁ ΒΓΔ· δεῖ δὴ ἀπὸ τοῦ Α σημείου τοῦ ΒΓΔ κύκλου ἐφ-<lb n="1799,001,,,,3,17,4"></lb> |
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2500 απτομένην εὐθεῖαν γραμμὴν ἀγαγεῖν.</s> <lb n="1799,001,,,,3,17,5"></lb> |
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2501 <s>Εἰλήφθω γὰρ τὸ κέντρον τοῦ κύκλου τὸ Ε, καὶ ἐπ-<lb n="1799,001,,,,3,17,6"></lb> |
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2502 εζεύχθω ἡ ΑΕ, καὶ κέντρῳ μὲν τῷ Ε διαστήματι δὲ τῷ <pb></pb> <lb n="1799,001,,,,3,17,7"></lb> |
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2503 ΕΑ κύκλος γεγράφθω ὁ ΑΖΗ, καὶ ἀπὸ τοῦ Δ τῇ ΕΑ <lb n="1799,001,,,,3,17,8"></lb> |
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2504 πρὸς ὀρθὰς ἤχθω ἡ ΔΖ, καὶ ἐπεζεύχθωσαν αἱ ΕΖ, ΑΒ· <lb n="1799,001,,,,3,17,9"></lb> |
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2505 λέγω, ὅτι ἀπὸ τοῦ Α σημείου τοῦ ΒΓΔ κύκλου ἐφαπτο-<lb n="1799,001,,,,3,17,10"></lb> |
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2506 μένη ἦκται ἡ ΑΒ.</s> <lb n="1799,001,,,,3,17,11"></lb> |
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2507 <s>Ἐπεὶ γὰρ τὸ Ε κέντρον ἐστὶ τῶν ΒΓΔ, ΑΖΗ κύκλων, <lb n="1799,001,,,,3,17,12"></lb> |
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2508 ἴση ἄρα ἐστὶν ἡ μὲν ΕΑ τῇ ΕΖ, ἡ δὲ ΕΔ τῇ ΕΒ· δύο <lb n="1799,001,,,,3,17,13"></lb> |
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2509 δὴ αἱ ΑΕ, ΕΒ δύο ταῖς ΖΕ, ΕΔ ἴσαι <lb n="1799,001,,,,3,17,14"></lb> |
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2510 εἰσίν· καὶ γωνίαν κοινὴν περιέχουσι τὴν <lb n="1799,001,,,,3,17,15"></lb> |
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2511 πρὸς τῷ Ε· βάσις ἄρα ἡ ΔΖ βάσει τῇ <lb n="1799,001,,,,3,17,16"></lb> |
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2512 ΑΒ ἴση ἐστίν, καὶ τὸ ΔΕΖ τρίγωνον <lb n="1799,001,,,,3,17,17"></lb> |
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2513 τῷ ΕΒΑ τριγώνῳ ἴσον ἐστίν, καὶ αἱ <lb n="1799,001,,,,3,17,18"></lb> |
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2514 λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις· ἴση <lb n="1799,001,,,,3,17,19"></lb> |
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2515 ἄρα ἡ ὑπὸ ΕΔΖ τῇ ὑπὸ ΕΒΑ.</s> <s>ὀρθὴ δὲ <lb n="1799,001,,,,3,17,20"></lb> |
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2516 ἡ ὑπὸ ΕΔΖ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΕΒΑ.</s> <lb n="1799,001,,,,3,17,21"></lb> |
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2517 <s>καί ἐστιν ἡ ΕΒ ἐκ τοῦ κέντρου· ἡ δὲ τῇ διαμέτρῳ τοῦ <lb n="1799,001,,,,3,17,22"></lb> |
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2518 κύκλου πρὸς ὀρθὰς ἀπ' ἄκρας ἀγομένη ἐφάπτεται τοῦ <lb n="1799,001,,,,3,17,23"></lb> |
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2519 κύκλου· ἡ ΑΒ ἄρα ἐφάπτεται τοῦ ΒΓΔ κύκλου.</s> <lb n="1799,001,,,,3,17,24"></lb> |
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2520 <s>Ἀπὸ τοῦ ἄρα δοθέντος σημείου τοῦ Α τοῦ δοθέντος <lb n="1799,001,,,,3,17,25"></lb> |
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2521 κύκλου τοῦ ΒΓΔ ἐφαπτομένη εὐθεῖα γραμμὴ ἦκται ἡ <lb n="1799,001,,,,3,17,26"></lb> |
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2522 ΑΒ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,3,17,27"></lb> |
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2523 <s>Ἐὰν κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τοῦ κέντρου <lb n="1799,001,,,,3,18,1"></lb> |
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2524 ἐπὶ τὴν ἁφὴν ἐπιζευχθῇ τις εὐθεῖα, ἡ ἐπιζευχθεῖσα κάθ-<lb n="1799,001,,,,3,18,2"></lb> |
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2525 ετος ἔσται ἐπὶ τὴν ἐφαπτομένην.</s> <s><pb></pb> <lb n="1799,001,,,,3,18,3"></lb> |
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2526 Κύκλου γὰρ τοῦ ΑΒΓ ἐφαπτέσθω τις εὐθεῖα ἡ ΔΕ <lb n="1799,001,,,,3,18,4"></lb> |
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2527 κατὰ τὸ Γ σημεῖον, καὶ εἰλήφθω τὸ κέντρον τοῦ ΑΒΓ κύ-<lb n="1799,001,,,,3,18,5"></lb> |
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2528 κλου τὸ Ζ, καὶ ἀπὸ τοῦ Ζ ἐπὶ τὸ Γ ἐπεζεύχθω ἡ ΖΓ· λέγω, <lb n="1799,001,,,,3,18,6"></lb> |
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2529 ὅτι ἡ ΖΓ κάθετός ἐστιν ἐπὶ τὴν ΔΕ.</s> <lb n="1799,001,,,,3,18,7"></lb> |
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2530 <s>Εἰ γὰρ μή, ἤχθω ἀπὸ τοῦ Ζ ἐπὶ τὴν ΔΕ κάθετος ἡ ΖΗ.</s> <lb n="1799,001,,,,3,18,8"></lb> |
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2531 <s>Ἐπεὶ οὖν ἡ ὑπὸ ΖΗΓ γωνία ὀρθή ἐστιν, ὀξεῖα ἄρα <lb n="1799,001,,,,3,18,9"></lb> |
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2532 ἐστὶν ἡ ὑπὸ ΖΓΗ· ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων <lb n="1799,001,,,,3,18,10"></lb> |
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2533 πλευρὰ ὑποτείνει· μείζων ἄρα ἡ <lb n="1799,001,,,,3,18,11"></lb> |
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2534 ΖΓ τῆς ΖΗ· ἴση δὲ ἡ ΖΓ τῇ <lb n="1799,001,,,,3,18,12"></lb> |
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2535 ΖΒ· μείζων ἄρα καὶ ἡ ΖΒ τῆς ΖΗ <lb n="1799,001,,,,3,18,13"></lb> |
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2536 ἡ ἐλάττων τῆς μείζονος· ὅπερ ἐστὶν <lb n="1799,001,,,,3,18,14"></lb> |
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2537 ἀδύνατον.</s> <s>οὐκ ἄρα ἡ ΖΗ κάθετός <lb n="1799,001,,,,3,18,15"></lb> |
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2538 ἐστιν ἐπὶ τὴν ΔΕ.</s> <s>ὁμοίως δὴ δεί-<lb n="1799,001,,,,3,18,16"></lb> |
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2539 ξομεν, ὅτι οὐδ' ἄλλη τις πλὴν τῆς <lb n="1799,001,,,,3,18,17"></lb> |
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2540 ΖΓ· ἡ ΖΓ ἄρα κάθετός ἐστιν ἐπὶ <lb n="1799,001,,,,3,18,18"></lb> |
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2541 τὴν ΔΕ.</s> <lb n="1799,001,,,,3,18,19"></lb> |
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2542 <s>Ἐὰν ἄρα κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τοῦ <lb n="1799,001,,,,3,18,20"></lb> |
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2543 κέντρου ἐπὶ τὴν ἁφὴν ἐπιζευχθῇ τις εὐθεῖα, ἡ ἐπιζευ-<lb n="1799,001,,,,3,18,21"></lb> |
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2544 χθεῖσα κάθετος ἔσται ἐπὶ τὴν ἐφαπτομένην· ὅπερ ἔδει <lb n="1799,001,,,,3,18,22"></lb> |
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2545 δεῖξαι.</s> <lb n="1799,001,,,,3,18,23"></lb> |
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2546 <s>Ἐὰν κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τῆς ἁφῆς τῇ <lb n="1799,001,,,,3,19,1"></lb> |
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2547 ἐφαπτομένῃ πρὸς ὀρθὰς [γωνίασ] εὐθεῖα γραμμὴ ἀχθῇ, <lb n="1799,001,,,,3,19,2"></lb> |
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2548 ἐπὶ τῆς ἀχθείσης ἔσται τὸ κέντρον τοῦ κύκλου.</s> <s><pb></pb> <lb n="1799,001,,,,3,19,3"></lb> |
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2549 Κύκλου γὰρ τοῦ ΑΒΓ ἐφαπτέσθω τις εὐθεῖα ἡ ΔΕ <lb n="1799,001,,,,3,19,4"></lb> |
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2550 κατὰ τὸ Γ σημεῖον, καὶ ἀπὸ τοῦ Γ τῇ ΔΕ πρὸς ὀρθὰς <lb n="1799,001,,,,3,19,5"></lb> |
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2551 ἤχθω ἡ ΓΑ· λέγω, ὅτι ἐπὶ τῆς ΑΓ ἐστι τὸ κέντρον τοῦ <lb n="1799,001,,,,3,19,6"></lb> |
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2552 κύκλου.</s> <lb n="1799,001,,,,3,19,7"></lb> |
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2553 <s>Μὴ γάρ, ἀλλ' εἰ δυνατόν, ἔστω τὸ Ζ, καὶ ἐπεζεύχθω <lb n="1799,001,,,,3,19,8"></lb> |
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2554 ἡ ΓΖ.</s> <lb n="1799,001,,,,3,19,9"></lb> |
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2555 <s>Ἐπεὶ [οὖν] κύκλου τοῦ ΑΒΓ ἐφάπτεταί τις εὐθεῖα ἡ <lb n="1799,001,,,,3,19,10"></lb> |
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2556 ΔΕ, ἀπὸ δὲ τοῦ κέντρου ἐπὶ τὴν ἁφὴν ἐπέζευκται ἡ ΖΓ, <lb n="1799,001,,,,3,19,11"></lb> |
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2557 ἡ ΖΓ ἄρα κάθετός ἐστιν ἐπὶ τὴν ΔΕ· <lb n="1799,001,,,,3,19,12"></lb> |
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2558 ὀρθὴ ἄρα ἐστὶν ἡ ὑπὸ ΖΓΕ.</s> <s>ἐστὶ δὲ <lb n="1799,001,,,,3,19,13"></lb> |
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2559 καὶ ἡ ὑπὸ ΑΓΕ ὀρθή· ἴση ἄρα ἐστὶν <lb n="1799,001,,,,3,19,14"></lb> |
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2560 ἡ ὑπὸ ΖΓΕ τῇ ὑπὸ ΑΓΕ ἡ ἐλάτ-<lb n="1799,001,,,,3,19,15"></lb> |
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2561 των τῇ μείζονι· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,3,19,16"></lb> |
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2562 <s>οὐκ ἄρα τὸ Ζ κέντρον ἐστὶ τοῦ ΑΒΓ <lb n="1799,001,,,,3,19,17"></lb> |
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2563 κύκλου.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι οὐδ' <lb n="1799,001,,,,3,19,18"></lb> |
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2564 ἄλλο τι πλὴν ἐπὶ τῆς ΑΓ.</s> <lb n="1799,001,,,,3,19,19"></lb> |
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2565 <s>Ἐὰν ἄρα κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τῆς <lb n="1799,001,,,,3,19,20"></lb> |
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2566 ἁφῆς τῇ ἐφαπτομένῃ πρὸς ὀρθὰς εὐθεῖα γραμμὴ ἀχθῇ, <lb n="1799,001,,,,3,19,21"></lb> |
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2567 ἐπὶ τῆς ἀχθείσης ἔσται τὸ κέντρον τοῦ κύκλου· ὅπερ ἔδει <lb n="1799,001,,,,3,19,22"></lb> |
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2568 δεῖξαι.</s> <lb n="1799,001,,,,3,19,23"></lb> |
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2569 <s>Ἐν κύκλῳ ἡ πρὸς τῷ κέντρῳ γωνία διπλασίων ἐστὶ τῆς <lb n="1799,001,,,,3,20,1"></lb> |
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2570 πρὸς τῇ περιφερείᾳ, ὅταν τὴν αὐτὴν περιφέρειαν βάσιν <lb n="1799,001,,,,3,20,2"></lb> |
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2571 ἔχωσιν αἱ γωνίαι.</s> <lb n="1799,001,,,,3,20,3"></lb> |
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2572 <s>Ἔστω κύκλος ὁ ΑΒΓ, καὶ πρὸς μὲν τῷ κέντρῳ αὐτοῦ <lb n="1799,001,,,,3,20,4"></lb> |
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2573 γωνία ἔστω ἡ ὑπὸ ΒΕΓ, πρὸς δὲ τῇ περιφερείᾳ ἡ ὑπὸ <pb></pb> <lb n="1799,001,,,,3,20,5"></lb> |
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2574 ΒΑΓ, ἐχέτωσαν δὲ τὴν αὐτὴν περιφέρειαν βάσιν τὴν ΒΓ· <lb n="1799,001,,,,3,20,6"></lb> |
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2575 λέγω, ὅτι διπλασίων ἐστὶν ἡ ὑπὸ ΒΕΓ γωνία τῆς ὑπὸ <lb n="1799,001,,,,3,20,7"></lb> |
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2576 ΒΑΓ.</s> <lb n="1799,001,,,,3,20,8"></lb> |
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2577 <s>Ἐπιζευχθεῖσα γὰρ ἡ ΑΕ διήχθω ἐπὶ τὸ Ζ.</s> <lb n="1799,001,,,,3,20,9"></lb> |
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2578 <s>Ἐπεὶ οὖν ἴση ἐστὶν ἡ ΕΑ τῇ ΕΒ, ἴση καὶ γωνία ἡ ὑπὸ <lb n="1799,001,,,,3,20,10"></lb> |
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2579 ΕΑΒ τῇ ὑπὸ ΕΒΑ· αἱ ἄρα ὑπὸ ΕΑΒ, ΕΒΑ γωνίαι τῆς <lb n="1799,001,,,,3,20,11"></lb> |
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2580 ὑπὸ ΕΑΒ διπλασίους εἰσίν.</s> <s>ἴση δὲ ἡ ὑπὸ ΒΕΖ ταῖς ὑπὸ <lb n="1799,001,,,,3,20,12"></lb> |
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2581 ΕΑΒ, ΕΒΑ· καὶ ἡ ὑπὸ ΒΕΖ ἄρα τῆς ὑπὸ ΕΑΒ ἐστι <lb n="1799,001,,,,3,20,13"></lb> |
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2582 διπλῆ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ ΖΕΓ τῆς ὑπὸ ΕΑΓ <lb n="1799,001,,,,3,20,14"></lb> |
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2583 ἐστι διπλῆ.</s> <s>ὅλη ἄρα ἡ ὑπὸ ΒΕΓ ὅλης <lb n="1799,001,,,,3,20,15"></lb> |
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2584 τῆς ὑπὸ ΒΑΓ ἐστι διπλῆ.</s> <lb n="1799,001,,,,3,20,16"></lb> |
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2585 <s>Κεκλάσθω δὴ πάλιν, καὶ ἔστω ἑτέρα <lb n="1799,001,,,,3,20,17"></lb> |
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2586 γωνία ἡ ὑπὸ ΒΔΓ, καὶ ἐπιζευχθεῖσα ἡ <lb n="1799,001,,,,3,20,18"></lb> |
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2587 ΔΕ ἐκβεβλήσθω ἐπὶ τὸ Η.</s> <s>ὁμοίως δὴ <lb n="1799,001,,,,3,20,19"></lb> |
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2588 δείξομεν, ὅτι διπλῆ ἐστιν ἡ ὑπὸ ΗΕΓ <lb n="1799,001,,,,3,20,20"></lb> |
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2589 γωνία τῆς ὑπὸ ΕΔΓ, ὧν ἡ ὑπὸ ΗΕΒ <lb n="1799,001,,,,3,20,21"></lb> |
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2590 διπλῆ ἐστι τῆς ὑπὸ ΕΔΒ· λοιπὴ ἄρα ἡ <lb n="1799,001,,,,3,20,22"></lb> |
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2591 ὑπὸ ΒΕΓ διπλῆ ἐστι τῆς ὑπὸ ΒΔΓ.</s> <lb n="1799,001,,,,3,20,23"></lb> |
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2592 <s>Ἐν κύκλῳ ἄρα ἡ πρὸς τῷ κέντρῳ γωνία διπλασίων ἐστὶ <lb n="1799,001,,,,3,20,24"></lb> |
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2593 τῆς πρὸς τῇ περιφερείᾳ, ὅταν τὴν αὐτὴν περιφέρειαν βάσιν <lb n="1799,001,,,,3,20,25"></lb> |
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2594 ἔχωσιν [αἱ γωνίαι]· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,20,26"></lb> |
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2595 <s>Ἐν κύκλῳ αἱ ἐν τῷ αὐτῷ τμήματι γωνίαι ἴσαι ἀλλή-<lb n="1799,001,,,,3,21,1"></lb> |
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2596 λαις εἰσίν.</s> <s><pb></pb> <lb n="1799,001,,,,3,21,2"></lb> |
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2597 Ἔστω κύκλος ὁ ΑΒΓΔ, καὶ ἐν τῷ αὐτῷ τμήματι τῷ <lb n="1799,001,,,,3,21,3"></lb> |
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2598 ΒΑΕΔ γωνίαι ἔστωσαν αἱ ὑπὸ ΒΑΔ, ΒΕΔ· λέγω, ὅτι <lb n="1799,001,,,,3,21,4"></lb> |
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2599 αἱ ὑπὸ ΒΑΔ, ΒΕΔ γωνίαι ἴσαι ἀλλήλαις εἰσίν.</s> <lb n="1799,001,,,,3,21,5"></lb> |
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2600 <s>Εἰλήφθω γὰρ τοῦ ΑΒΓΔ κύκλου <lb n="1799,001,,,,3,21,6"></lb> |
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2601 τὸ κέντρον, καὶ ἔστω τὸ Ζ, καὶ ἐπ-<lb n="1799,001,,,,3,21,7"></lb> |
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2602 εζεύχθωσαν αἱ ΒΖ, ΖΔ.</s> <lb n="1799,001,,,,3,21,8"></lb> |
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2603 <s>Καὶ ἐπεὶ ἡ μὲν ὑπὸ ΒΖΔ γωνία <lb n="1799,001,,,,3,21,9"></lb> |
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2604 πρὸς τῷ κέντρῳ ἐστίν, ἡ δὲ ὑπὸ ΒΑΔ <lb n="1799,001,,,,3,21,10"></lb> |
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2605 πρὸς τῇ περιφερείᾳ, καὶ ἔχουσι τὴν <lb n="1799,001,,,,3,21,11"></lb> |
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2606 αὐτὴν περιφέρειαν βάσιν τὴν ΒΓΔ, ἡ <lb n="1799,001,,,,3,21,12"></lb> |
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2607 ἄρα ὑπὸ ΒΖΔ γωνία διπλασίων ἐστὶ <lb n="1799,001,,,,3,21,13"></lb> |
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2608 τῆς ὑπὸ ΒΑΔ.</s> <s>διὰ τὰ αὐτὰ δὴ ἡ ὑπὸ ΒΖΔ καὶ τῆς ὑπὸ <lb n="1799,001,,,,3,21,14"></lb> |
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2609 ΒΕΔ ἐστι διπλασίων· ἴση ἄρα ἡ ὑπὸ ΒΑΔ τῇ ὑπὸ ΒΕΔ.</s> <lb n="1799,001,,,,3,21,15"></lb> |
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2610 <s>Ἐν κύκλῳ ἄρα αἱ ἐν τῷ αὐτῷ τμήματι γωνίαι ἴσαι ἀλλή-<lb n="1799,001,,,,3,21,16"></lb> |
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2611 λαις εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,21,17"></lb> |
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2612 <s>Τῶν ἐν τοῖς κύκλοις τετραπλεύρων αἱ ἀπεναντίον γω-<lb n="1799,001,,,,3,22,1"></lb> |
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2613 νίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <lb n="1799,001,,,,3,22,2"></lb> |
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2614 <s>Ἔστω κύκλος ὁ ΑΒΓΔ, καὶ ἐν αὐτῷ τετράπλευρον <lb n="1799,001,,,,3,22,3"></lb> |
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2615 ἔστω τὸ ΑΒΓΔ· λέγω, ὅτι αἱ ἀπεναντίον γωνίαι δυσὶν <lb n="1799,001,,,,3,22,4"></lb> |
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2616 ὀρθαῖς ἴσαι εἰσίν.</s> <lb n="1799,001,,,,3,22,5"></lb> |
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2617 <s>Ἐπεζεύχθωσαν αἱ ΑΓ, ΒΔ.</s> <lb n="1799,001,,,,3,22,6"></lb> |
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2618 <s>Ἐπεὶ οὖν παντὸς τριγώνου αἱ τρεῖς γωνίαι δυσὶν ὀρθαῖς <lb n="1799,001,,,,3,22,7"></lb> |
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2619 ἴσαι εἰσίν, τοῦ ΑΒΓ ἄρα τριγώνου αἱ τρεῖς γωνίαι αἱ ὑπὸ <pb></pb> <lb n="1799,001,,,,3,22,8"></lb> |
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2620 ΓΑΒ, ΑΒΓ, ΒΓΑ δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <s>ἴση δὲ ἡ μὲν <lb n="1799,001,,,,3,22,9"></lb> |
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2621 ὑπὸ ΓΑΒ τῇ ὑπὸ ΒΔΓ· ἐν γὰρ τῷ αὐτῷ τμήματί εἰσι τῷ <lb n="1799,001,,,,3,22,10"></lb> |
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2622 ΒΑΔΓ· ἡ δὲ ὑπὸ ΑΓΒ τῇ ὑπὸ ΑΔΒ· <lb n="1799,001,,,,3,22,11"></lb> |
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2623 ἐν γὰρ τῷ αὐτῷ τμήματί εἰσι τῷ ΑΔΓΒ· <lb n="1799,001,,,,3,22,12"></lb> |
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2624 ὅλη ἄρα ἡ ὑπὸ ΑΔΓ ταῖς ὑπὸ ΒΑΓ, <lb n="1799,001,,,,3,22,13"></lb> |
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2625 ΑΓΒ ἴση ἐστίν.</s> <s>κοινὴ προσκείσθω ἡ <lb n="1799,001,,,,3,22,14"></lb> |
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2626 ὑπὸ ΑΒΓ· αἱ ἄρα ὑπὸ ΑΒΓ, ΒΑΓ, <lb n="1799,001,,,,3,22,15"></lb> |
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2627 ΑΓΒ ταῖς ὑπὸ ΑΒΓ, ΑΔΓ ἴσαι εἰσίν.</s> <lb n="1799,001,,,,3,22,16"></lb> |
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2628 <s>ἀλλ' αἱ ὑπὸ ΑΒΓ, ΒΑΓ, ΑΓΒ δυσὶν <lb n="1799,001,,,,3,22,17"></lb> |
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2629 ὀρθαῖς ἴσαι εἰσίν.</s> <s>καὶ αἱ ὑπὸ ΑΒΓ, <lb n="1799,001,,,,3,22,18"></lb> |
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2630 ΑΔΓ ἄρα δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι <lb n="1799,001,,,,3,22,19"></lb> |
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2631 καὶ αἱ ὑπὸ ΒΑΔ, ΔΓΒ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <lb n="1799,001,,,,3,22,20"></lb> |
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2632 <s>Τῶν ἄρα ἐν τοῖς κύκλοις τετραπλεύρων αἱ ἀπεναντίον <lb n="1799,001,,,,3,22,21"></lb> |
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2633 γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,22,22"></lb> |
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2634 <s>Ἐπὶ τῆς αὐτῆς εὐθείας δύο τμήματα κύκλων ὅμοια <lb n="1799,001,,,,3,23,1"></lb> |
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2635 καὶ ἄνισα οὐ συσταθήσεται ἐπὶ τὰ αὐτὰ <lb n="1799,001,,,,3,23,2"></lb> |
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2636 μέρη.</s> <lb n="1799,001,,,,3,23,3"></lb> |
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2637 <s>Εἰ γὰρ δυνατόν, ἐπὶ τῆς αὐτῆς εὐθείας <lb n="1799,001,,,,3,23,4"></lb> |
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2638 τῆς ΑΒ δύο τμήματα κύκλων ὅμοια καὶ <lb n="1799,001,,,,3,23,5"></lb> |
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2639 ἄνισα συνεστάτω ἐπὶ τὰ αὐτὰ μέρη τὰ ΑΓΒ, <lb n="1799,001,,,,3,23,6"></lb> |
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2640 ΑΔΒ, καὶ διήχθω ἡ ΑΓΔ, καὶ ἐπεζεύ-<lb n="1799,001,,,,3,23,7"></lb> |
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2641 χθωσαν αἱ ΓΒ, ΔΒ.</s> <s><pb></pb> <lb n="1799,001,,,,3,23,8"></lb> |
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2642 Ἐπεὶ οὖν ὅμοιόν ἐστι τὸ ΑΓΒ τμῆμα τῷ ΑΔΒ τμή-<lb n="1799,001,,,,3,23,9"></lb> |
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2643 ματι, ὅμοια δὲ τμήματα κύκλων ἐστὶ τὰ δεχόμενα γωνίας <lb n="1799,001,,,,3,23,10"></lb> |
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2644 ἴσας, ἴση ἄρα ἐστὶν ἡ ὑπὸ ΑΓΒ γωνία τῇ ὑπὸ ΑΔΒ ἡ ἐκ-<lb n="1799,001,,,,3,23,11"></lb> |
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2645 τὸς τῇ ἐντός· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,3,23,12"></lb> |
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2646 <s>Οὐκ ἄρα ἐπὶ τῆς αὐτῆς εὐθείας δύο τμήματα κύκλων <lb n="1799,001,,,,3,23,13"></lb> |
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2647 ὅμοια καὶ ἄνισα συσταθήσεται ἐπὶ τὰ αὐτὰ μέρη· ὅπερ <lb n="1799,001,,,,3,23,14"></lb> |
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2648 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,23,15"></lb> |
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2649 <s>Τὰ ἐπὶ ἴσων εὐθειῶν ὅμοια τμήματα κύκλων ἴσα ἀλλή-<lb n="1799,001,,,,3,24,1"></lb> |
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2650 λοις ἐστίν.</s> <lb n="1799,001,,,,3,24,2"></lb> |
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2651 <s>Ἔστωσαν γὰρ ἐπὶ ἴσων εὐθειῶν τῶν ΑΒ, ΓΔ ὅμοια <lb n="1799,001,,,,3,24,3"></lb> |
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2652 τμήματα κύκλων τὰ ΑΕΒ, ΓΖΔ· λέγω, ὅτι ἴσον ἐστὶ <lb n="1799,001,,,,3,24,4"></lb> |
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2653 τὸ ΑΕΒ τμῆμα τῷ ΓΖΔ τμήματι.</s> <lb n="1799,001,,,,3,24,5"></lb> |
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2654 <s>Ἐφαρμοζομένου γὰρ τοῦ ΑΕΒ τμήματος ἐπὶ τὸ ΓΖΔ <lb n="1799,001,,,,3,24,6"></lb> |
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2655 καὶ τιθεμένου τοῦ μὲν Α σημείου ἐπὶ τὸ Γ τῆς δὲ ΑΒ <lb n="1799,001,,,,3,24,7"></lb> |
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2656 εὐθείας ἐπὶ τὴν ΓΔ, ἐφαρμόσει καὶ τὸ Β <lb n="1799,001,,,,3,24,8"></lb> |
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2657 σημεῖον ἐπὶ τὸ Δ σημεῖον διὰ τὸ ἴσην εἶναι <lb n="1799,001,,,,3,24,9"></lb> |
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2658 τὴν ΑΒ τῇ ΓΔ· τῆς δὲ ΑΒ ἐπὶ τὴν ΓΔ <lb n="1799,001,,,,3,24,10"></lb> |
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2659 ἐφαρμοσάσης ἐφαρμόσει καὶ τὸ ΑΕΒ <lb n="1799,001,,,,3,24,11"></lb> |
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2660 τμῆμα ἐπὶ τὸ ΓΖΔ.</s> <s>εἰ γὰρ ἡ ΑΒ εὐθεῖα <lb n="1799,001,,,,3,24,12"></lb> |
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2661 ἐπὶ τὴν ΓΔ ἐφαρμόσει, τὸ δὲ ΑΕΒ <lb n="1799,001,,,,3,24,13"></lb> |
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2662 τμῆμα ἐπὶ τὸ ΓΖΔ μὴ ἐφαρμόσει, ἤτοι <pb></pb> <lb n="1799,001,,,,3,24,14"></lb> |
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2663 ἐντὸς αὐτοῦ πεσεῖται ἢ ἐκτὸς ἢ παραλλάξει ὡς τὸ ΓΗΔ, <lb n="1799,001,,,,3,24,15"></lb> |
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2664 καὶ κύκλος κύκλον τέμνει κατὰ πλείονα σημεῖα ἢ δύο· <lb n="1799,001,,,,3,24,16"></lb> |
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2665 ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ἐφαρμοζομένης τῆς ΑΒ <lb n="1799,001,,,,3,24,17"></lb> |
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2666 εὐθείας ἐπὶ τὴν ΓΔ οὐκ ἐφαρμόσει καὶ τὸ ΑΕΒ τμῆμα ἐπὶ <lb n="1799,001,,,,3,24,18"></lb> |
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2667 τὸ ΓΖΔ· ἐφαρμόσει ἄρα, καὶ ἴσον αὐτῷ ἔσται.</s> <lb n="1799,001,,,,3,24,19"></lb> |
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2668 <s>Τὰ ἄρα ἐπὶ ἴσων εὐθειῶν ὅμοια τμήματα κύκλων ἴσα <lb n="1799,001,,,,3,24,20"></lb> |
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2669 ἀλλήλοις ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,24,21"></lb> |
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2670 <s>Κύκλου τμήματος δοθέντος προσαναγράψαι τὸν κύκλον, <lb n="1799,001,,,,3,25,1"></lb> |
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2671 οὗπέρ ἐστι τμῆμα.</s> <lb n="1799,001,,,,3,25,2"></lb> |
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2672 <s>Ἔστω τὸ δοθὲν τμῆμα κύκλου τὸ ΑΒΓ· δεῖ δὴ τοῦ <lb n="1799,001,,,,3,25,3"></lb> |
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2673 ΑΒΓ τμήματος προσαναγράψαι τὸν κύκλον, οὗπέρ ἐστι <lb n="1799,001,,,,3,25,4"></lb> |
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2674 τμῆμα.</s> <lb n="1799,001,,,,3,25,5"></lb> |
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2675 <s>Τετμήσθω γὰρ ἡ ΑΓ δίχα κατὰ τὸ Δ, καὶ ἤχθω ἀπὸ <lb n="1799,001,,,,3,25,6"></lb> |
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2676 τοῦ Δ σημείου τῇ ΑΓ πρὸς ὀρθὰς ἡ ΔΒ, καὶ ἐπεζεύχθω <lb n="1799,001,,,,3,25,7"></lb> |
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2677 ἡ ΑΒ· ἡ ὑπὸ ΑΒΔ γωνία ἄρα τῆς ὑπὸ ΒΑΔ ἤτοι μείζων <lb n="1799,001,,,,3,25,8"></lb> |
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2678 ἐστὶν ἢ ἴση ἢ ἐλάττων.</s> <lb n="1799,001,,,,3,25,9"></lb> |
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2679 <s>Ἔστω πρότερον μείζων, καὶ συνεστάτω πρὸς τῇ ΒΑ <lb n="1799,001,,,,3,25,10"></lb> |
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2680 εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Α τῇ ὑπὸ ΑΒΔ <pb></pb> <lb n="1799,001,,,,3,25,11"></lb> |
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2681 γωνίᾳ ἴση ἡ ὑπὸ ΒΑΕ, καὶ διήχθω ἡ ΔΒ ἐπὶ τὸ Ε, καὶ <lb n="1799,001,,,,3,25,12"></lb> |
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2682 ἐπεζεύχθω ἡ ΕΓ.</s> <s>ἐπεὶ οὖν ἴση ἐστὶν ἡ ὑπὸ ΑΒΕ γωνία <lb n="1799,001,,,,3,25,13"></lb> |
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2683 τῇ ὑπὸ ΒΑΕ, ἴση ἄρα ἐστὶ καὶ ἡ ΕΒ εὐθεῖα τῇ *εΑ.</s> <s>καὶ <lb n="1799,001,,,,3,25,14"></lb> |
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2684 ἐπεὶ ἴση ἐστὶν ἡ ΑΔ τῇ ΔΓ, κοινὴ δὲ <lb n="1799,001,,,,3,25,15"></lb> |
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2685 ἡ ΔΕ, δύο δὴ αἱ ΑΔ, ΔΕ δύο ταῖς ΓΔ, <lb n="1799,001,,,,3,25,16"></lb> |
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2686 ΔΕ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία <lb n="1799,001,,,,3,25,17"></lb> |
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2687 ἡ ὑπὸ ΑΔΕ γωνίᾳ τῇ ὑπὸ ΓΔΕ ἐστιν <lb n="1799,001,,,,3,25,18"></lb> |
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2688 ἴση· ὀρθὴ γὰρ ἑκατέρα· βάσις ἄρα ἡ ΑΕ <lb n="1799,001,,,,3,25,19"></lb> |
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2689 βάσει τῇ ΓΕ ἐστιν ἴση.</s> <s>ἀλλὰ ἡ ΑΕ τῇ <lb n="1799,001,,,,3,25,20"></lb> |
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2690 ΒΕ ἐδείχθη ἴση· καὶ ἡ ΒΕ ἄρα τῇ <lb n="1799,001,,,,3,25,21"></lb> |
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2691 ΓΕ ἐστιν ἴση· αἱ τρεῖς ἄρα αἱ ΑΕ, ΕΒ, ΕΓ ἴσαι ἀλλή-<lb n="1799,001,,,,3,25,22"></lb> |
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2692 λαις εἰσίν· ὁ ἄρα κέντρῳ τῷ Ε διαστήματι δὲ ἑνὶ τῶν ΑΕ, <lb n="1799,001,,,,3,25,23"></lb> |
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2693 ΕΒ, ΕΓ κύκλος γραφόμενος ἥξει καὶ διὰ τῶν λοιπῶν ση-<lb n="1799,001,,,,3,25,24"></lb> |
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2694 μείων καὶ ἔσται προσαναγεγραμμένος.</s> <s>κύκλου ἄρα τμή-<lb n="1799,001,,,,3,25,25"></lb> |
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2695 ματος δοθέντος προσαναγέγραπται ὁ κύκλος.</s> <s>καὶ δῆλον, <lb n="1799,001,,,,3,25,26"></lb> |
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2696 ὡς τὸ ΑΒΓ τμῆμα ἔλαττόν ἐστιν ἡμικυκλίου διὰ τὸ τὸ Ε <lb n="1799,001,,,,3,25,27"></lb> |
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2697 κέντρον ἐκτὸς αὐτοῦ τυγχάνειν.</s> <lb n="1799,001,,,,3,25,28"></lb> |
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2698 <s>Ὁμοίως [δὲ] κἂν ᾖ ἡ ὑπὸ ΑΒΔ γωνία ἴση τῇ ὑπὸ <lb n="1799,001,,,,3,25,29"></lb> |
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2699 ΒΑΔ, τῆς ΑΔ ἴσης γενομένης ἑκατέρᾳ τῶν ΒΔ, ΔΓ αἱ <lb n="1799,001,,,,3,25,30"></lb> |
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2700 τρεῖς αἱ ΔΑ, ΔΒ, ΔΓ ἴσαι ἀλλήλαις ἔσονται, καὶ ἔσται <lb n="1799,001,,,,3,25,31"></lb> |
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2701 τὸ Δ κέντρον τοῦ προσαναπεπληρωμένου κύκλου, καὶ δη-<lb n="1799,001,,,,3,25,32"></lb> |
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2702 λαδὴ ἔσται τὸ ΑΒΓ ἡμικύκλιον.</s> <s><pb></pb> <lb n="1799,001,,,,3,25,33"></lb> |
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2703 Ἐὰν δὲ ἡ ὑπὸ ΑΒΔ ἐλάττων ᾖ <lb n="1799,001,,,,3,25,34"></lb> |
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2704 τῆς ὑπὸ ΒΑΔ, καὶ συστησώμεθα <lb n="1799,001,,,,3,25,35"></lb> |
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2705 πρὸς τῇ ΒΑ εὐθείᾳ καὶ τῷ πρὸς <lb n="1799,001,,,,3,25,36"></lb> |
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2706 αὐτῇ σημείῳ τῷ Α τῇ ὑπὸ ΑΒΔ <lb n="1799,001,,,,3,25,37"></lb> |
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2707 γωνίᾳ ἴσην, ἐντὸς τοῦ ΑΒΓ τμή-<lb n="1799,001,,,,3,25,38"></lb> |
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2708 ματος πεσεῖται τὸ κέντρον ἐπὶ τῆς ΔΒ, καὶ ἔσται <lb n="1799,001,,,,3,25,39"></lb> |
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2709 δηλαδὴ τὸ ΑΒΓ τμῆμα μεῖζον ἡμικυκλίου.</s> <lb n="1799,001,,,,3,25,40"></lb> |
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2710 <s>Κύκλου ἄρα τμήματος δοθέντος προσαναγέγραπται ὁ <lb n="1799,001,,,,3,25,41"></lb> |
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2711 κύκλος· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,3,25,42"></lb> |
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2712 <s>Ἐν τοῖς ἴσοις κύκλοις αἱ ἴσαι γωνίαι ἐπὶ ἴσων περιφε-<lb n="1799,001,,,,3,26,1"></lb> |
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2713 ρειῶν βεβήκασιν, ἐάν τε πρὸς τοῖς κέντροις ἐάν τε πρὸς <lb n="1799,001,,,,3,26,2"></lb> |
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2714 ταῖς περιφερείαις ὦσι βεβηκυῖαι.</s> <lb n="1799,001,,,,3,26,3"></lb> |
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2715 <s>Ἔστωσαν ἴσοι κύκλοι οἱ ΑΒΓ, ΔΕΖ καὶ ἐν αὐτοῖς <lb n="1799,001,,,,3,26,4"></lb> |
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2716 ἴσαι γωνίαι ἔστωσαν πρὸς μὲν τοῖς κέντροις αἱ ὑπὸ ΒΗΓ, <lb n="1799,001,,,,3,26,5"></lb> |
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2717 ΕΘΖ, πρὸς δὲ ταῖς περιφερείαις αἱ ὑπὸ ΒΑΓ, ΕΔΖ· <lb n="1799,001,,,,3,26,6"></lb> |
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2718 λέγω, ὅτι ἴση ἐστὶν ἡ ΒΚΓ <lb n="1799,001,,,,3,26,7"></lb> |
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2719 περιφέρεια τῇ ΕΛΖ περιφε-<lb n="1799,001,,,,3,26,8"></lb> |
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2720 ρείᾳ.</s> <lb n="1799,001,,,,3,26,9"></lb> |
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2721 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΒΓ, <lb n="1799,001,,,,3,26,10"></lb> |
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2722 ΕΖ.</s> <lb n="1799,001,,,,3,26,11"></lb> |
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2723 <s>Καὶ ἐπεὶ ἴσοι εἰσὶν οἱ ΑΒΓ, <pb></pb> <lb n="1799,001,,,,3,26,12"></lb> |
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2724 ΔΕΖ κύκλοι, ἴσαι εἰσὶν αἱ ἐκ τῶν κέντρων· δύο δὴ αἱ <lb n="1799,001,,,,3,26,13"></lb> |
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2725 ΒΗ, ΗΓ δύο ταῖς ΕΘ, ΘΖ ἴσαι· καὶ γωνία ἡ πρὸς τῷ <lb n="1799,001,,,,3,26,14"></lb> |
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2726 Η γωνίᾳ τῇ πρὸς τῷ Θ ἴση· βάσις ἄρα ἡ ΒΓ βάσει τῇ <lb n="1799,001,,,,3,26,15"></lb> |
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2727 ΕΖ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ πρὸς τῷ Α γωνία τῇ <lb n="1799,001,,,,3,26,16"></lb> |
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2728 πρὸς τῷ Δ, ὅμοιον ἄρα ἐστὶ τὸ ΒΑΓ τμῆμα τῷ ΕΔΖ <lb n="1799,001,,,,3,26,17"></lb> |
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2729 τμήματι· καί εἰσιν ἐπὶ ἴσων εὐθειῶν [τῶν ΒΓ, ΕΖ]· <lb n="1799,001,,,,3,26,18"></lb> |
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2730 τὰ δὲ ἐπὶ ἴσων εὐθειῶν ὅμοια τμήματα κύκλων ἴσα <lb n="1799,001,,,,3,26,19"></lb> |
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2731 ἀλλήλοις ἐστίν· ἴσον ἄρα τὸ ΒΑΓ τμῆμα τῷ ΕΔΖ.</s> <lb n="1799,001,,,,3,26,20"></lb> |
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2732 <s>ἔστι δὲ καὶ ὅλος ὁ ΑΒΓ κύκλος ὅλῳ τῷ ΔΕΖ κύκλῳ <lb n="1799,001,,,,3,26,21"></lb> |
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2733 ἴσος· λοιπὴ ἄρα ἡ ΒΚΓ περιφέρεια τῇ ΕΛΖ περιφερείᾳ <lb n="1799,001,,,,3,26,22"></lb> |
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2734 ἐστὶν ἴση.</s> <lb n="1799,001,,,,3,26,23"></lb> |
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2735 <s>Ἐν ἄρα τοῖς ἴσοις κύκλοις αἱ ἴσαι γωνίαι ἐπὶ ἴσων περι-<lb n="1799,001,,,,3,26,24"></lb> |
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2736 φερειῶν βεβήκασιν, ἐάν τε πρὸς τοῖς κέντροις ἐάν τε πρὸς <lb n="1799,001,,,,3,26,25"></lb> |
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2737 ταῖς περιφερείαις ὦσι βεβηκυῖαι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,26,26"></lb> |
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2738 <s>Ἐν τοῖς ἴσοις κύκλοις αἱ ἐπὶ ἴσων περιφερειῶν βεβηκυῖαι <lb n="1799,001,,,,3,27,1"></lb> |
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2739 γωνίαι ἴσαι ἀλλήλαις εἰσίν, ἐάν τε πρὸς τοῖς κέντροις ἐάν <lb n="1799,001,,,,3,27,2"></lb> |
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2740 τε πρὸς ταῖς περιφερείαις ὦσι <lb n="1799,001,,,,3,27,3"></lb> |
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2741 βεβηκυῖαι.</s> <lb n="1799,001,,,,3,27,4"></lb> |
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2742 <s>Ἐν γὰρ ἴσοις κύκλοις τοῖς <lb n="1799,001,,,,3,27,5"></lb> |
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2743 ΑΒΓ, ΔΕΖ ἐπὶ ἴσων περι-<lb n="1799,001,,,,3,27,6"></lb> |
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2744 φερειῶν τῶν ΒΓ, ΕΖ πρὸς <pb></pb> <lb n="1799,001,,,,3,27,7"></lb> |
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2745 μὲν τοῖς Η, Θ κέντροις γωνίαι βεβηκέτωσαν αἱ ὑπὸ ΒΗΓ, <lb n="1799,001,,,,3,27,8"></lb> |
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2746 ΕΘΖ, πρὸς δὲ ταῖς περιφερείαις αἱ ὑπὸ ΒΑΓ, ΕΔΖ· <lb n="1799,001,,,,3,27,9"></lb> |
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2747 λέγω, ὅτι ἡ μὲν ὑπὸ ΒΗΓ γωνία τῇ ὑπὸ ΕΘΖ ἐστιν ἴση, <lb n="1799,001,,,,3,27,10"></lb> |
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2748 ἡ δὲ ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ ἐστιν ἴση.</s> <lb n="1799,001,,,,3,27,11"></lb> |
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2749 <s>Εἰ γὰρ ἄνισός ἐστιν ἡ ὑπὸ ΒΗΓ τῇ ὑπὸ ΕΘΖ, μία <lb n="1799,001,,,,3,27,12"></lb> |
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2750 αὐτῶν μείζων ἐστίν.</s> <s>ἔστω μείζων ἡ ὑπὸ ΒΗΓ, <lb n="1799,001,,,,3,27,13"></lb> |
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2751 καὶ συνεστάτω πρὸς τῇ ΒΗ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ <lb n="1799,001,,,,3,27,14"></lb> |
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2752 σημείῳ τῷ Η τῇ ὑπὸ ΕΘΖ γωνίᾳ ἴση ἡ ὑπὸ ΒΗΚ· αἱ <lb n="1799,001,,,,3,27,15"></lb> |
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2753 δὲ ἴσαι γωνίαι ἐπὶ ἴσων περιφερειῶν βεβήκασιν, ὅταν <lb n="1799,001,,,,3,27,16"></lb> |
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2754 πρὸς τοῖς κέντροις ὦσιν· ἴση ἄρα ἡ ΒΚ περιφέρεια τῇ <lb n="1799,001,,,,3,27,17"></lb> |
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2755 ΕΖ περιφερείᾳ.</s> <s>ἀλλὰ ἡ ΕΖ τῇ ΒΓ ἐστιν ἴση· καὶ ἡ ΒΚ <lb n="1799,001,,,,3,27,18"></lb> |
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2756 ἄρα τῇ ΒΓ ἐστιν ἴση ἡ ἐλάττων τῇ μείζονι· ὅπερ ἐστὶν <lb n="1799,001,,,,3,27,19"></lb> |
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2757 ἀδύνατον.</s> <s>οὐκ ἄρα ἄνισός ἐστιν ἡ ὑπὸ ΒΗΓ γωνία τῇ <lb n="1799,001,,,,3,27,20"></lb> |
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2758 ὑπὸ ΕΘΖ· ἴση ἄρα.</s> <s>καί ἐστι τῆς μὲν ὑπὸ ΒΗΓ ἡμίσεια <lb n="1799,001,,,,3,27,21"></lb> |
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2759 ἡ πρὸς τῷ Α, τῆς δὲ ὑπὸ ΕΘΖ ἡμίσεια ἡ πρὸς τῷ Δ· ἴση <lb n="1799,001,,,,3,27,22"></lb> |
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2760 ἄρα καὶ ἡ πρὸς τῷ Α γωνία τῇ πρὸς τῷ Δ.</s> <lb n="1799,001,,,,3,27,23"></lb> |
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2761 <s>Ἐν ἄρα τοῖς ἴσοις κύκλοις αἱ ἐπὶ ἴσων περιφερειῶν <lb n="1799,001,,,,3,27,24"></lb> |
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2762 βεβηκυῖαι γωνίαι ἴσαι ἀλλήλαις εἰσίν, ἐάν τε πρὸς τοῖς <lb n="1799,001,,,,3,27,25"></lb> |
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2763 κέντροις ἐάν τε πρὸς ταῖς περιφερείαις ὦσι βεβηκυῖαι· <lb n="1799,001,,,,3,27,26"></lb> |
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2764 ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,3,27,27"></lb> |
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2765 Ἐν τοῖς ἴσοις κύκλοις αἱ ἴσαι εὐθεῖαι ἴσας περιφερείας <lb n="1799,001,,,,3,28,1"></lb> |
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2766 ἀφαιροῦσι τὴν μὲν μείζονα τῇ μείζονι τὴν δὲ ἐλάττονα τῇ <lb n="1799,001,,,,3,28,2"></lb> |
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2767 ἐλάττονι.</s> <lb n="1799,001,,,,3,28,3"></lb> |
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2768 <s>Ἔστωσαν ἴσοι κύκλοι οἱ ΑΒΓ, ΔΕΖ, καὶ ἐν τοῖς κύ-<lb n="1799,001,,,,3,28,4"></lb> |
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2769 κλοις ἴσαι εὐθεῖαι ἔστωσαν αἱ ΑΒ, ΔΕ τὰς μὲν ΑΓΒ, <lb n="1799,001,,,,3,28,5"></lb> |
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2770 ΔΖΕ περιφερείας μείζονας ἀφαιροῦσαι τὰς δὲ ΑΗΒ, <lb n="1799,001,,,,3,28,6"></lb> |
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2771 ΔΘΕ ἐλάττονας· λέγω, ὅτι ἡ μὲν ΑΓΒ μείζων περιφέρεια <lb n="1799,001,,,,3,28,7"></lb> |
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2772 ἴση ἐστὶ τῇ ΔΖΕ μείζονι περιφερείᾳ, ἡ δὲ ΑΗΒ ἐλάτ-<lb n="1799,001,,,,3,28,8"></lb> |
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2773 των περιφέρεια τῇ ΔΘΕ. <lb n="1799,001,,,,3,28,9"></lb> |
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2774 Εἰλήφθω γὰρ τὰ κέντρα τῶν κύκλων τὰ Κ, Λ, καὶ ἐπ-<lb n="1799,001,,,,3,28,10"></lb> |
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2775 εζεύχθωσαν αἱ ΑΚ, ΚΒ, ΔΛ, ΛΕ.</s> <lb n="1799,001,,,,3,28,11"></lb> |
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2776 <s>Καὶ ἐπεὶ ἴσοι κύκλοι εἰσίν, ἴσαι εἰσὶ καὶ αἱ ἐκ τῶν κέν-<lb n="1799,001,,,,3,28,12"></lb> |
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2777 τρων· δύο δὴ αἱ ΑΚ, ΚΒ δυσὶ ταῖς ΔΛ, ΛΕ ἴσαι εἰσίν· <pb></pb> <lb n="1799,001,,,,3,28,13"></lb> |
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2778 καὶ βάσις ἡ ΑΒ βάσει τῇ ΔΕ ἴση· γωνία ἄρα ἡ ὑπὸ ΑΚΒ <lb n="1799,001,,,,3,28,14"></lb> |
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2779 γωνίᾳ τῇ ὑπὸ ΔΛΕ ἴση ἐστίν.</s> <s>αἱ δὲ ἴσαι γωνίαι ἐπὶ ἴσων <lb n="1799,001,,,,3,28,15"></lb> |
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2780 περιφερειῶν βεβήκασιν, ὅταν πρὸς τοῖς κέντροις ὦσιν· <lb n="1799,001,,,,3,28,16"></lb> |
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2781 ἴση ἄρα ἡ ΑΗΒ περιφέρεια τῇ ΔΘΕ.</s> <s>ἐστὶ δὲ καὶ ὅλος <lb n="1799,001,,,,3,28,17"></lb> |
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2782 ὁ ΑΒΓ κύκλος ὅλῳ τῷ ΔΕΖ κύκλῳ ἴσος· καὶ λοιπὴ ἄρα <lb n="1799,001,,,,3,28,18"></lb> |
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2783 ἡ ΑΓΒ περιφέρεια λοιπῇ τῇ ΔΖΕ περιφερείᾳ ἴση ἐστίν.</s> <lb n="1799,001,,,,3,28,19"></lb> |
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2784 <s>Ἐν ἄρα τοῖς ἴσοις κύκλοις αἱ ἴσαι εὐθεῖαι ἴσας περι-<lb n="1799,001,,,,3,28,20"></lb> |
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2785 φερείας ἀφαιροῦσι τὴν μὲν μείζονα τῇ μείζονι τὴν δὲ <lb n="1799,001,,,,3,28,21"></lb> |
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2786 ἐλάττονα τῇ ἐλάττονι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,28,22"></lb> |
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2787 <s>Ἐν τοῖς ἴσοις κύκλοις τὰς ἴσας περιφερείας ἴσαι εὐθεῖαι <lb n="1799,001,,,,3,29,1"></lb> |
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2788 ὑποτείνουσιν.</s> <lb n="1799,001,,,,3,29,2"></lb> |
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2789 <s>Ἔστωσαν ἴσοι κύκλοι οἱ ΑΒΓ, ΔΕΖ, καὶ ἐν αὐτοῖς <lb n="1799,001,,,,3,29,3"></lb> |
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2790 ἴσαι περιφέρειαι ἀπειλήφθωσαν αἱ ΒΗΓ, ΕΘΖ, καὶ <lb n="1799,001,,,,3,29,4"></lb> |
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2791 ἐπεζεύχθωσαν αἱ ΒΓ, ΕΖ εὐθεῖαι· λέγω, ὅτι ἴση ἐστὶν <lb n="1799,001,,,,3,29,5"></lb> |
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2792 ἡ ΒΓ τῇ ΕΖ.</s> <s><pb></pb> <lb n="1799,001,,,,3,29,6"></lb> |
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2793 Εἰλήφθω γὰρ τὰ κέντρα τῶν κύκλων, καὶ ἔστω τὰ Κ, <lb n="1799,001,,,,3,29,7"></lb> |
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2794 Λ, καὶ ἐπεζεύχθωσαν αἱ ΒΚ, ΚΓ, ΕΛ, ΛΖ. <lb n="1799,001,,,,3,29,8"></lb> |
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2795 Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΗΓ περιφέρεια τῇ ΕΘΖ περι-<lb n="1799,001,,,,3,29,9"></lb> |
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2796 φερείᾳ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΒΚΓ τῇ ὑπὸ ΕΛΖ.</s> <lb n="1799,001,,,,3,29,10"></lb> |
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2797 <s>καὶ ἐπεὶ ἴσοι εἰσὶν οἱ ΑΒΓ, ΔΕΖ κύκλοι, ἴσαι εἰσὶ καὶ <lb n="1799,001,,,,3,29,11"></lb> |
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2798 αἱ ἐκ τῶν κέντρων· δύο δὴ αἱ ΒΚ, ΚΓ δυσὶ ταῖς ΕΛ, ΛΖ <lb n="1799,001,,,,3,29,12"></lb> |
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2799 ἴσαι εἰσίν· καὶ γωνίας ἴσας περιέχουσιν· βάσις ἄρα ἡ ΒΓ <lb n="1799,001,,,,3,29,13"></lb> |
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2800 βάσει τῇ ΕΖ ἴση ἐστίν.</s> <lb n="1799,001,,,,3,29,14"></lb> |
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2801 <s>Ἐν ἄρα τοῖς ἴσοις κύκλοις τὰς ἴσας περιφερείας ἴσαι <lb n="1799,001,,,,3,29,15"></lb> |
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2802 εὐθεῖαι ὑποτείνουσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,29,16"></lb> |
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2803 <s>Τὴν δοθεῖσαν περιφέρειαν δίχα τεμεῖν.</s> <lb n="1799,001,,,,3,30,1"></lb> |
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2804 <s>Ἔστω ἡ δοθεῖσα περιφέρεια ἡ ΑΔΒ· δεῖ δὴ τὴν ΑΔΒ <lb n="1799,001,,,,3,30,2"></lb> |
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2805 περιφέρειαν δίχα τεμεῖν.</s> <s><pb></pb> <lb n="1799,001,,,,3,30,3"></lb> |
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2806 Ἐπεζεύχθω ἡ ΑΒ, καὶ τετμήσθω δίχα κατὰ τὸ Γ, καὶ <lb n="1799,001,,,,3,30,4"></lb> |
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2807 ἀπὸ τοῦ Γ σημείου τῇ ΑΒ εὐθείᾳ πρὸς ὀρθὰς ἤχθω ἡ ΓΔ, <lb n="1799,001,,,,3,30,5"></lb> |
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2808 καὶ ἐπεζεύχθωσαν αἱ ΑΔ, ΔΒ. <lb n="1799,001,,,,3,30,6"></lb> |
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2809 Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΓ τῇ ΓΒ, κοινὴ δὲ ἡ ΓΔ, δύο δὴ <lb n="1799,001,,,,3,30,7"></lb> |
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2810 αἱ ΑΓ, ΓΔ δυσὶ ταῖς ΒΓ, ΓΔ ἴσαι εἰσίν· καὶ γωνία ἡ ὑπὸ <lb n="1799,001,,,,3,30,8"></lb> |
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2811 ΑΓΔ γωνίᾳ τῇ ὑπὸ ΒΓΔ ἴση· ὀρθὴ γὰρ ἑκατέρα· βάσις <lb n="1799,001,,,,3,30,9"></lb> |
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2812 ἄρα ἡ ΑΔ βάσει τῇ ΔΒ ἴση ἐστίν.</s> <s>αἱ δὲ ἴσαι εὐθεῖαι ἴσας <lb n="1799,001,,,,3,30,10"></lb> |
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2813 περιφερείας ἀφαιροῦσι τὴν μὲν μείζονα τῇ μείζονι τὴν δὲ <lb n="1799,001,,,,3,30,11"></lb> |
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2814 ἐλάττονα τῇ ἐλάττονι· καί ἐστιν ἑκατέρα τῶν ΑΔ, ΔΒ <lb n="1799,001,,,,3,30,12"></lb> |
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2815 περιφερειῶν ἐλάττων ἡμικυκλίου· ἴση ἄρα ἡ ΑΔ περι-<lb n="1799,001,,,,3,30,13"></lb> |
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2816 φέρεια τῇ ΔΒ περιφερείᾳ.</s> <lb n="1799,001,,,,3,30,14"></lb> |
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2817 <s>Ἡ ἄρα δοθεῖσα περιφέρεια δίχα τέτμηται κατὰ τὸ Δ <lb n="1799,001,,,,3,30,15"></lb> |
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2818 σημεῖον· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,3,30,16"></lb> |
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2819 <s>Ἐν κύκλῳ ἡ μὲν ἐν τῷ ἡμικυκλίῳ γωνία ὀρθή ἐστιν, ἡ <lb n="1799,001,,,,3,31,1"></lb> |
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2820 δὲ ἐν τῷ μείζονι τμήματι ἐλάττων ὀρθῆς, ἡ δὲ ἐν τῷ ἐλάτ-<lb n="1799,001,,,,3,31,2"></lb> |
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2821 τονι τμήματι μείζων ὀρθῆς· καὶ ἔτι ἡ μὲν τοῦ μείζονος <pb></pb> <lb n="1799,001,,,,3,31,3"></lb> |
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2822 τμήματος γωνία μείζων ἐστὶν ὀρθῆς, ἡ δὲ τοῦ ἐλάττονος <lb n="1799,001,,,,3,31,4"></lb> |
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2823 τμήματος γωνία ἐλάττων ὀρθῆς.</s> <lb n="1799,001,,,,3,31,5"></lb> |
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2824 <s>Ἔστω κύκλος ὁ ΑΒΓΔ, διάμετρος δὲ αὐτοῦ ἔστω ἡ <lb n="1799,001,,,,3,31,6"></lb> |
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2825 ΒΓ, κέντρον δὲ τὸ Ε, καὶ ἐπεζεύχθωσαν αἱ ΒΑ, ΑΓ, <lb n="1799,001,,,,3,31,7"></lb> |
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2826 ΑΔ, ΔΓ· λέγω, ὅτι ἡ μὲν ἐν τῷ ΒΑΓ ἡμικυκλίῳ γωνία <lb n="1799,001,,,,3,31,8"></lb> |
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2827 ἡ ὑπὸ ΒΑΓ ὀρθή ἐστιν, ἡ δὲ ἐν τῷ <lb n="1799,001,,,,3,31,9"></lb> |
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2828 ΑΒΓ μείζονι τοῦ ἡμικυκλίου τμή-<lb n="1799,001,,,,3,31,10"></lb> |
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2829 ματι γωνία ἡ ὑπὸ ΑΒΓ ἐλάττων <lb n="1799,001,,,,3,31,11"></lb> |
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2830 ἐστὶν ὀρθῆς, ἡ δὲ ἐν τῷ ΑΔΓ ἐλάτ-<lb n="1799,001,,,,3,31,12"></lb> |
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2831 τονι τοῦ ἡμικυκλίου τμήματι γωνία <lb n="1799,001,,,,3,31,13"></lb> |
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2832 ἡ ὑπὸ ΑΔΓ μείζων ἐστὶν ὀρθῆς.</s> <lb n="1799,001,,,,3,31,14"></lb> |
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2833 <s>Ἐπεζεύχθω ἡ ΑΕ, καὶ διήχθω ἡ <lb n="1799,001,,,,3,31,15"></lb> |
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2834 ΒΑ ἐπὶ τὸ Ζ.</s> <lb n="1799,001,,,,3,31,16"></lb> |
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2835 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΕ τῇ ΕΑ, ἴση ἐστὶ καὶ γωνία ἡ <lb n="1799,001,,,,3,31,17"></lb> |
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2836 ὑπὸ ΑΒΕ τῇ ὑπὸ ΒΑΕ.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΓΕ τῇ <lb n="1799,001,,,,3,31,18"></lb> |
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2837 ΕΑ, ἴση ἐστὶ καὶ ἡ ὑπὸ ΑΓΕ τῇ ὑπὸ ΓΑΕ· ὅλη ἄρα ἡ <lb n="1799,001,,,,3,31,19"></lb> |
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2838 ὑπὸ ΒΑΓ δυσὶ ταῖς ὑπὸ ΑΒΓ, ΑΓΒ ἴση ἐστίν.</s> <s>ἐστὶ δὲ <lb n="1799,001,,,,3,31,20"></lb> |
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2839 καὶ ἡ ὑπὸ ΖΑΓ ἐκτὸς τοῦ ΑΒΓ τριγώνου δυσὶ ταῖς ὑπὸ <lb n="1799,001,,,,3,31,21"></lb> |
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2840 ΑΒΓ, ΑΓΒ γωνίαις ἴση· ἴση ἄρα καὶ ἡ ὑπὸ ΒΑΓ <lb n="1799,001,,,,3,31,22"></lb> |
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2841 γωνία τῇ ὑπὸ ΖΑΓ· ὀρθὴ ἄρα ἑκατέρα· ἡ ἄρα ἐν τῷ <lb n="1799,001,,,,3,31,23"></lb> |
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2842 ΒΑΓ ἡμικυκλίῳ γωνία ἡ ὑπὸ ΒΑΓ ὀρθή ἐστιν.</s> <s><pb></pb> <lb n="1799,001,,,,3,31,24"></lb> |
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2843 Καὶ ἐπεὶ τοῦ ΑΒΓ τριγώνου δύο γωνίαι αἱ ὑπὸ ΑΒΓ, <lb n="1799,001,,,,3,31,25"></lb> |
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2844 ΒΑΓ δύο ὀρθῶν ἐλάττονές εἰσιν, ὀρθὴ δὲ ἡ ὑπὸ ΒΑΓ, <lb n="1799,001,,,,3,31,26"></lb> |
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2845 ἐλάττων ἄρα ὀρθῆς ἐστιν ἡ ὑπὸ ΑΒΓ γωνία· καί ἐστιν <lb n="1799,001,,,,3,31,27"></lb> |
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2846 ἐν τῷ ΑΒΓ μείζονι τοῦ ἡμικυκλίου τμήματι.</s> <lb n="1799,001,,,,3,31,28"></lb> |
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2847 <s>Καὶ ἐπεὶ ἐν κύκλῳ τετράπλευρόν ἐστι τὸ ΑΒΓΔ, τῶν <lb n="1799,001,,,,3,31,29"></lb> |
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2848 δὲ ἐν τοῖς κύκλοις τετραπλεύρων αἱ ἀπεναντίον γωνίαι <lb n="1799,001,,,,3,31,30"></lb> |
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2849 δυσὶν ὀρθαῖς ἴσαι εἰσίν [αἱ ἄρα ὑπὸ ΑΒΓ, ΑΔΓ γωνίαι <lb n="1799,001,,,,3,31,31"></lb> |
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2850 δυσὶν ὀρθαῖς ἴσαι εἰσίν], καί ἐστιν ἡ ὑπὸ ΑΒΓ ἐλάττων <lb n="1799,001,,,,3,31,32"></lb> |
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2851 ὀρθῆς· λοιπὴ ἄρα ἡ ὑπὸ ΑΔΓ γωνία μείζων ὀρθῆς ἐστιν· <lb n="1799,001,,,,3,31,33"></lb> |
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2852 καί ἐστιν ἐν τῷ ΑΔΓ ἐλάττονι τοῦ ἡμικυκλίου τμήματι.</s> <lb n="1799,001,,,,3,31,34"></lb> |
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2853 <s>Λέγω, ὅτι καὶ ἡ μὲν τοῦ μείζονος τμήματος γωνία ἡ <lb n="1799,001,,,,3,31,35"></lb> |
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2854 περιεχομένη ὑπό [τε] τῆς ΑΒΓ περιφερείας καὶ τῆς ΑΓ <lb n="1799,001,,,,3,31,36"></lb> |
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2855 εὐθείας μείζων ἐστὶν ὀρθῆς, ἡ δὲ τοῦ ἐλάττονος τμήματος <lb n="1799,001,,,,3,31,37"></lb> |
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2856 γωνία ἡ περιεχομένη ὑπό [τε] τῆς ΑΔ[Γ] περιφερείας <lb n="1799,001,,,,3,31,38"></lb> |
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2857 καὶ τῆς ΑΓ εὐθείας ἐλάττων ἐστὶν ὀρθῆς.</s> <s>καί ἐστιν αὐτό-<lb n="1799,001,,,,3,31,39"></lb> |
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2858 θεν φανερόν.</s> <s>ἐπεὶ γὰρ ἡ ὑπὸ τῶν ΒΑ, ΑΓ εὐθειῶν ὀρθή <lb n="1799,001,,,,3,31,40"></lb> |
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2859 ἐστιν, ἡ ἄρα ὑπὸ τῆς ΑΒΓ περιφερείας καὶ τῆς ΑΓ <lb n="1799,001,,,,3,31,41"></lb> |
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2860 εὐθείας περιεχομένη μείζων ἐστὶν ὀρθῆς.</s> <s>πάλιν, ἐπεὶ ἡ <lb n="1799,001,,,,3,31,42"></lb> |
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2861 ὑπὸ τῶν ΑΓ, ΑΖ εὐθειῶν ὀρθή ἐστιν, ἡ ἄρα ὑπὸ τῆς ΓΑ <pb></pb> <lb n="1799,001,,,,3,31,43"></lb> |
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2862 εὐθείας καὶ τῆς ΑΔ[Γ] περιφερείας περιεχομένη ἐλάτ-<lb n="1799,001,,,,3,31,44"></lb> |
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2863 των ἐστὶν ὀρθῆς.</s> <lb n="1799,001,,,,3,31,45"></lb> |
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2864 <s>Ἐν κύκλῳ ἄρα ἡ μὲν ἐν τῷ ἡμικυκλίῳ γωνία ὀρθή ἐστιν, <lb n="1799,001,,,,3,31,46"></lb> |
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2865 ἡ δὲ ἐν τῷ μείζονι τμήματι ἐλάττων ὀρθῆς, ἡ δὲ ἐν τῷ <lb n="1799,001,,,,3,31,47"></lb> |
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2866 ἐλάττονι [τμήματι] μείζων ὀρθῆς, καὶ ἔτι ἡ μὲν τοῦ μεί-<lb n="1799,001,,,,3,31,48"></lb> |
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2867 ζονος τμήματος [γωνία] μείζων [ἐστὶν] ὀρθῆς, ἡ δὲ τοῦ <lb n="1799,001,,,,3,31,49"></lb> |
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2868 ἐλάττονος τμήματος [γωνία] ἐλάττων ὀρθῆς· ὅπερ ἔδει <lb n="1799,001,,,,3,31,50"></lb> |
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2869 δεῖξαι.</s> <lb n="1799,001,,,,3,31,51"></lb> |
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2870 <s>[Πόρισμα <lb n="1799,001,,,,3,31,52"></lb> |
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2871 Ἐκ δὴ τούτου φανερόν, ὅτι ἐὰν [ἡ] μία γωνία τριγώνου <lb n="1799,001,,,,3,31,53"></lb> |
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2872 ταῖς δυσὶν ἴση ᾖ, ὀρθή ἐστιν ἡ γωνία διὰ τὸ καὶ τὴν ἐκεί-<lb n="1799,001,,,,3,31,54"></lb> |
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2873 νης ἐκτὸς ταῖς αὐταῖς ἴσην εἶναι· ἐὰν δὲ αἱ ἐφεξῆς ἴσαι <lb n="1799,001,,,,3,31,55"></lb> |
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2874 ὦσιν, ὀρθαί εἰσιν.] <lb n="1799,001,,,,3,31,56"></lb> |
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2875 Ἐὰν κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τῆς ἁφῆς εἰς <lb n="1799,001,,,,3,32,1"></lb> |
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2876 τὸν κύκλον διαχθῇ τις εὐθεῖα τέμνουσα τὸν κύκλον, ἃς <lb n="1799,001,,,,3,32,2"></lb> |
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2877 ποιεῖ γωνίας πρὸς τῇ ἐφαπτομένῃ, ἴσαι <lb n="1799,001,,,,3,32,3"></lb> |
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2878 ἔσονται ταῖς ἐν τοῖς ἐναλλὰξ τοῦ κύκλου <lb n="1799,001,,,,3,32,4"></lb> |
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2879 τμήμασι γωνίαις.</s> <lb n="1799,001,,,,3,32,5"></lb> |
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2880 <s>Κύκλου γὰρ τοῦ ΑΒΓΔ ἐφαπτέσθω <lb n="1799,001,,,,3,32,6"></lb> |
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2881 τις εὐθεῖα ἡ ΕΖ κατὰ τὸ Β σημεῖον, <lb n="1799,001,,,,3,32,7"></lb> |
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2882 καὶ ἀπὸ τοῦ Β σημείου διήχθω τις <pb></pb> <lb n="1799,001,,,,3,32,8"></lb> |
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2883 εὐθεῖα εἰς τὸν ΑΒΓΔ κύκλον τέμνουσα αὐτὸν ἡ ΒΔ.</s> <s>λέγω, <lb n="1799,001,,,,3,32,9"></lb> |
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2884 ὅτι ἃς ποιεῖ γωνίας ἡ ΒΔ μετὰ τῆς ΕΖ ἐφαπτομένης, <lb n="1799,001,,,,3,32,10"></lb> |
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2885 ἴσαι ἔσονται ταῖς ἐν τοῖς ἐναλλὰξ τμήμασι τοῦ κύκλου <lb n="1799,001,,,,3,32,11"></lb> |
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2886 γωνίαις, τουτέστιν, ὅτι ἡ μὲν ὑπὸ ΖΒΔ γωνία ἴση ἐστὶ τῇ <lb n="1799,001,,,,3,32,12"></lb> |
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2887 ἐν τῷ ΒΑΔ τμήματι συνισταμένῃ γωνίᾳ, ἡ δὲ ὑπὸ ΕΒΔ <lb n="1799,001,,,,3,32,13"></lb> |
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2888 γωνία ἴση ἐστὶ τῇ ἐν τῷ ΔΓΒ τμήματι συνισταμένῃ <lb n="1799,001,,,,3,32,14"></lb> |
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2889 γωνίᾳ.</s> <lb n="1799,001,,,,3,32,15"></lb> |
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2890 <s>Ἤχθω γὰρ ἀπὸ τοῦ Β τῇ ΕΖ πρὸς ὀρθὰς ἡ ΒΑ, καὶ <lb n="1799,001,,,,3,32,16"></lb> |
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2891 εἰλήφθω ἐπὶ τῆς ΒΔ περιφερείας τυχὸν σημεῖον τὸ Γ, καὶ <lb n="1799,001,,,,3,32,17"></lb> |
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2892 ἐπεζεύχθωσαν αἱ ΑΔ, ΔΓ, ΓΒ.</s> <lb n="1799,001,,,,3,32,18"></lb> |
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2893 <s>Καὶ ἐπεὶ κύκλου τοῦ ΑΒΓΔ ἐφάπτεταί τις εὐθεῖα ἡ <lb n="1799,001,,,,3,32,19"></lb> |
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2894 ΕΖ κατὰ τὸ Β, καὶ ἀπὸ τῆς ἁφῆς ἦκται τῇ ἐφαπτομένῃ <lb n="1799,001,,,,3,32,20"></lb> |
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2895 πρὸς ὀρθὰς ἡ ΒΑ, ἐπὶ τῆς ΒΑ ἄρα τὸ κέντρον ἐστὶ τοῦ <lb n="1799,001,,,,3,32,21"></lb> |
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2896 ΑΒΓΔ κύκλου.</s> <s>ἡ ΒΑ ἄρα διάμετρός ἐστι τοῦ ΑΒΓΔ <lb n="1799,001,,,,3,32,22"></lb> |
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2897 κύκλου· ἡ ἄρα ὑπὸ ΑΔΒ γωνία ἐν ἡμικυκλίῳ οὖσα ὀρθή <lb n="1799,001,,,,3,32,23"></lb> |
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2898 ἐστιν.</s> <s>λοιπαὶ ἄρα αἱ ὑπὸ ΒΑΔ, ΑΒΔ μιᾷ ὀρθῇ ἴσαι εἰσίν.</s> <lb n="1799,001,,,,3,32,24"></lb> |
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2899 <s>ἐστὶ δὲ καὶ ἡ ὑπὸ ΑΒΖ ὀρθή· ἡ ἄρα ὑπὸ ΑΒΖ ἴση ἐστὶ <lb n="1799,001,,,,3,32,25"></lb> |
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2900 ταῖς ὑπὸ ΒΑΔ, ΑΒΔ.</s> <s>κοινὴ ἀφῃρήσθω ἡ ὑπὸ ΑΒΔ· <lb n="1799,001,,,,3,32,26"></lb> |
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2901 λοιπὴ ἄρα ἡ ὑπὸ ΔΒΖ γωνία ἴση ἐστὶ τῇ ἐν τῷ ἐναλλὰξ <lb n="1799,001,,,,3,32,27"></lb> |
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2902 τμήματι τοῦ κύκλου γωνίᾳ τῇ ὑπὸ ΒΑΔ.</s> <s>καὶ ἐπεὶ ἐν <lb n="1799,001,,,,3,32,28"></lb> |
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2903 κύκλῳ τετράπλευρόν ἐστι τὸ ΑΒΓΔ, αἱ ἀπεναντίον αὐτοῦ <pb></pb> <lb n="1799,001,,,,3,32,29"></lb> |
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2904 γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <s>εἰσὶ δὲ καὶ αἱ ὑπὸ ΔΒΖ, <lb n="1799,001,,,,3,32,30"></lb> |
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2905 ΔΒΕ δυσὶν ὀρθαῖς ἴσαι· αἱ ἄρα ὑπὸ ΔΒΖ, ΔΒΕ ταῖς <lb n="1799,001,,,,3,32,31"></lb> |
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2906 ὑπὸ ΒΑΔ, ΒΓΔ ἴσαι εἰσίν, ὧν ἡ ὑπὸ ΒΑΔ τῇ ὑπὸ ΔΒΖ <lb n="1799,001,,,,3,32,32"></lb> |
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2907 ἐδείχθη ἴση· λοιπὴ ἄρα ἡ ὑπὸ ΔΒΕ τῇ ἐν τῷ ἐναλλὰξ τοῦ <lb n="1799,001,,,,3,32,33"></lb> |
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2908 κύκλου τμήματι τῷ ΔΓΒ τῇ ὑπὸ ΔΓΒ γωνίᾳ ἐστὶν ἴση.</s> <lb n="1799,001,,,,3,32,34"></lb> |
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2909 <s>Ἐὰν ἄρα κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τῆς <lb n="1799,001,,,,3,32,35"></lb> |
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2910 ἁφῆς εἰς τὸν κύκλον διαχθῇ τις εὐθεῖα τέμνουσα τὸν κύ-<lb n="1799,001,,,,3,32,36"></lb> |
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2911 κλον, ἃς ποιεῖ γωνίας πρὸς τῇ ἐφαπτομένῃ, ἴσαι ἔσονται <lb n="1799,001,,,,3,32,37"></lb> |
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2912 ταῖς ἐν τοῖς ἐναλλὰξ τοῦ κύκλου τμήμασι γωνίαις· ὅπερ <lb n="1799,001,,,,3,32,38"></lb> |
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2913 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,32,39"></lb> |
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2914 <s>Ἐπὶ τῆς δοθείσης εὐθείας γράψαι τμῆμα κύκλου δεχό-<lb n="1799,001,,,,3,33,1"></lb> |
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2915 μενον γωνίαν ἴσην τῇ δοθείσῃ γωνίᾳ εὐθυγράμμῳ.</s> <lb n="1799,001,,,,3,33,2"></lb> |
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2916 <s>Ἔστω ἡ δοθεῖσα εὐθεῖα ἡ ΑΒ, ἡ δὲ δοθεῖσα γωνία <lb n="1799,001,,,,3,33,3"></lb> |
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2917 εὐθύγραμμος ἡ πρὸς τῷ Γ· δεῖ δὴ ἐπὶ τῆς δοθείσης εὐ-<lb n="1799,001,,,,3,33,4"></lb> |
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2918 θείας τῆς ΑΒ γράψαι τμῆμα κύκλου δεχόμενον γωνίαν <lb n="1799,001,,,,3,33,5"></lb> |
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2919 ἴσην τῇ πρὸς τῷ Γ.</s> <lb n="1799,001,,,,3,33,6"></lb> |
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2920 <s>Ἡ δὴ πρὸς τῷ Γ [γωνία] ἤτοι ὀξεῖά ἐστιν ἢ ὀρθὴ ἢ <lb n="1799,001,,,,3,33,7"></lb> |
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2921 ἀμβλεῖα· ἔστω πρότερον ὀξεῖα, καὶ ὡς ἐπὶ τῆς πρώτης <lb n="1799,001,,,,3,33,8"></lb> |
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2922 καταγραφῆς συνεστάτω πρὸς τῇ ΑΒ εὐθείᾳ καὶ τῷ <pb></pb> <lb n="1799,001,,,,3,33,9"></lb> |
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2923 Α σημείῳ τῇ πρὸς τῷ Γ γωνίᾳ ἴση ἡ ὑπὸ ΒΑΔ· ὀξεῖα <lb n="1799,001,,,,3,33,10"></lb> |
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2924 ἄρα ἐστὶ καὶ ἡ ὑπὸ ΒΑΔ.</s> <s>ἤχθω τῇ ΔΑ πρὸς ὀρθὰς ἡ <lb n="1799,001,,,,3,33,11"></lb> |
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2925 ΑΕ, καὶ τετμήσθω ἡ ΑΒ δίχα κατὰ <lb n="1799,001,,,,3,33,12"></lb> |
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2926 τὸ Ζ, καὶ ἤχθω ἀπὸ τοῦ Ζ σημείου τῇ <lb n="1799,001,,,,3,33,13"></lb> |
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2927 ΑΒ πρὸς ὀρθὰς ἡ ΖΗ, καὶ ἐπεζεύχθω <lb n="1799,001,,,,3,33,14"></lb> |
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2928 ἡ ΗΒ.</s> <lb n="1799,001,,,,3,33,15"></lb> |
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2929 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΖ τῇ ΖΒ, <lb n="1799,001,,,,3,33,16"></lb> |
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2930 κοινὴ δὲ ἡ ΖΗ, δύο δὴ αἱ ΑΖ, ΖΗ δύο <lb n="1799,001,,,,3,33,17"></lb> |
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2931 ταῖς ΒΖ, ΖΗ ἴσαι εἰσίν· καὶ γωνία ἡ <lb n="1799,001,,,,3,33,18"></lb> |
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2932 ὑπὸ ΑΖΗ [γωνίᾳ] τῇ ὑπὸ ΒΖΗ ἴση· <lb n="1799,001,,,,3,33,19"></lb> |
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2933 βάσις ἄρα ἡ ΑΗ βάσει τῇ ΒΗ ἴση <lb n="1799,001,,,,3,33,20"></lb> |
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2934 ἐστίν.</s> <s>ὁ ἄρα κέντρῳ μὲν τῷ Η διαστή-<lb n="1799,001,,,,3,33,21"></lb> |
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2935 ματι δὲ τῷ ΗΑ κύκλος γραφόμενος ἥξει καὶ διὰ τοῦ Β.</s> <lb n="1799,001,,,,3,33,22"></lb> |
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2936 <s>γεγράφθω καὶ ἔστω ὁ ΑΒΕ, καὶ ἐπεζεύχθω ἡ ΕΒ.</s> <s>ἐπεὶ <lb n="1799,001,,,,3,33,23"></lb> |
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2937 οὖν ἀπ' ἄκρας τῆς ΑΕ διαμέτρου ἀπὸ τοῦ Α τῇ ΑΕ πρὸς <lb n="1799,001,,,,3,33,24"></lb> |
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2938 ὀρθάς ἐστιν ἡ ΑΔ, ἡ ΑΔ ἄρα ἐφάπτεται τοῦ ΑΒΕ κύ-<lb n="1799,001,,,,3,33,25"></lb> |
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2939 κλου· ἐπεὶ οὖν κύκλου τοῦ ΑΒΕ ἐφάπτεταί τις εὐθεῖα ἡ <lb n="1799,001,,,,3,33,26"></lb> |
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2940 ΑΔ, καὶ ἀπὸ τῆς κατὰ τὸ Α ἁφῆς εἰς τὸν ΑΒΕ κύκλον <lb n="1799,001,,,,3,33,27"></lb> |
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2941 διῆκταί τις εὐθεῖα ἡ ΑΒ, ἡ ἄρα ὑπὸ ΔΑΒ γωνία ἴση ἐστὶ <lb n="1799,001,,,,3,33,28"></lb> |
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2942 τῇ ἐν τῷ ἐναλλὰξ τοῦ κύκλου τμήματι γωνίᾳ τῇ ὑπὸ ΑΕΒ.</s> <lb n="1799,001,,,,3,33,29"></lb> |
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2943 <s>ἀλλ' ἡ ὑπὸ ΔΑΒ τῇ πρὸς τῷ Γ ἐστιν ἴση· καὶ ἡ πρὸς τῷ <lb n="1799,001,,,,3,33,30"></lb> |
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2944 Γ ἄρα γωνία ἴση ἐστὶ τῇ ὑπὸ ΑΕΒ.</s> <lb n="1799,001,,,,3,33,31"></lb> |
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2945 <s>Ἐπὶ τῆς δοθείσης ἄρα εὐθείας τῆς ΑΒ τμῆμα κύκλου <pb></pb> <lb n="1799,001,,,,3,33,32"></lb> |
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2946 γέγραπται τὸ ΑΕΒ δεχόμενον γωνίαν τὴν ὑπὸ ΑΕΒ <lb n="1799,001,,,,3,33,33"></lb> |
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2947 ἴσην τῇ δοθείσῃ τῇ πρὸς τῷ Γ.</s> <lb n="1799,001,,,,3,33,34"></lb> |
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2948 <s>Ἀλλὰ δὴ ὀρθὴ ἔστω ἡ πρὸς τῷ Γ· καὶ δέον πάλιν ἔστω <lb n="1799,001,,,,3,33,35"></lb> |
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2949 ἐπὶ τῆς ΑΒ γράψαι τμῆμα κύκλου δεχόμενον γωνίαν ἴσην <lb n="1799,001,,,,3,33,36"></lb> |
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2950 τῇ πρὸς τῷ Γ ὀρθῇ [γωνίᾳ].</s> <s>συνεστάτω [πάλιν] τῇ πρὸς <lb n="1799,001,,,,3,33,37"></lb> |
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2951 τῷ Γ ὀρθῇ γωνίᾳ ἴση ἡ ὑπὸ ΒΑΔ, ὡς <lb n="1799,001,,,,3,33,38"></lb> |
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2952 ἔχει ἐπὶ τῆς δευτέρας καταγραφῆς, καὶ <lb n="1799,001,,,,3,33,39"></lb> |
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2953 τετμήσθω ἡ ΑΒ δίχα κατὰ τὸ Ζ, καὶ <lb n="1799,001,,,,3,33,40"></lb> |
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2954 κέντρῳ τῷ Ζ, διαστήματι δὲ ὁποτέρῳ <lb n="1799,001,,,,3,33,41"></lb> |
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2955 τῶν ΖΑ, ΖΒ, κύκλος γεγράφθω ὁ <lb n="1799,001,,,,3,33,42"></lb> |
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2956 ΑΕΒ.</s> <lb n="1799,001,,,,3,33,43"></lb> |
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2957 <s>Ἐφάπτεται ἄρα ἡ ΑΔ εὐθεῖα τοῦ <lb n="1799,001,,,,3,33,44"></lb> |
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2958 ΑΒΕ κύκλου διὰ τὸ ὀρθὴν εἶναι τὴν <lb n="1799,001,,,,3,33,45"></lb> |
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2959 πρὸς τῷ Α γωνίαν.</s> <s>καὶ ἴση ἐστὶν ἡ ὑπὸ <lb n="1799,001,,,,3,33,46"></lb> |
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2960 ΒΑΔ γωνία τῇ ἐν τῷ ΑΕΒ τμήματι· ὀρθὴ γὰρ καὶ αὐτὴ <lb n="1799,001,,,,3,33,47"></lb> |
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2961 ἐν ἡμικυκλίῳ οὖσα.</s> <s>ἀλλὰ καὶ ἡ ὑπὸ ΒΑΔ τῇ πρὸς τῷ Γ <lb n="1799,001,,,,3,33,48"></lb> |
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2962 ἴση ἐστίν.</s> <s>καὶ ἡ ἐν τῷ ΑΕΒ ἄρα ἴση ἐστὶ τῇ πρὸς τῷ Γ.</s> <lb n="1799,001,,,,3,33,49"></lb> |
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2963 <s>Γέγραπται ἄρα πάλιν ἐπὶ τῆς ΑΒ <lb n="1799,001,,,,3,33,50"></lb> |
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2964 τμῆμα κύκλου τὸ ΑΕΒ δεχόμενον γω-<lb n="1799,001,,,,3,33,51"></lb> |
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2965 νίαν ἴσην τῇ πρὸς τῷ Γ.</s> <lb n="1799,001,,,,3,33,52"></lb> |
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2966 <s>Ἀλλὰ δὴ ἡ πρὸς τῷ Γ ἀμβλεῖα ἔστω· <lb n="1799,001,,,,3,33,53"></lb> |
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2967 καὶ συνεστάτω αὐτῇ ἴση πρὸς τῇ ΑΒ <lb n="1799,001,,,,3,33,54"></lb> |
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2968 εὐθείᾳ καὶ τῷ Α σημείῳ ἡ ὑπὸ ΒΑΔ, <lb n="1799,001,,,,3,33,55"></lb> |
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2969 ὡς ἔχει ἐπὶ τῆς τρίτης καταγραφῆς, <lb n="1799,001,,,,3,33,56"></lb> |
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2970 καὶ τῇ ΑΔ πρὸς ὀρθὰς ἤχθω ἡ ΑΕ, <lb n="1799,001,,,,3,33,57"></lb> |
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2971 καὶ τετμήσθω πάλιν ἡ ΑΒ δίχα κατὰ <pb></pb> <lb n="1799,001,,,,3,33,58"></lb> |
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2972 τὸ Ζ, καὶ τῇ ΑΒ πρὸς ὀρθὰς ἤχθω ἡ ΖΗ, καὶ ἐπ-<lb n="1799,001,,,,3,33,59"></lb> |
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2973 εζεύχθω ἡ ΗΒ.</s> <lb n="1799,001,,,,3,33,60"></lb> |
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2974 <s>Καὶ ἐπεὶ πάλιν ἴση ἐστὶν ἡ ΑΖ τῇ ΖΒ, καὶ κοινὴ ἡ <lb n="1799,001,,,,3,33,61"></lb> |
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2975 ΖΗ, δύο δὴ αἱ ΑΖ, ΖΗ δύο ταῖς ΒΖ, ΖΗ ἴσαι εἰσίν· <lb n="1799,001,,,,3,33,62"></lb> |
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2976 καὶ γωνία ἡ ὑπὸ ΑΖΗ γωνίᾳ τῇ ὑπὸ ΒΖΗ ἴση· βάσις <lb n="1799,001,,,,3,33,63"></lb> |
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2977 ἄρα ἡ ΑΗ βάσει τῇ ΒΗ ἴση ἐστίν· ὁ ἄρα κέντρῳ μὲν τῷ <lb n="1799,001,,,,3,33,64"></lb> |
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2978 Η διαστήματι δὲ τῷ ΗΑ κύκλος γραφόμενος ἥξει καὶ <lb n="1799,001,,,,3,33,65"></lb> |
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2979 διὰ τοῦ Β.</s> <s>ἐρχέσθω ὡς ὁ ΑΕΒ.</s> <s>καὶ ἐπεὶ τῇ ΑΕ διαμέ-<lb n="1799,001,,,,3,33,66"></lb> |
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2980 τρῳ ἀπ' ἄκρας πρὸς ὀρθάς ἐστιν ἡ ΑΔ, ἡ ΑΔ ἄρα ἐφάπτε-<lb n="1799,001,,,,3,33,67"></lb> |
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2981 ται τοῦ ΑΕΒ κύκλου.</s> <s>καὶ ἀπὸ τῆς κατὰ τὸ Α ἐπαφῆς <lb n="1799,001,,,,3,33,68"></lb> |
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2982 διῆκται ἡ ΑΒ· ἡ ἄρα ὑπὸ ΒΑΔ γωνία ἴση ἐστὶ τῇ ἐν τῷ <lb n="1799,001,,,,3,33,69"></lb> |
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2983 ἐναλλὰξ τοῦ κύκλου τμήματι τῷ ΑΘΒ συνισταμένῃ γω-<lb n="1799,001,,,,3,33,70"></lb> |
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2984 νίᾳ.</s> <s>ἀλλ' ἡ ὑπὸ ΒΑΔ γωνία τῇ πρὸς τῷ Γ ἴση ἐστίν.</s> <s>καὶ <lb n="1799,001,,,,3,33,71"></lb> |
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2985 ἡ ἐν τῷ ΑΘΒ ἄρα τμήματι γωνία ἴση ἐστὶ τῇ πρὸς τῷ Γ.</s> <lb n="1799,001,,,,3,33,72"></lb> |
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2986 <s>Ἐπὶ τῆς ἄρα δοθείσης εὐθείας τῆς ΑΒ γέγραπται <lb n="1799,001,,,,3,33,73"></lb> |
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2987 τμῆμα κύκλου τὸ ΑΘΒ δεχόμενον γωνίαν ἴσην τῇ πρὸς <lb n="1799,001,,,,3,33,74"></lb> |
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2988 τῷ Γ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,3,33,75"></lb> |
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2989 <s>Ἀπὸ τοῦ δοθέντος κύκλου τμῆμα ἀφελεῖν δεχόμενον <lb n="1799,001,,,,3,34,1"></lb> |
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2990 γωνίαν ἴσην τῇ δοθείσῃ γωνίᾳ εὐθυγράμμῳ.</s> <lb n="1799,001,,,,3,34,2"></lb> |
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2991 <s>Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓ, ἡ δὲ δοθεῖσα γωνία <lb n="1799,001,,,,3,34,3"></lb> |
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2992 εὐθύγραμμος ἡ πρὸς τῷ Δ· δεῖ δὴ ἀπὸ τοῦ ΑΒΓ κύκλου <pb></pb> <lb n="1799,001,,,,3,34,4"></lb> |
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2993 τμῆμα ἀφελεῖν δεχόμενον γωνίαν ἴσην τῇ δοθείσῃ γωνίᾳ <lb n="1799,001,,,,3,34,5"></lb> |
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2994 εὐθυγράμμῳ τῇ πρὸς τῷ Δ.</s> <lb n="1799,001,,,,3,34,6"></lb> |
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2995 <s>Ἤχθω τοῦ ΑΒΓ ἐφαπτομένη ἡ ΕΖ κατὰ τὸ Β ση-<lb n="1799,001,,,,3,34,7"></lb> |
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2996 μεῖον, καὶ συνεστάτω πρὸς τῇ ΖΒ εὐθείᾳ καὶ τῷ πρὸς <lb n="1799,001,,,,3,34,8"></lb> |
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2997 αὐτῇ σημείῳ τῷ Β τῇ πρὸς τῷ Δ <lb n="1799,001,,,,3,34,9"></lb> |
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2998 γωνίᾳ ἴση ἡ ὑπὸ ΖΒΓ.</s> <lb n="1799,001,,,,3,34,10"></lb> |
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2999 <s>Ἐπεὶ οὖν κύκλου τοῦ ΑΒΓ ἐφ-<lb n="1799,001,,,,3,34,11"></lb> |
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3000 άπτεταί τις εὐθεῖα ἡ ΕΖ, καὶ <lb n="1799,001,,,,3,34,12"></lb> |
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3001 ἀπὸ τῆς κατὰ τὸ Β ἐπαφῆς δι-<lb n="1799,001,,,,3,34,13"></lb> |
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3002 ῆκται ἡ ΒΓ, ἡ ὑπὸ ΖΒΓ ἄρα <lb n="1799,001,,,,3,34,14"></lb> |
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3003 γωνία ἴση ἐστὶ τῇ ἐν τῷ ΒΑΓ <lb n="1799,001,,,,3,34,15"></lb> |
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3004 ἐναλλὰξ τμήματι συνισταμένῃ γω-<lb n="1799,001,,,,3,34,16"></lb> |
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3005 νίᾳ.</s> <s>ἀλλ' ἡ ὑπὸ ΖΒΓ τῇ πρὸς <lb n="1799,001,,,,3,34,17"></lb> |
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3006 τῷ Δ ἐστιν ἴση· καὶ ἡ ἐν τῷ ΒΑΓ ἄρα τμήματι ἴση <lb n="1799,001,,,,3,34,18"></lb> |
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3007 ἐστὶ τῇ πρὸς τῷ Δ [γωνίᾳ].</s> <lb n="1799,001,,,,3,34,19"></lb> |
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3008 <s>Ἀπὸ τοῦ δοθέντος ἄρα κύκλου τοῦ ΑΒΓ τμῆμα ἀφῄρη-<lb n="1799,001,,,,3,34,20"></lb> |
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3009 ται τὸ ΒΑΓ δεχόμενον γωνίαν ἴσην τῇ δοθείσῃ γωνίᾳ <lb n="1799,001,,,,3,34,21"></lb> |
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3010 εὐθυγράμμῳ τῇ πρὸς τῷ Δ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,3,34,22"></lb> |
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3011 <s>Ἐὰν ἐν κύκλῳ δύο εὐθεῖαι τέμνωσιν ἀλλήλας, τὸ ὑπὸ <lb n="1799,001,,,,3,35,1"></lb> |
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3012 τῶν τῆς μιᾶς τμημάτων περιεχόμενον ὀρθογώνιον ἴσον <lb n="1799,001,,,,3,35,2"></lb> |
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3013 ἐστὶ τῷ ὑπὸ τῶν τῆς ἑτέρας τμημάτων περιεχομένῳ <lb n="1799,001,,,,3,35,3"></lb> |
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3014 ὀρθογωνίῳ.</s> <s><pb></pb> <lb n="1799,001,,,,3,35,4"></lb> |
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3015 Ἐν γὰρ κύκλῳ τῷ ΑΒΓΔ δύο εὐθεῖαι αἱ ΑΓ, ΒΔ <lb n="1799,001,,,,3,35,5"></lb> |
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3016 τεμνέτωσαν ἀλλήλας κατὰ τὸ Ε σημεῖον· λέγω, ὅτι τὸ <lb n="1799,001,,,,3,35,6"></lb> |
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3017 ὑπὸ τῶν ΑΕ, ΕΓ περιεχόμενον ὀρθο- <lb n="1799,001,,,,3,35,7"></lb> |
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3018 γώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν ΔΕ, ΕΒ <lb n="1799,001,,,,3,35,8"></lb> |
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3019 περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,3,35,9"></lb> |
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3020 <s>Εἰ μὲν οὖν αἱ ΑΓ, ΒΔ διὰ τοῦ κέν-<lb n="1799,001,,,,3,35,10"></lb> |
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3021 τρου εἰσὶν ὥστε τὸ Ε κέντρον εἶναι τοῦ <lb n="1799,001,,,,3,35,11"></lb> |
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3022 ΑΒΓΔ κύκλου, φανερόν, ὅτι ἴσων <lb n="1799,001,,,,3,35,12"></lb> |
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3023 οὐσῶν τῶν ΑΕ, ΕΓ, ΔΕ, ΕΒ καὶ τὸ ὑπὸ τῶν ΑΕ, ΕΓ <lb n="1799,001,,,,3,35,13"></lb> |
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3024 περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν ΔΕ, <lb n="1799,001,,,,3,35,14"></lb> |
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3025 ΕΒ περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,3,35,15"></lb> |
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3026 <s>Μὴ ἔστωσαν δὴ αἱ ΑΓ, ΔΒ διὰ τοῦ κέντρου, καὶ εἰλή-<lb n="1799,001,,,,3,35,16"></lb> |
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3027 φθω τὸ κέντρον τοῦ ΑΒΓΔ, καὶ ἔστω τὸ Ζ, καὶ ἀπὸ τοῦ <lb n="1799,001,,,,3,35,17"></lb> |
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3028 Ζ ἐπὶ τὰς ΑΓ, ΔΒ εὐθείας κάθετοι ἤχθωσαν αἱ ΖΗ, <lb n="1799,001,,,,3,35,18"></lb> |
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3029 ΖΘ, καὶ ἐπεζεύχθωσαν αἱ ΖΒ, ΖΓ, <lb n="1799,001,,,,3,35,19"></lb> |
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3030 ΖΕ.</s> <lb n="1799,001,,,,3,35,20"></lb> |
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3031 <s>Καὶ ἐπεὶ εὐθει̂ά τις διὰ τοῦ κέντρου <lb n="1799,001,,,,3,35,21"></lb> |
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3032 ἡ ΗΖ εὐθεῖάν τινα μὴ διὰ τοῦ κέντρου <lb n="1799,001,,,,3,35,22"></lb> |
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3033 τὴν ΑΓ πρὸς ὀρθὰς τέμνει, καὶ δίχα <lb n="1799,001,,,,3,35,23"></lb> |
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3034 αὐτὴν τέμνει· ἴση ἄρα ἡ ΑΗ τῇ ΗΓ.</s> <lb n="1799,001,,,,3,35,24"></lb> |
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3035 <s>ἐπεὶ οὖν εὐθεῖα ἡ ΑΓ τέτμηται εἰς μὲν <lb n="1799,001,,,,3,35,25"></lb> |
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3036 ἴσα κατὰ τὸ Η, εἰς δὲ ἄνισα κατὰ <lb n="1799,001,,,,3,35,26"></lb> |
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3037 τὸ Ε, τὸ ἄρα ὑπὸ τῶν ΑΕ, ΕΓ περιεχόμενον ὀρθογώνιον <lb n="1799,001,,,,3,35,27"></lb> |
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3038 μετὰ τοῦ ἀπὸ τῆς ΕΗ τετραγώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,3,35,28"></lb> |
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3039 ΗΓ· [κοινὸν] προσκείσθω τὸ ἀπὸ τῆς ΗΖ· τὸ ἄρα ὑπὸ <pb></pb> <lb n="1799,001,,,,3,35,29"></lb> |
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3040 τῶν ΑΕ, ΕΓ μετὰ τῶν ἀπὸ τῶν ΗΕ, ΗΖ ἴσον ἐστὶ τοῖς <lb n="1799,001,,,,3,35,30"></lb> |
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3041 ἀπὸ τῶν ΓΗ, ΗΖ.</s> <s>ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΕΗ, ΗΖ ἴσον <lb n="1799,001,,,,3,35,31"></lb> |
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3042 ἐστὶ τὸ ἀπὸ τῆς ΖΕ, τοῖς δὲ ἀπὸ τῶν ΓΗ, ΗΖ ἴσον <lb n="1799,001,,,,3,35,32"></lb> |
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3043 ἐστὶ τὸ ἀπὸ τῆς ΖΓ· τὸ ἄρα ὑπὸ τῶν ΑΕ, ΕΓ μετὰ τοῦ <lb n="1799,001,,,,3,35,33"></lb> |
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3044 ἀπὸ τῆς ΖΕ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΖΓ.</s> <s>ἴση δὲ ἡ ΖΓ τῇ <lb n="1799,001,,,,3,35,34"></lb> |
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3045 ΖΒ· τὸ ἄρα ὑπὸ τῶν ΑΕ, ΕΓ μετὰ τοῦ ἀπὸ τῆς ΕΖ ἴσον <lb n="1799,001,,,,3,35,35"></lb> |
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3046 ἐστὶ τῷ ἀπὸ τῆς ΖΒ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ ὑπὸ τῶν ΔΕ, <lb n="1799,001,,,,3,35,36"></lb> |
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3047 ΕΒ μετὰ τοῦ ἀπὸ τῆς ΖΕ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΖΒ.</s> <lb n="1799,001,,,,3,35,37"></lb> |
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3048 <s>ἐδείχθη δὲ καὶ τὸ ὑπὸ τῶν ΑΕ, ΕΓ μετὰ τοῦ ἀπὸ τῆς <lb n="1799,001,,,,3,35,38"></lb> |
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3049 ΖΕ ἴσον τῷ ἀπὸ τῆς ΖΒ· τὸ ἄρα ὑπὸ τῶν ΑΕ, ΕΓ μετὰ <lb n="1799,001,,,,3,35,39"></lb> |
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3050 τοῦ ἀπὸ τῆς ΖΕ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΔΕ, ΕΒ μετὰ τοῦ <lb n="1799,001,,,,3,35,40"></lb> |
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3051 ἀπὸ τῆς ΖΕ.</s> <s>κοινὸν ἀφῃρήσθω τὸ ἀπὸ τῆς ΖΕ· λοιπὸν <lb n="1799,001,,,,3,35,41"></lb> |
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3052 ἄρα τὸ ὑπὸ τῶν ΑΕ, ΕΓ περιεχόμενον ὀρθογώνιον ἴσον <lb n="1799,001,,,,3,35,42"></lb> |
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3053 ἐστὶ τῷ ὑπὸ τῶν ΔΕ, ΕΒ περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,3,35,43"></lb> |
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3054 <s>Ἐὰν ἄρα ἐν κύκλῳ εὐθεῖαι δύο τέμνωσιν ἀλλήλας, τὸ <lb n="1799,001,,,,3,35,44"></lb> |
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3055 ὑπὸ τῶν τῆς μιᾶς τμημάτων περιεχόμενον ὀρθογώνιον <lb n="1799,001,,,,3,35,45"></lb> |
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3056 ἴσον ἐστὶ τῷ ὑπὸ τῶν τῆς ἑτέρας τμημάτων περιεχομένῳ <lb n="1799,001,,,,3,35,46"></lb> |
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3057 ὀρθογωνίῳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,35,47"></lb> |
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3058 <s>Ἐὰν κύκλου ληφθῇ τι σημεῖον ἐκτός, καὶ ἀπ' αὐτοῦ <lb n="1799,001,,,,3,36,1"></lb> |
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3059 πρὸς τὸν κύκλον προσπίπτωσι δύο εὐθεῖαι, καὶ ἡ μὲν <lb n="1799,001,,,,3,36,2"></lb> |
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3060 αὐτῶν τέμνῃ τὸν κύκλον, ἡ δὲ ἐφάπτηται, ἔσται τὸ ὑπὸ <lb n="1799,001,,,,3,36,3"></lb> |
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3061 ὅλης τῆς τεμνούσης καὶ τῆς ἐκτὸς ἀπολαμβανομένης <pb></pb> <lb n="1799,001,,,,3,36,4"></lb> |
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3062 μεταξὺ τοῦ τε σημείου καὶ τῆς κυρτῆς περιφερείας ἴσον <lb n="1799,001,,,,3,36,5"></lb> |
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3063 τῷ ἀπὸ τῆς ἐφαπτομένης τετραγώνῳ.</s> <lb n="1799,001,,,,3,36,6"></lb> |
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3064 <s>Κύκλου γὰρ τοῦ ΑΒΓ εἰλήφθω τι σημεῖον ἐκτὸς τὸ Δ, <lb n="1799,001,,,,3,36,7"></lb> |
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3065 καὶ ἀπὸ τοῦ Δ πρὸς τὸν ΑΒΓ κύκλον προσπιπτέτωσαν <lb n="1799,001,,,,3,36,8"></lb> |
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3066 δύο εὐθεῖαι αἱ ΔΓ[Α], ΔΒ· καὶ ἡ μὲν ΔΓΑ τεμνέτω τὸν <lb n="1799,001,,,,3,36,9"></lb> |
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3067 ΑΒΓ κύκλον, ἡ δὲ ΒΔ ἐφαπτέσθω· λέγω, ὅτι τὸ ὑπὸ <lb n="1799,001,,,,3,36,10"></lb> |
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3068 τῶν ΑΔ, ΔΓ περιεχόμενον ὀρθογώνιον <lb n="1799,001,,,,3,36,11"></lb> |
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3069 ἴσον ἐστὶ τῷ ἀπὸ τῆς ΔΒ τετραγώνῳ.</s> <lb n="1799,001,,,,3,36,12"></lb> |
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3070 <s>Ἡ ἄρα [Δ]ΓΑ ἤτοι διὰ τοῦ κέντρου ἐστὶν <lb n="1799,001,,,,3,36,13"></lb> |
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3071 ἢ οὔ.</s> <s>ἔστω πρότερον διὰ τοῦ κέντρου, καὶ <lb n="1799,001,,,,3,36,14"></lb> |
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3072 ἔστω τὸ Ζ κέντρον τοῦ ΑΒΓ κύκλου, καὶ <lb n="1799,001,,,,3,36,15"></lb> |
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3073 ἐπεζεύχθω ἡ ΖΒ· ὀρθὴ ἄρα ἐστὶν ἡ ὑπὸ <lb n="1799,001,,,,3,36,16"></lb> |
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3074 ΖΒΔ.</s> <s>καὶ ἐπεὶ εὐθεῖα ἡ ΑΓ δίχα τέτμηται <lb n="1799,001,,,,3,36,17"></lb> |
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3075 κατὰ τὸ Ζ, πρόσκειται δὲ αὐτῇ ἡ ΓΔ, τὸ ἄρα ὑπὸ τῶν ΑΔ, <lb n="1799,001,,,,3,36,18"></lb> |
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3076 ΔΓ μετὰ τοῦ ἀπὸ τῆς ΖΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΖΔ.</s> <s>ἴση δὲ ἡ <lb n="1799,001,,,,3,36,19"></lb> |
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3077 ΖΓ τῇ ΖΒ· τὸ ἄρα ὑπὸ τῶν ΑΔ, ΔΓ μετὰ τοῦ ἀπὸ τῆς <lb n="1799,001,,,,3,36,20"></lb> |
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3078 ΖΒ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΖΔ.</s> <s>τῷ δὲ ἀπὸ τῆς ΖΔ ἴσα <lb n="1799,001,,,,3,36,21"></lb> |
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3079 ἐστὶ τὰ ἀπὸ τῶν ΖΒ, ΒΔ· τὸ ἄρα ὑπὸ τῶν ΑΔ, ΔΓ μετὰ <lb n="1799,001,,,,3,36,22"></lb> |
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3080 τοῦ ἀπὸ τῆς ΖΒ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΖΒ, ΒΔ.</s> <s>κοινὸν <lb n="1799,001,,,,3,36,23"></lb> |
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3081 ἀφῃρήσθω τὸ ἀπὸ τῆς ΖΒ· λοιπὸν ἄρα τὸ ὑπὸ τῶν ΑΔ, <lb n="1799,001,,,,3,36,24"></lb> |
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3082 ΔΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΔΒ ἐφαπτο- <lb n="1799,001,,,,3,36,25"></lb> |
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3083 μένης.</s> <lb n="1799,001,,,,3,36,26"></lb> |
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3084 <s>ἀλλὰ δὴ ἡ ΔΓΑ μὴ ἔστω διὰ τοῦ <lb n="1799,001,,,,3,36,27"></lb> |
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3085 κέντρου τοῦ ΑΒΓ κύκλου, καὶ εἰλήφθω <lb n="1799,001,,,,3,36,28"></lb> |
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3086 τὸ κέντρον τὸ Ε, καὶ ἀπὸ τοῦ Ε ἐπὶ <pb></pb> <lb n="1799,001,,,,3,36,29"></lb> |
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3087 τὴν ΑΓ κάθετος ἤχθω ἡ ΕΖ, καὶ ἐπεζεύχθωσαν αἱ ΕΒ, <lb n="1799,001,,,,3,36,30"></lb> |
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3088 ΕΓ, ΕΔ· ὀρθὴ ἄρα ἐστὶν ἡ ὑπὸ ΕΒΔ.</s> <s>καὶ ἐπεὶ εὐθεῖά <lb n="1799,001,,,,3,36,31"></lb> |
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3089 τις διὰ τοῦ κέντρου ἡ ΕΖ εὐθεῖάν τινα μὴ διὰ τοῦ κέν-<lb n="1799,001,,,,3,36,32"></lb> |
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3090 τρου τὴν ΑΓ πρὸς ὀρθὰς τέμνει, καὶ δίχα αὐτὴν τέμνει· <lb n="1799,001,,,,3,36,33"></lb> |
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3091 ἡ ΑΖ ἄρα τῇ ΖΓ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ εὐθεῖα ἡ ΑΓ τέτμη-<lb n="1799,001,,,,3,36,34"></lb> |
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3092 ται δίχα κατὰ τὸ Ζ σημεῖον, πρόσκειται δὲ αὐτῇ ἡ ΓΔ, τὸ <lb n="1799,001,,,,3,36,35"></lb> |
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3093 ἄρα ὑπὸ τῶν ΑΔ, ΔΓ μετὰ τοῦ ἀπὸ τῆς ΖΓ ἴσον ἐστὶ τῷ <lb n="1799,001,,,,3,36,36"></lb> |
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3094 ἀπὸ τῆς ΖΔ.</s> <s>κοινὸν προσκείσθω τὸ ἀπὸ τῆς ΖΕ· τὸ ἄρα <lb n="1799,001,,,,3,36,37"></lb> |
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3095 ὑπὸ τῶν ΑΔ, ΔΓ μετὰ τῶν ἀπὸ τῶν ΓΖ, ΖΕ ἴσον ἐστὶ <lb n="1799,001,,,,3,36,38"></lb> |
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3096 τοῖς ἀπὸ τῶν ΖΔ, ΖΕ.</s> <s>τοῖς δὲ ἀπὸ τῶν ΓΖ, ΖΕ ἴσον <lb n="1799,001,,,,3,36,39"></lb> |
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3097 ἐστὶ τὸ ἀπὸ τῆς ΕΓ· ὀρθὴ γὰρ [ἐστιν] ἡ ὑπὸ ΕΖΓ [γω-<lb n="1799,001,,,,3,36,40"></lb> |
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3098 νία]· τοῖς δὲ ἀπὸ τῶν ΔΖ, ΖΕ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΕΔ· <lb n="1799,001,,,,3,36,41"></lb> |
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3099 τὸ ἄρα ὑπὸ τῶν ΑΔ, ΔΓ μετὰ τοῦ ἀπὸ τῆς ΕΓ ἴσον ἐστὶ <lb n="1799,001,,,,3,36,42"></lb> |
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3100 τῷ ἀπὸ τῆς ΕΔ.</s> <s>ἴση δὲ ἡ ΕΓ τῇ ΕΒ· τὸ ἄρα ὑπὸ τῶν <lb n="1799,001,,,,3,36,43"></lb> |
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3101 ΑΔ, ΔΓ μετὰ τοῦ ἀπὸ τῆς ΕΒ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΕΔ.</s> <lb n="1799,001,,,,3,36,44"></lb> |
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3102 <s>τῷ δὲ ἀπὸ τῆς ΕΔ ἴσα ἐστὶ τὰ ἀπὸ τῶν ΕΒ, ΒΔ· ὀρθὴ <lb n="1799,001,,,,3,36,45"></lb> |
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3103 γὰρ ἡ ὑπὸ ΕΒΔ γωνία· τὸ ἄρα ὑπὸ τῶν ΑΔ, ΔΓ μετὰ <lb n="1799,001,,,,3,36,46"></lb> |
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3104 τοῦ ἀπὸ τῆς ΕΒ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΕΒ, ΒΔ.</s> <s>κοινὸν <lb n="1799,001,,,,3,36,47"></lb> |
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3105 ἀφῃρήσθω τὸ ἀπὸ τῆς ΕΒ· λοιπὸν ἄρα τὸ ὑπὸ τῶν ΑΔ, <lb n="1799,001,,,,3,36,48"></lb> |
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3106 ΔΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΔΒ.</s> <lb n="1799,001,,,,3,36,49"></lb> |
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3107 <s>Ἐὰν ἄρα κύκλου ληφθῇ τι σημεῖον ἐκτός, καὶ ἀπ' αὐτοῦ <lb n="1799,001,,,,3,36,50"></lb> |
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3108 πρὸς τὸν κύκλον προσπίπτωσι δύο εὐθεῖαι, καὶ ἡ μὲν <lb n="1799,001,,,,3,36,51"></lb> |
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3109 αὐτῶν τέμνῃ τὸν κύκλον, ἡ δὲ ἐφάπτηται, ἔσται τὸ ὑπὸ <pb></pb> <lb n="1799,001,,,,3,36,52"></lb> |
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3110 ὅλης τῆς τεμνούσης καὶ τῆς ἐκτὸς ἀπολαμβανομένης <lb n="1799,001,,,,3,36,53"></lb> |
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3111 μεταξὺ τοῦ τε σημείου καὶ τῆς κυρτῆς περιφερείας ἴσον <lb n="1799,001,,,,3,36,54"></lb> |
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3112 τῷ ἀπὸ τῆς ἐφαπτομένης τετραγώνῳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,3,36,55"></lb> |
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3113 <s>Ἐὰν κύκλου ληφθῇ τι σημεῖον ἐκτός, ἀπὸ δὲ τοῦ ση-<lb n="1799,001,,,,3,37,1"></lb> |
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3114 μείου πρὸς τὸν κύκλον προσπίπτωσι δύο εὐθεῖαι, καὶ ἡ <lb n="1799,001,,,,3,37,2"></lb> |
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3115 μὲν αὐτῶν τέμνῃ τὸν κύκλον, ἡ δὲ προσπίπτῃ, ᾖ δὲ τὸ <lb n="1799,001,,,,3,37,3"></lb> |
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3116 ὑπὸ [τῆσ] ὅλης τῆς τεμνούσης καὶ τῆς ἐκτὸς ἀπολαμβανο-<lb n="1799,001,,,,3,37,4"></lb> |
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3117 μένης μεταξὺ τοῦ τε σημείου καὶ τῆς κυρτῆς περιφερείας <lb n="1799,001,,,,3,37,5"></lb> |
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3118 ἴσον τῷ ἀπὸ τῆς προσπιπτούσης, ἡ προσπίπτουσα ἐφ-<lb n="1799,001,,,,3,37,6"></lb> |
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3119 άψεται τοῦ κύκλου.</s> <lb n="1799,001,,,,3,37,7"></lb> |
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3120 <s>κύκλου γὰρ τοῦ ΑΒΓ εἰλήφθω τι σημεῖον ἐκτὸς τὸ Δ, <lb n="1799,001,,,,3,37,8"></lb> |
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3121 καὶ ἀπὸ τοῦ Δ πρὸς τὸν ΑΒΓ κύκλον προσπιπτέτωσαν <lb n="1799,001,,,,3,37,9"></lb> |
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3122 δύο εὐθεῖαι αἱ ΔΓΑ, ΔΒ, καὶ ἡ μὲν ΔΓΑ τεμνέτω τὸν <lb n="1799,001,,,,3,37,10"></lb> |
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3123 κύκλον, ἡ δὲ ΔΒ προσπιπτέτω, ἔστω <lb n="1799,001,,,,3,37,11"></lb> |
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3124 δὲ τὸ ὑπὸ τῶν ΑΔ, ΔΓ ἴσον τῷ ἀπὸ <lb n="1799,001,,,,3,37,12"></lb> |
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3125 τῆς ΔΒ.</s> <s>λέγω, ὅτι ἡ ΔΒ ἐφάπτεται <lb n="1799,001,,,,3,37,13"></lb> |
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3126 τοῦ ΑΒΓ κύκλου.</s> <lb n="1799,001,,,,3,37,14"></lb> |
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3127 <s>Ἤχθω γὰρ τοῦ ΑΒΓ ἐφαπτομένη ἡ <lb n="1799,001,,,,3,37,15"></lb> |
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3128 ΔΕ, καὶ εἰλήφθω τὸ κέντρον τοῦ ΑΒΓ <lb n="1799,001,,,,3,37,16"></lb> |
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3129 κύκλου, καὶ ἔστω τὸ Ζ, καὶ ἐπεζεύ-<lb n="1799,001,,,,3,37,17"></lb> |
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3130 χθωσαν αἱ ΖΕ, ΖΒ, ΖΔ.</s> <s>ἡ ἄρα ὑπὸ ΖΕΔ ὀρθή ἐστιν.</s> <s>καὶ <lb n="1799,001,,,,3,37,18"></lb> |
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3131 ἐπεὶ ἡ ΔΕ ἐφάπτεται τοῦ ΑΒΓ κύκλου, τέμνει δὲ ἡ ΔΓΑ, τὸ <lb n="1799,001,,,,3,37,19"></lb> |
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3132 ἄρα ὑπὸ τῶν ΑΔ, ΔΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΔΕ.</s> <s>ἦν δὲ καὶ <pb></pb> <lb n="1799,001,,,,3,37,20"></lb> |
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3133 τὸ ὑπὸ τῶν ΑΔ, ΔΓ ἴσον τῷ ἀπὸ τῆς ΔΒ· τὸ ἄρα ἀπὸ τῆς <lb n="1799,001,,,,3,37,21"></lb> |
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3134 ΔΕ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΔΒ· ἴση ἄρα ἡ ΔΕ τῇ ΔΒ.</s> <lb n="1799,001,,,,3,37,22"></lb> |
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3135 <s>ἐστὶ δὲ καὶ ἡ ΖΕ τῇ ΖΒ ἴση· δύο δὴ αἱ ΔΕ, ΕΖ δύο ταῖς <lb n="1799,001,,,,3,37,23"></lb> |
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3136 ΔΒ, ΒΖ ἴσαι εἰσίν· καὶ βάσις αὐτῶν κοινὴ ἡ ΖΔ· γωνία <lb n="1799,001,,,,3,37,24"></lb> |
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3137 ἄρα ἡ ὑπὸ ΔΕΖ γωνίᾳ τῇ ὑπὸ ΔΒΖ ἐστιν ἴση.</s> <s>ὀρθὴ δὲ <lb n="1799,001,,,,3,37,25"></lb> |
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3138 ἡ ὑπὸ ΔΕΖ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΔΒΖ.</s> <s>καί ἐστιν ἡ ΖΒ <lb n="1799,001,,,,3,37,26"></lb> |
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3139 ἐκβαλλομένη διάμετρος· ἡ δὲ τῇ διαμέτρῳ τοῦ κύκλου πρὸς <lb n="1799,001,,,,3,37,27"></lb> |
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3140 ὀρθὰς ἀπ' ἄκρας ἀγομένη ἐφάπτεται τοῦ κύκλου· ἡ ΔΒ <lb n="1799,001,,,,3,37,28"></lb> |
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3141 ἄρα ἐφάπτεται τοῦ ΑΒΓ κύκλου.</s> <s>ὁμοίως δὴ δειχθήσεται, <lb n="1799,001,,,,3,37,29"></lb> |
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3142 κἂν τὸ κέντρον ἐπὶ τῆς ΑΓ τυγχάνῃ.</s> <lb n="1799,001,,,,3,37,30"></lb> |
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3143 <s>Ἐὰν ἄρα κύκλου ληφθῇ τι σημεῖον ἐκτός, ἀπὸ δὲ τοῦ <lb n="1799,001,,,,3,37,31"></lb> |
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3144 σημείου πρὸς τὸν κύκλον προσπίπτωσι δύο εὐθεῖαι, καὶ <lb n="1799,001,,,,3,37,32"></lb> |
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3145 ἡ μὲν αὐτῶν τέμνῃ τὸν κύκλον, ἡ δὲ προσπίπτῃ, ᾖ δὲ τὸ <lb n="1799,001,,,,3,37,33"></lb> |
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3146 ὑπὸ ὅλης τῆς τεμνούσης καὶ τῆς ἐκτὸς ἀπολαμβανομένης <lb n="1799,001,,,,3,37,34"></lb> |
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3147 μεταξὺ τοῦ τε σημείου καὶ τῆς κυρτῆς περιφερείας ἴσον <lb n="1799,001,,,,3,37,35"></lb> |
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3148 τῷ ἀπὸ τῆς προσπιπτούσης, ἡ προσπίπτουσα ἐφάψεται <lb n="1799,001,,,,3,37,36"></lb> |
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3149 τοῦ κύκλου· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,3,37,37"></lb> |
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3150 </s></p> |
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3151 </chap> |
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3152 <chap><p type="main"><s>Σχῆμα εὐθύγραμμον εἰς σχῆμα εὐθύγραμμον ἐγ-<lb n="1799,001,,,,4;HOR,1,1"></lb> |
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3153 |
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3154 γράφεσθαι λέγεται, ὅταν ἑκάστη τῶν τοῦ ἐγγραφομένου <lb n="1799,001,,,,4;HOR,1,2"></lb> |
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3155 σχήματος γωνιῶν ἑκάστης πλευρᾶς τοῦ, εἰς ὃ ἐγγράφεται, <lb n="1799,001,,,,4;HOR,1,3"></lb> |
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3156 ἅπτηται.</s> <lb n="1799,001,,,,4;HOR,1,4"></lb> |
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3157 <s>Σχῆμα δὲ ὁμοίως περὶ σχῆμα περιγράφεσθαι λέγε-<lb n="1799,001,,,,4;HOR,2,1"></lb> |
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3158 ται, ὅταν ἑκάστη πλευρὰ τοῦ περιγραφομένου ἑκάστης <lb n="1799,001,,,,4;HOR,2,2"></lb> |
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3159 γωνίας τοῦ, περὶ ὃ περιγράφεται, ἅπτηται.</s> <lb n="1799,001,,,,4;HOR,2,3"></lb> |
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3160 <s>Σχῆμα εὐθύγραμμον εἰς κύκλον ἐγγράφεσθαι λέ-<lb n="1799,001,,,,4;HOR,3,1"></lb> |
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3161 γεται, ὅταν ἑκάστη γωνία τοῦ ἐγγραφομένου ἅπτηται <lb n="1799,001,,,,4;HOR,3,2"></lb> |
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3162 τῆς τοῦ κύκλου περιφερείας.</s> <lb n="1799,001,,,,4;HOR,3,3"></lb> |
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3163 <s>Σχῆμα δὲ εὐθύγραμμον περὶ κύκλον περιγράφεσθαι <lb n="1799,001,,,,4;HOR,4,1"></lb> |
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3164 λέγεται, ὅταν ἑκάστη πλευρὰ τοῦ περιγραφομένου ἐφάπτη-<lb n="1799,001,,,,4;HOR,4,2"></lb> |
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3165 ται τῆς τοῦ κύκλου περιφερείας.</s> <lb n="1799,001,,,,4;HOR,4,3"></lb> |
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3166 <s>Κύκλος δὲ εἰς σχῆμα ὁμοίως ἐγγράφεσθαι λέγεται, <lb n="1799,001,,,,4;HOR,5,1"></lb> |
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3167 ὅταν ἡ τοῦ κύκλου περιφέρεια ἑκάστης πλευρᾶς τοῦ, εἰς <lb n="1799,001,,,,4;HOR,5,2"></lb> |
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3168 ὃ ἐγγράφεται, ἅπτηται.</s> <lb n="1799,001,,,,4;HOR,5,3"></lb> |
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3169 <s>Κύκλος δὲ περὶ σχῆμα περιγράφεσθαι λέγεται, ὅταν <lb n="1799,001,,,,4;HOR,6,1"></lb> |
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3170 ἡ τοῦ κύκλου περιφέρεια ἑκάστης γωνίας τοῦ, περὶ ὃ <lb n="1799,001,,,,4;HOR,6,2"></lb> |
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3171 περιγράφεται, ἅπτηται.</s> <lb n="1799,001,,,,4;HOR,6,3"></lb> |
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3172 <s>Εὐθεῖα εἰς κύκλον ἐναρμόζεσθαι λέγεται, ὅταν τὰ <lb n="1799,001,,,,4;HOR,7,1"></lb> |
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3173 πέρατα αὐτῆς ἐπὶ τῆς περιφερείας ᾖ τοῦ κύκλου.</s> <s><pb></pb> <lb n="1799,001,,,,4;HOR,7,2"></lb> |
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3174 </s></p> |
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3175 </chap> |
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3176 <chap><p type="main"><s>Εἰς τὸν δοθέντα κύκλον τῇ δοθείσῃ εὐθείᾳ μὴ μείζονι <lb n="1799,001,,,,4,1,1"></lb> |
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3177 |
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3178 οὔσῃ τῆς τοῦ κύκλου διαμέτρου ἴσην εὐθεῖαν ἐναρμόσαι.</s> <lb n="1799,001,,,,4,1,2"></lb> |
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3179 <s>Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓ, ἡ δὲ δοθεῖσα εὐθεῖα μὴ <lb n="1799,001,,,,4,1,3"></lb> |
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3180 μείζων τῆς τοῦ κύκλου διαμέτρου ἡ Δ.</s> <s>δεῖ δὴ εἰς τὸν ΑΒΓ <lb n="1799,001,,,,4,1,4"></lb> |
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3181 κύκλον τῇ Δ εὐθείᾳ ἴσην εὐθεῖαν ἐναρμόσαι.</s> <lb n="1799,001,,,,4,1,5"></lb> |
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3182 <s>Ἤχθω τοῦ ΑΒΓ κύκλου διάμετρος ἡ <lb n="1799,001,,,,4,1,6"></lb> |
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3183 ΒΓ.</s> <s>εἰ μὲν οὖν ἴση ἐστὶν ἡ ΒΓ τῇ Δ, γε-<lb n="1799,001,,,,4,1,7"></lb> |
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3184 γονὸς ἂν εἴη τὸ ἐπιταχθέν· ἐνήρμοσται <lb n="1799,001,,,,4,1,8"></lb> |
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3185 γὰρ εἰς τὸν ΑΒΓ κύκλον τῇ Δ εὐθείᾳ ἴση <lb n="1799,001,,,,4,1,9"></lb> |
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3186 ἡ ΒΓ.</s> <s>εἰ δὲ μείζων ἐστὶν ἡ ΒΓ τῆς Δ, <lb n="1799,001,,,,4,1,10"></lb> |
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3187 κείσθω τῇ Δ ἴση ἡ ΓΕ, καὶ κέντρῳ τῷ <lb n="1799,001,,,,4,1,11"></lb> |
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3188 Γ διαστήματι δὲ τῷ ΓΕ κύκλος γεγρά-<lb n="1799,001,,,,4,1,12"></lb> |
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3189 φθω ὁ ΕΑΖ, καὶ ἐπεζεύχθω ἡ ΓΑ.</s> <lb n="1799,001,,,,4,1,13"></lb> |
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3190 <s>Ἐπεὶ οὖν τὸ Γ σημεῖον κέντρον ἐστὶ τοῦ ΕΑΖ κύκλου, <lb n="1799,001,,,,4,1,14"></lb> |
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3191 ἴση ἐστὶν ἡ ΓΑ τῇ ΓΕ.</s> <s>ἀλλὰ τῇ Δ ἡ ΓΕ ἐστιν ἴση· καὶ <lb n="1799,001,,,,4,1,15"></lb> |
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3192 ἡ Δ ἄρα τῇ ΓΑ ἐστιν ἴση.</s> <lb n="1799,001,,,,4,1,16"></lb> |
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3193 <s>Εἰς ἄρα τὸν δοθέντα κύκλον τὸν ΑΒΓ τῇ δοθείσῃ <lb n="1799,001,,,,4,1,17"></lb> |
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3194 εὐθείᾳ τῇ Δ ἴση ἐνήρμοσται ἡ ΓΑ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,1,18"></lb> |
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3195 <s>Εἰς τὸν δοθέντα κύκλον τῷ δοθέντι τριγώνῳ ἰσογώνιον <lb n="1799,001,,,,4,2,1"></lb> |
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3196 τρίγωνον ἐγγράψαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,2,2"></lb> |
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3197 Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓ, τὸ δὲ δοθὲν τρίγωνον <lb n="1799,001,,,,4,2,3"></lb> |
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3198 τὸ ΔΕΖ· δεῖ δὴ εἰς τὸν ΑΒΓ κύκλον τῷ ΔΕΖ τριγώνῳ <lb n="1799,001,,,,4,2,4"></lb> |
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3199 ἰσογώνιον τρίγωνον ἐγγράψαι.</s> <lb n="1799,001,,,,4,2,5"></lb> |
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3200 <s>Ἤχθω τοῦ ΑΒΓ κύκλου ἐφαπτομένη ἡ ΗΘ κατὰ τὸ <lb n="1799,001,,,,4,2,6"></lb> |
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3201 Α, καὶ συνεστάτω πρὸς τῇ ΑΘ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ <lb n="1799,001,,,,4,2,7"></lb> |
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3202 σημείῳ τῷ Α τῇ ὑπὸ ΔΕΖ γωνίᾳ <lb n="1799,001,,,,4,2,8"></lb> |
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3203 ἴση ἡ ὑπὸ ΘΑΓ, πρὸς δὲ τῇ ΑΗ <lb n="1799,001,,,,4,2,9"></lb> |
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3204 εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ <lb n="1799,001,,,,4,2,10"></lb> |
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3205 Α τῇ ὑπὸ ΔΖΕ [γωνίᾳ] ἴση ἡ <lb n="1799,001,,,,4,2,11"></lb> |
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3206 ὑπὸ ΗΑΒ, καὶ ἐπεζεύχθω ἡ ΒΓ.</s> <lb n="1799,001,,,,4,2,12"></lb> |
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3207 <s>Ἐπεὶ οὖν κύκλου τοῦ ΑΒΓ <lb n="1799,001,,,,4,2,13"></lb> |
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3208 ἐφάπτεταί τις εὐθεῖα ἡ ΑΘ, καὶ <lb n="1799,001,,,,4,2,14"></lb> |
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3209 ἀπὸ τῆς κατὰ τὸ Α ἐπαφῆς εἰς τὸν κύκλον διῆκται <lb n="1799,001,,,,4,2,15"></lb> |
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3210 εὐθεῖα ἡ ΑΓ, ἡ ἄρα ὑπὸ ΘΑΓ ἴση ἐστὶ τῇ ἐν τῷ ἐναλλὰξ <lb n="1799,001,,,,4,2,16"></lb> |
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3211 τοῦ κύκλου τμήματι γωνίᾳ τῇ ὑπὸ ΑΒΓ.</s> <s>ἀλλ' ἡ ὑπὸ <lb n="1799,001,,,,4,2,17"></lb> |
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3212 ΘΑΓ τῇ ὑπὸ ΔΕΖ ἐστιν ἴση· καὶ ἡ ὑπὸ ΑΒΓ ἄρα γωνία <lb n="1799,001,,,,4,2,18"></lb> |
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3213 τῇ ὑπὸ ΔΕΖ ἐστιν ἴση.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ ΑΓΒ <lb n="1799,001,,,,4,2,19"></lb> |
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3214 τῇ ὑπὸ ΔΖΕ ἐστιν ἴση· καὶ λοιπὴ ἄρα ἡ ὑπὸ ΒΑΓ λοιπῇ <lb n="1799,001,,,,4,2,20"></lb> |
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3215 τῇ ὑπὸ ΕΔΖ ἐστιν ἴση· [ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓ <lb n="1799,001,,,,4,2,21"></lb> |
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3216 τρίγωνον τῷ ΔΕΖ τριγώνῳ, καὶ ἐγγέγραπται εἰς τὸν <lb n="1799,001,,,,4,2,22"></lb> |
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3217 ΑΒΓ κύκλον].</s> <lb n="1799,001,,,,4,2,23"></lb> |
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3218 <s>Εἰς τὸν δοθέντα ἄρα κύκλον τῷ δοθέντι τριγώνῳ ἰσο-<lb n="1799,001,,,,4,2,24"></lb> |
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3219 γώνιον τρίγωνον ἐγγέγραπται· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,2,25"></lb> |
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3220 Περὶ τὸν δοθέντα κύκλον τῷ δοθέντι τριγώνῳ ἰσογώ-<lb n="1799,001,,,,4,3,1"></lb> |
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3221 νιον τρίγωνον περιγράψαι.</s> <lb n="1799,001,,,,4,3,2"></lb> |
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3222 <s>Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓ, τὸ δὲ δοθὲν τρίγωνον <lb n="1799,001,,,,4,3,3"></lb> |
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3223 τὸ ΔΕΖ· δεῖ δὴ περὶ τὸν ΑΒΓ κύκλον τῷ ΔΕΖ τρι-<lb n="1799,001,,,,4,3,4"></lb> |
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3224 γώνῳ ἰσογώνιον τρίγωνον περιγράψαι.</s> <lb n="1799,001,,,,4,3,5"></lb> |
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3225 <s>Ἐκβεβλήσθω ἡ ΕΖ ἐφ' ἑκάτερα τὰ μέρη κατὰ τὰ Η, <lb n="1799,001,,,,4,3,6"></lb> |
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3226 Θ σημεῖα, καὶ εἰλήφθω τοῦ ΑΒΓ κύκλου κέντρον τὸ Κ, <lb n="1799,001,,,,4,3,7"></lb> |
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3227 καὶ διήχθω, ὡς ἔτυχεν, εὐθεῖα ἡ ΚΒ, καὶ συνεστάτω <lb n="1799,001,,,,4,3,8"></lb> |
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3228 πρὸς τῇ ΚΒ εὐθείᾳ καὶ τῷ πρὸς <lb n="1799,001,,,,4,3,9"></lb> |
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3229 αὐτῇ σημείῳ τῷ Κ τῇ μὲν ὑπὸ <lb n="1799,001,,,,4,3,10"></lb> |
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3230 ΔΕΗ γωνίᾳ ἴση ἡ ὑπὸ ΒΚΑ, <lb n="1799,001,,,,4,3,11"></lb> |
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3231 τῇ δὲ ὑπὸ ΔΖΘ ἴση ἡ ὑπὸ <lb n="1799,001,,,,4,3,12"></lb> |
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3232 ΒΚΓ, καὶ διὰ τῶν Α, Β, Γ ση-<lb n="1799,001,,,,4,3,13"></lb> |
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3233 μείων ἤχθωσαν ἐφαπτόμεναι τοῦ <lb n="1799,001,,,,4,3,14"></lb> |
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3234 ΑΒΓ κύκλου αἱ ΛΑΜ, ΜΒΝ, <lb n="1799,001,,,,4,3,15"></lb> |
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3235 ΝΓΛ.</s> <lb n="1799,001,,,,4,3,16"></lb> |
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3236 <s>Καὶ ἐπεὶ ἐφάπτονται τοῦ ΑΒΓ κύκλου αἱ ΛΜ, ΜΝ, <lb n="1799,001,,,,4,3,17"></lb> |
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3237 ΝΛ κατὰ τὰ Α, Β, Γ σημεῖα, ἀπὸ δὲ τοῦ Κ κέντρου ἐπὶ <lb n="1799,001,,,,4,3,18"></lb> |
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3238 τὰ Α, Β, Γ σημεῖα ἐπεζευγμέναι εἰσὶν αἱ ΚΑ, ΚΒ, ΚΓ, <lb n="1799,001,,,,4,3,19"></lb> |
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3239 ὀρθαὶ ἄρα εἰσὶν αἱ πρὸς τοῖς Α, Β, Γ σημείοις γωνίαι.</s> <s>καὶ <lb n="1799,001,,,,4,3,20"></lb> |
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3240 ἐπεὶ τοῦ ΑΜΒΚ τετραπλεύρου αἱ τέσσαρες γωνίαι τέτρα-<lb n="1799,001,,,,4,3,21"></lb> |
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3241 σιν ὀρθαῖς ἴσαι εἰσίν, ἐπειδήπερ καὶ εἰς δύο τρίγωνα διαι- <pb></pb><lb n="1799,001,,,,4,3,22"></lb> |
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3242 ρεῖται τὸ ΑΜΒΚ, καί εἰσιν ὀρθαὶ αἱ ὑπὸ ΚΑΜ, ΚΒΜ <lb n="1799,001,,,,4,3,23"></lb> |
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3243 γωνίαι, λοιπαὶ ἄρα αἱ ὑπὸ ΑΚΒ, ΑΜΒ δυσὶν ὀρθαῖς <lb n="1799,001,,,,4,3,24"></lb> |
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3244 ἴσαι εἰσίν.</s> <s>εἰσὶ δὲ καὶ αἱ ὑπὸ ΔΕΗ, ΔΕΖ δυσὶν ὀρθαῖς <lb n="1799,001,,,,4,3,25"></lb> |
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3245 ἴσαι· αἱ ἄρα ὑπὸ ΑΚΒ, ΑΜΒ ταῖς ὑπὸ ΔΕΗ, ΔΕΖ <lb n="1799,001,,,,4,3,26"></lb> |
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3246 ἴσαι εἰσίν, ὧν ἡ ὑπὸ ΑΚΒ τῇ ὑπὸ ΔΕΗ ἐστιν ἴση· λοιπὴ <lb n="1799,001,,,,4,3,27"></lb> |
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3247 ἄρα ἡ ὑπὸ ΑΜΒ λοιπῇ τῇ ὑπὸ ΔΕΖ ἐστιν ἴση.</s> <s>ὁμοίως <lb n="1799,001,,,,4,3,28"></lb> |
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3248 δὴ δειχθήσεται, ὅτι καὶ ἡ ὑπὸ ΛΝΒ τῇ ὑπὸ ΔΖΕ ἐστιν <lb n="1799,001,,,,4,3,29"></lb> |
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3249 ἴση· καὶ λοιπὴ ἄρα ἡ ὑπὸ ΜΛΝ [λοιπῇ] τῇ ὑπὸ ΕΔΖ <lb n="1799,001,,,,4,3,30"></lb> |
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3250 ἐστιν ἴση.</s> <s>ἰσογώνιον ἄρα ἐστὶ τὸ ΛΜΝ τρίγωνον τῷ <lb n="1799,001,,,,4,3,31"></lb> |
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3251 ΔΕΖ τριγώνῳ· καὶ περιγέγραπται περὶ τὸν ΑΒΓ <lb n="1799,001,,,,4,3,32"></lb> |
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3252 κύκλον.</s> <lb n="1799,001,,,,4,3,33"></lb> |
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3253 <s>Περὶ τὸν δοθέντα ἄρα κύκλον τῷ δοθέντι τριγώνῳ ἰσο-<lb n="1799,001,,,,4,3,34"></lb> |
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3254 γώνιον τρίγωνον περιγέγραπται· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,3,35"></lb> |
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3255 <s>Εἰς τὸ δοθὲν τρίγωνον κύκλον ἐγγράψαι.</s> <lb n="1799,001,,,,4,4,1"></lb> |
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3256 <s>Ἔστω τὸ δοθὲν τρίγωνον τὸ ΑΒΓ· δεῖ δὴ εἰς τὸ ΑΒΓ <lb n="1799,001,,,,4,4,2"></lb> |
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3257 τρίγωνον κύκλον ἐγγράψαι.</s> <lb n="1799,001,,,,4,4,3"></lb> |
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3258 <s>Τετμήσθωσαν αἱ ὑπὸ ΑΒΓ, ΑΓΒ γωνίαι δίχα ταῖς <lb n="1799,001,,,,4,4,4"></lb> |
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3259 ΒΔ, ΓΔ εὐθείαις, καὶ συμβαλλέτωσαν ἀλλήλαις κατὰ τὸ Δ <pb></pb> <lb n="1799,001,,,,4,4,5"></lb> |
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3260 σημεῖον, καὶ ἤχθωσαν ἀπὸ τοῦ Δ ἐπὶ τὰς ΑΒ, ΒΓ, ΓΑ <lb n="1799,001,,,,4,4,6"></lb> |
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3261 εὐθείας κάθετοι αἱ ΔΕ, ΔΖ, ΔΗ.</s> <lb n="1799,001,,,,4,4,7"></lb> |
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3262 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ὑπὸ ΑΒΔ γωνία τῇ ὑπὸ ΓΒΔ, <lb n="1799,001,,,,4,4,8"></lb> |
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3263 ἐστὶ δὲ καὶ ὀρθὴ ἡ ὑπὸ ΒΕΔ ὀρθῇ τῇ ὑπὸ ΒΖΔ ἴση, δύο <lb n="1799,001,,,,4,4,9"></lb> |
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3264 δὴ τρίγωνά ἐστι τὰ ΕΒΔ, ΖΒΔ <lb n="1799,001,,,,4,4,10"></lb> |
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3265 τὰς δύο γωνίας ταῖς δυσὶ γωνίαις <lb n="1799,001,,,,4,4,11"></lb> |
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3266 ἴσας ἔχοντα καὶ μίαν πλευρὰν μιᾷ <lb n="1799,001,,,,4,4,12"></lb> |
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3267 πλευρᾷ ἴσην τὴν ὑποτείνουσαν ὑπὸ <lb n="1799,001,,,,4,4,13"></lb> |
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3268 μίαν τῶν ἴσων γωνιῶν κοινὴν <lb n="1799,001,,,,4,4,14"></lb> |
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3269 αὐτῶν τὴν ΒΔ· καὶ τὰς λοιπὰς <lb n="1799,001,,,,4,4,15"></lb> |
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3270 ἄρα πλευρὰς ταῖς λοιπαῖς πλευραῖς <lb n="1799,001,,,,4,4,16"></lb> |
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3271 ἴσας ἕξουσιν· ἴση ἄρα ἡ ΔΕ τῇ ΔΖ.</s> <s>διὰ τὰ αὐτὰ δὴ <lb n="1799,001,,,,4,4,17"></lb> |
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3272 καὶ ἡ ΔΗ τῇ ΔΖ ἐστιν ἴση.</s> <s>αἱ τρεῖς ἄρα εὐθεῖαι αἱ ΔΕ, <lb n="1799,001,,,,4,4,18"></lb> |
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3273 ΔΖ, ΔΗ ἴσαι ἀλλήλαις εἰσίν· ὁ ἄρα κέντρῳ τῷ Δ καὶ δια-<lb n="1799,001,,,,4,4,19"></lb> |
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3274 στήματι ἑνὶ τῶν Ε, Ζ, Η κύκλος γραφόμενος ἥξει καὶ διὰ <lb n="1799,001,,,,4,4,20"></lb> |
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3275 τῶν λοιπῶν σημείων καὶ ἐφάψεται τῶν ΑΒ, ΒΓ, ΓΑ <lb n="1799,001,,,,4,4,21"></lb> |
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3276 εὐθειῶν διὰ τὸ ὀρθὰς εἶναι τὰς πρὸς τοῖς Ε, Ζ, Η σημείοις <lb n="1799,001,,,,4,4,22"></lb> |
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3277 γωνίας.</s> <s>εἰ γὰρ τεμεῖ αὐτάς, ἔσται ἡ τῇ διαμέτρῳ τοῦ <lb n="1799,001,,,,4,4,23"></lb> |
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3278 κύκλου πρὸς ὀρθὰς ἀπ' ἄκρας ἀγομένη ἐντὸς πίπτουσα <lb n="1799,001,,,,4,4,24"></lb> |
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3279 τοῦ κύκλου· ὅπερ ἄτοπον ἐδείχθη· οὐκ ἄρα ὁ κέντρῳ τῷ Δ <pb></pb> <lb n="1799,001,,,,4,4,25"></lb> |
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3280 διαστήματι δὲ ἑνὶ τῶν Ε, Ζ, Η γραφόμενος κύκλος <lb n="1799,001,,,,4,4,26"></lb> |
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3281 τεμεῖ τὰς ΑΒ, ΒΓ, ΓΑ εὐθείας· ἐφάψεται ἄρα αὐτῶν, <lb n="1799,001,,,,4,4,27"></lb> |
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3282 καὶ ἔσται ὁ κύκλος ἐγγεγραμμένος εἰς τὸ ΑΒΓ τρίγωνον.</s> <lb n="1799,001,,,,4,4,28"></lb> |
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3283 <s>ἐγγεγράφθω ὡς ὁ ΖΗΕ.</s> <lb n="1799,001,,,,4,4,29"></lb> |
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3284 <s>Εἰς ἄρα τὸ δοθὲν τρίγωνον τὸ ΑΒΓ κύκλος ἐγγέγραπται <lb n="1799,001,,,,4,4,30"></lb> |
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3285 ὁ ΕΖΗ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,4,31"></lb> |
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3286 <s>Περὶ τὸ δοθὲν τρίγωνον κύκλον περιγράψαι.</s> <lb n="1799,001,,,,4,5,1"></lb> |
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3287 <s>Ἔστω τὸ δοθὲν τρίγωνον τὸ ΑΒΓ· δεῖ <ὴ>περὶ τὸ δο-<lb n="1799,001,,,,4,5,2"></lb> |
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3288 θὲν τρίγωνον τὸ ΑΒΓ κύκλον περιγράψαι.</s> <lb n="1799,001,,,,4,5,3"></lb> |
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3289 <s>Τετμήσθωσαν αἱ ΑΒ, ΑΓ εὐθεῖαι δίχα κατὰ τὰ Δ, Ε <lb n="1799,001,,,,4,5,4"></lb> |
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3290 σημεῖα, καὶ ἀπὸ τῶν Δ, Ε σημείων ταῖς ΑΒ, ΑΓ πρὸς <lb n="1799,001,,,,4,5,5"></lb> |
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3291 ὁρθὰς ἤχθωσαν αἱ ΔΖ, ΕΖ· συμπεσοῦνται δὴ ἤτοι ἐντὸς <lb n="1799,001,,,,4,5,6"></lb> |
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3292 τοῦ ΑΒΓ τριγώνου ἢ ἐπὶ τῆς ΒΓ εὐθείας ἢ ἐκτὸς τῆς <lb n="1799,001,,,,4,5,7"></lb> |
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3293 ΒΓ.</s> <lb n="1799,001,,,,4,5,8"></lb> |
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3294 <s>Συμπιπτέτωσαν πρότερον ἐντὸς κατὰ τὸ Ζ, καὶ ἐπεζεύ-<lb n="1799,001,,,,4,5,9"></lb> |
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3295 χθωσαν αἱ ΖΒ, ΖΓ, ΖΑ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΔ τῇ <lb n="1799,001,,,,4,5,10"></lb> |
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3296 ΔΒ, κοινὴ δὲ καὶ πρὸς ὀρθὰς ἡ ΔΖ, βάσις ἄρα ἡ ΑΖ <lb n="1799,001,,,,4,5,11"></lb> |
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3297 βάσει τῇ ΖΒ ἐστιν ἴση.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ ΓΖ <lb n="1799,001,,,,4,5,12"></lb> |
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3298 τῇ ΑΖ ἐστιν ἴση· ὥστε καὶ ἡ ΖΒ τῇ ΖΓ ἐστιν ἴση· αἱ <lb n="1799,001,,,,4,5,13"></lb> |
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3299 τρεῖς ἄρα αἱ ΖΑ, ΖΒ, ΖΓ ἴσαι ἀλλήλαις εἰσίν.</s> <s>ὁ ἄρα κέν- <pb></pb><lb n="1799,001,,,,4,5,14"></lb> |
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3300 τρῳ τῷ Ζ διαστήματι δὲ ἑνὶ τῶν Α, Β, Γ κύκλος γραφό-<lb n="1799,001,,,,4,5,15"></lb> |
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3301 μενος ἥξει καὶ διὰ τῶν λοιπῶν σημείων, καὶ ἔσται περι-<lb n="1799,001,,,,4,5,16"></lb> |
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3302 γεγραμμένος ὁ κύκλος περὶ τὸ ΑΒΓ τρίγωνον.</s> <s>περι-<lb n="1799,001,,,,4,5,17"></lb> |
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3303 γεγράφθω ὡς ὁ ΑΒΓ. <lb n="1799,001,,,,4,5,18"></lb> |
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3304 Ἀλλὰ δὴ αἱ ΔΖ, ΕΖ συμπιπτέτωσαν ἐπὶ τῆς ΒΓ εὐθείας <lb n="1799,001,,,,4,5,19"></lb> |
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3305 κατὰ τὸ Ζ, ὡς ἔχει ἐπὶ τῆς δευτέρας καταγραφῆς, καὶ <lb n="1799,001,,,,4,5,20"></lb> |
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3306 ἐπεζεύχθω ἡ ΑΖ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι τὸ Ζ σημεῖον <lb n="1799,001,,,,4,5,21"></lb> |
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3307 κέντρον ἐστὶ τοῦ περὶ τὸ ΑΒΓ τρίγωνον περιγραφομένου <lb n="1799,001,,,,4,5,22"></lb> |
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3308 κύκλου.</s> <lb n="1799,001,,,,4,5,23"></lb> |
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3309 <s>Ἀλλὰ δὴ αἱ ΔΖ, ΕΖ συμπιπτέτωσαν ἐκτὸς τοῦ ΑΒΓ <lb n="1799,001,,,,4,5,24"></lb> |
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3310 τριγώνου κατὰ τὸ Ζ πάλιν, ὡς ἔχει ἐπὶ τῆς τρίτης κατα-<lb n="1799,001,,,,4,5,25"></lb> |
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3311 γραφῆς, καὶ ἐπεζεύχθωσαν αἱ ΑΖ, ΒΖ, ΓΖ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,4,5,26"></lb> |
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3312 πάλιν ἴση ἐστὶν ἡ ΑΔ τῇ ΔΒ, κοινὴ δὲ καὶ πρὸς ὀρθὰς <lb n="1799,001,,,,4,5,27"></lb> |
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3313 ἡ ΔΖ, βάσις ἄρα ἡ ΑΖ βάσει τῇ ΒΖ ἐστιν ἴση.</s> <s>ὁμοίως <lb n="1799,001,,,,4,5,28"></lb> |
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3314 δὴ δείξομεν, ὅτι καὶ ἡ ΓΖ τῇ ΑΖ ἐστιν ἴση· ὥστε καὶ ἡ <lb n="1799,001,,,,4,5,29"></lb> |
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3315 ΒΖ τῇ ΖΓ ἐστιν ἴση· ὁ ἄρα [πάλιν] κέντρῳ τῷ Ζ δια- <pb></pb><lb n="1799,001,,,,4,5,30"></lb> |
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3316 στήματι δὲ ἑνὶ τῶν ΖΑ, ΖΒ, ΖΓ κύκλος γραφόμενος <lb n="1799,001,,,,4,5,31"></lb> |
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3317 ἥξει καὶ διὰ τῶν λοιπῶν σημείων, καὶ ἔσται περιγεγραμ-<lb n="1799,001,,,,4,5,32"></lb> |
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3318 μένος περὶ τὸ ΑΒΓ τρίγωνον.</s> <lb n="1799,001,,,,4,5,33"></lb> |
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3319 <s>Περὶ τὸ δοθὲν ἄρα τρίγωνον κύκλος περιγέγραπται· <lb n="1799,001,,,,4,5,34"></lb> |
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3320 ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,5,35"></lb> |
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3321 <s>[Πόρισμα] <lb n="1799,001,,,,4,5,36"></lb> |
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3322 Καὶ φανερόν, ὅτι, ὅτε μὲν ἐντὸς τοῦ τριγώνου πίπτει <lb n="1799,001,,,,4,5,37"></lb> |
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3323 τὸ κέντρον τοῦ κύκλου, ἡ ὑπὸ ΒΑΓ γωνία ἐν μείζονι τμή-<lb n="1799,001,,,,4,5,38"></lb> |
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3324 ματι τοῦ ἡμικυκλίου τυγχάνουσα ἐλάττων ἐστὶν ὀρθῆς· <lb n="1799,001,,,,4,5,39"></lb> |
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3325 ὅτε δὲ ἐπὶ τῆς ΒΓ εὐθείας τὸ κέντρον πίπτει, ἡ ὑπὸ ΒΑΓ <lb n="1799,001,,,,4,5,40"></lb> |
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3326 γωνία ἐν ἡμικυκλίῳ τυγχάνουσα ὀρθή ἐστιν· ὅτε δὲ τὸ <lb n="1799,001,,,,4,5,41"></lb> |
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3327 κέντρον τοῦ κύκλου ἐκτὸς τοῦ τριγώνου πίπτει, ἡ ὑπὸ <lb n="1799,001,,,,4,5,42"></lb> |
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3328 ΒΑΓ ἐν ἐλάττονι τμήματι τοῦ ἡμικυκλίου τυγχάνουσα <lb n="1799,001,,,,4,5,43"></lb> |
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3329 μείζων ἐστὶν ὀρθῆς.</s> <s>[ὥστε καὶ ὅταν ἐλάττων ὀρθῆς τυγ-<lb n="1799,001,,,,4,5,44"></lb> |
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3330 χάνῃ ἡ διδομένη γωνία, ἐντὸς τοῦ τριγώνου πεσοῦνται <lb n="1799,001,,,,4,5,45"></lb> |
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3331 αἱ ΔΖ, ΕΖ, ὅταν δὲ ὀρθή, ἐπὶ τῆς ΒΓ, ὅταν δὲ μείζων <lb n="1799,001,,,,4,5,46"></lb> |
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3332 ὀρθῆς, ἐκτὸς τῆς ΒΓ· ὅπερ ἔδει ποιῆσαι.] <pb></pb> <lb n="1799,001,,,,4,5,47"></lb> |
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3333 Εἰς τὸν δοθέντα κύκλον τετράγωνον ἐγγράψαι.</s> <lb n="1799,001,,,,4,6,1"></lb> |
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3334 <s>Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓΔ· δεῖ δὴ εἰς τὸν ΑΒΓΔ <lb n="1799,001,,,,4,6,2"></lb> |
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3335 κύκλον τετράγωνον ἐγγράψαι.</s> <lb n="1799,001,,,,4,6,3"></lb> |
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3336 <s>Ἤχθωσαν τοῦ ΑΒΓΔ κύκλου δύο διάμετροι πρὸς <lb n="1799,001,,,,4,6,4"></lb> |
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3337 ὀρθὰς ἀλλήλαις αἱ ΑΓ, ΒΔ, καὶ ἐπεζεύχθωσαν αἱ ΑΒ, <lb n="1799,001,,,,4,6,5"></lb> |
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3338 ΒΓ, ΓΔ, ΔΑ.</s> <lb n="1799,001,,,,4,6,6"></lb> |
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3339 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΕ τῇ ΕΔ· κέντρον γὰρ τὸ Ε· <lb n="1799,001,,,,4,6,7"></lb> |
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3340 κοινὴ δὲ καὶ πρὸς ὀρθὰς ἡ ΕΑ, βάσις ἄρα ἡ ΑΒ βάσει <lb n="1799,001,,,,4,6,8"></lb> |
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3341 τῇ ΑΔ ἴση ἐστίν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,4,6,9"></lb> |
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3342 ἑκατέρα τῶν ΒΓ, ΓΔ ἑκατέρᾳ τῶν ΑΒ, ΑΔ <lb n="1799,001,,,,4,6,10"></lb> |
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3343 ἴση ἐστίν· ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΒΓΔ <lb n="1799,001,,,,4,6,11"></lb> |
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3344 τετράπλευρον.</s> <s>λέγω δή, ὅτι καὶ ὀρθογώ-<lb n="1799,001,,,,4,6,12"></lb> |
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3345 νιον.</s> <s>ἐπεὶ γὰρ ἡ ΒΔ εὐθεῖα διάμετρός <lb n="1799,001,,,,4,6,13"></lb> |
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3346 ἐστι τοῦ ΑΒΓΔ κύκλου, ἡμικύκλιον ἄρα <lb n="1799,001,,,,4,6,14"></lb> |
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3347 ἐστὶ τὸ ΒΑΔ· ὀρθὴ ἄρα ἡ ὑπὸ ΒΑΔ <lb n="1799,001,,,,4,6,15"></lb> |
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3348 γωνία.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἑκάστη τῶν ὑπὸ ΑΒΓ, ΒΓΔ, <lb n="1799,001,,,,4,6,16"></lb> |
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3349 ΓΔΑ ὀρθή ἐστιν· ὀρθογώνιον ἄρα ἐστὶ τὸ ΑΒΓΔ τετρά-<lb n="1799,001,,,,4,6,17"></lb> |
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3350 πλευρον.</s> <s>ἐδείχθη δὲ καὶ ἰσόπλευρον· τετράγωνον ἄρα <lb n="1799,001,,,,4,6,18"></lb> |
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3351 ἐστίν.</s> <s>καὶ ἐγγέγραπται εἰς τὸν ΑΒΓΔ κύκλον.</s> <lb n="1799,001,,,,4,6,19"></lb> |
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3352 <s>Εἰς ἄρα τὸν δοθέντα κύκλον τετράγωνον ἐγγέγραπται <lb n="1799,001,,,,4,6,20"></lb> |
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3353 τὸ ΑΒΓΔ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,6,21"></lb> |
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3354 <s>Περὶ τὸν δοθέντα κύκλον τετράγωνον περιγράψαι.</s> <lb n="1799,001,,,,4,7,1"></lb> |
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3355 <s>Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓΔ· δεῖ δὴ περὶ τὸν <lb n="1799,001,,,,4,7,2"></lb> |
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3356 ΑΒΓΔ κύκλον τετράγωνον περιγράψαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,7,3"></lb> |
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3357 Ἤχθωσαν τοῦ ΑΒΓΔ κύκλου δύο διάμετροι πρὸς <lb n="1799,001,,,,4,7,4"></lb> |
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3358 ὀρθὰς ἀλλήλαις αἱ ΑΓ, ΒΔ, καὶ διὰ τῶν Α, Β, Γ, Δ ση-<lb n="1799,001,,,,4,7,5"></lb> |
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3359 μείων ἤχθωσαν ἐφαπτόμεναι τοῦ ΑΒΓΔ κύκλου αἱ ΖΗ, <lb n="1799,001,,,,4,7,6"></lb> |
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3360 ΗΘ, ΘΚ, ΚΖ.</s> <lb n="1799,001,,,,4,7,7"></lb> |
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3361 <s>Ἐπεὶ οὖν ἐφάπτεται ἡ ΖΗ τοῦ ΑΒΓΔ κύκλου, ἀπὸ δὲ <lb n="1799,001,,,,4,7,8"></lb> |
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3362 τοῦ Ε κέντρου ἐπὶ τὴν κατὰ τὸ Α ἐπαφὴν ἐπέζευκται ἡ <lb n="1799,001,,,,4,7,9"></lb> |
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3363 ΕΑ, αἱ ἄρα πρὸς τῷ Α γωνίαι ὀρθαί εἰσιν.</s> <s>διὰ τὰ αὐτὰ δὴ <lb n="1799,001,,,,4,7,10"></lb> |
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3364 καὶ αἱ πρὸς τοῖς Β, Γ, Δ σημείοις γωνίαι ὀρθαί εἰσιν.</s> <lb n="1799,001,,,,4,7,11"></lb> |
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3365 <s>καὶ ἐπεὶ ὀρθή ἐστιν ἡ ὑπὸ ΑΕΒ γωνία, <lb n="1799,001,,,,4,7,12"></lb> |
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3366 ἐστὶ δὲ ὀρθὴ καὶ ἡ ὑπὸ ΕΒΗ, παράλληλος <lb n="1799,001,,,,4,7,13"></lb> |
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3367 ἄρα ἐστὶν ἡ ΗΘ τῇ ΑΓ.</s> <s>διὰ τὰ αὐτὰ δὴ <lb n="1799,001,,,,4,7,14"></lb> |
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3368 καὶ ἡ ΑΓ τῇ ΖΚ ἐστι παράλληλος.</s> <s>ὥστε <lb n="1799,001,,,,4,7,15"></lb> |
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3369 καὶ ἡ ΗΘ τῇ ΖΚ ἐστι παράλληλος.</s> <lb n="1799,001,,,,4,7,16"></lb> |
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3370 <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἑκατέρα τῶν <lb n="1799,001,,,,4,7,17"></lb> |
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3371 ΗΖ, ΘΚ τῇ ΒΕΔ ἐστι παράλληλος.</s> <lb n="1799,001,,,,4,7,18"></lb> |
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3372 <s>παραλληλόγραμμα ἄρα ἐστὶ τὰ ΗΚ, ΗΓ, ΑΚ, ΖΒ, ΒΚ· <lb n="1799,001,,,,4,7,19"></lb> |
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3373 ἴση ἄρα ἐστὶν ἡ μὲν ΗΖ τῇ ΘΚ, ἡ δὲ ΗΘ τῇ ΖΚ.</s> <s>καὶ <lb n="1799,001,,,,4,7,20"></lb> |
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3374 ἐπεὶ ἴση ἐστὶν ἡ ΑΓ τῇ ΒΔ, ἀλλὰ καὶ ἡ μὲν ΑΓ ἑκατέρᾳ <lb n="1799,001,,,,4,7,21"></lb> |
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3375 τῶν ΗΘ, ΖΚ, ἡ δὲ ΒΔ ἑκατέρᾳ τῶν ΗΖ, ΘΚ ἐστιν <lb n="1799,001,,,,4,7,22"></lb> |
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3376 ἴση [καὶ ἑκατέρα ἄρα τῶν ΗΘ, ΖΚ ἑκατέρᾳ τῶν ΗΖ, <lb n="1799,001,,,,4,7,23"></lb> |
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3377 ΘΚ ἐστιν ἴση], ἰσόπλευρον ἄρα ἐστὶ τὸ ΖΗΘΚ τετρά-<lb n="1799,001,,,,4,7,24"></lb> |
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3378 πλευρον.</s> <s>λέγω δή, ὅτι καὶ ὀρθογώνιον.</s> <s>ἐπεὶ γὰρ παραλλη- <pb></pb><lb n="1799,001,,,,4,7,25"></lb> |
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3379 λόγραμμόν ἐστι τὸ ΗΒΕΑ, καί ἐστιν ὀρθὴ ἡ ὑπὸ ΑΕΒ, <lb n="1799,001,,,,4,7,26"></lb> |
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3380 ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΑΗΒ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ <lb n="1799,001,,,,4,7,27"></lb> |
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3381 αἱ πρὸς τοῖς Θ, Κ, Ζ γωνίαι ὀρθαί εἰσιν.</s> <s>ὀρθογώνιον ἄρα <lb n="1799,001,,,,4,7,28"></lb> |
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3382 ἐστὶ τὸ ΖΗΘΚ.</s> <s>ἐδείχθη δὲ καὶ ἰσόπλευρον· τετράγωνον <lb n="1799,001,,,,4,7,29"></lb> |
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3383 ἄρα ἐστίν.</s> <s>καὶ περιγέγραπται περὶ τὸν ΑΒΓΔ κύκλον.</s> <lb n="1799,001,,,,4,7,30"></lb> |
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3384 <s>Περὶ τὸν δοθέντα ἄρα κύκλον τετράγωνον περιγέγρα-<lb n="1799,001,,,,4,7,31"></lb> |
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3385 πται· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,7,32"></lb> |
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3386 <s>Εἰς τὸ δοθὲν τετράγωνον κύκλον ἐγγράψαι.</s> <lb n="1799,001,,,,4,8,1"></lb> |
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3387 <s>Ἔστω τὸ δοθὲν τετράγωνον τὸ ΑΒΓΔ· δεῖ δὴ εἰς τὸ <lb n="1799,001,,,,4,8,2"></lb> |
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3388 ΑΒΓΔ τετράγωνον κύκλον ἐγγράψαι.</s> <lb n="1799,001,,,,4,8,3"></lb> |
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3389 <s>Τετμήσθω ἑκατέρα τῶν ΑΔ, ΑΒ δίχα κατὰ τὰ Ε, Ζ <lb n="1799,001,,,,4,8,4"></lb> |
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3390 σημεῖα, καὶ διὰ μὲν τοῦ Ε ὁποτέρᾳ τῶν ΑΒ, ΓΔ παράλ-<lb n="1799,001,,,,4,8,5"></lb> |
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3391 ληλος ἤχθω ἡ ΕΘ, διὰ δὲ τοῦ Ζ ὁποτέρᾳ τῶν ΑΔ, ΒΓ <lb n="1799,001,,,,4,8,6"></lb> |
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3392 παράλληλος ἤχθω ἡ ΖΚ· παραλληλόγραμμον ἄρα ἐστὶν <lb n="1799,001,,,,4,8,7"></lb> |
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3393 ἕκαστον τῶν ΑΚ, ΚΒ, ΑΘ, ΘΔ, ΑΗ, <lb n="1799,001,,,,4,8,8"></lb> |
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3394 ΗΓ, ΒΗ, ΗΔ, καὶ αἱ ἀπεναντίον αὐτῶν <lb n="1799,001,,,,4,8,9"></lb> |
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3395 πλευραὶ δηλονότι ἴσαι [εἰσίν].</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,4,8,10"></lb> |
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3396 ἴση ἐστὶν ἡ ΑΔ τῇ ΑΒ, καί ἐστι τῆς μὲν <lb n="1799,001,,,,4,8,11"></lb> |
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3397 ΑΔ ἡμίσεια ἡ ΑΕ, τῆς δὲ ΑΒ ἡμίσεια <lb n="1799,001,,,,4,8,12"></lb> |
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3398 ἡ ΑΖ, ἴση ἄρα καὶ ἡ ΑΕ τῇ ΑΖ· ὥστε <lb n="1799,001,,,,4,8,13"></lb> |
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3399 καὶ αἱ ἀπεναντίον· ἴση ἄρα καὶ ἡ ΖΗ <pb></pb> <lb n="1799,001,,,,4,8,14"></lb> |
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3400 τῇ ΗΕ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἑκατέρα τῶν ΗΘ, <lb n="1799,001,,,,4,8,15"></lb> |
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3401 ΗΚ ἑκατέρᾳ τῶν ΖΗ, ΗΕ ἐστιν ἴση· αἱ τέσσαρες ἄρα <lb n="1799,001,,,,4,8,16"></lb> |
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3402 αἱ ΗΕ, ΗΖ, ΗΘ, ΗΚ ἴσαι ἀλλήλαις [εἰσίν].</s> <s>ὁ ἄρα <lb n="1799,001,,,,4,8,17"></lb> |
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3403 κέντρῳ μὲν τῷ Η διαστήματι δὲ ἑνὶ τῶν Ε, Ζ, Θ, Κ <lb n="1799,001,,,,4,8,18"></lb> |
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3404 κύκλος γραφόμενος ἥξει καὶ διὰ τῶν λοιπῶν σημείων· <lb n="1799,001,,,,4,8,19"></lb> |
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3405 καὶ ἐφάψεται τῶν ΑΒ, ΒΓ, ΓΔ, ΔΑ εὐθειῶν διὰ τὸ <lb n="1799,001,,,,4,8,20"></lb> |
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3406 ὀρθὰς εἶναι τὰς πρὸς τοῖς Ε, Ζ, Θ, Κ γωνίας· εἰ γὰρ τεμεῖ <lb n="1799,001,,,,4,8,21"></lb> |
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3407 ὁ κύκλος τὰς ΑΒ, ΒΓ, ΓΔ, ΔΑ, ἡ τῇ διαμέτρῳ τοῦ <lb n="1799,001,,,,4,8,22"></lb> |
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3408 κύκλου πρὸς ὀρθὰς ἀπ' ἄκρας ἀγομένη ἐντὸς πεσεῖται τοῦ <lb n="1799,001,,,,4,8,23"></lb> |
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3409 κύκλου· ὅπερ ἄτοπον ἐδείχθη.</s> <s>οὐκ ἄρα ὁ κέντρῳ τῷ Η <lb n="1799,001,,,,4,8,24"></lb> |
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3410 διαστήματι δὲ ἑνὶ τῶν Ε, Ζ, Θ, Κ κύκλος γραφόμενος <lb n="1799,001,,,,4,8,25"></lb> |
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3411 τεμεῖ τὰς ΑΒ, ΒΓ, ΓΔ, ΔΑ εὐθείας.</s> <s>ἐφάψεται ἄρα αὐ-<lb n="1799,001,,,,4,8,26"></lb> |
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3412 τῶν καὶ ἔσται ἐγγεγραμμένος εἰς τὸ ΑΒΓΔ τετράγωνον.</s> <lb n="1799,001,,,,4,8,27"></lb> |
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3413 <s>Εἰς ἄρα τὸ δοθὲν τετράγωνον κύκλος ἐγγέγραπται· <lb n="1799,001,,,,4,8,28"></lb> |
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3414 ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,8,29"></lb> |
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3415 <s>Περὶ τὸ δοθὲν τετράγωνον κύκλον περιγράψαι.</s> <lb n="1799,001,,,,4,9,1"></lb> |
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3416 <s>Ἔστω τὸ δοθὲν τετράγωνον τὸ ΑΒΓΔ· δεῖ δὴ περὶ τὸ <lb n="1799,001,,,,4,9,2"></lb> |
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3417 ΑΒΓΔ τετράγωνον κύκλον περιγράψαι.</s> <lb n="1799,001,,,,4,9,3"></lb> |
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3418 <s>Ἐπιζευχθεῖσαι γὰρ αἱ ΑΓ, ΒΔ τεμνέτωσαν ἀλλήλας <lb n="1799,001,,,,4,9,4"></lb> |
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3419 κατὰ τὸ Ε.</s> <lb n="1799,001,,,,4,9,5"></lb> |
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3420 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΔΑ τῇ ΑΒ, κοινὴ δὲ ἡ ΑΓ, δύο <lb n="1799,001,,,,4,9,6"></lb> |
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3421 δὴ αἱ ΔΑ, ΑΓ δυσὶ ταῖς ΒΑ, ΑΓ ἴσαι εἰσίν· καὶ βάσις <pb></pb> <lb n="1799,001,,,,4,9,7"></lb> |
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3422 ἡ ΔΓ βάσει τῇ ΒΓ ἴση· γωνία ἄρα ἡ ὑπὸ ΔΑΓ γωνίᾳ <lb n="1799,001,,,,4,9,8"></lb> |
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3423 τῇ ὑπὸ ΒΑΓ ἴση ἐστίν· ἡ ἄρα ὑπὸ ΔΑΒ γωνία δίχα <lb n="1799,001,,,,4,9,9"></lb> |
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3424 τέτμηται ὑπὸ τῆς ΑΓ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἑκάστη <lb n="1799,001,,,,4,9,10"></lb> |
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3425 τῶν ὑπὸ ΑΒΓ, ΒΓΔ, ΓΔΑ δίχα τέτμηται <lb n="1799,001,,,,4,9,11"></lb> |
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3426 ὑπὸ τῶν ΑΓ, ΔΒ εὐθειῶν.</s> <s>καὶ ἐπεὶ ἴση <lb n="1799,001,,,,4,9,12"></lb> |
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3427 ἐστὶν ἡ ὑπὸ ΔΑΒ γωνία τῇ ὑπὸ ΑΒΓ, <lb n="1799,001,,,,4,9,13"></lb> |
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3428 καί ἐστι τῆς μὲν ὑπὸ ΔΑΒ ἡμίσεια ἡ <lb n="1799,001,,,,4,9,14"></lb> |
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3429 ὑπὸ ΕΑΒ, τῆς δὲ ὑπὸ ΑΒΓ ἡμίσεια ἡ <lb n="1799,001,,,,4,9,15"></lb> |
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3430 ὑπὸ ΕΒΑ, καὶ ἡ ὑπὸ ΕΑΒ ἄρα τῇ ὑπὸ <lb n="1799,001,,,,4,9,16"></lb> |
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3431 ΕΒΑ ἐστιν ἴση· ὥστε καὶ πλευρὰ ἡ ΕΑ <lb n="1799,001,,,,4,9,17"></lb> |
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3432 τῇ ΕΒ ἐστιν ἴση.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἑκατέρα <lb n="1799,001,,,,4,9,18"></lb> |
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3433 τῶν ΕΑ, ΕΒ [εὐθειῶν] ἑκατέρᾳ τῶν ΕΓ, ΕΔ ἴση ἐστίν.</s> <lb n="1799,001,,,,4,9,19"></lb> |
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3434 <s>αἱ τέσσαρες ἄρα αἱ ΕΑ, ΕΒ, ΕΓ, ΕΔ ἴσαι ἀλλήλαις <lb n="1799,001,,,,4,9,20"></lb> |
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3435 εἰσίν.</s> <s>ὁ ἄρα κέντρῳ τῷ Ε καὶ διαστήματι ἑνὶ τῶν Α, Β, <lb n="1799,001,,,,4,9,21"></lb> |
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3436 Γ, Δ κύκλος γραφόμενος ἥξει καὶ διὰ τῶν λοιπῶν σημείων <lb n="1799,001,,,,4,9,22"></lb> |
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3437 καὶ ἔσται περιγεγραμμένος περὶ τὸ ΑΒΓΔ τετράγωνον.</s> <lb n="1799,001,,,,4,9,23"></lb> |
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3438 <s>περιγεγράφθω ὡς ὁ ΑΒΓΔ.</s> <lb n="1799,001,,,,4,9,24"></lb> |
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3439 <s>Περὶ τὸ δοθὲν ἄρα τετράγωνον κύκλος περιγέγραπται· <lb n="1799,001,,,,4,9,25"></lb> |
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3440 ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,9,26"></lb> |
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3441 Ἰσοσκελὲς τρίγωνον συστήσασθαι ἔχον ἑκατέραν τῶν <lb n="1799,001,,,,4,10,1"></lb> |
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3442 πρὸς τῇ βάσει γωνιῶν διπλασίονα τῆς λοιπῆς.</s> <lb n="1799,001,,,,4,10,2"></lb> |
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3443 <s>Ἐκκείσθω τις εὐθεῖα ἡ ΑΒ, καὶ τετμήσθω κατὰ τὸ Γ <lb n="1799,001,,,,4,10,3"></lb> |
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3444 σημεῖον, ὥστε τὸ ὑπὸ τῶν ΑΒ, ΒΓ περιεχόμενον ὀρθο-<lb n="1799,001,,,,4,10,4"></lb> |
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3445 γώνιον ἴσον εἶναι τῷ ἀπὸ τῆς ΓΑ <lb n="1799,001,,,,4,10,5"></lb> |
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3446 τετραγώνῳ· καὶ κέντρῳ τῷ Α καὶ <lb n="1799,001,,,,4,10,6"></lb> |
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3447 διαστήματι τῷ ΑΒ κύκλος γεγρά-<lb n="1799,001,,,,4,10,7"></lb> |
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3448 φθω ὁ Β*δΕ, καὶ ἐνηρμόσθω εἰς τὸν <lb n="1799,001,,,,4,10,8"></lb> |
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3449 ΒΔΕ κύκλον τῇ ΑΓ εὐθείᾳ μὴ μεί-<lb n="1799,001,,,,4,10,9"></lb> |
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3450 ζονι οὔσῃ τῆς τοῦ ΒΔΕ κύκλου δια-<lb n="1799,001,,,,4,10,10"></lb> |
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3451 μέτρου ἴση εὐθεῖα ἡ ΒΔ· καὶ ἐπεζεύ-<lb n="1799,001,,,,4,10,11"></lb> |
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3452 χθωσαν αἱ ΑΔ, ΔΓ, καὶ περιγεγράφθω περὶ τὸ ΑΓΔ <lb n="1799,001,,,,4,10,12"></lb> |
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3453 τρίγωνον κύκλος ὁ ΑΓΔ.</s> <lb n="1799,001,,,,4,10,13"></lb> |
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3454 <s>Καὶ ἐπεὶ τὸ ὑπὸ τῶν ΑΒ, ΒΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,4,10,14"></lb> |
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3455 ΑΓ, ἴση δὲ ἡ ΑΓ τῇ ΒΔ, τὸ ἄρα ὑπὸ τῶν ΑΒ, ΒΓ ἴσον <lb n="1799,001,,,,4,10,15"></lb> |
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3456 ἐστὶ τῷ ἀπὸ τῆς ΒΔ.</s> <s>καὶ ἐπεὶ κύκλου τοῦ ΑΓΔ εἴληπταί <lb n="1799,001,,,,4,10,16"></lb> |
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3457 τι σημεῖον ἐκτὸς τὸ Β, καὶ ἀπὸ τοῦ Β πρὸς τὸν ΑΓΔ <lb n="1799,001,,,,4,10,17"></lb> |
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3458 κύκλον προσπεπτώκασι δύο εὐθεῖαι αἱ ΒΑ, ΒΔ, καὶ ἡ <lb n="1799,001,,,,4,10,18"></lb> |
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3459 μὲν αὐτῶν τέμνει, ἡ δὲ προσπίπτει, καί ἐστι τὸ ὑπὸ τῶν <pb></pb> <lb n="1799,001,,,,4,10,19"></lb> |
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3460 ΑΒ, ΒΓ ἴσον τῷ ἀπὸ τῆς ΒΔ, ἡ ΒΔ ἄρα ἐφάπτεται τοῦ <lb n="1799,001,,,,4,10,20"></lb> |
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3461 ΑΓΔ κύκλου.</s> <s>ἐπεὶ οὖν ἐφάπτεται μὲν ἡ ΒΔ, ἀπὸ δὲ τῆς <lb n="1799,001,,,,4,10,21"></lb> |
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3462 κατὰ τὸ Δ ἐπαφῆς διῆκται ἡ ΔΓ, ἡ ἄρα ὑπὸ ΒΔΓ γωνία <lb n="1799,001,,,,4,10,22"></lb> |
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3463 ἴση ἐστὶ τῇ ἐν τῷ ἐναλλὰξ τοῦ κύκλου τμήματι γωνίᾳ τῇ <lb n="1799,001,,,,4,10,23"></lb> |
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3464 ὑπὸ ΔΑΓ.</s> <s>ἐπεὶ οὖν ἴση ἐστὶν ἡ ὑπὸ ΒΔΓ τῇ ὑπὸ ΔΑΓ, <lb n="1799,001,,,,4,10,24"></lb> |
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3465 κοινὴ προσκείσθω ἡ ὑπὸ ΓΔΑ· ὅλη ἄρα ἡ ὑπὸ ΒΔΑ ἴση <lb n="1799,001,,,,4,10,25"></lb> |
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3466 ἐστὶ δυσὶ ταῖς ὑπὸ ΓΔΑ, ΔΑΓ.</s> <s>ἀλλὰ ταῖς ὑπὸ ΓΔΑ, <lb n="1799,001,,,,4,10,26"></lb> |
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3467 ΔΑΓ ἴση ἐστὶν ἡ ἐκτὸς ἡ ὑπὸ ΒΓΔ· καὶ ἡ ὑπὸ ΒΔΑ ἄρα <lb n="1799,001,,,,4,10,27"></lb> |
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3468 ἴση ἐστὶ τῇ ὑπὸ ΒΓΔ.</s> <s>ἀλλὰ ἡ ὑπὸ ΒΔΑ τῇ ὑπὸ ΓΒΔ <lb n="1799,001,,,,4,10,28"></lb> |
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3469 ἐστιν ἴση, ἐπεὶ καὶ πλευρὰ ἡ ΑΔ τῇ ΑΒ ἐστιν ἴση· ὥστε <lb n="1799,001,,,,4,10,29"></lb> |
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3470 καὶ ἡ ὑπὸ ΔΒΑ τῇ ὑπὸ ΒΓΔ ἐστιν ἴση.</s> <s>αἱ τρεῖς ἄρα αἱ <lb n="1799,001,,,,4,10,30"></lb> |
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3471 ὑπὸ ΒΔΑ, ΔΒΑ, ΒΓΔ ἴσαι ἀλλήλαις εἰσίν.</s> <s>καὶ ἐπεὶ ἴση <lb n="1799,001,,,,4,10,31"></lb> |
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3472 ἐστὶν ἡ ὑπὸ ΔΒΓ γωνία τῇ ὑπὸ ΒΓΔ, ἴση ἐστὶ καὶ <lb n="1799,001,,,,4,10,32"></lb> |
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3473 πλευρὰ ἡ ΒΔ πλευρᾷ τῇ ΔΓ.</s> <s>ἀλλὰ ἡ ΒΔ τῇ ΓΑ ὑπόκειται <lb n="1799,001,,,,4,10,33"></lb> |
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3474 ἴση· καὶ ἡ ΓΑ ἄρα τῇ ΓΔ ἐστιν ἴση· ὥστε καὶ γωνία ἡ <lb n="1799,001,,,,4,10,34"></lb> |
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3475 ὑπὸ ΓΔΑ γωνίᾳ τῇ ὑπὸ ΔΑΓ ἐστιν ἴση· αἱ ἄρα ὑπὸ <lb n="1799,001,,,,4,10,35"></lb> |
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3476 ΓΔΑ, ΔΑΓ τῆς ὑπὸ ΔΑΓ εἰσι διπλασίους.</s> <s>ἴση δὲ ἡ <lb n="1799,001,,,,4,10,36"></lb> |
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3477 ὑπὸ ΒΓΔ ταῖς ὑπὸ ΓΔΑ, ΔΑΓ· καὶ ἡ ὑπὸ ΒΓΔ ἄρα τῆς <pb></pb> <lb n="1799,001,,,,4,10,37"></lb> |
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3478 ὑπὸ ΓΑΔ ἐστι διπλῆ.</s> <s>ἴση δὲ ἡ ὑπὸ ΒΓΔ ἑκατέρᾳ τῶν <lb n="1799,001,,,,4,10,38"></lb> |
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3479 ὑπὸ ΒΔΑ, ΔΒΑ· καὶ ἑκατέρα ἄρα τῶν ὑπὸ ΒΔΑ, ΔΒΑ <lb n="1799,001,,,,4,10,39"></lb> |
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3480 τῆς ὑπὸ ΔΑΒ ἐστι διπλῆ.</s> <lb n="1799,001,,,,4,10,40"></lb> |
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3481 <s>Ἰσοσκελὲς ἄρα τρίγωνον συνέσταται τὸ ΑΒΔ ἔχον <lb n="1799,001,,,,4,10,41"></lb> |
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3482 ἑκατέραν τῶν πρὸς τῇ ΔΒ βάσει γωνιῶν διπλασίονα τῆς <lb n="1799,001,,,,4,10,42"></lb> |
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3483 λοιπῆς· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,10,43"></lb> |
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3484 <s>Εἰς τὸν δοθέντα κύκλον πεντάγωνον ἰσόπλευρόν τε <lb n="1799,001,,,,4,11,1"></lb> |
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3485 καὶ ἰσογώνιον ἐγγράψαι.</s> <lb n="1799,001,,,,4,11,2"></lb> |
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3486 <s>Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓΔΕ· δεῖ δὴ εἰς τὸν <lb n="1799,001,,,,4,11,3"></lb> |
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3487 ΑΒΓΔΕ κύκλον πεντάγωνον ἰσόπλευρόν τε καὶ ἰσογώ-<lb n="1799,001,,,,4,11,4"></lb> |
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3488 νιον ἐγγράψαι.</s> <lb n="1799,001,,,,4,11,5"></lb> |
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3489 <s>Ἐκκείσθω τρίγωνον ἰσοσκελὲς τὸ ΖΗΘ διπλασίονα <lb n="1799,001,,,,4,11,6"></lb> |
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3490 ἔχον ἑκατέραν τῶν πρὸς τοῖς Η, Θ γωνιῶν τῆς πρὸς τῷ <lb n="1799,001,,,,4,11,7"></lb> |
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3491 Ζ, καὶ ἐγγεγράφθω εἰς τὸν ΑΒΓΔΕ κύκλον τῷ ΖΗΘ <lb n="1799,001,,,,4,11,8"></lb> |
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3492 τριγώνῳ ἰσογώνιον τρίγωνον τὸ ΑΓΔ, ὥστε τῇ μὲν πρὸς <lb n="1799,001,,,,4,11,9"></lb> |
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3493 τῷ Ζ γωνίᾳ ἴσην εἶναι τὴν ὑπὸ <lb n="1799,001,,,,4,11,10"></lb> |
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3494 ΓΑΔ, ἑκατέραν δὲ τῶν πρὸς τοῖς <lb n="1799,001,,,,4,11,11"></lb> |
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3495 Η, Θ ἴσην ἑκατέρᾳ τῶν ὑπὸ ΑΓΔ, <lb n="1799,001,,,,4,11,12"></lb> |
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3496 ΓΔΑ· καὶ ἑκατέρα ἄρα τῶν ὑπὸ <lb n="1799,001,,,,4,11,13"></lb> |
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3497 ΑΓΔ, ΓΔΑ τῆς ὑπὸ ΓΑΔ ἐστι <lb n="1799,001,,,,4,11,14"></lb> |
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3498 διπλῆ.</s> <s>τετμήσθω δὴ ἑκατέρα τῶν <pb></pb> <lb n="1799,001,,,,4,11,15"></lb> |
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3499 ὑπὸ ΑΓΔ, ΓΔΑ δίχα ὑπὸ ἑκατέρας τῶν ΓΕ, ΔΒ εὐθειῶν, <lb n="1799,001,,,,4,11,16"></lb> |
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3500 καὶ ἐπεζεύχθωσαν αἱ ΑΒ, ΒΓ, [ΓΔ], ΔΕ, ΕΑ.</s> <lb n="1799,001,,,,4,11,17"></lb> |
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3501 <s>Ἐπεὶ οὖν ἑκατέρα τῶν ὑπὸ ΑΓΔ, ΓΔΑ γωνιῶν διπλα-<lb n="1799,001,,,,4,11,18"></lb> |
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3502 σίων ἐστὶ τῆς ὑπὸ ΓΑΔ, καὶ τετμημέναι εἰσὶ δίχα ὑπὸ <lb n="1799,001,,,,4,11,19"></lb> |
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3503 τῶν ΓΕ, ΔΒ εὐθειῶν, αἱ πέντε ἄρα γωνίαι αἱ ὑπὸ ΔΑΓ, <lb n="1799,001,,,,4,11,20"></lb> |
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3504 ΑΓΕ, ΕΓΔ, ΓΔΒ, ΒΔΑ ἴσαι ἀλλήλαις εἰσίν.</s> <s>αἱ δὲ ἴσαι <lb n="1799,001,,,,4,11,21"></lb> |
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3505 γωνίαι ἐπὶ ἴσων περιφερειῶν βεβήκασιν· αἱ πέντε ἄρα <lb n="1799,001,,,,4,11,22"></lb> |
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3506 περιφέρειαι αἱ ΑΒ, ΒΓ, ΓΔ, ΔΕ, ΕΑ ἴσαι ἀλλήλαις <lb n="1799,001,,,,4,11,23"></lb> |
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3507 εἰσίν.</s> <s>ὑπὸ δὲ τὰς ἴσας περιφερείας ἴσαι εὐθεῖαι ὑποτείνου-<lb n="1799,001,,,,4,11,24"></lb> |
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3508 σιν· αἱ πέντε ἄρα εὐθεῖαι αἱ ΑΒ, ΒΓ, ΓΔ, ΔΕ, ΕΑ ἴσαι <lb n="1799,001,,,,4,11,25"></lb> |
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3509 ἀλλήλαις εἰσίν· ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΒΓΔΕ πεντά-<lb n="1799,001,,,,4,11,26"></lb> |
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3510 γωνον.</s> <s>λέγω δή, ὅτι καὶ ἰσογώνιον.</s> <s>ἐπεὶ γὰρ ἡ ΑΒ περι-<lb n="1799,001,,,,4,11,27"></lb> |
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3511 φέρεια τῇ ΔΕ περιφερείᾳ ἐστὶν ἴση, κοινὴ προσκείσθω <lb n="1799,001,,,,4,11,28"></lb> |
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3512 ἡ ΒΓΔ· ὅλη ἄρα ἡ ΑΒΓΔ περιφέρεια ὅλῃ τῇ ΕΔΓΒ <lb n="1799,001,,,,4,11,29"></lb> |
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3513 περιφερείᾳ ἐστὶν ἴση.</s> <s>καὶ βέβηκεν ἐπὶ μὲν τῆς ΑΒΓΔ <lb n="1799,001,,,,4,11,30"></lb> |
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3514 περιφερείας γωνία ἡ ὑπὸ ΑΕΔ, ἐπὶ δὲ τῆς ΕΔΓΒ περι-<lb n="1799,001,,,,4,11,31"></lb> |
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3515 φερείας γωνία ἡ ὑπὸ ΒΑΕ· καὶ ἡ ὑπὸ ΒΑΕ ἄρα γωνία <lb n="1799,001,,,,4,11,32"></lb> |
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3516 τῇ ὑπὸ ΑΕΔ ἐστιν ἴση.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἑκάστη τῶν <lb n="1799,001,,,,4,11,33"></lb> |
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3517 ὑπὸ ΑΒΓ, ΒΓΔ, ΓΔΕ γωνιῶν ἑκατέρᾳ τῶν ὑπὸ ΒΑΕ, <lb n="1799,001,,,,4,11,34"></lb> |
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3518 ΑΕΔ ἐστιν ἴση· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓΔΕ πεντά-<lb n="1799,001,,,,4,11,35"></lb> |
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3519 γωνον.</s> <s>ἐδείχθη δὲ καὶ ἰσόπλευρον.</s> <lb n="1799,001,,,,4,11,36"></lb> |
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3520 <s>Εἰς ἄρα τὸν δοθέντα κύκλον πεντάγωνον ἰσόπλευρόν τε <lb n="1799,001,,,,4,11,37"></lb> |
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3521 καὶ ἰσογώνιον ἐγγέγραπται· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,11,38"></lb> |
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3522 Περὶ τὸν δοθέντα κύκλον πεντάγωνον ἰσόπλευρόν τε <lb n="1799,001,,,,4,12,1"></lb> |
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3523 καὶ ἰσογώνιον περιγράψαι.</s> <lb n="1799,001,,,,4,12,2"></lb> |
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3524 <s>Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓΔΕ· δεῖ <ὴ>περὶ τὸν <lb n="1799,001,,,,4,12,3"></lb> |
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3525 ΑΒΓΔΕ κύκλον πεντάγωνον ἰσόπλευρόν τε καὶ ἰσογώνιον <lb n="1799,001,,,,4,12,4"></lb> |
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3526 περιγράψαι.</s> <lb n="1799,001,,,,4,12,5"></lb> |
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3527 <s>Νενοήσθω τοῦ ἐγγεγραμμένου πενταγώνου τῶν γω-<lb n="1799,001,,,,4,12,6"></lb> |
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3528 νιῶν σημεῖα τὰ Α, Β, Γ, Δ, Ε, ὥστε ἴσας εἶναι τὰς ΑΒ, <lb n="1799,001,,,,4,12,7"></lb> |
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3529 ΒΓ, ΓΔ, ΔΕ, ΕΑ περιφερείας· καὶ διὰ τῶν Α, Β, Γ, Δ, <lb n="1799,001,,,,4,12,8"></lb> |
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3530 Ε ἤχθωσαν τοῦ κύκλου ἐφαπτόμεναι αἱ ΗΘ, ΘΚ, ΚΛ, <lb n="1799,001,,,,4,12,9"></lb> |
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3531 ΛΜ, ΜΗ, καὶ εἰλήφθω τοῦ ΑΒΓΔΕ κύκλου κέντρον <lb n="1799,001,,,,4,12,10"></lb> |
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3532 τὸ Ζ, καὶ ἐπεζεύχθωσαν αἱ ΖΒ, ΖΚ, ΖΓ, ΖΛ, ΖΔ.</s> <lb n="1799,001,,,,4,12,11"></lb> |
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3533 <s>Καὶ ἐπεὶ ἡ μὲν ΚΛ εὐθεῖα ἐφάπτεται τοῦ <lb n="1799,001,,,,4,12,12"></lb> |
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3534 ΑΒΓΔΕ κατὰ τὸ Γ, ἀπὸ δὲ τοῦ Ζ κέντρου <lb n="1799,001,,,,4,12,13"></lb> |
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3535 ἐπὶ τὴν κατὰ τὸ Γ ἐπαφὴν ἐπέζευκται ἡ ΖΓ, <lb n="1799,001,,,,4,12,14"></lb> |
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3536 ἡ ΖΓ ἄρα κάθετός ἐστιν ἐπὶ τὴν ΚΛ· ὀρθὴ <lb n="1799,001,,,,4,12,15"></lb> |
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3537 ἄρα ἐστὶν ἑκατέρα τῶν πρὸς τῷ Γ γωνιῶν.</s> <lb n="1799,001,,,,4,12,16"></lb> |
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3538 <s>διὰ τὰ αὐτὰ δὴ καὶ αἱ πρὸς τοῖς Β, Δ σημείοις <lb n="1799,001,,,,4,12,17"></lb> |
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3539 γωνίαι ὀρθαί εἰσιν.</s> <s>καὶ ἐπεὶ ὀρθή ἐστιν ἡ ὑπὸ ΖΓΚ γωνία, <lb n="1799,001,,,,4,12,18"></lb> |
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3540 τὸ ἄρα ἀπὸ τῆς ΖΚ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΖΓ, ΓΚ.</s> <s>διὰ τὰ <lb n="1799,001,,,,4,12,19"></lb> |
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3541 αὐτὰ δὴ καὶ τοῖς ἀπὸ τῶν ΖΒ, ΒΚ ἴσον ἐστὶ τὸ ἀπὸ τῆς <lb n="1799,001,,,,4,12,20"></lb> |
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3542 ΖΚ· ὥστε τὰ ἀπὸ τῶν ΖΓ, ΓΚ τοῖς ἀπὸ τῶν ΖΒ, ΒΚ <pb></pb> <lb n="1799,001,,,,4,12,21"></lb> |
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3543 ἐστιν ἴσα, ὧν τὸ ἀπὸ τῆς ΖΓ τῷ ἀπὸ τῆς ΖΒ ἐστιν ἴσον· <lb n="1799,001,,,,4,12,22"></lb> |
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3544 λοιπὸν ἄρα τὸ ἀπὸ τῆς ΓΚ τῷ ἀπὸ τῆς ΒΚ ἐστιν ἴσον.</s> <lb n="1799,001,,,,4,12,23"></lb> |
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3545 <s>ἴση ἄρα ἡ ΒΚ τῇ ΓΚ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΖΒ τῇ ΖΓ, <lb n="1799,001,,,,4,12,24"></lb> |
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3546 καὶ κοινὴ ἡ ΖΚ, δύο δὴ αἱ ΒΖ, ΖΚ δυσὶ ταῖς ΓΖ, ΖΚ <lb n="1799,001,,,,4,12,25"></lb> |
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3547 ἴσαι εἰσίν· καὶ βάσις ἡ ΒΚ βάσει τῇ ΓΚ [ἐστιν] ἴση· <lb n="1799,001,,,,4,12,26"></lb> |
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3548 γωνία ἄρα ἡ μὲν ὑπὸ ΒΖΚ [γωνίᾳ] τῇ ὑπὸ ΚΖΓ ἐστιν <lb n="1799,001,,,,4,12,27"></lb> |
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3549 ἴση· ἡ δὲ ὑπὸ ΒΚΖ τῇ ὑπὸ ΖΚΓ· διπλῆ ἄρα ἡ μὲν ὑπὸ <lb n="1799,001,,,,4,12,28"></lb> |
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3550 ΒΖΓ τῆς ὑπὸ ΚΖΓ, ἡ δὲ ὑπὸ ΒΚΓ τῆς ὑπὸ ΖΚΓ.</s> <s>διὰ <lb n="1799,001,,,,4,12,29"></lb> |
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3551 τὰ αὐτὰ δὴ καὶ ἡ μὲν ὑπὸ ΓΖΔ τῆς ὑπὸ ΓΖΛ ἐστι διπλῆ, <lb n="1799,001,,,,4,12,30"></lb> |
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3552 ἡ δὲ ὑπὸ ΔΛΓ τῆς ὑπὸ ΖΛΓ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΓ <lb n="1799,001,,,,4,12,31"></lb> |
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3553 περιφέρεια τῇ ΓΔ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΒΖΓ τῇ <lb n="1799,001,,,,4,12,32"></lb> |
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3554 ὑπὸ ΓΖΔ.</s> <s>καί ἐστιν ἡ μὲν ὑπὸ ΒΖΓ τῆς ὑπὸ ΚΖΓ <lb n="1799,001,,,,4,12,33"></lb> |
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3555 διπλῆ, ἡ δὲ ὑπὸ ΔΖΓ τῆς ὑπὸ ΛΖΓ· ἴση ἄρα καὶ ἡ ὑπὸ <lb n="1799,001,,,,4,12,34"></lb> |
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3556 ΚΖΓ τῇ ὑπὸ ΛΖΓ· ἐστὶ δὲ καὶ ἡ ὑπὸ ΖΓΚ γωνία τῇ <lb n="1799,001,,,,4,12,35"></lb> |
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3557 ὑπὸ ΖΓΛ ἴση.</s> <s>δύο δὴ τρίγωνά ἐστι τὰ ΖΚΓ, ΖΛΓ τὰς <lb n="1799,001,,,,4,12,36"></lb> |
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3558 δύο γωνίας ταῖς δυσὶ γωνίαις ἴσας ἔχοντα καὶ μίαν πλευ-<lb n="1799,001,,,,4,12,37"></lb> |
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3559 ρὰν μιᾷ πλευρᾷ ἴσην κοινὴν αὐτῶν τὴν ΖΓ· καὶ τὰς <lb n="1799,001,,,,4,12,38"></lb> |
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3560 λοιπὰς ἄρα πλευρὰς ταῖς λοιπαῖς πλευραῖς ἴσας ἕξει καὶ τὴν <lb n="1799,001,,,,4,12,39"></lb> |
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3561 λοιπὴν γωνίαν τῇ λοιπῇ γωνίᾳ· ἴση ἄρα ἡ μὲν ΚΓ εὐθεῖα <pb></pb> <lb n="1799,001,,,,4,12,40"></lb> |
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3562 τῇ ΓΛ, ἡ δὲ ὑπὸ ΖΚΓ γωνία τῇ ὑπὸ ΖΛΓ.</s> <s>καὶ ἐπεὶ ἴση <lb n="1799,001,,,,4,12,41"></lb> |
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3563 ἐστὶν ἡ ΚΓ τῇ ΓΛ, διπλῆ ἄρα ἡ ΚΛ τῆς ΚΓ.</s> <s>διὰ τὰ <lb n="1799,001,,,,4,12,42"></lb> |
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3564 αὐτὰ δὴ δειχθήσεται καὶ ἡ ΘΚ τῆς ΒΚ διπλῆ.</s> <s>καί ἐστιν <lb n="1799,001,,,,4,12,43"></lb> |
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3565 ἡ ΒΚ τῇ ΚΓ ἴση· καὶ ἡ ΘΚ ἄρα τῇ ΚΛ ἐστιν ἴση.</s> <lb n="1799,001,,,,4,12,44"></lb> |
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3566 <s>ὁμοίως δὴ δειχθήσεται καὶ ἑκάστη τῶν ΘΗ, ΗΜ, ΜΛ <lb n="1799,001,,,,4,12,45"></lb> |
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3567 ἑκατέρᾳ τῶν ΘΚ, ΚΛ ἴση· ἰσόπλευρον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,4,12,46"></lb> |
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3568 ΗΘΚΛΜ πεντάγωνον.</s> <s>λέγω δή, ὅτι καὶ ἰσογώνιον.</s> <s>ἐπεὶ <lb n="1799,001,,,,4,12,47"></lb> |
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3569 γὰρ ἴση ἐστὶν ἡ ὑπὸ ΖΚΓ γωνία τῇ ὑπὸ ΖΛΓ, καὶ <lb n="1799,001,,,,4,12,48"></lb> |
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3570 ἐδείχθη τῆς μὲν ὑπὸ ΖΚΓ διπλῆ ἡ ὑπὸ ΘΚΛ, τῆς δὲ <lb n="1799,001,,,,4,12,49"></lb> |
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3571 ὑπὸ ΖΛΓ διπλῆ ἡ ὑπὸ ΚΛΜ, καὶ ἡ ὑπὸ ΘΚΛ ἄρα τῇ <lb n="1799,001,,,,4,12,50"></lb> |
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3572 ὑπὸ ΚΛΜ ἐστιν ἴση.</s> <s>ὁμοίως δὴ δειχθήσεται καὶ ἑκάστη <lb n="1799,001,,,,4,12,51"></lb> |
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3573 τῶν ὑπὸ ΚΘΗ, ΘΗΜ, ΗΜΛ ἑκατέρᾳ τῶν ὑπὸ ΘΚΛ, <lb n="1799,001,,,,4,12,52"></lb> |
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3574 ΚΛΜ ἴση· αἱ πέντε ἄρα γωνίαι αἱ ὑπὸ ΗΘΚ, ΘΚΛ, <lb n="1799,001,,,,4,12,53"></lb> |
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3575 ΚΛΜ, ΛΜΗ, ΜΗΘ ἴσαι ἀλλήλαις εἰσίν.</s> <s>ἰσογώνιον <lb n="1799,001,,,,4,12,54"></lb> |
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3576 ἄρα ἐστὶ τὸ ΗΘΚΛΜ πεντάγωνον.</s> <s>ἐδείχθη δὲ καὶ <lb n="1799,001,,,,4,12,55"></lb> |
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3577 ἰσόπλευρον, καὶ περιγέγραπται περὶ τὸν ΑΒΓΔΕ κύκλον.</s> <lb n="1799,001,,,,4,12,56"></lb> |
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3578 <s>[Περὶ τὸν δοθέντα ἄρα κύκλον πεντάγωνον ἰσόπλευρόν <lb n="1799,001,,,,4,12,57"></lb> |
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3579 τε καὶ ἰσογώνιον περιγέγραπται]· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,12,58"></lb> |
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3580 <s>Εἰς τὸ δοθὲν πεντάγωνον, ὅ ἐστιν ἰσόπλευρόν τε καὶ <lb n="1799,001,,,,4,13,1"></lb> |
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3581 ἰσογώνιον, κύκλον ἐγγράψαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,13,2"></lb> |
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3582 Ἔστω τὸ δοθὲν πεντάγωνον ἰσόπλευρόν τε καὶ ἰσογώνιον <lb n="1799,001,,,,4,13,3"></lb> |
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3583 τὸ ΑΒΓΔΕ· δεῖ δὴ εἰς τὸ ΑΒΓΔΕ πεντάγωνον κύκλον <lb n="1799,001,,,,4,13,4"></lb> |
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3584 ἐγγράψαι.</s> <lb n="1799,001,,,,4,13,5"></lb> |
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3585 <s>Τετμήσθω γὰρ ἑκατέρα τῶν ὑπὸ ΒΓΔ, ΓΔΕ γωνιῶν <lb n="1799,001,,,,4,13,6"></lb> |
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3586 δίχα ὑπὸ ἑκατέρας τῶν ΓΖ, ΔΖ εὐθειῶν· καὶ ἀπὸ τοῦ <lb n="1799,001,,,,4,13,7"></lb> |
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3587 Ζ σημείου, καθ' ὃ συμβάλλουσιν ἀλλήλαις αἱ ΓΖ, ΔΖ <lb n="1799,001,,,,4,13,8"></lb> |
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3588 εὐθεῖαι, ἐπεζεύχθωσαν αἱ ΖΒ, ΖΑ, ΖΕ εὐθεῖαι.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,4,13,9"></lb> |
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3589 ἴση ἐστὶν ἡ ΒΓ τῇ ΓΔ, κοινὴ δὲ ἡ ΓΖ, δύο δὴ αἱ ΒΓ, ΓΖ <lb n="1799,001,,,,4,13,10"></lb> |
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3590 δυσὶ ταῖς ΔΓ, ΓΖ ἴσαι εἰσίν· καὶ γωνία ἡ <lb n="1799,001,,,,4,13,11"></lb> |
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3591 ὑπὸ ΒΓΖ γωνίᾳ τῇ ὑπὸ ΔΓΖ [ἐστιν] ἴση· <lb n="1799,001,,,,4,13,12"></lb> |
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3592 βάσις ἄρα ἡ ΒΖ βάσει τῇ ΔΖ ἐστιν ἴση, <lb n="1799,001,,,,4,13,13"></lb> |
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3593 καὶ τὸ ΒΓΖ τρίγωνον τῷ ΔΓΖ τριγώνῳ <lb n="1799,001,,,,4,13,14"></lb> |
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3594 ἐστιν ἴσον, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοι-<lb n="1799,001,,,,4,13,15"></lb> |
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3595 παῖς γωνίαις ἴσαι ἔσονται, ὑφ' ἃς αἱ ἴσαι <lb n="1799,001,,,,4,13,16"></lb> |
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3596 πλευραὶ ὑποτείνουσιν· ἴση ἄρα ἡ ὑπὸ <lb n="1799,001,,,,4,13,17"></lb> |
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3597 ΓΒΖ γωνία τῇ ὑπὸ ΓΔΖ.</s> <s>καὶ ἐπεὶ διπλῆ ἐστιν ἡ ὑπὸ <lb n="1799,001,,,,4,13,18"></lb> |
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3598 ΓΔΕ τῆς ὑπὸ ΓΔΖ, ἴση δὲ ἡ μὲν ὑπὸ ΓΔΕ τῇ ὑπὸ ΑΒΓ, <lb n="1799,001,,,,4,13,19"></lb> |
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3599 ἡ δὲ ὑπὸ ΓΔΖ τῇ ὑπὸ ΓΒΖ, καὶ ἡ ὑπὸ ΓΒΑ ἄρα τῆς ὑπὸ <lb n="1799,001,,,,4,13,20"></lb> |
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3600 ΓΒΖ ἐστι διπλῆ· ἴση ἄρα ἡ ὑπὸ ΑΒΖ γωνία τῇ ὑπὸ <lb n="1799,001,,,,4,13,21"></lb> |
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3601 ΖΒΓ· ἡ ἄρα ὑπὸ ΑΒΓ γωνία δίχα τέτμηται ὑπὸ τῆς ΒΖ <lb n="1799,001,,,,4,13,22"></lb> |
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3602 εὐθείας.</s> <s>ὁμοίως δὴ δειχθήσεται, ὅτι καὶ ἑκατέρα τῶν <lb n="1799,001,,,,4,13,23"></lb> |
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3603 ὑπὸ ΒΑΕ, ΑΕΔ δίχα τέτμηται ὑπὸ ἑκατέρας τῶν ΖΑ, <lb n="1799,001,,,,4,13,24"></lb> |
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3604 ΖΕ εὐθειῶν.</s> <s>ἤχθωσαν δὴ ἀπὸ τοῦ Ζ σημείου ἐπὶ τὰς ΑΒ, <lb n="1799,001,,,,4,13,25"></lb> |
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3605 ΒΓ, ΓΔ, ΔΕ, ΕΑ εὐθείας κάθετοι αἱ ΖΗ, ΖΘ, ΖΚ, <lb n="1799,001,,,,4,13,26"></lb> |
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3606 ΖΛ, ΖΜ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ὑπὸ ΘΓΖ γωνία τῇ ὑπὸ <pb></pb> <lb n="1799,001,,,,4,13,27"></lb> |
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3607 ΚΓΖ, ἐστὶ δὲ καὶ ὀρθὴ ἡ ὑπὸ ΖΘΓ [ὀρθῇ] τῇ ὑπὸ ΖΚΓ <lb n="1799,001,,,,4,13,28"></lb> |
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3608 ἴση, δύο δὴ τρίγωνά ἐστι τὰ ΖΘΓ, ΖΚΓ τὰς δύο γωνίας <lb n="1799,001,,,,4,13,29"></lb> |
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3609 δυσὶ γωνίαις ἴσας ἔχοντα καὶ μίαν πλευρὰν μιᾷ πλευρᾷ <lb n="1799,001,,,,4,13,30"></lb> |
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3610 ἴσην κοινὴν αὐτῶν τὴν ΖΓ ὑποτείνουσαν ὑπὸ μίαν τῶν <lb n="1799,001,,,,4,13,31"></lb> |
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3611 ἴσων γωνιῶν· καὶ τὰς λοιπὰς ἄρα πλευρὰς ταῖς λοιπαῖς <lb n="1799,001,,,,4,13,32"></lb> |
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3612 πλευραῖς ἴσας ἕξει· ἴση ἄρα ἡ ΖΘ κάθετος τῇ ΖΚ καθ-<lb n="1799,001,,,,4,13,33"></lb> |
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3613 έτῳ.</s> <s>ὁμοίως δὴ δειχθήσεται, ὅτι καὶ ἑκάστη τῶν ΖΛ, <lb n="1799,001,,,,4,13,34"></lb> |
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3614 ΖΜ, ΖΗ ἑκατέρᾳ τῶν ΖΘ, ΖΚ ἴση ἐστίν· αἱ πέντε ἄρα <lb n="1799,001,,,,4,13,35"></lb> |
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3615 εὐθεῖαι αἱ ΖΗ, ΖΘ, ΖΚ, ΖΛ, ΖΜ ἴσαι ἀλλήλαις εἰσίν.</s> <lb n="1799,001,,,,4,13,36"></lb> |
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3616 <s>ὁ ἄρα κέντρῳ τῷ Ζ διαστήματι δὲ ἑνὶ τῶν Η, Θ, Κ, Λ, Μ <lb n="1799,001,,,,4,13,37"></lb> |
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3617 κύκλος γραφόμενος ἥξει καὶ διὰ τῶν λοιπῶν σημείων καὶ <lb n="1799,001,,,,4,13,38"></lb> |
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3618 ἐφάψεται τῶν ΑΒ, ΒΓ, ΓΔ, ΔΕ, ΕΑ εὐθειῶν διὰ τὸ <lb n="1799,001,,,,4,13,39"></lb> |
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3619 ὀρθὰς εἶναι τὰς πρὸς τοῖς Η, Θ, Κ, Λ, Μ σημείοις γω-<lb n="1799,001,,,,4,13,40"></lb> |
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3620 νίας.</s> <s>εἰ γὰρ οὐκ ἐφάψεται αὐτῶν, ἀλλὰ τεμεῖ αὐτάς, συμ-<lb n="1799,001,,,,4,13,41"></lb> |
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3621 βήσεται τὴν τῇ διαμέτρῳ τοῦ κύκλου πρὸς ὀρθὰς ἀπ' <lb n="1799,001,,,,4,13,42"></lb> |
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3622 ἄκρας ἀγομένην ἐντὸς πίπτειν τοῦ κύκλου· ὅπερ ἄτοπον <lb n="1799,001,,,,4,13,43"></lb> |
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3623 ἐδείχθη.</s> <s>οὐκ ἄρα ὁ κέντρῳ τῷ Ζ διαστήματι δὲ ἑνὶ τῶν <lb n="1799,001,,,,4,13,44"></lb> |
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3624 Η, Θ, Κ, Λ, Μ σημείων γραφόμενος κύκλος τεμεῖ τὰς <lb n="1799,001,,,,4,13,45"></lb> |
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3625 ΑΒ, ΒΓ, ΓΔ, ΔΕ, ΕΑ εὐθείας· ἐφάψεται ἄρα αὐτῶν.</s> <lb n="1799,001,,,,4,13,46"></lb> |
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3626 <s>γεγράφθω ὡς ὁ ΗΘΚΛΜ.</s> <lb n="1799,001,,,,4,13,47"></lb> |
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3627 <s>Εἰς ἄρα τὸ δοθὲν πεντάγωνον, ὅ ἐστιν ἰσόπλευρόν τε <lb n="1799,001,,,,4,13,48"></lb> |
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3628 καὶ ἰσογώνιον, κύκλος ἐγγέγραπται· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,13,49"></lb> |
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3629 <s>Περὶ τὸ δοθὲν πεντάγωνον, ὅ ἐστιν ἰσόπλευρόν τε καὶ <lb n="1799,001,,,,4,14,1"></lb> |
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3630 ἰσογώνιον, κύκλον περιγράψαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,14,2"></lb> |
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3631 Ἔστω τὸ δοθὲν πεντάγωνον, ὅ ἐστιν ἰσόπλευρόν τε καὶ <lb n="1799,001,,,,4,14,3"></lb> |
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3632 ἰσογώνιον, τὸ ΑΒΓΔΕ· δεῖ δὴ περὶ τὸ ΑΒΓΔΕ πεντά-<lb n="1799,001,,,,4,14,4"></lb> |
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3633 γωνον κύκλον περιγράψαι.</s> <lb n="1799,001,,,,4,14,5"></lb> |
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3634 <s>Τετμήσθω δὴ ἑκατέρα τῶν ὑπὸ ΒΓΔ, ΓΔΕ γωνιῶν <lb n="1799,001,,,,4,14,6"></lb> |
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3635 δίχα ὑπὸ ἑκατέρας τῶν ΓΖ, ΔΖ, καὶ ἀπὸ τοῦ Ζ σημείου, <lb n="1799,001,,,,4,14,7"></lb> |
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3636 καθ' ὃ συμβάλλουσιν αἱ εὐθεῖαι, ἐπὶ τὰ Β, Α, Ε σημεῖα <lb n="1799,001,,,,4,14,8"></lb> |
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3637 ἐπεζεύχθωσαν εὐθεῖαι αἱ ΖΒ, ΖΑ, ΖΕ.</s> <s>ὁμοίως δὴ τῷ πρὸ <lb n="1799,001,,,,4,14,9"></lb> |
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3638 τούτου δειχθήσεται, ὅτι καὶ ἑκάστη τῶν ὑπὸ ΓΒΑ, ΒΑΕ, <lb n="1799,001,,,,4,14,10"></lb> |
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3639 ΑΕΔ γωνιῶν δίχα τέτμηται ὑπὸ ἑκάστης τῶν ΖΒ, ΖΑ, <lb n="1799,001,,,,4,14,11"></lb> |
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3640 ΖΕ εὐθειῶν.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ὑπὸ ΒΓΔ γωνία τῇ ὑπὸ <lb n="1799,001,,,,4,14,12"></lb> |
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3641 ΓΔΕ, καί ἐστι τῆς μὲν ὑπὸ ΒΓΔ ἡμίσεια <lb n="1799,001,,,,4,14,13"></lb> |
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3642 ἡ ὑπὸ ΖΓΔ, τῆς δὲ ὑπὸ ΓΔΕ ἡμίσεια ἡ <lb n="1799,001,,,,4,14,14"></lb> |
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3643 ὑπὸ ΓΔΖ, καὶ ἡ ὑπὸ ΖΓΔ ἄρα τῇ ὑπὸ <lb n="1799,001,,,,4,14,15"></lb> |
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3644 ΖΔΓ ἐστιν ἴση· ὥστε καὶ πλευρὰ ἡ ΖΓ <lb n="1799,001,,,,4,14,16"></lb> |
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3645 πλευρᾷ τῇ ΖΔ ἐστιν ἴση.</s> <s>ὁμοίως δὴ δει-<lb n="1799,001,,,,4,14,17"></lb> |
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3646 χθήσεται, ὅτι καὶ ἑκάστη τῶν ΖΒ, ΖΑ, <lb n="1799,001,,,,4,14,18"></lb> |
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3647 ΖΕ ἑκατέρᾳ τῶν ΖΓ, ΖΔ ἐστιν ἴση· αἱ πέντε ἄρα <lb n="1799,001,,,,4,14,19"></lb> |
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3648 εὐθεῖαι αἱ ΖΑ, ΖΒ, ΖΓ, ΖΔ, ΖΕ ἴσαι ἀλλήλαις εἰσίν.</s> <lb n="1799,001,,,,4,14,20"></lb> |
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3649 <s>ὁ ἄρα κέντρῳ τῷ Ζ καὶ διαστήματι ἑνὶ τῶν ΖΑ, ΖΒ, <lb n="1799,001,,,,4,14,21"></lb> |
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3650 ΖΓ, ΖΔ, ΖΕ κύκλος γραφόμενος ἥξει καὶ διὰ τῶν λοι-<lb n="1799,001,,,,4,14,22"></lb> |
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3651 πῶν σημείων καὶ ἔσται περιγεγραμμένος.</s> <s>περιγεγράφθω <lb n="1799,001,,,,4,14,23"></lb> |
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3652 καὶ ἔστω ὁ ΑΒΓΔΕ.</s> <lb n="1799,001,,,,4,14,24"></lb> |
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3653 <s>Περὶ ἄρα τὸ δοθὲν πεντάγωνον, ὅ ἐστιν ἰσόπλευρόν τε <lb n="1799,001,,,,4,14,25"></lb> |
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3654 καὶ ἰσογώνιον, κύκλος περιγέγραπται· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,14,26"></lb> |
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3655 <s>Εἰς τὸν δοθέντα κύκλον ἑξάγωνον ἰσόπλευρόν τε καὶ <lb n="1799,001,,,,4,15,1"></lb> |
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3656 ἰσογώνιον ἐγγράψαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,15,2"></lb> |
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3657 Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓΔΕΖ· δεῖ δὴ εἰς τὸν <lb n="1799,001,,,,4,15,3"></lb> |
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3658 ΑΒΓΔΕΖ κύκλον ἑξάγωνον ἰσόπλευρόν τε καὶ ἰσογώ-<lb n="1799,001,,,,4,15,4"></lb> |
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3659 νιον ἐγγράψαι.</s> <lb n="1799,001,,,,4,15,5"></lb> |
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3660 <s>Ἤχθω τοῦ ΑΒΓΔΕΖ κύκλου διάμετρος ἡ ΑΔ, καὶ <lb n="1799,001,,,,4,15,6"></lb> |
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3661 εἰλήφθω τὸ κέντρον τοῦ κύκλου τὸ Η, καὶ κέντρῳ μὲν τῷ <lb n="1799,001,,,,4,15,7"></lb> |
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3662 Δ διαστήματι δὲ τῷ ΔΗ κύκλος γεγράφθω <lb n="1799,001,,,,4,15,8"></lb> |
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3663 ὁ ΕΗΓΘ, καὶ ἐπιζευχθεῖσαι αἱ ΕΗ, ΓΗ <lb n="1799,001,,,,4,15,9"></lb> |
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3664 διήχθωσαν ἐπὶ τὰ Β, Ζ σημεῖα, καὶ ἐπ-<lb n="1799,001,,,,4,15,10"></lb> |
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3665 εζεύχθωσαν αἱ ΑΒ, ΒΓ, ΓΔ, ΔΕ, ΕΖ, <lb n="1799,001,,,,4,15,11"></lb> |
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3666 ΖΑ· λέγω, ὅτι τὸ ΑΒΓΔΕΖ ἑξάγωνον <lb n="1799,001,,,,4,15,12"></lb> |
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3667 ἰσόπλευρόν τέ ἐστι καὶ ἰσογώνιον.</s> <lb n="1799,001,,,,4,15,13"></lb> |
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3668 <s>Ἐπεὶ γὰρ τὸ Η σημεῖον κέντρον ἐστὶ <lb n="1799,001,,,,4,15,14"></lb> |
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3669 τοῦ ΑΒΓΔΕΖ κύκλου, ἴση ἐστὶν ἡ <lb n="1799,001,,,,4,15,15"></lb> |
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3670 ΗΕ τῇ ΗΔ.</s> <s>πάλιν, ἐπεὶ τὸ Δ σημεῖον <lb n="1799,001,,,,4,15,16"></lb> |
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3671 κέντρον ἐστὶ τοῦ ΗΓΘ κύκλου, ἴση ἐστὶν ἡ ΔΕ τῇ ΔΗ.</s> <lb n="1799,001,,,,4,15,17"></lb> |
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3672 <s>ἀλλ' ἡ ΗΕ τῇ ΗΔ ἐδείχθη ἴση· καὶ ἡ ΗΕ ἄρα τῇ ΕΔ <lb n="1799,001,,,,4,15,18"></lb> |
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3673 ἴση ἐστίν· ἰσόπλευρον ἄρα ἐστὶ τὸ ΕΗΔ τρίγωνον· καὶ <lb n="1799,001,,,,4,15,19"></lb> |
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3674 αἱ τρεῖς ἄρα αὐτοῦ γωνίαι αἱ ὑπὸ ΕΗΔ, ΗΔΕ, ΔΕΗ <lb n="1799,001,,,,4,15,20"></lb> |
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3675 ἴσαι ἀλλήλαις εἰσίν, ἐπειδήπερ τῶν ἰσοσκελῶν τριγώνων <lb n="1799,001,,,,4,15,21"></lb> |
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3676 αἱ πρὸς τῇ βάσει γωνίαι ἴσαι ἀλλήλαις εἰσίν· καί εἰσιν αἱ <lb n="1799,001,,,,4,15,22"></lb> |
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3677 τρεῖς τοῦ τριγώνου γωνίαι δυσὶν ὀρθαῖς ἴσαι· ἡ ἄρα ὑπὸ <lb n="1799,001,,,,4,15,23"></lb> |
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3678 ΕΗΔ γωνία τρίτον ἐστὶ δύο ὀρθῶν.</s> <s>ὁμοίως δὴ δειχθήσε-<lb n="1799,001,,,,4,15,24"></lb> |
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3679 ται καὶ ἡ ὑπὸ ΔΗΓ τρίτον δύο ὀρθῶν.</s> <s>καὶ ἐπεὶ ἡ ΓΗ <lb n="1799,001,,,,4,15,25"></lb> |
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3680 εὐθεῖα ἐπὶ τὴν ΕΒ σταθεῖσα τὰς ἐφεξῆς γωνίας τὰς ὑπὸ <pb></pb> <lb n="1799,001,,,,4,15,26"></lb> |
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3681 ΕΗΓ, ΓΗΒ δυσὶν ὀρθαῖς ἴσας ποιεῖ, καὶ λοιπὴ ἄρα ἡ <lb n="1799,001,,,,4,15,27"></lb> |
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3682 ὑπὸ ΓΗΒ τρίτον ἐστὶ δύο ὀρθῶν· αἱ ἄρα ὑπὸ ΕΗΔ, <lb n="1799,001,,,,4,15,28"></lb> |
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3683 ΔΗΓ, ΓΗΒ γωνίαι ἴσαι ἀλλήλαις εἰσίν· ὥστε καὶ αἱ <lb n="1799,001,,,,4,15,29"></lb> |
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3684 κατὰ κορυφὴν αὐταῖς αἱ ὑπὸ ΒΗΑ, ΑΗΖ, ΖΗΕ ἴσαι <lb n="1799,001,,,,4,15,30"></lb> |
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3685 εἰσίν [ταῖς ὑπὸ ΕΗΔ, ΔΗΓ, ΓΗΒ].</s> <s>αἱ ἓξ ἄρα γωνίαι <lb n="1799,001,,,,4,15,31"></lb> |
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3686 αἱ ὑπὸ ΕΗΔ, ΔΗΓ, ΓΗΒ, ΒΗΑ, ΑΗΖ, ΖΗΕ ἴσαι <lb n="1799,001,,,,4,15,32"></lb> |
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3687 ἀλλήλαις εἰσίν.</s> <s>αἱ δὲ ἴσαι γωνίαι ἐπὶ ἴσων περιφερειῶν <lb n="1799,001,,,,4,15,33"></lb> |
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3688 βεβήκασιν· αἱ ἓξ ἄρα περιφέρειαι αἱ ΑΒ, ΒΓ, ΓΔ, ΔΕ, <lb n="1799,001,,,,4,15,34"></lb> |
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3689 ΕΖ, ΖΑ ἴσαι ἀλλήλαις εἰσίν.</s> <s>ὑπὸ δὲ τὰς ἴσας περιφερείας <lb n="1799,001,,,,4,15,35"></lb> |
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3690 αἱ ἴσαι εὐθεῖαι ὑποτείνουσιν· αἱ ἓξ ἄρα εὐθεῖαι ἴσαι ἀλλή-<lb n="1799,001,,,,4,15,36"></lb> |
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3691 λαις εἰσίν· ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΒΓΔΕΖ ἑξάγωνον.</s> <lb n="1799,001,,,,4,15,37"></lb> |
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3692 <s>λέγω δή, ὅτι καὶ ἰσογώνιον.</s> <s>ἐπεὶ γὰρ ἴση ἐστὶν ἡ ΖΑ <lb n="1799,001,,,,4,15,38"></lb> |
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3693 περιφέρεια τῇ ΕΔ περιφερείᾳ, κοινὴ προσκείσθω ἡ <lb n="1799,001,,,,4,15,39"></lb> |
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3694 ΑΒΓΔ περιφέρεια· ὅλη ἄρα ἡ ΖΑΒΓΔ ὅλῃ τῇ ΕΔΓΒΑ <lb n="1799,001,,,,4,15,40"></lb> |
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3695 ἐστιν ἴση· καὶ βέβηκεν ἐπὶ μὲν τῆς ΖΑΒΓΔ περιφερείας <lb n="1799,001,,,,4,15,41"></lb> |
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3696 ἡ ὑπὸ ΖΕΔ γωνία, ἐπὶ δὲ τῆς ΕΔΓΒΑ περιφερείας ἡ <lb n="1799,001,,,,4,15,42"></lb> |
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3697 ὑπὸ ΑΖΕ γωνία· ἴση ἄρα ἡ ὑπὸ ΑΖΕ γωνία τῇ ὑπὸ <lb n="1799,001,,,,4,15,43"></lb> |
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3698 ΔΕΖ.</s> <s>ὁμοίως δὴ δειχθήσεται, ὅτι καὶ αἱ λοιπαὶ γωνίαι <lb n="1799,001,,,,4,15,44"></lb> |
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3699 τοῦ ΑΒΓΔΕΖ ἑξαγώνου κατὰ μίαν ἴσαι εἰσὶν ἑκατέρᾳ <lb n="1799,001,,,,4,15,45"></lb> |
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3700 τῶν ὑπὸ ΑΖΕ, ΖΕΔ γωνιῶν· ἰσογώνιον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,4,15,46"></lb> |
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3701 ΑΒΓΔΕΖ ἑξάγωνον.</s> <s>ἐδείχθη δὲ καὶ ἰσόπλευρον· καὶ <lb n="1799,001,,,,4,15,47"></lb> |
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3702 ἐγγέγραπται εἰς τὸν ΑΒΓΔΕΖ κύκλον.</s> <s><pb></pb> <lb n="1799,001,,,,4,15,48"></lb> |
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3703 Εἰς ἄρα τὸν δοθέντα κύκλον ἑξάγωνον ἰσόπλευρόν τε <lb n="1799,001,,,,4,15,49"></lb> |
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3704 καὶ ἰσογώνιον ἐγγέγραπται· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,15,50"></lb> |
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3705 <s>Πόρισμα <lb n="1799,001,,,,4,15,51"></lb> |
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3706 Ἐκ δὴ τούτου φανερόν, ὅτι ἡ τοῦ ἑξαγώνου πλευρὰ ἴση <lb n="1799,001,,,,4,15,52"></lb> |
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3707 ἐστὶ τῇ ἐκ τοῦ κέντρου τοῦ κύκλου.</s> <lb n="1799,001,,,,4,15,53"></lb> |
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3708 <s>Ὁμοίως δὲ τοῖς ἐπὶ τοῦ πενταγώνου ἐὰν διὰ τῶν κατὰ <lb n="1799,001,,,,4,15,54"></lb> |
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3709 τὸν κύκλον διαιρέσεων ἐφαπτομένας τοῦ κύκλου ἀγάγω-<lb n="1799,001,,,,4,15,55"></lb> |
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3710 μεν, περιγραφήσεται περὶ τὸν κύκλον ἑξάγωνον ἰσόπλευρόν <lb n="1799,001,,,,4,15,56"></lb> |
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3711 τε καὶ ἰσογώνιον ἀκολούθως τοῖς ἐπὶ τοῦ πενταγώνου <lb n="1799,001,,,,4,15,57"></lb> |
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3712 εἰρημένοις.</s> <s>καὶ ἔτι διὰ τῶν ὁμοίων τοῖς ἐπὶ τοῦ πενταγώ-<lb n="1799,001,,,,4,15,58"></lb> |
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3713 νου εἰρημένοις εἰς τὸ δοθὲν ἑξάγωνον κύκλον ἐγγράψομέν <lb n="1799,001,,,,4,15,59"></lb> |
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3714 τε καὶ περιγράψομεν· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,4,15,60"></lb> |
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3715 <s>Εἰς τὸν δοθέντα κύκλον πεντεκαιδεκάγωνον ἰσόπλευρόν <lb n="1799,001,,,,4,16,1"></lb> |
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3716 τε καὶ ἰσογώνιον ἐγγράψαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,16,2"></lb> |
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3717 Ἔστω ὁ δοθεὶς κύκλος ὁ ΑΒΓΔ· δεῖ δὴ εἰς τὸν ΑΒΓΔ <lb n="1799,001,,,,4,16,3"></lb> |
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3718 κύκλον πεντεκαιδεκάγωνον ἰσόπλευρόν τε καὶ ἰσογώνιον <lb n="1799,001,,,,4,16,4"></lb> |
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3719 ἐγγράψαι.</s> <lb n="1799,001,,,,4,16,5"></lb> |
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3720 <s>Ἐγγεγράφθω εἰς τὸν ΑΒΓΔ κύκλον τριγώνου μὲν <lb n="1799,001,,,,4,16,6"></lb> |
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3721 ἰσοπλεύρου τοῦ εἰς αὐτὸν ἐγγραφομένου πλευρὰ ἡ ΑΓ, <lb n="1799,001,,,,4,16,7"></lb> |
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3722 πενταγώνου δὲ ἰσοπλεύρου ἡ ΑΒ· οἵων ἄρα ἐστὶν ὁ <lb n="1799,001,,,,4,16,8"></lb> |
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3723 ΑΒΓΔ κύκλος ἴσων τμημάτων δεκαπέντε, τοιούτων ἡ <lb n="1799,001,,,,4,16,9"></lb> |
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3724 μὲν ΑΒΓ περιφέρεια τρίτον οὖσα τοῦ κύκλου ἔσται <lb n="1799,001,,,,4,16,10"></lb> |
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3725 πέντε, ἡ δὲ ΑΒ περιφέρεια πέμπτον οὖσα τοῦ κύκλου ἔσται <lb n="1799,001,,,,4,16,11"></lb> |
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3726 τριῶν· λοιπὴ ἄρα ἡ ΒΓ τῶν ἴσων δύο.</s> <s>τετμήσθω ἡ ΒΓ <lb n="1799,001,,,,4,16,12"></lb> |
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3727 δίχα κατὰ τὸ Ε· ἑκατέρα <lb n="1799,001,,,,4,16,13"></lb> |
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3728 ἄρα τῶν ΒΕ, ΕΓ περιφε-<lb n="1799,001,,,,4,16,14"></lb> |
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3729 ρειῶν πεντεκαιδέκατόν ἐστι <lb n="1799,001,,,,4,16,15"></lb> |
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3730 τοῦ ΑΒΓΔ κύκλου.</s> <lb n="1799,001,,,,4,16,16"></lb> |
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3731 <s>Ἐὰν ἄρα ἐπιζεύξαντες τὰς <lb n="1799,001,,,,4,16,17"></lb> |
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3732 ΒΕ, ΕΓ ἴσας αὐταῖς κατὰ <lb n="1799,001,,,,4,16,18"></lb> |
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3733 τὸ συνεχὲς εὐθείας ἐναρμό-<lb n="1799,001,,,,4,16,19"></lb> |
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3734 σωμεν εἰς τὸν ΑΒΓΔ[Ε] <lb n="1799,001,,,,4,16,20"></lb> |
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3735 κύκλον, ἔσται εἰς αὐτὸν ἐγ-<lb n="1799,001,,,,4,16,21"></lb> |
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3736 γεγραμμένον πεντεκαιδεκά-<lb n="1799,001,,,,4,16,22"></lb> |
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3737 γωνον ἰσόπλευρόν τε καὶ ἰσο-<lb n="1799,001,,,,4,16,23"></lb> |
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3738 γώνιον· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,16,24"></lb> |
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3739 Ὁμοίως δὲ τοῖς ἐπὶ τοῦ πενταγώνου ἐὰν διὰ τῶν κατὰ <lb n="1799,001,,,,4,16,25"></lb> |
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3740 τὸν κύκλον διαιρέσεων ἐφαπτομένας τοῦ κύκλου ἀγάγω-<lb n="1799,001,,,,4,16,26"></lb> |
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3741 μεν, περιγραφήσεται περὶ τὸν κύκλον πεντεκαιδεκάγωνον <lb n="1799,001,,,,4,16,27"></lb> |
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3742 ἰσόπλευρόν τε καὶ ἰσογώνιον.</s> <s>ἔτι δὲ διὰ τῶν ὁμοίων τοῖς <lb n="1799,001,,,,4,16,28"></lb> |
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3743 ἐπὶ τοῦ πενταγώνου δείξεων καὶ εἰς τὸ δοθὲν πεντεκαι-<lb n="1799,001,,,,4,16,29"></lb> |
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3744 δεκάγωνον κύκλον ἐγγράψομέν τε καὶ περιγράψομεν· <lb n="1799,001,,,,4,16,30"></lb> |
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3745 ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,4,16,31"></lb> |
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3746 </s></p> |
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3747 </chap> |
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3748 <chap><p type="main"><s>Μέρος ἐστὶ μέγεθος μεγέθους τὸ ἔλασσον τοῦ <lb n="1799,001,,,,5;HOR,1,1"></lb> |
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3749 |
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3750 μείζονος, ὅταν καταμετρῇ τὸ μεῖζον.</s> <lb n="1799,001,,,,5;HOR,1,2"></lb> |
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3751 <s>Πολλαπλάσιον δὲ τὸ μεῖζον τοῦ ἐλάττονος, ὅταν <lb n="1799,001,,,,5;HOR,2,1"></lb> |
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3752 καταμετρῆται ὑπὸ τοῦ ἐλάττονος.</s> <lb n="1799,001,,,,5;HOR,2,2"></lb> |
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3753 <s>Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλι-<lb n="1799,001,,,,5;HOR,3,1"></lb> |
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3754 κότητά ποια σχέσις.</s> <lb n="1799,001,,,,5;HOR,3,2"></lb> |
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3755 <s>Λόγον ἔχειν πρὸς ἄλληλα μεγέθη λέγεται, ἃ <lb n="1799,001,,,,5;HOR,4,1"></lb> |
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3756 δύναται πολλαπλασιαζόμενα ἀλλήλων ὑπερέχειν.</s> <lb n="1799,001,,,,5;HOR,4,2"></lb> |
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3757 <s>Ἐν τῷ αὐτῷ λόγῳ μεγέθη λέγεται εἶναι πρῶτον <lb n="1799,001,,,,5;HOR,5,1"></lb> |
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3758 πρὸς δεύτερον καὶ τρίτον πρὸς τέταρτον, ὅταν τὰ τοῦ <lb n="1799,001,,,,5;HOR,5,2"></lb> |
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3759 πρώτου καὶ τρίτου ἰσάκις πολλαπλάσια τῶν τοῦ δευτέρου <lb n="1799,001,,,,5;HOR,5,3"></lb> |
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3760 καὶ τετάρτου ἰσάκις πολλαπλασίων καθ' ὁποιονοῦν <lb n="1799,001,,,,5;HOR,5,4"></lb> |
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3761 πολλαπλασιασμὸν ἑκάτερον ἑκατέρου ἢ ἅμα ὑπερέχῃ ἢ <lb n="1799,001,,,,5;HOR,5,5"></lb> |
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3762 ἅμα ἴσα ᾖ ἢ ἅμα ἐλλείπῃ ληφθέντα κατάλληλα.</s> <s><pb></pb> <lb n="1799,001,,,,5;HOR,5,6"></lb> |
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3763 Τὰ δὲ τὸν αὐτὸν ἔχοντα λόγον μεγέθη ἀνάλογον <lb n="1799,001,,,,5;HOR,6,1"></lb> |
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3764 καλείσθω.</s> <lb n="1799,001,,,,5;HOR,6,2"></lb> |
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3765 <s>Ὅταν δὲ τῶν ἰσάκις πολλαπλασίων τὸ μὲν τοῦ <lb n="1799,001,,,,5;HOR,7,1"></lb> |
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3766 πρώτου πολλαπλάσιον ὑπερέχῃ τοῦ τοῦ δευτέρου πολλα-<lb n="1799,001,,,,5;HOR,7,2"></lb> |
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3767 πλασίου, τὸ δὲ τοῦ τρίτου πολλαπλάσιον μὴ ὑπερέχῃ τοῦ <lb n="1799,001,,,,5;HOR,7,3"></lb> |
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3768 τοῦ τετάρτου πολλαπλασίου, τότε τὸ πρῶτον πρὸς τὸ <lb n="1799,001,,,,5;HOR,7,4"></lb> |
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3769 δεύτερον μείζονα λόγον ἔχειν λέγεται, ἤπερ τὸ τρίτον <lb n="1799,001,,,,5;HOR,7,5"></lb> |
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3770 πρὸς τὸ τέταρτον.</s> <lb n="1799,001,,,,5;HOR,7,6"></lb> |
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3771 <s>Ἀναλογία δὲ ἐν τρισὶν ὅροις ἐλαχίστη ἐστίν.</s> <lb n="1799,001,,,,5;HOR,8,1"></lb> |
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3772 <s>Ὅταν δὲ τρία μεγέθη ἀνάλογον ᾖ, τὸ πρῶτον πρὸς <lb n="1799,001,,,,5;HOR,9,1"></lb> |
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3773 τὸ τρίτον διπλασίονα λόγον ἔχειν λέγεται ἤπερ πρὸς τὸ <lb n="1799,001,,,,5;HOR,9,2"></lb> |
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3774 δεύτερον.</s> <lb n="1799,001,,,,5;HOR,9,3"></lb> |
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3775 <s>Ὅταν δὲ τέσσαρα μεγέθη ἀνάλογον ᾖ, τὸ πρῶτον <lb n="1799,001,,,,5;HOR,10,1"></lb> |
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3776 πρὸς τὸ τέταρτον τριπλασίονα λόγον ἔχειν λέγεται ἤπερ <lb n="1799,001,,,,5;HOR,10,2"></lb> |
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3777 πρὸς τὸ δεύτερον, καὶ ἀεὶ ἑξῆς ὁμοίως, ὡς ἂν ἡ ἀναλογία <lb n="1799,001,,,,5;HOR,10,3"></lb> |
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3778 ὑπάρχῃ.</s> <lb n="1799,001,,,,5;HOR,10,4"></lb> |
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3779 <s>Ὁμόλογα μεγέθη λέγεται τὰ μὲν ἡγούμενα τοῖς <lb n="1799,001,,,,5;HOR,11,1"></lb> |
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3780 ἡγουμένοις τὰ δὲ ἑπόμενα τοῖς ἑπομένοις.</s> <s><pb></pb> <lb n="1799,001,,,,5;HOR,11,2"></lb> |
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3781 Ἐναλλὰξ λόγος ἐστὶ λῆψις τοῦ ἡγουμένου πρὸς τὸ <lb n="1799,001,,,,5;HOR,12,1"></lb> |
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3782 ἡγούμενον καὶ τοῦ ἑπομένου πρὸς τὸ ἑπόμενον.</s> <lb n="1799,001,,,,5;HOR,12,2"></lb> |
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3783 <s>Ἀνάπαλιν λόγος ἐστὶ λῆψις τοῦ ἑπομένου ὡς <lb n="1799,001,,,,5;HOR,13,1"></lb> |
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3784 ἡγουμένου πρὸς τὸ ἡγούμενον ὡς ἑπόμενον.</s> <lb n="1799,001,,,,5;HOR,13,2"></lb> |
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3785 <s>Σύνθεσις λόγου ἐστὶ λῆψις τοῦ ἡγουμένου μετὰ <lb n="1799,001,,,,5;HOR,14,1"></lb> |
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3786 τοῦ ἑπομένου ὡς ἑνὸς πρὸς αὐτὸ τὸ ἑπόμενον.</s> <lb n="1799,001,,,,5;HOR,14,2"></lb> |
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3787 <s>Διαίρεσις λόγου ἐστὶ λῆψις τῆς ὑπεροχῆς, ᾗ <lb n="1799,001,,,,5;HOR,15,1"></lb> |
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3788 ὑπερέχει τὸ ἡγούμενον τοῦ ἑπομένου, πρὸς αὐτὸ τὸ <lb n="1799,001,,,,5;HOR,15,2"></lb> |
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3789 ἑπόμενον.</s> <lb n="1799,001,,,,5;HOR,15,3"></lb> |
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3790 <s>Ἀναστροφὴ λόγου ἐστὶ λῆψις τοῦ ἡγουμένου πρὸς <lb n="1799,001,,,,5;HOR,16,1"></lb> |
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3791 τὴν ὑπεροχήν, ᾗ ὑπερέχει τὸ ἡγούμενον τοῦ ἑπομένου.</s> <lb n="1799,001,,,,5;HOR,16,2"></lb> |
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3792 <s>Δι' ἴσου λόγος ἐστὶ πλειόνων ὄντων μεγεθῶν καὶ <lb n="1799,001,,,,5;HOR,17,1"></lb> |
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3793 ἄλλων αὐτοῖς ἴσων τὸ πλῆθος σύνδυο λαμβανομένων καὶ <lb n="1799,001,,,,5;HOR,17,2"></lb> |
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3794 ἐν τῷ αὐτῷ λόγῳ, ὅταν ᾖ ὡς ἐν τοῖς πρώτοις μεγέθεσι <lb n="1799,001,,,,5;HOR,17,3"></lb> |
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3795 τὸ πρῶτον πρὸς τὸ ἔσχατον, οὕτως ἐν τοῖς δευτέροις μεγέ-<lb n="1799,001,,,,5;HOR,17,4"></lb> |
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3796 θεσι τὸ πρῶτον πρὸς τὸ ἔσχατον· ἢ ἄλλως· Λῆψις τῶν <lb n="1799,001,,,,5;HOR,17,5"></lb> |
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3797 ἄκρων καθ' ὑπεξαίρεσιν τῶν μέσων.</s> <lb n="1799,001,,,,5;HOR,17,6"></lb> |
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3798 <s>Τεταραγμένη δὲ ἀναλογία ἐστίν, ὅταν τριῶν ὄντων <lb n="1799,001,,,,5;HOR,18,1"></lb> |
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3799 μεγεθῶν καὶ ἄλλων αὐτοῖς ἴσων τὸ πλῆθος γίνηται ὡς μὲν <lb n="1799,001,,,,5;HOR,18,2"></lb> |
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3800 ἐν τοῖς πρώτοις μεγέθεσιν ἡγούμενον πρὸς ἑπόμενον, <lb n="1799,001,,,,5;HOR,18,3"></lb> |
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3801 οὕτως ἐν τοῖς δευτέροις μεγέθεσιν ἡγούμενον πρὸς <lb n="1799,001,,,,5;HOR,18,4"></lb> |
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3802 ἑπόμενον, ὡς δὲ ἐν τοῖς πρώτοις μεγέθεσιν ἑπόμενον πρὸς <lb n="1799,001,,,,5;HOR,18,5"></lb> |
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3803 ἄλλο τι, οὕτως ἐν τοῖς δευτέροις ἄλλο τι πρὸς ἡγούμενον.</s> <s><pb></pb> <lb n="1799,001,,,,5;HOR,18,6"></lb> |
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3804 </s></p> |
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3805 </chap> |
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3806 <chap><p type="main"><s>Ἐὰν ᾖ ὁποσαοῦν μεγέθη ὁποσωνοῦν μεγεθῶν ἴσων τὸ <lb n="1799,001,,,,5,1,1"></lb> |
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3807 |
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3808 πλῆθος ἕκαστον ἑκάστου ἰσάκις πολλαπλάσιον, ὁσαπλάσιόν <lb n="1799,001,,,,5,1,2"></lb> |
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3809 ἐστιν ἓν τῶν μεγεθῶν ἑνός, τοσαυταπλάσια ἔσται καὶ τὰ <lb n="1799,001,,,,5,1,3"></lb> |
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3810 πάντα τῶν πάντων.</s> <lb n="1799,001,,,,5,1,4"></lb> |
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3811 <s>Ἔστω ὁποσαοῦν μεγέθη τὰ ΑΒ, ΓΔ ὁποσωνοῦν <lb n="1799,001,,,,5,1,5"></lb> |
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3812 μεγεθῶν τῶν Ε, Ζ ἴσων τὸ πλῆθος ἕκαστον ἑκάστου <lb n="1799,001,,,,5,1,6"></lb> |
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3813 ἰσάκις πολλαπλάσιον· <lb n="1799,001,,,,5,1,7"></lb> |
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3814 λέγω, ὅτι ὁσαπλάσιόν <lb n="1799,001,,,,5,1,8"></lb> |
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3815 ἐστι τὸ ΑΒ τοῦ Ε, το-<lb n="1799,001,,,,5,1,9"></lb> |
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3816 σαυταπλάσια ἔσται καὶ <lb n="1799,001,,,,5,1,10"></lb> |
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3817 τὰ ΑΒ, ΓΔ τῶν Ε, Ζ.</s> <lb n="1799,001,,,,5,1,11"></lb> |
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3818 <s>Ἐπεὶ γὰρ ἰσάκις ἐστὶ πολλαπλάσιον τὸ ΑΒ τοῦ Ε καὶ <lb n="1799,001,,,,5,1,12"></lb> |
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3819 τὸ ΓΔ τοῦ Ζ, ὅσα ἄρα ἐστὶν ἐν τῷ ΑΒ μεγέθη ἴσα τῷ Ε, <lb n="1799,001,,,,5,1,13"></lb> |
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3820 τοσαῦτα καὶ ἐν τῷ ΓΔ ἴσα τῷ Ζ.</s> <s>διῃρήσθω τὸ μὲν ΑΒ <lb n="1799,001,,,,5,1,14"></lb> |
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3821 εἰς τὰ τῷ Ε μεγέθη ἴσα τὰ ΑΗ, ΗΒ, τὸ δὲ ΓΔ εἰς τὰ τῷ <lb n="1799,001,,,,5,1,15"></lb> |
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3822 Ζ ἴσα τὰ ΓΘ, ΘΔ· ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΑΗ, <lb n="1799,001,,,,5,1,16"></lb> |
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3823 ΗΒ τῷ πλήθει τῶν ΓΘ, ΘΔ.</s> <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ μὲν <lb n="1799,001,,,,5,1,17"></lb> |
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3824 ΑΗ τῷ Ε, τὸ δὲ ΓΘ τῷ Ζ, ἴσον ἄρα τὸ ΑΗ τῷ Ε, καὶ τὰ <lb n="1799,001,,,,5,1,18"></lb> |
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3825 ΑΗ, ΓΘ τοῖς Ε, Ζ.</s> <s>διὰ τὰ αὐτὰ δὴ ἴσον ἐστὶ τὸ ΗΒ τῷ <lb n="1799,001,,,,5,1,19"></lb> |
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3826 Ε, καὶ τὰ ΗΒ, ΘΔ τοῖς Ε, Ζ· ὅσα ἄρα ἐστὶν ἐν τῷ ΑΒ <lb n="1799,001,,,,5,1,20"></lb> |
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3827 ἴσα τῷ Ε, τοσαῦτα καὶ ἐν τοῖς ΑΒ, ΓΔ ἴσα τοῖς Ε, Ζ· <lb n="1799,001,,,,5,1,21"></lb> |
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3828 ὁσαπλάσιον ἄρα ἐστὶ τὸ ΑΒ τοῦ Ε, τοσαυταπλάσια ἔσται <lb n="1799,001,,,,5,1,22"></lb> |
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3829 καὶ τὰ ΑΒ, ΓΔ τῶν Ε, Ζ.</s> <lb n="1799,001,,,,5,1,23"></lb> |
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3830 <s>Ἐὰν ἄρα ᾖ ὁποσαοῦν μεγέθη ὁποσωνοῦν μεγεθῶν ἴσων <lb n="1799,001,,,,5,1,24"></lb> |
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3831 τὸ πλῆθος ἕκαστον ἑκάστου ἰσάκις πολλαπλάσιον, <pb></pb> <lb n="1799,001,,,,5,1,25"></lb> |
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3832 ὁσαπλάσιόν ἐστιν ἓν τῶν μεγεθῶν ἑνός, τοσαυταπλάσια <lb n="1799,001,,,,5,1,26"></lb> |
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3833 ἔσται καὶ τὰ πάντα τῶν πάντων· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,1,27"></lb> |
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3834 <s>Ἐὰν πρῶτον δευτέρου ἰσάκις ᾖ πολλαπλάσιον καὶ <lb n="1799,001,,,,5,2,1"></lb> |
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3835 τρίτον τετάρτου, ᾖ δὲ καὶ πέμπτον δευτέρου ἰσάκις <lb n="1799,001,,,,5,2,2"></lb> |
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3836 πολλαπλάσιον καὶ ἕκτον τετάρτου, καὶ συντεθὲν πρῶτον <lb n="1799,001,,,,5,2,3"></lb> |
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3837 καὶ πέμπτον δευτέρου ἰσάκις ἔσται πολλαπλάσιον καὶ <lb n="1799,001,,,,5,2,4"></lb> |
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3838 τρίτον καὶ ἕκτον τετάρτου.</s> <lb n="1799,001,,,,5,2,5"></lb> |
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3839 <s>Πρῶτον γὰρ τὸ ΑΒ δευτέρου τοῦ Γ ἰσάκις ἔστω <lb n="1799,001,,,,5,2,6"></lb> |
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3840 πολλαπλάσιον καὶ τρίτον τὸ ΔΕ τετάρτου τοῦ Ζ, ἔστω <lb n="1799,001,,,,5,2,7"></lb> |
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3841 δὲ καὶ πέμπτον τὸ ΒΗ δευτέρου τοῦ Γ ἰσάκις πολλαπλά-<lb n="1799,001,,,,5,2,8"></lb> |
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3842 σιον καὶ ἕκτον τὸ ΕΘ <lb n="1799,001,,,,5,2,9"></lb> |
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3843 τετάρτου τοῦ Ζ· λέγω, <lb n="1799,001,,,,5,2,10"></lb> |
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3844 ὅτι καὶ συντεθὲν πρῶτον <lb n="1799,001,,,,5,2,11"></lb> |
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3845 καὶ πέμπτον τὸ ΑΗ <lb n="1799,001,,,,5,2,12"></lb> |
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3846 δευτέρου τοῦ Γ ἰσάκις <lb n="1799,001,,,,5,2,13"></lb> |
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3847 ἔσται πολλαπλάσιον καὶ <lb n="1799,001,,,,5,2,14"></lb> |
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3848 τρίτον καὶ ἕκτον τὸ ΔΘ <lb n="1799,001,,,,5,2,15"></lb> |
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3849 τετάρτου τοῦ Ζ.</s> <lb n="1799,001,,,,5,2,16"></lb> |
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3850 <s>Ἐπεὶ γὰρ ἰσάκις ἐστὶ πολλαπλάσιον τὸ ΑΒ τοῦ Γ καὶ <lb n="1799,001,,,,5,2,17"></lb> |
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3851 τὸ ΔΕ τοῦ Ζ, ὅσα ἄρα ἐστὶν ἐν τῷ ΑΒ ἴσα τῷ Γ, τοσαῦτα <lb n="1799,001,,,,5,2,18"></lb> |
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3852 καὶ ἐν τῷ ΔΕ ἴσα τῷ Ζ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὅσα ἐστὶν ἐν <lb n="1799,001,,,,5,2,19"></lb> |
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3853 τῷ ΒΗ ἴσα τῷ Γ, τοσαῦτα καὶ ἐν τῷ ΕΘ ἴσα τῷ Ζ· ὅσα <lb n="1799,001,,,,5,2,20"></lb> |
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3854 ἄρα ἐστὶν ἐν ὅλῳ τῷ ΑΗ ἴσα τῷ Γ, τοσαῦτα καὶ ἐν ὅλῳ <lb n="1799,001,,,,5,2,21"></lb> |
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3855 τῷ ΔΘ ἴσα τῷ Ζ· ὁσαπλάσιον ἄρα ἐστὶ τὸ ΑΗ τοῦ <lb n="1799,001,,,,5,2,22"></lb> |
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3856 Γ, τοσαυταπλάσιον ἔσται καὶ τὸ ΔΘ τοῦ Ζ.</s> <s>καὶ συν-<lb n="1799,001,,,,5,2,23"></lb> |
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3857 τεθὲν ἄρα πρῶτον καὶ πέμπτον τὸ ΑΗ δευτέρου τοῦ Γ <lb n="1799,001,,,,5,2,24"></lb> |
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3858 ἰσάκις ἔσται πολλαπλάσιον καὶ τρίτον καὶ ἕκτον τὸ ΔΘ <lb n="1799,001,,,,5,2,25"></lb> |
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3859 τετάρτου τοῦ Ζ.</s> <s><pb></pb> <lb n="1799,001,,,,5,2,26"></lb> |
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3860 Ἐὰν ἄρα πρῶτον δευτέρου ἰσάκις ᾖ πολλαπλάσιον καὶ <lb n="1799,001,,,,5,2,27"></lb> |
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3861 τρίτον τετάρτου, ᾖ δὲ καὶ πέμπτον δευτέρου ἰσάκις <lb n="1799,001,,,,5,2,28"></lb> |
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3862 πολλαπλάσιον καὶ ἕκτον τετάρτου, καὶ συντεθὲν πρῶτον <lb n="1799,001,,,,5,2,29"></lb> |
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3863 καὶ πέμπτον δευτέρου ἰσάκις ἔσται πολλαπλάσιον καὶ <lb n="1799,001,,,,5,2,30"></lb> |
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3864 τρίτον καὶ ἕκτον τετάρτου· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,2,31"></lb> |
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3865 <s>Ἐὰν πρῶτον δευτέρου ἰσάκις ᾖ πολλαπλάσιον καὶ τρίτον <lb n="1799,001,,,,5,3,1"></lb> |
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3866 τετάρτου, ληφθῇ δὲ ἰσάκις πολλαπλάσια τοῦ τε πρώτου <lb n="1799,001,,,,5,3,2"></lb> |
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3867 καὶ τρίτου, καὶ δι' ἴσου τῶν ληφθέντων ἑκάτερον ἑκατέ-<lb n="1799,001,,,,5,3,3"></lb> |
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3868 ρου ἰσάκις ἔσται πολλαπλάσιον τὸ μὲν τοῦ δευτέρου τὸ δὲ <lb n="1799,001,,,,5,3,4"></lb> |
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3869 τοῦ τετάρτου.</s> <lb n="1799,001,,,,5,3,5"></lb> |
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3870 <s>Πρῶτον γὰρ τὸ Α δευτέρου τοῦ Β ἰσάκις ἔστω πολλα-<lb n="1799,001,,,,5,3,6"></lb> |
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|
3871 πλάσιον καὶ τρίτον τὸ Γ τετάρτου τοῦ Δ, καὶ εἰλήφθω <lb n="1799,001,,,,5,3,7"></lb> |
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3872 τῶν Α, Γ ἰσάκις πολλαπλά-<lb n="1799,001,,,,5,3,8"></lb> |
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3873 σια τὰ ΕΖ, ΗΘ· λέγω, ὅτι <lb n="1799,001,,,,5,3,9"></lb> |
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3874 ἰσάκις ἐστὶ πολλαπλάσιον τὸ <lb n="1799,001,,,,5,3,10"></lb> |
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3875 ΕΖ τοῦ Β καὶ τὸ ΗΘ τοῦ Δ.</s> <lb n="1799,001,,,,5,3,11"></lb> |
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3876 <s>Ἐπεὶ γὰρ ἰσάκις ἐστὶ πολλα-<lb n="1799,001,,,,5,3,12"></lb> |
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3877 πλάσιον τὸ ΕΖ τοῦ Α καὶ τὸ <lb n="1799,001,,,,5,3,13"></lb> |
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3878 ΗΘ τοῦ Γ, ὅσα ἄρα ἐστὶν ἐν <lb n="1799,001,,,,5,3,14"></lb> |
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3879 τῷ ΕΖ ἴσα τῷ Α, τοσαῦτα καὶ <lb n="1799,001,,,,5,3,15"></lb> |
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3880 ἐν τῷ ΗΘ ἴσα τῷ Γ.</s> <s>διῃρή-<lb n="1799,001,,,,5,3,16"></lb> |
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3881 σθω τὸ μὲν ΕΖ εἰς τὰ τῷ <lb n="1799,001,,,,5,3,17"></lb> |
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3882 Α μεγέθη ἴσα τὰ ΕΚ, ΚΖ, τὸ δὲ ΗΘ εἰς τὰ τῷ Γ ἴσα <lb n="1799,001,,,,5,3,18"></lb> |
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3883 τὰ ΗΛ, ΛΘ· ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΕΚ, ΚΖ τῷ <lb n="1799,001,,,,5,3,19"></lb> |
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3884 πλήθει τῶν ΗΛ, ΛΘ.</s> <s>καὶ ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον <lb n="1799,001,,,,5,3,20"></lb> |
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3885 τὸ Α τοῦ Β καὶ τὸ Γ τοῦ Δ, ἴσον δὲ τὸ μὲν ΕΚ τῷ Α, τὸ <pb></pb> <lb n="1799,001,,,,5,3,21"></lb> |
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3886 δὲ ΗΛ τῷ Γ, ἰσάκις ἄρα ἐστὶ πολλαπλάσιον τὸ ΕΚ τοῦ Β <lb n="1799,001,,,,5,3,22"></lb> |
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3887 καὶ τὸ ΗΛ τοῦ Δ.</s> <s>διὰ τὰ αὐτὰ δὴ ἰσάκις ἐστὶ πολλαπλά-<lb n="1799,001,,,,5,3,23"></lb> |
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3888 σιον τὸ ΚΖ τοῦ Β καὶ τὸ ΛΘ τοῦ Δ.</s> <s>ἐπεὶ οὖν πρῶτον <lb n="1799,001,,,,5,3,24"></lb> |
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3889 τὸ ΕΚ δευτέρου τοῦ Β ἰσάκις ἐστὶ πολλαπλάσιον καὶ <lb n="1799,001,,,,5,3,25"></lb> |
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3890 τρίτον τὸ ΗΛ τετάρτου τοῦ Δ, ἔστι δὲ καὶ πέμπτον τὸ ΚΖ <lb n="1799,001,,,,5,3,26"></lb> |
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3891 δευτέρου τοῦ Β ἰσάκις πολλαπλάσιον καὶ ἕκτον τὸ ΛΘ <lb n="1799,001,,,,5,3,27"></lb> |
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3892 τετάρτου τοῦ Δ, καὶ συντεθὲν ἄρα πρῶτον καὶ πέμπτον τὸ <lb n="1799,001,,,,5,3,28"></lb> |
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3893 ΕΖ δευτέρου τοῦ Β ἰσάκις ἐστὶ πολλαπλάσιον καὶ τρίτον <lb n="1799,001,,,,5,3,29"></lb> |
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3894 καὶ ἕκτον τὸ ΗΘ τετάρτου τοῦ Δ.</s> <lb n="1799,001,,,,5,3,30"></lb> |
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3895 <s>Ἐὰν ἄρα πρῶτον δευτέρου ἰσάκις ᾖ πολλαπλάσιον καὶ <lb n="1799,001,,,,5,3,31"></lb> |
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3896 τρίτον τετάρτου, ληφθῇ δὲ τοῦ πρώτου καὶ τρίτου ἰσάκις <lb n="1799,001,,,,5,3,32"></lb> |
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3897 πολλαπλάσια, καὶ δι' ἴσου τῶν ληφθέντων ἑκάτερον <lb n="1799,001,,,,5,3,33"></lb> |
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3898 ἑκατέρου ἰσάκις ἔσται πολλαπλάσιον τὸ μὲν τοῦ δευτέρου <lb n="1799,001,,,,5,3,34"></lb> |
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3899 τὸ δὲ τοῦ τετάρτου· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,3,35"></lb> |
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3900 <s>Ἐὰν πρῶτον πρὸς δεύτερον τὸν αὐτὸν ἔχῃ λόγον καὶ <lb n="1799,001,,,,5,4,1"></lb> |
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3901 τρίτον πρὸς τέταρτον, καὶ τὰ ἰσάκις πολλαπλάσια τοῦ τε <lb n="1799,001,,,,5,4,2"></lb> |
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3902 πρώτου καὶ τρίτου πρὸς τὰ ἰσάκις πολλαπλάσια τοῦ <lb n="1799,001,,,,5,4,3"></lb> |
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3903 δευτέρου καὶ τετάρτου καθ' ὁποιονοῦν πολλαπλασιασμὸν <lb n="1799,001,,,,5,4,4"></lb> |
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3904 τὸν αὐτὸν ἕξει λόγον ληφθέντα κατάλληλα.</s> <lb n="1799,001,,,,5,4,5"></lb> |
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3905 <s>Πρῶτον γὰρ τὸ Α πρὸς δεύτερον τὸ Β τὸν αὐτὸν ἐχέτω <lb n="1799,001,,,,5,4,6"></lb> |
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3906 λόγον καὶ τρίτον τὸ Γ πρὸς τέταρτον τὸ Δ, καὶ εἰλήφθω <lb n="1799,001,,,,5,4,7"></lb> |
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3907 τῶν μὲν Α, Γ ἰσάκις πολλαπλάσια τὰ Ε, Ζ, τῶν δὲ Β, Δ <lb n="1799,001,,,,5,4,8"></lb> |
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3908 ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια τὰ Η, Θ· λέγω, ὅτι <lb n="1799,001,,,,5,4,9"></lb> |
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3909 ἐστὶν ὡς τὸ Ε πρὸς τὸ Η, οὕτως τὸ Ζ πρὸς τὸ Θ.</s> <s><pb></pb> <lb n="1799,001,,,,5,4,10"></lb> |
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3910 Εἰλήφθω γὰρ τῶν <lb n="1799,001,,,,5,4,11"></lb> |
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3911 μὲν Ε, Ζ ἰσάκις πολ-<lb n="1799,001,,,,5,4,12"></lb> |
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3912 λαπλάσια τὰ Κ, Λ, <lb n="1799,001,,,,5,4,13"></lb> |
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3913 τῶν δὲ Η, Θ ἄλλα, ἃ <lb n="1799,001,,,,5,4,14"></lb> |
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3914 ἔτυχεν, ἰσάκις πολλα-<lb n="1799,001,,,,5,4,15"></lb> |
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3915 πλάσια τὰ Μ, Ν.</s> <lb n="1799,001,,,,5,4,16"></lb> |
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3916 <s>[Καὶ] ἐπεὶ ἰσάκις <lb n="1799,001,,,,5,4,17"></lb> |
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3917 ἐστὶ πολλαπλάσιον τὸ <lb n="1799,001,,,,5,4,18"></lb> |
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3918 μὲν Ε τοῦ Α, τὸ δὲ <lb n="1799,001,,,,5,4,19"></lb> |
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3919 Ζ τοῦ Γ, καὶ εἴληπται <lb n="1799,001,,,,5,4,20"></lb> |
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3920 τῶν Ε, Ζ ἰσάκις πολ-<lb n="1799,001,,,,5,4,21"></lb> |
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3921 λαπλάσια τὰ Κ, Λ, <lb n="1799,001,,,,5,4,22"></lb> |
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3922 ἰσάκις ἄρα ἐστὶ πολ-<lb n="1799,001,,,,5,4,23"></lb> |
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3923 λαπλάσιον τὸ Κ τοῦ <lb n="1799,001,,,,5,4,24"></lb> |
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3924 Α καὶ τὸ Λ τοῦ Γ.</s> <lb n="1799,001,,,,5,4,25"></lb> |
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3925 <s>διὰ τὰ αὐτὰ δὴ ἰσάκις ἐστὶ πολλαπλάσιον τὸ Μ τοῦ <lb n="1799,001,,,,5,4,26"></lb> |
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3926 Β καὶ τὸ Ν τοῦ Λ.</s> <s>καὶ ἐπεί ἐστιν ὡς τὸ Α πρὸς τὸ Β, <lb n="1799,001,,,,5,4,27"></lb> |
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3927 οὕτως τὸ Γ πρὸς τὸ Δ, καὶ εἴληπται τῶν μὲν Α, Γ ἰσάκις <lb n="1799,001,,,,5,4,28"></lb> |
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3928 πολλαπλάσια τὰ Κ, Λ, τῶν δὲ Β, Δ ἄλλα ἃ ἔτυχεν, ἰσάκις <lb n="1799,001,,,,5,4,29"></lb> |
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3929 πολλαπλάσια τὰ Μ, Ν, εἰ ἄρα ὑπερέχει τὸ Κ τοῦ Μ, <lb n="1799,001,,,,5,4,30"></lb> |
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3930 ὑπερέχει καὶ τὸ Λ τοῦ Ν, καὶ εἰ ἴσον, ἴσον, καὶ εἰ ἔλαττον, <lb n="1799,001,,,,5,4,31"></lb> |
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3931 ἔλαττον.</s> <s>καί ἐστι τὰ μὲν Κ, Λ τῶν Ε, Ζ ἰσάκις πολλαπλά-<lb n="1799,001,,,,5,4,32"></lb> |
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3932 σια, τὰ δὲ Μ, Ν τῶν Η, Θ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλα-<lb n="1799,001,,,,5,4,33"></lb> |
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3933 πλάσια· ἔστιν ἄρα ὡς τὸ Ε πρὸς τὸ Η, οὕτως τὸ Ζ πρὸς <lb n="1799,001,,,,5,4,34"></lb> |
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3934 τὸ Θ.</s> <lb n="1799,001,,,,5,4,35"></lb> |
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3935 <s>Ἐὰν ἄρα πρῶτον πρὸς δεύτερον τὸν αὐτὸν ἔχῃ λόγον καὶ <lb n="1799,001,,,,5,4,36"></lb> |
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3936 τρίτον πρὸς τέταρτον, καὶ τὰ ἰσάκις πολλαπλάσια τοῦ <lb n="1799,001,,,,5,4,37"></lb> |
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3937 τε πρώτου καὶ τρίτου πρὸς τὰ ἰσάκις πολλαπλάσια τοῦ <pb></pb> <lb n="1799,001,,,,5,4,38"></lb> |
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3938 δευτέρου καὶ τετάρτου τὸν αὐτὸν ἕξει λόγον καθ' ὁποιονοῦν <lb n="1799,001,,,,5,4,39"></lb> |
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3939 πολλαπλασιασμὸν ληφθέντα κατάλληλα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,4,40"></lb> |
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3940 <s>Ἐὰν μέγεθος μεγέθους ἰσάκις ᾖ πολλαπλάσιον, ὅπερ <lb n="1799,001,,,,5,5,1"></lb> |
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3941 ἀφαιρεθὲν ἀφαιρεθέντος, καὶ τὸ λοιπὸν τοῦ λοιποῦ ἰσάκις <lb n="1799,001,,,,5,5,2"></lb> |
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3942 ἔσται πολλαπλάσιον, ὁσαπλάσιόν ἐστι τὸ ὅλον τοῦ ὅλου.</s> <lb n="1799,001,,,,5,5,3"></lb> |
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3943 <s>Μέγεθος γὰρ τὸ ΑΒ μεγέθους τοῦ ΓΔ ἰσάκις ἔστω <lb n="1799,001,,,,5,5,4"></lb> |
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3944 πολλαπλάσιον, ὅπερ ἀφαιρεθὲν τὸ ΑΕ ἀφαιρεθέντος τοῦ <lb n="1799,001,,,,5,5,5"></lb> |
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3945 ΓΖ· λέγω, ὅτι καὶ λοιπὸν <lb n="1799,001,,,,5,5,6"></lb> |
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3946 τὸ ΕΒ λοιποῦ τοῦ ΖΔ ἰσάκις <lb n="1799,001,,,,5,5,7"></lb> |
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3947 ἔσται πολλαπλάσιον, ὁσαπλά-<lb n="1799,001,,,,5,5,8"></lb> |
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3948 σιόν ἐστιν ὅλον τὸ ΑΒ ὅλου <lb n="1799,001,,,,5,5,9"></lb> |
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3949 τοῦ ΓΔ.</s> <lb n="1799,001,,,,5,5,10"></lb> |
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3950 <s>Ὁσαπλάσιον γάρ ἐστι τὸ ΑΕ τοῦ ΓΖ, τοσαυταπλάσιον <lb n="1799,001,,,,5,5,11"></lb> |
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3951 γεγονέτω καὶ τὸ ΕΒ τοῦ ΓΗ.</s> <lb n="1799,001,,,,5,5,12"></lb> |
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3952 <s>Καὶ ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον τὸ ΑΕ τοῦ ΓΖ καὶ <lb n="1799,001,,,,5,5,13"></lb> |
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3953 τὸ ΕΒ τοῦ ΗΓ, ἰσάκις ἄρα ἐστὶ πολλαπλάσιον τὸ ΑΕ τοῦ <lb n="1799,001,,,,5,5,14"></lb> |
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3954 ΓΖ καὶ τὸ ΑΒ τοῦ ΗΖ.</s> <s>κεῖται δὲ ἰσάκις πολλαπλάσιον τὸ <lb n="1799,001,,,,5,5,15"></lb> |
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3955 ΑΕ τοῦ ΓΖ καὶ τὸ ΑΒ τοῦ ΓΔ.</s> <s>ἰσάκις ἄρα ἐστὶ πολλαπλά- <pb></pb><lb n="1799,001,,,,5,5,16"></lb> |
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3956 σιον τὸ ΑΒ ἑκατέρου τῶν ΗΖ, ΓΔ· ἴσον ἄρα τὸ ΗΖ <lb n="1799,001,,,,5,5,17"></lb> |
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3957 τῷ ΓΔ.</s> <s>κοινὸν ἀφῃρήσθω τὸ ΓΖ· λοιπὸν ἄρα τὸ ΗΓ λοιπῷ <lb n="1799,001,,,,5,5,18"></lb> |
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3958 τῷ ΖΔ ἴσον ἐστίν.</s> <s>καὶ ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον τὸ <lb n="1799,001,,,,5,5,19"></lb> |
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3959 ΑΕ τοῦ ΓΖ καὶ τὸ ΕΒ τοῦ ΗΓ, ἴσον δὲ τὸ ΗΓ τῷ ΔΖ, <lb n="1799,001,,,,5,5,20"></lb> |
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3960 ἰσάκις ἄρα ἐστὶ πολλαπλάσιον τὸ ΑΕ τοῦ ΓΖ καὶ τὸ ΕΒ <lb n="1799,001,,,,5,5,21"></lb> |
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3961 τοῦ ΖΔ.</s> <s>ἰσάκις δὲ ὑπόκειται πολλαπλάσιον τὸ ΑΕ τοῦ <lb n="1799,001,,,,5,5,22"></lb> |
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3962 ΓΖ καὶ τὸ ΑΒ τοῦ ΓΔ· ἰσάκις ἄρα ἐστὶ πολλαπλάσιον <lb n="1799,001,,,,5,5,23"></lb> |
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3963 τὸ ΕΒ τοῦ ΖΔ καὶ τὸ ΑΒ τοῦ ΓΔ.</s> <s>καὶ λοιπὸν ἄρα τὸ <lb n="1799,001,,,,5,5,24"></lb> |
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3964 ΕΒ λοιποῦ τοῦ ΖΔ ἰσάκις ἔσται πολλαπλάσιον, ὁσαπλά-<lb n="1799,001,,,,5,5,25"></lb> |
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3965 σιόν ἐστιν ὅλον τὸ ΑΒ ὅλου τοῦ ΓΔ.</s> <lb n="1799,001,,,,5,5,26"></lb> |
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3966 <s>Ἐὰν ἄρα μέγεθος μεγέθους ἰσάκις ᾖ πολλαπλάσιον, <lb n="1799,001,,,,5,5,27"></lb> |
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3967 ὅπερ ἀφαιρεθὲν ἀφαιρεθέντος, καὶ τὸ λοιπὸν τοῦ λοιποῦ <lb n="1799,001,,,,5,5,28"></lb> |
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3968 ἰσάκις ἔσται πολλαπλάσιον, ὁσαπλάσιόν ἐστι καὶ τὸ ὅλον <lb n="1799,001,,,,5,5,29"></lb> |
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3969 τοῦ ὅλου· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,5,30"></lb> |
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3970 <s>Ἐὰν δύο μεγέθη δύο μεγεθῶν ἰσάκις ᾖ πολλαπλάσια, <lb n="1799,001,,,,5,6,1"></lb> |
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3971 καὶ ἀφαιρεθέντα τινὰ τῶν αὐτῶν ἰσάκις ᾖ πολλαπλάσια, <lb n="1799,001,,,,5,6,2"></lb> |
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3972 καὶ τὰ λοιπὰ τοῖς αὐτοῖς ἤτοι ἴσα ἐστὶν ἢ ἰσάκις αὐτῶν <lb n="1799,001,,,,5,6,3"></lb> |
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3973 πολλαπλάσια.</s> <lb n="1799,001,,,,5,6,4"></lb> |
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3974 <s>Δύο γὰρ μεγέθη τὰ ΑΒ, ΓΔ δύο μεγεθῶν τῶν Ε, Ζ <lb n="1799,001,,,,5,6,5"></lb> |
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3975 ἰσάκις ἔστω πολλαπλάσια, καὶ ἀφαιρεθέντα τὰ ΑΗ, ΓΘ <lb n="1799,001,,,,5,6,6"></lb> |
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3976 τῶν αὐτῶν τῶν Ε, Ζ ἰσάκις ἔστω πολλα-<lb n="1799,001,,,,5,6,7"></lb> |
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3977 πλάσια· λέγω, ὅτι καὶ λοιπὰ τὰ ΗΒ, <lb n="1799,001,,,,5,6,8"></lb> |
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3978 ΘΔ τοῖς Ε, Ζ ἤτοι ἴσα ἐστὶν ἢ ἰσάκις <lb n="1799,001,,,,5,6,9"></lb> |
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3979 αὐτῶν πολλαπλάσια.</s> <lb n="1799,001,,,,5,6,10"></lb> |
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3980 <s>Ἔστω γὰρ πρότερον τὸ ΗΒ τῷ Ε ἴσον.</s> <lb n="1799,001,,,,5,6,11"></lb> |
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3981 <s>λέγω, ὅτι καὶ τὸ ΘΔ τῷ Ζ ἴσον ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,5,6,12"></lb> |
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3982 Κείσθω γὰρ τῷ Ζ ἴσον τὸ ΓΚ.</s> <s>ἐπεὶ ἰσάκις ἐστὶ πολλα-<lb n="1799,001,,,,5,6,13"></lb> |
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3983 πλάσιον τὸ ΑΗ τοῦ Ε καὶ τὸ ΓΘ τοῦ Ζ, ἴσον δὲ τὸ μὲν <lb n="1799,001,,,,5,6,14"></lb> |
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3984 ΗΒ τῷ Ε, τὸ δὲ ΚΓ τῷ Ζ, ἰσάκις ἄρα ἐστὶ πολλαπλά-<lb n="1799,001,,,,5,6,15"></lb> |
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3985 σιον τὸ ΑΒ τοῦ Ε καὶ τὸ ΚΘ τοῦ Ζ.</s> <s>ἰσάκις δὲ ὑπόκειται <lb n="1799,001,,,,5,6,16"></lb> |
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3986 πολλαπλάσιον τὸ ΑΒ τοῦ Ε καὶ τὸ ΓΔ τοῦ Ζ· ἰσάκις <lb n="1799,001,,,,5,6,17"></lb> |
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3987 ἄρα ἐστὶ πολλαπλάσιον τὸ ΚΘ τοῦ Ζ καὶ τὸ ΓΔ τοῦ Ζ.</s> <lb n="1799,001,,,,5,6,18"></lb> |
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3988 <s>ἐπεὶ οὖν ἑκάτερον τῶν ΚΘ, ΓΔ τοῦ Ζ ἰσάκις ἐστὶ πολλα-<lb n="1799,001,,,,5,6,19"></lb> |
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3989 πλάσιον, ἴσον ἄρα ἐστὶ τὸ ΚΘ τῷ ΓΔ.</s> <s>κοινὸν ἀφῃρήσθω <lb n="1799,001,,,,5,6,20"></lb> |
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3990 τὸ ΓΘ· λοιπὸν ἄρα τὸ ΚΓ λοιπῷ τῷ ΘΔ ἴσον ἐστίν.</s> <s>ἀλλὰ τὸ <lb n="1799,001,,,,5,6,21"></lb> |
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3991 Ζ τῷ ΚΓ ἐστιν ἴσον· καὶ τὸ ΘΔ ἄρα τῷ Ζ ἴσον ἐστίν.</s> <s>ὥστε <lb n="1799,001,,,,5,6,22"></lb> |
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3992 εἰ τὸ ΗΒ τῷ Ε ἴσον ἐστίν, καὶ τὸ ΘΔ ἴσον ἔσται τῷ Ζ.</s> <lb n="1799,001,,,,5,6,23"></lb> |
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3993 <s>Ὁμοίως δὴ δείξομεν, ὅτι, κἂν πολλαπλάσιον ᾖ τὸ ΗΒ <lb n="1799,001,,,,5,6,24"></lb> |
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3994 τοῦ Ε, τοσαυταπλάσιον ἔσται καὶ τὸ ΘΔ τοῦ Ζ.</s> <lb n="1799,001,,,,5,6,25"></lb> |
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3995 <s>Ἐὰν ἄρα δύο μεγέθη δύο μεγεθῶν ἰσάκις ᾖ πολλα-<lb n="1799,001,,,,5,6,26"></lb> |
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3996 πλάσια, καὶ ἀφαιρεθέντα τινὰ τῶν αὐτῶν ἰσάκις ᾖ <lb n="1799,001,,,,5,6,27"></lb> |
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3997 πολλαπλάσια, καὶ τὰ λοιπὰ τοῖς αὐτοῖς ἤτοι ἴσα ἐστὶν ἢ <lb n="1799,001,,,,5,6,28"></lb> |
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3998 ἰσάκις αὐτῶν πολλαπλάσια· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,6,29"></lb> |
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3999 <s>Τὰ ἴσα πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον καὶ τὸ αὐτὸ <lb n="1799,001,,,,5,7,1"></lb> |
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4000 πρὸς τὰ ἴσα.</s> <lb n="1799,001,,,,5,7,2"></lb> |
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4001 <s>Ἔστω ἴσα μεγέθη τὰ Α, Β, ἄλλο δέ τι, ὃ ἔτυχεν, μέγεθος <lb n="1799,001,,,,5,7,3"></lb> |
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4002 τὸ Γ· λέγω, ὅτι ἑκάτερον <lb n="1799,001,,,,5,7,4"></lb> |
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4003 τῶν Α, Β πρὸς τὸ Γ τὸν <lb n="1799,001,,,,5,7,5"></lb> |
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4004 αὐτὸν ἔχει λόγον, καὶ τὸ Γ <lb n="1799,001,,,,5,7,6"></lb> |
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4005 πρὸς ἑκάτερον τῶν Α, Β.</s> <s><pb></pb> <lb n="1799,001,,,,5,7,7"></lb> |
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4006 Εἰλήφθω γὰρ τῶν μὲν Α, Β ἰσάκις πολλαπλάσια τὰ <lb n="1799,001,,,,5,7,8"></lb> |
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4007 Δ, Ε, τοῦ δὲ Γ ἄλλο, ὃ ἔτυχεν, πολλαπλάσιον τὸ Ζ.</s> <lb n="1799,001,,,,5,7,9"></lb> |
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4008 <s>Ἐπεὶ οὖν ἰσάκις ἐστὶ πολλαπλάσιον τὸ Δ τοῦ Α καὶ τὸ <lb n="1799,001,,,,5,7,10"></lb> |
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4009 Ε τοῦ Β, ἴσον δὲ τὸ Α τῷ Β, ἴσον ἄρα καὶ τὸ Δ τῷ Ε.</s> <s>ἄλλο <lb n="1799,001,,,,5,7,11"></lb> |
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4010 δέ, ὃ ἔτυχεν, τὸ Ζ.</s> <s>Εἰ ἄρα ὑπερέχει τὸ Δ τοῦ Ζ, ὑπερέχει <lb n="1799,001,,,,5,7,12"></lb> |
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4011 καὶ τὸ Ε τοῦ Ζ, καὶ εἰ ἴσον, ἴσον, καὶ εἰ ἔλαττον, ἔλαττον.</s> <lb n="1799,001,,,,5,7,13"></lb> |
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4012 <s>καί ἐστι τὰ μὲν Δ, Ε τῶν Α, Β ἰσάκις πολλαπλάσια, τὸ <lb n="1799,001,,,,5,7,14"></lb> |
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4013 δὲ Ζ τοῦ Γ ἄλλο, ὃ ἔτυχεν, πολλαπλάσιον· ἔστιν ἄρα ὡς <lb n="1799,001,,,,5,7,15"></lb> |
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4014 τὸ Α πρὸς τὸ Γ, οὕτως τὸ Β πρὸς τὸ Γ.</s> <lb n="1799,001,,,,5,7,16"></lb> |
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4015 <s>Λέγω [δή], ὅτι καὶ τὸ Γ πρὸς ἑκάτερον τῶν Α, Β τὸν <lb n="1799,001,,,,5,7,17"></lb> |
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4016 αὐτὸν ἔχει λόγον.</s> <lb n="1799,001,,,,5,7,18"></lb> |
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4017 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, <lb n="1799,001,,,,5,7,19"></lb> |
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4018 ὅτι ἴσον ἐστὶ τὸ Δ τῷ Ε· ἄλλο δέ τι τὸ Ζ· εἰ ἄρα ὑπερέχει <lb n="1799,001,,,,5,7,20"></lb> |
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4019 τὸ Ζ τοῦ Δ, ὑπερέχει καὶ τοῦ Ε, καὶ εἰ ἴσον, ἴσον, καὶ εἰ <lb n="1799,001,,,,5,7,21"></lb> |
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4020 ἔλαττον, ἔλαττον.</s> <s>καί ἐστι τὸ μὲν Ζ τοῦ Γ πολλαπλάσιον, <lb n="1799,001,,,,5,7,22"></lb> |
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4021 τὰ δὲ Δ, Ε τῶν Α, Β ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια· <lb n="1799,001,,,,5,7,23"></lb> |
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4022 ἔστιν ἄρα ὡς τὸ Γ πρὸς τὸ Α, οὕτως τὸ Γ πρὸς τὸ Β.</s> <lb n="1799,001,,,,5,7,24"></lb> |
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4023 <s>Τὰ ἴσα ἄρα πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον καὶ τὸ <lb n="1799,001,,,,5,7,25"></lb> |
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4024 αὐτὸ πρὸς τὰ ἴσα.</s> <lb n="1799,001,,,,5,7,26"></lb> |
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4025 <s>Πόρισμα <lb n="1799,001,,,,5,7,27"></lb> |
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4026 Ἐκ δὴ τούτου φανερόν, ὅτι ἐὰν μεγέθη τινὰ ἀνάλογον <lb n="1799,001,,,,5,7,28"></lb> |
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4027 ᾖ, καὶ ἀνάπαλιν ἀνάλογον ἔσται.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,5,7,29"></lb> |
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4028 Τῶν ἀνίσων μεγεθῶν τὸ μεῖζον πρὸς τὸ αὐτὸ μείζονα <lb n="1799,001,,,,5,8,1"></lb> |
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4029 λόγον ἔχει ἤπερ τὸ ἔλαττον.</s> <s>καὶ τὸ αὐτὸ πρὸς τὸ ἔλαττον <lb n="1799,001,,,,5,8,2"></lb> |
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4030 μείζονα λόγον ἔχει ἤπερ πρὸς τὸ μεῖζον.</s> <lb n="1799,001,,,,5,8,3"></lb> |
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4031 <s>Ἔστω ἄνισα μεγέθη τὰ ΑΒ, Γ, καὶ ἔστω μεῖζον τὸ <lb n="1799,001,,,,5,8,4"></lb> |
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4032 ΑΒ, ἄλλο δέ, ὃ ἔτυχεν, τὸ Δ· λέγω, ὅτι τὸ ΑΒ πρὸς τὸ Δ <lb n="1799,001,,,,5,8,5"></lb> |
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4033 μείζονα λόγον ἔχει ἤπερ τὸ Γ πρὸς τὸ Δ, καὶ τὸ Δ πρὸς τὸ <lb n="1799,001,,,,5,8,6"></lb> |
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4034 Γ μείζονα λόγον ἔχει ἤπερ πρὸς τὸ ΑΒ.</s> <lb n="1799,001,,,,5,8,7"></lb> |
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4035 <s>Ἐπεὶ γὰρ μεῖζόν ἐστι τὸ ΑΒ τοῦ Γ, κείσθω τῷ Γ ἴσον <lb n="1799,001,,,,5,8,8"></lb> |
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4036 τὸ ΒΕ· τὸ δὴ ἔλασσον τῶν ΑΕ, ΕΒ πολλαπλασιαζόμενον <lb n="1799,001,,,,5,8,9"></lb> |
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4037 ἔσται ποτὲ τοῦ Δ μεῖζον.</s> <s>ἔστω <lb n="1799,001,,,,5,8,10"></lb> |
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4038 πρότερον τὸ ΑΕ ἔλαττον τοῦ <lb n="1799,001,,,,5,8,11"></lb> |
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4039 ΕΒ, καὶ πεπολλαπλασιάσθω <lb n="1799,001,,,,5,8,12"></lb> |
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4040 τὸ ΑΕ, καὶ ἔστω αὐτοῦ πολλα-<lb n="1799,001,,,,5,8,13"></lb> |
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4041 πλάσιον τὸ ΖΗ μεῖζον ὂν τοῦ <lb n="1799,001,,,,5,8,14"></lb> |
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4042 Δ, καὶ ὁσαπλάσιόν ἐστι τὸ ΖΗ <lb n="1799,001,,,,5,8,15"></lb> |
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4043 τοῦ ΑΕ, τοσαυταπλάσιον γε-<lb n="1799,001,,,,5,8,16"></lb> |
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4044 γονέτω καὶ τὸ μὲν ΗΘ τοῦ ΕΒ <lb n="1799,001,,,,5,8,17"></lb> |
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4045 τὸ δὲ Κ τοῦ Γ· καὶ εἰλήφθω <lb n="1799,001,,,,5,8,18"></lb> |
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4046 τοῦ Δ διπλάσιον μὲν τὸ Λ, <lb n="1799,001,,,,5,8,19"></lb> |
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4047 τριπλάσιον δὲ τὸ Μ, καὶ ἑξῆς <lb n="1799,001,,,,5,8,20"></lb> |
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4048 ἑνὶ πλεῖον, ἕως ἂν τὸ λαμβανό-<lb n="1799,001,,,,5,8,21"></lb> |
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4049 μενον πολλαπλάσιον μὲν γένηται τοῦ Δ, πρώτως δὲ <lb n="1799,001,,,,5,8,22"></lb> |
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4050 μεῖζον τοῦ Κ.</s> <s>εἰλήφθω, καὶ ἔστω τὸ Ν τετραπλάσιον <lb n="1799,001,,,,5,8,23"></lb> |
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4051 μὲν τοῦ Δ, πρώτως δὲ μεῖζον τοῦ Κ.</s> <s><pb></pb> <lb n="1799,001,,,,5,8,24"></lb> |
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4052 Ἐπεὶ οὖν τὸ Κ τοῦ Ν πρώτως ἐστὶν ἔλαττον, τὸ Κ <lb n="1799,001,,,,5,8,25"></lb> |
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4053 ἄρα τοῦ Μ οὔκ ἐστιν ἔλαττον.</s> <s>καὶ ἐπεὶ ἰσάκις ἐστὶ πολλα-<lb n="1799,001,,,,5,8,26"></lb> |
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4054 πλάσιον τὸ ΖΗ τοῦ ΑΕ καὶ τὸ ΗΘ τοῦ ΕΒ, ἰσάκις ἄρα <lb n="1799,001,,,,5,8,27"></lb> |
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4055 ἐστὶ πολλαπλάσιον τὸ ΖΗ τοῦ ΑΕ καὶ τὸ ΖΘ τοῦ ΑΒ.</s> <lb n="1799,001,,,,5,8,28"></lb> |
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4056 <s>ἰσάκις δέ ἐστι πολλαπλάσιον τὸ ΖΗ τοῦ ΑΕ καὶ τὸ Κ τοῦ <lb n="1799,001,,,,5,8,29"></lb> |
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4057 Γ· ἰσάκις ἄρα ἐστὶ πολλαπλάσιον τὸ ΖΘ τοῦ ΑΒ καὶ τὸ Κ <lb n="1799,001,,,,5,8,30"></lb> |
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4058 τοῦ Γ.</s> <s>τὰ ΖΘ, Κ ἄρα τῶν ΑΒ, Γ ἰσάκις ἐστὶ πολλαπλάσια.</s> <lb n="1799,001,,,,5,8,31"></lb> |
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4059 <s>πάλιν, ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον τὸ ΗΘ τοῦ ΕΒ καὶ <lb n="1799,001,,,,5,8,32"></lb> |
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4060 τὸ Κ τοῦ Γ, ἴσον δὲ τὸ ΕΒ τῷ Γ, ἴσον ἄρα καὶ τὸ ΗΘ τῷ <lb n="1799,001,,,,5,8,33"></lb> |
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4061 Κ· τὸ δὲ Κ τοῦ Μ οὔκ ἐστιν ἔλαττον· οὐδ' ἄρα τὸ ΗΘ τοῦ <lb n="1799,001,,,,5,8,34"></lb> |
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4062 Μ ἔλαττόν ἐστιν.</s> <s>μεῖζον δὲ τὸ ΖΗ τοῦ Δ· ὅλον ἄρα τὸ <lb n="1799,001,,,,5,8,35"></lb> |
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4063 ΖΘ συναμφοτέρων τῶν Δ, Μ μεῖζόν ἐστιν.</s> <s>ἀλλὰ συναμ-<lb n="1799,001,,,,5,8,36"></lb> |
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4064 φότερα τὰ Δ, Μ τῷ Ν ἐστιν ἴσα, ἐπειδήπερ τὸ Μ τοῦ Δ <lb n="1799,001,,,,5,8,37"></lb> |
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4065 τριπλάσιόν ἐστιν, συναμφότερα δὲ τὰ Μ, Δ τοῦ Δ ἐστι <lb n="1799,001,,,,5,8,38"></lb> |
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4066 τετραπλάσια, ἔστι δὲ καὶ τὸ Ν τοῦ Δ τετραπλάσιον· <lb n="1799,001,,,,5,8,39"></lb> |
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4067 συναμφότερα ἄρα τὰ Μ, Δ τῷ Ν ἴσα ἐστίν.</s> <s>ἀλλὰ τὸ ΖΘ <lb n="1799,001,,,,5,8,40"></lb> |
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4068 τῶν Μ, Δ μεῖζόν ἐστιν· τὸ ΖΘ ἄρα τοῦ Ν ὑπερέχει· τὸ <lb n="1799,001,,,,5,8,41"></lb> |
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4069 δὲ Κ τοῦ Ν οὐχ ὑπερέχει.</s> <s>καί ἐστι τὰ μὲν ΖΘ, Κ τῶν <lb n="1799,001,,,,5,8,42"></lb> |
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4070 ΑΒ, Γ ἰσάκις πολλαπλάσια, τὸ δὲ Ν τοῦ Δ ἄλλο, ὃ ἔτυχεν, <lb n="1799,001,,,,5,8,43"></lb> |
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4071 πολλαπλάσιον· τὸ ΑΒ ἄρα πρὸς τὸ Δ μείζονα λόγον ἔχει <lb n="1799,001,,,,5,8,44"></lb> |
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4072 ἤπερ τὸ Γ πρὸς τὸ Δ.</s> <lb n="1799,001,,,,5,8,45"></lb> |
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4073 <s>Λέγω δή, ὅτι καὶ τὸ Δ πρὸς τὸ Γ μείζονα λόγον ἔχει <lb n="1799,001,,,,5,8,46"></lb> |
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4074 ἤπερ τὸ Δ πρὸς τὸ ΑΒ.</s> <lb n="1799,001,,,,5,8,47"></lb> |
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4075 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, <lb n="1799,001,,,,5,8,48"></lb> |
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4076 ὅτι τὸ μὲν Ν τοῦ Κ ὑπερέχει, τὸ δὲ Ν τοῦ ΖΘ οὐχ ὑπερ-<lb n="1799,001,,,,5,8,49"></lb> |
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4077 έχει.</s> <s>καί ἐστι τὸ μὲν Ν τοῦ Δ πολλαπλάσιον, τὰ δὲ ΖΘ, <pb></pb> <lb n="1799,001,,,,5,8,50"></lb> |
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4078 Κ τῶν ΑΒ, Γ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια· τὸ Δ <lb n="1799,001,,,,5,8,51"></lb> |
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4079 ἄρα πρὸς τὸ Γ μείζονα λόγον ἔχει ἤπερ τὸ Δ πρὸς τὸ ΑΒ.</s> <lb n="1799,001,,,,5,8,52"></lb> |
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4080 <s>Ἀλλὰ δὴ τὸ ΑΕ τοῦ ΕΒ <lb n="1799,001,,,,5,8,53"></lb> |
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4081 μεῖζον ἔστω.</s> <s>τὸ δὴ ἔλαττον <lb n="1799,001,,,,5,8,54"></lb> |
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4082 τὸ ΕΒ πολλαπλασιαζόμενον <lb n="1799,001,,,,5,8,55"></lb> |
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4083 ἔσται ποτὲ τοῦ Δ μεῖζον.</s> <lb n="1799,001,,,,5,8,56"></lb> |
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4084 <s>πεπολλαπλασιάσθω, καὶ ἔστω <lb n="1799,001,,,,5,8,57"></lb> |
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4085 τὸ ΗΘ πολλαπλάσιον μὲν <lb n="1799,001,,,,5,8,58"></lb> |
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4086 τοῦ ΕΒ, μεῖζον δὲ τοῦ Δ· <lb n="1799,001,,,,5,8,59"></lb> |
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4087 καὶ ὁσαπλάσιόν ἐστι τὸ ΗΘ <lb n="1799,001,,,,5,8,60"></lb> |
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4088 τοῦ ΕΒ, τοσαυταπλάσιον γε-<lb n="1799,001,,,,5,8,61"></lb> |
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4089 γονέτω καὶ τὸ μὲν ΖΗ τοῦ <lb n="1799,001,,,,5,8,62"></lb> |
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4090 ΑΕ, τὸ δὲ Κ τοῦ Γ.</s> <s>ὁμοίως <lb n="1799,001,,,,5,8,63"></lb> |
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4091 δὴ δείξομεν, ὅτι τὰ ΖΘ, Κ <lb n="1799,001,,,,5,8,64"></lb> |
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4092 τῶν ΑΒ, Γ ἰσάκις ἐστὶ πολλαπλάσια· καὶ εἰλήφθω <lb n="1799,001,,,,5,8,65"></lb> |
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4093 ὁμοίως τὸ Ν πολλαπλάσιον μὲν τοῦ Δ, πρώτως δὲ μεῖζον <lb n="1799,001,,,,5,8,66"></lb> |
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4094 τοῦ ΖΗ· ὥστε πάλιν τὸ ΖΗ τοῦ Μ οὔκ ἐστιν ἔλασσον.</s> <lb n="1799,001,,,,5,8,67"></lb> |
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4095 <s>μεῖζον δὲ τὸ ΗΘ τοῦ Δ· ὅλον ἄρα τὸ ΖΘ τῶν Δ, Μ, <lb n="1799,001,,,,5,8,68"></lb> |
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4096 τουτέστι τοῦ Ν, ὑπερέχει.</s> <s>τὸ δὲ Κ τοῦ Ν οὐχ ὑπερέχει, <lb n="1799,001,,,,5,8,69"></lb> |
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4097 ἐπειδήπερ καὶ τὸ ΖΗ μεῖζον ὂν τοῦ ΗΘ, τουτέστι τοῦ Κ, <lb n="1799,001,,,,5,8,70"></lb> |
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4098 τοῦ Ν οὐχ ὑπερέχει.</s> <s>καὶ ὡσαύτως κατακολουθοῦντες <lb n="1799,001,,,,5,8,71"></lb> |
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4099 τοῖς ἐπάνω περαίνομεν τὴν ἀπόδειξιν.</s> <lb n="1799,001,,,,5,8,72"></lb> |
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4100 <s>Τῶν ἄρα ἀνίσων μεγεθῶν τὸ μεῖζον πρὸς τὸ αὐτὸ <lb n="1799,001,,,,5,8,73"></lb> |
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4101 μείζονα λόγον ἔχει ἤπερ τὸ ἔλαττον· καὶ τὸ αὐτὸ πρὸς τὸ <lb n="1799,001,,,,5,8,74"></lb> |
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4102 ἔλαττον μείζονα λόγον ἔχει ἤπερ πρὸς τὸ μεῖζον· ὅπερ <lb n="1799,001,,,,5,8,75"></lb> |
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4103 ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,5,8,76"></lb> |
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4104 Τὰ πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχοντα λόγον ἴσα ἀλλήλοις <lb n="1799,001,,,,5,9,1"></lb> |
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4105 ἐστίν· καὶ πρὸς ἃ τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον, ἐκεῖνα <lb n="1799,001,,,,5,9,2"></lb> |
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4106 ἴσα ἐστίν.</s> <lb n="1799,001,,,,5,9,3"></lb> |
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4107 <s>Ἐχέτω γὰρ ἑκάτερον τῶν Α, Β πρὸς τὸ Γ τὸν αὐτὸν <lb n="1799,001,,,,5,9,4"></lb> |
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4108 λόγον· λέγω, ὅτι ἴσον ἐστὶ τὸ Α τῷ Β.</s> <lb n="1799,001,,,,5,9,5"></lb> |
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4109 <s>Εἰ γὰρ μή, οὐκ ἂν ἑκάτερον <lb n="1799,001,,,,5,9,6"></lb> |
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4110 τῶν Α, Β πρὸς τὸ Γ τὸν αὐτὸν <lb n="1799,001,,,,5,9,7"></lb> |
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4111 εἶχε λόγον· ἔχει δέ· ἴσον ἄρα ἐστὶ <lb n="1799,001,,,,5,9,8"></lb> |
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4112 τὸ Α τῷ Β.</s> <lb n="1799,001,,,,5,9,9"></lb> |
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4113 <s>Ἐχέτω δὴ πάλιν τὸ Γ πρὸς ἑκάτερον τῶν Α, Β τὸν <lb n="1799,001,,,,5,9,10"></lb> |
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4114 αὐτὸν λόγον· λέγω, ὅτι ἴσον ἐστὶ τὸ Α τῷ Β.</s> <lb n="1799,001,,,,5,9,11"></lb> |
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4115 <s>Εἰ γὰρ μή, οὐκ ἂν τὸ Γ πρὸς ἑκάτερον τῶν Α, Β τὸν <lb n="1799,001,,,,5,9,12"></lb> |
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4116 αὐτὸν εἶχε λόγον· ἔχει δέ· ἴσον ἄρα ἐστὶ τὸ Α τῷ Β.</s> <lb n="1799,001,,,,5,9,13"></lb> |
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4117 <s>Τὰ ἄρα πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχοντα λόγον ἴσα <lb n="1799,001,,,,5,9,14"></lb> |
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4118 ἀλλήλοις ἐστίν· καὶ πρὸς ἃ τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον, <lb n="1799,001,,,,5,9,15"></lb> |
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4119 ἐκεῖνα ἴσα ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,9,16"></lb> |
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4120 <s>Τῶν πρὸς τὸ αὐτὸ λόγον ἐχόντων τὸ μείζονα λόγον <lb n="1799,001,,,,5,10,1"></lb> |
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4121 ἔχον ἐκεῖνο μεῖζόν ἐστιν· πρὸς ὃ δὲ τὸ αὐτὸ μείζονα <lb n="1799,001,,,,5,10,2"></lb> |
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4122 λόγον ἔχει, ἐκεῖνο ἔλαττόν ἐστιν.</s> <lb n="1799,001,,,,5,10,3"></lb> |
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4123 <s>Ἐχέτω γὰρ τὸ Α πρὸς τὸ Γ μεί-<lb n="1799,001,,,,5,10,4"></lb> |
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4124 ζονα λόγον ἤπερ τὸ Β πρὸς τὸ Γ· <lb n="1799,001,,,,5,10,5"></lb> |
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4125 λέγω, ὅτι μεῖζόν ἐστι τὸ Α τοῦ Β.</s> <s><pb></pb> <lb n="1799,001,,,,5,10,6"></lb> |
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4126 Εἰ γὰρ μή, ἤτοι ἴσον ἐστὶ τὸ Α τῷ Β ἢ ἔλασσον.</s> <s>ἴσον <lb n="1799,001,,,,5,10,7"></lb> |
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4127 μὲν οὖν οὔκ ἐστι τὸ Α τῷ Β· ἑκάτερον γὰρ ἂν τῶν Α, Β <lb n="1799,001,,,,5,10,8"></lb> |
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4128 πρὸς τὸ Γ τὸν αὐτὸν εἶχε λόγον.</s> <s>οὐκ ἔχει δέ· οὐκ ἄρα <lb n="1799,001,,,,5,10,9"></lb> |
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4129 ἴσον ἐστὶ τὸ Α τῷ Β.</s> <s>οὐδὲ μὴν ἔλασσόν ἐστι τὸ Α τοῦ Β· <lb n="1799,001,,,,5,10,10"></lb> |
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4130 τὸ Α γὰρ ἂν πρὸς τὸ Γ ἐλάσσονα λόγον εἶχεν ἤπερ τὸ Β <lb n="1799,001,,,,5,10,11"></lb> |
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4131 πρὸς τὸ Γ.</s> <s>οὐκ ἔχει δέ· οὐκ ἄρα ἔλασσόν ἐστι τὸ Α τοῦ Β.</s> <lb n="1799,001,,,,5,10,12"></lb> |
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4132 <s>ἐδείχθη δὲ οὐδὲ ἴσον· μεῖζον ἄρα ἐστὶ τὸ Α τοῦ Β.</s> <lb n="1799,001,,,,5,10,13"></lb> |
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4133 <s>Ἐχέτω δὴ πάλιν τὸ Γ πρὸς τὸ Β μείζονα λόγον ἤπερ <lb n="1799,001,,,,5,10,14"></lb> |
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4134 τὸ Γ πρὸς τὸ Α· λέγω, ὅτι ἔλασσόν ἐστι τὸ Β τοῦ Α.</s> <lb n="1799,001,,,,5,10,15"></lb> |
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4135 <s>Εἰ γὰρ μή, ἤτοι ἴσον ἐστὶν ἢ μεῖζον.</s> <s>ἴσον μὲν οὖν οὔκ <lb n="1799,001,,,,5,10,16"></lb> |
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4136 ἐστι τὸ Β τῷ Α· τὸ Γ γὰρ ἂν πρὸς ἑκάτερον τῶν Α, Β <lb n="1799,001,,,,5,10,17"></lb> |
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4137 τὸν αὐτὸν εἶχε λόγον.</s> <s>οὐκ ἔχει δέ· οὐκ ἄρα ἴσον ἐστὶ τὸ Α <lb n="1799,001,,,,5,10,18"></lb> |
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4138 τῷ Β.</s> <s>οὐδὲ μὴν μεῖζόν ἐστι τὸ Β τοῦ Α· τὸ Γ γὰρ ἂν πρὸς <lb n="1799,001,,,,5,10,19"></lb> |
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4139 τὸ Β ἐλάσσονα λόγον εἶχεν ἤπερ πρὸς τὸ Α.</s> <s>οὐκ ἔχει δέ· <lb n="1799,001,,,,5,10,20"></lb> |
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4140 οὐκ ἄρα μεῖζόν ἐστι τὸ Β τοῦ Α.</s> <s>ἐδείχθη δέ, ὅτι οὐδὲ <lb n="1799,001,,,,5,10,21"></lb> |
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4141 ἴσον· ἔλαττον ἄρα ἐστὶ τὸ Β τοῦ Α.</s> <lb n="1799,001,,,,5,10,22"></lb> |
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4142 <s>Τῶν ἄρα πρὸς τὸ αὐτὸ λόγον ἐχόντων τὸ μείζονα λόγον <lb n="1799,001,,,,5,10,23"></lb> |
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4143 ἔχον μεῖζόν ἐστιν· καὶ πρὸς ὃ τὸ αὐτὸ μείζονα λόγον ἔχει, <lb n="1799,001,,,,5,10,24"></lb> |
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4144 ἐκεῖνο ἔλαττόν ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,10,25"></lb> |
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4145 <s>Οἱ τῷ αὐτῷ λόγῳ οἱ αὐτοὶ καὶ ἀλλήλοις εἰσὶν οἱ <lb n="1799,001,,,,5,11,1"></lb> |
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4146 αὐτοί.</s> <s><pb></pb> <lb n="1799,001,,,,5,11,2"></lb> |
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4147 Ἔστωσαν γὰρ ὡς μὲν τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ <lb n="1799,001,,,,5,11,3"></lb> |
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4148 πρὸς τὸ Δ, ὡς δὲ τὸ Γ πρὸς τὸ Δ, οὕτως τὸ Ε πρὸς τὸ Ζ· <lb n="1799,001,,,,5,11,4"></lb> |
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4149 λέγω, ὅτι ἐστὶν ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Ε πρὸς τὸ Ζ. <lb n="1799,001,,,,5,11,5"></lb> |
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4150 Εἰλήφθω γὰρ τῶν Α, Γ, Ε ἰσάκις πολλαπλάσια τὰ Η, <lb n="1799,001,,,,5,11,6"></lb> |
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4151 Θ, Κ, τῶν δὲ Β, Δ, Ζ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια <lb n="1799,001,,,,5,11,7"></lb> |
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4152 τὰ Λ, Μ, Ν.</s> <lb n="1799,001,,,,5,11,8"></lb> |
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4153 <s>Καὶ ἐπεί ἐστιν ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ πρὸς <lb n="1799,001,,,,5,11,9"></lb> |
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4154 τὸ Δ, καὶ εἴληπται τῶν μὲν Α, Γ ἰσάκις πολλαπλάσια <lb n="1799,001,,,,5,11,10"></lb> |
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4155 τὰ Η, Θ, τῶν δὲ Β, Δ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια <lb n="1799,001,,,,5,11,11"></lb> |
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4156 τὰ Λ, Μ, εἰ ἄρα ὑπερέχει τὸ Η τοῦ Λ, ὑπερέχει καὶ τὸ Θ <lb n="1799,001,,,,5,11,12"></lb> |
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4157 τοῦ Μ, καὶ εἰ ἴσον ἐστίν, ἴσον, καὶ εἰ ἐλλείπει, ἐλλείπει.</s> <lb n="1799,001,,,,5,11,13"></lb> |
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4158 <s>πάλιν, ἐπεί ἐστιν ὡς τὸ Γ πρὸς τὸ Δ, οὕτως τὸ Ε πρὸς <lb n="1799,001,,,,5,11,14"></lb> |
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4159 τὸ Ζ, καὶ εἴληπται τῶν Γ, Ε ἰσάκις πολλαπλάσια τὰ <lb n="1799,001,,,,5,11,15"></lb> |
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4160 Θ, Κ, τῶν δὲ Δ, Ζ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια <lb n="1799,001,,,,5,11,16"></lb> |
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4161 τὰ Μ, Ν, εἰ ἄρα ὑπερέχει τὸ Θ τοῦ Μ, ὑπερέχει καὶ τὸ <lb n="1799,001,,,,5,11,17"></lb> |
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4162 Κ τοῦ Ν, καὶ εἰ ἴσον, ἴσον, καὶ εἰ ἔλαττον, ἔλαττον.</s> <s>ἀλλὰ <lb n="1799,001,,,,5,11,18"></lb> |
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4163 εἰ ὑπερεῖχε τὸ Θ τοῦ Μ, ὑπερεῖχε καὶ τὸ Η τοῦ Λ, καὶ εἰ <lb n="1799,001,,,,5,11,19"></lb> |
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4164 ἴσον, ἴσον, καὶ εἰ ἔλαττον, ἔλαττον· ὥστε καὶ εἰ ὑπερέχει <lb n="1799,001,,,,5,11,20"></lb> |
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4165 τὸ Η τοῦ Λ, ὑπερέχει καὶ τὸ Κ τοῦ Ν, καὶ εἰ ἴσον, ἴσον, <pb></pb> <lb n="1799,001,,,,5,11,21"></lb> |
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4166 καὶ εἰ ἔλαττον, ἔλαττον.</s> <s>καί ἐστι τὰ μὲν Η, Κ τῶν Α, Ε <lb n="1799,001,,,,5,11,22"></lb> |
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4167 ἰσάκις πολλαπλάσια, τὰ δὲ Λ, Ν τῶν Β, Ζ ἄλλα, ἃ ἔτυχεν, <lb n="1799,001,,,,5,11,23"></lb> |
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4168 ἰσάκις πολλαπλάσια· ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Β, οὕτως <lb n="1799,001,,,,5,11,24"></lb> |
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4169 τὸ Ε πρὸς τὸ Ζ.</s> <lb n="1799,001,,,,5,11,25"></lb> |
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4170 <s>Οἱ ἄρα τῷ αὐτῷ λόγῳ οἱ αὐτοὶ καὶ ἀλλήλοις εἰσὶν οἱ <lb n="1799,001,,,,5,11,26"></lb> |
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4171 αὐτοί· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,11,27"></lb> |
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4172 <s>Ἐὰν ᾖ ὁποσαοῦν μεγέθη ἀνάλογον, ἔσται ὡς ἓν τῶν <lb n="1799,001,,,,5,12,1"></lb> |
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4173 ἡγουμένων πρὸς ἓν τῶν ἑπομένων, οὕτως ἅπαντα τὰ <lb n="1799,001,,,,5,12,2"></lb> |
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4174 ἡγούμενα πρὸς ἅπαντα τὰ ἑπόμενα.</s> <lb n="1799,001,,,,5,12,3"></lb> |
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4175 <s>Ἔστωσαν ὁποσαοῦν μεγέθη ἀνάλογον τὰ Α, Β, Γ, Δ, <lb n="1799,001,,,,5,12,4"></lb> |
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4176 Ε, Ζ, ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ πρὸς τὸ Δ, καὶ <lb n="1799,001,,,,5,12,5"></lb> |
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4177 τὸ Ε πρὸς τὸ Ζ· λέγω, ὅτι ἐστὶν ὡς τὸ Α πρὸς τὸ Β, <lb n="1799,001,,,,5,12,6"></lb> |
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4178 οὕτως τὰ Α, Γ, Ε πρὸς τὰ Β, Δ, Ζ. <lb n="1799,001,,,,5,12,7"></lb> |
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4179 Εἰλήφθω γὰρ τῶν μὲν Α, Γ, Ε ἰσάκις πολλαπλάσια <lb n="1799,001,,,,5,12,8"></lb> |
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4180 τὰ Η, Θ, Κ, τῶν δὲ Β, Δ, Ζ ἄλλα, ἃ ἔτυχεν, ἰσάκις <lb n="1799,001,,,,5,12,9"></lb> |
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4181 πολλαπλάσια τὰ Λ, Μ, Ν.</s> <lb n="1799,001,,,,5,12,10"></lb> |
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4182 <s>Καὶ ἐπεί ἐστιν ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ πρὸς <lb n="1799,001,,,,5,12,11"></lb> |
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4183 τὸ Δ, καὶ τὸ Ε πρὸς τὸ Ζ, καὶ εἴληπται τῶν μὲν Α, Γ, <lb n="1799,001,,,,5,12,12"></lb> |
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4184 Ε ἰσάκις πολλαπλάσια τὰ Η, Θ, Κ τῶν δὲ Β, Δ, Ζ ἄλλα, <pb></pb> <lb n="1799,001,,,,5,12,13"></lb> |
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4185 ἃ ἔτυχεν, ἰσάκις πολλαπλάσια τὰ Λ, Μ, Ν, εἰ ἄρα ὑπερ-<lb n="1799,001,,,,5,12,14"></lb> |
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4186 έχει τὸ Η τοῦ Λ, ὑπερέχει καὶ τὸ Θ τοῦ Μ, καὶ τὸ Κ <lb n="1799,001,,,,5,12,15"></lb> |
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4187 τοῦ Ν, καὶ εἰ ἴσον, ἴσον, καὶ εἰ ἔλαττον, ἔλαττον.</s> <s>ὥστε καὶ <lb n="1799,001,,,,5,12,16"></lb> |
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4188 εἰ ὑπερέχει τὸ Η τοῦ Λ, ὑπερέχει καὶ τὰ Η, Θ, Κ τῶν <lb n="1799,001,,,,5,12,17"></lb> |
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4189 Λ, Μ, Ν, καὶ εἰ ἴσον, ἴσα, καὶ εἰ ἔλαττον, ἐλάττονα.</s> <s>καί <lb n="1799,001,,,,5,12,18"></lb> |
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4190 ἐστι τὸ μὲν Η καὶ τὰ Η, Θ, Κ τοῦ Α καὶ τῶν Α, Γ, Ε <lb n="1799,001,,,,5,12,19"></lb> |
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4191 ἰσάκις πολλαπλάσια, ἐπειδήπερ ἐὰν ᾖ ὁποσαοῦν μεγέθη <lb n="1799,001,,,,5,12,20"></lb> |
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4192 ὁποσωνοῦν μεγεθῶν ἴσων τὸ πλῆθος ἕκαστον ἑκάστου <lb n="1799,001,,,,5,12,21"></lb> |
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4193 ἰσάκις πολλαπλάσιον, ὁσαπλάσιόν ἐστιν ἓν τῶν μεγεθῶν <lb n="1799,001,,,,5,12,22"></lb> |
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4194 ἑνός, τοσαυταπλάσια ἔσται καὶ τὰ πάντα τῶν πάντων.</s> <lb n="1799,001,,,,5,12,23"></lb> |
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4195 <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ Λ καὶ τὰ Λ, Μ, Ν τοῦ Β καὶ τῶν <lb n="1799,001,,,,5,12,24"></lb> |
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4196 Β, Δ, Ζ ἰσάκις ἐστὶ πολλαπλάσια· ἔστιν ἄρα ὡς τὸ Α <lb n="1799,001,,,,5,12,25"></lb> |
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4197 πρὸς τὸ Β, οὕτως τὰ Α, Γ, Ε πρὸς τὰ Β, Δ, Ζ.</s> <lb n="1799,001,,,,5,12,26"></lb> |
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4198 <s>Ἐὰν ἄρα ᾖ ὁποσαοῦν μεγέθη ἀνάλογον, ἔσται ὡς ἓν <lb n="1799,001,,,,5,12,27"></lb> |
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4199 τῶν ἡγουμένων πρὸς ἓν τῶν ἑπομένων, οὕτως ἅπαντα τὰ <lb n="1799,001,,,,5,12,28"></lb> |
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4200 ἡγούμενα πρὸς ἅπαντα τὰ ἑπόμενα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,12,29"></lb> |
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4201 <s>Ἐὰν πρῶτον πρὸς δεύτερον τὸν αὐτὸν ἔχῃ λόγον καὶ <lb n="1799,001,,,,5,13,1"></lb> |
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4202 τρίτον πρὸς τέταρτον, τρίτον δὲ πρὸς τέταρτον μείζονα <lb n="1799,001,,,,5,13,2"></lb> |
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4203 λόγον ἔχῃ ἢ πέμπτον πρὸς ἕκτον, καὶ πρῶτον πρὸς <lb n="1799,001,,,,5,13,3"></lb> |
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4204 δεύτερον μείζονα λόγον ἕξει ἢ πέμπτον πρὸς ἕκτον. <pb></pb> <lb n="1799,001,,,,5,13,4"></lb> |
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4205 Πρῶτον γὰρ τὸ Α πρὸς δεύτερον τὸ Β τὸν αὐτὸν ἐχέτω <lb n="1799,001,,,,5,13,5"></lb> |
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4206 λόγον καὶ τρίτον τὸ Γ πρὸς τέταρτον τὸ Δ, τρίτον δὲ τὸ Γ <lb n="1799,001,,,,5,13,6"></lb> |
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4207 πρὸς τέταρτον τὸ Δ μείζονα λόγον ἐχέτω ἢ πέμπτον τὸ Ε <lb n="1799,001,,,,5,13,7"></lb> |
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4208 πρὸς ἕκτον τὸ Ζ.</s> <s>λέγω, ὅτι καὶ πρῶτον τὸ Α πρὸς δεύτερον <lb n="1799,001,,,,5,13,8"></lb> |
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4209 τὸ Β μείζονα λόγον ἕξει ἤπερ πέμπτον τὸ Ε πρὸς ἕκτον <lb n="1799,001,,,,5,13,9"></lb> |
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4210 τὸ Ζ.</s> <lb n="1799,001,,,,5,13,10"></lb> |
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4211 <s>Ἐπεὶ γὰρ ἔστι τινὰ τῶν μὲν Γ, Ε ἰσάκις πολλαπλάσια, <lb n="1799,001,,,,5,13,11"></lb> |
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4212 τῶν δὲ Δ, Ζ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια, καὶ <lb n="1799,001,,,,5,13,12"></lb> |
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4213 τὸ μὲν τοῦ Γ πολλαπλάσιον τοῦ <lb n="1799,001,,,,5,13,13"></lb> |
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4214 τοῦ Δ πολλαπλασίου ὑπερέχει, τὸ <lb n="1799,001,,,,5,13,14"></lb> |
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4215 δὲ τοῦ Ε πολλαπλάσιον τοῦ τοῦ Ζ <lb n="1799,001,,,,5,13,15"></lb> |
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4216 πολλαπλασίου οὐχ ὑπερέχει, εἰλή-<lb n="1799,001,,,,5,13,16"></lb> |
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4217 φθω, καὶ ἔστω τῶν μὲν Γ, Ε ἰσά-<lb n="1799,001,,,,5,13,17"></lb> |
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4218 κις πολλαπλάσια τὰ Η, Θ, τῶν δὲ <lb n="1799,001,,,,5,13,18"></lb> |
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4219 Δ, Ζ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια τὰ Κ, Λ, ὥστε <lb n="1799,001,,,,5,13,19"></lb> |
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4220 τὸ μὲν Η τοῦ Κ ὑπερέχειν, τὸ δὲ Θ τοῦ Λ μὴ ὑπερέχειν· <lb n="1799,001,,,,5,13,20"></lb> |
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4221 καὶ ὁσαπλάσιον μέν ἐστι τὸ Η τοῦ Γ, τοσαυταπλάσιον <lb n="1799,001,,,,5,13,21"></lb> |
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4222 ἔστω καὶ τὸ Μ τοῦ Α, ὁσαπλάσιον δὲ τὸ Κ τοῦ Δ, <lb n="1799,001,,,,5,13,22"></lb> |
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4223 τοσαυταπλάσιον ἔστω καὶ τὸ Ν τοῦ Β.</s> <lb n="1799,001,,,,5,13,23"></lb> |
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4224 <s>Καὶ ἐπεί ἐστιν ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ πρὸς <lb n="1799,001,,,,5,13,24"></lb> |
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4225 τὸ Δ, καὶ εἴληπται τῶν μὲν Α, Γ ἰσάκις πολλαπλάσια τὰ <lb n="1799,001,,,,5,13,25"></lb> |
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4226 Μ, Η, τῶν δὲ Β, Δ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια <lb n="1799,001,,,,5,13,26"></lb> |
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4227 τὰ Ν, Κ, εἰ ἄρα ὑπερέχει τὸ Μ τοῦ Ν, ὑπερέχει καὶ τὸ Η <lb n="1799,001,,,,5,13,27"></lb> |
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4228 τοῦ Κ, καὶ εἰ ἴσον, ἴσον, καὶ εἰ ἔλαττον, ἔλαττον.</s> <s>ὑπερέχει <lb n="1799,001,,,,5,13,28"></lb> |
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4229 δὲ τὸ Η τοῦ Κ· ὑπερέχει ἄρα καὶ τὸ Μ τοῦ Ν.</s> <s>τὸ δὲ Θ <lb n="1799,001,,,,5,13,29"></lb> |
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4230 τοῦ Λ οὐχ ὑπερέχει· καί ἐστι τὰ μὲν Μ, Θ τῶν Α, Ε <pb></pb> <lb n="1799,001,,,,5,13,30"></lb> |
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4231 ἰσάκις πολλαπλάσια, τὰ δὲ Ν, Λ τῶν Β, Ζ ἄλλα, ἃ ἔτυχεν, <lb n="1799,001,,,,5,13,31"></lb> |
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4232 ἰσάκις πολλαπλάσια· τὸ ἄρα Α πρὸς τὸ Β μείζονα λόγον <lb n="1799,001,,,,5,13,32"></lb> |
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4233 ἔχει ἤπερ τὸ Ε πρὸς τὸ Ζ.</s> <lb n="1799,001,,,,5,13,33"></lb> |
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4234 <s>Ἐὰν ἄρα πρῶτον πρὸς δεύτερον τὸν αὐτὸν ἔχῃ λόγον <lb n="1799,001,,,,5,13,34"></lb> |
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4235 καὶ τρίτον πρὸς τέταρτον, τρίτον δὲ πρὸς τέταρτον μείζονα <lb n="1799,001,,,,5,13,35"></lb> |
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4236 λόγον ἔχῃ ἢ πέμπτον πρὸς ἕκτον, καὶ πρῶτον πρὸς <lb n="1799,001,,,,5,13,36"></lb> |
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4237 δεύτερον μείζονα λόγον ἕξει ἢ πέμπτον πρὸς ἕκτον· <lb n="1799,001,,,,5,13,37"></lb> |
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4238 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,13,38"></lb> |
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4239 <s>Ἐὰν πρῶτον πρὸς δεύτερον τὸν αὐτὸν ἔχῃ λόγον καὶ <lb n="1799,001,,,,5,14,1"></lb> |
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4240 τρίτον πρὸς τέταρτον, τὸ δὲ πρῶτον τοῦ τρίτου μεῖζον ᾖ, <lb n="1799,001,,,,5,14,2"></lb> |
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4241 καὶ τὸ δεύτερον τοῦ τετάρτου μεῖζον ἔσται, κἂν ἴσον, ἴσον, <lb n="1799,001,,,,5,14,3"></lb> |
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4242 κἂν ἔλαττον, ἔλαττον.</s> <lb n="1799,001,,,,5,14,4"></lb> |
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4243 <s>Πρῶτον γὰρ τὸ Α πρὸς δεύτερον τὸ Β τὸν αὐτὸν ἐχέτω <lb n="1799,001,,,,5,14,5"></lb> |
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4244 λόγον καὶ τρίτον τὸ Γ πρὸς τέταρτον τὸ Δ, μεῖζον δὲ ἔστω <lb n="1799,001,,,,5,14,6"></lb> |
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4245 τὸ Α τοῦ Γ· λέγω, ὅτι καὶ τὸ Β <lb n="1799,001,,,,5,14,7"></lb> |
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4246 τοῦ Δ μεῖζόν ἐστιν.</s> <lb n="1799,001,,,,5,14,8"></lb> |
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4247 <s>Ἐπεὶ γὰρ τὸ Α τοῦ Γ μεῖζόν <lb n="1799,001,,,,5,14,9"></lb> |
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4248 ἐστιν, ἄλλο δέ, ὃ ἔτυχεν, [μέγε-<lb n="1799,001,,,,5,14,10"></lb> |
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4249 θοσ] τὸ Β, τὸ Α ἄρα πρὸς τὸ Β μείζονα λόγον ἔχει ἤπερ <lb n="1799,001,,,,5,14,11"></lb> |
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4250 τὸ Γ πρὸς τὸ Β.</s> <s>ὡς δὲ τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ <lb n="1799,001,,,,5,14,12"></lb> |
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4251 πρὸς τὸ Δ· καὶ τὸ Γ ἄρα πρὸς τὸ Δ μείζονα λόγον ἔχει <lb n="1799,001,,,,5,14,13"></lb> |
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4252 ἤπερ τὸ Γ πρὸς τὸ Β.</s> <s>πρὸς ὃ δὲ τὸ αὐτὸ μείζονα λόγον <lb n="1799,001,,,,5,14,14"></lb> |
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4253 ἔχει, ἐκεῖνο ἔλασσόν ἐστιν· ἔλασσον ἄρα τὸ Δ τοῦ Β· <lb n="1799,001,,,,5,14,15"></lb> |
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4254 ὥστε μεῖζόν ἐστι τὸ Β τοῦ Δ.</s> <s><pb></pb> <lb n="1799,001,,,,5,14,16"></lb> |
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4255 Ὁμοίως δὴ δείξομεν, ὅτι κἂν ἴσον ᾖ τὸ Α τῷ Γ, ἴσον <lb n="1799,001,,,,5,14,17"></lb> |
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4256 ἔσται καὶ τὸ Β τῷ Δ, κἂν ἔλασσον ᾖ τὸ Α τοῦ Γ, ἔλασσον <lb n="1799,001,,,,5,14,18"></lb> |
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4257 ἔσται καὶ τὸ Β τοῦ Δ.</s> <lb n="1799,001,,,,5,14,19"></lb> |
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4258 <s>Ἐὰν ἄρα πρῶτον πρὸς δεύτερον τὸν αὐτὸν ἔχῃ λόγον καὶ <lb n="1799,001,,,,5,14,20"></lb> |
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4259 τρίτον πρὸς τέταρτον, τὸ δὲ πρῶτον τοῦ τρίτου μεῖζον ᾖ, <lb n="1799,001,,,,5,14,21"></lb> |
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4260 καὶ τὸ δεύτερον τοῦ τετάρτου μεῖζον ἔσται, κἂν ἴσον, ἴσον, <lb n="1799,001,,,,5,14,22"></lb> |
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4261 κἂν ἔλαττον, ἔλαττον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,14,23"></lb> |
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4262 <s>Τὰ μέρη τοῖς ὡσαύτως πολλαπλασίοις τὸν αὐτὸν ἔχει <lb n="1799,001,,,,5,15,1"></lb> |
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4263 λόγον ληφθέντα κατάλληλα.</s> <lb n="1799,001,,,,5,15,2"></lb> |
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4264 <s>Ἔστω γὰρ ἰσάκις πολλαπλάσιον τὸ ΑΒ τοῦ Γ καὶ τὸ <lb n="1799,001,,,,5,15,3"></lb> |
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4265 ΔΕ τοῦ Ζ· λέγω, ὅτι ἐστὶν ὡς τὸ Γ πρὸς τὸ Ζ, οὕτως <lb n="1799,001,,,,5,15,4"></lb> |
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4266 τὸ ΑΒ πρὸς τὸ ΔΕ.</s> <lb n="1799,001,,,,5,15,5"></lb> |
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4267 <s>Ἐπεὶ γὰρ ἰσάκις ἐστὶ πολλαπλάσιον τὸ ΑΒ τοῦ Γ <lb n="1799,001,,,,5,15,6"></lb> |
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4268 καὶ τὸ ΔΕ τοῦ Ζ, ὅσα ἄρα ἐστὶν ἐν τῷ ΑΒ μεγέθη ἴσα <lb n="1799,001,,,,5,15,7"></lb> |
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4269 τῷ Γ, τοσαῦτα καὶ ἐν τῷ ΔΕ ἴσα <lb n="1799,001,,,,5,15,8"></lb> |
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4270 τῷ Ζ.</s> <s>διῃρήσθω τὸ μὲν ΑΒ εἰς τὰ <lb n="1799,001,,,,5,15,9"></lb> |
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4271 τῷ Γ ἴσα τὰ ΑΗ, ΗΘ, ΘΒ, τὸ <lb n="1799,001,,,,5,15,10"></lb> |
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4272 δὲ ΔΕ εἰς τὰ τῷ Ζ ἴσα τὰ ΔΚ, <lb n="1799,001,,,,5,15,11"></lb> |
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4273 ΚΛ, ΛΕ· ἔσται δὴ ἴσον τὸ πλῆθος <lb n="1799,001,,,,5,15,12"></lb> |
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4274 τῶν ΑΗ, ΗΘ, ΘΒ τῷ πλήθει τῶν ΔΚ, ΚΛ, ΛΕ.</s> <s>καὶ <lb n="1799,001,,,,5,15,13"></lb> |
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4275 ἐπεὶ ἴσα ἐστὶ τὰ ΑΗ, ΗΘ, ΘΒ ἀλλήλοις, ἔστι δὲ καὶ τὰ <lb n="1799,001,,,,5,15,14"></lb> |
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4276 ΔΚ, ΚΛ, ΛΕ ἴσα ἀλλήλοις, ἔστιν ἄρα ὡς τὸ ΑΗ πρὸς <lb n="1799,001,,,,5,15,15"></lb> |
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4277 τὸ ΔΚ, οὕτως τὸ ΗΘ πρὸς τὸ ΚΛ, καὶ τὸ ΘΒ πρὸς τὸ <pb></pb> <lb n="1799,001,,,,5,15,16"></lb> |
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4278 ΛΕ.</s> <s>ἔσται ἄρα καὶ ὡς ἓν τῶν ἡγουμένων πρὸς ἓν τῶν <lb n="1799,001,,,,5,15,17"></lb> |
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4279 ἑπομένων, οὕτως ἅπαντα τὰ ἡγούμενα πρὸς ἅπαντα τὰ <lb n="1799,001,,,,5,15,18"></lb> |
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4280 ἑπόμενα· ἔστιν ἄρα ὡς τὸ ΑΗ πρὸς τὸ ΔΚ, οὕτως τὸ <lb n="1799,001,,,,5,15,19"></lb> |
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4281 ΑΒ πρὸς τὸ ΔΕ.</s> <s>ἴσον δὲ τὸ μὲν ΑΗ τῷ Γ, τὸ δὲ ΔΚ <lb n="1799,001,,,,5,15,20"></lb> |
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4282 τῷ Ζ· ἔστιν ἄρα ὡς τὸ Γ πρὸς τὸ Ζ οὕτως τὸ ΑΒ πρὸς <lb n="1799,001,,,,5,15,21"></lb> |
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4283 τὸ ΔΕ.</s> <lb n="1799,001,,,,5,15,22"></lb> |
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4284 <s>Τὰ ἄρα μέρη τοῖς ὡσαύτως πολλαπλασίοις τὸν αὐτὸν <lb n="1799,001,,,,5,15,23"></lb> |
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4285 ἔχει λόγον ληφθέντα κατάλληλα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,15,24"></lb> |
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4286 <s>Ἐὰν τέσσαρα μεγέθη ἀνάλογον ᾖ, καὶ ἐναλλὰξ ἀνάλογον <lb n="1799,001,,,,5,16,1"></lb> |
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4287 ἔσται.</s> <lb n="1799,001,,,,5,16,2"></lb> |
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4288 <s>Ἔστω τέσσαρα μεγέθη ἀνάλογον τὰ Α, Β, Γ, Δ, ὡς τὸ Α <lb n="1799,001,,,,5,16,3"></lb> |
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4289 πρὸς τὸ Β, οὕτως τὸ Γ πρὸς τὸ Δ· λέγω, ὅτι καὶ ἐναλλὰξ [ἀνά-<lb n="1799,001,,,,5,16,4"></lb> |
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4290 λογον] ἔσται, ὡς τὸ Α πρὸς τὸ <lb n="1799,001,,,,5,16,5"></lb> |
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4291 Γ, οὕτως τὸ Β πρὸς τὸ Δ.</s> <lb n="1799,001,,,,5,16,6"></lb> |
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4292 <s>Εἰλήφθω γὰρ τῶν μὲν Α, Β <lb n="1799,001,,,,5,16,7"></lb> |
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4293 ἰσάκις πολλαπλάσια τὰ Ε, Ζ, <lb n="1799,001,,,,5,16,8"></lb> |
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4294 τῶν δὲ Γ, Δ ἄλλα, ἃ ἔτυχεν, <lb n="1799,001,,,,5,16,9"></lb> |
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4295 ἰσάκις πολλαπλάσια τὰ Η, Θ.</s> <lb n="1799,001,,,,5,16,10"></lb> |
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4296 <s>Καὶ ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον τὸ Ε τοῦ Α καὶ τὸ <lb n="1799,001,,,,5,16,11"></lb> |
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4297 Ζ τοῦ Β, τὰ δὲ μέρη τοῖς ὡσαύτως πολλαπλασίοις τὸν <lb n="1799,001,,,,5,16,12"></lb> |
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4298 αὐτὸν ἔχει λόγον, ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ <lb n="1799,001,,,,5,16,13"></lb> |
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4299 Ε πρὸς τὸ Ζ.</s> <s>ὡς δὲ τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ πρὸς <lb n="1799,001,,,,5,16,14"></lb> |
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4300 τὸ Δ· καὶ ὡς ἄρα τὸ Γ πρὸς τὸ Δ, οὕτως τὸ Ε πρὸς τὸ Ζ.</s> <s><pb></pb> <lb n="1799,001,,,,5,16,15"></lb> |
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4301 πάλιν, ἐπεὶ τὰ Η, Θ τῶν Γ, Δ ἰσάκις ἐστὶ πολλαπλάσια, <lb n="1799,001,,,,5,16,16"></lb> |
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4302 ἔστιν ἄρα ὡς τὸ Γ πρὸς τὸ Δ, οὕτως τὸ Η πρὸς τὸ Θ.</s> <lb n="1799,001,,,,5,16,17"></lb> |
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4303 <s>ὡς δὲ τὸ Γ πρὸς τὸ Δ, [οὕτωσ] τὸ Ε πρὸς τὸ Ζ· καὶ ὡς <lb n="1799,001,,,,5,16,18"></lb> |
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4304 ἄρα τὸ Ε πρὸς τὸ Ζ, οὕτως τὸ Η πρὸς τὸ Θ.</s> <s>ἐὰν δὲ τέσ-<lb n="1799,001,,,,5,16,19"></lb> |
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4305 σαρα μεγέθη ἀνάλογον ᾖ, τὸ δὲ πρῶτον τοῦ τρίτου μεῖζον <lb n="1799,001,,,,5,16,20"></lb> |
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4306 ᾖ, καὶ τὸ δεύτερον τοῦ τετάρτου μεῖζον ἔσται, κἂν ἴσον, <lb n="1799,001,,,,5,16,21"></lb> |
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4307 ἴσον, κἂν ἔλαττον, ἔλαττον.</s> <s>εἰ ἄρα ὑπερέχει τὸ Ε τοῦ Η, <lb n="1799,001,,,,5,16,22"></lb> |
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4308 ὑπερέχει καὶ τὸ Ζ τοῦ Θ, καὶ εἰ ἴσον, ἴσον, καὶ εἰ ἔλαττον, <lb n="1799,001,,,,5,16,23"></lb> |
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4309 ἔλαττον.</s> <s>καί ἐστι τὰ μὲν Ε, Ζ τῶν Α, Β ἰσάκις πολλαπλά-<lb n="1799,001,,,,5,16,24"></lb> |
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4310 σια, τὰ δὲ Η, Θ τῶν Γ, Δ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλα-<lb n="1799,001,,,,5,16,25"></lb> |
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4311 πλάσια· ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Γ, οὕτως τὸ Β πρὸς <lb n="1799,001,,,,5,16,26"></lb> |
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4312 τὸ Δ.</s> <lb n="1799,001,,,,5,16,27"></lb> |
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4313 <s>Ἐὰν ἄρα τέσσαρα μεγέθη ἀνάλογον ᾖ, καὶ ἐναλλὰξ <lb n="1799,001,,,,5,16,28"></lb> |
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4314 ἀνάλογον ἔσται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,16,29"></lb> |
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4315 <s>Ἐὰν συγκείμενα μεγέθη ἀνάλογον ᾖ, καὶ διαιρεθέντα <lb n="1799,001,,,,5,17,1"></lb> |
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4316 ἀνάλογον ἔσται.</s> <lb n="1799,001,,,,5,17,2"></lb> |
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4317 <s>Ἔστω συγκείμενα μεγέθη ἀνάλογον τὰ ΑΒ, ΒΕ, <lb n="1799,001,,,,5,17,3"></lb> |
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4318 ΓΔ, ΔΖ, ὡς τὸ ΑΒ πρὸς τὸ ΒΕ, οὕτως τὸ ΓΔ πρὸς <lb n="1799,001,,,,5,17,4"></lb> |
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4319 τὸ ΔΖ· λέγω, ὅτι καὶ διαιρεθέντα ἀνάλογον ἔσται, ὡς <lb n="1799,001,,,,5,17,5"></lb> |
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4320 τὸ ΑΕ πρὸς τὸ ΕΒ, οὕτως τὸ ΓΖ πρὸς τὸ ΔΖ. <pb></pb> <lb n="1799,001,,,,5,17,6"></lb> |
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4321 Εἰλήφθω γὰρ τῶν μὲν ΑΕ, ΕΒ, ΓΖ, ΖΔ ἰσάκις <lb n="1799,001,,,,5,17,7"></lb> |
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4322 πολλαπλάσια τὰ ΗΘ, ΘΚ, ΛΜ, ΜΝ, τῶν δὲ ΕΒ, ΖΔ <lb n="1799,001,,,,5,17,8"></lb> |
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4323 ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια τὰ ΚΞ, ΝΠ.</s> <lb n="1799,001,,,,5,17,9"></lb> |
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4324 <s>Καὶ ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον τὸ ΗΘ τοῦ ΑΕ <lb n="1799,001,,,,5,17,10"></lb> |
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4325 καὶ τὸ ΘΚ τοῦ ΕΒ, ἰσάκις ἄρα ἐστὶ πολλαπλάσιον τὸ <lb n="1799,001,,,,5,17,11"></lb> |
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4326 ΗΘ τοῦ ΑΕ καὶ τὸ ΗΚ τοῦ ΑΒ.</s> <s>ἰσάκις δέ ἐστι πολλα-<lb n="1799,001,,,,5,17,12"></lb> |
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4327 πλάσιον τὸ ΗΘ τοῦ ΑΕ καὶ τὸ ΛΜ τοῦ ΓΖ· ἰσάκις ἄρα <lb n="1799,001,,,,5,17,13"></lb> |
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4328 ἐστὶ πολλαπλάσιον τὸ ΗΚ τοῦ ΑΒ καὶ τὸ ΛΜ τοῦ ΓΖ.</s> <lb n="1799,001,,,,5,17,14"></lb> |
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4329 <s>πάλιν, ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον τὸ ΛΜ τοῦ ΓΖ <lb n="1799,001,,,,5,17,15"></lb> |
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4330 καὶ τὸ ΜΝ τοῦ ΖΔ, ἰσάκις ἄρα ἐστὶ πολλαπλάσιον τὸ <lb n="1799,001,,,,5,17,16"></lb> |
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4331 ΛΜ τοῦ ΓΖ καὶ τὸ ΛΝ τοῦ ΓΔ.</s> <s>ἰσάκις δὲ ἦν πολλαπλάσιον <lb n="1799,001,,,,5,17,17"></lb> |
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4332 τὸ ΛΜ τοῦ ΓΖ καὶ τὸ ΗΚ τοῦ ΑΒ· ἰσάκις ἄρα ἐστὶ πολ-<lb n="1799,001,,,,5,17,18"></lb> |
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4333 λαπλάσιον τὸ ΗΚ τοῦ ΑΒ καὶ τὸ ΛΝ τοῦ ΓΔ.</s> <s>τὰ ΗΚ, <lb n="1799,001,,,,5,17,19"></lb> |
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4334 ΛΝ ἄρα τῶν ΑΒ, ΓΔ ἰσάκις ἐστὶ πολλαπλάσια.</s> <s>πάλιν, <lb n="1799,001,,,,5,17,20"></lb> |
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4335 ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον τὸ ΘΚ τοῦ ΕΒ καὶ τὸ <lb n="1799,001,,,,5,17,21"></lb> |
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4336 ΜΝ τοῦ ΖΔ, ἔστι δὲ καὶ τὸ ΚΞ τοῦ ΕΒ ἰσάκις πολλαπλά-<lb n="1799,001,,,,5,17,22"></lb> |
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4337 σιον καὶ τὸ ΝΠ τοῦ ΖΔ, καὶ συντεθὲν τὸ ΘΞ τοῦ ΕΒ <lb n="1799,001,,,,5,17,23"></lb> |
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4338 ἰσάκις ἐστὶ πολλαπλάσιον καὶ τὸ ΜΠ τοῦ ΖΔ.</s> <s>Καὶ ἐπεί <lb n="1799,001,,,,5,17,24"></lb> |
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4339 ἐστιν ὡς τὸ ΑΒ πρὸς τὸ ΒΕ, οὕτως τὸ ΓΔ πρὸς τὸ ΔΖ, <lb n="1799,001,,,,5,17,25"></lb> |
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4340 καὶ εἴληπται τῶν μὲν ΑΒ, ΓΔ ἰσάκις πολλαπλάσια τὰ <lb n="1799,001,,,,5,17,26"></lb> |
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4341 ΗΚ, ΛΝ, τῶν δὲ ΕΒ, ΖΔ ἰσάκις πολλαπλάσια τὰ ΘΞ, <lb n="1799,001,,,,5,17,27"></lb> |
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4342 ΜΠ, εἰ ἄρα ὑπερέχει τὸ ΗΚ τοῦ ΘΞ, ὑπερέχει καὶ τὸ <lb n="1799,001,,,,5,17,28"></lb> |
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4343 ΛΝ τοῦ ΜΠ, καὶ εἰ ἴσον, ἴσον, καὶ εἰ ἔλαττον, ἔλαττον.</s> <lb n="1799,001,,,,5,17,29"></lb> |
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4344 <s>ὑπερεχέτω δὴ τὸ ΗΚ τοῦ ΘΞ, καὶ κοινοῦ ἀφαιρεθέντος <pb></pb> <lb n="1799,001,,,,5,17,30"></lb> |
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4345 τοῦ ΘΚ ὑπερέχει ἄρα καὶ τὸ ΗΘ τοῦ ΚΞ.</s> <s>ἀλλὰ εἰ ὑπερεῖχε <lb n="1799,001,,,,5,17,31"></lb> |
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4346 τὸ ΗΚ τοῦ ΘΞ, ὑπερεῖχε καὶ τὸ ΛΝ τοῦ ΜΠ· ὑπερέχει <lb n="1799,001,,,,5,17,32"></lb> |
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4347 ἄρα καὶ τὸ ΛΝ τοῦ ΜΠ, καὶ κοινοῦ ἀφαιρεθέντος τοῦ <lb n="1799,001,,,,5,17,33"></lb> |
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4348 ΜΝ ὑπερέχει καὶ τὸ ΛΜ τοῦ ΝΠ· ὥστε εἰ ὑπερέχει τὸ <lb n="1799,001,,,,5,17,34"></lb> |
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4349 ΗΘ τοῦ ΚΞ, ὑπερέχει καὶ τὸ ΛΜ τοῦ ΝΠ.</s> <s>ὁμοίως δὴ <lb n="1799,001,,,,5,17,35"></lb> |
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4350 δείξομεν, ὅτι κἂν ἴσον ᾖ τὸ ΗΘ τῷ ΚΞ, ἴσον ἔσται καὶ <lb n="1799,001,,,,5,17,36"></lb> |
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4351 τὸ ΛΜ τῷ ΝΠ, κἂν ἔλαττον, ἔλαττον.</s> <s>καί ἐστι τὰ μὲν <lb n="1799,001,,,,5,17,37"></lb> |
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4352 ΗΘ, ΛΜ τῶν ΑΕ, ΓΖ ἰσάκις πολλαπλάσια, τὰ δὲ ΚΞ, <lb n="1799,001,,,,5,17,38"></lb> |
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4353 ΝΠ τῶν ΕΒ, ΖΔ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια· <lb n="1799,001,,,,5,17,39"></lb> |
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4354 ἔστιν ἄρα ὡς τὸ ΑΕ πρὸς τὸ ΕΒ, οὕτως τὸ ΓΖ πρὸς <lb n="1799,001,,,,5,17,40"></lb> |
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4355 τὸ ΖΔ.</s> <lb n="1799,001,,,,5,17,41"></lb> |
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4356 <s>Ἐὰν ἄρα συγκείμενα μεγέθη ἀνάλογον ᾖ, καὶ διαιρε-<lb n="1799,001,,,,5,17,42"></lb> |
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4357 θέντα ἀνάλογον ἔσται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,17,43"></lb> |
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4358 <s>Ἐὰν διῃρημένα μεγέθη ἀνάλογον ᾖ, καὶ συντεθέντα <lb n="1799,001,,,,5,18,1"></lb> |
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4359 ἀνάλογον ἔσται.</s> <lb n="1799,001,,,,5,18,2"></lb> |
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4360 <s>Ἔστω διῃρημένα μεγέθη ἀνάλογον τὰ ΑΕ, ΕΒ, ΓΖ, <lb n="1799,001,,,,5,18,3"></lb> |
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4361 ΖΔ, ὡς τὸ ΑΕ πρὸς τὸ ΕΒ, οὕτως τὸ ΓΖ πρὸς τὸ ΖΔ· <lb n="1799,001,,,,5,18,4"></lb> |
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4362 λέγω, ὅτι καὶ συντεθέντα ἀνά- <lb n="1799,001,,,,5,18,5"></lb> |
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4363 λογον ἔσται, ὡς τὸ ΑΒ πρὸς τὸ <lb n="1799,001,,,,5,18,6"></lb> |
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4364 ΒΕ, οὕτως τὸ ΓΔ πρὸς τὸ ΖΔ.</s> <lb n="1799,001,,,,5,18,7"></lb> |
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4365 <s>Εἰ γὰρ μή ἐστιν ὡς τὸ ΑΒ <lb n="1799,001,,,,5,18,8"></lb> |
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4366 πρὸς τὸ ΒΕ, οὕτως τὸ ΓΔ <pb></pb> <lb n="1799,001,,,,5,18,9"></lb> |
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4367 πρὸς τὸ ΔΖ, ἔσται ὡς τὸ ΑΒ πρὸς τὸ ΒΕ, οὕτως τὸ ΓΔ <lb n="1799,001,,,,5,18,10"></lb> |
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4368 ἤτοι πρὸς ἔλασσόν τι τοῦ ΔΖ ἢ πρὸς μεῖζον.</s> <lb n="1799,001,,,,5,18,11"></lb> |
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4369 <s>Ἔστω πρότερον πρὸς ἔλασσον τὸ ΔΗ.</s> <s>καὶ ἐπεί ἐστιν <lb n="1799,001,,,,5,18,12"></lb> |
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4370 ὡς τὸ ΑΒ πρὸς τὸ ΒΕ, οὕτως τὸ ΓΔ πρὸς τὸ ΔΗ, <lb n="1799,001,,,,5,18,13"></lb> |
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4371 συγκείμενα μεγέθη ἀνάλογόν ἐστιν· ὥστε καὶ διαιρεθέντα <lb n="1799,001,,,,5,18,14"></lb> |
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4372 ἀνάλογον ἔσται.</s> <s>ἔστιν ἄρα ὡς τὸ ΑΕ πρὸς τὸ ΕΒ, οὕτως <lb n="1799,001,,,,5,18,15"></lb> |
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4373 τὸ ΓΗ πρὸς τὸ ΗΔ.</s> <s>ὑπόκειται δὲ καὶ ὡς τὸ ΑΕ πρὸς τὸ <lb n="1799,001,,,,5,18,16"></lb> |
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4374 ΕΒ, οὕτως τὸ ΓΖ πρὸς τὸ ΖΔ.</s> <s>καὶ ὡς ἄρα τὸ ΓΗ πρὸς <lb n="1799,001,,,,5,18,17"></lb> |
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4375 τὸ ΗΔ, οὕτως τὸ ΓΖ πρὸς τὸ ΖΔ.</s> <s>μεῖζον δὲ τὸ πρῶτον <lb n="1799,001,,,,5,18,18"></lb> |
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4376 τὸ ΓΗ τοῦ τρίτου τοῦ ΓΖ· μεῖζον ἄρα καὶ τὸ δεύτερον <lb n="1799,001,,,,5,18,19"></lb> |
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4377 τὸ ΗΔ τοῦ τετάρτου τοῦ ΖΔ.</s> <s>ἀλλὰ καὶ ἔλαττον· ὅπερ <lb n="1799,001,,,,5,18,20"></lb> |
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4378 ἐστὶν ἀδύνατον· οὐκ ἄρα ἐστὶν ὡς τὸ ΑΒ πρὸς τὸ ΒΕ, <lb n="1799,001,,,,5,18,21"></lb> |
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4379 οὕτως τὸ ΓΔ πρὸς ἔλασσον τοῦ ΖΔ.</s> <s>ὁμοίως δὴ δείξομεν, <lb n="1799,001,,,,5,18,22"></lb> |
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4380 ὅτι οὐδὲ πρὸς μεῖζον· πρὸς αὐτὸ ἄρα.</s> <lb n="1799,001,,,,5,18,23"></lb> |
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4381 <s>Ἐὰν ἄρα διῃρημένα μεγέθη ἀνάλογον ᾖ, καὶ συν-<lb n="1799,001,,,,5,18,24"></lb> |
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4382 τεθέντα ἀνάλογον ἔσται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,18,25"></lb> |
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4383 <s>Ἐὰν ᾖ ὡς ὅλον πρὸς ὅλον, οὕτως ἀφαιρεθὲν πρὸς <lb n="1799,001,,,,5,19,1"></lb> |
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4384 ἀφαιρεθέν, καὶ τὸ λοιπὸν πρὸς τὸ λοιπὸν ἔσται ὡς ὅλον <lb n="1799,001,,,,5,19,2"></lb> |
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4385 πρὸς ὅλον.</s> <lb n="1799,001,,,,5,19,3"></lb> |
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4386 <s>Ἔστω γὰρ ὡς ὅλον τὸ ΑΒ πρὸς ὅλον τὸ ΓΔ, οὕτως <lb n="1799,001,,,,5,19,4"></lb> |
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4387 ἀφαιρεθὲν τὸ ΑΕ πρὸς ἀφαιρεθὲν τὸ ΓΖ· λέγω, ὅτι καὶ <pb></pb> <lb n="1799,001,,,,5,19,5"></lb> |
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4388 λοιπὸν τὸ ΕΒ πρὸς λοιπὸν τὸ ΖΔ ἔσται ὡς ὅλον τὸ ΑΒ <lb n="1799,001,,,,5,19,6"></lb> |
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4389 πρὸς ὅλον τὸ ΓΔ.</s> <lb n="1799,001,,,,5,19,7"></lb> |
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4390 <s>Ἐπεὶ γάρ ἐστιν ὡς τὸ ΑΒ πρὸς τὸ ΓΔ, οὕτως τὸ ΑΕ <lb n="1799,001,,,,5,19,8"></lb> |
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4391 πρὸς τὸ ΓΖ, καὶ ἐναλλὰξ ὡς τὸ ΒΑ πρὸς τὸ ΑΕ, οὕτως <lb n="1799,001,,,,5,19,9"></lb> |
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4392 τὸ ΔΓ πρὸς τὸ ΓΖ.</s> <s>καὶ <lb n="1799,001,,,,5,19,10"></lb> |
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4393 ἐπεὶ συγκείμενα μεγέθη <lb n="1799,001,,,,5,19,11"></lb> |
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4394 ἀνάλογόν ἐστιν, καὶ διαιρε-<lb n="1799,001,,,,5,19,12"></lb> |
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4395 θέντα ἀνάλογον ἔσται, ὡς <lb n="1799,001,,,,5,19,13"></lb> |
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4396 τὸ ΒΕ πρὸς τὸ ΕΑ, οὕτως <lb n="1799,001,,,,5,19,14"></lb> |
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4397 τὸ ΔΖ πρὸς τὸ ΓΖ· καὶ ἐναλλάξ, ὡς τὸ ΒΕ πρὸς τὸ <lb n="1799,001,,,,5,19,15"></lb> |
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4398 ΔΖ, οὕτως τὸ ΕΑ πρὸς τὸ ΖΓ.</s> <s>ὡς δὲ τὸ ΑΕ πρὸς τὸ <lb n="1799,001,,,,5,19,16"></lb> |
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4399 ΓΖ, οὕτως ὑπόκειται ὅλον τὸ ΑΒ πρὸς ὅλον τὸ ΓΔ.</s> <s>καὶ <lb n="1799,001,,,,5,19,17"></lb> |
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4400 λοιπὸν ἄρα τὸ ΕΒ πρὸς λοιπὸν τὸ ΖΔ ἔσται ὡς ὅλον τὸ <lb n="1799,001,,,,5,19,18"></lb> |
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4401 ΑΒ πρὸς ὅλον τὸ ΓΔ.</s> <lb n="1799,001,,,,5,19,19"></lb> |
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4402 <s>Ἐὰν ἄρα ᾖ ὡς ὅλον πρὸς ὅλον, οὕτως ἀφαιρεθὲν πρὸς <lb n="1799,001,,,,5,19,20"></lb> |
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4403 ἀφαιρεθέν, καὶ τὸ λοιπὸν πρὸς τὸ λοιπὸν ἔσται ὡς ὅλον <lb n="1799,001,,,,5,19,21"></lb> |
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4404 πρὸς ὅλον [ὅπερ ἔδει δεῖξαι].</s> <lb n="1799,001,,,,5,19,22"></lb> |
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4405 <s>[Καὶ ἐπεὶ ἐδείχθη ὡς τὸ ΑΒ πρὸς τὸ ΓΔ, οὕτως τὸ <lb n="1799,001,,,,5,19,23"></lb> |
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4406 ΕΒ πρὸς τὸ ΖΔ, καὶ ἐναλλὰξ ὡς τὸ ΑΒ πρὸς τὸ ΒΕ <lb n="1799,001,,,,5,19,24"></lb> |
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4407 οὕτως τὸ ΓΔ πρὸς τὸ ΖΔ, συγκείμενα ἄρα μεγέθη <lb n="1799,001,,,,5,19,25"></lb> |
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4408 ἀνάλογόν ἐστιν· ἐδείχθη δὲ ὡς τὸ ΒΑ πρὸς τὸ ΑΕ, οὕτως <lb n="1799,001,,,,5,19,26"></lb> |
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4409 τὸ ΔΓ πρὸς τὸ ΓΖ· καί ἐστιν ἀναστρέψαντι].</s> <lb n="1799,001,,,,5,19,27"></lb> |
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4410 <s>Πόρισμα <lb n="1799,001,,,,5,19,28"></lb> |
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4411 Ἐκ δὴ τούτου φανερόν, ὅτι ἐὰν συγκείμενα μεγέθη ἀνάλο-<lb n="1799,001,,,,5,19,29"></lb> |
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4412 γον ᾖ, καὶ ἀναστρέψαντι ἀνάλογον ἔσται· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,5,19,30"></lb> |
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4413 Ἐὰν ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος, <lb n="1799,001,,,,5,20,1"></lb> |
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4414 σύνδυο λαμβανόμενα καὶ ἐν τῷ αὐτῷ λόγῳ, δι' ἴσου δὲ τὸ <lb n="1799,001,,,,5,20,2"></lb> |
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4415 πρῶτον τοῦ τρίτου μεῖζον ᾖ, καὶ τὸ τέταρτον τοῦ ἕκτου <lb n="1799,001,,,,5,20,3"></lb> |
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4416 μεῖζον ἔσται, κἂν ἴσον, ἴσον, κἂν ἔλαττον, ἔλαττον.</s> <lb n="1799,001,,,,5,20,4"></lb> |
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4417 <s>Ἔστω τρία μεγέθη τὰ Α, Β, Γ, καὶ ἄλλα αὐτοῖς ἴσα <lb n="1799,001,,,,5,20,5"></lb> |
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4418 τὸ πλῆθος τὰ Δ, Ε, Ζ, σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ <lb n="1799,001,,,,5,20,6"></lb> |
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4419 λόγῳ, ὡς μὲν τὸ Α πρὸς τὸ Β, οὕτως τὸ Δ πρὸς τὸ Ε, ὡς <lb n="1799,001,,,,5,20,7"></lb> |
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4420 δὲ τὸ Β πρὸς τὸ Γ, οὕτως τὸ Ε <lb n="1799,001,,,,5,20,8"></lb> |
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4421 πρὸς τὸ Ζ, δι' ἴσου δὲ μεῖζον ἔστω <lb n="1799,001,,,,5,20,9"></lb> |
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4422 τὸ Α τοῦ Γ· λέγω, ὅτι καὶ τὸ Δ τοῦ <lb n="1799,001,,,,5,20,10"></lb> |
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4423 Ζ μεῖζον ἔσται, κἂν ἴσον, ἴσον, κἂν <lb n="1799,001,,,,5,20,11"></lb> |
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4424 ἔλαττον, ἔλαττον.</s> <lb n="1799,001,,,,5,20,12"></lb> |
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4425 <s>Ἐπεὶ γὰρ μεῖζόν ἐστι τὸ Α τοῦ Γ, ἄλλο δέ τι τὸ Β, τὸ <lb n="1799,001,,,,5,20,13"></lb> |
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4426 δὲ μεῖζον πρὸς τὸ αὐτὸ μείζονα λόγον ἔχει ἤπερ τὸ <lb n="1799,001,,,,5,20,14"></lb> |
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4427 ἔλαττον, τὸ Α ἄρα πρὸς τὸ Β μείζονα λόγον ἔχει ἤπερ τὸ <lb n="1799,001,,,,5,20,15"></lb> |
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4428 Γ πρὸς τὸ Β.</s> <s>ἀλλ' ὡς μὲν τὸ Α πρὸς τὸ Β, [οὕτωσ] τὸ Δ <lb n="1799,001,,,,5,20,16"></lb> |
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4429 πρὸς τὸ Ε, ὡς δὲ τὸ Γ πρὸς τὸ Β, ἀνάπαλιν οὕτως τὸ Ζ <lb n="1799,001,,,,5,20,17"></lb> |
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4430 πρὸς τὸ Ε· καὶ τὸ Δ ἄρα πρὸς τὸ Ε μείζονα λόγον ἔχει <lb n="1799,001,,,,5,20,18"></lb> |
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4431 ἤπερ τὸ Ζ πρὸς τὸ Ε.</s> <s>τῶν δὲ πρὸς τὸ αὐτὸ λόγον ἐχόντων <pb></pb> <lb n="1799,001,,,,5,20,19"></lb> |
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4432 τὸ μείζονα λόγον ἔχον μεῖζόν ἐστιν.</s> <s>μεῖζον ἄρα τὸ Δ <lb n="1799,001,,,,5,20,20"></lb> |
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4433 τοῦ Ζ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι κἂν ἴσον ᾖ τὸ Α τῷ Γ, <lb n="1799,001,,,,5,20,21"></lb> |
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4434 ἴσον ἔσται καὶ τὸ Δ τῷ Ζ, κἂν ἔλαττον, ἔλαττον.</s> <lb n="1799,001,,,,5,20,22"></lb> |
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4435 <s>Ἐὰν ἄρα ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος, <lb n="1799,001,,,,5,20,23"></lb> |
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4436 σύνδυο λαμβανόμενα καὶ ἐν τῷ αὐτῷ λόγῳ, δι' ἴσου δὲ <lb n="1799,001,,,,5,20,24"></lb> |
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4437 τὸ πρῶτον τοῦ τρίτου μεῖζον ᾖ, καὶ τὸ τέταρτον τοῦ <lb n="1799,001,,,,5,20,25"></lb> |
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4438 ἕκτου μεῖζον ἔσται, κἂν ἴσον, ἴσον, κἂν ἔλαττον, ἔλαττον· <lb n="1799,001,,,,5,20,26"></lb> |
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4439 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,20,27"></lb> |
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4440 <s>Ἐὰν ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος <lb n="1799,001,,,,5,21,1"></lb> |
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4441 σύνδυο λαμβανόμενα καὶ ἐν τῷ αὐτῷ λόγῳ, ᾖ δὲ τετα-<lb n="1799,001,,,,5,21,2"></lb> |
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4442 ραγμένη αὐτῶν ἡ ἀναλογία, δι' ἴσου δὲ τὸ πρῶτον τοῦ <lb n="1799,001,,,,5,21,3"></lb> |
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4443 τρίτου μεῖζον ᾖ, καὶ τὸ τέταρτον τοῦ ἕκτου μεῖζον ἔσται, <lb n="1799,001,,,,5,21,4"></lb> |
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4444 κἂν ἴσον, ἴσον, κἂν ἔλαττον, ἔλαττον.</s> <lb n="1799,001,,,,5,21,5"></lb> |
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4445 <s>Ἔστω τρία μεγέθη τὰ Α, Β, Γ καὶ ἄλλα αὐτοῖς ἴσα τὸ <lb n="1799,001,,,,5,21,6"></lb> |
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4446 πλῆθος τὰ Δ, Ε, Ζ, σύνδυο λαμβανόμενα καὶ ἐν τῷ αὐτῷ <lb n="1799,001,,,,5,21,7"></lb> |
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4447 λόγῳ, ἔστω δὲ τεταραγμένη <lb n="1799,001,,,,5,21,8"></lb> |
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4448 αὐτῶν ἡ ἀναλογία, ὡς μὲν <lb n="1799,001,,,,5,21,9"></lb> |
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4449 τὸ Α πρὸς τὸ Β, οὕτως τὸ <lb n="1799,001,,,,5,21,10"></lb> |
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4450 Ε πρὸς τὸ Ζ, ὡς δὲ τὸ Β <lb n="1799,001,,,,5,21,11"></lb> |
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4451 πρὸς τὸ Γ, οὕτως τὸ Δ πρὸς <lb n="1799,001,,,,5,21,12"></lb> |
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4452 τὸ Ε, δι' ἴσου δὲ τὸ Α τοῦ Γ μεῖζον ἔστω· λέγω, ὅτι καὶ τὸ <lb n="1799,001,,,,5,21,13"></lb> |
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4453 Δ τοῦ Ζ μεῖζον ἔσται, κἂν ἴσον, ἴσον, κἂν ἔλαττον, ἔλαττον.</s> <lb n="1799,001,,,,5,21,14"></lb> |
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4454 <s>Ἐπεὶ γὰρ μεῖζόν ἐστι τὸ Α τοῦ Γ, ἄλλο δέ τι τὸ Β, <lb n="1799,001,,,,5,21,15"></lb> |
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4455 τὸ Α ἄρα πρὸς τὸ Β μείζονα λόγον ἔχει ἤπερ τὸ Γ πρὸς <pb></pb> <lb n="1799,001,,,,5,21,16"></lb> |
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4456 τὸ Β.</s> <s>ἀλλ' ὡς μὲν τὸ Α πρὸς τὸ Β, οὕτως τὸ Ε πρὸς τὸ Ζ, <lb n="1799,001,,,,5,21,17"></lb> |
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4457 ὡς δὲ τὸ Γ πρὸς τὸ Β, ἀνάπαλιν οὕτως τὸ Ε πρὸς τὸ Δ.</s> <s>καὶ <lb n="1799,001,,,,5,21,18"></lb> |
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4458 τὸ Ε ἄρα πρὸς τὸ Ζ μείζονα λόγον ἔχει ἤπερ τὸ Ε πρὸς <lb n="1799,001,,,,5,21,19"></lb> |
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4459 τὸ Δ.</s> <s>πρὸς ὃ δὲ τὸ αὐτὸ μείζονα λόγον ἔχει, ἐκεῖνο ἔλασσόν <lb n="1799,001,,,,5,21,20"></lb> |
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4460 ἐστιν· ἔλασσον ἄρα ἐστὶ τὸ Ζ τοῦ Δ· μεῖζον ἄρα ἐστὶ τὸ Δ <lb n="1799,001,,,,5,21,21"></lb> |
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4461 τοῦ Ζ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι κἂν ἴσον ᾖ τὸ Α τῷ Γ, <lb n="1799,001,,,,5,21,22"></lb> |
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4462 ἴσον ἔσται καὶ τὸ Δ τῷ Ζ, κἂν ἔλαττον, ἔλαττον.</s> <lb n="1799,001,,,,5,21,23"></lb> |
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4463 <s>Ἐὰν ἄρα ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος, <lb n="1799,001,,,,5,21,24"></lb> |
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4464 σύνδυο λαμβανόμενα καὶ ἐν τῷ αὐτῷ λόγῳ, ᾖ δὲ τεταραγ-<lb n="1799,001,,,,5,21,25"></lb> |
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4465 μένη αὐτῶν ἡ ἀναλογία, δι' ἴσου δὲ τὸ πρῶτον τοῦ τρίτου <lb n="1799,001,,,,5,21,26"></lb> |
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4466 μεῖζον ᾖ, καὶ τὸ τέταρτον τοῦ ἕκτου μεῖζον ἔσται, κἂν <lb n="1799,001,,,,5,21,27"></lb> |
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4467 ἴσον, ἴσον, κἂν ἔλαττον, ἔλαττον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,21,28"></lb> |
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4468 <s>Ἐὰν ᾖ ὁποσαοῦν μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ <lb n="1799,001,,,,5,22,1"></lb> |
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4469 πλῆθος, σύνδυο λαμβανόμενα καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ <lb n="1799,001,,,,5,22,2"></lb> |
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4470 δι' ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσται. <lb n="1799,001,,,,5,22,3"></lb> |
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4471 Ἔστω ὁποσαοῦν μεγέθη τὰ Α, Β, Γ καὶ ἄλλα αὐτοῖς <lb n="1799,001,,,,5,22,4"></lb> |
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4472 ἴσα τὸ πλῆθος τὰ Δ, Ε, Ζ, σύνδυο λαμβανόμενα ἐν τῷ <lb n="1799,001,,,,5,22,5"></lb> |
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4473 αὐτῷ λόγῳ, ὡς μὲν τὸ Α πρὸς τὸ Β, οὕτως τὸ Δ πρὸς <lb n="1799,001,,,,5,22,6"></lb> |
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4474 τὸ Ε, ὡς δὲ τὸ Β πρὸς τὸ Γ, οὕτως τὸ Ε πρὸς τὸ Ζ· λέγω, <lb n="1799,001,,,,5,22,7"></lb> |
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4475 ὅτι καὶ δι' ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσται.</s> <s><pb></pb> <lb n="1799,001,,,,5,22,8"></lb> |
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4476 Εἰλήφθω γὰρ τῶν μὲν Α, Δ ἰσάκις πολλαπλάσια τὰ <lb n="1799,001,,,,5,22,9"></lb> |
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4477 Η, Θ, τῶν δὲ Β, Ε ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια <lb n="1799,001,,,,5,22,10"></lb> |
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4478 τὰ Κ, Λ, καὶ ἔτι τῶν Γ, Ζ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλα-<lb n="1799,001,,,,5,22,11"></lb> |
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4479 πλάσια τὰ Μ, Ν.</s> <lb n="1799,001,,,,5,22,12"></lb> |
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4480 <s>Καὶ ἐπεί ἐστιν ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Δ πρὸς <lb n="1799,001,,,,5,22,13"></lb> |
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4481 τὸ Ε, καὶ εἴληπται τῶν μὲν Α, Δ ἰσάκις πολλαπλάσια τὰ <lb n="1799,001,,,,5,22,14"></lb> |
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4482 Η, Θ, τῶν δὲ Β, Ε ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια <lb n="1799,001,,,,5,22,15"></lb> |
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4483 τὰ Κ, Λ, ἔστιν ἄρα ὡς τὸ Η πρὸς τὸ Κ, οὕτως τὸ Θ πρὸς <lb n="1799,001,,,,5,22,16"></lb> |
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4484 τὸ Λ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὡς τὸ Κ πρὸς τὸ Μ, οὕτως τὸ Λ <lb n="1799,001,,,,5,22,17"></lb> |
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4485 πρὸς τὸ Ν.</s> <s>ἐπεὶ οὖν τρία μεγέθη ἐστὶ τὰ Η, Κ, Μ, καὶ <lb n="1799,001,,,,5,22,18"></lb> |
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4486 ἄλλα αὐτοῖς ἴσα τὸ πλῆθος τὰ Θ, Λ, Ν, σύνδυο λαμβανό-<lb n="1799,001,,,,5,22,19"></lb> |
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4487 μενα καὶ ἐν τῷ αὐτῷ λόγῳ, δι' ἴσου ἄρα, εἰ ὑπερέχει τὸ <lb n="1799,001,,,,5,22,20"></lb> |
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4488 Η τοῦ Μ, ὑπερέχει καὶ τὸ Θ τοῦ Ν, καὶ εἰ ἴσον, ἴσον, καὶ <lb n="1799,001,,,,5,22,21"></lb> |
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4489 εἰ ἔλαττον, ἔλαττον.</s> <s>καί ἐστι τὰ μὲν Η, Θ τῶν Α, Δ <lb n="1799,001,,,,5,22,22"></lb> |
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4490 ἰσάκις πολλαπλάσια, τὰ δὲ Μ, Ν τῶν Γ, Ζ ἄλλα, ἃ ἔτυχεν, <lb n="1799,001,,,,5,22,23"></lb> |
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4491 ἰσάκις πολλαπλάσια.</s> <s>ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Γ, οὕτως <lb n="1799,001,,,,5,22,24"></lb> |
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4492 τὸ Δ πρὸς τὸ Ζ.</s> <lb n="1799,001,,,,5,22,25"></lb> |
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4493 <s>Ἐὰν ἄρα ᾖ ὁποσαοῦν μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ <lb n="1799,001,,,,5,22,26"></lb> |
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4494 πλῆθος, σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ, καὶ δι' <lb n="1799,001,,,,5,22,27"></lb> |
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4495 ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,22,28"></lb> |
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4496 <s>Ἐὰν ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος <lb n="1799,001,,,,5,23,1"></lb> |
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4497 σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ, ᾖ δὲ τεταραγμένη <lb n="1799,001,,,,5,23,2"></lb> |
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4498 αὐτῶν ἡ ἀναλογία, καὶ δι' ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσται.</s> <s><pb></pb> <lb n="1799,001,,,,5,23,3"></lb> |
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4499 Ἔστω τρία μεγέθη τὰ Α, Β, Γ καὶ ἄλλα αὐτοῖς ἴσα <lb n="1799,001,,,,5,23,4"></lb> |
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4500 τὸ πλῆθος σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ τὰ Δ, <lb n="1799,001,,,,5,23,5"></lb> |
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4501 Ε, Ζ, ἔστω δὲ τεταραγμένη αὐτῶν ἡ ἀναλογία, ὡς μὲν τὸ <lb n="1799,001,,,,5,23,6"></lb> |
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4502 Α πρὸς τὸ Β, οὕτως τὸ Ε πρὸς τὸ Ζ, ὡς δὲ τὸ Β πρὸς <lb n="1799,001,,,,5,23,7"></lb> |
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4503 τὸ Γ, οὕτως τὸ Δ πρὸς τὸ Ε· λέγω, ὅτι ἐστὶν ὡς τὸ Α <lb n="1799,001,,,,5,23,8"></lb> |
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4504 πρὸς τὸ Γ, οὕτως τὸ Δ πρὸς τὸ Ζ.</s> <lb n="1799,001,,,,5,23,9"></lb> |
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4505 <s>Εἰλήφθω τῶν μὲν Α, Β, Δ ἰσάκις πολλαπλάσια τὰ Η, <lb n="1799,001,,,,5,23,10"></lb> |
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4506 Θ, Κ, τῶν δὲ Γ, Ε, Ζ ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια <lb n="1799,001,,,,5,23,11"></lb> |
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4507 τὰ Λ, Μ, Ν.</s> <lb n="1799,001,,,,5,23,12"></lb> |
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4508 <s>Καὶ ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσια τὰ Η, Θ τῶν Α, Β, <lb n="1799,001,,,,5,23,13"></lb> |
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4509 τὰ δὲ μέρη τοῖς ὡσαύτως πολλαπλασίοις τὸν αὐτὸν ἔχει <lb n="1799,001,,,,5,23,14"></lb> |
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4510 λόγον, ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Η πρὸς <lb n="1799,001,,,,5,23,15"></lb> |
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4511 τὸ Θ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὡς τὸ Ε πρὸς τὸ Ζ, οὕτως τὸ <lb n="1799,001,,,,5,23,16"></lb> |
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4512 Μ πρὸς τὸ Ν· καί ἐστιν ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ Ε <lb n="1799,001,,,,5,23,17"></lb> |
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4513 πρὸς τὸ Ζ· καὶ ὡς ἄρα τὸ Η πρὸς τὸ Θ, οὕτως τὸ Μ <lb n="1799,001,,,,5,23,18"></lb> |
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4514 πρὸς τὸ Ν.</s> <s>καὶ ἐπεί ἐστιν ὡς τὸ Β πρὸς τὸ Γ, οὕτως τὸ <lb n="1799,001,,,,5,23,19"></lb> |
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4515 Δ πρὸς τὸ Ε, καὶ ἐναλλὰξ ὡς τὸ Β πρὸς τὸ Δ, οὕτως <lb n="1799,001,,,,5,23,20"></lb> |
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4516 τὸ Γ πρὸς τὸ Ε.</s> <s>καὶ ἐπεὶ τὰ Θ, Κ τῶν Β, Δ ἰσάκις ἐστὶ <lb n="1799,001,,,,5,23,21"></lb> |
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4517 πολλαπλάσια, τὰ δὲ μέρη τοῖς ἰσάκις πολλαπλασίοις τὸν <pb></pb> <lb n="1799,001,,,,5,23,22"></lb> |
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4518 αὐτὸν ἔχει λόγον, ἔστιν ἄρα ὡς τὸ Β πρὸς τὸ Δ, οὕτως <lb n="1799,001,,,,5,23,23"></lb> |
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4519 τὸ Θ πρὸς τὸ Κ.</s> <s>ἀλλ' ὡς τὸ Β πρὸς τὸ Δ, οὕτως τὸ Γ <lb n="1799,001,,,,5,23,24"></lb> |
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4520 πρὸς τὸ Ε· καὶ ὡς ἄρα τὸ Θ πρὸς τὸ Κ, οὕτως τὸ Γ πρὸς <lb n="1799,001,,,,5,23,25"></lb> |
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4521 τὸ Ε.</s> <s>πάλιν, ἐπεὶ τὰ Λ, Μ τῶν Γ, Ε ἰσάκις ἐστι πολλα-<lb n="1799,001,,,,5,23,26"></lb> |
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4522 πλάσια, ἔστιν ἄρα ὡς τὸ Γ πρὸς τὸ Ε, οὕτως τὸ Λ πρὸς <lb n="1799,001,,,,5,23,27"></lb> |
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4523 τὸ Μ.</s> <s>ἀλλ' ὡς τὸ Γ πρὸς τὸ Ε, οὕτως τὸ Θ πρὸς τὸ Κ· <lb n="1799,001,,,,5,23,28"></lb> |
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4524 καὶ ὡς ἄρα τὸ Θ πρὸς τὸ Κ, οὕτως τὸ Λ πρὸς τὸ Μ, καὶ <lb n="1799,001,,,,5,23,29"></lb> |
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4525 ἐναλλὰξ ὡς τὸ Θ πρὸς τὸ Λ, τὸ Κ πρὸς τὸ Μ.</s> <s>ἐδείχθη δὲ <lb n="1799,001,,,,5,23,30"></lb> |
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4526 καὶ ὡς τὸ Η πρὸς τὸ Θ, οὕτως τὸ Μ πρὸς τὸ Ν.</s> <s>ἐπεὶ οὖν <lb n="1799,001,,,,5,23,31"></lb> |
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4527 τρία μεγέθη ἐστὶ τὰ Η, Θ, Λ, καὶ ἄλλα αὐτοῖς ἴσα τὸ <lb n="1799,001,,,,5,23,32"></lb> |
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4528 πλῆθος τὰ Κ, Μ, Ν σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ, <lb n="1799,001,,,,5,23,33"></lb> |
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4529 καί ἐστιν αὐτῶν τεταραγμένη ἡ ἀναλογία, δι' ἴσου ἄρα, εἰ <lb n="1799,001,,,,5,23,34"></lb> |
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4530 ὑπερέχει τὸ Η τοῦ Λ, ὑπερέχει καὶ τὸ Κ τοῦ Ν, καὶ εἰ <lb n="1799,001,,,,5,23,35"></lb> |
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4531 ἴσον, ἴσον, καὶ εἰ ἔλαττον, ἔλαττον.</s> <s>καί ἐστι τὰ μὲν Η, Κ <lb n="1799,001,,,,5,23,36"></lb> |
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4532 τῶν Α, Δ ἰσάκις πολλαπλάσια, τὰ δὲ Λ, Ν τῶν Γ, Ζ.</s> <lb n="1799,001,,,,5,23,37"></lb> |
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4533 <s>ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Γ, οὕτως τὸ Δ πρὸς τὸ Ζ.</s> <lb n="1799,001,,,,5,23,38"></lb> |
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4534 <s>Ἐὰν ἄρα ᾖ τρία μεγέθη καὶ ἄλλα αὐτοῖς ἴσα τὸ πλῆθος <lb n="1799,001,,,,5,23,39"></lb> |
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4535 σύνδυο λαμβανόμενα ἐν τῷ αὐτῷ λόγῳ, ᾖ δὲ τεταραγμένη <lb n="1799,001,,,,5,23,40"></lb> |
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4536 αὐτῶν ἡ ἀναλογία, καὶ δι' ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσται· <lb n="1799,001,,,,5,23,41"></lb> |
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4537 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,5,23,42"></lb> |
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4538 <s>Ἐὰν πρῶτον πρὸς δεύτερον τὸν αὐτὸν ἔχῃ λόγον καὶ <lb n="1799,001,,,,5,24,1"></lb> |
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4539 τρίτον πρὸς τέταρτον, ἔχῃ δὲ καὶ πέμπτον πρὸς δεύτερον <lb n="1799,001,,,,5,24,2"></lb> |
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4540 τὸν αὐτὸν λόγον καὶ ἕκτον πρὸς τέταρτον, καὶ συντεθὲν <pb></pb> <lb n="1799,001,,,,5,24,3"></lb> |
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4541 πρῶτον καὶ πέμπτον πρὸς δεύτερον τὸν αὐτὸν ἕξει λόγον <lb n="1799,001,,,,5,24,4"></lb> |
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4542 καὶ τρίτον καὶ ἕκτον πρὸς τέταρτον.</s> <lb n="1799,001,,,,5,24,5"></lb> |
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4543 <s>Πρῶτον γὰρ τὸ ΑΒ πρὸς δεύτερον τὸ Γ τὸν αὐτὸν <lb n="1799,001,,,,5,24,6"></lb> |
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4544 ἐχέτω λόγον καὶ τρίτον τὸ ΔΕ πρὸς τέταρτον τὸ Ζ, <lb n="1799,001,,,,5,24,7"></lb> |
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4545 ἐχέτω δὲ καὶ πέμπτον τὸ ΒΗ πρὸς δεύτερον τὸ Γ τὸν <lb n="1799,001,,,,5,24,8"></lb> |
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4546 αὐτὸν λόγον καὶ ἕκτον τὸ ΕΘ <lb n="1799,001,,,,5,24,9"></lb> |
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4547 πρὸς τέταρτον τὸ Ζ· λέγω, ὅτι <lb n="1799,001,,,,5,24,10"></lb> |
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4548 καὶ συντεθὲν πρῶτον καὶ πέμπ-<lb n="1799,001,,,,5,24,11"></lb> |
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4549 τον τὸ ΑΗ πρὸς δεύτερον τὸ Γ <lb n="1799,001,,,,5,24,12"></lb> |
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4550 τὸν αὐτὸν ἕξει λόγον, καὶ τρίτον <lb n="1799,001,,,,5,24,13"></lb> |
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4551 καὶ ἕκτον τὸ ΔΘ πρὸς τέταρτον <lb n="1799,001,,,,5,24,14"></lb> |
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4552 τὸ Ζ.</s> <lb n="1799,001,,,,5,24,15"></lb> |
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4553 <s>Ἐπεὶ γάρ ἐστιν ὡς τὸ ΒΗ πρὸς τὸ Γ, οὕτως τὸ ΕΘ <lb n="1799,001,,,,5,24,16"></lb> |
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4554 πρὸς τὸ Ζ, ἀνάπαλιν ἄρα ὡς τὸ Γ πρὸς τὸ ΒΗ, οὕτως τὸ <lb n="1799,001,,,,5,24,17"></lb> |
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4555 Ζ πρὸς τὸ ΕΘ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς τὸ ΑΒ πρὸς τὸ Γ, <lb n="1799,001,,,,5,24,18"></lb> |
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4556 οὕτως τὸ ΔΕ πρὸς τὸ Ζ, ὡς δὲ τὸ Γ πρὸς τὸ ΒΗ, οὕτως <lb n="1799,001,,,,5,24,19"></lb> |
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4557 τὸ Ζ πρὸς τὸ ΕΘ, δι' ἴσου ἄρα ἐστὶν ὡς τὸ ΑΒ πρὸς τὸ <lb n="1799,001,,,,5,24,20"></lb> |
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4558 ΒΗ, οὕτως τὸ ΔΕ πρὸς τὸ ΕΘ.</s> <s>καὶ ἐπεὶ διῃρημένα <lb n="1799,001,,,,5,24,21"></lb> |
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4559 μεγέθη ἀνάλογόν ἐστιν, καὶ συντεθέντα ἀνάλογον ἔσται· <lb n="1799,001,,,,5,24,22"></lb> |
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4560 ἔστιν ἄρα ὡς τὸ ΑΗ πρὸς τὸ ΗΒ, οὕτως τὸ ΔΘ πρὸς τὸ <lb n="1799,001,,,,5,24,23"></lb> |
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4561 ΘΕ.</s> <s>ἔστι δὲ καὶ ὡς τὸ ΒΗ πρὸς τὸ Γ, οὕτως τὸ ΕΘ πρὸς <lb n="1799,001,,,,5,24,24"></lb> |
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4562 τὸ Ζ· δι' ἴσου ἄρα ἐστὶν ὡς τὸ ΑΗ πρὸς τὸ Γ, οὕτως τὸ <lb n="1799,001,,,,5,24,25"></lb> |
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4563 ΔΘ πρὸς τὸ Ζ.</s> <lb n="1799,001,,,,5,24,26"></lb> |
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4564 <s>Ἐὰν ἄρα πρῶτον πρὸς δεύτερον τὸν αὐτὸν ἔχῃ λόγον <lb n="1799,001,,,,5,24,27"></lb> |
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4565 καὶ τρίτον πρὸς τέταρτον, ἔχῃ δὲ καὶ πέμπτον πρὸς δεύτε-<lb n="1799,001,,,,5,24,28"></lb> |
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4566 ρον τὸν αὐτὸν λόγον καὶ ἕκτον πρὸς τέταρτον, καὶ συντεθὲν <lb n="1799,001,,,,5,24,29"></lb> |
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4567 πρῶτον καὶ πέμπτον πρὸς δεύτερον τὸν αὐτὸν ἕξει λόγον <lb n="1799,001,,,,5,24,30"></lb> |
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4568 καὶ τρίτον καὶ ἕκτον πρὸς τέταρτον· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,5,24,31"></lb> |
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4569 Ἐὰν τέσσαρα μεγέθη ἀνάλογον ᾖ, τὸ μέγιστον [αὐτῶν] <lb n="1799,001,,,,5,25,1"></lb> |
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4570 καὶ τὸ ἐλάχιστον δύο τῶν λοιπῶν μείζονά ἐστιν.</s> <lb n="1799,001,,,,5,25,2"></lb> |
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4571 <s>Ἔστω τέσσαρα μεγέθη ἀνάλογον τὰ ΑΒ, ΓΔ, Ε, Ζ, <lb n="1799,001,,,,5,25,3"></lb> |
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4572 ὡς τὸ ΑΒ πρὸς τὸ ΓΔ, οὕτως τὸ Ε πρὸς τὸ Ζ, ἔστω δὲ <lb n="1799,001,,,,5,25,4"></lb> |
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4573 μέγιστον μὲν αὐτῶν τὸ ΑΒ, ἐλάχιστον <lb n="1799,001,,,,5,25,5"></lb> |
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4574 δὲ τὸ Ζ· λέγω, ὅτι τὰ ΑΒ, Ζ τῶν <lb n="1799,001,,,,5,25,6"></lb> |
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4575 ΓΔ, Ε μείζονά ἐστιν.</s> <lb n="1799,001,,,,5,25,7"></lb> |
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4576 <s>Κείσθω γὰρ τῷ μὲν Ε ἴσον τὸ <lb n="1799,001,,,,5,25,8"></lb> |
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4577 ΑΗ, τῷ δὲ Ζ ἴσον τὸ ΓΘ.</s> <lb n="1799,001,,,,5,25,9"></lb> |
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4578 <s>Ἐπεὶ [οὖν] ἐστιν ὡς τὸ ΑΒ πρὸς <lb n="1799,001,,,,5,25,10"></lb> |
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4579 τὸ ΓΔ, οὕτως τὸ Ε πρὸς τὸ Ζ, <lb n="1799,001,,,,5,25,11"></lb> |
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4580 ἴσον δὲ τὸ μὲν Ε τῷ ΑΗ, τὸ δὲ Ζ τῷ ΓΘ, ἔστιν <lb n="1799,001,,,,5,25,12"></lb> |
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4581 ἄρα ὡς τὸ ΑΒ πρὸς τὸ ΓΔ, οὕτως τὸ ΑΗ πρὸς τὸ <lb n="1799,001,,,,5,25,13"></lb> |
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4582 ΓΘ.</s> <s>καὶ ἐπεί ἐστιν ὡς ὅλον τὸ ΑΒ πρὸς ὅλον τὸ ΓΔ, <lb n="1799,001,,,,5,25,14"></lb> |
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4583 οὕτως ἀφαιρεθὲν τὸ ΑΗ πρὸς ἀφαιρεθὲν τὸ ΓΘ, καὶ λοιπὸν <lb n="1799,001,,,,5,25,15"></lb> |
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4584 ἄρα τὸ ΗΒ πρὸς λοιπὸν τὸ ΘΔ ἔσται ὡς ὅλον τὸ ΑΒ <lb n="1799,001,,,,5,25,16"></lb> |
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4585 πρὸς ὅλον τὸ ΓΔ.</s> <s>μεῖζον δὲ τὸ ΑΒ τοῦ ΓΔ· μεῖζον ἄρα <lb n="1799,001,,,,5,25,17"></lb> |
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4586 καὶ τὸ ΗΒ τοῦ ΘΔ.</s> <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ μὲν ΑΗ <lb n="1799,001,,,,5,25,18"></lb> |
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4587 τῷ Ε, τὸ δὲ ΓΘ τῷ Ζ, τὰ ἄρα ΑΗ, Ζ ἴσα ἐστὶ τοῖς ΓΘ, <lb n="1799,001,,,,5,25,19"></lb> |
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4588 Ε.</s> <s>Καὶ [ἐπεὶ] ἐὰν [ἀνίσοις ἴσα προστεθῇ, τὰ ὅλα ἄνισά <lb n="1799,001,,,,5,25,20"></lb> |
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4589 ἐστιν, ἐὰν ἄρα] τῶν ΗΒ, ΘΔ ἀνίσων ὄντων καὶ μείζονος <pb></pb> <lb n="1799,001,,,,5,25,21"></lb> |
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4590 τοῦ ΗΒ τῷ μὲν ΗΒ προστεθῇ τὰ ΑΗ, Ζ, τῷ δὲ ΘΔ <lb n="1799,001,,,,5,25,22"></lb> |
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4591 προστεθῇ τὰ ΓΘ, Ε, συνάγεται τὰ ΑΒ, Ζ μείζονα τῶν <lb n="1799,001,,,,5,25,23"></lb> |
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4592 ΓΔ, Ε.</s> <lb n="1799,001,,,,5,25,24"></lb> |
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4593 <s>Ἐὰν ἄρα τέσσαρα μεγέθη ἀνάλογον ᾖ, τὸ μέγιστον <lb n="1799,001,,,,5,25,25"></lb> |
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4594 αὐτῶν καὶ τὸ ἐλάχιστον δύο τῶν λοιπῶν μείζονά ἐστιν· <lb n="1799,001,,,,5,25,26"></lb> |
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4595 ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,5,25,27"></lb> |
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4596 </s></p> |
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4597 </chap> |
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4598 <chap><p type="main"><s>Ὅμοια σχήματα εὐθύγραμμά ἐστιν, ὅσα τάς τε <lb n="1799,001,,,,6;HOR,1,1"></lb> |
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4599 |
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4600 γωνίας ἴσας ἔχει κατὰ μίαν καὶ τὰς περὶ τὰς ἴσας γωνίας <lb n="1799,001,,,,6;HOR,1,2"></lb> |
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4601 πλευρὰς ἀνάλογον.</s> <lb n="1799,001,,,,6;HOR,1,3"></lb> |
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4602 <s>[Ἀντιπεπονθότα δὲ σχήματά ἐστιν, ὅταν ἐν ἑκα-<lb n="1799,001,,,,6;HOR,2*,1"></lb> |
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4603 τέρῳ τῶν σχημάτων ἡγούμενοί τε καὶ ἑπόμενοι λόγοι <lb n="1799,001,,,,6;HOR,2*,2"></lb> |
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4604 ὦσιν.] <lb n="1799,001,,,,6;HOR,2*,3"></lb> |
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4605 Ἄκρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, <lb n="1799,001,,,,6;HOR,3,1"></lb> |
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4606 ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον <lb n="1799,001,,,,6;HOR,3,2"></lb> |
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4607 πρὸς τὸ ἔλαττον.</s> <lb n="1799,001,,,,6;HOR,3,3"></lb> |
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4608 <s>Ὕψος ἐστὶ παντὸς σχήματος ἡ ἀπὸ τῆς κορυφῆς <lb n="1799,001,,,,6;HOR,4,1"></lb> |
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4609 ἐπὶ τὴν βάσιν κάθετος ἀγομένη.</s> <s><pb></pb> <lb n="1799,001,,,,6;HOR,4,2"></lb> |
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4610 [Λόγος ἐκ λόγων συγκεῖσθαι λέγεται, ὅταν αἱ τῶν <lb n="1799,001,,,,6;HOR,5*,1"></lb> |
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4611 λόγων πηλικότητες ἐφ' ἑαυτὰς πολλαπλασιασθεῖσαι <lb n="1799,001,,,,6;HOR,5*,2"></lb> |
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4612 ποιῶσί τινα.] <lb n="1799,001,,,,6;HOR,5*,3"></lb> |
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4613 </s></p> |
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4614 </chap> |
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4615 <chap><p type="main"><s>Τὰ τρίγωνα καὶ τὰ παραλληλόγραμμα, τὰ ὑπὸ τὸ <lb n="1799,001,,,,6,1,1"></lb> |
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4616 |
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4617 αὐτὸ ὕψος ὄντα πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις.</s> <lb n="1799,001,,,,6,1,2"></lb> |
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4618 <s>Ἔστω τρίγωνα μὲν τὰ <lb n="1799,001,,,,6,1,3"></lb> |
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4619 ΑΒΓ, ΑΓΔ, παραλληλό-<lb n="1799,001,,,,6,1,4"></lb> |
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4620 γραμμα δὲ τὰ ΕΓ, ΓΖ <lb n="1799,001,,,,6,1,5"></lb> |
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4621 ὑπὸ τὸ αὐτὸ ὕψος τὸ ΑΓ· <lb n="1799,001,,,,6,1,6"></lb> |
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4622 λέγω, ὅτι ἐστὶν ὡς ἡ ΒΓ <lb n="1799,001,,,,6,1,7"></lb> |
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4623 βάσις πρὸς τὴν ΓΔ βάσιν, <lb n="1799,001,,,,6,1,8"></lb> |
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4624 οὕτως τὸ ΑΒΓ τρίγωνον <lb n="1799,001,,,,6,1,9"></lb> |
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4625 πρὸς τὸ ΑΓΔ τρίγωνον, καὶ τὸ ΕΓ παραλληλόγραμμον <lb n="1799,001,,,,6,1,10"></lb> |
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4626 πρὸς τὸ ΓΖ παραλληλόγραμμον.</s> <lb n="1799,001,,,,6,1,11"></lb> |
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4627 <s>Ἐκβεβλήσθω γὰρ ἡ ΒΔ ἐφ' ἑκάτερα τὰ μέρη ἐπὶ <lb n="1799,001,,,,6,1,12"></lb> |
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4628 τὰ Θ, Λ σημεῖα, καὶ κείσθωσαν τῇ μὲν ΒΓ βάσει ἴσαι <lb n="1799,001,,,,6,1,13"></lb> |
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4629 [ὁσαιδηποτοῦν] αἱ ΒΗ, ΗΘ, τῇ δὲ ΓΔ βάσει ἴσαι <lb n="1799,001,,,,6,1,14"></lb> |
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4630 ὁσαιδηποτοῦν αἱ ΔΚ, ΚΛ, καὶ ἐπεζεύχθωσαν αἱ ΑΗ, <lb n="1799,001,,,,6,1,15"></lb> |
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4631 ΑΘ, ΑΚ, ΑΛ.</s> <lb n="1799,001,,,,6,1,16"></lb> |
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4632 <s>Καὶ ἐπεὶ ἴσαι εἰσὶν αἱ ΓΒ, ΒΗ, ΗΘ ἀλλήλαις, ἴσα <lb n="1799,001,,,,6,1,17"></lb> |
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4633 ἐστὶ καὶ τὰ ΑΘΗ, ΑΗΒ, ΑΒΓ τρίγωνα ἀλλήλοις.</s> <s><pb></pb> <lb n="1799,001,,,,6,1,18"></lb> |
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4634 ὁσαπλασίων ἄρα ἐστὶν ἡ ΘΓ βάσις τῆς ΒΓ βάσεως, <lb n="1799,001,,,,6,1,19"></lb> |
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4635 τοσαυταπλάσιόν ἐστι καὶ τὸ ΑΘΓ τρίγωνον τοῦ ΑΒΓ <lb n="1799,001,,,,6,1,20"></lb> |
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4636 τριγώνου.</s> <s>διὰ τὰ αὐτὰ δὴ ὁσαπλασίων ἐστὶν ἡ ΛΓ βάσις <lb n="1799,001,,,,6,1,21"></lb> |
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4637 τῆς ΓΔ βάσεως, τοσαυταπλάσιόν ἐστι καὶ τὸ ΑΛΓ <lb n="1799,001,,,,6,1,22"></lb> |
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4638 τρίγωνον τοῦ ΑΓΔ τριγώνου· καὶ εἰ ἴση ἐστὶν ἡ ΘΓ <lb n="1799,001,,,,6,1,23"></lb> |
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4639 βάσις τῇ ΓΛ βάσει, ἴσον ἐστὶ καὶ τὸ ΑΘΓ τρίγωνον τῷ <lb n="1799,001,,,,6,1,24"></lb> |
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4640 ΑΓΛ τριγώνῳ, καὶ εἰ ὑπερέχει ἡ ΘΓ βάσις τῆς ΓΛ <lb n="1799,001,,,,6,1,25"></lb> |
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4641 βάσεως, ὑπερέχει καὶ τὸ ΑΘΓ τρίγωνον τοῦ ΑΓΛ <lb n="1799,001,,,,6,1,26"></lb> |
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4642 τριγώνου, καὶ εἰ ἐλάσσων, ἔλασσον.</s> <s>τεσσάρων δὴ ὄντων <lb n="1799,001,,,,6,1,27"></lb> |
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4643 μεγεθῶν δύο μὲν βάσεων τῶν ΒΓ, ΓΔ, δύο δὲ τριγώνων <lb n="1799,001,,,,6,1,28"></lb> |
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4644 τῶν ΑΒΓ, ΑΓΔ εἴληπται ἰσάκις πολλαπλάσια τῆσ μὲν <lb n="1799,001,,,,6,1,29"></lb> |
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4645 ΒΓ βάσεως καὶ τοῦ ΑΒΓ τριγώνου ἥ τε ΘΓ βάσις καὶ τὸ <lb n="1799,001,,,,6,1,30"></lb> |
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4646 ΑΘΓ τρίγωνον, τῆς δὲ ΓΔ βάσεως καὶ τοῦ ΑΔΓ τρι-<lb n="1799,001,,,,6,1,31"></lb> |
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4647 γώνου ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια ἥ τε ΛΓ <lb n="1799,001,,,,6,1,32"></lb> |
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4648 βάσις καὶ τὸ ΑΛΓ τρίγωνον· καὶ δέδεικται, ὅτι, εἰ ὑπερ-<lb n="1799,001,,,,6,1,33"></lb> |
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4649 έχει ἡ ΘΓ βάσις τῆς ΓΛ βάσεως, ὑπερέχει καὶ τὸ ΑΘΓ <lb n="1799,001,,,,6,1,34"></lb> |
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4650 τρίγωνον τοῦ ΑΛΓ τριγώνου, καὶ εἰ ἴση, ἴσον, καὶ εἰ <lb n="1799,001,,,,6,1,35"></lb> |
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4651 ἐλάσσων, ἔλασσον· ἔστιν ἄρα ὡς ἡ ΒΓ βάσις πρὸς τὴν <lb n="1799,001,,,,6,1,36"></lb> |
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4652 ΓΔ βάσιν, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΑΓΔ <lb n="1799,001,,,,6,1,37"></lb> |
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4653 τρίγωνον.</s> <lb n="1799,001,,,,6,1,38"></lb> |
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4654 <s>Καὶ ἐπεὶ τοῦ μὲν ΑΒΓ τριγώνου διπλάσιόν ἐστι τὸ <lb n="1799,001,,,,6,1,39"></lb> |
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4655 ΕΓ παραλληλόγραμμον, τοῦ δὲ ΑΓΔ τριγώνου διπλάσιόν <lb n="1799,001,,,,6,1,40"></lb> |
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4656 ἐστι τὸ ΖΓ παραλληλόγραμμον, τὰ δὲ μέρη τοῖς ὡσαύτως <lb n="1799,001,,,,6,1,41"></lb> |
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4657 πολλαπλασίοις τὸν αὐτὸν ἔχει λόγον, ἔστιν ἄρα ὡς τὸ <pb></pb> <lb n="1799,001,,,,6,1,42"></lb> |
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4658 ΑΒΓ τρίγωνον πρὸς τὸ ΑΓΔ τρίγωνον, οὕτως τὸ ΕΓ <lb n="1799,001,,,,6,1,43"></lb> |
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4659 παραλληλόγραμμον πρὸς τὸ ΖΓ παραλληλόγραμμον.</s> <lb n="1799,001,,,,6,1,44"></lb> |
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4660 <s>ἐπεὶ οὖν ἐδείχθη, ὡς μὲν ἡ ΒΓ βάσις πρὸς τὴν ΓΔ, <lb n="1799,001,,,,6,1,45"></lb> |
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4661 οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΑΓΔ τρίγωνον, ὡς δὲ <lb n="1799,001,,,,6,1,46"></lb> |
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4662 τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΑΓΔ τρίγωνον, οὕτως τὸ ΕΓ <lb n="1799,001,,,,6,1,47"></lb> |
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4663 παραλληλόγραμμον πρὸς τὸ ΓΖ παραλληλόγραμμον, καὶ <lb n="1799,001,,,,6,1,48"></lb> |
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4664 ὡς ἄρα ἡ ΒΓ βάσις πρὸς τὴν ΓΔ βάσιν, οὕτως τὸ ΕΓ <lb n="1799,001,,,,6,1,49"></lb> |
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4665 παραλληλόγραμμον πρὸς τὸ ΖΓ παραλληλόγραμμον.</s> <lb n="1799,001,,,,6,1,50"></lb> |
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4666 <s>Τὰ ἄρα τρίγωνα καὶ τὰ παραλληλόγραμμα τὰ ὑπὸ τὸ <lb n="1799,001,,,,6,1,51"></lb> |
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4667 αὐτὸ ὕψος ὄντα πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις· ὅπερ <lb n="1799,001,,,,6,1,52"></lb> |
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4668 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,1,53"></lb> |
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4669 <s>Ἐὰν τριγώνου παρὰ μίαν τῶν πλευρῶν ἀχθῇ τις <lb n="1799,001,,,,6,2,1"></lb> |
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4670 εὐθεῖα, ἀνάλογον τεμεῖ τὰς τοῦ τριγώνου πλευράς· καὶ <lb n="1799,001,,,,6,2,2"></lb> |
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4671 ἐὰν αἱ τοῦ τριγώνου πλευραὶ ἀνάλογον τμηθῶσιν, ἡ ἐπὶ τὰς <lb n="1799,001,,,,6,2,3"></lb> |
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4672 τομὰς ἐπιζευγνυμένη εὐθεῖα παρὰ <lb n="1799,001,,,,6,2,4"></lb> |
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4673 τὴν λοιπὴν ἔσται τοῦ τριγώνου <lb n="1799,001,,,,6,2,5"></lb> |
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4674 πλευράν.</s> <lb n="1799,001,,,,6,2,6"></lb> |
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4675 <s>Τριγώνου γὰρ τοῦ ΑΒΓ <lb n="1799,001,,,,6,2,7"></lb> |
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4676 παράλληλος μιᾷ τῶν πλευρῶν <lb n="1799,001,,,,6,2,8"></lb> |
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4677 τῇ ΒΓ ἤχθω ἡ ΔΕ· λέγω, ὅτι <pb></pb> <lb n="1799,001,,,,6,2,9"></lb> |
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4678 ἐστὶν ὡς ἡ ΒΔ πρὸς τὴν ΔΑ, οὕτως ἡ ΓΕ πρὸς <lb n="1799,001,,,,6,2,10"></lb> |
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4679 τὴν ΕΑ.</s> <lb n="1799,001,,,,6,2,11"></lb> |
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4680 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΒΕ, ΓΔ.</s> <lb n="1799,001,,,,6,2,12"></lb> |
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4681 <s>Ἴσον ἄρα ἐστὶ ΒΔΕ τρίγωνον τῷ ΓΔΕ τριγώνῳ· ἐπὶ <lb n="1799,001,,,,6,2,13"></lb> |
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4682 γὰρ τῆς αὐτῆς βάσεώς ἐστι τῆς ΔΕ καὶ ἐν ταῖς αὐταῖς <lb n="1799,001,,,,6,2,14"></lb> |
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4683 παραλλήλοις ταῖς ΔΕ, ΒΓ· ἄλλο δέ τι τὸ ΑΔΕ τρίγωνον.</s> <lb n="1799,001,,,,6,2,15"></lb> |
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4684 <s>τὰ δὲ ἴσα πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον· ἔστιν ἄρα <lb n="1799,001,,,,6,2,16"></lb> |
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4685 ὡς τὸ ΒΔΕ τρίγωνον πρὸς τὸ ΑΔΕ [τρίγωνον], οὕτως <lb n="1799,001,,,,6,2,17"></lb> |
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4686 τὸ ΓΔΕ τρίγωνον πρὸς τὸ ΑΔΕ τρίγωνον.</s> <s>ἀλλ' ὡς μὲν τὸ <lb n="1799,001,,,,6,2,18"></lb> |
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4687 ΒΔΕ τρίγωνον πρὸς τὸ ΑΔΕ, οὕτως ἡ ΒΔ πρὸς τὴν <lb n="1799,001,,,,6,2,19"></lb> |
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4688 ΔΑ· ὑπὸ γὰρ τὸ αὐτὸ ὕψος ὄντα τὴν ἀπὸ τοῦ Ε ἐπὶ <lb n="1799,001,,,,6,2,20"></lb> |
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4689 τὴν ΑΒ κάθετον ἀγομένην πρὸς ἄλληλά εἰσιν ὡς αἱ <lb n="1799,001,,,,6,2,21"></lb> |
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4690 βάσεις.</s> <s>διὰ τὰ αὐτὰ δὴ ὡς τὸ ΓΔΕ τρίγωνον πρὸς τὸ <lb n="1799,001,,,,6,2,22"></lb> |
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4691 ΑΔΕ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΑ· καὶ ὡς ἄρα ἡ ΒΔ πρὸς <lb n="1799,001,,,,6,2,23"></lb> |
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4692 τὴν ΔΑ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΑ.</s> <lb n="1799,001,,,,6,2,24"></lb> |
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4693 <s>Ἀλλὰ δὴ αἱ τοῦ ΑΒΓ τριγώνου πλευραὶ αἱ ΑΒ, ΑΓ <lb n="1799,001,,,,6,2,25"></lb> |
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4694 ἀνάλογον τετμήσθωσαν, ὡς ἡ ΒΔ πρὸς τὴν ΔΑ, οὕτως ἡ <lb n="1799,001,,,,6,2,26"></lb> |
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4695 ΓΕ πρὸς τὴν ΕΑ, καὶ ἐπεζεύχθω ἡ ΔΕ· λέγω, ὅτι <lb n="1799,001,,,,6,2,27"></lb> |
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4696 παράλληλός ἐστιν ἡ ΔΕ τῇ ΒΓ.</s> <lb n="1799,001,,,,6,2,28"></lb> |
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4697 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεί ἐστιν ὡς ἡ <lb n="1799,001,,,,6,2,29"></lb> |
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4698 ΒΔ πρὸς τὴν ΔΑ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΑ, ἀλλ' ὡς <lb n="1799,001,,,,6,2,30"></lb> |
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4699 μὲν ἡ ΒΔ πρὸς τὴν ΔΑ, οὕτως τὸ ΒΔΕ τρίγωνον πρὸς τὸ <lb n="1799,001,,,,6,2,31"></lb> |
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4700 ΑΔΕ τρίγωνον, ὡς δὲ ἡ ΓΕ πρὸς τὴν ΕΑ, οὕτως τὸ <pb></pb> <lb n="1799,001,,,,6,2,32"></lb> |
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4701 ΓΔΕ τρίγωνον πρὸς τὸ ΑΔΕ τρίγωνον, καὶ ὡς ἄρα τὸ <lb n="1799,001,,,,6,2,33"></lb> |
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4702 ΒΔΕ τρίγωνον πρὸς τὸ ΑΔΕ τρίγωνον, οὕτως τὸ ΓΔΕ <lb n="1799,001,,,,6,2,34"></lb> |
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4703 τρίγωνον πρὸς τὸ ΑΔΕ τρίγωνον.</s> <s>ἑκάτερον ἄρα τῶν <lb n="1799,001,,,,6,2,35"></lb> |
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4704 ΒΔΕ, ΓΔΕ τριγώνων πρὸς τὸ ΑΔΕ τὸν αὐτὸν ἔχει <lb n="1799,001,,,,6,2,36"></lb> |
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4705 λόγον.</s> <s>ἴσον ἄρα ἐστὶ τὸ ΒΔΕ τρίγωνον τῷ ΓΔΕ τριγώνῳ· <lb n="1799,001,,,,6,2,37"></lb> |
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4706 καί εἰσιν ἐπὶ τῆς αὐτῆς βάσεως τῆς ΔΕ.</s> <s>τὰ δὲ ἴσα <lb n="1799,001,,,,6,2,38"></lb> |
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4707 τρίγωνα καὶ ἐπὶ τῆς αὐτῆς βάσεως ὄντα καὶ ἐν ταῖς αὐταῖς <lb n="1799,001,,,,6,2,39"></lb> |
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4708 παραλλήλοις ἐστίν.</s> <s>παράλληλος ἄρα ἐστὶν ἡ ΔΕ τῇ ΒΓ.</s> <lb n="1799,001,,,,6,2,40"></lb> |
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4709 <s>Ἐὰν ἄρα τριγώνου παρὰ μίαν τῶν πλευρῶν ἀχθῇ τις <lb n="1799,001,,,,6,2,41"></lb> |
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4710 εὐθεῖα, ἀνάλογον τεμεῖ τὰς τοῦ τριγώνου πλευράς· καὶ <lb n="1799,001,,,,6,2,42"></lb> |
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4711 ἐὰν αἱ τοῦ τριγώνου πλευραὶ ἀνάλογον τμηθῶσιν, ἡ ἐπὶ <lb n="1799,001,,,,6,2,43"></lb> |
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4712 τὰς τομὰς ἐπιζευγνυμένη εὐθεῖα παρὰ τὴν λοιπὴν ἔσται <lb n="1799,001,,,,6,2,44"></lb> |
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4713 τοῦ τριγώνου πλευράν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,2,45"></lb> |
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4714 <s>*̓εὰν τριγώνου ἡ γωνία δίχα τμηθῇ, ἡ δὲ τέμνουσα τὴν <lb n="1799,001,,,,6,3,1"></lb> |
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4715 γωνίαν εὐθεῖα τέμνῃ καὶ τὴν βάσιν, τὰ τῆς βάσεως <lb n="1799,001,,,,6,3,2"></lb> |
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4716 τμήματα τὸν αὐτὸν ἕξει λόγον ταῖς λοιπαῖς τοῦ τριγώνου <lb n="1799,001,,,,6,3,3"></lb> |
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4717 πλευραῖς· καὶ ἐὰν τὰ τῆς βάσεως τμήματα τὸν αὐτὸν ἔχῃ <lb n="1799,001,,,,6,3,4"></lb> |
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4718 λόγον ταῖς λοιπαῖς τοῦ τριγώνου πλευραῖς, ἡ ἀπὸ τῆς <lb n="1799,001,,,,6,3,5"></lb> |
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4719 κορυφῆς ἐπὶ τὴν τομὴν ἐπιζευγνυμένη εὐθεῖα δίχα τεμεῖ <lb n="1799,001,,,,6,3,6"></lb> |
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4720 τὴν τοῦ τριγώνου γωνίαν.</s> <s><pb></pb> <lb n="1799,001,,,,6,3,7"></lb> |
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4721 Ἔστω τρίγωνον τὸ ΑΒΓ, καὶ τετμήσθω ἡ ὑπὸ ΒΑΓ <lb n="1799,001,,,,6,3,8"></lb> |
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4722 γωνία δίχα ὑπὸ τῆς ΑΔ εὐθείας· λέγω, ὅτι ἐστὶν ὡς ἡ ΒΔ <lb n="1799,001,,,,6,3,9"></lb> |
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4723 πρὸς τὴν ΓΔ, οὕτως ἡ ΒΑ πρὸς τὴν ΑΓ.</s> <lb n="1799,001,,,,6,3,10"></lb> |
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4724 <s>Ἤχθω γὰρ διὰ τοῦ Γ τῇ ΔΑ παράλληλος ἡ ΓΕ καὶ <lb n="1799,001,,,,6,3,11"></lb> |
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4725 διαχθεῖσα ἡ ΒΑ συμπιπτέτω αὐτῇ κατὰ τὸ Ε.</s> <lb n="1799,001,,,,6,3,12"></lb> |
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4726 <s>Καὶ ἐπεὶ εἰς παραλλήλους <lb n="1799,001,,,,6,3,13"></lb> |
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4727 τὰς ΑΔ, ΕΓ εὐθεῖα ἐνέπεσεν <lb n="1799,001,,,,6,3,14"></lb> |
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4728 ἡ ΑΓ, ἡ ἄρα ὑπὸ ΑΓΕ γω-<lb n="1799,001,,,,6,3,15"></lb> |
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4729 νία ἴση ἐστὶ τῇ ὑπὸ ΓΑΔ.</s> <lb n="1799,001,,,,6,3,16"></lb> |
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4730 <s>ἀλλ' ἡ ὑπὸ ΓΑΔ τῇ ὑπὸ <lb n="1799,001,,,,6,3,17"></lb> |
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4731 ΒΑΔ ὑπόκειται ἴση· καὶ ἡ <lb n="1799,001,,,,6,3,18"></lb> |
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4732 ὑπὸ ΒΑΔ ἄρα τῇ ὑπὸ ΑΓΕ <lb n="1799,001,,,,6,3,19"></lb> |
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4733 ἐστιν ἴση.</s> <s>πάλιν, ἐπεὶ εἰς <lb n="1799,001,,,,6,3,20"></lb> |
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4734 παραλλήλους τὰς ΑΔ, ΕΓ εὐθεῖα ἐνέπεσεν ἡ ΒΑΕ, ἡ <lb n="1799,001,,,,6,3,21"></lb> |
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4735 ἐκτὸς γωνία ἡ ὑπὸ ΒΑΔ ἴση ἐστὶ τῇ ἐντὸς τῇ ὑπὸ ΑΕΓ.</s> <lb n="1799,001,,,,6,3,22"></lb> |
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4736 <s>ἐδείχθη δὲ καὶ ἡ ὑπὸ ΑΓΕ τῇ ὑπὸ ΒΑΔ ἴση· καὶ ἡ ὑπὸ <lb n="1799,001,,,,6,3,23"></lb> |
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4737 ΑΓΕ ἄρα γωνία τῇ ὑπὸ ΑΕΓ ἐστιν ἴση· ὥστε καὶ <lb n="1799,001,,,,6,3,24"></lb> |
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4738 πλευρὰ ἡ ΑΕ πλευρᾷ τῇ ΑΓ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ τριγώνου <lb n="1799,001,,,,6,3,25"></lb> |
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4739 τοῦ ΒΓΕ παρὰ μίαν τῶν πλευρῶν τὴν ΕΓ ἦκται ἡ ΑΔ, <lb n="1799,001,,,,6,3,26"></lb> |
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4740 ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΒΔ πρὸς τὴν ΔΓ, οὕτως ἡ ΒΑ <lb n="1799,001,,,,6,3,27"></lb> |
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4741 πρὸς τὴν ΑΕ.</s> <s>ἴση δὲ ἡ ΑΕ τῇ ΑΓ· ὡς ἄρα ἡ ΒΔ πρὸς <lb n="1799,001,,,,6,3,28"></lb> |
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4742 τὴν ΔΓ, οὕτως ἡ ΒΑ πρὸς τὴν ΑΓ.</s> <lb n="1799,001,,,,6,3,29"></lb> |
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4743 <s>Ἀλλὰ δὴ ἔστω ὡς ἡ ΒΔ πρὸς τὴν ΔΓ, οὕτως ἡ ΒΑ <lb n="1799,001,,,,6,3,30"></lb> |
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4744 πρὸς τὴν ΑΓ, καὶ ἐπεζεύχθω ἡ ΑΔ· λέγω, ὅτι δίχα <lb n="1799,001,,,,6,3,31"></lb> |
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4745 τέτμηται ἡ ὑπὸ ΒΑΓ γωνία ὑπὸ τῆς ΑΔ εὐθείας.</s> <s><pb></pb> <lb n="1799,001,,,,6,3,32"></lb> |
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4746 Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεί ἐστιν ὡς ἡ <lb n="1799,001,,,,6,3,33"></lb> |
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4747 ΒΔ πρὸς τὴν ΔΓ, οὕτως ἡ ΒΑ πρὸς τὴν ΑΓ, ἀλλὰ καὶ <lb n="1799,001,,,,6,3,34"></lb> |
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4748 ὡς ἡ ΒΔ πρὸς τὴν ΔΓ, οὕτως ἐστὶν ἡ ΒΑ πρὸς τὴν ΑΕ· <lb n="1799,001,,,,6,3,35"></lb> |
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4749 τριγώνου γὰρ τοῦ ΒΓΕ παρὰ μίαν τὴν ΕΓ ἦκται ἡ ΑΔ· <lb n="1799,001,,,,6,3,36"></lb> |
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4750 καὶ ὡς ἄρα ἡ ΒΑ πρὸς τὴν ΑΓ, οὕτως ἡ ΒΑ πρὸς τὴν <lb n="1799,001,,,,6,3,37"></lb> |
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4751 ΑΕ.</s> <s>ἴση ἄρα ἡ ΑΓ τῇ ΑΕ· ὥστε καὶ γωνία ἡ ὑπὸ ΑΕΓ <lb n="1799,001,,,,6,3,38"></lb> |
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4752 τῇ ὑπὸ ΑΓΕ ἐστιν ἴση.</s> <s>ἀλλ' ἡ μὲν ὑπὸ ΑΕΓ τῇ ἐκτὸς τῇ <lb n="1799,001,,,,6,3,39"></lb> |
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4753 ὑπὸ ΒΑΔ [ἐστιν] ἴση, ἡ δὲ ὑπὸ ΑΓΕ τῇ ἐναλλὰξ τῇ <lb n="1799,001,,,,6,3,40"></lb> |
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4754 ὑπὸ ΓΑΔ ἐστιν ἴση· καὶ ἡ ὑπὸ ΒΑΔ ἄρα τῇ ὑπὸ ΓΑΔ <lb n="1799,001,,,,6,3,41"></lb> |
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4755 ἐστιν ἴση.</s> <s>ἡ ἄρα ὑπὸ ΒΑΓ γωνία δίχα τέτμηται ὑπὸ τῆς <lb n="1799,001,,,,6,3,42"></lb> |
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4756 ΑΔ εὐθείας.</s> <lb n="1799,001,,,,6,3,43"></lb> |
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4757 <s>Ἐὰν ἄρα τριγώνου ἡ γωνία δίχα τμηθῇ, ἡ δὲ τέμνουσα <lb n="1799,001,,,,6,3,44"></lb> |
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4758 τὴν γωνίαν εὐθεῖα τέμνῃ καὶ τὴν βάσιν, τὰ τῆς βάσεως <lb n="1799,001,,,,6,3,45"></lb> |
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4759 τμήματα τὸν αὐτὸν ἕξει λόγον ταῖς λοιπαῖς τοῦ τριγώνου <lb n="1799,001,,,,6,3,46"></lb> |
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4760 πλευραῖς· καὶ ἐὰν τὰ τῆς βάσεως τμήματα τὸν αὐτὸν <lb n="1799,001,,,,6,3,47"></lb> |
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4761 ἔχῃ λόγον ταῖς λοιπαῖς τοῦ τριγώνου πλευραῖς, ἡ ἀπὸ τῆς <lb n="1799,001,,,,6,3,48"></lb> |
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4762 κορυφῆς ἐπὶ τὴν τομὴν ἐπιζευγνυμένη εὐθεῖα δίχα τέμνει <lb n="1799,001,,,,6,3,49"></lb> |
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4763 τὴν τοῦ τριγώνου γωνίαν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,3,50"></lb> |
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4764 <s>Τῶν ἰσογωνίων τριγώνων ἀνάλογόν εἰσιν αἱ πλευραὶ <lb n="1799,001,,,,6,4,1"></lb> |
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4765 αἱ περὶ τὰς ἴσας γωνίας καὶ ὁμόλογοι αἱ ὑπὸ τὰς ἴσας <lb n="1799,001,,,,6,4,2"></lb> |
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4766 γωνίας ὑποτείνουσαι.</s> <s><pb></pb> <lb n="1799,001,,,,6,4,3"></lb> |
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4767 Ἔστω ἰσογώνια τρίγωνα τὰ ΑΒΓ, ΔΓΕ ἴσην ἔχοντα <lb n="1799,001,,,,6,4,4"></lb> |
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4768 τὴν μὲν ὑπὸ ΑΒΓ γωνίαν τῇ ὑπὸ ΔΓΕ, τὴν δὲ ὑπὸ <lb n="1799,001,,,,6,4,5"></lb> |
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4769 ΒΑΓ τῇ ὑπὸ ΓΔΕ καὶ ἔτι τὴν ὑπὸ ΑΓΒ τῇ ὑπὸ ΓΕΔ· <lb n="1799,001,,,,6,4,6"></lb> |
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4770 λέγω, ὅτι τῶν ΑΒΓ, ΔΓΕ τριγώνων ἀνάλογόν εἰσιν αἱ <lb n="1799,001,,,,6,4,7"></lb> |
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4771 πλευραὶ αἱ περὶ τὰς ἴσας γωνίας <lb n="1799,001,,,,6,4,8"></lb> |
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4772 καὶ ὁμόλογοι αἱ ὑπὸ τὰς ἴσας <lb n="1799,001,,,,6,4,9"></lb> |
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4773 γωνίας ὑποτείνουσαι.</s> <lb n="1799,001,,,,6,4,10"></lb> |
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4774 <s>Κείσθω γὰρ ἐπ' εὐθείας ἡ <lb n="1799,001,,,,6,4,11"></lb> |
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4775 ΒΓ τῇ ΓΕ.</s> <s>καὶ ἐπεὶ αἱ ὑπὸ <lb n="1799,001,,,,6,4,12"></lb> |
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4776 ΑΒΓ, ΑΓΒ γωνίαι δύο ὀρθῶν <lb n="1799,001,,,,6,4,13"></lb> |
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4777 ἐλάττονές εἰσιν, ἴση δὲ ἡ ὑπὸ <lb n="1799,001,,,,6,4,14"></lb> |
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4778 ΑΓΒ τῇ ὑπὸ ΔΕΓ, αἱ ἄρα <lb n="1799,001,,,,6,4,15"></lb> |
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4779 ὑπὸ ΑΒΓ, ΔΕΓ δύο ὀρθῶν ἐλάττονές εἰσιν· αἱ ΒΑ, ΕΔ <lb n="1799,001,,,,6,4,16"></lb> |
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4780 ἄρα ἐκβαλλόμεναι συμπεσοῦνται.</s> <s>ἐκβεβλήσθωσαν καὶ συμ-<lb n="1799,001,,,,6,4,17"></lb> |
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4781 πιπτέτωσαν κατὰ τὸ Ζ.</s> <lb n="1799,001,,,,6,4,18"></lb> |
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4782 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ὑπὸ ΔΓΕ γωνία τῇ ὑπὸ ΑΒΓ, <lb n="1799,001,,,,6,4,19"></lb> |
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4783 παράλληλός ἐστιν ἡ ΒΖ τῇ ΓΔ.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν ἡ <lb n="1799,001,,,,6,4,20"></lb> |
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4784 ὑπὸ ΑΓΒ τῇ ὑπὸ ΔΕΓ, παράλληλός ἐστιν ἡ ΑΓ τῇ ΖΕ.</s> <lb n="1799,001,,,,6,4,21"></lb> |
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4785 <s>παραλληλόγραμμον ἄρα ἐστὶ τὸ ΖΑΓΔ· ἴση ἄρα ἡ μὲν <lb n="1799,001,,,,6,4,22"></lb> |
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4786 ΖΑ τῇ ΔΓ, ἡ δὲ ΑΓ τῇ ΖΔ.</s> <s>καὶ ἐπεὶ τριγώνου τοῦ <lb n="1799,001,,,,6,4,23"></lb> |
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4787 ΖΒΕ παρὰ μίαν τὴν ΖΕ ἦκται ἡ ΑΓ, ἔστιν ἄρα ὡς ἡ <lb n="1799,001,,,,6,4,24"></lb> |
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4788 ΒΑ πρὸς τὴν ΑΖ, οὕτως ἡ ΒΓ πρὸς τὴν ΓΕ.</s> <s>ἴση δὲ ἡ <lb n="1799,001,,,,6,4,25"></lb> |
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4789 ΑΖ τῇ ΓΔ· ὡς ἄρα ἡ ΒΑ πρὸς τὴν ΓΔ, οὕτως ἡ ΒΓ πρὸς <pb></pb> <lb n="1799,001,,,,6,4,26"></lb> |
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4790 τὴν ΓΕ, καὶ ἐναλλὰξ ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως ἡ <lb n="1799,001,,,,6,4,27"></lb> |
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4791 ΔΓ πρὸς τὴν ΓΕ.</s> <s>πάλιν, ἐπεὶ παράλληλός ἐστιν ἡ ΓΔ <lb n="1799,001,,,,6,4,28"></lb> |
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4792 τῇ ΒΖ, ἔστιν ἄρα ὡς ἡ ΒΓ πρὸς τὴν ΓΕ, οὕτως ἡ ΖΔ <lb n="1799,001,,,,6,4,29"></lb> |
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4793 πρὸς τὴν ΔΕ.</s> <s>ἴση δὲ ἡ ΖΔ τῇ ΑΓ· ὡς ἄρα ἡ ΒΓ πρὸς τὴν <lb n="1799,001,,,,6,4,30"></lb> |
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4794 ΓΕ, οὕτως ἡ ΑΓ πρὸς τὴν ΔΕ, καὶ ἐναλλὰξ ὡς ἡ ΒΓ <lb n="1799,001,,,,6,4,31"></lb> |
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4795 πρὸς τὴν ΓΑ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΔ.</s> <s>ἐπεὶ οὖν ἐδείχθη <lb n="1799,001,,,,6,4,32"></lb> |
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4796 ὡς μὲν ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως ἡ ΔΓ πρὸς τὴν ΓΕ, ὡς <lb n="1799,001,,,,6,4,33"></lb> |
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4797 δὲ ἡ ΒΓ πρὸς τὴν ΓΑ, οὕτως ἡ ΓΕ πρὸς τὴν ΕΔ, δι' <lb n="1799,001,,,,6,4,34"></lb> |
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4798 ἴσου ἄρα ὡς ἡ ΒΑ πρὸς τὴν ΑΓ, οὕτως ἡ ΓΔ πρὸς <lb n="1799,001,,,,6,4,35"></lb> |
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4799 τὴν ΔΕ.</s> <lb n="1799,001,,,,6,4,36"></lb> |
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4800 <s>Τῶν ἄρα ἰσογωνίων τριγώνων ἀνάλογόν εἰσιν αἱ <lb n="1799,001,,,,6,4,37"></lb> |
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4801 πλευραὶ αἱ περὶ τὰς ἴσας γωνίας καὶ ὁμόλογοι αἱ ὑπὸ τὰς <lb n="1799,001,,,,6,4,38"></lb> |
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4802 ἴσας γωνίας ὑποτείνουσαι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,4,39"></lb> |
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4803 <s>Ἐὰν δύο τρίγωνα τὰς πλευρὰς ἀνάλογον ἔχῃ, ἰσογώ-<lb n="1799,001,,,,6,5,1"></lb> |
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4804 νια ἔσται τὰ τρίγωνα καὶ ἴσας ἕξει τὰς γωνίας, ὑφ' <lb n="1799,001,,,,6,5,2"></lb> |
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4805 ἃς αἱ ὁμόλογοι πλευραὶ ὑποτεί-<lb n="1799,001,,,,6,5,3"></lb> |
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4806 νουσιν.</s> <lb n="1799,001,,,,6,5,4"></lb> |
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4807 <s>Ἔστω δύο τρίγωνα τὰ ΑΒΓ, <lb n="1799,001,,,,6,5,5"></lb> |
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4808 ΔΕΖ τὰς πλευρὰς ἀνάλογον <lb n="1799,001,,,,6,5,6"></lb> |
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4809 ἔχοντα, ὡς μὲν τὴν ΑΒ πρὸς <lb n="1799,001,,,,6,5,7"></lb> |
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4810 τὴν ΒΓ, οὕτως τὴν ΔΕ πρὸς <lb n="1799,001,,,,6,5,8"></lb> |
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4811 τὴν ΕΖ, ὡς δὲ τὴν ΒΓ πρὸς τὴν <pb></pb> <lb n="1799,001,,,,6,5,9"></lb> |
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4812 ΓΑ, οὕτως τὴν ΕΖ πρὸς τὴν ΖΔ, καὶ ἔτι ὡς τὴν ΒΑ <lb n="1799,001,,,,6,5,10"></lb> |
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4813 πρὸς τὴν ΑΓ, οὕτως τὴν ΕΔ πρὸς τὴν ΔΖ.</s> <s>λέγω, ὅτι <lb n="1799,001,,,,6,5,11"></lb> |
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4814 ἰσογώνιόν ἐστι τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ καὶ <lb n="1799,001,,,,6,5,12"></lb> |
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4815 ἴσας ἕξουσι τὰς γωνίας, ὑφ' ἃς αἱ ὁμόλογοι πλευραὶ ὑπο-<lb n="1799,001,,,,6,5,13"></lb> |
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4816 τείνουσιν, τὴν μὲν ὑπὸ ΑΒΓ τῇ ὑπὸ ΔΕΖ, τὴν δὲ ὑπὸ <lb n="1799,001,,,,6,5,14"></lb> |
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4817 ΒΓΑ τῇ ὑπὸ ΕΖΔ καὶ ἔτι τὴν ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ.</s> <lb n="1799,001,,,,6,5,15"></lb> |
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4818 <s>Συνεστάτω γὰρ πρὸς τῇ ΕΖ εὐθείᾳ καὶ τοῖς πρὸς αὐτῇ <lb n="1799,001,,,,6,5,16"></lb> |
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4819 σημείοις τοῖς Ε, Ζ τῇ μὲν ὑπὸ ΑΒΓ γωνίᾳ ἴση ἡ ὑπὸ <lb n="1799,001,,,,6,5,17"></lb> |
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4820 ΖΕΗ, τῇ δὲ ὑπὸ ΑΓΒ ἴση ἡ ὑπὸ ΕΖΗ· λοιπὴ ἄρα ἡ <lb n="1799,001,,,,6,5,18"></lb> |
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4821 πρὸς τῷ Α λοιπῇ τῇ πρὸς τῷ Η ἐστιν ἴση.</s> <lb n="1799,001,,,,6,5,19"></lb> |
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4822 <s>ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΕΗΖ <lb n="1799,001,,,,6,5,20"></lb> |
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4823 [τριγώνῳ].</s> <s>τῶν ἄρα ΑΒΓ, ΕΗΖ τριγώνων ἀνάλογόν <lb n="1799,001,,,,6,5,21"></lb> |
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4824 εἰσιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας καὶ ὁμόλογοι <lb n="1799,001,,,,6,5,22"></lb> |
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4825 αἱ ὑπὸ τὰς ἴσας γωνίας ὑποτείνουσαι· ἔστιν ἄρα ὡς ἡ ΑΒ <lb n="1799,001,,,,6,5,23"></lb> |
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4826 πρὸς τὴν ΒΓ, [οὕτωσ] ἡ ΗΕ πρὸς τὴν ΕΖ.</s> <s>ἀλλ' ὡς ἡ <lb n="1799,001,,,,6,5,24"></lb> |
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4827 ΑΒ πρὸς τὴν ΒΓ, οὕτως ὑπόκειται ἡ ΔΕ πρὸς τὴν ΕΖ· <lb n="1799,001,,,,6,5,25"></lb> |
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4828 ὡς ἄρα ἡ ΔΕ πρὸς τὴν ΕΖ, οὕτως ἡ ΗΕ πρὸς τὴν ΕΖ.</s> <lb n="1799,001,,,,6,5,26"></lb> |
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4829 <s>ἑκατέρα ἄρα τῶν ΔΕ, ΗΕ πρὸς τὴν ΕΖ τὸν αὐτὸν ἔχει <lb n="1799,001,,,,6,5,27"></lb> |
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4830 λόγον· ἴση ἄρα ἐστὶν ἡ ΔΕ τῇ ΗΕ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ <lb n="1799,001,,,,6,5,28"></lb> |
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4831 ΔΖ τῇ ΗΖ ἐστιν ἴση.</s> <s>ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΕ τῇ ΕΗ, <lb n="1799,001,,,,6,5,29"></lb> |
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4832 κοινὴ δὲ ἡ ΕΖ, δύο δὴ αἱ ΔΕ, ΕΖ δυσὶ ταῖς ΗΕ, ΕΖ <lb n="1799,001,,,,6,5,30"></lb> |
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4833 ἴσαι εἰσίν· καὶ βάσις ἡ ΔΖ βάσει τῇ ΖΗ [ἐστιν] ἴση· <pb></pb> <lb n="1799,001,,,,6,5,31"></lb> |
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4834 γωνία ἄρα ἡ ὑπὸ ΔΕΖ γωνίᾳ τῇ ὑπὸ ΗΕΖ ἐστιν ἴση, <lb n="1799,001,,,,6,5,32"></lb> |
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4835 καὶ τὸ ΔΕΖ τρίγωνον τῷ ΗΕΖ τριγώνῳ ἴσον, καὶ αἱ <lb n="1799,001,,,,6,5,33"></lb> |
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4836 λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι, ὑφ' ἃς αἱ ἴσαι <lb n="1799,001,,,,6,5,34"></lb> |
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4837 πλευραὶ ὑποτείνουσιν.</s> <s>ἴση ἄρα ἐστὶ καὶ ἡ μὲν ὑπὸ ΔΖΕ <lb n="1799,001,,,,6,5,35"></lb> |
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4838 γωνία τῇ ὑπὸ ΗΖΕ, ἡ δὲ ὑπὸ ΕΔΖ τῇ ὑπὸ ΕΗΖ.</s> <s>καὶ <lb n="1799,001,,,,6,5,36"></lb> |
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4839 ἐπεὶ ἡ μὲν ὑπὸ ΖΕΔ τῇ ὑπὸ ΗΕΖ ἐστιν ἴση, ἀλλ' ἡ ὑπὸ <lb n="1799,001,,,,6,5,37"></lb> |
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4840 ΗΕΖ τῇ ὑπὸ ΑΒΓ, καὶ ἡ ὑπὸ ΑΒΓ ἄρα γωνία τῇ ὑπὸ <lb n="1799,001,,,,6,5,38"></lb> |
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4841 ΔΕΖ ἐστιν ἴση.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ ΑΓΒ τῇ ὑπὸ <lb n="1799,001,,,,6,5,39"></lb> |
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4842 ΔΖΕ ἐστιν ἴση, καὶ ἔτι ἡ πρὸς τῷ Α τῇ πρὸς τῷ Δ· ἰσο-<lb n="1799,001,,,,6,5,40"></lb> |
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4843 γώνιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ.</s> <lb n="1799,001,,,,6,5,41"></lb> |
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4844 <s>Ἐὰν ἄρα δύο τρίγωνα τὰς πλευρὰς ἀνάλογον ἔχῃ, <lb n="1799,001,,,,6,5,42"></lb> |
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4845 ἰσογώνια ἔσται τὰ τρίγωνα καὶ ἴσας ἕξει τὰς γωνίας, ὑφ' <lb n="1799,001,,,,6,5,43"></lb> |
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4846 ἃς αἱ ὁμόλογοι πλευραὶ ὑποτείνουσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,5,44"></lb> |
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4847 <s>Ἐὰν δύο τρίγωνα μίαν γωνίαν μιᾷ γωνίᾳ ἴσην ἔχῃ, περὶ <lb n="1799,001,,,,6,6,1"></lb> |
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4848 δὲ τὰς ἴσας γωνίας τὰς πλευρὰς ἀνάλογον, ἰσογώνια ἔσται <lb n="1799,001,,,,6,6,2"></lb> |
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4849 τὰ τρίγωνα καὶ ἴσας ἕξει τὰς γω-<lb n="1799,001,,,,6,6,3"></lb> |
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4850 νίας, ὑφ' ἃς αἱ ὁμόλογοι πλευραὶ <lb n="1799,001,,,,6,6,4"></lb> |
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4851 ὑποτείνουσιν.</s> <lb n="1799,001,,,,6,6,5"></lb> |
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4852 <s>Ἔστω δύο τρίγωνα τὰ ΑΒΓ, <lb n="1799,001,,,,6,6,6"></lb> |
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4853 ΔΕΖ μίαν γωνίαν τὴν ὑπὸ ΒΑΓ <lb n="1799,001,,,,6,6,7"></lb> |
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4854 μιᾷ γωνίᾳ τῇ ὑπὸ ΕΔΖ ἴσην <lb n="1799,001,,,,6,6,8"></lb> |
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4855 ἔχοντα, περὶ δὲ τὰς ἴσας γωνίας τὰς <lb n="1799,001,,,,6,6,9"></lb> |
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4856 πλευρὰς ἀνάλογον, ὡς τὴν ΒΑ πρὸς <lb n="1799,001,,,,6,6,10"></lb> |
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4857 τὴν ΑΓ, οὕτως τὴν ΕΔ πρὸς τὴν ΔΖ· λέγω, ὅτι ἰσογώ-<lb n="1799,001,,,,6,6,11"></lb> |
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4858 νιόν ἐστι τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ καὶ ἴσην <pb></pb> <lb n="1799,001,,,,6,6,12"></lb> |
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4859 ἕξει τὴν ὑπὸ ΑΒΓ γωνίαν τῇ ὑπὸ ΔΕΖ, τὴν δὲ ὑπὸ <lb n="1799,001,,,,6,6,13"></lb> |
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4860 ΑΓΒ τῇ ὑπὸ ΔΖΕ.</s> <lb n="1799,001,,,,6,6,14"></lb> |
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4861 <s>Συνεστάτω γὰρ πρὸς τῇ ΔΖ εὐθείᾳ καὶ τοῖς πρὸς <lb n="1799,001,,,,6,6,15"></lb> |
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4862 αὐτῇ σημείοις τοῖς Δ, Ζ ὁποτέρᾳ μὲν τῶν ὑπὸ ΒΑΓ, <lb n="1799,001,,,,6,6,16"></lb> |
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4863 ΕΔΖ ἴση ἡ ὑπὸ ΖΔΗ, τῇ δὲ ὑπὸ ΑΓΒ ἴση ἡ ὑπὸ ΔΖΗ· <lb n="1799,001,,,,6,6,17"></lb> |
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4864 λοιπὴ ἄρα ἡ πρὸς τῷ Β γωνία λοιπῇ τῇ πρὸς τῷ Η ἴση <lb n="1799,001,,,,6,6,18"></lb> |
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4865 ἐστίν.</s> <lb n="1799,001,,,,6,6,19"></lb> |
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4866 <s>Ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΗΖ <lb n="1799,001,,,,6,6,20"></lb> |
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4867 τριγώνῳ.</s> <s>ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΒΑ πρὸς τὴν ΑΓ, <lb n="1799,001,,,,6,6,21"></lb> |
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4868 οὕτως ἡ ΗΔ πρὸς τὴν ΔΖ.</s> <s>ὑπόκειται δὲ καὶ ὡς ἡ ΒΑ <lb n="1799,001,,,,6,6,22"></lb> |
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4869 πρὸς τὴν ΑΓ, οὕτως ἡ ΕΔ πρὸς τὴν ΔΖ· καὶ ὡς ἄρα ἡ <lb n="1799,001,,,,6,6,23"></lb> |
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4870 ΕΔ πρὸς τὴν ΔΖ, οὕτως ἡ ΗΔ πρὸς τὴν ΔΖ.</s> <s>ἴση ἄρα ἡ <lb n="1799,001,,,,6,6,24"></lb> |
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4871 ΕΔ τῇ ΔΗ· καὶ κοινὴ ἡ ΔΖ· δύο δὴ αἱ ΕΔ, ΔΖ δυσὶ ταῖς <lb n="1799,001,,,,6,6,25"></lb> |
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4872 ΗΔ, ΔΖ ἴσαι εἰσίν· καὶ γωνία ἡ ὑπὸ ΕΔΖ γωνίᾳ τῇ <lb n="1799,001,,,,6,6,26"></lb> |
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4873 ὑπὸ ΗΔΖ [ἐστιν] ἴση· βάσις ἄρα ἡ ΕΖ βάσει τῇ ΗΖ <lb n="1799,001,,,,6,6,27"></lb> |
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4874 ἐστιν ἴση, καὶ τὸ ΔΕΖ τρίγωνον τῷ ΗΔΖ τριγώνῳ ἴσον <lb n="1799,001,,,,6,6,28"></lb> |
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4875 ἐστίν, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι <lb n="1799,001,,,,6,6,29"></lb> |
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4876 ἔσονται, ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑποτείνουσιν.</s> <s>ἴση ἄρα <lb n="1799,001,,,,6,6,30"></lb> |
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4877 ἐστὶν ἡ μὲν ὑπὸ ΔΖΗ τῇ ὑπὸ ΔΖΕ, ἡ δὲ ὑπὸ ΔΗΖ τῇ <lb n="1799,001,,,,6,6,31"></lb> |
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4878 ὑπὸ ΔΕΖ.</s> <s>ἀλλ' ἡ ὑπὸ ΔΖΗ τῇ ὑπὸ ΑΓΒ ἐστιν ἴση· <lb n="1799,001,,,,6,6,32"></lb> |
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4879 καὶ ἡ ὑπὸ ΑΓΒ ἄρα τῇ ὑπὸ ΔΖΕ ἐστιν ἴση.</s> <s>ὑπόκειται <pb></pb> <lb n="1799,001,,,,6,6,33"></lb> |
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4880 δὲ καὶ ἡ ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ ἴση· καὶ λοιπὴ ἄρα ἡ <lb n="1799,001,,,,6,6,34"></lb> |
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4881 πρὸς τῷ Β λοιπῇ τῇ πρὸς τῷ Ε ἴση ἐστίν· ἰσογώνιον ἄρα <lb n="1799,001,,,,6,6,35"></lb> |
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4882 ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ.</s> <lb n="1799,001,,,,6,6,36"></lb> |
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4883 <s>Ἐὰν ἄρα δύο τρίγωνα μίαν γωνίαν μιᾷ γωνίᾳ ἴσην ἔχῃ, <lb n="1799,001,,,,6,6,37"></lb> |
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4884 περὶ δὲ τὰς ἴσας γωνίας τὰς πλευρὰς ἀνάλογον, ἰσογώνια <lb n="1799,001,,,,6,6,38"></lb> |
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4885 ἔσται τὰ τρίγωνα καὶ ἴσας ἕξει τὰς γωνίας, ὑφ' ἃς αἱ <lb n="1799,001,,,,6,6,39"></lb> |
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4886 ὁμόλογοι πλευραὶ ὑποτείνουσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,6,40"></lb> |
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4887 <s>Ἐὰν δύο τρίγωνα μίαν γωνίαν μιᾷ γωνίᾳ ἴσην ἔχῃ, περὶ <lb n="1799,001,,,,6,7,1"></lb> |
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4888 δὲ ἄλλας γωνίας τὰς πλευρὰς ἀνάλογον, τῶν δὲ λοιπῶν <lb n="1799,001,,,,6,7,2"></lb> |
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4889 ἑκατέραν ἅμα ἤτοι ἐλάσσονα ἢ μὴ ἐλάσσονα ὀρθῆς, <lb n="1799,001,,,,6,7,3"></lb> |
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4890 ἰσογώνια ἔσται τὰ τρίγωνα καὶ ἴσας ἕξει τὰς γωνίας, περὶ <lb n="1799,001,,,,6,7,4"></lb> |
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4891 ἃς ἀνάλογόν εἰσιν αἱ πλευραί.</s> <lb n="1799,001,,,,6,7,5"></lb> |
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4892 <s>Ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ μίαν γωνίαν μιᾷ <lb n="1799,001,,,,6,7,6"></lb> |
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4893 γωνίᾳ ἴσην ἔχοντα τὴν ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ, περὶ δὲ <lb n="1799,001,,,,6,7,7"></lb> |
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4894 ἄλλας γωνίας τὰς ὑπὸ ΑΒΓ, ΔΕΖ τὰς πλευρὰς ἀνάλογον, <lb n="1799,001,,,,6,7,8"></lb> |
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4895 ὡς τὴν ΑΒ πρὸς τὴν *βΓ, οὕτως τὴν ΔΕ πρὸς τὴν ΕΖ, <lb n="1799,001,,,,6,7,9"></lb> |
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4896 τῶν δὲ λοιπῶν τῶν πρὸς τοῖς Γ, Ζ πρότερον ἑκατέραν <lb n="1799,001,,,,6,7,10"></lb> |
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4897 ἅμα ἐλάσσονα ὀρθῆς· λέγω, ὅτι ἰσο-<lb n="1799,001,,,,6,7,11"></lb> |
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4898 γώνιόν ἐστι τὸ ΑΒΓ τρίγωνον τῷ <lb n="1799,001,,,,6,7,12"></lb> |
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4899 ΔΕΖ τριγώνῳ, καὶ ἴση ἔσται ἡ ὑπὸ <lb n="1799,001,,,,6,7,13"></lb> |
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4900 ΑΒΓ γωνία τῇ ὑπὸ ΔΕΖ, καὶ λοιπὴ <lb n="1799,001,,,,6,7,14"></lb> |
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4901 δηλονότι ἡ πρὸς τῷ Γ λοιπῇ τῇ <lb n="1799,001,,,,6,7,15"></lb> |
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4902 πρὸς τῷ Ζ ἴση.</s> <lb n="1799,001,,,,6,7,16"></lb> |
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4903 <s>Εἰ γὰρ ἄνισός ἐστιν ἡ ὑπὸ ΑΒΓ <lb n="1799,001,,,,6,7,17"></lb> |
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4904 γωνία τῇ ὑπὸ ΔΕΖ, μία αὐτῶν <pb></pb> <lb n="1799,001,,,,6,7,18"></lb> |
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4905 μείζων ἐστίν.</s> <s>ἔστω μείζων ἡ ὑπὸ ΑΒΓ.</s> <s>καὶ συνεστάτω <lb n="1799,001,,,,6,7,19"></lb> |
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4906 πρὸς τῇ ΑΒ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Β τῇ <lb n="1799,001,,,,6,7,20"></lb> |
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4907 ὑπὸ ΔΕΖ γωνίᾳ ἴση ἡ ὑπὸ ΑΒΗ.</s> <lb n="1799,001,,,,6,7,21"></lb> |
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4908 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ μὲν Α γωνία τῇ Δ, ἡ δὲ ὑπὸ ΑΒΗ <lb n="1799,001,,,,6,7,22"></lb> |
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4909 τῇ ὑπὸ ΔΕΖ, λοιπὴ ἄρα ἡ ὑπὸ ΑΗΒ λοιπῇ τῇ ὑπὸ <lb n="1799,001,,,,6,7,23"></lb> |
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4910 ΔΖΕ ἐστιν ἴση.</s> <s>ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΗ τρίγωνον <lb n="1799,001,,,,6,7,24"></lb> |
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4911 τῷ ΔΕΖ τριγώνῳ.</s> <s>ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΒΗ, <lb n="1799,001,,,,6,7,25"></lb> |
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4912 οὕτως ἡ ΔΕ πρὸς τὴν ΕΖ.</s> <s>ὡς δὲ ἡ ΔΕ πρὸς τὴν ΕΖ, <lb n="1799,001,,,,6,7,26"></lb> |
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4913 [οὕτωσ] ὑπόκειται ἡ ΑΒ πρὸς τὴν ΒΓ· ἡ ΑΒ ἄρα πρὸς <lb n="1799,001,,,,6,7,27"></lb> |
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4914 ἑκατέραν τῶν ΒΓ, ΒΗ τὸν αὐτὸν ἔχει λόγον· ἴση ἄρα ἡ <lb n="1799,001,,,,6,7,28"></lb> |
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4915 ΒΓ τῇ ΒΗ.</s> <s>ὥστε καὶ γωνία ἡ πρὸς τῷ Γ γωνίᾳ τῇ ὑπὸ <lb n="1799,001,,,,6,7,29"></lb> |
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4916 ΒΗΓ ἐστιν ἴση.</s> <s>ἐλάττων δὲ ὀρθῆς ὑπόκειται ἡ πρὸς τῷ <lb n="1799,001,,,,6,7,30"></lb> |
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4917 Γ· ἐλάττων ἄρα ἐστὶν ὀρθῆς καὶ ἡ ὑπὸ ΒΗΓ· ὥστε <lb n="1799,001,,,,6,7,31"></lb> |
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4918 ἡ ἐφεξῆς αὐτῇ γωνία ἡ ὑπὸ ΑΗΒ μείζων ἐστὶν ὀρθῆς.</s> <lb n="1799,001,,,,6,7,32"></lb> |
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4919 <s>καὶ ἐδείχθη ἴση οὖσα τῇ πρὸς τῷ Ζ· καὶ ἡ πρὸς τῷ Ζ ἄρα <lb n="1799,001,,,,6,7,33"></lb> |
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4920 μείζων ἐστὶν ὀρθῆς.</s> <s>ὑπόκειται δὲ ἐλάσσων ὀρθῆς· ὅπερ <lb n="1799,001,,,,6,7,34"></lb> |
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4921 ἐστὶν ἄτοπον.</s> <s>οὐκ ἄρα ἄνισός ἐστιν ἡ ὑπὸ ΑΒΓ γωνία τῇ <lb n="1799,001,,,,6,7,35"></lb> |
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4922 ὑπὸ ΔΕΖ· ἴση ἄρα.</s> <s>ἔστι δὲ καὶ ἡ πρὸς τῷ Α ἴση τῇ πρὸς <lb n="1799,001,,,,6,7,36"></lb> |
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4923 τῷ Δ· καὶ λοιπὴ ἄρα ἡ πρὸς τῷ Γ λοιπῇ τῇ πρὸς τῷ Ζ <lb n="1799,001,,,,6,7,37"></lb> |
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4924 ἴση ἐστίν.</s> <s>ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ <lb n="1799,001,,,,6,7,38"></lb> |
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4925 ΔΕΖ τριγώνῳ.</s> <lb n="1799,001,,,,6,7,39"></lb> |
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4926 <s>Ἀλλὰ δὴ πάλιν ὑποκείσθω ἑκατέρα τῶν πρὸς τοῖς Γ, <pb></pb> <lb n="1799,001,,,,6,7,40"></lb> |
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4927 Ζ μὴ ἐλάσσων ὀρθῆς· λέγω πάλιν, ὅτι καὶ οὕτως ἐστὶν <lb n="1799,001,,,,6,7,41"></lb> |
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4928 ἰσογώνιον τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ.</s> <lb n="1799,001,,,,6,7,42"></lb> |
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4929 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων <lb n="1799,001,,,,6,7,43"></lb> |
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4930 ὁμοίως δείξομεν, ὅτι ἴση ἐστὶν ἡ ΒΓ <lb n="1799,001,,,,6,7,44"></lb> |
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4931 τῇ ΒΗ· ὥστε καὶ γωνία ἡ πρὸς τῷ <lb n="1799,001,,,,6,7,45"></lb> |
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4932 Γ τῇ ὑπὸ ΒΗΓ ἴση ἐστίν.</s> <s>οὐκ ἐλάττων <lb n="1799,001,,,,6,7,46"></lb> |
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4933 δὲ ὀρθῆς ἡ πρὸς τῷ Γ· οὐκ ἐλάττων <lb n="1799,001,,,,6,7,47"></lb> |
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4934 ἄρα ὀρθῆς οὐδὲ ἡ ὑπὸ ΒΗΓ.</s> <s>τριγώνου <lb n="1799,001,,,,6,7,48"></lb> |
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4935 δὴ τοῦ ΒΗΓ αἱ δύο γωνίαι δύο ὀρθῶν <lb n="1799,001,,,,6,7,49"></lb> |
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4936 οὔκ εἰσιν ἐλάττονες· ὅπερ ἐστὶν ἀδύνα-<lb n="1799,001,,,,6,7,50"></lb> |
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4937 τον.</s> <s>οὐκ ἄρα πάλιν ἄνισός ἐστιν ἡ <lb n="1799,001,,,,6,7,51"></lb> |
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4938 ὑπὸ ΑΒΓ γωνία τῇ ὑπὸ ΔΕΖ· ἴση <lb n="1799,001,,,,6,7,52"></lb> |
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4939 ἄρα.</s> <s>ἔστι δὲ καὶ ἡ πρὸς τῷ Α τῇ πρὸς τῷ Δ ἴση· λοιπὴ <lb n="1799,001,,,,6,7,53"></lb> |
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4940 ἄρα ἡ πρὸς τῷ Γ λοιπῇ τῇ πρὸς τῷ Ζ ἴση ἐστίν.</s> <s>ἰσο-<lb n="1799,001,,,,6,7,54"></lb> |
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4941 γώνιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ.</s> <lb n="1799,001,,,,6,7,55"></lb> |
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4942 <s>Ἐὰν ἄρα δύο τρίγωνα μίαν γωνίαν μιᾷ γωνίᾳ ἴσην <lb n="1799,001,,,,6,7,56"></lb> |
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4943 ἔχῃ, περὶ δὲ ἄλλας γωνίας τὰς πλευρὰς ἀνάλογον, τῶν δὲ <lb n="1799,001,,,,6,7,57"></lb> |
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4944 λοιπῶν ἑκατέραν ἅμα ἐλάττονα ἢ μὴ ἐλάττονα ὀρθῆς, <lb n="1799,001,,,,6,7,58"></lb> |
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4945 ἰσογώνια ἔσται τὰ τρίγωνα καὶ ἴσας ἕξει τὰς γωνίας, <lb n="1799,001,,,,6,7,59"></lb> |
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4946 περὶ ἃς ἀνάλογόν εἰσιν αἱ πλευραί· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,7,60"></lb> |
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4947 <s>Ἐὰν ἐν ὀρθογωνίῳ τριγώνῳ ἀπὸ τῆς ὀρθῆς γωνίας ἐπὶ <lb n="1799,001,,,,6,8,1"></lb> |
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4948 τὴν βάσιν κάθετος ἀχθῇ, τὰ πρὸς τῇ καθέτῳ τρίγωνα <lb n="1799,001,,,,6,8,2"></lb> |
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4949 ὅμοιά ἐστι τῷ τε ὅλῳ καὶ ἀλλήλοις.</s> <lb n="1799,001,,,,6,8,3"></lb> |
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4950 <s>Ἔστω τρίγωνον ὀρθογώνιον τὸ ΑΒΓ ὀρθὴν ἔχον τὴν <lb n="1799,001,,,,6,8,4"></lb> |
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4951 ὑπὸ ΒΑΓ γωνίαν, καὶ ἤχθω ἀπὸ τοῦ Α ἐπὶ τὴν ΒΓ <pb></pb> <lb n="1799,001,,,,6,8,5"></lb> |
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4952 κάθετος ἡ ΑΔ· λέγω, ὅτι ὅμοιόν ἐστιν ἑκάτερον τῶν <lb n="1799,001,,,,6,8,6"></lb> |
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4953 ΑΒΔ, ΑΔΓ τριγώνων ὅλῳ τῷ ΑΒΓ καὶ ἔτι ἀλλήλοις.</s> <lb n="1799,001,,,,6,8,7"></lb> |
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4954 <s>Ἐπεὶ γὰρ ἴση ἐστὶν ἡ ὑπὸ ΒΑΓ <lb n="1799,001,,,,6,8,8"></lb> |
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4955 τῇ ὑπὸ ΑΔΒ· ὀρθὴ γὰρ ἑκατέρα· <lb n="1799,001,,,,6,8,9"></lb> |
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4956 καὶ κοινὴ τῶν δύο τριγώνων τοῦ <lb n="1799,001,,,,6,8,10"></lb> |
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4957 τε ΑΒΓ καὶ τοῦ ΑΒΔ ἡ πρὸς <lb n="1799,001,,,,6,8,11"></lb> |
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4958 τῷ Β, λοιπὴ ἄρα ἡ ὑπὸ ΑΓΒ <lb n="1799,001,,,,6,8,12"></lb> |
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4959 λοιπῇ τῇ ὑπὸ ΒΑΔ ἐστιν ἴση· <lb n="1799,001,,,,6,8,13"></lb> |
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4960 ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓ <lb n="1799,001,,,,6,8,14"></lb> |
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4961 τρίγωνον τῷ ΑΒΔ τριγώνῳ.</s> <s>ἔστιν ἄρα ὡς ἡ ΒΓ ὑπο-<lb n="1799,001,,,,6,8,15"></lb> |
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4962 τείνουσα τὴν ὀρθὴν τοῦ ΑΒΓ τριγώνου πρὸς τὴν ΒΑ <lb n="1799,001,,,,6,8,16"></lb> |
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4963 ὑποτείνουσαν τὴν ὀρθὴν τοῦ ΑΒΔ τριγώνου, οὕτως αὐτὴ <lb n="1799,001,,,,6,8,17"></lb> |
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4964 ἡ ΑΒ ὑποτείνουσα τὴν πρὸς τῷ Γ γωνίαν τοῦ ΑΒΓ τριγώ-<lb n="1799,001,,,,6,8,18"></lb> |
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4965 νου πρὸς τὴν ΒΔ ὑποτείνουσαν τὴν ἴσην τὴν ὑπὸ ΒΑΔ <lb n="1799,001,,,,6,8,19"></lb> |
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4966 τοῦ ΑΒΔ τριγώνου, καὶ ἔτι ἡ ΑΓ πρὸς τὴν ΑΔ ὑποτεί-<lb n="1799,001,,,,6,8,20"></lb> |
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4967 νουσαν τὴν πρὸς τῷ Β γωνίαν κοινὴν τῶν δύο τριγώνων.</s> <lb n="1799,001,,,,6,8,21"></lb> |
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4968 <s>τὸ ΑΒΓ ἄρα τρίγωνον τῷ ΑΒΔ τριγώνῳ ἰσογώνιόν τέ <lb n="1799,001,,,,6,8,22"></lb> |
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4969 ἐστι καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον ἔχει.</s> <lb n="1799,001,,,,6,8,23"></lb> |
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4970 <s>ὅμοιον ἄρα [ἐστὶ] τὸ ΑΒΓ τρίγωνον τῷ ΑΒΔ τριγώνῳ.</s> <lb n="1799,001,,,,6,8,24"></lb> |
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4971 <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ τῷ ΑΔΓ τριγώνῳ ὅμοιόν <lb n="1799,001,,,,6,8,25"></lb> |
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4972 ἐστι τὸ ΑΒΓ τρίγωνον· ἑκάτερον ἄρα τῶν ΑΒΔ, ΑΔΓ <lb n="1799,001,,,,6,8,26"></lb> |
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4973 [τριγώνων] ὅμοιόν ἐστιν ὅλῳ τῷ ΑΒΓ.</s> <s><pb></pb> <lb n="1799,001,,,,6,8,27"></lb> |
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4974 Λέγω δή, ὅτι καὶ ἀλλήλοις ἐστὶν ὅμοια τὰ ΑΒΔ, ΑΔΓ <lb n="1799,001,,,,6,8,28"></lb> |
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4975 τρίγωνα.</s> <lb n="1799,001,,,,6,8,29"></lb> |
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4976 <s>Ἐπεὶ γὰρ ὀρθὴ ἡ ὑπὸ ΒΔΑ ὀρθῇ τῇ ὑπὸ ΑΔΓ ἐστιν <lb n="1799,001,,,,6,8,30"></lb> |
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4977 ἴση, ἀλλὰ μὴν καὶ ἡ ὑπὸ ΒΑΔ τῇ πρὸς τῷ Γ ἐδείχθη <lb n="1799,001,,,,6,8,31"></lb> |
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4978 ἴση, καὶ λοιπὴ ἄρα ἡ πρὸς τῷ Β λοιπῇ τῇ ὑπὸ ΔΑΓ ἐστιν <lb n="1799,001,,,,6,8,32"></lb> |
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4979 ἴση· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΔ τρίγωνον τῷ ΑΔΓ <lb n="1799,001,,,,6,8,33"></lb> |
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4980 τριγώνῳ.</s> <s>ἔστιν ἄρα ὡς ἡ ΒΔ τοῦ ΑΒΔ τριγώνου ὑπο-<lb n="1799,001,,,,6,8,34"></lb> |
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4981 τείνουσα τὴν ὑπὸ ΒΑΔ πρὸς τὴν ΔΑ τοῦ ΑΔΓ τριγώνου <lb n="1799,001,,,,6,8,35"></lb> |
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4982 ὑποτείνουσαν τὴν πρὸς τῷ Γ ἴσην τῇ ὑπὸ ΒΑΔ, οὕτως <lb n="1799,001,,,,6,8,36"></lb> |
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4983 αὐτὴ ἡ ΑΔ τοῦ ΑΒΔ τριγώνου ὑποτείνουσα τὴν πρὸς τῷ <lb n="1799,001,,,,6,8,37"></lb> |
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4984 Β γωνίαν πρὸς τὴν ΔΓ ὑποτείνουσαν τὴν ὑπὸ ΔΑΓ τοῦ <lb n="1799,001,,,,6,8,38"></lb> |
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4985 ΑΔΓ τριγώνου ἴσην τῇ πρὸς τῷ Β, καὶ ἔτι ἡ ΒΑ πρὸς τὴν <lb n="1799,001,,,,6,8,39"></lb> |
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4986 ΑΓ ὑποτείνουσαι τὰς ὀρθάσ· ὅμοιον ἄρα ἐστὶ τὸ ΑΒΔ <lb n="1799,001,,,,6,8,40"></lb> |
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4987 τρίγωνον τῷ ΑΔΓ τριγώνῳ.</s> <lb n="1799,001,,,,6,8,41"></lb> |
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4988 <s>Ἐὰν ἄρα ἐν ὀρθογωνίῳ τριγώνῳ ἀπὸ τῆς ὀρθῆς γωνίας <lb n="1799,001,,,,6,8,42"></lb> |
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4989 ἐπὶ τὴν βάσιν κάθετος ἀχθῇ, τὰ πρὸς τῇ καθέτῳ τρίγωνα <lb n="1799,001,,,,6,8,43"></lb> |
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4990 ὅμοιά ἐστι τῷ τε ὅλῳ καὶ ἀλλήλοις [ὅπερ ἔδει δεῖξαι].</s> <lb n="1799,001,,,,6,8,44"></lb> |
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4991 <s>Πόρισμα <lb n="1799,001,,,,6,8,45"></lb> |
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4992 Ἐκ δὴ τούτου φανερόν, ὅτι ἐὰν ἐν ὀρθογωνίῳ τριγώνῳ <lb n="1799,001,,,,6,8,46"></lb> |
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4993 ἀπὸ τῆς ὀρθῆς ἐπὶ τὴν βάσιν κάθετος ἀχθῇ, ἡ ἀχθεῖσα <lb n="1799,001,,,,6,8,47"></lb> |
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4994 τῶν τῆς βάσεως τμημάτων μέση ἀνάλογόν ἐστιν· ὅπερ <pb></pb> <lb n="1799,001,,,,6,8,48"></lb> |
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4995 ἔδει δεῖξαι [καὶ ἔτι τῆς βάσεως καὶ ἑνὸς ὁποιουοῦν τῶν <lb n="1799,001,,,,6,8,49"></lb> |
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4996 τμημάτων ἡ πρὸς τῷ τμήματι πλευρὰ μέση ἀνάλογόν <lb n="1799,001,,,,6,8,50"></lb> |
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4997 ἐστιν].</s> <lb n="1799,001,,,,6,8,51"></lb> |
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4998 <s>Τῆς δοθείσης εὐθείας τὸ προσταχθὲν μέρος ἀφελεῖν.</s> <lb n="1799,001,,,,6,9,1"></lb> |
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4999 <s>Ἔστω ἡ δοθεῖσα εὐθεῖα ἡ ΑΒ· δεῖ δὴ τῆς ΑΒ τὸ <lb n="1799,001,,,,6,9,2"></lb> |
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5000 προσταχθὲν μέρος ἀφελεῖν.</s> <lb n="1799,001,,,,6,9,3"></lb> |
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5001 <s>Ἐπιτετάχθω δὴ τὸ τρίτον.</s> <s>[καὶ] διήχθω τις ἀπὸ τοῦ <lb n="1799,001,,,,6,9,4"></lb> |
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5002 Α εὐθεῖα ἡ ΑΓ γωνίαν περιέχουσα μετὰ τῆς ΑΒ τυχοῦ-<lb n="1799,001,,,,6,9,5"></lb> |
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5003 σαν· καὶ εἰλήφθω τυχὸν σημεῖον <lb n="1799,001,,,,6,9,6"></lb> |
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5004 ἐπὶ τῆς ΑΓ τὸ Δ, καὶ κείσθωσαν <lb n="1799,001,,,,6,9,7"></lb> |
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5005 τῇ ΑΔ ἴσαι αἱ ΔΕ, ΕΓ.</s> <s>καὶ <lb n="1799,001,,,,6,9,8"></lb> |
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5006 ἐπεζεύχθω ἡ ΒΓ, καὶ διὰ τοῦ Δ <lb n="1799,001,,,,6,9,9"></lb> |
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5007 παράλληλος αὐτῇ ἤχθω ἡ ΔΖ.</s> <lb n="1799,001,,,,6,9,10"></lb> |
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5008 <s>Ἐπεὶ οὖν τριγώνου τοῦ ΑΒΓ <lb n="1799,001,,,,6,9,11"></lb> |
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5009 παρὰ μίαν τῶν πλευρῶν τὴν <lb n="1799,001,,,,6,9,12"></lb> |
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5010 ΒΓ ἦκται ἡ ΖΔ, ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΓΔ πρὸς <lb n="1799,001,,,,6,9,13"></lb> |
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5011 τὴν ΔΑ, οὕτως ἡ ΒΖ πρὸς τὴν ΖΑ.</s> <s>διπλῆ δὲ ἡ ΓΔ τῆς <lb n="1799,001,,,,6,9,14"></lb> |
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5012 ΔΑ· διπλῆ ἄρα καὶ ἡ ΒΖ τῆς ΖΑ· τριπλῆ ἄρα ἡ ΒΑ <lb n="1799,001,,,,6,9,15"></lb> |
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5013 τῆς ΑΖ.</s> <lb n="1799,001,,,,6,9,16"></lb> |
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5014 <s>Τῆς ἄρα δοθείσης εὐθείας τῆς ΑΒ τὸ ἐπιταχθὲν τρίτον <lb n="1799,001,,,,6,9,17"></lb> |
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5015 μέρος ἀφῄρηται τὸ ΑΖ· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,6,9,18"></lb> |
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5016 Τὴν δοθεῖσαν εὐθεῖαν ἄτμητον τῇ δοθείσῃ τετμημένῃ <lb n="1799,001,,,,6,10,1"></lb> |
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5017 ὁμοίως τεμεῖν.</s> <lb n="1799,001,,,,6,10,2"></lb> |
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5018 <s>Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἄτμητος ἡ ΑΒ, ἡ δὲ <lb n="1799,001,,,,6,10,3"></lb> |
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5019 τετμημένη ἡ ΑΓ κατὰ τὰ Δ, Ε σημεῖα, καὶ κείσθωσαν <lb n="1799,001,,,,6,10,4"></lb> |
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5020 ὥστε γωνίαν τυχοῦσαν περιέχειν, <lb n="1799,001,,,,6,10,5"></lb> |
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5021 καὶ ἐπεζεύχθω ἡ ΓΒ, καὶ διὰ τῶν <lb n="1799,001,,,,6,10,6"></lb> |
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5022 Δ, Ε τῇ ΒΓ παράλληλοι ἤχθωσαν <lb n="1799,001,,,,6,10,7"></lb> |
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5023 αἱ ΔΖ, ΕΗ, διὰ δὲ τοῦ Δ τῇ ΑΒ <lb n="1799,001,,,,6,10,8"></lb> |
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5024 παράλληλος ἤχθω ἡ ΔΘΚ.</s> <lb n="1799,001,,,,6,10,9"></lb> |
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5025 <s>Παραλληλόγραμμον ἄρα ἐστὶν <lb n="1799,001,,,,6,10,10"></lb> |
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5026 ἑκάτερον τῶν ΖΘ, ΘΒ· ἴση ἄρα <lb n="1799,001,,,,6,10,11"></lb> |
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5027 ἡ μὲν ΔΘ τῇ ΖΗ, ἡ δὲ ΘΚ τῇ <lb n="1799,001,,,,6,10,12"></lb> |
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5028 ΗΒ.</s> <s>καὶ ἐπεὶ τριγώνου τοῦ ΔΚΓ <lb n="1799,001,,,,6,10,13"></lb> |
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5029 παρὰ μίαν τῶν πλευρῶν τὴν ΚΓ <lb n="1799,001,,,,6,10,14"></lb> |
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5030 εὐθεῖα ἦκται ἡ ΘΕ, ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΓΕ πρὸς <lb n="1799,001,,,,6,10,15"></lb> |
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5031 τὴν ΕΔ, οὕτως ἡ ΚΘ πρὸς τὴν ΘΔ.</s> <s>ἴση δὲ ἡ μὲν ΚΘ <lb n="1799,001,,,,6,10,16"></lb> |
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5032 τῇ ΒΗ, ἡ δὲ ΘΔ τῇ ΗΖ.</s> <s>ἔστιν ἄρα ὡς ἡ ΓΕ πρὸς τὴν <lb n="1799,001,,,,6,10,17"></lb> |
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5033 ΕΔ, οὕτως ἡ ΒΗ πρὸς τὴν ΗΖ.</s> <s>πάλιν, ἐπεὶ τριγώνου τοῦ <lb n="1799,001,,,,6,10,18"></lb> |
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5034 ΑΗΕ παρὰ μίαν τῶν πλευρῶν τὴν ΗΕ ἦκται ἡ ΖΔ, ἀνά-<lb n="1799,001,,,,6,10,19"></lb> |
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5035 λογον ἄρα ἐστὶν ὡς ἡ ΕΔ πρὸς τὴν ΔΑ, οὕτως ἡ ΗΖ <lb n="1799,001,,,,6,10,20"></lb> |
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5036 πρὸς τὴν ΖΑ.</s> <s>ἐδείχθη δὲ καὶ ὡς ἡ ΓΕ πρὸς τὴν ΕΔ, <lb n="1799,001,,,,6,10,21"></lb> |
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5037 οὕτως ἡ ΒΗ πρὸς τὴν ΗΖ· ἔστιν ἄρα ὡς μὲν ἡ ΓΕ <pb></pb> <lb n="1799,001,,,,6,10,22"></lb> |
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5038 πρὸς τὴν ΕΔ, οὕτως ἡ ΒΗ πρὸς τὴν ΗΖ, ὡς δὲ ἡ ΕΔ <lb n="1799,001,,,,6,10,23"></lb> |
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5039 πρὸς τὴν ΔΑ, οὕτως ἡ ΗΖ πρὸς τὴν ΖΑ.</s> <lb n="1799,001,,,,6,10,24"></lb> |
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5040 <s>Ἡ ἄρα δοθεῖσα εὐθεῖα ἄτμητος ἡ ΑΒ τῇ δοθείσῃ <lb n="1799,001,,,,6,10,25"></lb> |
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5041 εὐθείᾳ τετμημένῃ τῇ ΑΓ ὁμοίως τέτμηται· ὅπερ ἔδει <lb n="1799,001,,,,6,10,26"></lb> |
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5042 ποιῆσαι.</s> <lb n="1799,001,,,,6,10,27"></lb> |
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5043 <s>Δύο δοθεισῶν εὐθειῶν τρίτην ἀνάλογον προσευρεῖν.</s> <lb n="1799,001,,,,6,11,1"></lb> |
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5044 <s>Ἔστωσαν αἱ δοθεῖσαι [δύο εὐθεῖαι] αἱ ΒΑ, ΑΓ καὶ <lb n="1799,001,,,,6,11,2"></lb> |
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5045 κείσθωσαν γωνίαν περιέχουσαι τυχοῦσαν.</s> <s>δεῖ δὴ τῶν <lb n="1799,001,,,,6,11,3"></lb> |
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5046 ΒΑ, ΑΓ τρίτην ἀνάλογον προσευρεῖν.</s> <s>ἐκ- <lb n="1799,001,,,,6,11,4"></lb> |
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5047 βεβλήσθωσαν γὰρ ἐπὶ τὰ Δ, Ε σημεῖα, <lb n="1799,001,,,,6,11,5"></lb> |
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5048 καὶ κείσθω τῇ ΑΓ ἴση ἡ ΒΔ, καὶ ἐπεζεύ-<lb n="1799,001,,,,6,11,6"></lb> |
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5049 χθω ἡ ΒΓ, καὶ διὰ τοῦ Δ παράλληλος <lb n="1799,001,,,,6,11,7"></lb> |
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5050 αὐτῇ ἤχθω ἡ ΔΕ.</s> <lb n="1799,001,,,,6,11,8"></lb> |
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5051 <s>Ἐπεὶ οὖν τριγώνου τοῦ ΑΔΕ παρὰ μίαν <lb n="1799,001,,,,6,11,9"></lb> |
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5052 τῶν πλευρῶν τὴν ΔΕ ἦκται ἡ ΒΓ, ἀνάλο-<lb n="1799,001,,,,6,11,10"></lb> |
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5053 γόν ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΒΔ, οὕτως <lb n="1799,001,,,,6,11,11"></lb> |
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5054 ἡ ΑΓ πρὸς τὴν ΓΕ.</s> <s>ἴση δὲ ἡ ΒΔ τῇ ΑΓ.</s> <lb n="1799,001,,,,6,11,12"></lb> |
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5055 <s>ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΑΓ, οὕτως ἡ <lb n="1799,001,,,,6,11,13"></lb> |
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5056 ΑΓ πρὸς τὴν ΓΕ.</s> <lb n="1799,001,,,,6,11,14"></lb> |
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5057 <s>Δύο ἄρα δοθεισῶν εὐθειῶν τῶν ΑΒ, ΑΓ τρίτη ἀνά-<lb n="1799,001,,,,6,11,15"></lb> |
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5058 λογον αὐταῖς προσεύρηται ἡ ΓΕ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,6,11,16"></lb> |
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5059 <s>Τριῶν δοθεισῶν εὐθειῶν τετάρτην ἀνάλογον προσευρεῖν.</s> <s><pb></pb> <lb n="1799,001,,,,6,12,1"></lb> |
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5060 Ἔστωσαν αἱ δοθεῖσαι τρεῖς εὐθεῖαι αἱ Α, Β, Γ· δεῖ <lb n="1799,001,,,,6,12,2"></lb> |
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5061 δὴ τῶν Α, Β, Γ τετάρτην ἀνάλογον προσευρεῖν.</s> <lb n="1799,001,,,,6,12,3"></lb> |
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5062 <s>Ἐκκείσθωσαν δύο εὐθεῖαι αἱ ΔΕ, ΔΖ γωνίαν περι-<lb n="1799,001,,,,6,12,4"></lb> |
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5063 έχουσαι [τυχοῦσαν] τὴν ὑπὸ ΕΔΖ· καὶ κείσθω τῇ μὲν Α <lb n="1799,001,,,,6,12,5"></lb> |
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5064 ἴση ἡ ΔΗ, τῇ δὲ Β ἴση ἡ ΗΕ, καὶ ἔτι τῇ Γ ἴση ἡ ΔΘ· <lb n="1799,001,,,,6,12,6"></lb> |
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5065 καὶ ἐπιζευχθείσης τῆς ΗΘ παράλληλος αὐτῇ ἤχθω διὰ <lb n="1799,001,,,,6,12,7"></lb> |
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5066 τοῦ Ε ἡ ΕΖ.</s> <lb n="1799,001,,,,6,12,8"></lb> |
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5067 <s>Ἐπεὶ οὖν τριγώνου τοῦ ΔΕΖ παρὰ μίαν τὴν ΕΖ ἦκται <lb n="1799,001,,,,6,12,9"></lb> |
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5068 ἡ ΗΘ, ἔστιν ἄρα ὡς ἡ ΔΗ πρὸς τὴν ΗΕ, οὕτως ἡ ΔΘ <lb n="1799,001,,,,6,12,10"></lb> |
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5069 πρὸς τὴν ΘΖ.</s> <s>ἴση δὲ ἡ μὲν ΔΗ τῇ Α, ἡ δὲ ΗΕ τῇ Β, <lb n="1799,001,,,,6,12,11"></lb> |
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5070 ἡ δὲ ΔΘ τῇ Γ· ἔστιν ἄρα ὡς ἡ Α πρὸς τὴν Β, οὕτως ἡ Γ <lb n="1799,001,,,,6,12,12"></lb> |
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5071 πρὸς τὴν ΘΖ.</s> <lb n="1799,001,,,,6,12,13"></lb> |
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5072 <s>Τριῶν ἄρα δοθεισῶν εὐθειῶν τῶν Α, Β, Γ τετάρτη <lb n="1799,001,,,,6,12,14"></lb> |
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5073 ἀνάλογον προσεύρηται ἡ ΘΖ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,6,12,15"></lb> |
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5074 <s>Δύο δοθεισῶν εὐθειῶν μέσην ἀνάλογον προσευρεῖν.</s> <s><pb></pb> <lb n="1799,001,,,,6,13,1"></lb> |
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5075 Ἔστωσαν αἱ δοθεῖσαι δύο εὐθεῖαι αἱ ΑΒ, ΒΓ· δεῖ δὴ <lb n="1799,001,,,,6,13,2"></lb> |
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5076 τῶν ΑΒ, ΒΓ μέσην ἀνάλογον προσευρεῖν.</s> <lb n="1799,001,,,,6,13,3"></lb> |
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5077 <s>Κείσθωσαν ἐπ' εὐθείας, καὶ γεγράφθω ἐπὶ τῆς ΑΓ <lb n="1799,001,,,,6,13,4"></lb> |
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5078 ἡμικύκλιον τὸ ΑΔΓ, καὶ ἤχθω ἀπὸ τοῦ Β σημείου τῇ <lb n="1799,001,,,,6,13,5"></lb> |
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5079 ΑΓ εὐθείᾳ πρὸς ὀρθὰς ἡ ΒΔ, καὶ <lb n="1799,001,,,,6,13,6"></lb> |
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5080 ἐπεζεύχθωσαν αἱ ΑΔ, ΔΓ.</s> <lb n="1799,001,,,,6,13,7"></lb> |
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5081 <s>Ἐπεὶ ἐν ἡμικυκλίῳ γωνία <lb n="1799,001,,,,6,13,8"></lb> |
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5082 ἐστὶν ἡ ὑπὸ ΑΔΓ, ὀρθή ἐστιν.</s> <lb n="1799,001,,,,6,13,9"></lb> |
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5083 <s>καὶ ἐπεὶ ἐν ὀρθογωνίῳ τριγώνῳ <lb n="1799,001,,,,6,13,10"></lb> |
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5084 τῷ ΑΔΓ ἀπὸ τῆς ὀρθῆς γωνίας <lb n="1799,001,,,,6,13,11"></lb> |
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5085 ἐπὶ τὴν βάσιν κάθετος ἦκται ἡ <lb n="1799,001,,,,6,13,12"></lb> |
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5086 ΔΒ, ἡ ΔΒ ἄρα τῶν τῆς βάσεως τμημάτων τῶν ΑΒ, <lb n="1799,001,,,,6,13,13"></lb> |
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5087 ΒΓ μέση ἀνάλογόν ἐστιν.</s> <lb n="1799,001,,,,6,13,14"></lb> |
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5088 <s>Δύο ἄρα δοθεισῶν εὐθειῶν τῶν ΑΒ, ΒΓ μέση ἀνάλογον <lb n="1799,001,,,,6,13,15"></lb> |
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5089 προσεύρηται ἡ ΔΒ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,6,13,16"></lb> |
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5090 <s>Τῶν ἴσων τε καὶ ἰσογωνίων παραλληλογράμμων <lb n="1799,001,,,,6,14,1"></lb> |
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5091 ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας· καὶ <lb n="1799,001,,,,6,14,2"></lb> |
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5092 ὧν ἰσογωνίων παραλληλογράμμων ἀντιπεπόνθασιν αἱ <lb n="1799,001,,,,6,14,3"></lb> |
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5093 πλευραὶ αἱ περὶ τὰς ἴσας γωνίας, ἴσα ἐστὶν ἐκεῖνα.</s> <lb n="1799,001,,,,6,14,4"></lb> |
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5094 <s>Ἔστω ἴσα τε καὶ ἰσογώνια παραλληλόγραμμα τὰ ΑΒ, <lb n="1799,001,,,,6,14,5"></lb> |
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5095 ΒΓ ἴσας ἔχοντα τὰς πρὸς τῷ Β γωνίας, καὶ κείσθωσαν <pb></pb> <lb n="1799,001,,,,6,14,6"></lb> |
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5096 ἐπ' εὐθείας αἱ ΔΒ, ΒΕ· ἐπ' εὐθείας ἄρα εἰσὶ καὶ αἱ <lb n="1799,001,,,,6,14,7"></lb> |
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5097 ΖΒ, ΒΗ.</s> <s>λέγω, ὅτι τῶν ΑΒ, ΒΓ ἀντιπεπόνθασιν αἱ <lb n="1799,001,,,,6,14,8"></lb> |
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5098 πλευραὶ αἱ περὶ τὰς ἴσας γωνίας, τουτέστιν, ὅτι ἐστὶν ὡς <lb n="1799,001,,,,6,14,9"></lb> |
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5099 ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως ἡ <lb n="1799,001,,,,6,14,10"></lb> |
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5100 ΗΒ πρὸς τὴν ΒΖ.</s> <lb n="1799,001,,,,6,14,11"></lb> |
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5101 <s>Συμπεπληρώσθω γὰρ τὸ ΖΕ <lb n="1799,001,,,,6,14,12"></lb> |
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5102 παραλληλόγραμμον.</s> <s>ἐπεὶ οὖν ἴσον <lb n="1799,001,,,,6,14,13"></lb> |
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5103 ἐστὶ τὸ ΑΒ παραλληλόγραμμον τῷ <lb n="1799,001,,,,6,14,14"></lb> |
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5104 ΒΓ παραλληλογράμμῳ, ἄλλο δέ <lb n="1799,001,,,,6,14,15"></lb> |
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5105 τι τὸ ΖΕ, ἔστιν ἄρα ὡς τὸ ΑΒ <lb n="1799,001,,,,6,14,16"></lb> |
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5106 πρὸς τὸ ΖΕ, οὕτως τὸ ΒΓ πρὸς <lb n="1799,001,,,,6,14,17"></lb> |
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5107 τὸ ΖΕ.</s> <s>ἀλλ' ὡς μὲν τὸ ΑΒ πρὸς <lb n="1799,001,,,,6,14,18"></lb> |
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5108 τὸ ΖΕ, οὕτως ἡ ΔΒ πρὸς τὴν ΒΕ, <lb n="1799,001,,,,6,14,19"></lb> |
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5109 ὡς δὲ τὸ ΒΓ πρὸς τὸ ΖΕ, οὕτως ἡ ΗΒ πρὸς τὴν ΒΖ· <lb n="1799,001,,,,6,14,20"></lb> |
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5110 καὶ ὡς ἄρα ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως ἡ ΗΒ πρὸς τὴν <lb n="1799,001,,,,6,14,21"></lb> |
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5111 ΒΖ.</s> <s>τῶν ἄρα ΑΒ, ΒΓ παραλληλογράμμων ἀντιπεπόνθα-<lb n="1799,001,,,,6,14,22"></lb> |
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5112 σιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας.</s> <lb n="1799,001,,,,6,14,23"></lb> |
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5113 <s>Ἀλλὰ δὴ ἔστω ὡς ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως ἡ ΗΒ <lb n="1799,001,,,,6,14,24"></lb> |
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5114 πρὸς τὴν ΒΖ· λέγω, ὅτι ἴσον ἐστὶ τὸ ΑΒ παραλληλό-<lb n="1799,001,,,,6,14,25"></lb> |
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5115 γραμμον τῷ ΒΓ παραλληλογράμμῳ.</s> <lb n="1799,001,,,,6,14,26"></lb> |
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5116 <s>Ἐπεὶ γάρ ἐστιν ὡς ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως ἡ ΗΒ <lb n="1799,001,,,,6,14,27"></lb> |
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5117 πρὸς τὴν ΒΖ, ἀλλ' ὡς μὲν ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως τὸ <lb n="1799,001,,,,6,14,28"></lb> |
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5118 ΑΒ παραλληλόγραμμον πρὸς τὸ ΖΕ παραλληλόγραμμον, <lb n="1799,001,,,,6,14,29"></lb> |
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5119 ὡς δὲ ἡ ΗΒ πρὸς τὴν ΒΖ, οὕτως τὸ ΒΓ παραλληλό- <pb></pb><lb n="1799,001,,,,6,14,30"></lb> |
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5120 γραμμον πρὸς τὸ ΖΕ παραλληλόγραμμον, καὶ ὡς ἄρα τὸ <lb n="1799,001,,,,6,14,31"></lb> |
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5121 ΑΒ πρὸς τὸ ΖΕ, οὕτως τὸ ΒΓ πρὸς τὸ ΖΕ· ἴσον ἄρα ἐστὶ <lb n="1799,001,,,,6,14,32"></lb> |
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5122 τὸ ΑΒ παραλληλόγραμμον τῷ ΒΓ παραλληλογράμμῳ.</s> <lb n="1799,001,,,,6,14,33"></lb> |
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5123 <s>Τῶν ἄρα ἴσων τε καὶ ἰσογωνίων παραλληλογράμμων <lb n="1799,001,,,,6,14,34"></lb> |
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5124 ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας· καὶ <lb n="1799,001,,,,6,14,35"></lb> |
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5125 ὧν ἰσογωνίων παραλληλογράμμων ἀντιπεπόνθασιν αἱ <lb n="1799,001,,,,6,14,36"></lb> |
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5126 πλευραὶ αἱ περὶ τὰς ἴσας γωνίας, ἴσα ἐστὶν ἐκεῖνα· ὅπερ <lb n="1799,001,,,,6,14,37"></lb> |
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5127 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,14,38"></lb> |
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5128 <s>Τῶν ἴσων καὶ μίαν μιᾷ ἴσην ἐχόντων γωνίαν τριγώνων <lb n="1799,001,,,,6,15,1"></lb> |
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5129 ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας· καὶ <lb n="1799,001,,,,6,15,2"></lb> |
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5130 ὧν μίαν μιᾷ ἴσην ἐχόντων γωνίαν τριγώνων ἀντιπεπόν-<lb n="1799,001,,,,6,15,3"></lb> |
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5131 θασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας, ἴσα ἐστὶν <lb n="1799,001,,,,6,15,4"></lb> |
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5132 ἐκεῖνα.</s> <lb n="1799,001,,,,6,15,5"></lb> |
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5133 <s>Ἔστω ἴσα τρίγωνα τὰ ΑΒΓ, ΑΔΕ μίαν μιᾷ ἴσην <lb n="1799,001,,,,6,15,6"></lb> |
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5134 ἔχοντα γωνίαν τὴν ὑπὸ ΒΑΓ τῇ ὑπὸ ΔΑΕ· λέγω, ὅτι <lb n="1799,001,,,,6,15,7"></lb> |
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5135 τῶν ΑΒΓ, ΑΔΕ τριγώνων ἀντιπεπόνθασιν αἱ πλευραὶ <lb n="1799,001,,,,6,15,8"></lb> |
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5136 αἱ περὶ τὰς ἴσας γωνίας, τουτέστιν, ὅτι ἐστὶν ὡς ἡ ΓΑ <lb n="1799,001,,,,6,15,9"></lb> |
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5137 πρὸς τὴν ΑΔ, οὕτως ἡ ΕΑ πρὸς τὴν ΑΒ.</s> <lb n="1799,001,,,,6,15,10"></lb> |
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5138 <s>Κείσθω γὰρ ὥστε ἐπ' εὐθείας εἶναι τὴν ΓΑ τῇ ΑΔ· <lb n="1799,001,,,,6,15,11"></lb> |
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5139 ἐπ' εὐθείας ἄρα ἐστὶ καὶ ἡ ΕΑ τῇ ΑΒ.</s> <s>καὶ ἐπεζεύχθω <lb n="1799,001,,,,6,15,12"></lb> |
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5140 ἡ ΒΔ.</s> <lb n="1799,001,,,,6,15,13"></lb> |
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5141 <s>Ἐπεὶ οὖν ἴσον ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΑΔΕ <pb></pb> <lb n="1799,001,,,,6,15,14"></lb> |
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5142 τριγώνῳ, ἄλλο δέ τι τὸ ΒΑΔ, ἔστιν ἄρα ὡς τὸ ΓΑΒ <lb n="1799,001,,,,6,15,15"></lb> |
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5143 τρίγωνον πρὸς τὸ ΒΑΔ τρίγωνον, οὕτως τὸ ΕΑΔ <lb n="1799,001,,,,6,15,16"></lb> |
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5144 τρίγωνον πρὸς τὸ ΒΑΔ τρίγωνον.</s> <s>ἀλλ' ὡς μὲν τὸ ΓΑΒ <lb n="1799,001,,,,6,15,17"></lb> |
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5145 πρὸς τὸ ΒΑΔ, οὕτως ἡ ΓΑ πρὸς τὴν ΑΔ, ὡς δὲ τὸ ΕΑΔ <lb n="1799,001,,,,6,15,18"></lb> |
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5146 πρὸς τὸ ΒΑΔ, οὕτως ἡ ΕΑ πρὸς τὴν ΑΒ.</s> <s>καὶ ὡς ἄρα ἡ <lb n="1799,001,,,,6,15,19"></lb> |
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5147 ΓΑ πρὸς τὴν ΑΔ, οὕτως ἡ ΕΑ πρὸς τὴν ΑΒ.</s> <s>τῶν ΑΒΓ, <lb n="1799,001,,,,6,15,20"></lb> |
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5148 ΑΔΕ ἄρα τριγώνων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ <lb n="1799,001,,,,6,15,21"></lb> |
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5149 τὰς ἴσας γωνίας.</s> <lb n="1799,001,,,,6,15,22"></lb> |
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5150 <s>Ἀλλὰ δὴ ἀντιπεπονθέτωσαν αἱ πλευραὶ τῶν ΑΒΓ, <lb n="1799,001,,,,6,15,23"></lb> |
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5151 ΑΔΕ τριγώνων, καὶ ἔστω ὡς ἡ ΓΑ πρὸς τὴν ΑΔ, οὕτως <lb n="1799,001,,,,6,15,24"></lb> |
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5152 ἡ ΕΑ πρὸς τὴν ΑΒ· λέγω, ὅτι ἴσον ἐστὶ τὸ ΑΒΓ τρίγωνον <lb n="1799,001,,,,6,15,25"></lb> |
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5153 τῷ ΑΔΕ τριγώνῳ.</s> <lb n="1799,001,,,,6,15,26"></lb> |
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5154 <s>Ἐπιζευχθείσης γὰρ πάλιν τῆς ΒΔ, ἐπεί ἐστιν ὡς ἡ <lb n="1799,001,,,,6,15,27"></lb> |
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5155 ΓΑ πρὸς τὴν ΑΔ, οὕτως ἡ ΕΑ πρὸς τὴν ΑΒ, ἀλλ' ὡς <lb n="1799,001,,,,6,15,28"></lb> |
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5156 μὲν ἡ ΓΑ πρὸς τὴν ΑΔ, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς <pb></pb> <lb n="1799,001,,,,6,15,29"></lb> |
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5157 τὸ ΒΑΔ τρίγωνον, ὡς δὲ ἡ ΕΑ πρὸς τὴν ΑΒ, οὕτως τὸ <lb n="1799,001,,,,6,15,30"></lb> |
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5158 ΕΑΔ τρίγωνον πρὸς τὸ ΒΑΔ τρίγωνον, ὡς ἄρα τὸ ΑΒΓ <lb n="1799,001,,,,6,15,31"></lb> |
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5159 τρίγωνον πρὸς τὸ ΒΑΔ τρίγωνον, οὕτως τὸ ΕΑΔ <lb n="1799,001,,,,6,15,32"></lb> |
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5160 τρίγωνον πρὸς τὸ ΒΑΔ τρίγωνον.</s> <s>ἑκάτερον ἄρα τῶν <lb n="1799,001,,,,6,15,33"></lb> |
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5161 ΑΒΓ, ΕΑΔ πρὸς τὸ ΒΑΔ τὸν αὐτὸν ἔχει λόγον.</s> <s>ἴσον <lb n="1799,001,,,,6,15,34"></lb> |
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5162 ἄρα ἐστὶ τὸ ΑΒΓ [τρίγωνον] τῷ ΕΑΔ τριγώνῳ.</s> <lb n="1799,001,,,,6,15,35"></lb> |
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5163 <s>Τῶν ἄρα ἴσων καὶ μίαν μιᾷ ἴσην ἐχόντων γωνίαν <lb n="1799,001,,,,6,15,36"></lb> |
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5164 τριγώνων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας <lb n="1799,001,,,,6,15,37"></lb> |
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5165 γωνίας· καὶ ὧν μίαν μιᾷ ἴσην ἐχόντων γωνίαν τριγώνων <lb n="1799,001,,,,6,15,38"></lb> |
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5166 ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας, <lb n="1799,001,,,,6,15,39"></lb> |
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5167 ἐκεῖνα ἴσα ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,15,40"></lb> |
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5168 <s>Ἐὰν τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, τὸ ὑπὸ τῶν <lb n="1799,001,,,,6,16,1"></lb> |
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5169 ἄκρων περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν <lb n="1799,001,,,,6,16,2"></lb> |
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5170 μέσων περιεχομένῳ ὀρθογωνίῳ· κἂν τὸ ὑπὸ τῶν ἄκρων <lb n="1799,001,,,,6,16,3"></lb> |
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5171 περιεχόμενον ὀρθογώνιον ἴσον ᾖ τῷ ὑπὸ τῶν μέσων <lb n="1799,001,,,,6,16,4"></lb> |
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5172 περιεχομένῳ ὀρθογωνίῳ, αἱ τέσσαρες εὐθεῖαι ἀνάλογον <lb n="1799,001,,,,6,16,5"></lb> |
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5173 ἔσονται.</s> <lb n="1799,001,,,,6,16,6"></lb> |
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5174 <s>Ἔστωσαν τέσσαρες εὐθεῖαι ἀνάλογον αἱ ΑΒ, ΓΔ, Ε, <lb n="1799,001,,,,6,16,7"></lb> |
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5175 Ζ, ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ· λέγω, <pb></pb> <lb n="1799,001,,,,6,16,8"></lb> |
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5176 ὅτι τὸ ὑπὸ τῶν ΑΒ, Ζ περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ <lb n="1799,001,,,,6,16,9"></lb> |
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5177 τῷ ὑπὸ τῶν ΓΔ, Ε περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,6,16,10"></lb> |
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5178 <s>Ἤχθωσαν [γὰρ] ἀπὸ τῶν Α, Γ σημείων ταῖς ΑΒ, ΓΔ <lb n="1799,001,,,,6,16,11"></lb> |
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5179 εὐθείαις πρὸς ὀρθὰς αἱ ΑΗ, ΓΘ, καὶ κείσθω τῇ μὲν Ζ <lb n="1799,001,,,,6,16,12"></lb> |
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5180 ἴση ἡ ΑΓ, τῇ δὲ Ε ἴση ἡ ΓΘ.</s> <s>καὶ συμπεπληρώσθω τὰ <lb n="1799,001,,,,6,16,13"></lb> |
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5181 ΒΗ, ΔΘ παραλληλόγραμμα.</s> <lb n="1799,001,,,,6,16,14"></lb> |
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5182 <s>Καὶ ἐπεί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε <lb n="1799,001,,,,6,16,15"></lb> |
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5183 πρὸς τὴν Ζ, ἴση δὲ ἡ μὲν Ε τῇ ΓΘ, ἡ δὲ Ζ τῇ ΑΗ, ἔστιν <lb n="1799,001,,,,6,16,16"></lb> |
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5184 ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΓΘ πρὸς τὴν ΑΗ.</s> <lb n="1799,001,,,,6,16,17"></lb> |
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5185 <s>τῶν ΒΗ, ΔΘ ἄρα παραλληλογράμμων ἀντιπεπόνθασιν αἱ <lb n="1799,001,,,,6,16,18"></lb> |
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5186 πλευραὶ αἱ περὶ τὰς ἴσας γωνίας.</s> <s>ὧν δὲ ἰσογωνίων <lb n="1799,001,,,,6,16,19"></lb> |
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5187 παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ <lb n="1799,001,,,,6,16,20"></lb> |
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5188 τὰς ἴσας γωνίας, ἴσα ἐστὶν ἐκεῖνα· ἴσον ἄρα ἐστὶ τὸ ΒΗ <lb n="1799,001,,,,6,16,21"></lb> |
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5189 παραλληλόγραμμον τῷ ΔΘ παραλληλογράμμῳ.</s> <s>καί ἐστι <lb n="1799,001,,,,6,16,22"></lb> |
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5190 τὸ μὲν ΒΗ τὸ ὑπὸ τῶν ΑΒ, Ζ· ἴση γὰρ ἡ ΑΗ τῇ Ζ· τὸ <lb n="1799,001,,,,6,16,23"></lb> |
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5191 δὲ ΔΘ τὸ ὑπὸ τῶν ΓΔ, Ε· ἴση γὰρ ἡ Ε τῇ ΓΘ· τὸ ἄρα <lb n="1799,001,,,,6,16,24"></lb> |
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5192 ὑπὸ τῶν ΑΒ, Ζ περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ <lb n="1799,001,,,,6,16,25"></lb> |
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5193 ὑπὸ τῶν ΓΔ, Ε περιεχομένῳ ὀρθογωνίῳ.</s> <lb n="1799,001,,,,6,16,26"></lb> |
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5194 <s>Ἀλλὰ δὴ τὸ ὑπὸ τῶν ΑΒ, Ζ περιεχόμενον ὀρθογώνιον <lb n="1799,001,,,,6,16,27"></lb> |
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5195 ἴσον ἔστω τῷ ὑπὸ τῶν ΓΔ, Ε περιεχομένῳ ὀρθογωνίῳ· <lb n="1799,001,,,,6,16,28"></lb> |
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5196 λέγω, ὅτι αἱ τέσσαρες εὐθεῖαι ἀνάλογον ἔσονται, ὡς ἡ <lb n="1799,001,,,,6,16,29"></lb> |
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5197 ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ.</s> <lb n="1799,001,,,,6,16,30"></lb> |
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5198 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεὶ τὸ ὑπὸ τῶν <lb n="1799,001,,,,6,16,31"></lb> |
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5199 ΑΒ, Ζ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΓΔ, Ε, καί ἐστι τὸ μὲν ὑπὸ <lb n="1799,001,,,,6,16,32"></lb> |
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5200 τῶν ΑΒ, Ζ τὸ ΒΗ· ἴση γάρ ἐστιν ἡ ΑΗ τῇ Ζ· τὸ δὲ ὑπὸ <lb n="1799,001,,,,6,16,33"></lb> |
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5201 τῶν ΓΔ, Ε τὸ ΔΘ· ἴση γὰρ ἡ ΓΘ τῇ Ε· τὸ ἄρα ΒΗ ἴσον <pb></pb> <lb n="1799,001,,,,6,16,34"></lb> |
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5202 ἐστὶ τῷ ΔΘ.</s> <s>καί ἐστιν ἰσογώνια.</s> <s>τῶν δὲ ἴσων καὶ ἰσογω-<lb n="1799,001,,,,6,16,35"></lb> |
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5203 νίων παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ <lb n="1799,001,,,,6,16,36"></lb> |
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5204 περὶ τὰς ἴσας γωνίας.</s> <s>ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, <lb n="1799,001,,,,6,16,37"></lb> |
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5205 οὕτως ἡ ΓΘ πρὸς τὴν ΑΗ.</s> <s>ἴση δὲ ἡ μὲν ΓΘ τῇ Ε, ἡ δὲ <lb n="1799,001,,,,6,16,38"></lb> |
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5206 ΑΗ τῇ Ζ· ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε <lb n="1799,001,,,,6,16,39"></lb> |
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5207 πρὸς τὴν Ζ.</s> <lb n="1799,001,,,,6,16,40"></lb> |
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5208 <s>Ἐὰν ἄρα τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, τὸ ὑπὸ τῶν <lb n="1799,001,,,,6,16,41"></lb> |
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5209 ἄκρων περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν <lb n="1799,001,,,,6,16,42"></lb> |
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5210 μέσων περιεχομένῳ ὀρθογωνίῳ· κἂν τὸ ὑπὸ τῶν ἄκρων <lb n="1799,001,,,,6,16,43"></lb> |
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5211 περιεχόμενον ὀρθογώνιον ἴσον ᾖ τῷ ὑπὸ τῶν μέσων <lb n="1799,001,,,,6,16,44"></lb> |
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5212 περιεχομένῳ ὀρθογωνίῳ, αἱ τέσσαρες εὐθεῖαι ἀνάλογον <lb n="1799,001,,,,6,16,45"></lb> |
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5213 ἔσονται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,16,46"></lb> |
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5214 <s>Ἐὰν τρεῖς εὐθεῖαι ἀνάλογον ὦσιν, τὸ ὑπὸ τῶν ἄκρων <lb n="1799,001,,,,6,17,1"></lb> |
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5215 περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ἀπὸ τῆς μέσης <lb n="1799,001,,,,6,17,2"></lb> |
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5216 τετραγώνῳ· κἂν τὸ ὑπὸ τῶν ἄκρων περιεχόμενον ὀρθο-<lb n="1799,001,,,,6,17,3"></lb> |
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5217 γώνιον ἴσον ᾖ τῷ ἀπὸ τῆς μέσης τετραγώνῳ, αἱ τρεῖς <lb n="1799,001,,,,6,17,4"></lb> |
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5218 εὐθεῖαι ἀνάλογον ἔσονται.</s> <lb n="1799,001,,,,6,17,5"></lb> |
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5219 <s>Ἔστωσαν τρεῖς εὐθεῖαι ἀνάλογον αἱ Α, Β, Γ, ὡς ἡ Α <lb n="1799,001,,,,6,17,6"></lb> |
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5220 πρὸς τὴν Β, οὕτως ἡ Β πρὸς τὴν Γ· λέγω, ὅτι τὸ ὑπὸ τῶν <lb n="1799,001,,,,6,17,7"></lb> |
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5221 Α, Γ περιεχόμενον ὀρθογώ- <lb n="1799,001,,,,6,17,8"></lb> |
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5222 νιον ἴσον ἐστὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,6,17,9"></lb> |
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5223 Β τετραγώνῳ.</s> <lb n="1799,001,,,,6,17,10"></lb> |
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5224 <s>Κείσθω τῇ Β ἴση ἡ Δ.</s> <s><pb></pb> <lb n="1799,001,,,,6,17,11"></lb> |
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5225 Καὶ ἐπεί ἐστιν ὡς ἡ Α πρὸς τὴν Β, οὕτως ἡ Β πρὸς τὴν <lb n="1799,001,,,,6,17,12"></lb> |
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5226 Γ, ἴση δὲ ἡ Β τῇ Δ, ἔστιν ἄρα ὡς ἡ Α πρὸς τὴν Β, ἡ Δ πρὸς <lb n="1799,001,,,,6,17,13"></lb> |
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5227 τὴν Γ.</s> <s>ἐὰν δὲ τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, τὸ ὑπὸ <lb n="1799,001,,,,6,17,14"></lb> |
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5228 τῶν ἄκρων περιεχόμενον [ὀρθογώνιον] ἴσον ἐστὶ τῷ ὑπὸ <lb n="1799,001,,,,6,17,15"></lb> |
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5229 τῶν μέσων περιεχομένῳ ὀρθογωνίῳ.</s> <s>τὸ ἄρα ὑπὸ τῶν Α, Γ <lb n="1799,001,,,,6,17,16"></lb> |
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5230 ἴσον ἐστὶ τῷ ὑπὸ τῶν Β, Δ.</s> <s>ἀλλὰ τὸ ὑπὸ τῶν Β, Δ τὸ <lb n="1799,001,,,,6,17,17"></lb> |
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5231 ἀπὸ τῆς Β ἐστιν· ἴση γὰρ ἡ Β τῇ Δ· τὸ ἄρα ὑπὸ τῶν Α, Γ <lb n="1799,001,,,,6,17,18"></lb> |
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5232 περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ἀπὸ τῆς Β τετρα-<lb n="1799,001,,,,6,17,19"></lb> |
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5233 γώνῳ.</s> <lb n="1799,001,,,,6,17,20"></lb> |
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5234 <s>Ἀλλὰ δὴ τὸ ὑπὸ τῶν Α, Γ ἴσον ἔστω τῷ ἀπὸ τῆς Β· <lb n="1799,001,,,,6,17,21"></lb> |
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5235 λέγω, ὅτι ἐστὶν ὡς ἡ Α πρὸς τὴν Β, οὕτως ἡ Β πρὸς τὴν Γ.</s> <lb n="1799,001,,,,6,17,22"></lb> |
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5236 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεὶ τὸ ὑπὸ τῶν <lb n="1799,001,,,,6,17,23"></lb> |
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5237 Α, Γ ἴσον ἐστὶ τῷ ἀπὸ τῆς Β, ἀλλὰ τὸ ἀπὸ τῆς Β τὸ ὑπὸ <lb n="1799,001,,,,6,17,24"></lb> |
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5238 τῶν Β, Δ ἐστιν· ἴση γὰρ ἡ Β τῇ Δ· τὸ ἄρα ὑπὸ τῶν <lb n="1799,001,,,,6,17,25"></lb> |
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5239 Α, Γ ἴσον ἐστὶ τῷ ὑπὸ τῶν Β, Δ.</s> <s>ἐὰν δὲ τὸ ὑπὸ τῶν <lb n="1799,001,,,,6,17,26"></lb> |
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5240 ἄκρων ἴσον ᾖ τῷ ὑπὸ τῶν μέσων, αἱ τέσσαρες εὐθεῖαι <lb n="1799,001,,,,6,17,27"></lb> |
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5241 ἀνάλογόν εἰσιν.</s> <s>ἔστιν ἄρα ὡς ἡ Α πρὸς τὴν Β, οὕτως ἡ Δ <lb n="1799,001,,,,6,17,28"></lb> |
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5242 πρὸς τὴν Γ.</s> <s>ἴση δὲ ἡ Β τῇ Δ· ὡς ἄρα ἡ Α πρὸς τὴν Β, <lb n="1799,001,,,,6,17,29"></lb> |
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5243 οὕτως ἡ Β πρὸς τὴν Γ.</s> <lb n="1799,001,,,,6,17,30"></lb> |
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5244 <s>Ἐὰν ἄρα τρεῖς εὐθεῖαι ἀνάλογον ὦσιν, τὸ ὑπὸ τῶν ἄκρων <lb n="1799,001,,,,6,17,31"></lb> |
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5245 περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ἀπὸ τῆς μέσης <lb n="1799,001,,,,6,17,32"></lb> |
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5246 τετραγώνῳ· κἂν τὸ ὑπὸ τῶν ἄκρων περιεχόμενον ὀρθογώ-<lb n="1799,001,,,,6,17,33"></lb> |
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5247 νιον ἴσον ᾖ τῷ ἀπὸ τῆς μέσης τετραγώνῳ, αἱ τρεῖς εὐθεῖαι <lb n="1799,001,,,,6,17,34"></lb> |
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5248 ἀνάλογον ἔσονται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,17,35"></lb> |
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5249 <s>Ἀπὸ τῆς δοθείσης εὐθείας τῷ δοθέντι εὐθυγράμμῳ <lb n="1799,001,,,,6,18,1"></lb> |
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5250 ὅμοιόν τε καὶ ὁμοίως κείμενον εὐθύγραμμον ἀναγράψαι.</s> <s><pb></pb> <lb n="1799,001,,,,6,18,2"></lb> |
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5251 Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ δοθὲν εὐθύ-<lb n="1799,001,,,,6,18,3"></lb> |
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5252 γραμμον τὸ ΓΕ· δεῖ δὴ ἀπὸ τῆς ΑΒ εὐθείας τῷ ΓΕ <lb n="1799,001,,,,6,18,4"></lb> |
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5253 εὐθυγράμμῳ ὅμοιόν τε καὶ <lb n="1799,001,,,,6,18,5"></lb> |
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5254 ὁμοίως κείμενον εὐθύγραμμον <lb n="1799,001,,,,6,18,6"></lb> |
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5255 ἀναγράψαι.</s> <lb n="1799,001,,,,6,18,7"></lb> |
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5256 <s>Ἐπεζεύχθω ἡ ΔΖ, καὶ συν-<lb n="1799,001,,,,6,18,8"></lb> |
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5257 εστάτω πρὸς τῇ ΑΒ εὐθείᾳ <lb n="1799,001,,,,6,18,9"></lb> |
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5258 καὶ τοῖς πρὸς αὐτῇ σημείοις <lb n="1799,001,,,,6,18,10"></lb> |
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5259 τοῖς Α, Β τῇ μὲν πρὸς τῷ <lb n="1799,001,,,,6,18,11"></lb> |
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5260 Γ γωνίᾳ ἴση ἡ ὑπὸ ΗΑΒ, τῇ δὲ ὑπὸ ΓΔΖ ἴση ἡ ὑπὸ <lb n="1799,001,,,,6,18,12"></lb> |
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5261 ΑΒΗ.</s> <s>λοιπὴ ἄρα ἡ ὑπὸ ΓΖΔ τῇ ὑπὸ ΑΗΒ ἐστιν ἴση· <lb n="1799,001,,,,6,18,13"></lb> |
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5262 ἰσογώνιον ἄρα ἐστὶ τὸ ΖΓΔ τρίγωνον τῷ ΗΑΒ τριγώνῳ.</s> <lb n="1799,001,,,,6,18,14"></lb> |
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5263 <s>ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΖΔ πρὸς τὴν ΗΒ, οὕτως ἡ <lb n="1799,001,,,,6,18,15"></lb> |
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5264 ΖΓ πρὸς τὴν ΗΑ, καὶ ἡ ΓΔ πρὸς τὴν ΑΒ.</s> <s>πάλιν συν-<lb n="1799,001,,,,6,18,16"></lb> |
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5265 εστάτω πρὸς τῇ ΒΗ εὐθείᾳ καὶ τοῖς πρὸς αὐτῇ σημείοις <lb n="1799,001,,,,6,18,17"></lb> |
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5266 τοῖς Β, Η τῇ μὲν ὑπὸ ΔΖΕ γωνίᾳ ἴση ἡ ὑπὸ ΒΗΘ, τῇ <lb n="1799,001,,,,6,18,18"></lb> |
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5267 δὲ ὑπὸ ΖΔΕ ἴση ἡ ὑπὸ ΗΒΘ.</s> <s>λοιπὴ ἄρα ἡ πρὸς τῷ Ε <lb n="1799,001,,,,6,18,19"></lb> |
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5268 λοιπῇ τῇ πρὸς τῷ Θ ἐστιν ἴση· ἰσογώνιον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,6,18,20"></lb> |
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5269 ΖΔΕ τρίγωνον τῷ ΗΘΒ τριγώνῳ· ἀνάλογον ἄρα ἐστὶν <lb n="1799,001,,,,6,18,21"></lb> |
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5270 ὡς ἡ ΖΔ πρὸς τὴν ΗΒ, οὕτως ἡ ΖΕ πρὸς τὴν ΗΘ καὶ <lb n="1799,001,,,,6,18,22"></lb> |
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5271 ἡ ΕΔ πρὸς τὴν ΘΒ.</s> <s>ἐδείχθη δὲ καὶ ὡς ἡ ΖΔ πρὸς τὴν <lb n="1799,001,,,,6,18,23"></lb> |
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5272 ΗΒ, οὕτως ἡ ΖΓ πρὸς τὴν ΗΑ καὶ ἡ ΓΔ πρὸς τὴν ΑΒ· <lb n="1799,001,,,,6,18,24"></lb> |
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5273 καὶ ὡς ἄρα ἡ ΖΓ πρὸς τὴν ΑΗ, οὕτως ἥ τε ΓΔ πρὸς <lb n="1799,001,,,,6,18,25"></lb> |
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5274 τὴν ΑΒ καὶ ἡ ΖΕ πρὸς τὴν ΗΘ καὶ ἔτι ἡ ΕΔ πρὸς τὴν <pb></pb> <lb n="1799,001,,,,6,18,26"></lb> |
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5275 ΘΒ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ μὲν ὑπὸ ΓΖΔ γωνία τῇ ὑπὸ <lb n="1799,001,,,,6,18,27"></lb> |
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5276 ΑΗΒ, ἡ δὲ ὑπὸ ΔΖΕ τῇ ὑπὸ ΒΗΘ, ὅλη ἄρα ἡ ὑπὸ <lb n="1799,001,,,,6,18,28"></lb> |
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5277 ΓΖΕ ὅλῃ τῇ ὑπὸ ΑΗΘ ἐστιν ἴση.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,6,18,29"></lb> |
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5278 ἡ ὑπὸ ΓΔΕ τῇ ὑπὸ ΑΒΘ ἐστιν ἴση.</s> <s>ἔστι δὲ καὶ ἡ μὲν <lb n="1799,001,,,,6,18,30"></lb> |
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5279 πρὸς τῷ Γ τῇ πρὸς τῷ Α ἴση, ἡ δὲ πρὸς τῷ Ε τῇ πρὸς τῷ <lb n="1799,001,,,,6,18,31"></lb> |
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5280 Θ.</s> <s>ἰσογώνιον ἄρα ἐστὶ τὸ ΑΘ τῷ ΓΕ· καὶ τὰς περὶ τὰς <lb n="1799,001,,,,6,18,32"></lb> |
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5281 ἴσας γωνίας αὐτῶν πλευρὰς ἀνάλογον ἔχει· ὅμοιον ἄρα ἐστὶ <lb n="1799,001,,,,6,18,33"></lb> |
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5282 τὸ ΑΘ εὐθύγραμμον τῷ ΓΕ εὐθυγράμμῳ.</s> <lb n="1799,001,,,,6,18,34"></lb> |
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5283 <s>Ἀπὸ τῆς δοθείσης ἄρα εὐθείας τῆς ΑΒ τῷ δοθέντι <lb n="1799,001,,,,6,18,35"></lb> |
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5284 εὐθυγράμμῳ τῷ ΓΕ ὅμοιόν τε καὶ ὁμοίως κείμενον <lb n="1799,001,,,,6,18,36"></lb> |
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5285 εὐθύγραμμον ἀναγέγραπται τὸ ΑΘ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,6,18,37"></lb> |
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5286 <s>Τὰ ὅμοια τρίγωνα πρὸς ἄλληλα ἐν διπλασίονι λόγῳ ἐστὶ <lb n="1799,001,,,,6,19,1"></lb> |
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5287 τῶν ὁμολόγων πλευρῶν.</s> <lb n="1799,001,,,,6,19,2"></lb> |
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5288 <s>*̓́εστω ὅμοια τρίγωνα τὰ ΑΒΓ, ΔΕΖ ἴσην ἔχοντα τὴν <lb n="1799,001,,,,6,19,3"></lb> |
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5289 πρὸς τῷ Β γωνίαν τῇ πρὸς τῷ Ε, ὡς δὲ τὴν ΑΒ πρὸς τὴν <lb n="1799,001,,,,6,19,4"></lb> |
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5290 ΒΓ, οὕτως τὴν ΔΕ πρὸς <lb n="1799,001,,,,6,19,5"></lb> |
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5291 τὴν ΕΖ, ὥστε ὁμόλογον εἶναι <lb n="1799,001,,,,6,19,6"></lb> |
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5292 τὴν ΒΓ τῇ ΕΖ· λέγω, ὅτι <lb n="1799,001,,,,6,19,7"></lb> |
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5293 τὸ ΑΒΓ τρίγωνον πρὸς τὸ <lb n="1799,001,,,,6,19,8"></lb> |
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5294 ΔΕΖ τρίγωνον διπλασίονα <lb n="1799,001,,,,6,19,9"></lb> |
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5295 λόγον ἔχει ἤπερ ἡ ΒΓ πρὸς <lb n="1799,001,,,,6,19,10"></lb> |
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5296 τὴν ΕΖ.</s> <s><pb></pb> <lb n="1799,001,,,,6,19,11"></lb> |
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5297 Εἰλήφθω γὰρ τῶν ΒΓ, ΕΖ τρίτη ἀνάλογον ἡ ΒΗ, <lb n="1799,001,,,,6,19,12"></lb> |
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5298 ὥστε εἶναι ὡς τὴν ΒΓ πρὸς τὴν ΕΖ, οὕτως τὴν ΕΖ <lb n="1799,001,,,,6,19,13"></lb> |
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5299 πρὸς τὴν ΒΗ· καὶ ἐπεζεύχθω ἡ ΑΗ.</s> <lb n="1799,001,,,,6,19,14"></lb> |
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5300 <s>Ἐπεὶ οὖν ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως ἡ ΔΕ <lb n="1799,001,,,,6,19,15"></lb> |
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5301 πρὸς τὴν ΕΖ, ἐναλλὰξ ἄρα ἐστὶν ὡς ἡ ΑΒ πρὸς τὴν ΔΕ, <lb n="1799,001,,,,6,19,16"></lb> |
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5302 οὕτως ἡ ΒΓ πρὸς τὴν ΕΖ.</s> <s>ἀλλ' ὡς ἡ ΒΓ πρὸς ΕΖ, <lb n="1799,001,,,,6,19,17"></lb> |
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5303 οὕτως ἐστὶν ἡ ΕΖ πρὸς ΒΗ.</s> <s>καὶ ὡς ἄρα ἡ ΑΒ πρὸς <lb n="1799,001,,,,6,19,18"></lb> |
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5304 ΔΕ, οὕτως ἡ ΕΖ πρὸς ΒΗ· τῶν ΑΒΗ, ΔΕΖ ἄρα <lb n="1799,001,,,,6,19,19"></lb> |
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5305 τριγώνων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας <lb n="1799,001,,,,6,19,20"></lb> |
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5306 γωνίας.</s> <s>ὧν δὲ μίαν μιᾷ ἴσην ἐχόντων γωνίαν τριγώνων <lb n="1799,001,,,,6,19,21"></lb> |
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5307 ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας, ἴσα <lb n="1799,001,,,,6,19,22"></lb> |
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5308 ἐστὶν ἐκεῖνα.</s> <s>ἴσον ἄρα ἐστὶ τὸ ΑΒΗ τρίγωνον τῷ ΔΕΖ <lb n="1799,001,,,,6,19,23"></lb> |
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5309 τριγώνῳ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ ΒΓ πρὸς τὴν ΕΖ, οὕτως <lb n="1799,001,,,,6,19,24"></lb> |
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5310 ἡ ΕΖ πρὸς τὴν ΒΗ, ἐὰν δὲ τρεῖς εὐθεῖαι ἀνάλογον ὦσιν, <lb n="1799,001,,,,6,19,25"></lb> |
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5311 ἡ πρώτη πρὸς τὴν τρίτην διπλασίονα λόγον ἔχει ἤπερ <lb n="1799,001,,,,6,19,26"></lb> |
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5312 πρὸς τὴν δευτέραν, ἡ ΒΓ ἄρα πρὸς τὴν ΒΗ διπλασίονα <lb n="1799,001,,,,6,19,27"></lb> |
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5313 λόγον ἔχει ἤπερ ἡ ΓΒ πρὸς τὴν ΕΖ.</s> <s>ὡς δὲ ἡ ΓΒ πρὸς τὴν <lb n="1799,001,,,,6,19,28"></lb> |
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5314 ΒΗ, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΑΒΗ τρίγωνον· <lb n="1799,001,,,,6,19,29"></lb> |
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5315 καὶ τὸ ΑΒΓ ἄρα τρίγωνον πρὸς τὸ ΑΒΗ διπλασίονα <lb n="1799,001,,,,6,19,30"></lb> |
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5316 λόγον ἔχει ἤπερ ἡ ΒΓ πρὸς τὴν ΕΖ.</s> <s>ἴσον δὲ τὸ ΑΒΗ <lb n="1799,001,,,,6,19,31"></lb> |
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5317 τρίγωνον τῷ ΔΕΖ τριγώνῳ· καὶ τὸ ΑΒΓ ἄρα τρίγωνον <lb n="1799,001,,,,6,19,32"></lb> |
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5318 πρὸς τὸ ΔΕΖ τρίγωνον διπλασίονα λόγον ἔχει ἤπερ ἡ <lb n="1799,001,,,,6,19,33"></lb> |
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5319 ΒΓ πρὸς τὴν ΕΖ.</s> <lb n="1799,001,,,,6,19,34"></lb> |
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5320 <s>Τὰ ἄρα ὅμοια τρίγωνα πρὸς ἄλληλα ἐν διπλασίονι <lb n="1799,001,,,,6,19,35"></lb> |
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5321 λόγῳ ἐστὶ τῶν ὁμολόγων πλευρῶν· [ὅπερ ἔδει δεῖξαι].</s> <s><pb></pb> <lb n="1799,001,,,,6,19,36"></lb> |
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5322 Πόρισμα <lb n="1799,001,,,,6,19,37"></lb> |
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5323 Ἐκ δὴ τούτου φανερόν, ὅτι, ἐὰν τρεῖς εὐθεῖαι ἀνάλογον <lb n="1799,001,,,,6,19,38"></lb> |
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5324 ὦσιν, ἔστιν ὡς ἡ πρώτη πρὸς τὴν τρίτην, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,6,19,39"></lb> |
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5325 τῆς πρώτης εἶδος πρὸς τὸ ἀπὸ τῆς δευτέρας τὸ ὅμοιον καὶ <lb n="1799,001,,,,6,19,40"></lb> |
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5326 ὁμοίως ἀναγραφόμενον [ἐπείπερ ἐδείχθη, ὡς ἡ ΓΒ πρὸς <lb n="1799,001,,,,6,19,41"></lb> |
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5327 ΒΗ, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΑΒΗ τρίγωνον, <lb n="1799,001,,,,6,19,42"></lb> |
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5328 τουτέστι τὸ ΔΕΖ]· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,19,43"></lb> |
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5329 <s>Τὰ ὅμοια πολύγωνα εἴς τε ὅμοια τρίγωνα διαιρεῖται <lb n="1799,001,,,,6,20,1"></lb> |
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5330 καὶ εἰς ἴσα τὸ πλῆθος καὶ ὁμόλογα τοῖς ὅλοις, καὶ τὸ <lb n="1799,001,,,,6,20,2"></lb> |
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5331 πολύγωνον πρὸς τὸ πολύγωνον διπλασίονα λόγον ἔχει <lb n="1799,001,,,,6,20,3"></lb> |
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5332 ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον πλευράν.</s> <lb n="1799,001,,,,6,20,4"></lb> |
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5333 <s>Ἔστω ὅμοια πολύγωνα τὰ ΑΒΓΔΕ, ΖΗΘΚΛ, <lb n="1799,001,,,,6,20,5"></lb> |
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5334 ὁμόλογος δὲ ἔστω ἡ ΑΒ τῇ ΖΗ· λέγω, ὅτι τὰ ΑΒΓΔΕ, <pb></pb> <lb n="1799,001,,,,6,20,6"></lb> |
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5335 ΖΗΘΚΛ πολύγωνα εἴς τε ὅμοια τρίγωνα διαιρεῖται <lb n="1799,001,,,,6,20,7"></lb> |
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5336 καὶ εἰς ἴσα τὸ πλῆθος καὶ ὁμόλογα τοῖς ὅλοις, καὶ τὸ <lb n="1799,001,,,,6,20,8"></lb> |
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5337 ΑΒΓΔΕ πολύγωνον πρὸς τὸ ΖΗΘΚΛ πολύγωνον <lb n="1799,001,,,,6,20,9"></lb> |
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5338 διπλασίονα λόγον ἔχει ἤπερ ἡ ΑΒ πρὸς τὴν ΖΗ.</s> <lb n="1799,001,,,,6,20,10"></lb> |
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5339 <s>Ἐπεζεύχθωσαν αἱ ΒΕ, ΕΓ, <lb n="1799,001,,,,6,20,11"></lb> |
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5340 ΗΛ, ΛΘ.</s> <lb n="1799,001,,,,6,20,12"></lb> |
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5341 <s>Καὶ ἐπεὶ ὅμοιόν ἐστι τὸ <lb n="1799,001,,,,6,20,13"></lb> |
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5342 ΑΒΓΔΕ πολύγωνον τῷ <lb n="1799,001,,,,6,20,14"></lb> |
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5343 ΖΗΘΚΛ πολυγώνῳ, ἴση <lb n="1799,001,,,,6,20,15"></lb> |
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5344 ἐστὶν ἡ ὑπὸ ΒΑΕ γωνία τῇ <lb n="1799,001,,,,6,20,16"></lb> |
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5345 ὑπὸ ΗΖΛ.</s> <s>καί ἐστιν ὡς ἡ <lb n="1799,001,,,,6,20,17"></lb> |
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5346 ΒΑ πρὸς ΑΕ, οὕτως ἡ ΗΖ <lb n="1799,001,,,,6,20,18"></lb> |
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5347 πρὸς ΖΛ.</s> <s>ἐπεὶ οὖν δύο τρίγωνά ἐστι τὰ ΑΒΕ, ΖΗΛ μίαν <lb n="1799,001,,,,6,20,19"></lb> |
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5348 γωνίαν μιᾷ γωνίᾳ ἴσην ἔχοντα, περὶ δὲ τὰς ἴσας γωνίας τὰς <lb n="1799,001,,,,6,20,20"></lb> |
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5349 πλευρὰς ἀνάλογον, ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΕ τρίγωνον <lb n="1799,001,,,,6,20,21"></lb> |
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5350 τῷ ΖΗΛ τριγώνῳ· ὥστε καὶ ὅμοιον· ἴση ἄρα ἐστὶν ἡ <lb n="1799,001,,,,6,20,22"></lb> |
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5351 ὑπὸ ΑΒΕ γωνία τῇ ὑπὸ ΖΗΛ.</s> <s>ἔστι δὲ καὶ ὅλη ἡ ὑπὸ <lb n="1799,001,,,,6,20,23"></lb> |
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5352 ΑΒΓ ὅλῃ τῇ ὑπὸ ΖΗΘ ἴση διὰ τὴν ὁμοιότητα τῶν <lb n="1799,001,,,,6,20,24"></lb> |
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5353 πολυγώνων· λοιπὴ ἄρα ἡ ὑπὸ ΕΒΓ γωνία τῇ ὑπὸ ΛΗΘ <lb n="1799,001,,,,6,20,25"></lb> |
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5354 ἐστιν ἴση.</s> <s>καὶ ἐπεὶ διὰ τὴν ὁμοιότητα τῶν ΑΒΕ, ΖΗΛ <lb n="1799,001,,,,6,20,26"></lb> |
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5355 τριγώνων ἐστὶν ὡς ἡ ΕΒ πρὸς ΒΑ, οὕτως ἡ ΛΗ πρὸς <lb n="1799,001,,,,6,20,27"></lb> |
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5356 ΗΖ, ἀλλὰ μὴν καὶ διὰ τὴν ὁμοιότητα τῶν πολυγώνων <lb n="1799,001,,,,6,20,28"></lb> |
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5357 ἐστὶν ὡς ἡ ΑΒ πρὸς ΒΓ, οὕτως ἡ ΖΗ πρὸς ΗΘ, δι' <lb n="1799,001,,,,6,20,29"></lb> |
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5358 ἴσου ἄρα ἐστὶν ὡς ἡ ΕΒ πρὸσ ΒΓ, οὕτως ἡ ΛΗ πρὸς <lb n="1799,001,,,,6,20,30"></lb> |
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5359 ΗΘ, καὶ περὶ τὰς ἴσας γωνίας τὰς ὑπὸ ΕΒΓ, ΛΗΘ αἱ <pb></pb> <lb n="1799,001,,,,6,20,31"></lb> |
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5360 πλευραὶ ἀνάλογόν εἰσιν· ἰσογώνιον ἄρα ἐστὶ τὸ ΕΒΓ <lb n="1799,001,,,,6,20,32"></lb> |
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5361 τρίγωνον τῷ ΛΗΘ τριγώνῳ· ὥστε καὶ ὅμοιόν ἐστι τὸ <lb n="1799,001,,,,6,20,33"></lb> |
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5362 ΕΒΓ τρίγωνον τῷ ΛΗΘ τριγώνῳ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ <lb n="1799,001,,,,6,20,34"></lb> |
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5363 ΕΓΔ τρίγωνον ὅμοιόν ἐστι τῷ ΛΘΚ τριγώνῳ.</s> <s>τὰ ἄρα <lb n="1799,001,,,,6,20,35"></lb> |
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5364 ὅμοια πολύγωνα τὰ ΑΒΓΔΕ, ΖΗΘΚΛ εἴς τε ὅμοια <lb n="1799,001,,,,6,20,36"></lb> |
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5365 τρίγωνα διῄρηται καὶ εἰς ἴσα τὸ πλῆθος.</s> <lb n="1799,001,,,,6,20,37"></lb> |
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5366 <s>Λέγω, ὅτι καὶ ὁμόλογα τοῖς ὅλοις, τουτέστιν ὥστε <lb n="1799,001,,,,6,20,38"></lb> |
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5367 ἀνάλογον εἶναι τὰ τρίγωνα, καὶ ἡγούμενα μὲν εἶναι τὰ <lb n="1799,001,,,,6,20,39"></lb> |
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5368 ΑΒΕ, ΕΒΓ, ΕΓΔ, ἑπόμενα δὲ αὐτῶν τὰ ΖΗΛ, ΛΗΘ, <lb n="1799,001,,,,6,20,40"></lb> |
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5369 ΛΘΚ, καὶ ὅτι τὸ ΑΒΓΔΕ πολύγωνον πρὸς τὸ ΖΗΘΚΛ <lb n="1799,001,,,,6,20,41"></lb> |
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5370 πολύγωνον διπλασίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος <lb n="1799,001,,,,6,20,42"></lb> |
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5371 πλευρὰ πρὸς τὴν ὁμόλογον πλευράν, τουτέστιν ἡ ΑΒ πρὸς <lb n="1799,001,,,,6,20,43"></lb> |
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5372 τὴν ΖΗ.</s> <lb n="1799,001,,,,6,20,44"></lb> |
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5373 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΑΓ, ΖΘ.</s> <s>καὶ ἐπεὶ διὰ τὴν <lb n="1799,001,,,,6,20,45"></lb> |
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5374 ὁμοιότητα τῶν πολυγώνων ἴση ἐστὶν ἡ ὑπὸ ΑΒΓ γωνία <lb n="1799,001,,,,6,20,46"></lb> |
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5375 τῇ ὑπὸ ΖΗΘ, καί ἐστιν ὡς ἡ ΑΒ πρὸς ΒΓ, οὕτως ἡ ΖΗ <lb n="1799,001,,,,6,20,47"></lb> |
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5376 πρὸς ΗΘ, ἰσογώνιόν ἐστι τὸ ΑΒΓ τρίγωνον τῷ ΖΗΘ <lb n="1799,001,,,,6,20,48"></lb> |
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5377 τριγώνῳ· ἴση ἄρα ἐστὶν ἡ μὲν ὑπὸ ΒΑΓ γωνία τῇ ὑπὸ <lb n="1799,001,,,,6,20,49"></lb> |
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5378 ΗΖΘ, ἡ δὲ ὑπὸ ΒΓΑ τῇ ὑπὸ ΗΘΖ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν <pb></pb> <lb n="1799,001,,,,6,20,50"></lb> |
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5379 ἡ ὑπὸ ΒΑΜ γωνία τῇ ὑπὸ ΗΖΝ, ἔστι δὲ καὶ ἡ ὑπὸ ΑΒΜ <lb n="1799,001,,,,6,20,51"></lb> |
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5380 τῇ ὑπὸ ΖΗΝ ἴση, καὶ λοιπὴ ἄρα ἡ ὑπὸ ΑΜΒ λοιπῇ τῇ <lb n="1799,001,,,,6,20,52"></lb> |
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5381 ὑπὸ ΖΝΗ ἴση ἐστίν· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΜ <lb n="1799,001,,,,6,20,53"></lb> |
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5382 τρίγωνον τῷ ΖΗΝ τριγώνῳ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ <lb n="1799,001,,,,6,20,54"></lb> |
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5383 τὸ ΒΜΓ τρίγωνον ἰσογώνιόν ἐστι τῷ ΗΝΘ τριγώνῳ.</s> <lb n="1799,001,,,,6,20,55"></lb> |
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5384 <s>ἀνάλογον ἄρα ἐστίν, ὡς μὲν ἡ ΑΜ πρὸς ΜΒ, οὕτως ἡ <lb n="1799,001,,,,6,20,56"></lb> |
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5385 ΖΝ πρὸς ΝΗ, ὡς δὲ ἡ ΒΜ πρὸς ΜΓ, οὕτως ἡ ΗΝ <lb n="1799,001,,,,6,20,57"></lb> |
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5386 πρὸς ΝΘ· ὥστε καὶ δι' ἴσου, ὡς ἡ ΑΜ πρὸς ΜΓ, οὕτως <lb n="1799,001,,,,6,20,58"></lb> |
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5387 ἡ ΖΝ πρὸς ΝΘ.</s> <s>ἀλλ' ὡς ἡ ΑΜ πρὸς ΜΓ, οὕτως τὸ <lb n="1799,001,,,,6,20,59"></lb> |
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5388 ΑΒΜ [τρίγωνον] πρὸς τὸ ΜΒΓ, καὶ τὸ ΑΜΕ πρὸς τὸ <lb n="1799,001,,,,6,20,60"></lb> |
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5389 ΕΜΓ· πρὸς ἄλληλα γάρ εἰσιν ὡς αἱ βάσεις.</s> <s>καὶ ὡς ἄρα <lb n="1799,001,,,,6,20,61"></lb> |
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5390 ἓν τῶν ἡγουμένων πρὸς ἓν τῶν ἑπομένων, οὕτως ἅπαντα <lb n="1799,001,,,,6,20,62"></lb> |
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5391 τὰ ἡγούμενα πρὸς ἅπαντα τὰ ἑπόμενα· ὡς ἄρα τὸ ΑΜΒ <lb n="1799,001,,,,6,20,63"></lb> |
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5392 τρίγωνον πρὸς τὸ ΒΜΓ, οὕτως τὸ ΑΒΕ πρὸς τὸ ΓΒΕ.</s> <lb n="1799,001,,,,6,20,64"></lb> |
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5393 <s>ἀλλ' ὡς τὸ ΑΜΒ πρὸς τὸ ΒΜΓ, οὕτως ἡ ΑΜ πρὸς ΜΓ· <lb n="1799,001,,,,6,20,65"></lb> |
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5394 καὶ ὡς ἄρα ἡ ΑΜ πρὸς ΜΓ, οὕτως τὸ ΑΒΕ τρίγωνον <lb n="1799,001,,,,6,20,66"></lb> |
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5395 πρὸς τὸ ΕΒΓ τρίγωνον.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὡς ἡ ΖΝ <lb n="1799,001,,,,6,20,67"></lb> |
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5396 πρὸς ΝΘ, οὕτως τὸ ΖΗΛ τρίγωνον πρὸς τὸ ΗΛΘ <lb n="1799,001,,,,6,20,68"></lb> |
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5397 τρίγωνον.</s> <s>καί ἐστιν ὡς ἡ ΑΜ πρὸς ΜΓ, οὕτως ἡ ΖΝ <lb n="1799,001,,,,6,20,69"></lb> |
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5398 πρὸς ΝΘ· καὶ ὡς ἄρα τὸ ΑΒΕ τρίγωνον πρὸς τὸ ΒΕΓ <lb n="1799,001,,,,6,20,70"></lb> |
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5399 τρίγωνον, οὕτως τὸ ΖΗΛ τρίγωνον πρὸς τὸ ΗΛΘ <lb n="1799,001,,,,6,20,71"></lb> |
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5400 τρίγωνον, καὶ ἐναλλὰξ, ὡς τὸ ΑΒΕ τρίγωνον πρὸς τὸ <lb n="1799,001,,,,6,20,72"></lb> |
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5401 ΖΗΛ τρίγωνον, οὕτως τὸ ΒΕΓ τρίγωνον πρὸς τὸ ΗΛΘ <pb></pb> <lb n="1799,001,,,,6,20,73"></lb> |
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5402 τρίγωνον.</s> <s>ὁμοίως δὴ δείξομεν ἐπιζευχθεισῶν τῶν ΒΔ, <lb n="1799,001,,,,6,20,74"></lb> |
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5403 ΗΚ, ὅτι καὶ ὡς τὸ ΒΕΓ τρίγωνον πρὸς τὸ ΛΗΘ <lb n="1799,001,,,,6,20,75"></lb> |
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5404 τρίγωνον, οὕτως τὸ ΕΓΔ τρίγωνον πρὸς τὸ ΛΘΚ <lb n="1799,001,,,,6,20,76"></lb> |
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5405 τρίγωνον.</s> <s>καὶ ἐπεί ἐστιν ὡς τὸ ΑΒΕ τρίγωνον πρὸς τὸ <lb n="1799,001,,,,6,20,77"></lb> |
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5406 ΖΗΛ τρίγωνον, οὕτως τὸ ΕΒΓ πρὸς τὸ ΛΗΘ, καὶ ἔτι <lb n="1799,001,,,,6,20,78"></lb> |
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5407 τὸ ΕΓΔ πρὸς τὸ ΛΘΚ, καὶ ὡς ἄρα ἓν τῶν ἡγουμένων <lb n="1799,001,,,,6,20,79"></lb> |
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5408 πρὸς ἓν τῶν ἑπομένων, οὕτως ἅπαντα τὰ ἡγούμενα πρὸς <lb n="1799,001,,,,6,20,80"></lb> |
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5409 ἅπαντα τὰ ἑπόμενα· ἔστιν ἄρα ὡς τὸ ΑΒΕ τρίγωνον <lb n="1799,001,,,,6,20,81"></lb> |
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5410 πρὸς τὸ ΖΗΛ τρίγωνον, οὕτως τὸ ΑΒΓΔΕ πολύγωνον <lb n="1799,001,,,,6,20,82"></lb> |
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5411 πρὸς τὸ ΖΗΘΚΛ πολύγωνον.</s> <s>ἀλλὰ τὸ ΑΒΕ τρίγωνον <lb n="1799,001,,,,6,20,83"></lb> |
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5412 πρὸς τὸ ΖΗΛ τρίγωνον διπλασίονα λόγον ἔχει ἤπερ ἡ <lb n="1799,001,,,,6,20,84"></lb> |
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5413 ΑΒ ὁμόλογος πλευρὰ πρὸς τὴν ΖΗ ὁμόλογον πλευράν· <lb n="1799,001,,,,6,20,85"></lb> |
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5414 τὰ γὰρ ὅμοια τρίγωνα ἐν διπλασίονι λόγῳ ἐστὶ τῶν <lb n="1799,001,,,,6,20,86"></lb> |
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5415 ὁμολόγων πλευρῶν.</s> <s>καὶ τὸ ΑΒΓΔΕ ἄρα πολύγωνον πρὸς <lb n="1799,001,,,,6,20,87"></lb> |
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5416 τὸ ΖΗΘΚΛ πολύγωνον διπλασίονα λόγον ἔχει ἤπερ ἡ <lb n="1799,001,,,,6,20,88"></lb> |
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5417 ΑΒ ὁμόλογος πλευρὰ πρὸς τὴν ΖΗ ὁμόλογον πλευράν.</s> <lb n="1799,001,,,,6,20,89"></lb> |
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5418 <s>Τὰ ἄρα ὅμοια πολύγωνα εἴς τε ὅμοια τρίγωνα διαιρεῖται <lb n="1799,001,,,,6,20,90"></lb> |
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5419 καὶ εἰς ἴσα τὸ πλῆθος καὶ ὁμόλογα τοῖς ὅλοις, καὶ τὸ <lb n="1799,001,,,,6,20,91"></lb> |
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5420 πολύγωνον πρὸς τὸ πολύγωνον διπλασίονα λόγον ἔχει <lb n="1799,001,,,,6,20,92"></lb> |
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5421 ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον πλευράν· <lb n="1799,001,,,,6,20,93"></lb> |
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5422 [ὅπερ ἔδει δεῖξαι].</s> <lb n="1799,001,,,,6,20,94"></lb> |
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5423 <s>Πόρισμα <lb n="1799,001,,,,6,20,95"></lb> |
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5424 Ὡσαύτως δὲ καὶ ἐπὶ τῶν [ὁμοίων] τετραπλεύρων <lb n="1799,001,,,,6,20,96"></lb> |
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5425 δειχθήσεται, ὅτι ἐν διπλασίονι λόγῳ εἰσὶ τῶν ὁμολόγων <lb n="1799,001,,,,6,20,97"></lb> |
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5426 πλευρῶν.</s> <s>ἐδείχθη δὲ καὶ ἐπὶ τῶν τριγώνων· ὥστε καὶ καθ-<lb n="1799,001,,,,6,20,98"></lb> |
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5427 όλου τὰ ὅμοια εὐθύγραμμα σχήματα πρὸς ἄλληλα ἐν διπλα-<lb n="1799,001,,,,6,20,99"></lb> |
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5428 σίονι λόγῳ εἰσὶ τῶν ὁμολόγων πλευρῶν.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,6,20,100"></lb> |
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5429 [Πόρισμα β# <lb n="1799,001,,,,6,20,101"></lb> |
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5430 Καὶ ἐὰν τῶν ΑΒ, ΖΗ τρίτην ἀνάλογον λάβωμεν τὴν <lb n="1799,001,,,,6,20,102"></lb> |
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5431 Ξ, ἡ ΒΑ πρὸς τὴν Ξ διπλασίονα λόγον ἔχει ἤπερ ἡ ΑΒ <lb n="1799,001,,,,6,20,103"></lb> |
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5432 πρὸς τὴν ΖΗ.</s> <s>ἔχει δὲ καὶ τὸ πολύγωνον πρὸς τὸ πολύ-<lb n="1799,001,,,,6,20,104"></lb> |
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5433 γωνον ἢ τὸ τετράπλευρον πρὸς τὸ τετράπλευρον διπλα-<lb n="1799,001,,,,6,20,105"></lb> |
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5434 σίονα λόγον ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον <lb n="1799,001,,,,6,20,106"></lb> |
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5435 πλευράν, τουτέστιν ἡ ΑΒ πρὸς τὴν ΖΗ· ἐδείχθη δὲ τοῦτο <lb n="1799,001,,,,6,20,107"></lb> |
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5436 καὶ ἐπὶ τῶν τριγώνων· ὥστε καὶ καθόλου φανερόν, ὅτι, <lb n="1799,001,,,,6,20,108"></lb> |
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5437 ἐὰν τρεῖς εὐθεῖαι ἀνάλογον ὦσιν, ἔσται ὡς ἡ πρώτη πρὸς <lb n="1799,001,,,,6,20,109"></lb> |
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5438 τὴν τρίτην, οὕτως τὸ ἀπὸ τῆς πρώτης εἶδος πρὸς τὸ ἀπὸ <lb n="1799,001,,,,6,20,110"></lb> |
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5439 τῆς δευτέρας τὸ ὅμοιον καὶ ὁμοίως ἀναγραφόμενον.] <lb n="1799,001,,,,6,20,111"></lb> |
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5440 Τὰ τῷ αὐτῷ εὐθυγράμμῳ ὅμοια καὶ ἀλλήλοις ἐστὶν <lb n="1799,001,,,,6,21,1"></lb> |
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5441 ὅμοια.</s> <lb n="1799,001,,,,6,21,2"></lb> |
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5442 <s>Ἔστω γὰρ ἑκάτερον τῶν Α, Β εὐθυγράμμων τῷ Γ <lb n="1799,001,,,,6,21,3"></lb> |
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5443 ὅμοιον· λέγω, ὅτι καὶ τὸ Α τῷ Β ἐστιν ὅμοιον.</s> <lb n="1799,001,,,,6,21,4"></lb> |
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5444 <s>Ἐπεὶ γὰρ ὅμοιόν ἐστι τὸ Α τῷ Γ, ἰσογώνιόν τέ ἐστιν <lb n="1799,001,,,,6,21,5"></lb> |
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5445 αὐτῷ καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον <lb n="1799,001,,,,6,21,6"></lb> |
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5446 ἔχει.</s> <s>πάλιν, ἐπεὶ ὅμοιόν ἐστι τὸ Β τῷ Γ, ἰσογώνιόν τέ <pb></pb> <lb n="1799,001,,,,6,21,7"></lb> |
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5447 ἐστιν αὐτῷ καὶ τὰς περὶ τὰς ἴσας γωνίας <lb n="1799,001,,,,6,21,8"></lb> |
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5448 πλευρὰς ἀνάλογον ἔχει.</s> <s>ἑκάτερον ἄρα <lb n="1799,001,,,,6,21,9"></lb> |
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5449 τῶν Α, Β τῷ Γ ἰσογώνιόν τέ ἐστι καὶ <lb n="1799,001,,,,6,21,10"></lb> |
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5450 τὰς περὶ τὰς ἴσας γωνίας πλευρὰς <lb n="1799,001,,,,6,21,11"></lb> |
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5451 ἀνάλογον ἔχει [ὥστε καὶ τὸ Α τῷ Β <lb n="1799,001,,,,6,21,12"></lb> |
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5452 ἰσογώνιόν τέ ἐστι καὶ τὰς περὶ τὰς ἴσας <lb n="1799,001,,,,6,21,13"></lb> |
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5453 γωνίας πλευρὰς ἀνάλογον ἔχει].</s> <s>ὅμοιον <lb n="1799,001,,,,6,21,14"></lb> |
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5454 ἄρα ἐστὶ τὸ Α τῷ Β· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,21,15"></lb> |
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5455 <s>Ἐὰν τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, καὶ τὰ ἀπ' <lb n="1799,001,,,,6,22,1"></lb> |
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5456 αὐτῶν εὐθύγραμμα ὅμοιά τε καὶ ὁμοίως ἀναγεγραμμένα <lb n="1799,001,,,,6,22,2"></lb> |
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5457 ἀνάλογον ἔσται· κἂν τὰ ἀπ' αὐτῶν εὐθύγραμμα ὅμοιά τε <lb n="1799,001,,,,6,22,3"></lb> |
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5458 καὶ ὁμοίως ἀναγεγραμμένα ἀνάλογον ᾖ, καὶ αὐταὶ αἱ <lb n="1799,001,,,,6,22,4"></lb> |
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5459 εὐθεῖαι ἀνάλογον ἔσονται.</s> <lb n="1799,001,,,,6,22,5"></lb> |
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5460 <s>Ἔστωσαν τέσσαρες εὐθεῖαι ἀνάλογον αἱ ΑΒ, ΓΔ, <lb n="1799,001,,,,6,22,6"></lb> |
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5461 ΕΖ, ΗΘ, ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν <lb n="1799,001,,,,6,22,7"></lb> |
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5462 ΗΘ, καὶ ἀναγεγράφθωσαν ἀπὸ μὲν τῶν <lb n="1799,001,,,,6,22,8"></lb> |
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5463 ΑΒ, ΓΔ ὅμοιά τε καὶ ὁμοίως κείμενα <lb n="1799,001,,,,6,22,9"></lb> |
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5464 εὐθύγραμμα τὰ ΚΑΒ, ΛΓΔ, ἀπὸ δὲ <lb n="1799,001,,,,6,22,10"></lb> |
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5465 τῶν ΕΖ, ΗΘ ὅμοιά τε καὶ ὁμοίως κεί-<lb n="1799,001,,,,6,22,11"></lb> |
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5466 μενα εὐθύγραμμα τὰ ΜΖ, ΝΘ· λέγω, <lb n="1799,001,,,,6,22,12"></lb> |
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5467 ὅτι ἐστὶν ὡς τὸ ΚΑΒ πρὸς τὸ ΛΓΔ, <lb n="1799,001,,,,6,22,13"></lb> |
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5468 οὕτως τὸ ΜΖ πρὸς τὸ ΝΘ.</s> <s><pb></pb> <lb n="1799,001,,,,6,22,14"></lb> |
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5469 Εἰλήφθω γὰρ τῶν μὲν ΑΒ, ΓΔ τρίτη ἀνάλογον ἡ Ξ, <lb n="1799,001,,,,6,22,15"></lb> |
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5470 τῶν δὲ ΕΖ, ΗΘ τρίτη ἀνάλογον ἡ Ο.</s> <s>καὶ ἐπεί ἐστιν ὡς <lb n="1799,001,,,,6,22,16"></lb> |
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5471 μὲν ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΗΘ, ὡς <lb n="1799,001,,,,6,22,17"></lb> |
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5472 δὲ ἡ ΓΔ πρὸς τὴν Ξ, οὕτως ἡ ΗΘ πρὸς τὴν Ο, δι' ἴσου <lb n="1799,001,,,,6,22,18"></lb> |
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5473 ἄρα ἐστὶν ὡς ἡ ΑΒ πρὸς τὴν Ξ, οὕτως ἡ ΕΖ <lb n="1799,001,,,,6,22,19"></lb> |
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5474 πρὸς τὴν Ο.</s> <s>ἀλλ' ὡς μὲν ἡ ΑΒ πρὸς τὴν Ξ, <lb n="1799,001,,,,6,22,20"></lb> |
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5475 οὕτως [καὶ] τὸ ΚΑΒ πρὸς τὸ ΛΓΔ, ὡς δὲ ἡ <lb n="1799,001,,,,6,22,21"></lb> |
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5476 ΕΖ πρὸς τὴν Ο, οὕτως τὸ ΜΖ πρὸς τὸ ΝΘ· <lb n="1799,001,,,,6,22,22"></lb> |
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5477 καὶ ὡς ἄρα τὸ ΚΑΒ πρὸς τὸ ΛΓΔ, οὕτως <lb n="1799,001,,,,6,22,23"></lb> |
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5478 τὸ ΜΖ πρὸς τὸ ΝΘ.</s> <lb n="1799,001,,,,6,22,24"></lb> |
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5479 <s>Ἀλλὰ δὴ ἔστω ὡς τὸ ΚΑΒ πρὸς τὸ ΛΓΔ, οὕτως τὸ <lb n="1799,001,,,,6,22,25"></lb> |
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5480 ΜΖ πρὸς τὸ ΝΘ· λέγω, ὅτι ἐστὶ καὶ ὡς ἡ ΑΒ πρὸς τὴν <lb n="1799,001,,,,6,22,26"></lb> |
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5481 ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΗΘ.</s> <s>εἰ γὰρ μή ἐστιν, ὡς ἡ <lb n="1799,001,,,,6,22,27"></lb> |
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5482 ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΗΘ, ἔστω ὡς <lb n="1799,001,,,,6,22,28"></lb> |
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5483 ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΠΡ, καὶ <lb n="1799,001,,,,6,22,29"></lb> |
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5484 ἀναγεγράφθω ἀπὸ τῆς ΠΡ ὁποτέρῳ τῶν ΜΖ, ΝΘ <lb n="1799,001,,,,6,22,30"></lb> |
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5485 ὅμοιόν τε καὶ ὁμοίως κείμενον εὐθύγραμμον τὸ ΣΡ.</s> <lb n="1799,001,,,,6,22,31"></lb> |
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5486 <s>Ἐπεὶ οὖν ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ <lb n="1799,001,,,,6,22,32"></lb> |
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5487 πρὸς τὴν ΠΡ, καὶ ἀναγέγραπται ἀπὸ μὲν τῶν ΑΒ, ΓΔ <lb n="1799,001,,,,6,22,33"></lb> |
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5488 ὅμοιά τε καὶ ὁμοίως κείμενα τὰ ΚΑΒ, ΛΓΔ, ἀπὸ δὲ τῶν <lb n="1799,001,,,,6,22,34"></lb> |
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5489 ΕΖ, ΠΡ ὅμοιά τε καὶ ὁμοίως κείμενα τὰ ΜΖ, ΣΡ, ἔστιν <lb n="1799,001,,,,6,22,35"></lb> |
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5490 ἄρα ὡς τὸ ΚΑΒ πρὸς τὸ ΛΓΔ, οὕτως τὸ ΜΖ πρὸς τὸ <pb></pb> <lb n="1799,001,,,,6,22,36"></lb> |
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5491 ΣΡ.</s> <s>ὑπόκειται δὲ καὶ ὡς τὸ ΚΑΒ πρὸς τὸ ΛΓΔ, οὕτως <lb n="1799,001,,,,6,22,37"></lb> |
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5492 τὸ ΜΖ πρὸς τὸ ΝΘ· καὶ ὡς ἄρα τὸ ΜΖ πρὸς τὸ ΣΡ, <lb n="1799,001,,,,6,22,38"></lb> |
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5493 οὕτως τὸ ΜΖ πρὸς τὸ ΝΘ.</s> <s>τὸ ΜΖ ἄρα πρὸς ἑκάτερον <lb n="1799,001,,,,6,22,39"></lb> |
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5494 τῶν ΝΘ, ΣΡ τὸν αὐτὸν ἔχει λόγον· ἴσον ἄρα ἐστὶ τὸ ΝΘ <lb n="1799,001,,,,6,22,40"></lb> |
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5495 τῷ ΣΡ.</s> <s>ἔστι δὲ αὐτῷ καὶ ὅμοιον καὶ ὁμοίως κείμενον· <lb n="1799,001,,,,6,22,41"></lb> |
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5496 ἴση ἄρα ἡ ΗΘ τῇ ΠΡ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν <lb n="1799,001,,,,6,22,42"></lb> |
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5497 ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΠΡ, ἴση δὲ ἡ ΠΡ τῇ ΗΘ, <lb n="1799,001,,,,6,22,43"></lb> |
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5498 ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς <lb n="1799,001,,,,6,22,44"></lb> |
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5499 τὴν ΗΘ.</s> <lb n="1799,001,,,,6,22,45"></lb> |
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5500 <s>Ἐὰν ἄρα τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, καὶ τὰ ἀπ' <lb n="1799,001,,,,6,22,46"></lb> |
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5501 αὐτῶν εὐθύγραμμα ὅμοιά τε καὶ ὁμοίως ἀναγεγραμμένα <lb n="1799,001,,,,6,22,47"></lb> |
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5502 ἀνάλογον ἔσται· κἂν τὰ ἀπ' αὐτῶν εὐθύγραμμα ὅμοιά τε <lb n="1799,001,,,,6,22,48"></lb> |
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5503 καὶ ὁμοίως ἀναγεγραμμένα ἀνάλογον ᾖ, καὶ αὐταὶ αἱ <lb n="1799,001,,,,6,22,49"></lb> |
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5504 εὐθεῖαι ἀνάλογον ἔσονται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,22,50"></lb> |
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5505 <s>[Λῆμμα] <lb n="1799,001,,,,6,22,51"></lb> |
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5506 [Ὅτι δέ, ἐὰν εὐθύγραμμα ἴσα ᾖ καὶ ὅμοια, αἱ ὁμόλογοι <lb n="1799,001,,,,6,22,52"></lb> |
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5507 αὐτῶν πλευραὶ ἴσαι ἀλλήλαις εἰσίν, δείξομεν οὕτως.</s> <lb n="1799,001,,,,6,22,53"></lb> |
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5508 <s>Ἔστω ἴσα καὶ ὅμοια εὐθύγραμμα τὰ ΝΘ, ΣΡ, καὶ <lb n="1799,001,,,,6,22,54"></lb> |
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5509 ἔστω ὡς ἡ ΘΗ πρὸς τὴν ΗΝ, οὕτως ἡ ΡΠ πρὸς τὴν <lb n="1799,001,,,,6,22,55"></lb> |
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5510 ΠΣ· λέγω, ὅτι ἴση ἐστὶν ἡ ΡΠ τῇ ΘΗ.</s> <lb n="1799,001,,,,6,22,56"></lb> |
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5511 <s>Εἰ γὰρ ἄνισοί εἰσιν, μία αὐτῶν μείζων ἐστίν.</s> <s>ἔστω <pb></pb> <lb n="1799,001,,,,6,22,57"></lb> |
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5512 μείζων ἡ ΡΠ τῆς ΘΗ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ ΡΠ πρὸς <lb n="1799,001,,,,6,22,58"></lb> |
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5513 ΠΣ, οὕτως ἡ ΘΗ πρὸς τὴν ΗΝ, καὶ ἐναλλάξ, ὡς ἡ ΡΠ <lb n="1799,001,,,,6,22,59"></lb> |
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5514 πρὸς τὴν ΘΗ, οὕτως ἡ ΠΣ πρὸς τὴν ΗΝ, μείζων δὲ ἡ <lb n="1799,001,,,,6,22,60"></lb> |
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5515 ΠΡ τῆς ΘΗ, μείζων ἄρα καὶ ἡ ΠΣ τῆς ΗΝ· ὥστε καὶ <lb n="1799,001,,,,6,22,61"></lb> |
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5516 τὸ ΡΣ μεῖζόν ἐστι τοῦ ΘΝ.</s> <s>ἀλλὰ καὶ ἴσον· ὅπερ ἀδύνατον.</s> <lb n="1799,001,,,,6,22,62"></lb> |
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5517 <s>οὐκ ἄρα ἄνισός ἐστιν ἡ ΠΡ τῇ ΗΘ· ἴση ἄρα· ὅπερ ἔδει <lb n="1799,001,,,,6,22,63"></lb> |
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5518 δεῖξαι.] <lb n="1799,001,,,,6,22,64"></lb> |
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5519 Τὰ ἰσογώνια παραλληλόγραμμα πρὸς ἄλληλα λόγον <lb n="1799,001,,,,6,23,1"></lb> |
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5520 ἔχει τὸν συγκείμενον ἐκ τῶν πλευρῶν.</s> <lb n="1799,001,,,,6,23,2"></lb> |
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5521 <s>Ἔστω ἰσογώνια παραλληλό- <lb n="1799,001,,,,6,23,3"></lb> |
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5522 γραμμα τὰ ΑΓ, ΓΖ ἴσην <lb n="1799,001,,,,6,23,4"></lb> |
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5523 ἔχοντα τὴν ὑπὸ ΒΓΔ γωνίαν <lb n="1799,001,,,,6,23,5"></lb> |
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5524 τῇ ὑπὸ ΕΓΗ· λέγω, ὅτι τὸ <lb n="1799,001,,,,6,23,6"></lb> |
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5525 ΑΓ παραλληλόγραμμον πρὸς <lb n="1799,001,,,,6,23,7"></lb> |
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5526 τὸ ΓΖ παραλληλόγραμμον <lb n="1799,001,,,,6,23,8"></lb> |
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5527 λόγον ἔχει τὸν συγκείμενον ἐκ <lb n="1799,001,,,,6,23,9"></lb> |
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5528 τῶν πλευρῶν.</s> <lb n="1799,001,,,,6,23,10"></lb> |
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5529 <s>Κείσθω γὰρ ὥστε ἐπ' εὐθείας εἶναι τὴν ΒΓ τῇ ΓΗ· <lb n="1799,001,,,,6,23,11"></lb> |
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5530 ἐπ' εὐθείας ἄρα ἐστὶ καὶ ἡ ΔΓ τῇ ΓΕ.</s> <s>καὶ συμπε-<lb n="1799,001,,,,6,23,12"></lb> |
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5531 πληρώσθω τὸ ΔΗ παραλληλόγραμμον, καὶ ἐκκείσθω τις <pb></pb> <lb n="1799,001,,,,6,23,13"></lb> |
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5532 εὐθεῖα ἡ Κ, καὶ γεγονέτω ὡς μὲν ἡ ΒΓ πρὸς τὴν ΓΗ, <lb n="1799,001,,,,6,23,14"></lb> |
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5533 οὕτως ἡ Κ πρὸς τὴν Λ, ὡς δὲ ἡ ΔΓ πρὸς τὴν ΓΕ, οὕτως <lb n="1799,001,,,,6,23,15"></lb> |
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5534 ἡ Λ πρὸς τὴν Μ.</s> <lb n="1799,001,,,,6,23,16"></lb> |
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5535 <s>Οἱ ἄρα λόγοι τῆς τε Κ πρὸς τὴν Λ καὶ τῆς Λ πρὸς τὴν <lb n="1799,001,,,,6,23,17"></lb> |
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5536 Μ οἱ αὐτοί εἰσι τοῖς λόγοις τῶν πλευρῶν, τῆς τε ΒΓ πρὸς <lb n="1799,001,,,,6,23,18"></lb> |
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5537 τὴν ΓΗ καὶ τῆς ΔΓ πρὸς τὴν ΓΕ.</s> <s>ἀλλ' ὁ τῆς Κ πρὸς Μ <lb n="1799,001,,,,6,23,19"></lb> |
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5538 λόγος σύγκειται ἔκ τε τοῦ τῆς Κ πρὸς Λ λόγου καὶ τοῦ <lb n="1799,001,,,,6,23,20"></lb> |
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5539 τῆς Λ πρὸς Μ· ὥστε καὶ ἡ Κ πρὸς τὴν Μ λόγον ἔχει τὸν <lb n="1799,001,,,,6,23,21"></lb> |
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5540 συγκείμενον ἐκ τῶν πλευρῶν.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ ΒΓ <lb n="1799,001,,,,6,23,22"></lb> |
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5541 πρὸς τὴν ΓΗ, οὕτως τὸ ΑΓ παραλληλόγραμμον πρὸς τὸ <lb n="1799,001,,,,6,23,23"></lb> |
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5542 ΓΘ, ἀλλ' ὡς ἡ ΒΓ πρὸς τὴν ΓΗ, οὕτως ἡ Κ πρὸς τὴν <lb n="1799,001,,,,6,23,24"></lb> |
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5543 Λ, καὶ ὡς ἄρα ἡ Κ πρὸς τὴν Λ, οὕτως τὸ ΑΓ πρὸς τὸ ΓΘ.</s> <lb n="1799,001,,,,6,23,25"></lb> |
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5544 <s>πάλιν, ἐπεί ἐστιν ὡς ἡ ΔΓ πρὸς τὴν ΓΕ, οὕτως τὸ ΓΘ <lb n="1799,001,,,,6,23,26"></lb> |
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5545 παραλληλόγραμμον πρὸς τὸ ΓΖ, ἀλλ' ὡς ἡ ΔΓ πρὸς τὴν <lb n="1799,001,,,,6,23,27"></lb> |
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5546 ΓΕ, οὕτως ἡ Λ πρὸς τὴν Μ, καὶ ὡς ἄρα ἡ Λ πρὸς τὴν Μ, <lb n="1799,001,,,,6,23,28"></lb> |
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5547 οὕτως τὸ ΓΘ παραλληλόγραμμον πρὸς τὸ ΓΖ παραλλη-<lb n="1799,001,,,,6,23,29"></lb> |
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5548 λόγραμμον.</s> <s>ἐπεὶ οὖν ἐδείχθη, ὡς μὲν ἡ Κ πρὸς τὴν Λ, <lb n="1799,001,,,,6,23,30"></lb> |
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5549 οὕτως τὸ ΑΓ παραλληλόγραμμον πρὸς τὸ ΓΘ παραλ-<lb n="1799,001,,,,6,23,31"></lb> |
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5550 ληλόγραμμον, ὡς δὲ ἡ Λ πρὸς τὴν Μ, οὕτως τὸ ΓΘ <lb n="1799,001,,,,6,23,32"></lb> |
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5551 παραλληλόγραμμον πρὸς τὸ ΓΖ παραλληλόγραμμον, δι' <lb n="1799,001,,,,6,23,33"></lb> |
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5552 ἴσου ἄρα ἐστὶν ὡς ἡ Κ πρὸς τὴν Μ, οὕτως τὸ ΑΓ πρὸς τὸ <lb n="1799,001,,,,6,23,34"></lb> |
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5553 ΓΖ παραλληλόγραμμον.</s> <s>ἡ δὲ Κ πρὸς τὴν Μ λόγον ἔχει <lb n="1799,001,,,,6,23,35"></lb> |
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5554 τὸν συγκείμενον ἐκ τῶν πλευρῶν· καὶ τὸ ΑΓ ἄρα πρὸς τὸ <lb n="1799,001,,,,6,23,36"></lb> |
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5555 ΓΖ λόγον ἔχει τὸν συγκείμενον ἐκ τῶν πλευρῶν.</s> <s><pb></pb> <lb n="1799,001,,,,6,23,37"></lb> |
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5556 Τὰ ἄρα ἰσογώνια παραλληλόγραμμα πρὸς ἄλληλα λόγον <lb n="1799,001,,,,6,23,38"></lb> |
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5557 ἔχει τὸν συγκείμενον ἐκ τῶν πλευρῶν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,23,39"></lb> |
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5558 <s>Παντὸς παραλληλογράμμου τὰ περὶ τὴν διάμετρον <lb n="1799,001,,,,6,24,1"></lb> |
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5559 παραλληλόγραμμα ὅμοιά ἐστι τῷ τε ὅλῳ καὶ ἀλλήλοις.</s> <lb n="1799,001,,,,6,24,2"></lb> |
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5560 <s>Ἔστω παραλληλόγραμμον τὸ ΑΒΓΔ, διάμετρος δὲ <lb n="1799,001,,,,6,24,3"></lb> |
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5561 αὐτοῦ ἡ ΑΓ, περὶ δὲ τὴν ΑΓ παραλληλόγραμμα ἔστω <lb n="1799,001,,,,6,24,4"></lb> |
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5562 τὰ ΕΗ, ΘΚ· λέγω, ὅτι ἑκάτερον τῶν ΕΗ, ΘΚ παραλλη-<lb n="1799,001,,,,6,24,5"></lb> |
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5563 λογράμμων ὅμοιόν ἐστι ὅλῳ τῷ ΑΒΓΔ καὶ ἀλλήλοις.</s> <lb n="1799,001,,,,6,24,6"></lb> |
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5564 <s>Ἐπεὶ γὰρ τριγώνου τοῦ ΑΒΓ <lb n="1799,001,,,,6,24,7"></lb> |
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5565 παρὰ μίαν τῶν πλευρῶν τὴν ΒΓ <lb n="1799,001,,,,6,24,8"></lb> |
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5566 ἦκται ἡ ΕΖ, ἀνάλογόν ἐστιν ὡς ἡ <lb n="1799,001,,,,6,24,9"></lb> |
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5567 ΒΕ πρὸς τὴν ΕΑ, οὕτως, ἡ ΓΖ <lb n="1799,001,,,,6,24,10"></lb> |
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5568 πρὸς τὴν ΖΑ.</s> <s>πάλιν, ἐπεὶ τριγώ-<lb n="1799,001,,,,6,24,11"></lb> |
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5569 νου τοῦ ΑΓΔ παρὰ μίαν τὴν ΓΔ <lb n="1799,001,,,,6,24,12"></lb> |
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5570 ἦκται ἡ ΖΗ, ἀνάλογόν ἐστιν ὡς <lb n="1799,001,,,,6,24,13"></lb> |
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5571 ἡ ΓΖ πρὸς τὴν ΖΑ, οὕτως ἡ ΔΗ πρὸς τὴν ΗΑ.</s> <s>ἀλλ' <lb n="1799,001,,,,6,24,14"></lb> |
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5572 ὡς ἡ ΓΖ πρὸς τὴν ΖΑ, οὕτως ἐδείχθη καὶ ἡ ΒΕ πρὸς τὴν <lb n="1799,001,,,,6,24,15"></lb> |
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5573 ΕΑ· καὶ ὡς ἄρα ἡ ΒΕ πρὸς τὴν ΕΑ, οὕτως ἡ ΔΗ πρὸς <lb n="1799,001,,,,6,24,16"></lb> |
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5574 τὴν ΗΑ, καὶ συνθέντι ἄρα ὡς ἡ ΒΑ πρὸς ΑΕ, οὕτως <lb n="1799,001,,,,6,24,17"></lb> |
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5575 ἡ ΔΑ πρὸς ΑΗ, καὶ ἐναλλὰξ ὡς ἡ ΒΑ πρὸς τὴν ΑΔ, <lb n="1799,001,,,,6,24,18"></lb> |
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5576 οὕτως ἡ ΕΑ πρὸς τὴν ΑΗ.</s> <s>τῶν ἄρα ΑΒΓΔ, ΕΗ παραλ- <pb></pb><lb n="1799,001,,,,6,24,19"></lb> |
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5577 ληλογράμμων ἀνάλογόν εἰσιν αἱ πλευραὶ αἱ περὶ τὴν <lb n="1799,001,,,,6,24,20"></lb> |
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5578 κοινὴν γωνίαν τὴν ὑπὸ ΒΑΔ.</s> <s>καὶ ἐπεὶ παράλληλός ἐστιν <lb n="1799,001,,,,6,24,21"></lb> |
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5579 ἡ ΗΖ τῇ ΔΓ, ἴση ἐστὶν ἡ μὲν ὑπὸ ΑΖΗ γωνία τῇ ὑπὸ <lb n="1799,001,,,,6,24,22"></lb> |
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5580 ΔΓΑ· καὶ κοινὴ τῶν δύο τριγώνων τῶν ΑΔΓ, ΑΗΖ <lb n="1799,001,,,,6,24,23"></lb> |
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5581 ἡ ὑπὸ ΔΑΓ γωνία· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΔΓ τρίγωνον <lb n="1799,001,,,,6,24,24"></lb> |
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5582 τῷ ΑΗΖ τριγώνῳ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ ΑΓΒ τρίγωνον <lb n="1799,001,,,,6,24,25"></lb> |
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5583 ἰσογώνιόν ἐστι τῷ ΑΖΕ τριγώνῳ, καὶ ὅλον τὸ ΑΒΓΔ <lb n="1799,001,,,,6,24,26"></lb> |
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5584 παραλληλόγραμμον τῷ ΕΗ παραλληλογράμμῳ ἰσογώνιόν <lb n="1799,001,,,,6,24,27"></lb> |
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5585 ἐστιν.</s> <s>ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΑΔ πρὸς τὴν ΔΓ, οὕτως <lb n="1799,001,,,,6,24,28"></lb> |
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5586 ἡ ΑΗ πρὸς τὴν ΗΖ, ὡς δὲ ἡ ΔΓ πρὸς τὴν ΓΑ, οὕτως ἡ <lb n="1799,001,,,,6,24,29"></lb> |
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5587 ΗΖ πρὸς τὴν ΖΑ, ὡς δὲ ἡ ΑΓ πρὸς τὴν ΓΒ, οὕτως ἡ <lb n="1799,001,,,,6,24,30"></lb> |
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5588 ΑΖ πρὸς τὴν ΖΕ, καὶ ἔτι ὡς ἡ ΓΒ πρὸς τὴν ΒΑ, οὕτως <lb n="1799,001,,,,6,24,31"></lb> |
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5589 ἡ ΖΕ πρὸς τὴν ΕΑ.</s> <s>καὶ ἐπεὶ ἐδείχθη ὡς μὲν ἡ ΔΓ πρὸς <lb n="1799,001,,,,6,24,32"></lb> |
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5590 τὴν ΓΑ, οὕτως ἡ ΗΖ πρὸς τὴν ΖΑ, ὡς δὲ ἡ ΑΓ πρὸς τὴν <lb n="1799,001,,,,6,24,33"></lb> |
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5591 ΓΒ, οὕτως ἡ ΑΖ πρὸς τὴν ΖΕ, δι' ἴσου ἄρα ἐστὶν ὡς ἡ <lb n="1799,001,,,,6,24,34"></lb> |
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5592 ΔΓ πρὸς τὴν ΓΒ, οὕτως ἡ ΗΖ πρὸς τὴν ΖΕ.</s> <s>τῶν ἄρα <lb n="1799,001,,,,6,24,35"></lb> |
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5593 ΑΒΓΔ, ΕΗ παραλληλογράμμων ἀνάλογόν εἰσιν αἱ <lb n="1799,001,,,,6,24,36"></lb> |
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5594 πλευραὶ αἱ περὶ τὰς ἴσας γωνίας· ὅμοιον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,6,24,37"></lb> |
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5595 ΑΒΓΔ παραλληλόγραμμον τῷ ΕΗ παραλληλογράμμῳ.</s> <s><pb></pb> <lb n="1799,001,,,,6,24,38"></lb> |
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5596 διὰ τὰ αὐτὰ δὴ τὸ ΑΒΓΔ παραλληλόγραμμον καὶ τῷ ΚΘ <lb n="1799,001,,,,6,24,39"></lb> |
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5597 παραλληλογράμμῳ ὅμοιόν ἐστιν· ἑκάτερον ἄρα τῶν ΕΗ, <lb n="1799,001,,,,6,24,40"></lb> |
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5598 ΘΚ παραλληλογράμμων τῷ ΑΒΓΔ [παραλληλογράμμῳ] <lb n="1799,001,,,,6,24,41"></lb> |
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5599 ὅμοιόν ἐστιν.</s> <s>τὰ δὲ τῷ αὐτῷ εὐθυγράμμῳ ὅμοια καὶ <lb n="1799,001,,,,6,24,42"></lb> |
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5600 ἀλλήλοις ἐστὶν ὅμοια· καὶ τὸ ΕΗ ἄρα παραλληλόγραμμον <lb n="1799,001,,,,6,24,43"></lb> |
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5601 τῷ ΘΚ παραλληλογράμμῳ ὅμοιόν ἐστιν.</s> <lb n="1799,001,,,,6,24,44"></lb> |
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5602 <s>Παντὸς ἄρα παραλληλογράμμου τὰ περὶ τὴν διάμετρον <lb n="1799,001,,,,6,24,45"></lb> |
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5603 παραλληλόγραμμα ὅμοιά ἐστι τῷ τε ὅλῳ καὶ ἀλλήλοις· <lb n="1799,001,,,,6,24,46"></lb> |
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5604 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,24,47"></lb> |
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5605 <s>Τῷ δοθέντι εὐθυγράμμῳ ὅμοιον καὶ ἄλλῳ τῷ δοθέντι <lb n="1799,001,,,,6,25,1"></lb> |
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5606 ἴσον τὸ αὐτὸ συστήσασθαι.</s> <lb n="1799,001,,,,6,25,2"></lb> |
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5607 <s>Ἔστω τὸ μὲν δοθὲν εὐθύγραμμον, ᾧ δεῖ ὅμοιον συστή-<lb n="1799,001,,,,6,25,3"></lb> |
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5608 σασθαι, τὸ ΑΒΓ, ᾧ δὲ δεῖ ἴσον, τὸ Δ· δεῖ δὴ τῷ μὲν <lb n="1799,001,,,,6,25,4"></lb> |
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5609 ΑΒΓ ὅμοιον, τῷ δὲ Δ ἴσον τὸ αὐτὸ συστήσασθαι.</s> <lb n="1799,001,,,,6,25,5"></lb> |
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5610 <s>Παραβεβλήσθω γὰρ παρὰ μὲν τὴν ΒΓ τῷ ΑΒΓ τρι-<lb n="1799,001,,,,6,25,6"></lb> |
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5611 γώνῳ ἴσον παραλληλόγραμμον τὸ ΒΕ, παρὰ δὲ τὴν ΓΕ τῷ <lb n="1799,001,,,,6,25,7"></lb> |
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5612 Δ ἴσον παραλληλόγραμμον τὸ ΓΜ ἐν γωνίᾳ τῇ ὑπὸ ΖΓΕ, <pb></pb> <lb n="1799,001,,,,6,25,8"></lb> |
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5613 ἥ ἐστιν ἴση τῇ ὑπὸ ΓΒΛ.</s> <s>ἐπ' εὐθείας ἄρα ἐστὶν ἡ μὲν ΒΓ <lb n="1799,001,,,,6,25,9"></lb> |
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5614 τῇ ΓΖ, ἡ δὲ ΛΕ τῇ ΕΜ.</s> <s>καὶ εἰλήφθω τῶν ΒΓ, ΓΖ μέση <lb n="1799,001,,,,6,25,10"></lb> |
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5615 ἀνάλογον ἡ ΗΘ, καὶ ἀναγεγράφθω ἀπὸ τῆς ΗΘ τῷ ΑΒΓ <lb n="1799,001,,,,6,25,11"></lb> |
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5616 ὅμοιόν τε καὶ ὁμοίως κείμενον τὸ ΚΗΘ.</s> <lb n="1799,001,,,,6,25,12"></lb> |
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5617 <s>Καὶ ἐπεί ἐστιν ὡς ἡ ΒΓ πρὸς τὴν ΗΘ, οὕτως ἡ ΗΘ <lb n="1799,001,,,,6,25,13"></lb> |
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5618 πρὸς τὴν ΓΖ, ἐὰν δὲ τρεῖς εὐθεῖαι ἀνάλογον ὦσιν, ἔστιν <lb n="1799,001,,,,6,25,14"></lb> |
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5619 ὡς ἡ πρώτη πρὸς τὴν τρίτην, οὕτως τὸ ἀπὸ τῆς πρώτης <lb n="1799,001,,,,6,25,15"></lb> |
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5620 εἶδος πρὸς τὸ ἀπὸ τῆς δευτέρας τὸ ὅμοιον καὶ ὁμοίως <lb n="1799,001,,,,6,25,16"></lb> |
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5621 ἀναγραφόμενον, ἔστιν ἄρα ὡς ἡ ΒΓ πρὸς τὴν ΓΖ, οὕτως <lb n="1799,001,,,,6,25,17"></lb> |
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5622 τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΚΗΘ τρίγωνον.</s> <s>ἀλλὰ καὶ <lb n="1799,001,,,,6,25,18"></lb> |
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5623 ὡς ἡ ΒΓ πρὸς τὴν ΓΖ, οὕτως τὸ ΒΕ παραλληλόγραμμον <lb n="1799,001,,,,6,25,19"></lb> |
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5624 πρὸς τὸ ΕΖ παραλληλόγραμμον.</s> <s>καὶ ὡς ἄρα τὸ ΑΒΓ <lb n="1799,001,,,,6,25,20"></lb> |
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5625 τρίγωνον πρὸς τὸ ΚΗΘ τρίγωνον, οὕτως τὸ ΒΕ παραλ-<lb n="1799,001,,,,6,25,21"></lb> |
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5626 ληλόγραμμον πρὸς τὸ ΕΖ παραλληλόγραμμον· ἐναλλὰξ <lb n="1799,001,,,,6,25,22"></lb> |
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5627 ἄρα ὡς τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΒΕ παραλληλό-<lb n="1799,001,,,,6,25,23"></lb> |
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5628 γραμμον, οὕτως τὸ ΚΗΘ τρίγωνον πρὸς τὸ ΕΖ παραλ-<lb n="1799,001,,,,6,25,24"></lb> |
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5629 ληλόγραμμον.</s> <s>ἴσον δὲ τὸ ΑΒΓ τρίγωνον τῷ ΒΕ παραλ-<lb n="1799,001,,,,6,25,25"></lb> |
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5630 ληλογράμμῳ· ἴσον ἄρα καὶ τὸ ΚΗΘ τρίγωνον τῷ ΕΖ <lb n="1799,001,,,,6,25,26"></lb> |
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5631 παραλληλογράμμῳ.</s> <s>ἀλλὰ τὸ ΕΖ παραλληλόγραμμον τῷ <lb n="1799,001,,,,6,25,27"></lb> |
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5632 Δ ἐστιν ἴσον· καὶ τὸ ΚΗΘ ἄρα τῷ Δ ἐστιν ἴσον.</s> <s>ἔστι δὲ <lb n="1799,001,,,,6,25,28"></lb> |
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5633 τὸ ΚΗΘ καὶ τῷ ΑΒΓ ὅμοιον.</s> <lb n="1799,001,,,,6,25,29"></lb> |
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5634 <s>Τῷ ἄρα δοθέντι εὐθυγράμμῳ τῷ ΑΒΓ ὅμοιον καὶ <lb n="1799,001,,,,6,25,30"></lb> |
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5635 ἄλλῳ τῷ δοθέντι τῷ Δ ἴσον τὸ αὐτὸ συνέσταται τὸ <lb n="1799,001,,,,6,25,31"></lb> |
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5636 ΚΗΘ· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,6,25,32"></lb> |
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5637 Ἐὰν ἀπὸ παραλληλογράμμου παραλληλόγραμμον ἀφ-<lb n="1799,001,,,,6,26,1"></lb> |
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5638 αιρεθῇ ὅμοιόν τε τῷ ὅλῳ καὶ ὁμοίως κείμενον κοινὴν <lb n="1799,001,,,,6,26,2"></lb> |
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5639 γωνίαν ἔχον αὐτῷ, περὶ τὴν αὐτὴν διάμετρόν ἐστι τῷ ὅλῳ.</s> <lb n="1799,001,,,,6,26,3"></lb> |
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5640 <s>Ἀπὸ γὰρ παραλληλογράμ- <lb n="1799,001,,,,6,26,4"></lb> |
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5641 μου τοῦ ΑΒΓΔ παραλληλό-<lb n="1799,001,,,,6,26,5"></lb> |
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5642 γραμμον ἀφῃρήσθω τὸ ΑΖ <lb n="1799,001,,,,6,26,6"></lb> |
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5643 ὅμοιον τῷ ΑΒΓΔ καὶ ὁμοίως <lb n="1799,001,,,,6,26,7"></lb> |
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5644 κείμενον κοινὴν γωνίαν ἔχον <lb n="1799,001,,,,6,26,8"></lb> |
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5645 αὐτῷ τὴν ὑπὸ ΔΑΒ· λέγω, <lb n="1799,001,,,,6,26,9"></lb> |
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5646 ὅτι περὶ τὴν αὐτὴν διάμετρόν <lb n="1799,001,,,,6,26,10"></lb> |
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5647 ἐστι τὸ ΑΒΓΔ τῷ ΑΖ.</s> <lb n="1799,001,,,,6,26,11"></lb> |
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5648 <s>Μὴ γάρ, ἀλλ' εἰ δυνατόν, ἔστω [αὐτῶν] διάμετρος ἡ <lb n="1799,001,,,,6,26,12"></lb> |
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5649 ΑΘΓ, καὶ ἐκβληθεῖσα ἡ ΗΖ διήχθω ἐπὶ τὸ Θ, καὶ <lb n="1799,001,,,,6,26,13"></lb> |
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5650 ἤχθω διὰ τοῦ Θ ὁποτέρᾳ τῶν ΑΔ, ΒΓ παράλληλος <lb n="1799,001,,,,6,26,14"></lb> |
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5651 ἡ ΘΚ.</s> <lb n="1799,001,,,,6,26,15"></lb> |
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5652 <s>Ἐπεὶ οὖν περὶ τὴν αὐτὴν διάμετρόν ἐστι τὸ ΑΒΓΔ <lb n="1799,001,,,,6,26,16"></lb> |
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5653 τῷ ΚΗ, ἔστιν ἄρα ὡς ἡ ΔΑ πρὸς τὴν ΑΒ, οὕτως ἡ ΗΑ <lb n="1799,001,,,,6,26,17"></lb> |
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5654 πρὸς τὴν ΑΚ.</s> <s>ἔστι δὲ καὶ διὰ τὴν ὁμοιότητα τῶν ΑΒΓΔ, <lb n="1799,001,,,,6,26,18"></lb> |
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5655 ΕΗ καὶ ὡς ἡ ΔΑ πρὸς τὴν ΑΒ, οὕτως ἡ ΗΑ πρὸς τὴν <pb></pb> <lb n="1799,001,,,,6,26,19"></lb> |
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5656 ΑΕ· καὶ ὡς ἄρα ἡ ΗΑ πρὸς τὴν ΑΚ, οὕτως ἡ ΗΑ πρὸς <lb n="1799,001,,,,6,26,20"></lb> |
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5657 τὴν ΑΕ.</s> <s>ἡ ΗΑ ἄρα πρὸς ἑκατέραν τῶν ΑΚ, ΑΕ τὸν <lb n="1799,001,,,,6,26,21"></lb> |
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5658 αὐτὸν ἔχει λόγον.</s> <s>ἴση ἄρα ἐστὶν ἡ ΑΕ τῇ ΑΚ ἡ ἐλάττων <lb n="1799,001,,,,6,26,22"></lb> |
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5659 τῇ μείζονι· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα οὔκ ἐστι περὶ <lb n="1799,001,,,,6,26,23"></lb> |
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5660 τὴν αὐτὴν διάμετρον τὸ ΑΒΓΔ τῷ ΑΖ· περὶ τὴν αὐτὴν <lb n="1799,001,,,,6,26,24"></lb> |
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5661 ἄρα ἐστὶ διάμετρον τὸ ΑΒΓΔ παραλληλόγραμμον τῷ <lb n="1799,001,,,,6,26,25"></lb> |
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5662 ΑΖ παραλληλογράμμῳ.</s> <lb n="1799,001,,,,6,26,26"></lb> |
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5663 <s>Ἐὰν ἄρα ἀπὸ παραλληλογράμμου παραλληλόγραμμον <lb n="1799,001,,,,6,26,27"></lb> |
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5664 ἀφαιρεθῇ ὅμοιόν τε τῷ ὅλῳ καὶ ὁμοίως κείμενον κοινὴν <lb n="1799,001,,,,6,26,28"></lb> |
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5665 γωνίαν ἔχον αὐτῷ, περὶ τὴν αὐτὴν διάμετρόν ἐστι τῷ <lb n="1799,001,,,,6,26,29"></lb> |
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5666 ὅλῳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,26,30"></lb> |
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5667 <s>Πάντων τῶν παρὰ τὴν αὐτὴν εὐθεῖαν παραβαλλομένων <lb n="1799,001,,,,6,27,1"></lb> |
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5668 παραλληλογράμμων καὶ ἐλλειπόντων εἴδεσι παραλληλο-<lb n="1799,001,,,,6,27,2"></lb> |
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5669 γράμμοις ὁμοίοις τε καὶ ὁμοίως κειμένοις τῷ ἀπὸ τῆς <lb n="1799,001,,,,6,27,3"></lb> |
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5670 ἡμισείας ἀναγραφομένῳ μέγιστόν ἐστι τὸ ἀπὸ τῆς <lb n="1799,001,,,,6,27,4"></lb> |
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5671 ἡμισείας παραβαλλόμενον [παραλληλόγραμμον] ὅμοιον ὂν <lb n="1799,001,,,,6,27,5"></lb> |
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5672 τῷ ἐλλείμματι.</s> <lb n="1799,001,,,,6,27,6"></lb> |
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5673 <s>Ἔστω εὐθεῖα ἡ ΑΒ καὶ τετμήσθω δίχα κατὰ τὸ Γ, καὶ <lb n="1799,001,,,,6,27,7"></lb> |
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5674 παραβεβλήσθω παρὰ τὴν ΑΒ εὐθεῖαν τὸ ΑΔ παραλληλό-<lb n="1799,001,,,,6,27,8"></lb> |
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5675 γραμμον ἐλλεῖπον εἴδει παραλληλογράμμῳ τῷ ΔΒ ἀνα- <pb></pb><lb n="1799,001,,,,6,27,9"></lb> |
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5676 γραφέντι ἀπὸ τῆς ἡμισείας τῆς ΑΒ, τουτέστι τῆς ΓΒ· <lb n="1799,001,,,,6,27,10"></lb> |
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5677 λέγω, ὅτι πάντων τῶν παρὰ τὴν ΑΒ παραβαλλομένων <lb n="1799,001,,,,6,27,11"></lb> |
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5678 παραλληλογράμμων καὶ ἐλλειπόντων εἴδεσι [παραλληλο-<lb n="1799,001,,,,6,27,12"></lb> |
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5679 γράμμοισ] ὁμοίοις τε καὶ ὁμοίως κειμένοις τῷ ΔΒ μέγιστόν <lb n="1799,001,,,,6,27,13"></lb> |
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5680 ἐστι τὸ ΑΔ.</s> <s>παραβεβλήσθω γὰρ παρὰ <lb n="1799,001,,,,6,27,14"></lb> |
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5681 τὴν ΑΒ εὐθεῖαν τὸ ΑΖ παραλληλό-<lb n="1799,001,,,,6,27,15"></lb> |
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5682 γραμμον ἐλλεῖπον εἴδει παραλληλο-<lb n="1799,001,,,,6,27,16"></lb> |
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5683 γράμμῳ τῷ ΖΒ ὁμοίῳ τε καὶ ὁμοίως <lb n="1799,001,,,,6,27,17"></lb> |
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5684 κειμένῳ τῷ ΔΒ· λέγω, ὅτι μεῖζόν <lb n="1799,001,,,,6,27,18"></lb> |
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5685 ἐστι τὸ ΑΔ τοῦ ΑΖ.</s> <lb n="1799,001,,,,6,27,19"></lb> |
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5686 <s>Ἐπεὶ γὰρ ὅμοιόν ἐστι τὸ ΔΒ παραλληλόγραμμον τῷ <lb n="1799,001,,,,6,27,20"></lb> |
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5687 ΖΒ παραλληλογράμμῳ, περὶ τὴν αὐτήν εἰσι διάμετρον.</s> <lb n="1799,001,,,,6,27,21"></lb> |
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5688 <s>ἤχθω αὐτῶν διάμετρος ἡ ΔΒ, καὶ καταγεγράφθω τὸ <lb n="1799,001,,,,6,27,22"></lb> |
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5689 σχῆμα.</s> <lb n="1799,001,,,,6,27,23"></lb> |
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5690 <s>Ἐπεὶ οὖν ἴσον ἐστὶ τὸ ΓΖ τῷ ΖΕ, κοινὸν δὲ τὸ ΖΒ, <lb n="1799,001,,,,6,27,24"></lb> |
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5691 ὅλον ἄρα τὸ ΓΘ ὅλῳ τῷ ΚΕ ἐστιν ἴσον.</s> <s>ἀλλὰ τὸ ΓΘ τῷ <lb n="1799,001,,,,6,27,25"></lb> |
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5692 ΓΗ ἐστιν ἴσον, ἐπεὶ καὶ ἡ ΑΓ τῇ ΓΒ.</s> <s>καὶ τὸ ΗΓ ἄρα τῷ <lb n="1799,001,,,,6,27,26"></lb> |
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5693 ΕΚ ἐστιν ἴσον.</s> <s>κοινὸν προσκείσθω τὸ ΓΖ· ὅλον ἄρα τὸ <lb n="1799,001,,,,6,27,27"></lb> |
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5694 ΑΖ τῷ ΛΜΝ γνώμονί ἐστιν ἴσον· ὥστε τὸ ΔΒ παραλλη-<lb n="1799,001,,,,6,27,28"></lb> |
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5695 λόγραμμον, τουτέστι τὸ ΑΔ, τοῦ ΑΖ παραλληλογράμμου <lb n="1799,001,,,,6,27,29"></lb> |
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5696 μεῖζόν ἐστιν.</s> <lb n="1799,001,,,,6,27,30"></lb> |
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5697 <s>Πάντων ἄρα τῶν παρὰ τὴν αὐτὴν εὐθεῖαν παρα-<lb n="1799,001,,,,6,27,31"></lb> |
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5698 βαλλομένων παραλληλογράμμων καὶ ἐλλειπόντων εἴδεσι <lb n="1799,001,,,,6,27,32"></lb> |
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5699 παραλληλογράμμοις ὁμοίοις τε καὶ ὁμοίως κειμένοις τῷ <pb></pb> <lb n="1799,001,,,,6,27,33"></lb> |
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5700 ἀπὸ τῆς ἡμισείας ἀναγραφομένῳ μέγιστόν ἐστι τὸ ἀπὸ τῆς <lb n="1799,001,,,,6,27,34"></lb> |
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5701 ἡμισείας παραβληθέν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,6,27,35"></lb> |
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5702 <s>Παρὰ τὴν δοθεῖσαν εὐθεῖαν τῷ δοθέντι εὐθυγράμμῳ <lb n="1799,001,,,,6,28,1"></lb> |
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5703 ἴσον παραλληλόγραμμον παραβαλεῖν ἐλλεῖπον εἴδει παρ-<lb n="1799,001,,,,6,28,2"></lb> |
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5704 αλληλογράμμῳ ὁμοίῳ τῷ δοθέντι· δεῖ δὲ τὸ διδόμενον <lb n="1799,001,,,,6,28,3"></lb> |
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5705 εὐθύγραμμον [ᾧ δεῖ ἴσον παραβαλεῖν] μὴ μεῖζον εἶναι τοῦ <lb n="1799,001,,,,6,28,4"></lb> |
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5706 ἀπὸ τῆς ἡμισείας ἀναγραφομένου ὁμοίου τῷ ἐλλείμματι <lb n="1799,001,,,,6,28,5"></lb> |
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5707 [τοῦ τε ἀπὸ τῆς ἡμισείας καὶ ᾧ δεῖ ὅμοιον ἐλλείπειν].</s> <lb n="1799,001,,,,6,28,6"></lb> |
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5708 <s>Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ δοθὲν εὐθύ-<lb n="1799,001,,,,6,28,7"></lb> |
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5709 γραμμον, ᾧ δεῖ ἴσον παρὰ τὴν ΑΒ παραβαλεῖν, τὸ Γ μὴ <lb n="1799,001,,,,6,28,8"></lb> |
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5710 μεῖζον [ὂν] τοῦ ἀπὸ τῆς ἡμισείας τῆς ΑΒ ἀναγραφομένου <pb></pb> <lb n="1799,001,,,,6,28,9"></lb> |
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5711 ὁμοίου τῷ ἐλλείμματι, ᾧ δὲ δεῖ ὅμοιον ἐλλείπειν, τὸ Δ· <lb n="1799,001,,,,6,28,10"></lb> |
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5712 δεῖ δὴ παρὰ τὴν δοθεῖσαν εὐθεῖαν τὴν ΑΒ τῷ δοθέντι <lb n="1799,001,,,,6,28,11"></lb> |
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5713 εὐθυγράμμῳ τῷ Γ ἴσον παραλληλόγραμμον παραβαλεῖν <lb n="1799,001,,,,6,28,12"></lb> |
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5714 ἐλλεῖπον εἴδει παραλληλογράμμῳ ὁμοίῳ ὄντι τῷ Δ. <lb n="1799,001,,,,6,28,13"></lb> |
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5715 Τετμήσθω ἡ ΑΒ δίχα κατὰ τὸ Ε σημεῖον, καὶ ἀνα-<lb n="1799,001,,,,6,28,14"></lb> |
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5716 γεγράφθω ἀπὸ τῆς ΕΒ τῷ Δ ὅμοιον καὶ ὁμοίως κείμενον <lb n="1799,001,,,,6,28,15"></lb> |
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5717 τὸ ΕΒΖΗ, καὶ συμπεπληρώσθω τὸ ΑΗ παραλληλό-<lb n="1799,001,,,,6,28,16"></lb> |
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5718 γραμμον.</s> <lb n="1799,001,,,,6,28,17"></lb> |
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5719 <s>Εἰ μὲν οὖν ἴσον ἐστὶ τὸ ΑΗ τῷ Γ, γεγονὸς ἂν εἴη τὸ <lb n="1799,001,,,,6,28,18"></lb> |
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5720 ἐπιταχθέν· παραβέβληται γὰρ παρὰ τὴν δοθεῖσαν εὐθεῖαν <lb n="1799,001,,,,6,28,19"></lb> |
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5721 τὴν ΑΒ τῳ̂ δοθέντι εὐθυγράμμῳ τῷ Γ ἴσον παραλληλό-<lb n="1799,001,,,,6,28,20"></lb> |
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5722 γραμμον τὸ ΑΗ ἐλλεῖπον εἴδει παραλληλογράμμῳ τῷ ΗΒ <lb n="1799,001,,,,6,28,21"></lb> |
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5723 ὁμοίῳ ὄντι τῷ Δ.</s> <s>εἰ δὲ οὔ, μεῖζον ἔστω τὸ ΘΕ τοῦ Γ.</s> <s>ἴσον <lb n="1799,001,,,,6,28,22"></lb> |
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5724 δὲ τὸ ΘΕ τῷ ΗΒ· μεῖζον ἄρα καὶ τὸ ΗΒ τοῦ Γ.</s> <s>ᾧ δὴ <lb n="1799,001,,,,6,28,23"></lb> |
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5725 μεῖζόν ἐστι τὸ ΗΒ τοῦ Γ, ταύτῃ τῇ ὑπεροχῇ ἴσον, τῷ δὲ Δ <lb n="1799,001,,,,6,28,24"></lb> |
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5726 ὅμοιον καὶ ὁμοίως κείμενον τὸ αὐτὸ συνεστάτω τὸ ΚΛΜΝ.</s> <s><pb></pb> <lb n="1799,001,,,,6,28,25"></lb> |
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5727 ἀλλὰ τὸ Δ τῷ ΗΒ [ἐστιν] ὅμοιον· καὶ τὸ ΚΜ ἄρα τῷ <lb n="1799,001,,,,6,28,26"></lb> |
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5728 ΗΒ ἐστιν ὅμοιον.</s> <s>ἔστω οὖν ὁμόλογος ἡ μὲν ΚΛ τῇ ΗΕ, <lb n="1799,001,,,,6,28,27"></lb> |
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5729 ἡ δὲ ΛΜ τῇ ΗΖ.</s> <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΗΒ τοῖς Γ, ΚΜ, <lb n="1799,001,,,,6,28,28"></lb> |
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5730 μεῖζον ἄρα ἐστὶ τὸ ΗΒ τοῦ ΚΜ· μείζων ἄρα ἐστὶ καὶ <lb n="1799,001,,,,6,28,29"></lb> |
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5731 ἡ μὲν ΗΕ τῆς ΚΛ, ἡ δὲ ΗΖ τῆς ΛΜ.</s> <s>κείσθω τῇ μὲν ΚΛ <lb n="1799,001,,,,6,28,30"></lb> |
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5732 ἴση ἡ ΗΞ, τῇ δὲ ΛΜ ἴση ἡ ΗΟ, καὶ συμπεπληρώσθω τὸ <lb n="1799,001,,,,6,28,31"></lb> |
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5733 ΞΗΟΠ παραλληλόγραμμον· ἴσον ἄρα καὶ ὅμοιόν ἐστι <lb n="1799,001,,,,6,28,32"></lb> |
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5734 [τὸ ΗΠ] τῷ ΚΜ [ἀλλὰ τὸ ΚΜ τῷ ΗΒ ὅμοιόν ἐστιν].</s> <lb n="1799,001,,,,6,28,33"></lb> |
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5735 <s>καὶ τὸ ΗΠ ἄρα τῷ ΗΒ ὅμοιόν ἐστιν· περὶ τὴν αὐτὴν ἄρα <lb n="1799,001,,,,6,28,34"></lb> |
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5736 διάμετρόν ἐστι τὸ ΗΠ τῷ ΗΒ.</s> <s>ἔστω αὐτῶν διάμετρος <lb n="1799,001,,,,6,28,35"></lb> |
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5737 ἡ ΗΠΒ, καὶ καταγεγράφθω τὸ σχῆμα.</s> <lb n="1799,001,,,,6,28,36"></lb> |
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5738 <s>Ἐπεὶ οὖν ἴσον ἐστὶ τὸ ΒΗ τοῖς Γ, ΚΜ, ὧν τὸ ΗΠ τῷ <lb n="1799,001,,,,6,28,37"></lb> |
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5739 ΚΜ ἐστιν ἴσον, λοιπὸς ἄρα ὁ ΥΧΦ γνώμων λοιπῷ τῷ <lb n="1799,001,,,,6,28,38"></lb> |
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5740 Γ ἴσος ἐστίν.</s> <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΟΡ τῷ ΞΣ, κοινὸν <lb n="1799,001,,,,6,28,39"></lb> |
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5741 προσκείσθω τὸ Π*β· ὅλον ἄρα τὸ ΟΒ ὅλῳ τῷ ΞΒ ἴσον <lb n="1799,001,,,,6,28,40"></lb> |
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5742 ἐστίν.</s> <s>ἀλλὰ τὸ ΞΒ τῷ ΤΕ ἐστιν ἴσον, ἐπεὶ καὶ πλευρὰ ἡ <lb n="1799,001,,,,6,28,41"></lb> |
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5743 ΑΕ πλευρᾷ τῇ ΕΒ ἐστιν ἴση· καὶ τὸ ΤΕ ἄρα τῷ ΟΒ ἐστιν <lb n="1799,001,,,,6,28,42"></lb> |
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5744 ἴσον.</s> <s>κοινὸν προσκείσθω τὸ ΞΣ· ὅλον ἄρα τὸ ΤΣ ὅλῳ τῷ <lb n="1799,001,,,,6,28,43"></lb> |
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5745 ΦΧΥ γνώμονί ἐστιν ἴσον.</s> <s>ἀλλ' ὁ ΦΧΥ γνώμων τῷ Γ <lb n="1799,001,,,,6,28,44"></lb> |
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5746 ἐδείχθη ἴσος· καὶ τὸ ΤΣ ἄρα τῷ Γ ἐστιν ἴσον.</s> <s><pb></pb> <lb n="1799,001,,,,6,28,45"></lb> |
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5747 Παρὰ τὴν δοθεῖσαν ἄρα εὐθεῖαν τὴν ΑΒ τῷ δοθέντι <lb n="1799,001,,,,6,28,46"></lb> |
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5748 εὐθυγράμμῳ τῷ Γ ἴσον παραλληλόγραμμον παραβέβληται <lb n="1799,001,,,,6,28,47"></lb> |
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5749 τὸ ΣΤ ἐλλεῖπον εἴδει παραλληλογράμμῳ τῷ ΠΒ ὁμοίῳ <lb n="1799,001,,,,6,28,48"></lb> |
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5750 ὄντι τῷ Δ [ἐπειδήπερ τὸ ΠΒ τῷ ΗΠ ὅμοιόν ἐστιν]· ὅπερ <lb n="1799,001,,,,6,28,49"></lb> |
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5751 ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,6,28,50"></lb> |
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5752 <s>Παρὰ τὴν δοθεῖσαν εὐθεῖαν τῷ δοθέντι εὐθυγράμμῳ <lb n="1799,001,,,,6,29,1"></lb> |
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5753 ἴσον παραλληλόγραμμον παραβαλεῖν ὑπερβάλλον εἴδει <lb n="1799,001,,,,6,29,2"></lb> |
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5754 παραλληλογράμμῳ ὁμοίῳ τῷ δοθέντι.</s> <lb n="1799,001,,,,6,29,3"></lb> |
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5755 <s>Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ δοθὲν εὐθύ-<lb n="1799,001,,,,6,29,4"></lb> |
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5756 γραμμον, ᾧ δεῖ ἴσον παρὰ τὴν ΑΒ παραβαλεῖν, τὸ Γ, <lb n="1799,001,,,,6,29,5"></lb> |
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5757 ᾧ δὲ δεῖ ὅμοιον ὑπερβάλλειν, τὸ Δ· δεῖ δὴ παρὰ τὴν ΑΒ <lb n="1799,001,,,,6,29,6"></lb> |
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5758 εὐθεῖαν τῷ Γ εὐθυγράμμῳ ἴσον παραλληλόγραμμον <lb n="1799,001,,,,6,29,7"></lb> |
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5759 παραβαλεῖν ὑπερβάλλον εἴδει παραλληλογράμμῳ ὁμοίῳ <lb n="1799,001,,,,6,29,8"></lb> |
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5760 τῷ Δ.</s> <lb n="1799,001,,,,6,29,9"></lb> |
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5761 <s>Τετμήσθω ἡ ΑΒ δίχα κατὰ τὸ Ε, καὶ ἀναγεγράφθω <lb n="1799,001,,,,6,29,10"></lb> |
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5762 ἀπὸ τῆς ΕΒ τῷ Δ ὅμοιον καὶ ὁμοίως κείμενον παραλλη-<lb n="1799,001,,,,6,29,11"></lb> |
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5763 λόγραμμον τὸ ΒΖ, καὶ συναμφοτέροις μὲν τοῖς ΒΖ, Γ <lb n="1799,001,,,,6,29,12"></lb> |
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5764 ἴσον, τῷ δὲ Δ ὅμοιον καὶ ὁμοίως κείμενον τὸ αὐτὸ <lb n="1799,001,,,,6,29,13"></lb> |
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5765 συνεστάτω τὸ ΗΘ.</s> <s>ὁμόλογος δὲ ἔστω ἡ μὲν ΚΘ τῇ ΖΛ, <lb n="1799,001,,,,6,29,14"></lb> |
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5766 ἡ δὲ ΚΗ τῇ ΖΕ.</s> <s>καὶ ἐπεὶ μεῖζόν ἐστι τὸ ΗΘ τοῦ ΖΒ, <pb></pb> <lb n="1799,001,,,,6,29,15"></lb> |
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5767 μείζων ἄρα ἐστὶ καὶ ἡ μὲν ΚΘ τῆς ΖΛ, ἡ δὲ ΚΗ τῆς <lb n="1799,001,,,,6,29,16"></lb> |
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5768 ΖΕ.</s> <s>ἐκβεβλήσθωσαν αἱ ΖΛ, ΖΕ, καὶ τῇ μὲν ΚΘ ἴση <lb n="1799,001,,,,6,29,17"></lb> |
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5769 ἔστω ἡ ΖΛΜ, τῇ δὲ ΚΗ ἴση ἡ ΖΕΝ, καὶ συμπεπλη-<lb n="1799,001,,,,6,29,18"></lb> |
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5770 ρώσθω τὸ ΜΝ· τὸ ΜΝ ἄρα τῷ ΗΘ ἴσον τέ ἐστι καὶ <lb n="1799,001,,,,6,29,19"></lb> |
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5771 ὅμοιον.</s> <s>ἀλλὰ τὸ ΗΘ τῷ ΕΛ ἐστιν ὅμοιον· καὶ τὸ ΜΝ <lb n="1799,001,,,,6,29,20"></lb> |
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5772 ἄρα τῷ ΕΛ ὅμοιόν ἐστιν· περὶ τὴν αὐτὴν ἄρα διάμετρόν <lb n="1799,001,,,,6,29,21"></lb> |
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5773 ἐστι τὸ ΕΛ τῷ ΜΝ.</s> <s>ἤχθω αὐτῶν διάμετρος ἡ ΖΞ, καὶ <lb n="1799,001,,,,6,29,22"></lb> |
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5774 καταγεγράφθω τὸ σχῆμα.</s> <lb n="1799,001,,,,6,29,23"></lb> |
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5775 <s>Ἐπεὶ ἴσον ἐστὶ τὸ ΗΘ τοῖς ΕΛ, Γ, ἀλλὰ τὸ ΗΘ τῷ <lb n="1799,001,,,,6,29,24"></lb> |
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5776 ΜΝ ἴσον ἐστίν, καὶ τὸ ΜΝ ἄρα τοῖς ΕΛ, Γ ἴσον ἐστίν.</s> <lb n="1799,001,,,,6,29,25"></lb> |
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5777 <s>κοινὸν ἀφῃρήσθω τὸ ΕΛ· λοιπὸς ἄρα ὁ ΨΧΦ γνώμων <lb n="1799,001,,,,6,29,26"></lb> |
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5778 τῷ Γ ἐστιν ἴσος.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΕ τῇ ΕΒ, ἴσον <lb n="1799,001,,,,6,29,27"></lb> |
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5779 ἐστὶ καὶ τὸ ΑΝ τῷ ΝΒ, τουτέστι τῷ ΛΟ.</s> <s>κοινὸν προσ- <pb></pb><lb n="1799,001,,,,6,29,28"></lb> |
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5780 κείσθω τὸ ΕΞ· ὅλον ἄρα τὸ ΑΞ ἴσον ἐστὶ τῷ ΦΧΨ <lb n="1799,001,,,,6,29,29"></lb> |
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5781 γνώμονι.</s> <s>ἀλλὰ ὁ ΦΧΨ γνώμων τῷ Γ ἴσος ἐστίν· καὶ τὸ <lb n="1799,001,,,,6,29,30"></lb> |
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5782 ΑΞ ἄρα τῷ Γ ἴσον ἐστίν.</s> <lb n="1799,001,,,,6,29,31"></lb> |
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5783 <s>Παρὰ τὴν δοθεῖσαν ἄρα εὐθεῖαν τὴν ΑΒ τῷ δοθέντι <lb n="1799,001,,,,6,29,32"></lb> |
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5784 εὐθυγράμμῳ τῷ Γ ἴσον παραλληλόγραμμον παραβέβληται <lb n="1799,001,,,,6,29,33"></lb> |
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5785 τὸ ΑΞ ὑπερβάλλον εἴδει παραλληλογράμμῳ τῷ ΠΟ <lb n="1799,001,,,,6,29,34"></lb> |
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5786 ὁμοίῳ ὄντι τῷ Δ, ἐπεὶ καὶ τῷ ΕΛ ἐστιν ὅμοιον τὸ ΟΠ· <lb n="1799,001,,,,6,29,35"></lb> |
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5787 ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,6,29,36"></lb> |
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5788 <s>Τὴν δοθεῖσαν εὐθεῖαν πεπερασμένην ἄκρον καὶ μέσον <lb n="1799,001,,,,6,30,1"></lb> |
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5789 λόγον τεμεῖν.</s> <lb n="1799,001,,,,6,30,2"></lb> |
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5790 <s>Ἔστω ἡ δοθεῖσα εὐθεῖα πεπερασμένη ἡ ΑΒ· δεῖ δὴ <lb n="1799,001,,,,6,30,3"></lb> |
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5791 τὴν ΑΒ εὐθεῖαν ἄκρον καὶ μέσον λόγον τεμεῖν. <lb n="1799,001,,,,6,30,4"></lb> |
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5792 Ἀναγεγράφθω ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΒΓ, καὶ <lb n="1799,001,,,,6,30,5"></lb> |
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5793 παραβεβλήσθω παρὰ τὴν ΑΓ τῇ ΒΓ ἴσον παραλληλό-<lb n="1799,001,,,,6,30,6"></lb> |
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5794 γραμμον τὸ ΓΔ ὑπερβάλλον εἴδει τῷ ΑΔ ὁμοίῳ τῷ <lb n="1799,001,,,,6,30,7"></lb> |
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5795 ΒΓ.</s> <lb n="1799,001,,,,6,30,8"></lb> |
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5796 <s>Τετράγωνον δέ ἐστι τὸ ΒΓ· τετράγωνον ἄρα ἐστὶ καὶ <lb n="1799,001,,,,6,30,9"></lb> |
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5797 τὸ ΑΔ.</s> <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΒΓ τῷ ΓΔ, κοινὸν ἀφῃρή- <pb></pb><lb n="1799,001,,,,6,30,10"></lb> |
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5798 σθω τὸ ΓΕ· λοιπὸν ἄρα τὸ ΒΖ λοιπῷ τῷ ΑΔ ἐστιν <lb n="1799,001,,,,6,30,11"></lb> |
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5799 ἴσον.</s> <s>ἔστι δὲ αὐτῷ καὶ ἰσογώνιον· τῶν ΒΖ, ΑΔ ἄρα <lb n="1799,001,,,,6,30,12"></lb> |
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5800 ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας· ἔστιν <lb n="1799,001,,,,6,30,13"></lb> |
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5801 ἄρα ὡς ἡ ΖΕ πρὸς τὴν ΕΔ, οὕτως ἡ ΑΕ πρὸς τὴν ΕΒ.</s> <lb n="1799,001,,,,6,30,14"></lb> |
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5802 <s>ἴση δὲ ἡ μὲν ΖΕ τῇ ΑΒ, ἡ δὲ ΕΔ τῇ ΑΕ.</s> <s>ἔστιν ἄρα ὡς <lb n="1799,001,,,,6,30,15"></lb> |
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5803 ἡ ΒΑ πρὸς τὴν ΑΕ, οὕτως ἡ ΑΕ πρὸς τὴν ΕΒ.</s> <s>μείζων <lb n="1799,001,,,,6,30,16"></lb> |
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5804 δὲ ἡ ΑΒ τῆς ΑΕ· μείζων ἄρα καὶ ἡ ΑΕ τῆς ΕΒ.</s> <lb n="1799,001,,,,6,30,17"></lb> |
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5805 <s>Ἡ ἄρα ΑΒ εὐθεῖα ἄκρον καὶ μέσον λόγον τέτμηται <lb n="1799,001,,,,6,30,18"></lb> |
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5806 κατὰ τὸ Ε, καὶ τὸ μεῖζον αὐτῆς τμῆμά ἐστι τὸ ΑΕ· <lb n="1799,001,,,,6,30,19"></lb> |
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5807 ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,6,30,20"></lb> |
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5808 <s>Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν <lb n="1799,001,,,,6,31,1"></lb> |
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5809 γωνίαν ὑποτεινούσης πλευρᾶς εἶδος ἴσον ἐστὶ τοῖς ἀπὸ <lb n="1799,001,,,,6,31,2"></lb> |
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5810 τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν εἴδεσι τοῖς <lb n="1799,001,,,,6,31,3"></lb> |
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5811 ὁμοίοις τε καὶ ὁμοίως ἀναγραφομένοις.</s> <lb n="1799,001,,,,6,31,4"></lb> |
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5812 <s>Ἔστω τρίγωνον ὀρθογώνιον τὸ ΑΒΓ ὀρθὴν ἔχον τὴν <lb n="1799,001,,,,6,31,5"></lb> |
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5813 ὑπὸ ΒΑΓ γωνίαν· λέγω, ὅτι τὸ ἀπὸ τῆς ΒΓ εἶδος ἴσον <pb></pb> <lb n="1799,001,,,,6,31,6"></lb> |
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5814 ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ εἴδεσι τοῖς ὁμοίοις τε καὶ <lb n="1799,001,,,,6,31,7"></lb> |
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5815 ὁμοίως ἀναγραφομένοις.</s> <lb n="1799,001,,,,6,31,8"></lb> |
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5816 <s>Ἤχθω κάθετος ἡ ΑΔ.</s> <lb n="1799,001,,,,6,31,9"></lb> |
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5817 <s>Ἐπεὶ οὖν ἐν ὀρθογωνίῳ τριγώνῳ τῷ ΑΒΓ ἀπὸ τῆς πρὸς <lb n="1799,001,,,,6,31,10"></lb> |
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5818 τῷ Α ὀρθῆς γωνίας ἐπὶ τὴν ΒΓ βάσιν κάθετος ἦκται ἡ ΑΔ, <lb n="1799,001,,,,6,31,11"></lb> |
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5819 τὰ ΑΒΔ, ΑΔΓ πρὸς τῇ καθέτῳ τρίγωνα ὅμοιά ἐστι τῷ <lb n="1799,001,,,,6,31,12"></lb> |
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5820 τε ὅλῳ τῷ ΑΒΓ καὶ ἀλλήλοις.</s> <s>καὶ ἐπεὶ ὅμοιόν ἐστι τὸ <lb n="1799,001,,,,6,31,13"></lb> |
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5821 ΑΒΓ τῷ ΑΒΔ, ἔστιν ἄρα ὡς ἡ ΓΒ πρὸς τὴν ΒΑ, <lb n="1799,001,,,,6,31,14"></lb> |
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5822 οὕτως ἡ ΑΒ πρὸς τὴν ΒΔ.</s> <s>καὶ ἐπεὶ τρεῖς εὐθεῖαι <lb n="1799,001,,,,6,31,15"></lb> |
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5823 ἀνάλογόν εἰσιν, ἔστιν ὡς ἡ πρώτη πρὸς τὴν τρίτην, οὕτως <lb n="1799,001,,,,6,31,16"></lb> |
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5824 τὸ ἀπὸ τῆς πρώτης εἶδος πρὸς τὸ ἀπὸ τῆς δευτέρας τὸ <lb n="1799,001,,,,6,31,17"></lb> |
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5825 ὅμοιον καὶ ὁμοίως ἀναγραφόμενον.</s> <s>ὡς ἄρα ἡ ΓΒ πρὸς τὴν <lb n="1799,001,,,,6,31,18"></lb> |
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5826 ΒΔ, οὕτως τὸ ἀπὸ τῆς ΓΒ εἶδος πρὸς τὸ ἀπὸ τῆς ΒΑ <lb n="1799,001,,,,6,31,19"></lb> |
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5827 τὸ ὅμοιον καὶ ὁμοίως ἀναγραφόμενον.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,6,31,20"></lb> |
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5828 ὡς ἡ ΒΓ πρὸς τὴν ΓΔ, οὕτως τὸ ἀπὸ τῆς ΒΓ εἶδος πρὸς <lb n="1799,001,,,,6,31,21"></lb> |
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5829 τὸ ἀπὸ τῆς ΓΑ.</s> <s>ὥστε καὶ ὡς ἡ ΒΓ πρὸς τὰς ΒΔ, ΔΓ, <lb n="1799,001,,,,6,31,22"></lb> |
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5830 οὕτως τὸ ἀπὸ τῆς ΒΓ εἶδος πρὸς τὰ ἀπὸ τῶν ΒΑ, ΑΓ τὰ <lb n="1799,001,,,,6,31,23"></lb> |
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5831 ὅμοια καὶ ὁμοίως ἀναγραφόμενα.</s> <s>ἴση δὲ ἡ ΒΓ ταῖς ΒΔ, <lb n="1799,001,,,,6,31,24"></lb> |
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5832 ΔΓ· ἴσον ἄρα καὶ τὸ ἀπὸ τῆς ΒΓ εἶδος τοῖς ἀπὸ τῶν ΒΑ, <lb n="1799,001,,,,6,31,25"></lb> |
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5833 ΑΓ εἴδεσι τοῖς ὁμοίοις τε καὶ ὁμοίως ἀναγραφομένοις.</s> <lb n="1799,001,,,,6,31,26"></lb> |
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5834 <s>Ἐν ἄρα τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν <pb></pb> <lb n="1799,001,,,,6,31,27"></lb> |
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5835 ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς εἶδος ἴσον ἐστὶ τοῖς <lb n="1799,001,,,,6,31,28"></lb> |
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5836 ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν εἴδεσι <lb n="1799,001,,,,6,31,29"></lb> |
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5837 τοῖς ὁμοίοις τε καὶ ὁμοίως ἀναγραφομένοις· ὅπερ ἔδει <lb n="1799,001,,,,6,31,30"></lb> |
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5838 δεῖξαι.</s> <lb n="1799,001,,,,6,31,31"></lb> |
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5839 <s>Ἐὰν δύο τρίγωνα συντεθῇ κατὰ μίαν γωνίαν τὰς δύο <lb n="1799,001,,,,6,32,1"></lb> |
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5840 πλευρὰς ταῖς δυσὶ πλευραῖς ἀνάλογον ἔχοντα ὥστε τὰς <lb n="1799,001,,,,6,32,2"></lb> |
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5841 ὁμολόγους αὐτῶν πλευρὰς καὶ παραλλήλους εἶναι, αἱ <lb n="1799,001,,,,6,32,3"></lb> |
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5842 λοιπαὶ τῶν τριγώνων πλευραὶ ἐπ' εὐθείας ἔσονται.</s> <lb n="1799,001,,,,6,32,4"></lb> |
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5843 <s>Ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΓΕ τὰς δύο πλευρὰς <lb n="1799,001,,,,6,32,5"></lb> |
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5844 τὰς ΒΑ, ΑΓ ταῖς δυσὶ πλευραῖς ταῖς ΔΓ, ΔΕ ἀνάλογον <lb n="1799,001,,,,6,32,6"></lb> |
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5845 ἔχοντα, ὡς μὲν τὴν ΑΒ πρὸς τὴν ΑΓ, οὕτως τὴν ΔΓ πρὸς <lb n="1799,001,,,,6,32,7"></lb> |
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5846 τὴν ΔΕ, παράλληλον δὲ τὴν μὲν ΑΒ τῇ ΔΓ, τὴν δὲ ΑΓ <lb n="1799,001,,,,6,32,8"></lb> |
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5847 τῇ ΔΕ· λέγω, ὅτι ἐπ' εὐθείας ἐστὶν ἡ ΒΓ τῇ ΓΕ. <lb n="1799,001,,,,6,32,9"></lb> |
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5848 Ἐπεὶ γὰρ παράλληλός ἐστιν ἡ ΑΒ τῇ ΔΓ, καὶ εἰς αὐτὰς <lb n="1799,001,,,,6,32,10"></lb> |
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5849 ἐμπέπτωκεν εὐθεῖα ἡ ΑΓ, αἱ ἐναλλὰξ γωνίαι αἱ ὑπὸ ΒΑΓ, <lb n="1799,001,,,,6,32,11"></lb> |
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5850 ΑΓΔ ἴσαι ἀλλήλαις εἰσίν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ <lb n="1799,001,,,,6,32,12"></lb> |
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5851 ΓΔΕ τῇ ὑπὸ ΑΓΔ ἴση ἐστίν.</s> <s>ὥστε καὶ ἡ ὑπὸ ΒΑΓ τῇ <pb></pb> <lb n="1799,001,,,,6,32,13"></lb> |
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5852 ὑπὸ ΓΔΕ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ δύο τρίγωνά ἐστι τὰ ΑΒΓ, <lb n="1799,001,,,,6,32,14"></lb> |
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5853 ΔΓΕ μίαν γωνίαν τὴν πρὸς τῷ Α μιᾷ γωνίᾳ τῇ πρὸς τῷ <lb n="1799,001,,,,6,32,15"></lb> |
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5854 Δ ἴσην ἔχοντα, περὶ δὲ τὰς ἴσας γωνίας τὰς πλευρὰς ἀνά-<lb n="1799,001,,,,6,32,16"></lb> |
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5855 λογον, ὡς τὴν ΒΑ πρὸς τὴν ΑΓ, οὕτως τὴν ΓΔ πρὸς τὴν <lb n="1799,001,,,,6,32,17"></lb> |
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5856 ΔΕ, ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΓΕ <lb n="1799,001,,,,6,32,18"></lb> |
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5857 τριγώνῳ· ἴση ἄρα ἡ ὑπὸ ΑΒΓ γωνία τῇ ὑπὸ ΔΓΕ.</s> <lb n="1799,001,,,,6,32,19"></lb> |
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5858 <s>ἐδείχθη δὲ καὶ ἡ ὑπὸ ΑΓΔ τῇ ὑπὸ ΒΑΓ ἴση· ὅλη ἄρα <lb n="1799,001,,,,6,32,20"></lb> |
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5859 ἡ ὑπὸ ΑΓΕ δυσὶ ταῖς ὑπὸ ΑΒΓ, ΒΑΓ ἴση ἐστίν.</s> <s>κοινὴ <lb n="1799,001,,,,6,32,21"></lb> |
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5860 προσκείσθω ἡ ὑπὸ ΑΓΒ· αἱ ἄρα ὑπὸ ΑΓΕ, ΑΓΒ ταῖς <lb n="1799,001,,,,6,32,22"></lb> |
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5861 ὑπὸ ΒΑΓ, ΑΓΒ, ΓΒΑ ἴσαι εἰσίν.</s> <s>ἀλλ' αἱ ὑπὸ ΒΑΓ, <lb n="1799,001,,,,6,32,23"></lb> |
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5862 ΑΒΓ, ΑΓΒ δυσὶν ὀρθαῖς ἴσαι εἰσίν· καὶ αἱ ὑπὸ ΑΓΕ, <lb n="1799,001,,,,6,32,24"></lb> |
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5863 ΑΓΒ ἄρα δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <s>πρὸς δή τινι εὐθείᾳ <lb n="1799,001,,,,6,32,25"></lb> |
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5864 τῇ ΑΓ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Γ δύο εὐθεῖαι αἱ <lb n="1799,001,,,,6,32,26"></lb> |
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5865 ΒΓ, ΓΕ μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς γωνίας <lb n="1799,001,,,,6,32,27"></lb> |
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5866 τὰς ὑπὸ ΑΓΕ, ΑΓΒ δυσὶν ὀρθαῖς ἴσας ποιοῦσιν· ἐπ' <lb n="1799,001,,,,6,32,28"></lb> |
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5867 εὐθείας ἄρα ἐστὶν ἡ ΒΓ τῇ ΓΕ.</s> <lb n="1799,001,,,,6,32,29"></lb> |
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5868 <s>Ἐὰν ἄρα δύο τρίγωνα συντεθῇ κατὰ μίαν γωνίαν τὰς δύο <lb n="1799,001,,,,6,32,30"></lb> |
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5869 πλευρὰς ταῖς δυσὶ πλευραῖς ἀνάλογον ἔχοντα ὥστε τὰς <lb n="1799,001,,,,6,32,31"></lb> |
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5870 ὁμολόγους αὐτῶν πλευρὰς καὶ παραλλήλους εἶναι, αἱ <lb n="1799,001,,,,6,32,32"></lb> |
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5871 λοιπαὶ τῶν τριγώνων πλευραὶ ἐπ' εὐθείας ἔσονται· ὅπερ <lb n="1799,001,,,,6,32,33"></lb> |
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5872 ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,6,32,34"></lb> |
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5873 Ἐν τοῖς ἴσοις κύκλοις αἱ γωνίαι τὸν αὐτὸν ἔχουσι λόγον <lb n="1799,001,,,,6,33,1"></lb> |
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5874 ταῖς περιφερείαις, ἐφ' ὧν βεβήκασιν, ἐάν τε πρὸς τοῖς <lb n="1799,001,,,,6,33,2"></lb> |
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5875 κέντροις ἐάν τε πρὸς ταῖς περιφερείαις ὦσι βεβηκυῖαι. <lb n="1799,001,,,,6,33,3"></lb> |
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5876 Ἔστωσαν ἴσοι κύκλοι οἱ ΑΒΓ, ΔΕΖ, καὶ πρὸς μὲν <lb n="1799,001,,,,6,33,4"></lb> |
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5877 τοῖς κέντροις αὐτῶν τοῖς Η, Θ γωνίαι ἔστωσαν αἱ ὑπὸ <lb n="1799,001,,,,6,33,5"></lb> |
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5878 ΒΗΓ, ΕΘΖ, πρὸς δὲ ταῖς περιφερείαις αἱ ὑπὸ ΒΑΓ, <lb n="1799,001,,,,6,33,6"></lb> |
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5879 ΕΔΖ· λέγω, ὅτι ἐστὶν ὡς ἡ ΒΓ περιφέρεια πρὸς τὴν ΕΖ <lb n="1799,001,,,,6,33,7"></lb> |
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5880 περιφέρειαν, οὕτως ἥ τε ὑπὸ ΒΗΓ γωνία πρὸς τὴν ὑπὸ <lb n="1799,001,,,,6,33,8"></lb> |
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5881 ΕΘΖ καὶ ἡ ὑπὸ ΒΑΓ πρὸς τὴν ὑπὸ ΕΔΖ.</s> <lb n="1799,001,,,,6,33,9"></lb> |
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5882 <s>Κείσθωσαν γὰρ τῇ μὲν ΒΓ περιφερείᾳ ἴσαι κατὰ τὸ <lb n="1799,001,,,,6,33,10"></lb> |
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5883 ἑξῆς ὁσαιδηποτοῦν αἱ ΓΚ, ΚΛ, τῇ δὲ ΕΖ περιφερείᾳ ἴσαι <lb n="1799,001,,,,6,33,11"></lb> |
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5884 ὁσαιδηποτοῦν αἱ ΖΜ, ΜΝ, καὶ ἐπεζεύχθωσαν αἱ ΗΚ, <lb n="1799,001,,,,6,33,12"></lb> |
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5885 ΗΛ, ΘΜ, ΘΝ.</s> <s><pb></pb> <lb n="1799,001,,,,6,33,13"></lb> |
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5886 Ἐπεὶ οὖν ἴσαι εἰσὶν αἱ ΒΓ, ΓΚ, ΚΛ περιφέρειαι <lb n="1799,001,,,,6,33,14"></lb> |
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5887 ἀλλήλαις, ἴσαι εἰσὶ καὶ αἱ ὑπὸ ΒΗΓ, ΓΗΚ, ΚΗΛ <lb n="1799,001,,,,6,33,15"></lb> |
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5888 γωνίαι ἀλλήλαις· ὁσαπλασίων ἄρα ἐστὶν ἡ ΒΛ περι-<lb n="1799,001,,,,6,33,16"></lb> |
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5889 φέρεια τῆς ΒΓ, τοσαυταπλασίων ἐστὶ καὶ ἡ ὑπὸ ΒΗΛ <lb n="1799,001,,,,6,33,17"></lb> |
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5890 γωνία τῆς ὑπὸ ΒΗΓ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὁσαπλασίων <lb n="1799,001,,,,6,33,18"></lb> |
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5891 ἐστὶν ἡ ΝΕ περιφέρεια τῆς ΕΖ, τοσαυταπλασίων ἐστὶ <lb n="1799,001,,,,6,33,19"></lb> |
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5892 καὶ ἡ ὑπὸ ΝΘΕ γωνία τῆς ὑπὸ ΕΘΖ.</s> <s>εἰ ἄρα ἴση ἐστὶν <lb n="1799,001,,,,6,33,20"></lb> |
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5893 ἡ ΒΛ περιφέρεια τῇ ΕΝ περιφερείᾳ, ἴση ἐστὶ καὶ γωνία <lb n="1799,001,,,,6,33,21"></lb> |
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5894 ἡ ὑπὸ ΒΗΛ τῇ ὑπὸ ΕΘΝ, καὶ εἰ μείζων ἐστὶν <lb n="1799,001,,,,6,33,22"></lb> |
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5895 ἡ ΒΛ περιφέρεια τῆς ΕΝ περιφερείας, μείζων ἐστὶ καὶ <lb n="1799,001,,,,6,33,23"></lb> |
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5896 ἡ ὑπὸ ΒΗΛ γωνία τῆς ὑπὸ ΕΘΝ, καὶ εἰ ἐλάσσων, <lb n="1799,001,,,,6,33,24"></lb> |
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5897 ἐλάσσων.</s> <s>τεσσάρων δὴ ὄντων μεγεθῶν, δύο μὲν περι-<lb n="1799,001,,,,6,33,25"></lb> |
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5898 φερειῶν τῶν ΒΓ, ΕΖ, δύο δὲ γωνιῶν τῶν ὑπὸ ΒΗΓ, <lb n="1799,001,,,,6,33,26"></lb> |
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5899 ΕΘΖ, εἴληπται τῆς μὲν ΒΓ περιφερείας καὶ τῆς ὑπὸ <lb n="1799,001,,,,6,33,27"></lb> |
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5900 ΒΗΓ γωνίας ἰσάκις πολλαπλασίων ἥ τε ΒΛ περιφέρεια <lb n="1799,001,,,,6,33,28"></lb> |
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5901 καὶ ἡ ὑπὸ ΒΗΛ γωνία, τῆς δὲ ΕΖ περιφερείας καὶ τῆς <lb n="1799,001,,,,6,33,29"></lb> |
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5902 ὑπὸ ΕΘΖ γωνίας ἥ τε ΕΝ περιφέρεια καὶ ἡ ὑπὸ ΕΘΝ <lb n="1799,001,,,,6,33,30"></lb> |
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5903 γωνία.</s> <s>καὶ δέδεικται, ὅτι εἰ ὑπερέχει ἡ ΒΛ περιφέρεια τῆς <lb n="1799,001,,,,6,33,31"></lb> |
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5904 ΕΝ περιφερείας, ὑπερέχει καὶ ἡ ὑπὸ ΒΗΛ γωνία τῆς ὑπὸ <lb n="1799,001,,,,6,33,32"></lb> |
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5905 ΕΘΝ γωνίας, καὶ εἰ ἴση, ἴση, καὶ εἰ ἐλάσσων, ἐλάσσων.</s> <lb n="1799,001,,,,6,33,33"></lb> |
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5906 <s>ἔστιν ἄρα, ὡς ἡ ΒΓ περιφέρεια πρὸς τὴν ΕΖ, οὕτως ἡ <lb n="1799,001,,,,6,33,34"></lb> |
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5907 ὑπὸ ΒΗΓ γωνία πρὸς τὴν ὑπὸ ΕΘΖ.</s> <s>ἀλλ' ὡς ἡ ὑπὸ <lb n="1799,001,,,,6,33,35"></lb> |
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5908 ΒΗΓ γωνία πρὸς τὴν ὑπὸ ΕΘΖ, οὕτως ἡ ὑπὸ ΒΑΓ <lb n="1799,001,,,,6,33,36"></lb> |
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5909 πρὸς τὴν ὑπὸ ΕΔΖ· διπλασία γὰρ ἑκατέρα ἑκατέρας.</s> <s>καὶ <lb n="1799,001,,,,6,33,37"></lb> |
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5910 ὡς ἄρα ἡ ΒΓ περιφέρεια πρὸς τὴν ΕΖ περιφέρειαν, οὕτως <lb n="1799,001,,,,6,33,38"></lb> |
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5911 ἥ τε ὑπὸ ΒΗΓ γωνία πρὸς τὴν ὑπὸ ΕΘΖ καὶ ἡ ὑπὸ <lb n="1799,001,,,,6,33,39"></lb> |
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5912 ΒΑΓ πρὸς τὴν ὑπὸ ΕΔΖ.</s> <s><pb></pb> <lb n="1799,001,,,,6,33,40"></lb> |
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5913 Ἐν ἄρα τοῖς ἴσοις κύκλοις αἱ γωνίαι τὸν αὐτὸν ἔχουσι <lb n="1799,001,,,,6,33,41"></lb> |
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5914 λόγον ταῖς περιφερείαις, ἐφ' ὧν βεβήκασιν, ἐάν τε πρὸς <lb n="1799,001,,,,6,33,42"></lb> |
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5915 τοῖς κέντροις ἐάν τε πρὸς ταῖς περιφερείαις ὦσι βεβηκυῖαι· <lb n="1799,001,,,,6,33,43"></lb> |
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5916 ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,6,33,44"></lb> |
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5917 </s></p> |
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5918 </chap> |
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5919 <chap><p type="main"><s>Μονάς ἐστιν, καθ' ἣν ἕκαστον τῶν ὄντων ἓν <lb n="1799,001,,,,7;HOR,1,1"></lb> |
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5920 |
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5921 λέγεται.</s> <lb n="1799,001,,,,7;HOR,1,2"></lb> |
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5922 <s>Ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος.</s> <lb n="1799,001,,,,7;HOR,2,1"></lb> |
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5923 <s>Μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μεί-<lb n="1799,001,,,,7;HOR,3,1"></lb> |
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5924 ζονος, ὅταν καταμετρῇ τὸν μείζονα.</s> <lb n="1799,001,,,,7;HOR,3,2"></lb> |
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5925 <s>Μέρη δέ, ὅταν μὴ καταμετρῇ.</s> <lb n="1799,001,,,,7;HOR,4,1"></lb> |
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5926 <s>Πολλαπλάσιος δὲ ὁ μείζων τοῦ ἐλάσσονος, ὅταν <lb n="1799,001,,,,7;HOR,5,1"></lb> |
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5927 καταμετρῆται ὑπὸ τοῦ ἐλάσσονος.</s> <lb n="1799,001,,,,7;HOR,5,2"></lb> |
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5928 <s>Ἄρτιος ἀριθμός ἐστιν ὁ δίχα διαιρούμενος.</s> <lb n="1799,001,,,,7;HOR,6,1"></lb> |
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5929 <s>Περισσὸς δὲ ὁ μὴ διαιρούμενος δίχα ἢ [ὁ] μονάδι <lb n="1799,001,,,,7;HOR,7,1"></lb> |
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5930 διαφέρων ἀρτίου ἀριθμοῦ.</s> <lb n="1799,001,,,,7;HOR,7,2"></lb> |
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5931 <s>Ἀρτιάκις ἄρτιος ἀριθμός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ <lb n="1799,001,,,,7;HOR,8,1"></lb> |
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5932 μετρούμενος κατὰ ἄρτιον ἀριθμόν.</s> <lb n="1799,001,,,,7;HOR,8,2"></lb> |
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5933 <s>Ἀρτιάκις δὲ περισσός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ <lb n="1799,001,,,,7;HOR,9,1"></lb> |
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5934 μετρούμενος κατὰ περισσὸν ἀριθμόν.</s> <s><pb></pb> <lb n="1799,001,,,,7;HOR,9,2"></lb> |
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5935 [Περισσάκις ἀρτιός ἐστιν ὁ ὑπὸ περισσοῦ ἀριθμοῦ <lb n="1799,001,,,,7;HOR,10*,1"></lb> |
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5936 μετρούμενος κατὰ ἄρτιον ἀριθμόν].</s> <lb n="1799,001,,,,7;HOR,10*,2"></lb> |
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5937 <s>Περισσάκις δὲ περισσὸς ἀριθμός ἐστιν ὁ ὑπὸ <lb n="1799,001,,,,7;HOR,11,1"></lb> |
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5938 περισσοῦ ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν.</s> <lb n="1799,001,,,,7;HOR,11,2"></lb> |
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5939 <s>Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος.</s> <lb n="1799,001,,,,7;HOR,12,1"></lb> |
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5940 <s>Πρῶτοι πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ μονάδι <lb n="1799,001,,,,7;HOR,13,1"></lb> |
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5941 μόνῃ μετρούμενοι κοινῷ μέτρῳ.</s> <lb n="1799,001,,,,7;HOR,13,2"></lb> |
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5942 <s>Σύνθετος ἀριθμός ἐστιν ὁ ἀριθμῷ τινι μετρού-<lb n="1799,001,,,,7;HOR,14,1"></lb> |
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5943 μενος.</s> <lb n="1799,001,,,,7;HOR,14,2"></lb> |
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5944 <s>Σύνθετοι δὲ πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ ἀριθμῷ <lb n="1799,001,,,,7;HOR,15,1"></lb> |
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5945 τινι μετρούμενοι κοινῷ μέτρῳ.</s> <lb n="1799,001,,,,7;HOR,15,2"></lb> |
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5946 <s>Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, <lb n="1799,001,,,,7;HOR,16,1"></lb> |
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5947 ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ <lb n="1799,001,,,,7;HOR,16,2"></lb> |
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5948 πολλαπλασιαζόμενος, καὶ γένηταί τις.</s> <lb n="1799,001,,,,7;HOR,16,3"></lb> |
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5949 <s>Ὅταν δὲ δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους <lb n="1799,001,,,,7;HOR,17,1"></lb> |
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5950 ποιῶσί τινα, ὁ γενόμενος ἐπίπεδος καλεῖται, πλευραὶ δὲ <lb n="1799,001,,,,7;HOR,17,2"></lb> |
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5951 αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.</s> <lb n="1799,001,,,,7;HOR,17,3"></lb> |
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5952 <s>Ὅταν δὲ τρεῖς ἀριθμοὶ πολλαπλασιάσαντες ἀλλή-<lb n="1799,001,,,,7;HOR,18,1"></lb> |
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5953 λους ποιῶσί τινα, ὁ γενόμενος στερεός ἐστιν, πλευραὶ δὲ <lb n="1799,001,,,,7;HOR,18,2"></lb> |
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5954 αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.</s> <lb n="1799,001,,,,7;HOR,18,3"></lb> |
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5955 <s>Τετράγωνος ἀριθμός ἐστιν ὁ ἰσάκις ἴσος ἢ [ὁ] ὑπὸ <lb n="1799,001,,,,7;HOR,19,1"></lb> |
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5956 δύο ἴσων ἀριθμῶν περιεχόμενος.</s> <lb n="1799,001,,,,7;HOR,19,2"></lb> |
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5957 <s>Κύβος δὲ ὁ ἰσάκις ἴσος ἰσάκις ἢ [ὁ] ὑπὸ τριῶν ἴσων <lb n="1799,001,,,,7;HOR,20,1"></lb> |
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5958 ἀριθμῶν περιεχόμενος.</s> <lb n="1799,001,,,,7;HOR,20,2"></lb> |
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5959 <s>Ἀριθμοὶ ἀνάλογόν εἰσιν, ὅταν ὁ πρῶτος τοῦ <pb></pb> <lb n="1799,001,,,,7;HOR,21,1"></lb> |
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5960 δευτέρου καὶ ὁ τρίτος τοῦ τετάρτου ἰσάκις ᾖ πολλαπλάσιος <lb n="1799,001,,,,7;HOR,21,2"></lb> |
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5961 ἢ τὸ αὐτὸ μέρος ἢ τὰ αὐτὰ μέρη ὦσιν.</s> <lb n="1799,001,,,,7;HOR,21,3"></lb> |
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5962 <s>Ὅμοιοι ἐπίπεδοι καὶ στερεοὶ ἀριθμοί εἰσιν οἱ ἀνά-<lb n="1799,001,,,,7;HOR,22,1"></lb> |
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5963 λογον ἔχοντες τὰς πλευράς.</s> <lb n="1799,001,,,,7;HOR,22,2"></lb> |
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5964 <s>Τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν <lb n="1799,001,,,,7;HOR,23,1"></lb> |
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5965 ἴσος ὤν.</s> <lb n="1799,001,,,,7;HOR,23,2"></lb> |
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5966 <s></s></p> |
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5967 </chap> |
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5968 <chap><p type="main"><s>Δύο ἀριθμῶν ἀνίσων ἐκκειμένων, ἀνθυφαιρουμένου δὲ <lb n="1799,001,,,,7,1,1"></lb> |
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5969 |
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5970 ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος, ἐὰν ὁ λειπόμενος <lb n="1799,001,,,,7,1,2"></lb> |
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5971 μηδέποτε καταμετρῇ τὸν πρὸ ἑαυτοῦ, ἕως οὗ λειφθῇ μονάς, <lb n="1799,001,,,,7,1,3"></lb> |
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5972 οἱ ἐξ ἀρχῆς ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ἔσονται.</s> <lb n="1799,001,,,,7,1,4"></lb> |
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5973 <s>Δύο γὰρ [ἀνίσων] ἀριθμῶν τῶν ΑΒ, ΓΔ <lb n="1799,001,,,,7,1,5"></lb> |
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5974 ἀνθυφαιρουμένου ἀεὶ τοῦ ἐλάσσονος ἀπὸ <lb n="1799,001,,,,7,1,6"></lb> |
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5975 τοῦ μείζονος ὁ λειπόμενος μηδέποτε κατα-<lb n="1799,001,,,,7,1,7"></lb> |
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5976 μετρείτω τὸν πρὸ ἑαυτοῦ, ἕως οὗ λειφθῇ <lb n="1799,001,,,,7,1,8"></lb> |
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5977 μονάς· λέγω, ὅτι οἱ ΑΒ, ΓΔ πρῶτοι πρὸς <lb n="1799,001,,,,7,1,9"></lb> |
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5978 ἀλλήλους εἰσίν, τουτέστιν ὅτι τοὺς ΑΒ, <lb n="1799,001,,,,7,1,10"></lb> |
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5979 ΓΔ μονὰς μόνη μετρεῖ.</s> <lb n="1799,001,,,,7,1,11"></lb> |
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5980 <s>Εἰ γὰρ μή εἰσιν οἱ ΑΒ, ΓΔ πρῶτοι πρὸς ἀλλήλους, <lb n="1799,001,,,,7,1,12"></lb> |
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5981 μετρήσει τις αὐτοὺς ἀριθμός.</s> <s>μετρείτω, καὶ ἔστω ὁ Ε· <lb n="1799,001,,,,7,1,13"></lb> |
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5982 καὶ ὁ μὲν ΓΔ τὸν ΒΖ μετρῶν λειπέτω ἑαυτοῦ ἐλάσσονα <pb></pb> <lb n="1799,001,,,,7,1,14"></lb> |
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5983 τὸν ΖΑ, ὁ δὲ ΑΖ τὸν ΔΗ μετρῶν λειπέτω ἑαυτοῦ <lb n="1799,001,,,,7,1,15"></lb> |
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5984 ἐλάσσονα τὸν ΗΓ, ὁ δὲ ΗΓ τὸν ΖΘ μετρῶν λειπέτω <lb n="1799,001,,,,7,1,16"></lb> |
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5985 μονάδα τὴν ΘΑ.</s> <lb n="1799,001,,,,7,1,17"></lb> |
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5986 <s>Ἐπεὶ οὖν ὁ Ε τὸν ΓΔ μετρεῖ, ὁ δὲ ΓΔ τὸν ΒΖ μετρεῖ <lb n="1799,001,,,,7,1,18"></lb> |
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5987 καὶ ὁ Ε ἄρα τὸν ΒΖ μετρεῖ· μετρεῖ δὲ καὶ ὅλον τὸν ΒΑ· <lb n="1799,001,,,,7,1,19"></lb> |
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5988 καὶ λοιπὸν ἄρα τὸν ΑΖ μετρήσει.</s> <s>ὁ δὲ ΑΖ τὸν ΔΗ <lb n="1799,001,,,,7,1,20"></lb> |
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5989 μετρεῖ· καὶ ὁ Ε ἄρα τὸν ΔΗ μετρεῖ· μετρεῖ δὲ καὶ ὅλον <lb n="1799,001,,,,7,1,21"></lb> |
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5990 τὸν ΔΓ· καὶ λοιπὸν ἄρα τὸν ΓΗ μετρήσει.</s> <s>ὁ δὲ ΓΗ τὸν <lb n="1799,001,,,,7,1,22"></lb> |
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5991 ΖΘ μετρεῖ· καὶ ὁ Ε ἄρα τὸν ΖΘ μετρεῖ· μετρεῖ δὲ καὶ <lb n="1799,001,,,,7,1,23"></lb> |
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5992 ὅλον τὸν ΖΑ· καὶ λοιπὴν ἄρα τὴν ΑΘ μονάδα μετρήσει <lb n="1799,001,,,,7,1,24"></lb> |
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5993 ἀριθμὸς ὤν· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τοὺς ΑΒ, <lb n="1799,001,,,,7,1,25"></lb> |
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5994 ΓΔ ἀριθμοὺς μετρήσει τις ἀριθμός· οἱ ΑΒ, ΓΔ ἄρα <lb n="1799,001,,,,7,1,26"></lb> |
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5995 πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,1,27"></lb> |
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5996 <s>Δύο ἀριθμῶν δοθέντων μὴ πρώτων πρὸς ἀλλήλους τὸ <lb n="1799,001,,,,7,2,1"></lb> |
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5997 μέγιστον αὐτῶν κοινὸν μέτρον εὑρεῖν.</s> <lb n="1799,001,,,,7,2,2"></lb> |
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5998 <s>Ἔστωσαν οἱ δοθέντες δύο ἀριθμοὶ μὴ πρῶτοι πρὸς <lb n="1799,001,,,,7,2,3"></lb> |
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5999 ἀλλήλους οἱ ΑΒ, ΓΔ.</s> <s>δεῖ δὴ τῶν ΑΒ, ΓΔ τὸ μέγιστον <lb n="1799,001,,,,7,2,4"></lb> |
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6000 κοινὸν μέτρον εὑρεῖν.</s> <lb n="1799,001,,,,7,2,5"></lb> |
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6001 <s>Εἰ μὲν οὖν ὁ ΓΔ τὸν ΑΒ μετρεῖ, μετρεῖ δὲ καὶ ἑαυτόν, <lb n="1799,001,,,,7,2,6"></lb> |
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6002 ὁ ΓΔ ἄρα τῶν ΓΔ, ΑΒ κοινὸν μέτρον ἐστίν.</s> <s>καὶ φανερόν, <pb></pb> <lb n="1799,001,,,,7,2,7"></lb> |
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6003 ὅτι καὶ μέγιστον· οὐδεὶς γὰρ μείζων τοῦ ΓΔ τὸν ΓΔ <lb n="1799,001,,,,7,2,8"></lb> |
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6004 μετρήσει.</s> <lb n="1799,001,,,,7,2,9"></lb> |
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6005 <s>Εἰ δὲ οὐ μετρεῖ ὁ ΓΔ τὸν ΑΒ, τῶν ΑΒ, ΓΔ ἀνθυφαι-<lb n="1799,001,,,,7,2,10"></lb> |
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6006 ρουμένου ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος λειφθήσεταί <lb n="1799,001,,,,7,2,11"></lb> |
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6007 τις ἀριθμός, ὃς μετρήσει τὸν πρὸ ἑαυτοῦ. <lb n="1799,001,,,,7,2,12"></lb> |
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6008 μονὰς μὲν γὰρ οὐ λειφθήσεται· εἰ δὲ μή, <lb n="1799,001,,,,7,2,13"></lb> |
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6009 ἔσονται οἱ ΑΒ, ΓΔ πρῶτοι πρὸς ἀλλήλους· <lb n="1799,001,,,,7,2,14"></lb> |
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6010 ὅπερ οὐχ ὑπόκειται.</s> <s>λειφθήσεταί τις ἄρα <lb n="1799,001,,,,7,2,15"></lb> |
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6011 ἀριθμός, ὃς μετρήσει τὸν πρὸ ἑαυτοῦ.</s> <s>καὶ <lb n="1799,001,,,,7,2,16"></lb> |
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6012 ὁ μὲν ΓΔ τὸν ΒΕ μετρῶν λειπέτω ἑαυ-<lb n="1799,001,,,,7,2,17"></lb> |
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6013 τοῦ ἐλάσσονα τὸν ΕΑ, ὁ δὲ ΕΑ τὸν ΔΖ <lb n="1799,001,,,,7,2,18"></lb> |
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6014 μετρῶν λειπέτω ἑαυτοῦ ἐλάσσονα τὸν ΖΓ, ὁ δὲ ΓΖ <lb n="1799,001,,,,7,2,19"></lb> |
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6015 τὸν ΑΕ μετρείτω.</s> <s>ἐπεὶ οὖν ὁ ΓΖ τὸν ΑΕ μετρεῖ, ὁ δὲ <lb n="1799,001,,,,7,2,20"></lb> |
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6016 ΑΕ τὸν ΔΖ μετρεῖ, καὶ ὁ ΓΖ ἄρα τὸν ΔΖ μετρήσει· <lb n="1799,001,,,,7,2,21"></lb> |
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6017 μετρεῖ δὲ καὶ ἑαυτόν· καὶ ὅλον ἄρα τὸν ΓΔ μετρήσει.</s> <s>ὁ δὲ <lb n="1799,001,,,,7,2,22"></lb> |
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6018 ΓΔ τὸν ΒΕ μετρεῖ· καὶ ὁ ΓΖ ἄρα τὸν ΒΕ μετρεῖ· μετρεῖ <lb n="1799,001,,,,7,2,23"></lb> |
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6019 δὲ καὶ τὸν ΕΑ· καὶ ὅλον ἄρα τὸν ΒΑ μετρήσει· μετρεῖ δὲ <lb n="1799,001,,,,7,2,24"></lb> |
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6020 καὶ τὸν ΓΔ· ὁ ΓΖ ἄρα τοὺς ΑΒ, ΓΔ μετρεῖ.</s> <s>ὁ ΓΖ ἄρα <lb n="1799,001,,,,7,2,25"></lb> |
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6021 τῶν ΑΒ, ΓΔ κοινὸν μέτρον ἐστίν.</s> <s>λέγω δή, ὅτι καὶ <lb n="1799,001,,,,7,2,26"></lb> |
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6022 μέγιστον.</s> <s>εἰ γὰρ μή ἐστιν ὁ ΓΖ τῶν ΑΒ, ΓΔ μέγιστον <lb n="1799,001,,,,7,2,27"></lb> |
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6023 κοινὸν μέτρον, μετρήσει τις τοὺς ΑΒ, ΓΔ ἀριθμοὺς <lb n="1799,001,,,,7,2,28"></lb> |
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6024 ἀριθμὸς μείζων ὢν τοῦ ΓΖ.</s> <s>μετρείτω, καὶ ἔστω ὁ Η.</s> <lb n="1799,001,,,,7,2,29"></lb> |
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6025 <s>καὶ ἐπεὶ ὁ Η τὸν ΓΔ μετρεῖ, ὁ δὲ ΓΔ τὸν ΒΕ μετρεῖ, καὶ <lb n="1799,001,,,,7,2,30"></lb> |
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6026 ὁ Η ἄρα τὸν ΒΕ μετρεῖ· μετρεῖ δὲ καὶ ὅλον τὸν ΒΑ· καὶ <lb n="1799,001,,,,7,2,31"></lb> |
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6027 λοιπὸν ἄρα τὸν ΑΕ μετρήσει.</s> <s>ὁ δὲ ΑΕ τὸν ΔΖ μετρεῖ· <lb n="1799,001,,,,7,2,32"></lb> |
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6028 καὶ ὁ Η ἄρα τὸν ΔΖ μετρήσει· μετρεῖ δὲ καὶ ὅλον τὸν <pb></pb> <lb n="1799,001,,,,7,2,33"></lb> |
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6029 ΔΓ· καὶ λοιπὸν ἄρα τὸν ΓΖ μετρήσει ὁ μείζων τὸν ἐλάσ-<lb n="1799,001,,,,7,2,34"></lb> |
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6030 σονα· ὅπερ ἐστὶν ἀδύνατον· οὐκ ἄρα τοὺς ΑΒ, ΓΔ ἀριθμοὺς <lb n="1799,001,,,,7,2,35"></lb> |
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6031 ἀριθμός τις μετρήσει μείζων ὢν τοῦ ΓΖ· ὁ ΓΖ ἄρα τῶν <lb n="1799,001,,,,7,2,36"></lb> |
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6032 ΑΒ, ΓΔ μέγιστόν ἐστι κοινὸν μέτρον· [ὅπερ ἔδει δεῖξαι].</s> <lb n="1799,001,,,,7,2,37"></lb> |
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6033 <s>Πόρισμα <lb n="1799,001,,,,7,2,38"></lb> |
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6034 Ἐκ δὴ τούτου φανερόν, ὅτι ἐὰν ἀριθμὸς δύο ἀριθμοὺς <lb n="1799,001,,,,7,2,39"></lb> |
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6035 μετρῇ, καὶ τὸ μέγιστον αὐτῶν κοινὸν μέτρον μετρήσει· <lb n="1799,001,,,,7,2,40"></lb> |
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6036 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,2,41"></lb> |
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6037 <s>Τριῶν ἀριθμῶν δοθέντων μὴ πρώτων πρὸς ἀλλήλους <lb n="1799,001,,,,7,3,1"></lb> |
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6038 τὸ μέγιστον αὐτῶν κοινὸν μέτρον εὑρεῖν.</s> <lb n="1799,001,,,,7,3,2"></lb> |
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6039 <s>Ἔστωσαν οἱ δοθέντες τρεῖς ἀρι-<lb n="1799,001,,,,7,3,3"></lb> |
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6040 θμοὶ μὴ πρῶτοι πρὸς ἀλλήλους οἱ <lb n="1799,001,,,,7,3,4"></lb> |
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6041 Α, Β, Γ· δεῖ δὴ τῶν Α, Β, Γ τὸ <lb n="1799,001,,,,7,3,5"></lb> |
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6042 μέγιστον κοινὸν μέτρον εὑρεῖν.</s> <lb n="1799,001,,,,7,3,6"></lb> |
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6043 <s>Εἰλήφθω γὰρ δύο τῶν Α, Β τὸ <lb n="1799,001,,,,7,3,7"></lb> |
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6044 μέγιστον κοινὸν μέτρον ὁ Δ· ὁ <lb n="1799,001,,,,7,3,8"></lb> |
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6045 δὴ Δ τὸν Γ ἤτοι μετρεῖ ἢ οὐ μετρεῖ.</s> <s>μετρείτω πρό-<lb n="1799,001,,,,7,3,9"></lb> |
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6046 τερον· μετρεῖ δὲ καὶ τοὺς Α, Β· ὁ Δ ἄρα τοὺς Α, Β, Γ <lb n="1799,001,,,,7,3,10"></lb> |
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6047 μετρεῖ· ὁ Δ ἄρα τῶν Α, Β, Γ κοινὸν μέτρον ἐστίν.</s> <s>λέγω δή, <lb n="1799,001,,,,7,3,11"></lb> |
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6048 ὅτι καὶ μέγιστον.</s> <s>εἰ γὰρ μή ἐστιν ὁ Δ τῶν Α, Β, Γ μέγι-<lb n="1799,001,,,,7,3,12"></lb> |
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6049 στον κοινὸν μέτρον, μετρήσει τις τοὺς Α, Β, Γ ἀριθμοὺς <lb n="1799,001,,,,7,3,13"></lb> |
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6050 ἀριθμὸς μείζων ὢν τοῦ Δ.</s> <s>μετρείτω, καὶ ἔστω ὁ Ε.</s> <s>ἐπεὶ <lb n="1799,001,,,,7,3,14"></lb> |
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6051 οὖν ὁ Ε τοὺς Α, Β, Γ μετρεῖ, καὶ τοὺς Α, Β ἄρα μετρήσει· <lb n="1799,001,,,,7,3,15"></lb> |
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6052 καὶ τὸ τῶν Α, Β ἄρα μέγιστον κοινὸν μέτρον μετρήσει.</s> <s><pb></pb> <lb n="1799,001,,,,7,3,16"></lb> |
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6053 τὸ δὲ τῶν Α, Β μέγιστον κοινὸν μέτρον ἐστὶν ὁ Δ· ὁ Ε <lb n="1799,001,,,,7,3,17"></lb> |
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6054 ἄρα τὸν Δ μετρεῖ ὁ μείζων τὸν ἐλάσσονα· ὅπερ ἐστὶν <lb n="1799,001,,,,7,3,18"></lb> |
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6055 ἀδύνατον.</s> <s>οὐκ ἄρα τοὺς Α, Β, Γ ἀριθμοὺς ἀριθμός τις <lb n="1799,001,,,,7,3,19"></lb> |
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6056 μετρήσει μείζων ὢν τοῦ Δ· ὁ Δ ἄρα τῶν Α, Β, Γ μέγι-<lb n="1799,001,,,,7,3,20"></lb> |
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6057 στόν ἐστι κοινὸν μέτρον.</s> <lb n="1799,001,,,,7,3,21"></lb> |
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6058 <s>Μὴ μετρείτω δὴ ὁ Δ τὸν Γ· λέγω πρῶτον, ὅτι οἱ Γ, Δ <lb n="1799,001,,,,7,3,22"></lb> |
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6059 οὔκ εἰσι πρῶτοι πρὸς ἀλλήλους.</s> <s>ἐπεὶ γὰρ οἱ Α, Β, Γ οὔκ <lb n="1799,001,,,,7,3,23"></lb> |
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6060 εἰσι πρῶτοι πρὸς ἀλλήλους, μετρήσει τις αὐτοὺς ἀριθμός.</s> <lb n="1799,001,,,,7,3,24"></lb> |
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6061 <s>ὁ δὴ τοὺς Α, Β, Γ μετρῶν καὶ τοὺς Α, Β μετρήσει, καὶ τὸ <lb n="1799,001,,,,7,3,25"></lb> |
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6062 τῶν Α, Β μέγιστον κοινὸν μέτρον τὸν Δ μετρήσει· μετρεῖ <lb n="1799,001,,,,7,3,26"></lb> |
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6063 δὲ καὶ τὸν Γ· τοὺς Δ, Γ ἄρα ἀριθμοὺς ἀριθμός τις μετρήσει· <lb n="1799,001,,,,7,3,27"></lb> |
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6064 οἱ Δ, Γ ἄρα οὔκ εἰσι πρῶτοι πρὸς ἀλλήλους.</s> <s>εἰλήφθω <lb n="1799,001,,,,7,3,28"></lb> |
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6065 οὖν αὐτῶν τὸ μέγιστον κοινὸν μέτρον ὁ Ε.</s> <s>καὶ ἐπεὶ ὁ Ε <lb n="1799,001,,,,7,3,29"></lb> |
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6066 τὸν Δ μετρεῖ, ὁ δὲ Δ τοὺς Α, Β μετρεῖ, καὶ ὁ Ε ἄρα τοὺς <lb n="1799,001,,,,7,3,30"></lb> |
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6067 Α, Β μετρεῖ· μετρεῖ δὲ καὶ τὸν Γ· ὁ Ε ἄρα τοὺς Α, Β, Γ <lb n="1799,001,,,,7,3,31"></lb> |
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6068 μετρεῖ· ὁ Ε ἄρα τῶν Α, Β, Γ κοινόν ἐστι μέτρον.</s> <s>λέγω δή, <lb n="1799,001,,,,7,3,32"></lb> |
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6069 ὅτι καὶ μέγιστον.</s> <s>εἰ γὰρ μή ἐστιν ὁ Ε τῶν Α, Β, Γ τὸ <lb n="1799,001,,,,7,3,33"></lb> |
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6070 μέγιστον κοινὸν μέτρον, μετρήσει τις τοὺς Α, Β, Γ ἀρι-<lb n="1799,001,,,,7,3,34"></lb> |
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6071 θμοὺς ἀριθμὸς μείζων ὢν τοῦ Ε.</s> <s>μετρείτω, καὶ ἔστω ὁ Ζ.</s> <lb n="1799,001,,,,7,3,35"></lb> |
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6072 <s>καὶ ἐπεὶ ὁ Ζ τοὺς Α, Β, Γ μετρεῖ, καὶ τοὺς Α, Β μετρεῖ· <lb n="1799,001,,,,7,3,36"></lb> |
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6073 καὶ τὸ τῶν Α, Β ἄρα μέγιστον κοινὸν μέτρον μετρήσει.</s> <s>τὸ <lb n="1799,001,,,,7,3,37"></lb> |
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6074 δὲ τῶν Α, Β μέγιστον κοινὸν μέτρον ἐστὶν ὁ Δ· ὁ Ζ ἄρα τὸν <lb n="1799,001,,,,7,3,38"></lb> |
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6075 Δ μετρεῖ· μετρεῖ δὲ καὶ τὸν Γ· ὁ Ζ ἄρα τοὺς Δ, Γ μετρεῖ· <lb n="1799,001,,,,7,3,39"></lb> |
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6076 καὶ τὸ τῶν Δ, Γ ἄρα μέγιστον κοινὸν μέτρον μετρήσει.</s> <s>τὸ <lb n="1799,001,,,,7,3,40"></lb> |
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6077 δὲ τῶν Δ, Γ μέγιστον κοινὸν μέτρον ἐστὶν ὁ Ε· ὁ Ζ ἄρα τὸν <lb n="1799,001,,,,7,3,41"></lb> |
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6078 Ε μετρεῖ ὁ μείζων τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον.</s> <s><pb></pb> <lb n="1799,001,,,,7,3,42"></lb> |
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6079 οὐκ ἄρα τοὺς Α, Β, Γ ἀριθμοὺς ἀριθμός τις μετρήσει <lb n="1799,001,,,,7,3,43"></lb> |
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6080 μείζων ὢν τοῦ Ε· ὁ Ε ἄρα τῶν Α, Β, Γ μέγιστόν ἐστι <lb n="1799,001,,,,7,3,44"></lb> |
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6081 κοινὸν μέτρον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,3,45"></lb> |
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6082 <s>Ἅπας ἀριθμὸς παντὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος <lb n="1799,001,,,,7,4,1"></lb> |
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6083 ἤτοι μέρος ἐστὶν ἢ μέρη.</s> <lb n="1799,001,,,,7,4,2"></lb> |
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6084 <s>Ἔστωσαν δύο ἀριθμοὶ οἱ Α, ΒΓ, καὶ ἔστω ἐλάσσων <lb n="1799,001,,,,7,4,3"></lb> |
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6085 ὁ ΒΓ· λέγω, ὅτι ὁ ΒΓ τοῦ Α ἤτοι μέρος ἐστὶν ἢ μέρη.</s> <lb n="1799,001,,,,7,4,4"></lb> |
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6086 <s>Οἱ Α, ΒΓ γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ.</s> <lb n="1799,001,,,,7,4,5"></lb> |
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6087 <s>ἔστωσαν πρότερον οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλήλους.</s> <lb n="1799,001,,,,7,4,6"></lb> |
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6088 <s>διαιρεθέντος δὴ τοῦ ΒΓ εἰς τὰς ἐν αὐτῷ μο-<lb n="1799,001,,,,7,4,7"></lb> |
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6089 νάδας ἔσται ἑκάστη μονὰς τῶν ἐν τῷ ΒΓ μέ-<lb n="1799,001,,,,7,4,8"></lb> |
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6090 ρος τι τοῦ Α· ὥστε μέρη ἐστὶν ὁ ΒΓ τοῦ Α.</s> <lb n="1799,001,,,,7,4,9"></lb> |
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6091 <s>Μὴ ἔστωσαν δὴ οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλή-<lb n="1799,001,,,,7,4,10"></lb> |
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6092 λους· ὁ δὴ ΒΓ τὸν Α ἤτοι μετρεῖ ἢ οὐ μετρεῖ.</s> <lb n="1799,001,,,,7,4,11"></lb> |
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6093 <s>εἰ μὲν οὖν ὁ ΒΓ τὸν Α μετρεῖ, μέρος ἐστὶν ὁ <lb n="1799,001,,,,7,4,12"></lb> |
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6094 ΒΓ τοῦ Α.</s> <s>εἰ δὲ οὔ, εἰλήφθω τῶν Α, ΒΓ μέ-<lb n="1799,001,,,,7,4,13"></lb> |
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6095 γιστον κοινὸν μέτρον ὁ Δ, καὶ διῃρήσθω ὁ ΒΓ <pb></pb> <lb n="1799,001,,,,7,4,14"></lb> |
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6096 εἰς τοὺς τῷ Δ ἴσους τοὺς ΒΕ, ΕΖ, ΖΓ.</s> <s>καὶ ἐπεὶ ὁ Δ τὸν <lb n="1799,001,,,,7,4,15"></lb> |
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6097 Α μετρεῖ, μέρος ἐστὶν ὁ Δ τοῦ Α· ἴσος δὲ ὁ Δ ἑκάστῳ <lb n="1799,001,,,,7,4,16"></lb> |
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6098 τῶν ΒΕ, ΕΖ, ΖΓ· καὶ ἕκαστος ἄρα τῶν ΒΕ, ΕΖ, ΖΓ <lb n="1799,001,,,,7,4,17"></lb> |
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6099 τοῦ Α μέρος ἐστίν· ὥστε μέρη ἐστὶν ὁ ΒΓ τοῦ Α.</s> <lb n="1799,001,,,,7,4,18"></lb> |
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6100 <s>Ἅπας ἄρα ἀριθμὸς παντὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ <lb n="1799,001,,,,7,4,19"></lb> |
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6101 μείζονος ἤτοι μέρος ἐστὶν ἢ μέρη· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,4,20"></lb> |
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6102 <s>Ἐὰν ἀριθμός ἀριθμοῦ μέρος ᾖ, καὶ ἕτερος ἑτέρου τὸ <lb n="1799,001,,,,7,5,1"></lb> |
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6103 αὐτὸ μέρος ᾖ, καὶ συναμφότερος συναμφοτέρου τὸ αὐτὸ <lb n="1799,001,,,,7,5,2"></lb> |
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6104 μέρος ἔσται, ὅπερ ὁ εἷς τοῦ ἑνός.</s> <lb n="1799,001,,,,7,5,3"></lb> |
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6105 <s>Ἀριθμὸς γὰρ ὁ Α [ἀριθμοῦ] τοῦ ΒΓ <lb n="1799,001,,,,7,5,4"></lb> |
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6106 μέρος ἔστω, καὶ ἕτερος ὁ Δ ἑτέρου τοῦ <lb n="1799,001,,,,7,5,5"></lb> |
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6107 ΕΖ τὸ αὐτὸ μέρος, ὅπερ ὁ Α τοῦ ΒΓ· <lb n="1799,001,,,,7,5,6"></lb> |
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6108 λέγω, ὅτι καὶ συναμφότερος ὁ Α, Δ συν-<lb n="1799,001,,,,7,5,7"></lb> |
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6109 αμφοτέρου τοῦ ΒΓ, ΕΖ τὸ αὐτὸ μέρος <lb n="1799,001,,,,7,5,8"></lb> |
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6110 ἐστίν, ὅπερ ὁ Α τοῦ ΒΓ.</s> <lb n="1799,001,,,,7,5,9"></lb> |
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6111 <s>Ἐπεὶ γάρ, ὃ μέρος ἐστὶν ὁ Α τοῦ ΒΓ, τὸ αὐτὸ μέρος <lb n="1799,001,,,,7,5,10"></lb> |
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6112 ἐστὶ καὶ ὁ Δ τοῦ ΕΖ, ὅσοι ἄρα εἰσὶν ἐν τῷ ΒΓ ἀριθμοὶ <lb n="1799,001,,,,7,5,11"></lb> |
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6113 ἴσοι τῷ Α, τοσοῦτοί εἰσι καὶ ἐν τῷ ΕΖ ἀριθμοὶ ἴσοι τῷ <lb n="1799,001,,,,7,5,12"></lb> |
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6114 Δ.</s> <s>διῃρήσθω ὁ μὲν ΒΓ εἰς τοὺς τῷ Α ἴσους τοὺς ΒΗ, <lb n="1799,001,,,,7,5,13"></lb> |
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6115 ΗΓ, ὁ δὲ ΕΖ εἰς τοὺς τῷ Δ ἴσους τοὺς ΕΘ, ΘΖ· ἔσται <lb n="1799,001,,,,7,5,14"></lb> |
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6116 δὴ ἴσον τὸ πλῆθος τῶν ΒΗ, ΗΓ τῷ πλήθει τῶν ΕΘ, <lb n="1799,001,,,,7,5,15"></lb> |
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6117 ΘΖ.</s> <s>καὶ ἐπεὶ ἴσος ἐστὶν ὁ μὲν ΒΗ τῷ Α, ὁ δὲ ΕΘ τῷ Δ, <lb n="1799,001,,,,7,5,16"></lb> |
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6118 καὶ οἱ ΒΗ, ΕΘ ἄρα τοῖς Α, Δ ἴσοι.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <pb></pb> <lb n="1799,001,,,,7,5,17"></lb> |
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6119 οἱ ΗΓ, ΘΖ τοῖς Α, Δ.</s> <s>ὅσοι ἄρα [εἰσὶν] ἐν τῷ ΒΓ ἀριθμοὶ <lb n="1799,001,,,,7,5,18"></lb> |
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6120 ἴσοι τῷ Α, τοσοῦτοί εἰσι καὶ ἐν τοῖς ΒΓ, ΕΖ ἴσοι τοῖς <lb n="1799,001,,,,7,5,19"></lb> |
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6121 Α, Δ.</s> <s>ὁσαπλασίων ἄρα ἐστὶν ὁ ΒΓ τοῦ Α, τοσαυταπλα-<lb n="1799,001,,,,7,5,20"></lb> |
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6122 σίων ἐστὶ καὶ συναμφότερος ὁ ΒΓ, ΕΖ συναμφοτέρου <lb n="1799,001,,,,7,5,21"></lb> |
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6123 τοῦ Α, Δ.</s> <s>ὃ ἄρα μέρος ἐστὶν ὁ Α τοῦ ΒΓ, τὸ αὐτὸ μέρος <lb n="1799,001,,,,7,5,22"></lb> |
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6124 ἐστὶ καὶ συναμφότερος ὁ Α, Δ συναμφοτέρου τοῦ ΒΓ, <lb n="1799,001,,,,7,5,23"></lb> |
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6125 ΕΖ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,5,24"></lb> |
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6126 <s>Ἐὰν ἀριθμὸς ἀριθμοῦ μέρη ᾖ, καὶ ἕτερος ἑτέρου τὰ <lb n="1799,001,,,,7,6,1"></lb> |
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6127 αὐτὰ μέρη ᾖ, καὶ συναμφότερος συναμφοτέρου τὰ αὐτὰ <lb n="1799,001,,,,7,6,2"></lb> |
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6128 μέρη ἔσται, ὅπερ ὁ εἷς τοῦ ἑνός.</s> <lb n="1799,001,,,,7,6,3"></lb> |
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6129 <s>Ἀριθμὸς γὰρ ὁ ΑΒ ἀριθμοῦ τοῦ Γ μέρη ἔστω, καὶ <lb n="1799,001,,,,7,6,4"></lb> |
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6130 ἕτερος ὁ ΔΕ ἑτέρου τοῦ Ζ τὰ αὐτὰ μέρη, ἅπερ ὁ ΑΒ τοῦ <lb n="1799,001,,,,7,6,5"></lb> |
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6131 Γ· λέγω, ὅτι καὶ συναμφότερος ὁ ΑΒ, ΔΕ συναμφοτέρου <lb n="1799,001,,,,7,6,6"></lb> |
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6132 τοῦ Γ, Ζ τὰ αὐτὰ μέρη ἐστίν, ἅπερ ὁ ΑΒ τοῦ Γ.</s> <lb n="1799,001,,,,7,6,7"></lb> |
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6133 <s>Ἐπεὶ γάρ, ἃ μέρη ἐστὶν ὁ ΑΒ τοῦ Γ, τὰ αὐτὰ μέρη καὶ <lb n="1799,001,,,,7,6,8"></lb> |
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6134 ὁ ΔΕ τοῦ Ζ, ὅσα ἄρα ἐστὶν ἐν τῷ ΑΒ μέρη τοῦ Γ, το-<lb n="1799,001,,,,7,6,9"></lb> |
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6135 σαῦτά ἐστι καὶ ἐν τῷ ΔΕ μέρη τοῦ Ζ.</s> <s>διῃρήσθω ὁ μὲν <lb n="1799,001,,,,7,6,10"></lb> |
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6136 ΑΒ εἰς τὰ τοῦ Γ μέρη τὰ ΑΗ, ΗΒ, ὁ δὲ ΔΕ εἰς τὰ τοῦ <lb n="1799,001,,,,7,6,11"></lb> |
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6137 Ζ μέρη τὰ ΔΘ, ΘΕ· ἔσται δὴ ἴσον τὸ <lb n="1799,001,,,,7,6,12"></lb> |
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6138 πλῆθος τῶν ΑΗ, ΗΒ τῷ πλήθει τῶν ΔΘ, <lb n="1799,001,,,,7,6,13"></lb> |
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6139 ΘΕ.</s> <s>καὶ ἐπεί, ὃ μέρος ἐστὶν ὁ ΑΗ τοῦ Γ, <lb n="1799,001,,,,7,6,14"></lb> |
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6140 τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΔΘ τοῦ Ζ, ὃ <lb n="1799,001,,,,7,6,15"></lb> |
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6141 ἄρα μέρος ἐστὶν ὁ ΑΗ τοῦ Γ, τὸ αὐτὸ μέ-<lb n="1799,001,,,,7,6,16"></lb> |
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6142 ρος ἐστὶ καὶ συναμφότερος ὁ ΑΗ, ΔΘ συν- <pb></pb><lb n="1799,001,,,,7,6,17"></lb> |
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6143 αμφοτέρου τοῦ Γ, Ζ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὃ μέρος ἐστὶν <lb n="1799,001,,,,7,6,18"></lb> |
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6144 ὁ ΗΒ τοῦ Γ, τὸ αὐτὸ μέρος ἐστὶ καὶ συναμφότερος ὁ <lb n="1799,001,,,,7,6,19"></lb> |
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6145 ΗΒ, ΘΕ συναμφοτέρου τοῦ Γ, Ζ.</s> <s>ἃ ἄρα μέρη ἐστὶν ὁ <lb n="1799,001,,,,7,6,20"></lb> |
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6146 ΑΒ τοῦ Γ, τὰ αὐτὰ μέρη ἐστὶ καὶ συναμφότερος ὁ ΑΒ, <lb n="1799,001,,,,7,6,21"></lb> |
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6147 ΔΕ συναμφοτέρου τοῦ Γ, Ζ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,6,22"></lb> |
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6148 <s>Ἐὰν ἀριθμὸς ἀριθμοῦ μέρος ᾖ, ὅπερ ἀφαιρεθεὶς ἀφαιρε-<lb n="1799,001,,,,7,7,1"></lb> |
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6149 θέντος, καὶ ὁ λοιπὸς τοῦ λοιποῦ τὸ αὐτὸ μέρος ἔσται, <lb n="1799,001,,,,7,7,2"></lb> |
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6150 ὅπερ ὁ ὅλος τοῦ ὅλου.</s> <lb n="1799,001,,,,7,7,3"></lb> |
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6151 <s>Ἀριθμὸς γὰρ ὁ ΑΒ ἀριθμοῦ τοῦ ΓΔ μέρος ἔστω, ὅπερ <lb n="1799,001,,,,7,7,4"></lb> |
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6152 ἀφαιρεθεὶς ὁ ΑΕ ἀφαιρεθέντος τοῦ ΓΖ· λέγω, ὅτι καὶ <lb n="1799,001,,,,7,7,5"></lb> |
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6153 λοιπὸς ὁ ΕΒ λοιποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος ἐστίν, ὅπερ <lb n="1799,001,,,,7,7,6"></lb> |
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6154 ὅλος ὁ ΑΒ ὅλου τοῦ ΓΔ.</s> <lb n="1799,001,,,,7,7,7"></lb> |
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6155 <s>Ὃ γὰρ μέρος ἐστὶν ὁ ΑΕ τοῦ ΓΖ, τὸ αὐτὸ μέρος ἔστω <lb n="1799,001,,,,7,7,8"></lb> |
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6156 καὶ ὁ ΕΒ τοῦ ΓΗ.</s> <s>καὶ ἐπεί, ὃ μέρος ἐστὶν ὁ ΑΕ τοῦ <lb n="1799,001,,,,7,7,9"></lb> |
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6157 ΓΖ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΕΒ τοῦ ΓΗ, ὃ ἄρα μέρος <lb n="1799,001,,,,7,7,10"></lb> |
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6158 ἐστὶν ὁ ΑΕ τοῦ ΓΖ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΑΒ τοῦ <lb n="1799,001,,,,7,7,11"></lb> |
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6159 ΗΖ.</s> <s>ὃ δὲ μέρος ἐστὶν ὁ ΑΕ τοῦ ΓΖ, τὸ αὐτὸ μέρος <lb n="1799,001,,,,7,7,12"></lb> |
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6160 ὑπόκειται καὶ ὁ ΑΒ τοῦ ΓΔ· ὃ ἄρα μέρος ἐστὶ καὶ ὁ ΑΒ <lb n="1799,001,,,,7,7,13"></lb> |
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6161 τοῦ ΗΖ, τὸ αὐτὸ μέρος ἐστὶ καὶ τοῦ ΓΔ· ἴσος ἄρα ἐστὶν <pb></pb> <lb n="1799,001,,,,7,7,14"></lb> |
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6162 ὁ ΗΖ τῷ ΓΔ.</s> <s>κοινὸς ἀφῃρήσθω ὁ ΓΖ· λοιπὸς ἄρα ὁ <lb n="1799,001,,,,7,7,15"></lb> |
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6163 ΗΓ λοιπῷ τῷ ΖΔ ἐστιν ἴσος.</s> <s>καὶ ἐπεί, ὃ μέρος ἐστὶν ὁ <lb n="1799,001,,,,7,7,16"></lb> |
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6164 ΑΕ τοῦ ΓΖ, τὸ αὐτὸ μέρος [ἐστὶ] καὶ ὁ ΕΒ τοῦ ΗΓ, ἴσος <lb n="1799,001,,,,7,7,17"></lb> |
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6165 δὲ ὁ ΗΓ τῷ ΖΔ, ὃ ἄρα μέρος ἐστὶν ὁ ΑΕ τοῦ ΓΖ, τὸ <lb n="1799,001,,,,7,7,18"></lb> |
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6166 αὐτὸ μέρος ἐστὶ καὶ ὁ ΕΒ τοῦ ΖΔ.</s> <s>ἀλλὰ ὃ μέρος ἐστὶν ὁ <lb n="1799,001,,,,7,7,19"></lb> |
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6167 ΑΕ τοῦ ΓΖ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΑΒ τοῦ ΓΔ· καὶ <lb n="1799,001,,,,7,7,20"></lb> |
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6168 λοιπὸς ἄρα ὁ ΕΒ λοιποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος ἐστίν, <lb n="1799,001,,,,7,7,21"></lb> |
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6169 ὅπερ ὅλος ὁ ΑΒ ὅλου τοῦ ΓΔ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,7,22"></lb> |
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6170 <s>Ἐὰν ἀριθμὸς ἀριθμοῦ μέρη ᾖ, ἅπερ ἀφαιρεθεὶς ἀφαιρε-<lb n="1799,001,,,,7,8,1"></lb> |
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6171 θέντος, καὶ ὁ λοιπὸς τοῦ λοιποῦ τὰ αὐτὰ μέρη ἔσται, <lb n="1799,001,,,,7,8,2"></lb> |
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6172 ἅπερ ὁ ὅλος τοῦ ὅλου.</s> <lb n="1799,001,,,,7,8,3"></lb> |
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6173 <s>Ἀριθμὸς γὰρ ὁ ΑΒ ἀριθμοῦ τοῦ ΓΔ μέρη ἔστω, ἅπερ <lb n="1799,001,,,,7,8,4"></lb> |
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6174 ἀφαιρεθεὶς ὁ ΑΕ ἀφαιρεθέντος τοῦ ΓΖ· λέγω, ὅτι καὶ <lb n="1799,001,,,,7,8,5"></lb> |
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6175 λοιπὸς ὁ ΕΒ λοιποῦ τοῦ ΖΔ τὰ αὐτὰ μέρη ἐστίν, ἅπερ <lb n="1799,001,,,,7,8,6"></lb> |
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6176 ὅλος ὁ ΑΒ ὅλου τοῦ ΓΔ.</s> <lb n="1799,001,,,,7,8,7"></lb> |
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6177 <s>Κείσθω γὰρ τῷ ΑΒ ἴσος ὁ ΗΘ.</s> <s>ἃ ἄρα μέρη ἐστὶν ὁ <lb n="1799,001,,,,7,8,8"></lb> |
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6178 ΗΘ τοῦ ΓΔ, τὰ αὐτὰ μέρη ἐστὶ καὶ ὁ ΑΕ τοῦ ΓΖ.</s> <lb n="1799,001,,,,7,8,9"></lb> |
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6179 <s>διῃρήσθω ὁ μὲν ΗΘ εἰς τὰ τοῦ ΓΔ <lb n="1799,001,,,,7,8,10"></lb> |
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6180 μέρη τὰ ΗΚ, ΚΘ, ὁ δὲ ΑΕ εἰς τὰ <lb n="1799,001,,,,7,8,11"></lb> |
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6181 τοῦ ΓΖ μέρη τὰ ΑΛ, ΛΕ· ἔσται δὴ <lb n="1799,001,,,,7,8,12"></lb> |
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6182 ἴσον τὸ πλῆθος τῶν ΗΚ, ΚΘ τῷ <lb n="1799,001,,,,7,8,13"></lb> |
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6183 πλήθει τῶν ΑΛ, ΛΕ.</s> <s>καὶ ἐπεί, ὃ <lb n="1799,001,,,,7,8,14"></lb> |
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6184 μέρος ἐστὶν ὁ ΗΚ τοῦ ΓΔ, τὸ αὐτὸ <pb></pb> <lb n="1799,001,,,,7,8,15"></lb> |
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6185 μέρος ἐστὶ καὶ ὁ ΑΛ τοῦ ΓΖ, μείζων δὲ ὁ ΓΔ τοῦ ΓΖ, <lb n="1799,001,,,,7,8,16"></lb> |
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6186 μείζων ἄρα καὶ ὁ ΗΚ τοῦ ΑΛ.</s> <s>κείσθω τῷ ΑΛ ἴσος ὁ <lb n="1799,001,,,,7,8,17"></lb> |
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6187 ΗΜ.</s> <s>ὃ ἄρα μέρος ἐστὶν ὁ ΗΚ τοῦ ΓΔ, τὸ αὐτὸ μέρος ἐστὶ <lb n="1799,001,,,,7,8,18"></lb> |
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6188 καὶ ὁ ΗΜ τοῦ ΓΖ· καὶ λοιπὸς ἄρα ὁ ΜΚ λοιποῦ τοῦ ΖΔ <lb n="1799,001,,,,7,8,19"></lb> |
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6189 τὸ αὐτὸ μέρος ἐστίν, ὅπερ ὅλος ὁ ΗΚ ὅλου τοῦ ΓΔ.</s> <s>πάλιν <lb n="1799,001,,,,7,8,20"></lb> |
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6190 ἐπεί, ὃ μέρος ἐστὶν ὁ ΚΘ τοῦ ΓΔ, τὸ αὐτὸ μέρος ἐστὶ καὶ <lb n="1799,001,,,,7,8,21"></lb> |
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6191 ὁ ΕΛ τοῦ ΓΖ, μείζων δὲ ὁ ΓΔ τοῦ ΓΖ, μείζων ἄρα καὶ ὁ <lb n="1799,001,,,,7,8,22"></lb> |
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6192 ΘΚ τοῦ ΕΛ.</s> <s>κείσθω τῷ ΕΛ ἴσος ὁ ΚΝ.</s> <s>ὃ ἄρα μέρος <lb n="1799,001,,,,7,8,23"></lb> |
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6193 ἐστὶν ὁ ΚΘ τοῦ ΓΔ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΚΝ τοῦ <lb n="1799,001,,,,7,8,24"></lb> |
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6194 ΓΖ· καὶ λοιπὸς ἄρα ὁ ΝΘ λοιποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος <lb n="1799,001,,,,7,8,25"></lb> |
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6195 ἐστίν, ὅπερ ὅλος ὁ ΚΘ ὅλου τοῦ ΓΔ.</s> <s>ἐδείχθη δὲ καὶ <lb n="1799,001,,,,7,8,26"></lb> |
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6196 λοιπὸς ὁ ΜΚ λοιποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος ὤν, ὅπερ ὅλος <lb n="1799,001,,,,7,8,27"></lb> |
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6197 ὁ ΗΚ ὅλου τοῦ ΓΔ· καὶ συναμφότερος ἄρα ὁ ΜΚ, ΝΘ <lb n="1799,001,,,,7,8,28"></lb> |
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6198 τοῦ ΔΖ τὰ αὐτὰ μέρη ἐστίν, ἅπερ ὅλος ὁ ΘΗ ὅλου <lb n="1799,001,,,,7,8,29"></lb> |
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6199 τοῦ ΓΔ.</s> <s>ἴσος δὲ συναμφότερος μὲν ὁ ΜΚ, ΝΘ τῷ ΕΒ, <lb n="1799,001,,,,7,8,30"></lb> |
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6200 ὁ δὲ ΘΗ τῷ ΒΑ· καὶ λοιπὸς ἄρα ὁ ΕΒ λοιποῦ τοῦ ΖΔ <lb n="1799,001,,,,7,8,31"></lb> |
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6201 τὰ αὐτὰ μέρη ἐστίν, ἅπερ ὅλος ὁ ΑΒ ὅλου τοῦ ΓΔ· ὅπερ <lb n="1799,001,,,,7,8,32"></lb> |
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6202 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,8,33"></lb> |
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6203 <s>Ἐὰν ἀριθμὸς ἀριθμοῦ μέρος ᾖ, καὶ ἕτερος ἑτέρου τὸ <lb n="1799,001,,,,7,9,1"></lb> |
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6204 αὐτὸ μέρος ᾖ, καὶ ἐναλλάξ, ὃ μέρος ἐστὶν ἢ μέρη ὁ πρῶτος <pb></pb> <lb n="1799,001,,,,7,9,2"></lb> |
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6205 τοῦ τρίτου, τὸ αὐτὸ μέρος ἔσται ἢ τὰ αὐτὰ μέρη καὶ ὁ <lb n="1799,001,,,,7,9,3"></lb> |
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6206 δεύτερος τοῦ τετάρτου.</s> <lb n="1799,001,,,,7,9,4"></lb> |
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6207 <s>Ἀριθμὸς γὰρ ὁ Α ἀριθμοῦ τοῦ ΒΓ μέρος ἔστω, καὶ <lb n="1799,001,,,,7,9,5"></lb> |
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6208 ἕτερος ὁ Δ ἑτέρου τοῦ ΕΖ τὸ αὐτὸ μέρος, ὅπερ ὁ Α τοῦ <lb n="1799,001,,,,7,9,6"></lb> |
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6209 ΒΓ· λέγω, ὅτι καὶ ἐναλλάξ, ὃ μέρος ἐστὶν ὁ Α τοῦ Δ <lb n="1799,001,,,,7,9,7"></lb> |
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6210 ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΒΓ τοῦ ΕΖ ἢ μέρη.</s> <lb n="1799,001,,,,7,9,8"></lb> |
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6211 <s>Ἐπεὶ γὰρ ὃ μέρος ἐστὶν ὁ Α τοῦ ΒΓ, τὸ αὐτὸ μέρος ἐστὶ <lb n="1799,001,,,,7,9,9"></lb> |
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6212 καὶ ὁ Δ τοῦ ΕΖ, ὅσοι ἄρα εἰσὶν ἐν τῷ <lb n="1799,001,,,,7,9,10"></lb> |
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6213 ΒΓ ἀριθμοὶ ἴσοι τῷ Α, τοσοῦτοί εἰσι <lb n="1799,001,,,,7,9,11"></lb> |
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6214 καὶ ἐν τῷ ΕΖ ἴσοι τῷ Δ.</s> <s>διῃρήσθω ὁ <lb n="1799,001,,,,7,9,12"></lb> |
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6215 μὲν ΒΓ εἰς τοὺς τῷ Α ἴσους τοὺς ΒΗ, <lb n="1799,001,,,,7,9,13"></lb> |
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6216 ΗΓ, ὁ δὲ ΕΖ εἰς τοὺς τῷ Δ ἴσους <lb n="1799,001,,,,7,9,14"></lb> |
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6217 τοὺς ΕΘ, ΘΖ· ἔσται δὴ ἴσον τὸ πλῆ-<lb n="1799,001,,,,7,9,15"></lb> |
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6218 θος τῶν ΒΗ, ΗΓ τῷ πλήθει τῶν ΕΘ, <lb n="1799,001,,,,7,9,16"></lb> |
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6219 ΘΖ.</s> <lb n="1799,001,,,,7,9,17"></lb> |
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6220 <s>Καὶ ἐπεὶ ἴσοι εἰσὶν οἱ ΒΗ, ΗΓ ἀριθμοὶ ἀλλήλοις, εἰσὶ <lb n="1799,001,,,,7,9,18"></lb> |
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6221 δὲ καὶ οἱ ΕΘ, ΘΖ ἀριθμοὶ ἴσοι ἀλλήλοις, καί ἐστιν ἴσον <lb n="1799,001,,,,7,9,19"></lb> |
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6222 τὸ πλῆθος τῶν ΒΗ, ΗΓ τῷ πλήθει τῶν ΕΘ, ΘΖ, ὃ <lb n="1799,001,,,,7,9,20"></lb> |
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6223 ἄρα μέρος ἐστὶν ὁ ΒΗ τοῦ ΕΘ ἢ μέρη, τὸ αὐτὸ μέρος <lb n="1799,001,,,,7,9,21"></lb> |
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6224 ἐστὶ καὶ ὁ ΗΓ τοῦ ΘΖ ἢ τὰ αὐτὰ μέρη· ὥστε καὶ ὃ μέρος <lb n="1799,001,,,,7,9,22"></lb> |
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6225 ἐστὶν ὁ ΒΗ τοῦ ΕΘ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ <lb n="1799,001,,,,7,9,23"></lb> |
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6226 συναμφότερος ὁ ΒΓ συναμφοτέρου τοῦ ΕΖ ἢ τὰ αὐτὰ <lb n="1799,001,,,,7,9,24"></lb> |
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6227 μέρη.</s> <s>ἴσος δὲ ὁ μὲν ΒΗ τῷ Α, ὁ δὲ ΕΘ τῷ Δ· ὃ ἄρα <lb n="1799,001,,,,7,9,25"></lb> |
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6228 μέρος ἐστὶν ὁ Α τοῦ Δ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ <lb n="1799,001,,,,7,9,26"></lb> |
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6229 ὁ ΒΓ τοῦ ΕΖ ἢ τὰ αὐτὰ μέρη· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,9,27"></lb> |
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6230 <s>Ἐὰν ἀριθμὸς ἀριθμοῦ μέρη ᾖ, καὶ ἕτερος ἑτέρου τὰ <lb n="1799,001,,,,7,10,1"></lb> |
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6231 αὐτὰ μέρη ᾖ, καὶ ἐναλλάξ, ἃ μέρη ἐστὶν ὁ πρῶτος τοῦ <pb></pb> <lb n="1799,001,,,,7,10,2"></lb> |
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6232 τρίτου ἢ μέρος, τὰ αὐτὰ μέρη ἔσται καὶ ὁ δεύτερος τοῦ <lb n="1799,001,,,,7,10,3"></lb> |
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6233 τετάρτου ἢ τὸ αὐτὸ μέρος.</s> <lb n="1799,001,,,,7,10,4"></lb> |
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6234 <s>Ἀριθμὸς γὰρ ὁ ΑΒ ἀριθμοῦ τοῦ Γ μέρη ἔστω, καὶ <lb n="1799,001,,,,7,10,5"></lb> |
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6235 ἕτερος ὁ ΔΕ ἑτέρου τοῦ Ζ τὰ αὐτὰ μέρη· λέγω, ὅτι καὶ <lb n="1799,001,,,,7,10,6"></lb> |
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6236 ἐναλλάξ, ἃ μέρη ἐστὶν ὁ ΑΒ τοῦ ΔΕ ἢ μέρος, τὰ αὐτὰ <lb n="1799,001,,,,7,10,7"></lb> |
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6237 μέρη ἐστὶ καὶ ὁ Γ τοῦ Ζ ἢ τὸ αὐτὸ μέρος. <lb n="1799,001,,,,7,10,8"></lb> |
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6238 Ἐπεὶ γάρ, ἃ μέρη ἐστὶν ὁ ΑΒ τοῦ Γ, <lb n="1799,001,,,,7,10,9"></lb> |
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6239 τὰ αὐτὰ μέρη ἐστὶ καὶ ὁ ΔΕ τοῦ Ζ, ὅσα <lb n="1799,001,,,,7,10,10"></lb> |
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6240 ἄρα ἐστὶν ἐν τῷ ΑΒ μέρη τοῦ Γ, τοσαῦτα <lb n="1799,001,,,,7,10,11"></lb> |
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6241 καὶ ἐν τῷ ΔΕ μέρη τοῦ Ζ.</s> <s>διῃρήσθω ὁ μὲν <lb n="1799,001,,,,7,10,12"></lb> |
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6242 ΑΒ εἰς τὰ τοῦ Γ μέρη τὰ ΑΗ, ΗΒ, ὁ δὲ <lb n="1799,001,,,,7,10,13"></lb> |
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6243 ΔΕ εἰς τὰ τοῦ Ζ μέρη τὰ ΔΘ, ΘΕ· ἔσται <lb n="1799,001,,,,7,10,14"></lb> |
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6244 δὴ ἴσον τὸ πλῆθος τῶν ΑΗ, ΗΒ τῷ πλή-<lb n="1799,001,,,,7,10,15"></lb> |
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6245 θει τῶν ΔΘ, ΘΕ.</s> <s>καὶ ἐπεί, ὃ μέρος ἐστὶν <lb n="1799,001,,,,7,10,16"></lb> |
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6246 ὁ ΑΗ τοῦ Γ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΔΘ τοῦ Ζ, καὶ <lb n="1799,001,,,,7,10,17"></lb> |
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6247 ἐναλλάξ, ὃ μέρος ἐστὶν ὁ ΑΗ τοῦ ΔΘ ἢ μέρη, τὸ αὐτὸ <lb n="1799,001,,,,7,10,18"></lb> |
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6248 μέρος ἐστὶ καὶ ὁ Γ τοῦ Ζ ἢ τὰ αὐτὰ μέρη.</s> <s>διὰ τὰ αὐτὰ δὴ <lb n="1799,001,,,,7,10,19"></lb> |
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6249 καί, ὃ μέρος ἐστὶν ὁ ΗΒ τοῦ ΘΕ ἢ μέρη, τὸ αὐτὸ μέρος <lb n="1799,001,,,,7,10,20"></lb> |
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6250 ἐστὶ καὶ ὁ Γ τοῦ Ζ ἢ τὰ αὐτὰ μέρη· ὥστε καί [ὃ μέρος <lb n="1799,001,,,,7,10,21"></lb> |
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6251 ἐστὶν ὁ ΑΗ τοῦ ΔΘ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ <lb n="1799,001,,,,7,10,22"></lb> |
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6252 ΗΒ τοῦ ΘΕ ἢ τὰ αὐτὰ μέρη· καὶ ὃ ἄρα μέρος ἐστὶν ὁ <pb></pb> <lb n="1799,001,,,,7,10,23"></lb> |
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6253 ΑΗ τοῦ ΔΘ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΑΒ τοῦ <lb n="1799,001,,,,7,10,24"></lb> |
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6254 ΔΕ ἢ τὰ αὐτὰ μέρη· ἀλλ' ὃ μέρος ἐστὶν ὁ ΑΗ τοῦ ΔΘ <lb n="1799,001,,,,7,10,25"></lb> |
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6255 ἢ μέρη, τὸ αὐτὸ μέρος ἐδείχθη καὶ ὁ Γ τοῦ Ζ ἢ τὰ αὐτὰ <lb n="1799,001,,,,7,10,26"></lb> |
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6256 μέρη, καὶ] ἃ [ἄρα] μέρη ἐστὶν ὁ ΑΒ τοῦ ΔΕ ἢ μέρος, τὰ <lb n="1799,001,,,,7,10,27"></lb> |
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6257 αὐτὰ μέρη ἐστὶ καὶ ὁ Γ τοῦ Ζ ἢ τὸ αὐτὸ μέρος· ὅπερ ἔδει <lb n="1799,001,,,,7,10,28"></lb> |
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6258 δεῖξαι.</s> <lb n="1799,001,,,,7,10,29"></lb> |
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6259 <s>Ἐὰν ᾖ ὡς ὅλος πρὸς ὅλον, οὕτως ἀφαιρεθεὶς πρὸς <lb n="1799,001,,,,7,11,1"></lb> |
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6260 ἀφαιρεθέντα, καὶ ὁ λοιπὸς πρὸς τὸν λοιπὸν ἔσται, ὡς <lb n="1799,001,,,,7,11,2"></lb> |
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6261 ὅλος πρὸς ὅλον.</s> <lb n="1799,001,,,,7,11,3"></lb> |
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6262 <s>Ἔστω ὡς ὅλος ὁ ΑΒ πρὸς ὅλον τὸν ΓΔ, οὕτως ἀφαιρε-<lb n="1799,001,,,,7,11,4"></lb> |
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6263 θεὶς ὁ ΑΕ πρὸς ἀφαιρεθέντα τὸν ΓΖ· λέγω, ὅτι καὶ <lb n="1799,001,,,,7,11,5"></lb> |
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6264 λοιπὸς ὁ ΕΒ πρὸς λοιπὸν τὸν ΖΔ ἐστιν, ὡς ὅλος ὁ ΑΒ <lb n="1799,001,,,,7,11,6"></lb> |
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6265 πρὸς ὅλον τὸν ΓΔ.</s> <lb n="1799,001,,,,7,11,7"></lb> |
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6266 <s>Ἐπεί ἐστιν ὡς ὁ ΑΒ πρὸς τὸν ΓΔ, οὕτως ὁ <lb n="1799,001,,,,7,11,8"></lb> |
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6267 ΑΕ πρὸς τὸν ΓΖ, ὃ ἄρα μέρος ἐστὶν ὁ ΑΒ τοῦ <lb n="1799,001,,,,7,11,9"></lb> |
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6268 ΓΔ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΑΕ τοῦ <lb n="1799,001,,,,7,11,10"></lb> |
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6269 ΓΖ ἢ τὰ αὐτὰ μέρη.</s> <s>καὶ λοιπὸς ἄρα ὁ ΕΒ λοι-<lb n="1799,001,,,,7,11,11"></lb> |
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6270 ποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος ἐστὶν ἢ μέρη, ἅπερ ὁ <lb n="1799,001,,,,7,11,12"></lb> |
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6271 ΑΒ τοῦ ΓΔ.</s> <s>ἔστιν ἄρα ὡς ὁ ΕΒ πρὸς τὸν ΖΔ, <lb n="1799,001,,,,7,11,13"></lb> |
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6272 οὕτως ὁ ΑΒ πρὸς τὸν ΓΔ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,7,11,14"></lb> |
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6273 Ἐὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ ἀνάλογον, ἔσται ὡς εἷς <lb n="1799,001,,,,7,12,1"></lb> |
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6274 τῶν ἡγουμένων πρὸς ἕνα τῶν ἑπομένων, οὕτως ἅπαντες <lb n="1799,001,,,,7,12,2"></lb> |
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6275 οἱ ἡγούμενοι πρὸς ἅπαντας τοὺς ἑπομένους.</s> <lb n="1799,001,,,,7,12,3"></lb> |
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6276 <s>Ἔστωσαν ὁποσοιοῦν ἀριθμοὶ ἀνάλογον οἱ Α, Β, Γ, Δ, <lb n="1799,001,,,,7,12,4"></lb> |
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6277 ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ· λέγω, ὅτι ἐστὶν <lb n="1799,001,,,,7,12,5"></lb> |
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6278 ὡς ὁ Α πρὸς τὸν Β, οὕτως οἱ Α, Γ πρὸς τοὺς Β, Δ.</s> <lb n="1799,001,,,,7,12,6"></lb> |
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6279 <s>Ἐπεὶ γάρ ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς <lb n="1799,001,,,,7,12,7"></lb> |
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6280 τὸν Δ, ὃ ἄρα μέρος ἐστὶν ὁ Α τοῦ Β ἢ μέρη, <lb n="1799,001,,,,7,12,8"></lb> |
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6281 τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Γ τοῦ Δ ἢ μέρη.</s> <lb n="1799,001,,,,7,12,9"></lb> |
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6282 <s>καὶ συναμφότερος ἄρα ὁ Α, Γ συναμφοτέ-<lb n="1799,001,,,,7,12,10"></lb> |
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6283 ρου τοῦ Β, Δ τὸ αὐτὸ μέρος ἐστὶν ἢ τὰ <lb n="1799,001,,,,7,12,11"></lb> |
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6284 αὐτὰ μέρη, ἅπερ ὁ Α τοῦ Β.</s> <s>ἔστιν ἄρα ὡς <lb n="1799,001,,,,7,12,12"></lb> |
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6285 ὁ Α πρὸς τὸν Β, οὕτως οἱ Α, Γ πρὸς τοὺς <lb n="1799,001,,,,7,12,13"></lb> |
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6286 Β, Δ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,12,14"></lb> |
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6287 <s>Ἐὰν τέσσαρες ἀριθμοὶ ἀνάλογον ὦσιν, καὶ ἐναλλὰξ <lb n="1799,001,,,,7,13,1"></lb> |
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6288 ἀνάλογον ἔσονται.</s> <lb n="1799,001,,,,7,13,2"></lb> |
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6289 <s>Ἔστωσαν τέσσαρες ἀριθμοὶ ἀνάλογον οἱ Α, Β, Γ, Δ, <lb n="1799,001,,,,7,13,3"></lb> |
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6290 ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ· λέγω, ὅτι καὶ <lb n="1799,001,,,,7,13,4"></lb> |
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6291 ἐναλλὰξ ἀνάλογον ἔσονται, ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Β <lb n="1799,001,,,,7,13,5"></lb> |
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6292 πρὸς τὸν Δ.</s> <s><pb></pb> <lb n="1799,001,,,,7,13,6"></lb> |
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6293 Ἐπεὶ γάρ ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ <lb n="1799,001,,,,7,13,7"></lb> |
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6294 Γ πρὸς τὸν Δ, ὃ ἄρα μέρος ἐστὶν ὁ Α τοῦ Β <lb n="1799,001,,,,7,13,8"></lb> |
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6295 ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Γ τοῦ Δ ἢ <lb n="1799,001,,,,7,13,9"></lb> |
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6296 τὰ αὐτὰ μέρη.</s> <s>ἐναλλὰξ ἄρα, ὃ μέρος ἐστὶν ὁ Α <lb n="1799,001,,,,7,13,10"></lb> |
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6297 τοῦ Γ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Β τοῦ <lb n="1799,001,,,,7,13,11"></lb> |
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6298 Δ ἢ τὰ αὐτὰ μέρη.</s> <s>ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν <lb n="1799,001,,,,7,13,12"></lb> |
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6299 Γ, οὕτως ὁ Β πρὸς τὸν Δ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,13,13"></lb> |
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6300 <s>Ἐὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ καὶ ἄλλοι αὐτοῖς ἴσοι τὸ <lb n="1799,001,,,,7,14,1"></lb> |
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6301 πλῆθος σύνδυο λαμβανόμενοι καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ <lb n="1799,001,,,,7,14,2"></lb> |
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6302 δι' ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσονται.</s> <lb n="1799,001,,,,7,14,3"></lb> |
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6303 <s>Ἔστωσαν ὁποσοιοῦν ἀριθμοὶ οἱ Α, Β, Γ καὶ ἄλλοι <lb n="1799,001,,,,7,14,4"></lb> |
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6304 αὐτοῖς ἴσοι τὸ πλῆθος σύνδυο λαμβανόμενοι ἐν τῷ αὐτῷ <lb n="1799,001,,,,7,14,5"></lb> |
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6305 λόγῳ οἱ Δ, Ε, Ζ, ὡς μὲν ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς <lb n="1799,001,,,,7,14,6"></lb> |
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6306 τὸν Ε, ὡς δὲ ὁ Β πρὸς τὸν Γ, οὕτως ὁ Ε πρὸς τὸν Ζ· λέγω, <lb n="1799,001,,,,7,14,7"></lb> |
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6307 ὅτι καὶ δι' ἴσου ἐστὶν ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς <lb n="1799,001,,,,7,14,8"></lb> |
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6308 τὸν Ζ.</s> <lb n="1799,001,,,,7,14,9"></lb> |
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6309 <s>Ἐπεὶ γάρ ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν <lb n="1799,001,,,,7,14,10"></lb> |
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6310 Ε, ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Δ, οὕτως ὁ Β πρὸς <lb n="1799,001,,,,7,14,11"></lb> |
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6311 τὸν Ε.</s> <s>πάλιν, ἐπεί ἐστιν ὡς ὁ Β πρὸς τὸν Γ, οὕτως ὁ Ε <lb n="1799,001,,,,7,14,12"></lb> |
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6312 πρὸς τὸν Ζ, ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Β πρὸς τὸν Ε, οὕτως <pb></pb> <lb n="1799,001,,,,7,14,13"></lb> |
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6313 ὁ Γ πρὸς τὸν Ζ.</s> <s>ὡς δὲ ὁ Β πρὸς τὸν Ε, οὕτως ὁ Α πρὸς τὸν <lb n="1799,001,,,,7,14,14"></lb> |
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6314 Δ· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Δ, οὕτως ὁ Γ πρὸς τὸν Ζ· <lb n="1799,001,,,,7,14,15"></lb> |
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6315 ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς <lb n="1799,001,,,,7,14,16"></lb> |
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6316 τὸν Ζ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,14,17"></lb> |
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6317 <s>Ἐὰν μονὰς ἀριθμόν τινα μετρῇ, ἰσάκις δὲ ἕτερος <lb n="1799,001,,,,7,15,1"></lb> |
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6318 ἀριθμὸς ἄλλον τινὰ ἀριθμὸν μετρῇ, καὶ ἐναλλὰξ ἰσάκις ἡ <lb n="1799,001,,,,7,15,2"></lb> |
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6319 μονὰς τὸν τρίτον ἀριθμὸν μετρήσει καὶ ὁ δεύτερος τὸν <lb n="1799,001,,,,7,15,3"></lb> |
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6320 τέταρτον.</s> <lb n="1799,001,,,,7,15,4"></lb> |
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6321 <s>Μονὰς γὰρ ἡ Α ἀριθμόν τινα τὸν ΒΓ μετρείτω, <lb n="1799,001,,,,7,15,5"></lb> |
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6322 ἰσάκις δὲ ἕτερος ἀριθμὸς ὁ Δ ἄλλον τινὰ ἀριθμὸν τὸν <lb n="1799,001,,,,7,15,6"></lb> |
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6323 ΕΖ μετρείτω· λέγω, ὅτι καὶ ἐναλλὰξ ἰσάκις ἡ Α μονὰς <lb n="1799,001,,,,7,15,7"></lb> |
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6324 τὸν Δ ἀριθμὸν μετρεῖ καὶ ὁ ΒΓ τὸν ΕΖ. <lb n="1799,001,,,,7,15,8"></lb> |
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6325 Ἐπεὶ γὰρ ἰσάκις ἡ Α μονὰς τὸν ΒΓ ἀριθμὸν μετρεῖ <lb n="1799,001,,,,7,15,9"></lb> |
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6326 καὶ ὁ Δ τὸν ΕΖ, ὅσαι ἄρα εἰσὶν ἐν τῷ ΒΓ μονάδες, <lb n="1799,001,,,,7,15,10"></lb> |
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6327 τοσοῦτοί εἰσι καὶ ἐν τῷ ΕΖ ἀριθμοὶ ἴσοι τῷ Δ.</s> <s>διῃρήσθω <lb n="1799,001,,,,7,15,11"></lb> |
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6328 ὁ μὲν ΒΓ εἰς τὰς ἐν ἑαυτῷ μονάδας τὰς ΒΗ, ΗΘ, ΘΓ, <lb n="1799,001,,,,7,15,12"></lb> |
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6329 ὁ δὲ ΕΖ εἰς τοὺς τῷ Δ ἴσους τοὺς ΕΚ, ΚΛ, ΛΖ.</s> <s>ἔσται <pb></pb> <lb n="1799,001,,,,7,15,13"></lb> |
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6330 δὴ ἴσον τὸ πλῆθος τῶν ΒΗ, ΗΘ, ΘΓ τῷ πλήθει τῶν <lb n="1799,001,,,,7,15,14"></lb> |
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6331 ΕΚ, ΚΛ, ΛΖ.</s> <s>καὶ ἐπεὶ ἴσαι εἰσὶν αἱ ΒΗ, ΗΘ, ΘΓ <lb n="1799,001,,,,7,15,15"></lb> |
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6332 μονάδες ἀλλήλαις, εἰσὶ δὲ καὶ οἱ ΕΚ, ΚΛ, ΛΖ ἀριθμοὶ <lb n="1799,001,,,,7,15,16"></lb> |
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6333 ἴσοι ἀλλήλοις, καί ἐστιν ἴσον τὸ πλῆθος τῶν ΒΗ, ΗΘ, <lb n="1799,001,,,,7,15,17"></lb> |
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6334 ΘΓ μονάδων τῷ πλήθει τῶν ΕΚ, ΚΛ, ΛΖ ἀριθμῶν, <lb n="1799,001,,,,7,15,18"></lb> |
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6335 ἔσται ἄρα ὡς ἡ ΒΗ μονὰς πρὸς τὸν ΕΚ ἀριθμόν, οὕτως <lb n="1799,001,,,,7,15,19"></lb> |
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6336 ἡ ΗΘ μονὰς πρὸς τὸν ΚΛ ἀριθμὸν καὶ ἡ ΘΓ μονὰς πρὸς <lb n="1799,001,,,,7,15,20"></lb> |
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6337 τὸν ΛΖ ἀριθμόν.</s> <s>ἔσται ἄρα καὶ ὡς εἷς τῶν ἡγουμένων <lb n="1799,001,,,,7,15,21"></lb> |
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6338 πρὸς ἕνα τῶν ἑπομένων, οὕτως ἅπαντες οἱ ἡγούμενοι <lb n="1799,001,,,,7,15,22"></lb> |
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6339 πρὸς ἅπαντας τοὺς ἑπομένους· ἔστιν ἄρα ὡς ἡ ΒΗ μονὰς <lb n="1799,001,,,,7,15,23"></lb> |
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6340 πρὸς τὸν ΕΚ ἀριθμόν, οὕτως ὁ ΒΓ πρὸς τὸν ΕΖ.</s> <s>ἴση δὲ <lb n="1799,001,,,,7,15,24"></lb> |
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6341 ἡ ΒΗ μονὰς τῇ Α μονάδι, ὁ δὲ ΕΚ ἀριθμὸς τῷ Δ ἀριθμῷ.</s> <lb n="1799,001,,,,7,15,25"></lb> |
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6342 <s>ἔστιν ἄρα ὡς ἡ Α μονὰς πρὸς τὸν Δ ἀριθμόν, οὕτως ὁ ΒΓ <lb n="1799,001,,,,7,15,26"></lb> |
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6343 πρὸς τὸν ΕΖ.</s> <s>ἰσάκις ἄρα ἡ Α μονὰς τὸν Δ ἀριθμὸν με-<lb n="1799,001,,,,7,15,27"></lb> |
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6344 τρεῖ καὶ ὁ ΒΓ τὸν ΕΖ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,15,28"></lb> |
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6345 <s>Ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί <lb n="1799,001,,,,7,16,1"></lb> |
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6346 τινας, οἱ γενόμενοι ἐξ αὐτῶν ἴσοι ἀλλήλοις ἔσονται.</s> <lb n="1799,001,,,,7,16,2"></lb> |
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6347 <s>Ἔστωσαν δύο ἀριθμοὶ οἱ Α, <lb n="1799,001,,,,7,16,3"></lb> |
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6348 Β, καὶ ὁ μὲν Α τὸν Β πολλα-<lb n="1799,001,,,,7,16,4"></lb> |
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6349 πλασιάσας τὸν Γ ποιείτω, ὁ δὲ <lb n="1799,001,,,,7,16,5"></lb> |
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6350 Β τὸν Α πολλαπλασιάσας τὸν Δ <lb n="1799,001,,,,7,16,6"></lb> |
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6351 ποιείτω· λέγω, ὅτι ἴσος ἐστὶν <lb n="1799,001,,,,7,16,7"></lb> |
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6352 ὁ Γ τῷ Δ.</s> <s><pb></pb> <lb n="1799,001,,,,7,16,8"></lb> |
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6353 Ἐπεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν, <lb n="1799,001,,,,7,16,9"></lb> |
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6354 ὁ Β ἄρα τὸν Γ μετρεῖ κατὰ τὰς ἐν τῷ Α μονάδας.</s> <s>μετρεῖ <lb n="1799,001,,,,7,16,10"></lb> |
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6355 δὲ καὶ ἡ Ε μονὰς τὸν Α ἀριθμὸν κατὰ τὰς ἐν αὐτῷ <lb n="1799,001,,,,7,16,11"></lb> |
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6356 μονάδας· ἰσάκις ἄρα ἡ Ε μονὰς τὸν Α ἀριθμὸν μετρεῖ καὶ <lb n="1799,001,,,,7,16,12"></lb> |
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6357 ὁ Β τὸν Γ.</s> <s>ἐναλλὰξ ἄρα ἰσάκις ἡ Ε μονὰς τὸν Β ἀριθμὸν <lb n="1799,001,,,,7,16,13"></lb> |
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6358 μετρεῖ καὶ ὁ Α τὸν Γ.</s> <s>πάλιν, ἐπεὶ ὁ Β τὸν Α πολλα-<lb n="1799,001,,,,7,16,14"></lb> |
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6359 πλασιάσας τὸν Δ πεποίηκεν, ὁ Α ἄρα τὸν Δ μετρεῖ κατὰ <lb n="1799,001,,,,7,16,15"></lb> |
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6360 τὰς ἐν τῷ Β μονάδας.</s> <s>μετρεῖ δὲ καὶ ἡ Ε μονὰς τὸν Β <lb n="1799,001,,,,7,16,16"></lb> |
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6361 κατὰ τὰς ἐν αὐτῷ μονάδας· ἰσάκις ἄρα ἡ Ε μονὰς τὸν Β <lb n="1799,001,,,,7,16,17"></lb> |
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6362 ἀριθμὸν μετρεῖ καὶ ὁ Α τὸν Δ.</s> <s>ἰσάκις δὲ ἡ Ε μονὰς τὸν Β <lb n="1799,001,,,,7,16,18"></lb> |
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6363 ἀριθμὸν ἐμέτρει καὶ ὁ Α τὸν Γ· ἰσάκις ἄρα ὁ Α ἑκάτερον <lb n="1799,001,,,,7,16,19"></lb> |
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6364 τῶν Γ, Δ μετρεῖ.</s> <s>ἴσος ἄρα ἐστὶν ὁ Γ τῷ Δ· ὅπερ ἔδει <lb n="1799,001,,,,7,16,20"></lb> |
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6365 δεῖξαι.</s> <lb n="1799,001,,,,7,16,21"></lb> |
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6366 <s>Ἐὰν ἀριθμὸς δύο ἀριθμοὺς πολλαπλασιάσας ποιῇ <lb n="1799,001,,,,7,17,1"></lb> |
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6367 τινας, οἱ γενόμενοι ἐξ αὐτῶν τὸν αὐτὸν ἕξουσι λόγον τοῖς <lb n="1799,001,,,,7,17,2"></lb> |
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6368 πολλαπλασιασθεῖσιν.</s> <lb n="1799,001,,,,7,17,3"></lb> |
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6369 <s>Ἀριθμὸς γὰρ ὁ Α δύο ἀριθμοὺς τοὺς Β, Γ πολλα-<lb n="1799,001,,,,7,17,4"></lb> |
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6370 πλασιάσας τοὺς Δ, Ε ποιείτω· λέγω, ὅτι ἐστὶν ὡς ὁ Β <lb n="1799,001,,,,7,17,5"></lb> |
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6371 πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς τὸν Ε. <pb></pb> <lb n="1799,001,,,,7,17,6"></lb> |
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6372 Ἐπεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Δ πεποίηκεν, <lb n="1799,001,,,,7,17,7"></lb> |
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6373 ὁ Β ἄρα τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Α μονάδας.</s> <s>μετρεῖ <lb n="1799,001,,,,7,17,8"></lb> |
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6374 δὲ καὶ ἡ Ζ μονὰς τὸν Α ἀριθμὸν κατὰ τὰς ἐν αὐτῷ μονά-<lb n="1799,001,,,,7,17,9"></lb> |
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6375 δας· ἰσάκις ἄρα ἡ Ζ μονὰς τὸν Α ἀριθμὸν μετρεῖ καὶ ὁ Β <lb n="1799,001,,,,7,17,10"></lb> |
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6376 τὸν Δ.</s> <s>ἔστιν ἄρα ὡς ἡ Ζ μονὰς πρὸς τὸν Α ἀριθμόν, οὕτως <lb n="1799,001,,,,7,17,11"></lb> |
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6377 ὁ Β πρὸς τὸν Δ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὡς ἡ Ζ μονὰς πρὸς <lb n="1799,001,,,,7,17,12"></lb> |
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6378 τὸν Α ἀριθμόν, οὕτως ὁ Γ πρὸς τὸν Ε· καὶ ὡς ἄρα ὁ Β <lb n="1799,001,,,,7,17,13"></lb> |
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6379 πρὸς τὸν Δ, οὕτως ὁ Γ πρὸς τὸν Ε.</s> <s>ἐναλλὰξ ἄρα ἐστὶν ὡς <lb n="1799,001,,,,7,17,14"></lb> |
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6380 ὁ Β πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς τὸν Ε· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,17,15"></lb> |
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6381 <s>Ἐὰν δύο ἀριθμοὶ ἀριθμόν τινα πολλαπλασιάσαντες <lb n="1799,001,,,,7,18,1"></lb> |
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6382 ποιῶσί τινας, οἱ γενόμενοι ἐξ αὐτῶν τὸν αὐτὸν ἕξουσι <lb n="1799,001,,,,7,18,2"></lb> |
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6383 λόγον τοῖς πολλαπλασιάσασιν.</s> <lb n="1799,001,,,,7,18,3"></lb> |
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6384 <s>Δύο γὰρ ἀριθμοὶ οἱ Α, Β ἀριθμόν τινα τὸν Γ πολλα-<lb n="1799,001,,,,7,18,4"></lb> |
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6385 πλασιάσαντες τοὺς Δ, Ε ποιείτωσαν· λέγω, ὅτι ἐστὶν <lb n="1799,001,,,,7,18,5"></lb> |
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6386 ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν Ε.</s> <lb n="1799,001,,,,7,18,6"></lb> |
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6387 <s>Ἐπεὶ γὰρ ὁ Α τὸν Γ πολλαπλασιάσας τὸν Δ πεποίηκεν, <lb n="1799,001,,,,7,18,7"></lb> |
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6388 καὶ ὁ Γ ἄρα τὸν Α πολλαπλασιάσας τὸν Δ πεποίηκεν.</s> <s>διὰ <lb n="1799,001,,,,7,18,8"></lb> |
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6389 τὰ αὐτὰ δὴ καὶ ὁ Γ τὸν Β πολλαπλασιάσας τὸν Ε πεποίη-<lb n="1799,001,,,,7,18,9"></lb> |
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6390 κεν.</s> <s>ἀριθμὸς δὴ ὁ Γ δύο <lb n="1799,001,,,,7,18,10"></lb> |
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6391 ἀριθμοὺς τοὺς Α, Β πολλα-<lb n="1799,001,,,,7,18,11"></lb> |
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6392 πλασιάσας τοὺς Δ, Ε πεποίη-<lb n="1799,001,,,,7,18,12"></lb> |
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6393 κεν.</s> <s>ἔστιν ἄρα ὡς ὁ Α πρὸς <lb n="1799,001,,,,7,18,13"></lb> |
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6394 τὸν Β, οὕτως ὁ Δ πρὸς τὸν <lb n="1799,001,,,,7,18,14"></lb> |
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6395 Ε· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,7,18,15"></lb> |
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6396 Ἐὰν τέσσαρες ἀριθμοὶ ἀνάλογον ὦσιν, ὁ ἐκ πρώτου καὶ <lb n="1799,001,,,,7,19,1"></lb> |
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6397 τετάρτου γενόμενος ἀριθμὸς ἴσος ἔσται τῷ ἐκ δευτέρου καὶ <lb n="1799,001,,,,7,19,2"></lb> |
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6398 τρίτου γενομένῳ ἀριθμῷ· καὶ ἐὰν ὁ ἐκ πρώτου καὶ τετάρ-<lb n="1799,001,,,,7,19,3"></lb> |
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6399 του γενόμενος ἀριθμὸς ἴσος ᾖ τῷ ἐκ δευτέρου καὶ τρίτου, <lb n="1799,001,,,,7,19,4"></lb> |
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6400 οἱ τέσσαρες ἀριθμοὶ ἀνάλογον ἔσονται.</s> <lb n="1799,001,,,,7,19,5"></lb> |
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6401 <s>Ἔστωσαν τέσσαρες ἀριθμοὶ ἀνά- <lb n="1799,001,,,,7,19,6"></lb> |
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6402 λογον οἱ Α, Β, Γ, Δ, ὡς ὁ Α πρὸς <lb n="1799,001,,,,7,19,7"></lb> |
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6403 τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ, καὶ <lb n="1799,001,,,,7,19,8"></lb> |
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6404 ὁ μὲν Α τὸν Δ πολλαπλασιάσας <lb n="1799,001,,,,7,19,9"></lb> |
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6405 τὸν Ε ποιείτω, ὁ δὲ Β τὸν Γ πολ-<lb n="1799,001,,,,7,19,10"></lb> |
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6406 λαπλασιάσας τὸν Ζ ποιείτω· λέγω, <lb n="1799,001,,,,7,19,11"></lb> |
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6407 ὅτι ἴσος ἐστὶν ὁ Ε τῷ Ζ.</s> <lb n="1799,001,,,,7,19,12"></lb> |
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6408 <s>Ὁ γὰρ Α τὸν Γ πολλαπλασιάσας <lb n="1799,001,,,,7,19,13"></lb> |
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6409 τὸν Η ποιείτω.</s> <s>ἐπεὶ οὖν ὁ Α τὸν Γ <lb n="1799,001,,,,7,19,14"></lb> |
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6410 πολλαπλασιάσας τὸν Η πεποίηκεν, <lb n="1799,001,,,,7,19,15"></lb> |
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6411 τὸν δὲ Δ πολλαπλασιάσας τὸν Ε πεποίηκεν, ἀριθμὸς δὴ ὁ <lb n="1799,001,,,,7,19,16"></lb> |
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6412 Α δύο ἀριθμοὺς τοὺς Γ, Δ πολλαπλασιάσας τοὺς Η, Ε <lb n="1799,001,,,,7,19,17"></lb> |
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6413 πεποίηκεν.</s> <s>ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Η <lb n="1799,001,,,,7,19,18"></lb> |
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6414 πρὸς τὸν Ε.</s> <s>ἀλλ' ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς <lb n="1799,001,,,,7,19,19"></lb> |
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6415 τὸν Β· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς τὸν <lb n="1799,001,,,,7,19,20"></lb> |
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6416 Ε.</s> <s>πάλιν, ἐπεὶ ὁ Α τὸν Γ πολλαπλασιάσας τὸν Η πε-<lb n="1799,001,,,,7,19,21"></lb> |
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6417 ποίηκεν, ἀλλὰ μὴν καὶ ὁ Β τὸν Γ πολλαπλασιάσας τὸν Ζ <lb n="1799,001,,,,7,19,22"></lb> |
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6418 πεποίηκεν, δύο δὴ ἀριθμοὶ οἱ Α, Β ἀριθμόν τινα τὸν Γ <pb></pb> <lb n="1799,001,,,,7,19,23"></lb> |
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6419 πολλαπλασιάσαντες τοὺς Η, Ζ πεποιήκασιν.</s> <s>ἔστιν ἄρα <lb n="1799,001,,,,7,19,24"></lb> |
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6420 ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς τὸν Ζ.</s> <s>ἀλλὰ μὴν καὶ <lb n="1799,001,,,,7,19,25"></lb> |
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6421 ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς τὸν Ε· καὶ ὡς ἄρα <lb n="1799,001,,,,7,19,26"></lb> |
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6422 ὁ Η πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν Ζ.</s> <s>ὁ Η ἄρα πρὸς <lb n="1799,001,,,,7,19,27"></lb> |
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6423 ἑκάτερον τῶν Ε, Ζ τὸν αὐτὸν ἔχει λόγον· ἴσος ἄρα ἐστὶν <lb n="1799,001,,,,7,19,28"></lb> |
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6424 ὁ Ε τῷ Ζ.</s> <lb n="1799,001,,,,7,19,29"></lb> |
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6425 <s>Ἔστω δὴ πάλιν ἴσος ὁ Ε τῷ Ζ· λέγω, ὅτι ἐστὶν ὡς ὁ Α <lb n="1799,001,,,,7,19,30"></lb> |
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6426 πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ.</s> <lb n="1799,001,,,,7,19,31"></lb> |
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6427 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεὶ ἴσος ἐστὶν ὁ <lb n="1799,001,,,,7,19,32"></lb> |
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6428 Ε τῷ Ζ, ἔστιν ἄρα ὡς ὁ Η πρὸς τὸν Ε, οὕτως ὁ Η πρὸς <lb n="1799,001,,,,7,19,33"></lb> |
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6429 τὸν Ζ.</s> <s>ἀλλ' ὡς μὲν ὁ Η πρὸς τὸν Ε, οὕτως ὁ Γ πρὸς τὸν <lb n="1799,001,,,,7,19,34"></lb> |
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6430 Δ, ὡς δὲ ὁ Η πρὸς τὸν Ζ, οὕτως ὁ Α πρὸς τὸν Β.</s> <s>καὶ ὡς <lb n="1799,001,,,,7,19,35"></lb> |
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6431 ἄρα ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ· ὅπερ ἔδει <lb n="1799,001,,,,7,19,36"></lb> |
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6432 δεῖξαι.</s> <lb n="1799,001,,,,7,19,37"></lb> |
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6433 <s>Οἱ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων <lb n="1799,001,,,,7,20,1"></lb> |
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6434 αὐτοῖς μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε <lb n="1799,001,,,,7,20,2"></lb> |
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6435 μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα.</s> <s><pb></pb> <lb n="1799,001,,,,7,20,3"></lb> |
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6436 Ἔστωσαν γὰρ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐ- <lb n="1799,001,,,,7,20,4"></lb> |
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6437 τὸν λόγον ἐχόντων τοῖς Α, Β οἱ ΓΔ, ΕΖ· λέγω, <lb n="1799,001,,,,7,20,5"></lb> |
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6438 ὅτι ἰσάκις ὁ ΓΔ τὸν Α μετρεῖ καὶ ὁ ΕΖ τὸν Β.</s> <lb n="1799,001,,,,7,20,6"></lb> |
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6439 <s>Ὁ ΓΔ γὰρ τοῦ Α οὔκ ἐστι μέρη.</s> <s>εἰ γὰρ δυνα-<lb n="1799,001,,,,7,20,7"></lb> |
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6440 τόν, ἔστω· καὶ ὁ ΕΖ ἄρα τοῦ Β τὰ αὐτὰ μέρη <lb n="1799,001,,,,7,20,8"></lb> |
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6441 ἐστίν, ἅπερ ὁ ΓΔ τοῦ Α.</s> <s>ὅσα ἄρα ἐστὶν ἐν τῷ <lb n="1799,001,,,,7,20,9"></lb> |
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6442 ΓΔ μέρη τοῦ Α, τοσαῦτά ἐστι καὶ ἐν τῷ ΕΖ μέρη <lb n="1799,001,,,,7,20,10"></lb> |
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6443 τοῦ Β.</s> <s>διῃρήσθω ὁ μὲν ΓΔ εἰς τὰ τοῦ Α μέρη <lb n="1799,001,,,,7,20,11"></lb> |
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6444 τὰ ΓΗ, ΗΔ, ὁ δὲ ΕΖ εἰς τὰ τοῦ Β μέρη τὰ <lb n="1799,001,,,,7,20,12"></lb> |
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6445 ΕΘ, ΘΖ· ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΓΗ, <lb n="1799,001,,,,7,20,13"></lb> |
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6446 ΗΔ τῷ πλήθει τῶν ΕΘ, ΘΖ.</s> <s>καὶ ἐπεὶ ἴσοι <lb n="1799,001,,,,7,20,14"></lb> |
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6447 εἰσὶν οἱ ΓΗ, ΗΔ ἀριθμοὶ ἀλλήλοις, εἰσὶ δὲ καὶ <lb n="1799,001,,,,7,20,15"></lb> |
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6448 οἱ ΕΘ, ΘΖ ἀριθμοὶ ἴσοι ἀλλήλοις, καί ἐστιν <lb n="1799,001,,,,7,20,16"></lb> |
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6449 ἴσον τὸ πλῆθος τῶν ΓΗ, ΗΔ τῷ πλήθει τῶν <lb n="1799,001,,,,7,20,17"></lb> |
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6450 ΕΘ, ΘΖ, ἔστιν ἄρα ὡς ὁ ΓΗ πρὸς τὸν ΕΘ, οὕτως ὁ <lb n="1799,001,,,,7,20,18"></lb> |
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6451 ΗΔ πρὸς τὸν ΘΖ.</s> <s>ἔσται ἄρα καὶ ὡς εἷς τῶν ἡγουμένων <lb n="1799,001,,,,7,20,19"></lb> |
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6452 πρὸς ἕνα τῶν ἑπομένων, οὕτως ἅπαντες οἱ ἡγούμενοι πρὸς <lb n="1799,001,,,,7,20,20"></lb> |
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6453 ἅπαντας τοὺς ἑπομένους.</s> <s>ἔστιν ἄρα ὡς ὁ ΓΗ πρὸς τὸν <lb n="1799,001,,,,7,20,21"></lb> |
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6454 ΕΘ, οὕτως ὁ ΓΔ πρὸς τὸν ΕΖ· οἱ ΓΗ, ΕΘ ἄρα τοῖς <lb n="1799,001,,,,7,20,22"></lb> |
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6455 ΓΔ, ΕΖ ἐν τῷ αὐτῷ λόγῳ εἰσὶν ἐλάσσονες ὄντες αὐτῶν· <lb n="1799,001,,,,7,20,23"></lb> |
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6456 ὅπερ ἐστὶν ἀδύνατον· ὑπόκεινται γὰρ οἱ ΓΔ, ΕΖ ἐλά-<lb n="1799,001,,,,7,20,24"></lb> |
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6457 χιστοι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.</s> <s>οὐκ ἄρα μέρη <lb n="1799,001,,,,7,20,25"></lb> |
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6458 ἐστὶν ὁ ΓΔ τοῦ Α· μέρος ἄρα.</s> <s>καὶ ὁ ΕΖ τοῦ Β τὸ αὐτὸ <pb></pb> <lb n="1799,001,,,,7,20,26"></lb> |
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6459 μέρος ἐστίν, ὅπερ ὁ ΓΔ τοῦ Α· ἰσάκις ἄρα ὁ ΓΔ τὸν Α <lb n="1799,001,,,,7,20,27"></lb> |
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6460 μετρεῖ καὶ ὁ ΕΖ τὸν Β· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,20,28"></lb> |
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6461 <s>Οἱ πρῶτοι πρὸς ἀλλήλους ἀριθμοὶ ἐλάχιστοί εἰσι τῶν <lb n="1799,001,,,,7,21,1"></lb> |
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6462 τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.</s> <lb n="1799,001,,,,7,21,2"></lb> |
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6463 <s>Ἔστωσαν πρῶτοι πρὸς ἀλλήλους ἀριθμοὶ <lb n="1799,001,,,,7,21,3"></lb> |
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6464 οἱ Α, Β· λέγω, ὅτι οἱ Α, Β ἐλάχιστοί εἰσι <lb n="1799,001,,,,7,21,4"></lb> |
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6465 τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.</s> <lb n="1799,001,,,,7,21,5"></lb> |
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6466 <s>Εἰ γὰρ μή, ἔσονταί τινες τῶν Α, Β ἐλάσ-<lb n="1799,001,,,,7,21,6"></lb> |
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6467 σονες ἀριθμοὶ ἐν τῷ αὐτῷ λόγῳ ὄντες <lb n="1799,001,,,,7,21,7"></lb> |
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6468 τοῖς Α, Β.</s> <s>ἔστωσαν οἱ Γ, Δ.</s> <lb n="1799,001,,,,7,21,8"></lb> |
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6469 <s>Ἐπεὶ οὖν οἱ ἐλάχιστοι ἀριθμοὶ τῶν τὸν <lb n="1799,001,,,,7,21,9"></lb> |
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6470 αὐτὸν λόγον ἐχόντων <ὐτοῖσ>μετροῦσι τοὺς <lb n="1799,001,,,,7,21,10"></lb> |
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6471 τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε μείζων <lb n="1799,001,,,,7,21,11"></lb> |
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6472 τὸν μείζονα καὶ ὁ ἐλάττων τὸν ἐλάττονα, <lb n="1799,001,,,,7,21,12"></lb> |
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6473 τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος <lb n="1799,001,,,,7,21,13"></lb> |
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6474 τὸν ἑπόμενον, ἰσάκις ἄρα ὁ Γ τὸν Α μετρεῖ καὶ ὁ Δ τὸν Β.</s> <lb n="1799,001,,,,7,21,14"></lb> |
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6475 <s>ὁσάκις δὴ ὁ Γ τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν <lb n="1799,001,,,,7,21,15"></lb> |
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6476 ἐν τῷ Ε.</s> <s>καὶ ὁ Δ ἄρα τὸν Β μετρεῖ κατὰ τὰς ἐν τῷ Ε <pb></pb> <lb n="1799,001,,,,7,21,16"></lb> |
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6477 μονάδας.</s> <s>καὶ ἐπεὶ ὁ Γ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Ε <lb n="1799,001,,,,7,21,17"></lb> |
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6478 μονάδας, καὶ ὁ Ε ἄρα τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Γ <lb n="1799,001,,,,7,21,18"></lb> |
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6479 μονάδας.</s> <s>διὰ τὰ αὐτὰ δὴ ὁ Ε καὶ τὸν Β μετρεῖ κατὰ τὰς <lb n="1799,001,,,,7,21,19"></lb> |
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6480 ἐν τῷ Δ μονάδας.</s> <s>ὁ Ε ἄρα τοὺς Α, Β μετρεῖ πρώτους <lb n="1799,001,,,,7,21,20"></lb> |
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6481 ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα <lb n="1799,001,,,,7,21,21"></lb> |
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6482 ἔσονταί τινες τῶν Α, Β ἐλάσσονες ἀριθμοὶ ἐν τῷ αὐτῷ <lb n="1799,001,,,,7,21,22"></lb> |
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6483 λόγῳ ὄντες τοῖς Α, Β.</s> <s>οἱ Α, Β ἄρα ἐλάχιστοί εἰσι τῶν τὸν <lb n="1799,001,,,,7,21,23"></lb> |
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6484 αὐτὸν λόγον ἐχόντων αὐτοῖς· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,21,24"></lb> |
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6485 <s>Οἱ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων <lb n="1799,001,,,,7,22,1"></lb> |
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6486 αὐτοῖς πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,7,22,2"></lb> |
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6487 <s>Ἔστωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον <lb n="1799,001,,,,7,22,3"></lb> |
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6488 ἐχόντων αὐτοῖς οἱ Α, Β· λέγω, ὅτι οἱ Α, Β πρῶτοι πρὸς <lb n="1799,001,,,,7,22,4"></lb> |
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6489 ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,7,22,5"></lb> |
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6490 <s>Εἰ γὰρ μή εἰσι πρῶτοι πρὸς ἀλλήλους, <lb n="1799,001,,,,7,22,6"></lb> |
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6491 μετρήσει τις αὐτοὺς ἀριθμός.</s> <s>μετρείτω, <lb n="1799,001,,,,7,22,7"></lb> |
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6492 καὶ ἔστω ὁ Γ.</s> <s>καὶ ὁσάκις μὲν ὁ Γ τὸν <lb n="1799,001,,,,7,22,8"></lb> |
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6493 Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν <lb n="1799,001,,,,7,22,9"></lb> |
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6494 τῷ Δ, ὁσάκις δὲ ὁ Γ τὸν Β μετρεῖ, το-<lb n="1799,001,,,,7,22,10"></lb> |
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6495 σαῦται μονάδες ἔστωσαν ἐν τῷ Ε.</s> <lb n="1799,001,,,,7,22,11"></lb> |
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6496 <s>Ἐπεὶ ὁ Γ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας, ὁ Γ <lb n="1799,001,,,,7,22,12"></lb> |
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6497 ἄρα τὸν Δ πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <s>διὰ τὰ <pb></pb> <lb n="1799,001,,,,7,22,13"></lb> |
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6498 αὐτὰ δὴ καὶ ὁ Γ τὸν Ε πολλαπλασιάσας τὸν Β πεποίηκεν.</s> <lb n="1799,001,,,,7,22,14"></lb> |
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6499 <s>ἀριθμὸς δὴ ὁ Γ δύο ἀριθμοὺς τοὺς Δ, Ε πολλαπλασιάσας <lb n="1799,001,,,,7,22,15"></lb> |
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6500 τοὺς Α, Β πεποίηκεν· ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Ε, <lb n="1799,001,,,,7,22,16"></lb> |
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6501 οὕτως ὁ Α πρὸς τὸν Β· οἱ Δ, Ε ἄρα τοῖς Α, Β ἐν τῷ αὐτῷ <lb n="1799,001,,,,7,22,17"></lb> |
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6502 λόγῳ εἰσὶν ἐλάσσονες ὄντες αὐτῶν· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,7,22,18"></lb> |
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6503 <s>οὐκ ἄρα τοὺς Α, Β ἀριθμοὺς ἀριθμός τις μετρήσει.</s> <s>οἱ <lb n="1799,001,,,,7,22,19"></lb> |
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6504 Α, Β ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,22,20"></lb> |
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6505 <s>Ἐὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, ὁ τὸν ἕνα <lb n="1799,001,,,,7,23,1"></lb> |
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6506 αὐτῶν μετρῶν ἀριθμὸς πρὸς τὸν λοιπὸν πρῶτος ἔσται.</s> <lb n="1799,001,,,,7,23,2"></lb> |
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6507 <s>Ἔστωσαν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλ-<lb n="1799,001,,,,7,23,3"></lb> |
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6508 λήλους οἱ Α, Β, τὸν δὲ Α μετρείτω τις <lb n="1799,001,,,,7,23,4"></lb> |
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6509 ἀριθμὸς ὁ Γ· λέγω, ὅτι καὶ οἱ Γ, Β πρῶ-<lb n="1799,001,,,,7,23,5"></lb> |
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6510 τοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,7,23,6"></lb> |
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6511 <s>Εἰ γὰρ μή εἰσιν οἱ Γ, Β πρῶτοι πρὸς ἀλ-<lb n="1799,001,,,,7,23,7"></lb> |
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6512 λήλους, μετρήσει [τισ] τοὺς Γ, Β ἀριθμός.</s> <lb n="1799,001,,,,7,23,8"></lb> |
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6513 <s>μετρείτω, καὶ ἔστω ὁ Δ.</s> <s>ἐπεὶ ὁ Δ τὸν Γ <lb n="1799,001,,,,7,23,9"></lb> |
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6514 μετρεῖ, ὁ δὲ Γ τὸν Α μετρεῖ, καὶ ὁ Δ ἄρα <lb n="1799,001,,,,7,23,10"></lb> |
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6515 τὸν Α μετρεῖ.</s> <s>μετρεῖ δὲ καὶ τὸν Β· ὁ Δ ἄρα τοὺς Α, Β <lb n="1799,001,,,,7,23,11"></lb> |
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6516 μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν ἀδύνα-<lb n="1799,001,,,,7,23,12"></lb> |
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6517 τον.</s> <s>οὐκ ἄρα τοὺς Γ, Β ἀριθμοὺς ἀριθμός τις μετρήσει.</s> <lb n="1799,001,,,,7,23,13"></lb> |
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6518 <s>οἱ Γ, Β ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,7,23,14"></lb> |
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6519 Ἐὰν δύο ἀριθμοὶ πρός τινα ἀριθμὸν πρῶτοι ὦσιν, καὶ <lb n="1799,001,,,,7,24,1"></lb> |
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6520 ὁ ἐξ αὐτῶν γενόμενος πρὸς τὸν αὐτὸν πρῶτος ἔσται.</s> <lb n="1799,001,,,,7,24,2"></lb> |
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6521 <s>Δύο γὰρ ἀριθμοὶ οἱ Α, Β πρός τινα <lb n="1799,001,,,,7,24,3"></lb> |
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6522 ἀριθμὸν τὸν Γ πρῶτοι ἔστωσαν, καὶ ὁ <lb n="1799,001,,,,7,24,4"></lb> |
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6523 Α τὸν Β πολλαπλασιάσας τὸν Δ ποιείτω· <lb n="1799,001,,,,7,24,5"></lb> |
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6524 λέγω, ὅτι οἱ Γ, Δ πρῶτοι πρὸς ἀλλή-<lb n="1799,001,,,,7,24,6"></lb> |
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6525 λους εἰσίν.</s> <lb n="1799,001,,,,7,24,7"></lb> |
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6526 <s>Εἰ γὰρ μή εἰσιν οἱ Γ, Δ πρῶτοι πρὸς <lb n="1799,001,,,,7,24,8"></lb> |
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6527 ἀλλήλους, μετρήσει [τισ] τοὺς Γ, Δ ἀριθ-<lb n="1799,001,,,,7,24,9"></lb> |
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6528 μός.</s> <s>μετρείτω, καὶ ἔστω ὁ Ε.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,7,24,10"></lb> |
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6529 οἱ Γ, Α πρῶτοι πρὸς ἀλλήλους εἰσίν, τὸν δὲ Γ μετρεῖ τις <lb n="1799,001,,,,7,24,11"></lb> |
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6530 ἀριθμὸς ὁ Ε, οἱ Α, Ε ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,7,24,12"></lb> |
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6531 <s>ὁσάκις δὴ ὁ Ε τὸν Δ μετρεῖ, τοσαῦται μονάδες ἔστωσαν <lb n="1799,001,,,,7,24,13"></lb> |
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6532 ἐν τῷ Ζ· καὶ ὁ Ζ ἄρα τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Ε μονά-<lb n="1799,001,,,,7,24,14"></lb> |
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6533 δας.</s> <s>ὁ Ε ἄρα τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν.</s> <s>ἀλλὰ <lb n="1799,001,,,,7,24,15"></lb> |
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6534 μὴν καὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Δ πεποίηκεν· ἴσος <lb n="1799,001,,,,7,24,16"></lb> |
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6535 ἄρα ἐστὶν ὁ ἐκ τῶν Ε, Ζ τῷ ἐκ τῶν Α, Β.</s> <s>ἐὰν δὲ ὁ ὑπὸ <lb n="1799,001,,,,7,24,17"></lb> |
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6536 τῶν ἄκρων ἴσος ᾖ τῷ ὑπὸ τῶν μέσων, οἱ τέσσαρες ἀριθμοὶ <lb n="1799,001,,,,7,24,18"></lb> |
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6537 ἀνάλογόν εἰσιν· ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Α, οὕτως ὁ Β <lb n="1799,001,,,,7,24,19"></lb> |
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6538 πρὸς τὸν Ζ.</s> <s>οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, <lb n="1799,001,,,,7,24,20"></lb> |
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6539 οἱ δὲ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων <lb n="1799,001,,,,7,24,21"></lb> |
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6540 αὐτοῖς μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις <lb n="1799,001,,,,7,24,22"></lb> |
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6541 ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, <pb></pb> <lb n="1799,001,,,,7,24,23"></lb> |
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6542 τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος <lb n="1799,001,,,,7,24,24"></lb> |
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6543 τὸν ἑπόμενον· ὁ Ε ἄρα τὸν Β μετρεῖ.</s> <s>μετρεῖ δὲ καὶ τὸν Γ· <lb n="1799,001,,,,7,24,25"></lb> |
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6544 ὁ Ε ἄρα τοὺς Β, Γ μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· <lb n="1799,001,,,,7,24,26"></lb> |
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6545 ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τοὺς Γ, Δ ἀριθμοὺς ἀριθμός <lb n="1799,001,,,,7,24,27"></lb> |
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6546 τις μετρήσει.</s> <s>οἱ Γ, Δ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· <lb n="1799,001,,,,7,24,28"></lb> |
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6547 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,24,29"></lb> |
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6548 <s>Ἐὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, ὁ ἐκ τοῦ <lb n="1799,001,,,,7,25,1"></lb> |
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6549 ἑνὸς αὐτῶν γενόμενος πρὸς τὸν λοιπὸν πρῶτος ἔσται.</s> <lb n="1799,001,,,,7,25,2"></lb> |
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6550 <s>Ἔστωσαν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλή-<lb n="1799,001,,,,7,25,3"></lb> |
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6551 λους οἱ Α, Β, καὶ ὁ Α ἑαυτὸν πολλαπλασιά-<lb n="1799,001,,,,7,25,4"></lb> |
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6552 σας τὸν Γ ποιείτω· λέγω, ὅτι οἱ Β, Γ πρῶ-<lb n="1799,001,,,,7,25,5"></lb> |
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6553 τοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,7,25,6"></lb> |
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6554 <s>Κείσθω γὰρ τῷ Α ἴσος ὁ Δ.</s> <s>ἐπεὶ οἱ Α, Β <lb n="1799,001,,,,7,25,7"></lb> |
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6555 πρῶτοι πρὸς ἀλλήλους εἰσίν, ἴσος δὲ ὁ Α τῷ <lb n="1799,001,,,,7,25,8"></lb> |
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6556 Δ, καὶ οἱ Δ, Β ἄρα πρῶτοι πρὸς ἀλλήλους εἰ-<lb n="1799,001,,,,7,25,9"></lb> |
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6557 σίν.</s> <s>ἑκάτερος ἄρα τῶν Δ, Α πρὸς τὸν Β πρῶτός ἐστιν· <lb n="1799,001,,,,7,25,10"></lb> |
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6558 καὶ ὁ ἐκ τῶν Δ, Α ἄρα γενόμενος πρὸς τὸν Β πρῶτος <lb n="1799,001,,,,7,25,11"></lb> |
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6559 ἔσται.</s> <s>ὁ δὲ ἐκ τῶν Δ, Α γενόμενος ἀριθμός ἐστιν ὁ Γ.</s> <lb n="1799,001,,,,7,25,12"></lb> |
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6560 <s>οἱ Γ, Β ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,25,13"></lb> |
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6561 <s>Ἐὰν δύο ἀριθμοὶ πρὸς δύο ἀριθμοὺς ἀμφότεροι πρὸς <lb n="1799,001,,,,7,26,1"></lb> |
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6562 ἑκάτερον πρῶτοι ὦσιν, καὶ οἱ ἐξ αὐτῶν γενόμενοι πρῶτοι <lb n="1799,001,,,,7,26,2"></lb> |
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6563 πρὸς ἀλλήλους ἔσονται.</s> <s><pb></pb> <lb n="1799,001,,,,7,26,3"></lb> |
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6564 Δύο γὰρ ἀριθμοὶ οἱ Α, Β πρὸς δύο ἀριθμοὺς τοὺς <lb n="1799,001,,,,7,26,4"></lb> |
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6565 Γ, Δ ἀμφότεροι πρὸς ἑκάτερον πρῶτοι ἔστωσαν, καὶ ὁ μὲν <lb n="1799,001,,,,7,26,5"></lb> |
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6566 Α τὸν Β πολλαπλασιάσας τὸν Ε ποιείτω, ὁ δὲ Γ τὸν Δ <lb n="1799,001,,,,7,26,6"></lb> |
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6567 πολλαπλασιάσας τὸν Ζ ποιείτω· λέγω, ὅτι οἱ Ε, Ζ πρῶτοι <lb n="1799,001,,,,7,26,7"></lb> |
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6568 πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,7,26,8"></lb> |
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6569 <s>Ἐπεὶ γὰρ ἑκάτερος τῶν Α, Β πρὸς τὸν Γ πρῶτός ἐστιν, <lb n="1799,001,,,,7,26,9"></lb> |
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6570 καὶ ὁ ἐκ τῶν Α, Β ἄρα γενόμενος πρὸς τὸν Γ πρῶτος <lb n="1799,001,,,,7,26,10"></lb> |
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6571 ἔσται.</s> <s>ὁ δὲ ἐκ τῶν Α, Β γενόμενός ἐστιν ὁ Ε· οἱ Ε, Γ ἄρα <lb n="1799,001,,,,7,26,11"></lb> |
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6572 πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ οἱ Δ, Ε <lb n="1799,001,,,,7,26,12"></lb> |
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6573 πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>ἑκάτερος ἄρα τῶν Γ, Δ <lb n="1799,001,,,,7,26,13"></lb> |
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6574 πρὸς τὸν Ε πρῶτός ἐστιν.</s> <s>καὶ ὁ ἐκ τῶν Γ, Δ ἄρα γενόμενος <lb n="1799,001,,,,7,26,14"></lb> |
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6575 πρὸς τὸν Ε πρῶτος ἔσται.</s> <s>ὁ δὲ ἐκ τῶν Γ, Δ γενόμενός <lb n="1799,001,,,,7,26,15"></lb> |
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6576 ἐστιν ὁ Ζ.</s> <s>οἱ Ε, Ζ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ <lb n="1799,001,,,,7,26,16"></lb> |
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6577 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,26,17"></lb> |
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6578 <s>Ἐὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, καὶ <lb n="1799,001,,,,7,27,1"></lb> |
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6579 πολλαπλασιάσας ἑκάτερος ἑαυτὸν ποιῇ τινα, οἱ γενόμενοι <lb n="1799,001,,,,7,27,2"></lb> |
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6580 ἐξ αὐτῶν πρῶτοι πρὸς ἀλλήλους ἔσονται, κἂν οἱ ἐξ ἀρχῆς <pb></pb> <lb n="1799,001,,,,7,27,3"></lb> |
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6581 τοὺς γενομένους πολλαπλασιάσαντες ποιῶσί τινας, κἀ-<lb n="1799,001,,,,7,27,4"></lb> |
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6582 κεῖνοι πρῶτοι πρὸς ἀλλήλους ἔσονται [καὶ ἀεὶ περὶ τοὺς <lb n="1799,001,,,,7,27,5"></lb> |
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6583 ἄκρους τοῦτο συμβαίνει].</s> <lb n="1799,001,,,,7,27,6"></lb> |
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6584 <s>Ἔστωσαν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους οἱ Α, <lb n="1799,001,,,,7,27,7"></lb> |
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6585 Β, καὶ ὁ Α ἑαυτὸν μὲν πολλαπλασιάσας τὸν Γ ποιείτω, <lb n="1799,001,,,,7,27,8"></lb> |
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6586 τὸν δὲ Γ πολλαπλασιάσας τὸν Δ ποιείτω, ὁ δὲ Β ἑαυτὸν <lb n="1799,001,,,,7,27,9"></lb> |
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6587 μὲν πολλαπλασιάσας τὸν Ε ποιείτω, τὸν δὲ Ε πολλαπλα-<lb n="1799,001,,,,7,27,10"></lb> |
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6588 σιάσας τὸν Ζ ποιείτω· λέγω, ὅτι οἵ τε Γ, Ε καὶ οἱ Δ, Ζ <lb n="1799,001,,,,7,27,11"></lb> |
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6589 πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,7,27,12"></lb> |
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6590 <s>Ἐπεὶ γὰρ οἱ Α, Β πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,7,27,13"></lb> |
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6591 εἰσίν, καὶ ὁ Α ἑαυτὸν πολλαπλασιάσας τὸν <lb n="1799,001,,,,7,27,14"></lb> |
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6592 Γ πεποίηκεν, οἱ Γ, Β ἄρα πρῶτοι πρὸς <lb n="1799,001,,,,7,27,15"></lb> |
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6593 ἀλλήλους εἰσίν.</s> <s>ἐπεὶ οὖν οἱ Γ, Β πρῶτοι <lb n="1799,001,,,,7,27,16"></lb> |
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6594 πρὸς ἀλλήλους εἰσίν, καὶ ὁ Β ἑαυτὸν πολ-<lb n="1799,001,,,,7,27,17"></lb> |
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6595 λαπλασιάσας τὸν Ε πεποίηκεν, οἱ Γ, Ε <lb n="1799,001,,,,7,27,18"></lb> |
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6596 ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>πάλιν, <lb n="1799,001,,,,7,27,19"></lb> |
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6597 ἐπεὶ οἱ Α, Β πρῶτοι πρὸς ἀλλήλους εἰσίν, <lb n="1799,001,,,,7,27,20"></lb> |
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6598 καὶ ὁ Β ἑαυτὸν πολλαπλασιάσας τὸν Ε πεποίηκεν, οἱ Α, Ε <lb n="1799,001,,,,7,27,21"></lb> |
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6599 ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>ἐπεὶ οὖν δύο ἀριθμοὶ οἱ Α, <lb n="1799,001,,,,7,27,22"></lb> |
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6600 Γ πρὸς δύο ἀριθμοὺς τοὺς Β, Ε ἀμφότεροι πρὸς ἑκάτερον <pb></pb> <lb n="1799,001,,,,7,27,23"></lb> |
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6601 πρῶτοί εἰσιν, καὶ ὁ ἐκ τῶν Α, Γ ἄρα γενόμενος πρὸς τὸν ἐκ <lb n="1799,001,,,,7,27,24"></lb> |
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6602 τῶν Β, Ε πρῶτός ἐστιν.</s> <s>καί ἐστιν ὁ μὲν ἐκ τῶν Α, Γ ὁ Δ, <lb n="1799,001,,,,7,27,25"></lb> |
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6603 ὁ δὲ ἐκ τῶν Β, Ε ὁ Ζ.</s> <s>οἱ Δ, Ζ ἄρα πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,7,27,26"></lb> |
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6604 εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,27,27"></lb> |
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6605 <s>Ἐὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, καὶ <lb n="1799,001,,,,7,28,1"></lb> |
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6606 συναμφότερος πρὸς ἑκάτερον αὐτῶν πρῶτος ἔσται· καὶ <lb n="1799,001,,,,7,28,2"></lb> |
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6607 ἐὰν συναμφότερος πρὸς ἕνα τινὰ αὐτῶν πρῶτος ᾖ, καὶ οἱ <lb n="1799,001,,,,7,28,3"></lb> |
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6608 ἐξ ἀρχῆς ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ἔσονται.</s> <lb n="1799,001,,,,7,28,4"></lb> |
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6609 <s>Συγκείσθωσαν γὰρ δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,7,28,5"></lb> |
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6610 οἱ ΑΒ, ΒΓ· λέγω, ὅτι καὶ συναμφότερος ὁ ΑΓ πρὸς <lb n="1799,001,,,,7,28,6"></lb> |
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6611 ἑκάτερον τῶν ΑΒ, ΒΓ πρῶτός ἐστιν.</s> <lb n="1799,001,,,,7,28,7"></lb> |
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6612 <s>Εἰ γὰρ μή εἰσιν οἱ ΓΑ, ΑΒ πρῶτοι πρὸς ἀλλήλους, <lb n="1799,001,,,,7,28,8"></lb> |
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6613 μετρήσει τις τοὺς ΓΑ, ΑΒ ἀριθμός.</s> <s>μετρείτω, καὶ <lb n="1799,001,,,,7,28,9"></lb> |
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6614 ἔστω ὁ Δ.</s> <s>ἐπεὶ οὖν ὁ Δ τοὺς ΓΑ, ΑΒ μετρεῖ, καὶ λοιπὸν <lb n="1799,001,,,,7,28,10"></lb> |
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6615 ἄρα τὸν ΒΓ μετρήσει.</s> <s>μετρεῖ δὲ καὶ τὸν ΒΑ· ὁ Δ ἄρα <lb n="1799,001,,,,7,28,11"></lb> |
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6616 τοὺς ΑΒ, ΒΓ μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· <lb n="1799,001,,,,7,28,12"></lb> |
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6617 ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τοὺς ΓΑ, ΑΒ ἀριθμοὺς <lb n="1799,001,,,,7,28,13"></lb> |
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6618 ἀριθμός τις μετρήσει· οἱ ΓΑ, ΑΒ ἄρα πρῶτοι πρὸς <lb n="1799,001,,,,7,28,14"></lb> |
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6619 ἀλλήλους εἰσίν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,7,28,15"></lb> |
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6620 οἱ ΑΓ, ΓΒ πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,7,28,16"></lb> |
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6621 εἰσίν.</s> <s>ὁ ΓΑ ἄρα πρὸς ἑκάτερον τῶν <lb n="1799,001,,,,7,28,17"></lb> |
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6622 ΑΒ, ΒΓ πρῶτός ἐστιν.</s> <s><pb></pb> <lb n="1799,001,,,,7,28,18"></lb> |
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6623 Ἔστωσαν δὴ πάλιν οἱ ΓΑ, ΑΒ πρῶτοι πρὸς ἀλλήλους· <lb n="1799,001,,,,7,28,19"></lb> |
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6624 λέγω, ὅτι καὶ οἱ ΑΒ, ΒΓ πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,7,28,20"></lb> |
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6625 <s>Εἰ γὰρ μή εἰσιν οἱ ΑΒ, ΒΓ πρῶτοι πρὸς ἀλλήλους, <lb n="1799,001,,,,7,28,21"></lb> |
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6626 μετρήσει τις τοὺς ΑΒ, ΒΓ ἀριθμός.</s> <s>μετρείτω, καὶ ἔστω <lb n="1799,001,,,,7,28,22"></lb> |
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6627 ὁ Δ.</s> <s>καὶ ἐπεὶ ὁ Δ ἑκάτερον τῶν ΑΒ, ΒΓ μετρεῖ, καὶ ὅλον <lb n="1799,001,,,,7,28,23"></lb> |
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6628 ἄρα τὸν ΓΑ μετρήσει.</s> <s>μετρεῖ δὲ καὶ τὸν ΑΒ· ὁ Δ ἄρα <lb n="1799,001,,,,7,28,24"></lb> |
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6629 τοὺς ΓΑ, ΑΒ μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ <lb n="1799,001,,,,7,28,25"></lb> |
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6630 ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τοὺς ΑΒ, ΒΓ ἀριθμοὺς ἀριθμός <lb n="1799,001,,,,7,28,26"></lb> |
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6631 τις μετρήσει.</s> <s>οἱ ΑΒ, ΒΓ ἄρα πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,7,28,27"></lb> |
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6632 εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,28,28"></lb> |
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6633 <s>Ἅπας πρῶτος ἀριθμὸς πρὸς ἅπαντα ἀριθμόν, ὃν μὴ <lb n="1799,001,,,,7,29,1"></lb> |
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6634 μετρεῖ, πρῶτός ἐστιν.</s> <lb n="1799,001,,,,7,29,2"></lb> |
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6635 <s>Ἔστω πρῶτος ἀριθμὸς ὁ Α καὶ τὸν Β μὴ μετρείτω· <lb n="1799,001,,,,7,29,3"></lb> |
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6636 λέγω, ὅτι οἱ Β, Α πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,7,29,4"></lb> |
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6637 <s>Εἰ γὰρ μή εἰσιν οἱ Β, Α πρῶτοι <lb n="1799,001,,,,7,29,5"></lb> |
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6638 πρὸς ἀλλήλους, μετρήσει τις αὐτοὺς <lb n="1799,001,,,,7,29,6"></lb> |
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6639 ἀριθμός.</s> <s>μετρείτω ὁ Γ.</s> <s>ἐπεὶ ὁ Γ τὸν <lb n="1799,001,,,,7,29,7"></lb> |
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6640 Β μετρεῖ, ὁ δὲ Α τὸν Β οὐ μετρεῖ, ὁ Γ <lb n="1799,001,,,,7,29,8"></lb> |
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6641 ἄρα τῷ Α οὔκ ἐστιν ὁ αὐτός.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,7,29,9"></lb> |
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6642 ὁ Γ τοὺς Β, Α μετρεῖ, καὶ τὸν Α ἄρα μετρεῖ πρῶτον ὄντα <lb n="1799,001,,,,7,29,10"></lb> |
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6643 μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τοὺς <lb n="1799,001,,,,7,29,11"></lb> |
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6644 Β, Α μετρήσει τις ἀριθμός.</s> <s>οἱ Α, Β ἄρα πρῶτοι πρὸς <lb n="1799,001,,,,7,29,12"></lb> |
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6645 ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,7,29,13"></lb> |
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6646 Ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί <lb n="1799,001,,,,7,30,1"></lb> |
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6647 τινα, τὸν δὲ γενόμενον ἐξ αὐτῶν μετρῇ τις πρῶτος <lb n="1799,001,,,,7,30,2"></lb> |
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6648 ἀριθμός, καὶ ἕνα τῶν ἐξ ἀρχῆς μετρήσει.</s> <lb n="1799,001,,,,7,30,3"></lb> |
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6649 <s>Δύο γὰρ ἀριθμοὶ οἱ Α, Β πολ- <lb n="1799,001,,,,7,30,4"></lb> |
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6650 λαπλασιάσαντες ἀλλήλους τὸν Γ <lb n="1799,001,,,,7,30,5"></lb> |
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6651 ποιείτωσαν, τὸν δὲ Γ μετρείτω τις <lb n="1799,001,,,,7,30,6"></lb> |
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6652 πρῶτος ἀριθμὸς ὁ Δ· λέγω, ὅτι ὁ <lb n="1799,001,,,,7,30,7"></lb> |
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6653 Δ ἕνα τῶν Α, Β μετρεῖ.</s> <lb n="1799,001,,,,7,30,8"></lb> |
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6654 <s>Τὸν γὰρ Α μὴ μετρείτω· καί <lb n="1799,001,,,,7,30,9"></lb> |
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6655 ἐστι πρῶτος ὁ Δ· οἱ Α, Δ ἄρα πρῶ-<lb n="1799,001,,,,7,30,10"></lb> |
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6656 τοι πρὸς ἀλλήλους εἰσίν.</s> <s>καὶ ὁσάκις ὁ Δ τὸν Γ μετρεῖ, το-<lb n="1799,001,,,,7,30,11"></lb> |
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6657 σαῦται μονάδες ἔστωσαν ἐν τῷ Ε.</s> <s>ἐπεὶ οὖν ὁ Δ τὸν Γ <lb n="1799,001,,,,7,30,12"></lb> |
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6658 μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας, ὁ Δ ἄρα τὸν Ε πολ-<lb n="1799,001,,,,7,30,13"></lb> |
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6659 λαπλασιάσας τὸν Γ πεποίηκεν.</s> <s>ἀλλὰ μὴν καὶ ὁ Α τὸν <lb n="1799,001,,,,7,30,14"></lb> |
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6660 Β πολλαπλασιάσας τὸν Γ πεποίηκεν· ἴσος ἄρα ἐστὶν ὁ ἐκ <lb n="1799,001,,,,7,30,15"></lb> |
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6661 τῶν Δ, Ε τῷ ἐκ τῶν Α, Β.</s> <s>ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Α, <lb n="1799,001,,,,7,30,16"></lb> |
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6662 οὕτως ὁ Β πρὸς τὸν Ε.</s> <s>οἱ δὲ Δ, Α πρῶτοι, οἱ δὲ πρῶτοι <lb n="1799,001,,,,7,30,17"></lb> |
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6663 καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν <lb n="1799,001,,,,7,30,18"></lb> |
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6664 λόγον ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ <lb n="1799,001,,,,7,30,19"></lb> |
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6665 ἐλάσσων τὸν ἐλάσσονα, τουτέστιν ὅ τε ἡγούμενος τὸν <lb n="1799,001,,,,7,30,20"></lb> |
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6666 ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον· ὁ Δ ἄρα τὸν <lb n="1799,001,,,,7,30,21"></lb> |
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6667 Β μετρεῖ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἐὰν τὸν Β μὴ μετρῇ, <lb n="1799,001,,,,7,30,22"></lb> |
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6668 τὸν Α μετρήσει.</s> <s>ὁ Δ ἄρα ἕνα τῶν Α, Β μετρεῖ· ὅπερ ἔδει <lb n="1799,001,,,,7,30,23"></lb> |
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6669 δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,7,30,24"></lb> |
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6670 Ἅπας σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ <lb n="1799,001,,,,7,31,1"></lb> |
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6671 μετρεῖται.</s> <lb n="1799,001,,,,7,31,2"></lb> |
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6672 <s>Ἔστω σύνθετος ἀριθμὸς ὁ Α· λέγω, ὅτι ὁ Α ὑπὸ <lb n="1799,001,,,,7,31,3"></lb> |
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6673 πρώτου τινὸς ἀριθμοῦ μετρεῖται.</s> <lb n="1799,001,,,,7,31,4"></lb> |
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6674 <s>Ἐπεὶ γὰρ σύνθετός ἐστιν ὁ Α, μετρήσει <lb n="1799,001,,,,7,31,5"></lb> |
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6675 τις αὐτὸν ἀριθμός.</s> <s>μετρείτω, καὶ ἔστω ὁ <lb n="1799,001,,,,7,31,6"></lb> |
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6676 Β.</s> <s>καὶ εἰ μὲν πρῶτός ἐστιν ὁ Β, γεγονὸς <lb n="1799,001,,,,7,31,7"></lb> |
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6677 ἂν εἴη τὸ ἐπιταχθέν.</s> <s>εἰ δὲ σύνθετος, με-<lb n="1799,001,,,,7,31,8"></lb> |
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6678 τρήσει τις αὐτὸν ἀριθμός.</s> <s>μετρείτω, καὶ ἔστω ὁ Γ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,7,31,9"></lb> |
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6679 ὁ Γ τὸν Β μετρεῖ, ὁ δὲ Β τὸν Α μετρεῖ, καὶ ὁ Γ ἄρα τὸν Α <lb n="1799,001,,,,7,31,10"></lb> |
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6680 μετρεῖ.</s> <s>καὶ εἰ μὲν πρῶτός ἐστιν ὁ Γ, γεγονὸς ἂν εἴη τὸ <lb n="1799,001,,,,7,31,11"></lb> |
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6681 ἐπιταχθέν.</s> <s>εἰ δὲ σύνθετος, μετρήσει τις αὐτὸν ἀριθμός.</s> <lb n="1799,001,,,,7,31,12"></lb> |
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6682 <s>τοιαύτης δὴ γινομένης ἐπισκέψεως ληφθήσεταί τις πρῶτος <lb n="1799,001,,,,7,31,13"></lb> |
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6683 ἀριθμός, ὃς μετρήσει.</s> <s>εἰ γὰρ οὐ ληφθήσεται, μετρήσουσι <lb n="1799,001,,,,7,31,14"></lb> |
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6684 τὸν Α ἀριθμὸν ἄπειροι ἀριθμοί, ὧν ἕτερος ἑτέρου ἐλάσσων <lb n="1799,001,,,,7,31,15"></lb> |
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6685 ἐστίν· ὅπερ ἐστὶν ἀδύνατον ἐν ἀριθμοῖς.</s> <s>ληφθήσεταί τις <lb n="1799,001,,,,7,31,16"></lb> |
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6686 ἄρα πρῶτος ἀριθμός, ὃς μετρήσει τὸν πρὸ ἑαυτοῦ, ὃς καὶ <lb n="1799,001,,,,7,31,17"></lb> |
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6687 τὸν Α μετρήσει.</s> <lb n="1799,001,,,,7,31,18"></lb> |
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6688 <s>Ἅπας ἄρα σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ <lb n="1799,001,,,,7,31,19"></lb> |
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6689 μετρεῖται· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,7,31,20"></lb> |
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6690 Ἅπας ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς <lb n="1799,001,,,,7,32,1"></lb> |
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6691 ἀριθμοῦ μετρεῖται.</s> <lb n="1799,001,,,,7,32,2"></lb> |
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6692 <s>Ἔστω ἀριθμὸς ὁ Α· λέγω, ὅτι ὁ Α ἤτοι πρῶτός ἐστιν <lb n="1799,001,,,,7,32,3"></lb> |
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6693 ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται.</s> <lb n="1799,001,,,,7,32,4"></lb> |
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6694 <s>Εἰ μὲν οὖν πρῶτός ἐστιν ὁ Α, γεγονὸς ἂν εἴη <lb n="1799,001,,,,7,32,5"></lb> |
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6695 τὸ ἐπιταχθέν.</s> <s>εἰ δὲ σύνθετος, μετρήσει τις αὐτὸν <lb n="1799,001,,,,7,32,6"></lb> |
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6696 πρῶτος ἀριθμός.</s> <lb n="1799,001,,,,7,32,7"></lb> |
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6697 <s>Ἅπας ἄρα ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ <lb n="1799,001,,,,7,32,8"></lb> |
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6698 πρώτου τινὸς ἀριθμοῦ μετρεῖται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,32,9"></lb> |
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6699 <s>Ἀριθμῶν δοθέντων ὁποσωνοῦν εὑρεῖν τοὺς ἐλαχίστους <lb n="1799,001,,,,7,33,1"></lb> |
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6700 τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.</s> <lb n="1799,001,,,,7,33,2"></lb> |
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6701 <s>Ἔστωσαν οἱ δοθέντες ὁποσοιοῦν ἀριθμοὶ οἱ Α, Β, Γ· <lb n="1799,001,,,,7,33,3"></lb> |
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6702 δεῖ δὴ εὑρεῖν τοὺς ἐλαχίστους τῶν τὸν αὐτὸν λόγον ἐχόντων <lb n="1799,001,,,,7,33,4"></lb> |
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6703 τοῖς Α, Β, Γ.</s> <lb n="1799,001,,,,7,33,5"></lb> |
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6704 <s>Οἱ Α, Β, Γ γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν <lb n="1799,001,,,,7,33,6"></lb> |
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6705 ἢ οὔ.</s> <s>εἰ μὲν οὖν οἱ Α, Β, Γ πρῶτοι πρὸς ἀλλήλους εἰσίν, <lb n="1799,001,,,,7,33,7"></lb> |
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6706 ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς. <pb></pb> <lb n="1799,001,,,,7,33,8"></lb> |
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6707 Εἰ δὲ οὔ, εἰλήφθω τῶν Α, Β, Γ τὸ μέγιστον κοινὸν <lb n="1799,001,,,,7,33,9"></lb> |
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6708 μέτρον ὁ Δ, καὶ ὁσάκις ὁ Δ ἕκαστον τῶν Α, Β, Γ μετρεῖ, <lb n="1799,001,,,,7,33,10"></lb> |
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6709 τοσαῦται μονάδες ἔστωσαν ἐν ἑκάστῳ τῶν Ε, Ζ, Η.</s> <s>καὶ <lb n="1799,001,,,,7,33,11"></lb> |
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6710 ἕκαστος ἄρα τῶν Ε, Ζ, Η ἕκαστον τῶν Α, Β, Γ μετρεῖ κατὰ <lb n="1799,001,,,,7,33,12"></lb> |
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6711 τὰς ἐν τῷ Δ μονάδας.</s> <s>οἱ Ε, Ζ, Η ἄρα τοὺς Α, Β, Γ ἰσάκις <lb n="1799,001,,,,7,33,13"></lb> |
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6712 μετροῦσιν· οἱ Ε, Ζ, Η ἄρα τοῖς Α, Β, Γ ἐν τῷ αὐτῷ λόγῳ <lb n="1799,001,,,,7,33,14"></lb> |
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6713 εἰσίν.</s> <s>λέγω δή, ὅτι καὶ ἐλάχιστοι.</s> <s>εἰ γὰρ μή εἰσιν οἱ Ε, <lb n="1799,001,,,,7,33,15"></lb> |
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6714 Ζ, Η ἐλάχιστοι τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β, <lb n="1799,001,,,,7,33,16"></lb> |
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6715 Γ, ἔσονται [τινεσ] τῶν Ε, Ζ, Η ἐλάσσονες ἀριθμοὶ ἐν <lb n="1799,001,,,,7,33,17"></lb> |
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6716 τῷ αὐτῷ λόγῳ ὄντες τοῖς Α, Β, Γ.</s> <s>ἔστωσαν οἱ Θ, Κ, Λ· <lb n="1799,001,,,,7,33,18"></lb> |
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6717 ἰσάκις ἄρα ὁ Θ τὸν Α μετρεῖ καὶ ἑκάτερος τῶν Κ, Λ <lb n="1799,001,,,,7,33,19"></lb> |
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6718 ἑκάτερον τῶν Β, Γ.</s> <s>ὁσάκις δὲ ὁ Θ τὸν Α μετρεῖ, τοσαῦται <lb n="1799,001,,,,7,33,20"></lb> |
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6719 μονάδες ἔστωσαν ἐν τῷ Μ· καὶ ἑκάτερος ἄρα τῶν Κ, Λ <lb n="1799,001,,,,7,33,21"></lb> |
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6720 ἑκάτερον τῶν Β, Γ μετρεῖ κατὰ τὰς ἐν τῷ Μ μονάδας.</s> <lb n="1799,001,,,,7,33,22"></lb> |
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6721 <s>καὶ ἐπεὶ ὁ Θ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Μ μονάδας, <lb n="1799,001,,,,7,33,23"></lb> |
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6722 καὶ ὁ Μ ἄρα τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Θ μονάδας.</s> <s>διὰ <lb n="1799,001,,,,7,33,24"></lb> |
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6723 τὰ αὐτὰ δὴ ὁ Μ καὶ ἑκάτερον τῶν Β, Γ μετρεῖ κατὰ τὰς <lb n="1799,001,,,,7,33,25"></lb> |
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6724 ἐν ἑκατέρῳ τῶν Κ, Λ μονάδας· ὁ Μ ἄρα τοὺς Α, Β, Γ <lb n="1799,001,,,,7,33,26"></lb> |
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6725 μετρεῖ.</s> <s>καὶ ἐπεὶ ὁ Θ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Μ <lb n="1799,001,,,,7,33,27"></lb> |
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6726 μονάδας, ὁ Θ ἄρα τὸν Μ πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <lb n="1799,001,,,,7,33,28"></lb> |
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6727 <s>διὰ τὰ αὐτὰ δὴ καὶ ὁ Ε τὸν Δ πολλαπλασιάσας τὸν Α <lb n="1799,001,,,,7,33,29"></lb> |
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6728 πεποίηκεν.</s> <s>ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Ε, Δ τῷ ἐκ τῶν Θ, <lb n="1799,001,,,,7,33,30"></lb> |
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6729 Μ.</s> <s>ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Μ πρὸς τὸν Δ.</s> <lb n="1799,001,,,,7,33,31"></lb> |
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6730 <s>μείζων δὲ ὁ Ε τοῦ Θ· μείζων ἄρα καὶ ὁ Μ τοῦ Δ.</s> <s>καὶ μετρεῖ <lb n="1799,001,,,,7,33,32"></lb> |
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6731 τοὺς Α, Β, Γ· ὅπερ ἐστὶν ἀδύνατον· ὑπόκειται γὰρ ὁ Δ <lb n="1799,001,,,,7,33,33"></lb> |
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6732 τῶν Α, Β, Γ τὸ μέγιστον κοινὸν μέτρον.</s> <s>οὐκ ἄρα ἔσονταί <lb n="1799,001,,,,7,33,34"></lb> |
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6733 τινες τῶν Ε, Ζ, Η ἐλάσσονες ἀριθμοὶ ἐν τῷ αὐτῷ λόγῳ <pb></pb> <lb n="1799,001,,,,7,33,35"></lb> |
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6734 ὄντες τοῖς Α, Β, Γ.</s> <s>οἱ Ε, Ζ, Η ἄρα ἐλάχιστοί εἰσι τῶν <lb n="1799,001,,,,7,33,36"></lb> |
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6735 τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β, Γ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,33,37"></lb> |
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6736 <s>Δύο ἀριθμῶν δοθέντων εὑρεῖν, ὃν ἐλάχιστον μετροῦσιν <lb n="1799,001,,,,7,34,1"></lb> |
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6737 ἀριθμόν.</s> <lb n="1799,001,,,,7,34,2"></lb> |
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6738 <s>Ἔστωσαν οἱ δοθέντες δύο ἀριθμοὶ οἱ Α, Β· δεῖ δὴ <lb n="1799,001,,,,7,34,3"></lb> |
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6739 εὑρεῖν, ὃν ἐλάχιστον μετροῦσιν ἀριθμόν.</s> <lb n="1799,001,,,,7,34,4"></lb> |
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6740 <s>Οἱ Α, Β γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ.</s> <lb n="1799,001,,,,7,34,5"></lb> |
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6741 <s>ἔστωσαν πρότερον οἱ Α, Β πρῶτοι πρὸς ἀλλήλους, καὶ <lb n="1799,001,,,,7,34,6"></lb> |
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6742 ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ ποιείτω· καὶ ὁ Β ἄρα <lb n="1799,001,,,,7,34,7"></lb> |
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6743 τὸν Α πολλαπλασιάσας τὸν Γ πεποίηκεν.</s> <s>οἱ Α, Β ἄρα <lb n="1799,001,,,,7,34,8"></lb> |
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6744 τὸν Γ μετροῦσιν.</s> <s>λέγω δή, ὅτι καὶ ἐλάχιστον.</s> <s>εἰ γὰρ μή, <lb n="1799,001,,,,7,34,9"></lb> |
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6745 μετρήσουσί τινα ἀριθμὸν οἱ Α, Β ἐλάσσονα ὄντα τοῦ Γ.</s> <lb n="1799,001,,,,7,34,10"></lb> |
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6746 <s>μετρείτωσαν τὸν Δ.</s> <s>καὶ ὁσάκις ὁ Α τὸν Δ μετρεῖ, τοσαῦται <lb n="1799,001,,,,7,34,11"></lb> |
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6747 μονάδες ἔστωσαν ἐν τῷ Ε, ὁσάκις δὲ ὁ Β τὸν Δ μετρεῖ, <lb n="1799,001,,,,7,34,12"></lb> |
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6748 τοσαῦται μονάδες ἔστωσαν ἐν τῷ Ζ· ὁ μὲν Α ἄρα τὸν Ε <lb n="1799,001,,,,7,34,13"></lb> |
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6749 πολλαπλασιάσας τὸν Δ πεποίηκεν, ὁ δὲ Β τὸν Ζ πολλα-<lb n="1799,001,,,,7,34,14"></lb> |
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6750 πλασιάσας τὸν Δ πεποίηκεν· ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Α, <lb n="1799,001,,,,7,34,15"></lb> |
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6751 Ε τῷ ἐκ τῶν Β, Ζ.</s> <s>ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως <lb n="1799,001,,,,7,34,16"></lb> |
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6752 ὁ Ζ πρὸς τὸν Ε.</s> <s>οἱ δὲ Α, Β πρῶτοι, οἱ δὲ πρῶτοι καὶ <pb></pb> <lb n="1799,001,,,,7,34,17"></lb> |
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6753 ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον <lb n="1799,001,,,,7,34,18"></lb> |
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6754 ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν <lb n="1799,001,,,,7,34,19"></lb> |
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6755 ἐλάσσονα· ὁ Β ἄρα τὸν Ε μετρεῖ, ὡς ἑπόμενος ἑπόμενον.</s> <lb n="1799,001,,,,7,34,20"></lb> |
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6756 <s>καὶ ἐπεὶ ὁ Α τοὺς Β, Ε πολλαπλασιάσας τοὺς Γ, Δ πε-<lb n="1799,001,,,,7,34,21"></lb> |
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6757 ποίηκεν, ἔστιν ἄρα ὡς ὁ Β πρὸς τὸν Ε, οὕτως ὁ Γ πρὸς <lb n="1799,001,,,,7,34,22"></lb> |
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6758 τὸν Δ.</s> <s>μετρεῖ δὲ ὁ Β τὸν Ε· μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ <lb n="1799,001,,,,7,34,23"></lb> |
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6759 ὁ μείζων τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα <lb n="1799,001,,,,7,34,24"></lb> |
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6760 οἱ Α, Β μετροῦσί τινα ἀριθμὸν ἐλάσσονα ὄντα τοῦ Γ.</s> <s>ὁ Γ <lb n="1799,001,,,,7,34,25"></lb> |
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6761 ἄρα ἐλάχιστος ὢν ὑπὸ τῶν Α, Β μετρεῖται. <lb n="1799,001,,,,7,34,26"></lb> |
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6762 Μὴ ἔστωσαν δὴ οἱ Α, Β πρῶτοι πρὸς ἀλλήλους, καὶ <lb n="1799,001,,,,7,34,27"></lb> |
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6763 εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον <lb n="1799,001,,,,7,34,28"></lb> |
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6764 ἐχόντων τοῖς Α, Β οἱ Ζ, Ε· ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Α, Ε <lb n="1799,001,,,,7,34,29"></lb> |
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6765 τῷ ἐκ τῶν Β, Ζ.</s> <s>καὶ ὁ Α τὸν Ε πολλαπλασιάσας τὸν Γ <lb n="1799,001,,,,7,34,30"></lb> |
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6766 ποιείτω· καὶ ὁ Β ἄρα τὸν Ζ πολλαπλασιάσας τὸν Γ <lb n="1799,001,,,,7,34,31"></lb> |
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6767 πεποίηκεν· οἱ Α, Β ἄρα τὸν Γ μετροῦσιν.</s> <s>λέγω δή, ὅτι καὶ <lb n="1799,001,,,,7,34,32"></lb> |
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6768 ἐλάχιστον.</s> <s>εἰ γὰρ μή, μετρήσουσί τινα ἀριθμὸν οἱ Α, Β <lb n="1799,001,,,,7,34,33"></lb> |
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6769 ἐλάσσονα ὄντα τοῦ Γ.</s> <s>μετρείτωσαν τὸν Δ.</s> <s>καὶ ὁσάκις <lb n="1799,001,,,,7,34,34"></lb> |
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6770 μὲν ὁ Α τὸν Δ μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν <pb></pb> <lb n="1799,001,,,,7,34,35"></lb> |
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6771 τῷ Η, ὁσάκις δὲ ὁ Β τὸν Δ μετρεῖ, τοσαῦται μονάδες <lb n="1799,001,,,,7,34,36"></lb> |
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6772 ἔστωσαν ἐν τῷ Θ.</s> <s>ὁ μὲν Α ἄρα τὸν Η πολλαπλασιάσας <lb n="1799,001,,,,7,34,37"></lb> |
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6773 τὸν Δ πεποίηκεν, ὁ δὲ Β τὸν Θ πολλαπλασιάσας τὸν Δ <lb n="1799,001,,,,7,34,38"></lb> |
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6774 πεποίηκεν.</s> <s>ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Α, Η τῷ ἐκ τῶν Β, Θ· <lb n="1799,001,,,,7,34,39"></lb> |
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6775 ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Θ πρὸς τὸν Η.</s> <lb n="1799,001,,,,7,34,40"></lb> |
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6776 <s>ὡς δὲ ὁ Α πρὸς τὸν Β, οὕτως ὁ Ζ πρὸς τὸν Ε· καὶ ὡς <lb n="1799,001,,,,7,34,41"></lb> |
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6777 ἄρα ὁ Ζ πρὸς τὸν Ε, οὕτως ὁ Θ πρὸς τὸν Η.</s> <s>οἱ δὲ Ζ, <lb n="1799,001,,,,7,34,42"></lb> |
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6778 Ε ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν <lb n="1799,001,,,,7,34,43"></lb> |
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6779 λόγον ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ <lb n="1799,001,,,,7,34,44"></lb> |
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6780 ἐλάσσων τὸν ἐλάσσονα· ὁ Ε ἄρα τὸν Η μετρεῖ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,7,34,45"></lb> |
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6781 ὁ Α τοὺς Ε, Η πολλαπλασιάσας τοὺς Γ, Δ πεποίηκεν, <lb n="1799,001,,,,7,34,46"></lb> |
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6782 ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Η, οὕτως ὁ Γ πρὸς τὸν Δ.</s> <lb n="1799,001,,,,7,34,47"></lb> |
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6783 <s>ὁ δὲ Ε τὸν Η μετρεῖ· καὶ ὁ Γ ἄρα τὸν Δ μετρεῖ ὁ μείζων <lb n="1799,001,,,,7,34,48"></lb> |
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6784 τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα οἱ Α, Β <lb n="1799,001,,,,7,34,49"></lb> |
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6785 μετρήσουσί τινα ἀριθμὸν ἐλάσσονα ὄντα τοῦ Γ.</s> <s>ὁ Γ ἄρα <lb n="1799,001,,,,7,34,50"></lb> |
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6786 ἐλάχιστος ὢν ὑπὸ τῶν Α, Β μετρεῖται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,34,51"></lb> |
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6787 <s>Ἐὰν δύο ἀριθμοὶ ἀριθμόν τινα μετρῶσιν, καὶ ὁ ἐλάχιστος <lb n="1799,001,,,,7,35,1"></lb> |
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6788 ὑπ' αὐτῶν μετρούμενος τὸν αὐτὸν μετρήσει.</s> <lb n="1799,001,,,,7,35,2"></lb> |
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6789 <s>Δύο γὰρ ἀριθμοὶ οἱ Α, Β ἀριθμόν τινα τὸν ΓΔ μετρεί-<lb n="1799,001,,,,7,35,3"></lb> |
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6790 τωσαν, ἐλάχιστον δὲ τὸν Ε· λέγω, ὅτι καὶ ὁ Ε τὸν ΓΔ <lb n="1799,001,,,,7,35,4"></lb> |
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6791 μετρεῖ.</s> <s><pb></pb> <lb n="1799,001,,,,7,35,5"></lb> |
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6792 Εἰ γὰρ οὐ μετρεῖ ὁ Ε τὸν ΓΔ, ὁ Ε τὸν ΔΖ μετρῶν <lb n="1799,001,,,,7,35,6"></lb> |
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6793 λειπέτω ἑαυτοῦ ἐλάσσονα τὸν ΓΖ.</s> <s>καὶ ἐπεὶ οἱ Α, Β τὸν <lb n="1799,001,,,,7,35,7"></lb> |
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6794 Ε μετροῦσιν, ὁ δὲ Ε τὸν ΔΖ μετρεῖ, καὶ οἱ Α, Β ἄρα τὸν <lb n="1799,001,,,,7,35,8"></lb> |
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6795 ΔΖ μετρήσουσιν.</s> <s>μετροῦσι δὲ καὶ ὅλον τὸν ΓΔ· καὶ <lb n="1799,001,,,,7,35,9"></lb> |
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6796 λοιπὸν ἄρα τὸν ΓΖ μετρήσουσιν ἐλάσσονα ὄντα τοῦ Ε· <lb n="1799,001,,,,7,35,10"></lb> |
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6797 ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα οὐ μετρεῖ ὁ Ε τὸν ΓΔ· <lb n="1799,001,,,,7,35,11"></lb> |
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6798 μετρεῖ ἄρα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,35,12"></lb> |
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6799 <s>Τριῶν ἀριθμῶν δοθέντων εὑρεῖν, ὃν ἐλάχιστον μετροῦ-<lb n="1799,001,,,,7,36,1"></lb> |
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6800 σιν ἀριθμόν.</s> <lb n="1799,001,,,,7,36,2"></lb> |
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6801 <s>Ἔστωσαν οἱ δοθέντες τρεῖς ἀριθμοὶ οἱ Α, Β, Γ· δεῖ <lb n="1799,001,,,,7,36,3"></lb> |
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6802 δὴ εὑρεῖν, ὃν ἐλάχιστον μετροῦσιν ἀριθμόν. <lb n="1799,001,,,,7,36,4"></lb> |
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6803 Εἰλήφθω γὰρ ὑπὸ δύο τῶν Α, Β ἐλάχιστος μετρούμενος <lb n="1799,001,,,,7,36,5"></lb> |
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6804 ὁ Δ.</s> <s>ὁ δὴ Γ τὸν Δ ἤτοι μετρεῖ ἢ οὐ μετρεῖ.</s> <s>μετρείτω <lb n="1799,001,,,,7,36,6"></lb> |
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6805 πρότερον.</s> <s>μετροῦσι δὲ καὶ οἱ Α, Β τὸν Δ· οἱ Α, Β, Γ <pb></pb> <lb n="1799,001,,,,7,36,7"></lb> |
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6806 ἄρα τὸν Δ μετροῦσιν.</s> <s>λέγω δή, ὅτι καὶ ἐλάχιστον.</s> <s>εἰ γὰρ <lb n="1799,001,,,,7,36,8"></lb> |
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6807 μή, μετρήσουσιν [τινα] ἀριθμὸν οἱ Α, Β, Γ ἐλάσσονα ὄντα <lb n="1799,001,,,,7,36,9"></lb> |
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6808 τοῦ Δ.</s> <s>μετρείτωσαν τὸν Ε.</s> <s>ἐπεὶ οἱ Α, Β, Γ τὸν Ε μετροῦσιν, <lb n="1799,001,,,,7,36,10"></lb> |
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6809 καὶ οἱ Α, Β ἄρα τὸν Ε μετροῦσιν.</s> <s>καὶ ὁ ἐλάχιστος ἄρα <lb n="1799,001,,,,7,36,11"></lb> |
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6810 ὑπὸ τῶν Α, Β μετρούμενος [τὸν Ε] μετρήσει.</s> <s>ἐλάχιστος <lb n="1799,001,,,,7,36,12"></lb> |
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6811 δὲ ὑπὸ τῶν Α, Β μετρούμενός ἐστιν ὁ Δ· ὁ Δ ἄρα τὸν Ε <lb n="1799,001,,,,7,36,13"></lb> |
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6812 μετρήσει ὁ μείζων τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,7,36,14"></lb> |
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6813 <s>οὐκ ἄρα οἱ Α, Β, Γ μετρήσουσί τινα ἀριθμὸν ἐλάσσονα <lb n="1799,001,,,,7,36,15"></lb> |
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6814 ὄντα τοῦ Δ· οἱ Α, Β, Γ ἄρα ἐλάχιστον τὸν Δ μετροῦσιν. <lb n="1799,001,,,,7,36,16"></lb> |
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6815 Μὴ μετρείτω δὴ πάλιν ὁ Γ τὸν Δ, καὶ εἰλήφθω ὑπὸ <lb n="1799,001,,,,7,36,17"></lb> |
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6816 τῶν Γ, Δ ἐλάχιστος μετρούμενος ἀριθμὸς ὁ Ε.</s> <s>ἐπεὶ οἱ <lb n="1799,001,,,,7,36,18"></lb> |
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6817 Α, Β τὸν Δ μετροῦσιν, ὁ δὲ Δ τὸν Ε μετρεῖ, καὶ οἱ Α, Β <lb n="1799,001,,,,7,36,19"></lb> |
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6818 ἄρα τὸν Ε μετροῦσιν.</s> <s>μετρεῖ δὲ καὶ ὁ Γ [τὸν Ε· καὶ] οἱ <lb n="1799,001,,,,7,36,20"></lb> |
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6819 Α, Β, Γ ἄρα τὸν Ε μετροῦσιν.</s> <s>λέγω δή, ὅτι καὶ ἐλάχιστον.</s> <lb n="1799,001,,,,7,36,21"></lb> |
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6820 <s>εἰ γὰρ μή, μετρήσουσί τινα οἱ Α, Β, Γ ἐλάσσονα ὄντα τοῦ <pb></pb> <lb n="1799,001,,,,7,36,22"></lb> |
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6821 Ε.</s> <s>μετρείτωσαν τὸν Ζ.</s> <s>ἐπεὶ οἱ Α, Β, Γ τὸν Ζ μετροῦσιν, <lb n="1799,001,,,,7,36,23"></lb> |
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6822 καὶ οἱ Α, Β ἄρα τὸν Ζ μετροῦσιν· καὶ ὁ ἐλάχιστος ἄρα <lb n="1799,001,,,,7,36,24"></lb> |
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6823 ὑπὸ τῶν Α, Β μετρούμενος τὸν Ζ μετρήσει.</s> <s>ἐλάχιστος <lb n="1799,001,,,,7,36,25"></lb> |
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6824 δὲ ὑπὸ τῶν Α, Β μετρούμενός ἐστιν ὁ Δ· ὁ Δ ἄρα τὸν Ζ <lb n="1799,001,,,,7,36,26"></lb> |
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6825 μετρεῖ.</s> <s>μετρεῖ δὲ καὶ ὁ Γ τὸν Ζ· οἱ Δ, Γ ἄρα τὸν Ζ <lb n="1799,001,,,,7,36,27"></lb> |
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6826 μετροῦσιν· ὥστε καὶ ὁ ἐλάχιστος ὑπὸ τῶν Δ, Γ μετρού-<lb n="1799,001,,,,7,36,28"></lb> |
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6827 μενος τὸν Ζ μετρήσει.</s> <s>ὁ δὲ ἐλάχιστος ὑπὸ τῶν Γ, Δ <lb n="1799,001,,,,7,36,29"></lb> |
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6828 μετρούμενός ἐστιν ὁ Ε· ὁ Ε ἄρα τὸν Ζ μετρεῖ ὁ μείζων <lb n="1799,001,,,,7,36,30"></lb> |
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6829 τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα οἱ Α, Β, Γ <lb n="1799,001,,,,7,36,31"></lb> |
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6830 μετρήσουσί τινα ἀριθμὸν ἐλάσσονα ὄντα τοῦ Ε.</s> <s>ὁ Ε ἄρα <lb n="1799,001,,,,7,36,32"></lb> |
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6831 ἐλάχιστος ὢν ὑπὸ τῶν Α, Β, Γ μετρεῖται· ὅπερ ἔδει <lb n="1799,001,,,,7,36,33"></lb> |
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6832 δεῖξαι.</s> <lb n="1799,001,,,,7,36,34"></lb> |
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6833 <s>Ἐὰν ἀριθμὸς ὑπό τινος ἀριθμοῦ μετρῆται, ὁ μετρούμενος <lb n="1799,001,,,,7,37,1"></lb> |
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6834 ὁμώνυμον μέρος ἕξει τῷ μετροῦντι.</s> <lb n="1799,001,,,,7,37,2"></lb> |
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6835 <s>Ἀριθμὸς γὰρ ὁ Α ὑπό τινος ἀριθμοῦ τοῦ Β μετρείσθω· <lb n="1799,001,,,,7,37,3"></lb> |
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6836 λέγω, ὅτι ὁ Α ὁμώνυμον μέρος ἔχει τῷ Β. <lb n="1799,001,,,,7,37,4"></lb> |
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6837 Ὁσάκις γὰρ ὁ Β τὸν Α μετρεῖ, τοσαῦται μονάδες <lb n="1799,001,,,,7,37,5"></lb> |
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6838 ἔστωσαν ἐν τῷ Γ.</s> <s>ἐπεὶ ὁ Β τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ <lb n="1799,001,,,,7,37,6"></lb> |
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6839 Γ μονάδας, μετρεῖ δὲ καὶ ἡ Δ μονὰς τὸν Γ ἀριθμὸν κατὰ <pb></pb> <lb n="1799,001,,,,7,37,7"></lb> |
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6840 τὰς ἐν αὐτῷ μονάδας, ἰσάκις ἄρα ἡ Δ μονὰς τὸν Γ ἀριθμὸν <lb n="1799,001,,,,7,37,8"></lb> |
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6841 μετρεῖ καὶ ὁ Β τὸν Α.</s> <s>ἐναλλὰξ ἄρα ἰσάκις ἡ Δ μονὰς τὸν Β <lb n="1799,001,,,,7,37,9"></lb> |
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6842 ἀριθμὸν μετρεῖ καὶ ὁ Γ τὸν Α· ὃ ἄρα μέρος ἐστὶν ἡ Δ <lb n="1799,001,,,,7,37,10"></lb> |
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6843 μονὰς τοῦ Β ἀριθμοῦ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Γ τοῦ Α.</s> <lb n="1799,001,,,,7,37,11"></lb> |
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6844 <s>ἡ δὲ Δ μονὰς τοῦ Β ἀριθμοῦ μέρος ἐστὶν ὁμώνυμον αὐτῷ· <lb n="1799,001,,,,7,37,12"></lb> |
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6845 καὶ ὁ Γ ἄρα τοῦ Α μέρος ἐστὶν ὁμώνυμον τῷ Β.</s> <s>ὥστε ὁ Α <lb n="1799,001,,,,7,37,13"></lb> |
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6846 μέρος ἔχει τὸν Γ ὁμώνυμον ὄντα τῷ Β· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,7,37,14"></lb> |
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6847 <s>Ἐὰν ἀριθμὸς μέρος ἔχῃ ὁτιοῦν, ὑπὸ ὁμωνύμου ἀριθμοῦ <lb n="1799,001,,,,7,38,1"></lb> |
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6848 μετρηθήσεται τῷ μέρει. <lb n="1799,001,,,,7,38,2"></lb> |
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6849 Ἀριθμὸς γὰρ ὁ Α μέρος ἐχέτω ὁτιοῦν τὸν Β, καὶ τῷ Β <lb n="1799,001,,,,7,38,3"></lb> |
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6850 μέρει ὁμώνυμος ἔστω [ἀριθμὸσ] ὁ Γ· λέγω, ὅτι ὁ Γ <lb n="1799,001,,,,7,38,4"></lb> |
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6851 τὸν Α μετρεῖ.</s> <lb n="1799,001,,,,7,38,5"></lb> |
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6852 <s>Ἐπεὶ γὰρ ὁ Β τοῦ Α μέρος ἐστὶν ὁμώνυμον τῷ Γ, ἔστι <lb n="1799,001,,,,7,38,6"></lb> |
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6853 δὲ καὶ ἡ Δ μονὰς τοῦ Γ μέρος ὁμώνυμον αὐτῷ, ὃ ἄρα μέρος <lb n="1799,001,,,,7,38,7"></lb> |
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6854 ἐστὶν ἡ Δ μονὰς τοῦ Γ ἀριθμοῦ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ <lb n="1799,001,,,,7,38,8"></lb> |
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6855 Β τοῦ Α· ἰσάκις ἄρα ἡ Δ μονὰς τὸν Γ ἀριθμὸν μετρεῖ καὶ <lb n="1799,001,,,,7,38,9"></lb> |
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6856 ὁ Β τὸν Α.</s> <s>ἐναλλὰξ ἄρα ἰσάκις ἡ Δ μονὰς τὸν Β ἀριθμὸν <lb n="1799,001,,,,7,38,10"></lb> |
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6857 μετρεῖ καὶ ὁ Γ τὸν Α.</s> <s>ὁ Γ ἄρα τὸν Α μετρεῖ· ὅπερ ἔδει <lb n="1799,001,,,,7,38,11"></lb> |
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6858 δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,7,38,12"></lb> |
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6859 Ἀριθμὸν εὑρεῖν, ὃς ἐλάχιστος ὢν ἕξει τὰ δοθέντα μέρη.</s> <lb n="1799,001,,,,7,39,1"></lb> |
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6860 <s>Ἔστω τὰ δοθέντα μέρη τὰ Α, Β, Γ· δεῖ δὴ ἀριθμὸν <lb n="1799,001,,,,7,39,2"></lb> |
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6861 εὑρεῖν, ὃς ἐλάχιστος ὢν ἕξει τὰ Α, Β, Γ μέρη.</s> <lb n="1799,001,,,,7,39,3"></lb> |
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6862 <s>Ἔστωσαν γὰρ τοῖς Α, Β, Γ <lb n="1799,001,,,,7,39,4"></lb> |
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6863 μέρεσιν ὁμώνυμοι ἀριθμοὶ οἱ Δ, <lb n="1799,001,,,,7,39,5"></lb> |
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6864 Ε, Ζ, καὶ εἰλήφθω ὑπὸ τῶν Δ, <lb n="1799,001,,,,7,39,6"></lb> |
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6865 Ε, Ζ ἐλάχιστος μετρούμενος <lb n="1799,001,,,,7,39,7"></lb> |
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6866 ἀριθμὸς ὁ Η.</s> <lb n="1799,001,,,,7,39,8"></lb> |
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6867 <s>Ὁ Η ἄρα ὁμώνυμα μέρη ἔχει <lb n="1799,001,,,,7,39,9"></lb> |
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6868 τοῖς Δ, Ε, Ζ.</s> <s>τοῖς δὲ Δ, Ε, Ζ <lb n="1799,001,,,,7,39,10"></lb> |
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6869 ὁμώνυμα μέρη ἐστὶ τὰ Α, Β, Γ· <lb n="1799,001,,,,7,39,11"></lb> |
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6870 ὁ Η ἄρα ἔχει τὰ Α, Β, Γ μέρη.</s> <s>λέγω δή, ὅτι καὶ ἐλάχιστος <lb n="1799,001,,,,7,39,12"></lb> |
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6871 ὤν.</s> <s>εἰ γὰρ μή, ἔσται τις τοῦ Η ἐλάσσων ἀριθμός, ὃς ἕξει <lb n="1799,001,,,,7,39,13"></lb> |
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6872 τὰ Α, Β, Γ μέρη.</s> <s>ἔστω ὁ Θ.</s> <s>ἐπεὶ ὁ Θ ἔχει τὰ Α, Β, Γ μέρη, <lb n="1799,001,,,,7,39,14"></lb> |
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6873 ὁ Θ ἄρα ὑπὸ ὁμωνύμων ἀριθμῶν μετρηθήσεται τοῖς Α, Β, <lb n="1799,001,,,,7,39,15"></lb> |
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6874 Γ μέρεσιν.</s> <s>τοῖς δὲ Α, Β, Γ μέρεσιν ὁμώνυμοι ἀριθμοί εἰ-<lb n="1799,001,,,,7,39,16"></lb> |
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6875 σιν οἱ Δ, Ε, Ζ· ὁ Θ ἄρα ὑπὸ τῶν Δ, Ε, Ζ μετρεῖται.</s> <s>καί <lb n="1799,001,,,,7,39,17"></lb> |
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6876 ἐστιν ἐλάσσων τοῦ Η· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα <lb n="1799,001,,,,7,39,18"></lb> |
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6877 ἔσται τις τοῦ Η ἐλάσσων ἀριθμός, ὃς ἕξει τὰ Α, Β, Γ <lb n="1799,001,,,,7,39,19"></lb> |
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6878 μέρη· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,7,39,20"></lb> |
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6879 </s></p> |
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6880 </chap> |
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6881 <chap><p type="main"><s>Ἐὰν ὦσιν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, οἱ δὲ <lb n="1799,001,,,,8,1,1"></lb> |
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6882 |
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6883 ἄκροι αὐτῶν πρῶτοι πρὸς ἀλλήλους ὦσιν, ἐλάχιστοί εἰσι <lb n="1799,001,,,,8,1,2"></lb> |
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6884 τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.</s> <lb n="1799,001,,,,8,1,3"></lb> |
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6885 <s>Ἔστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, <lb n="1799,001,,,,8,1,4"></lb> |
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6886 Δ, οἱ δὲ ἄκροι αὐτῶν οἱ Α, Δ πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,8,1,5"></lb> |
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6887 ἔστωσαν· λέγω, ὅτι οἱ Α, Β, Γ, Δ ἐλάχιστοί εἰσι τῶν τὸν <lb n="1799,001,,,,8,1,6"></lb> |
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6888 αὐτὸν λόγον ἐχόντων αὐτοῖς. <lb n="1799,001,,,,8,1,7"></lb> |
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6889 Εἰ γὰρ μή, ἔστωσαν ἐλάττονες τῶν Α, Β, Γ, Δ οἱ <lb n="1799,001,,,,8,1,8"></lb> |
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6890 Ε, Ζ, Η, Θ ἐν τῷ αὐτῷ λόγῳ ὄντες αὐτοῖς.</s> <s>καὶ ἐπεὶ οἱ <lb n="1799,001,,,,8,1,9"></lb> |
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6891 Α, Β, Γ, Δ ἐν τῷ αὐτῷ λόγῳ εἰσὶ τοῖς Ε, Ζ, Η, Θ, καί <lb n="1799,001,,,,8,1,10"></lb> |
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6892 ἐστιν ἴσον τὸ πλῆθος [τῶν Α, Β, Γ, Δ] τῷ πλήθει [τῶν <lb n="1799,001,,,,8,1,11"></lb> |
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6893 Ε, Ζ, Η, Θ], δι' ἴσου ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Δ, ὁ Ε <lb n="1799,001,,,,8,1,12"></lb> |
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6894 πρὸς τὸν Θ.</s> <s>οἱ δὲ Α, Δ πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, <lb n="1799,001,,,,8,1,13"></lb> |
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6895 οἱ δὲ ἐλάχιστοι ἀριθμοὶ μετροῦσι τοὺς τὸν αὐτὸν λόγον <pb></pb> <lb n="1799,001,,,,8,1,14"></lb> |
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6896 ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων <lb n="1799,001,,,,8,1,15"></lb> |
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6897 τὸν ἐλάσσονα, τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον <lb n="1799,001,,,,8,1,16"></lb> |
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6898 καὶ ὁ ἑπόμενος τὸν ἑπόμενον.</s> <s>μετρεῖ ἄρα ὁ Α τὸν Ε ὁ μεί-<lb n="1799,001,,,,8,1,17"></lb> |
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6899 ζων τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα οἱ Ε, Ζ, <lb n="1799,001,,,,8,1,18"></lb> |
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6900 Η, Θ ἐλάσσονες ὄντες τῶν Α, Β, Γ, Δ ἐν τῷ αὐτῷ λόγῳ <lb n="1799,001,,,,8,1,19"></lb> |
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6901 εἰσὶν αὐτοῖς.</s> <s>οἱ Α, Β, Γ, Δ ἄρα ἐλάχιστοί εἰσι τῶν τὸν <lb n="1799,001,,,,8,1,20"></lb> |
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6902 αὐτὸν λόγον ἐχόντων αὐτοῖς· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,1,21"></lb> |
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6903 <s>Ἀριθμοὺς εὑρεῖν ἑξῆς ἀνάλογον ἐλαχίστους, ὅσους ἂν <lb n="1799,001,,,,8,2,1"></lb> |
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6904 ἐπιτάξῃ τις, ἐν τῷ δοθέντι λόγῳ.</s> <lb n="1799,001,,,,8,2,2"></lb> |
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6905 <s>Ἔστω ὁ δοθεὶς λόγος ἐν ἐλαχίστοις ἀριθμοῖς ὁ τοῦ Α <lb n="1799,001,,,,8,2,3"></lb> |
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6906 πρὸς τὸν Β· δεῖ δὴ ἀριθμοὺς εὑρεῖν ἑξῆς ἀνάλογον ἐλαχίσ-<lb n="1799,001,,,,8,2,4"></lb> |
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6907 τους, ὅσους ἄν τις ἐπιτάξῃ, ἐν τῷ τοῦ Α πρὸς τὸν Β λόγῳ. <lb n="1799,001,,,,8,2,5"></lb> |
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6908 Ἐπιτετάχθωσαν δὴ τέσσαρες, καὶ ὁ Α ἑαυτὸν πολλα-<lb n="1799,001,,,,8,2,6"></lb> |
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6909 πλασιάσας τὸν Γ ποιείτω, τὸν δὲ Β πολλαπλασιάσας τὸν <lb n="1799,001,,,,8,2,7"></lb> |
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6910 Δ ποιείτω, καὶ ἔτι ὁ Β ἑαυτὸν πολλαπλασιάσας τὸν Ε <lb n="1799,001,,,,8,2,8"></lb> |
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6911 ποιείτω, καὶ ἔτι ὁ Α τοὺς Γ, Δ, Ε πολλαπλασιάσας τοὺς <pb></pb> <lb n="1799,001,,,,8,2,9"></lb> |
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6912 Ζ, Η, Θ ποιείτω, ὁ δὲ Β τὸν Ε πολλαπλασιάσας τὸν Κ <lb n="1799,001,,,,8,2,10"></lb> |
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6913 ποιείτω.</s> <lb n="1799,001,,,,8,2,11"></lb> |
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6914 <s>Καὶ ἐπεὶ ὁ Α ἑαυτὸν μὲν πολλαπλασιάσας τὸν Γ <lb n="1799,001,,,,8,2,12"></lb> |
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6915 πεποίηκεν, τὸν δὲ Β πολλαπλασιάσας τὸν Δ πεποίηκεν, <lb n="1799,001,,,,8,2,13"></lb> |
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6916 ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, [οὕτωσ] ὁ Γ πρὸς τὸν Δ.</s> <lb n="1799,001,,,,8,2,14"></lb> |
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6917 <s>πάλιν, ἐπεὶ ὁ μὲν Α τὸν Β πολλαπλασιάσας τὸν Δ πε-<lb n="1799,001,,,,8,2,15"></lb> |
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6918 ποίηκεν, ὁ δὲ Β ἑαυτὸν πολλαπλασιάσας τὸν Ε πεποίηκεν, <lb n="1799,001,,,,8,2,16"></lb> |
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6919 ἑκάτερος ἄρα τῶν Α, Β τὸν Β πολλαπλασιάσας ἑκάτερον <lb n="1799,001,,,,8,2,17"></lb> |
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6920 τῶν Δ, Ε πεποίηκεν.</s> <s>ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως <lb n="1799,001,,,,8,2,18"></lb> |
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6921 ὁ Δ πρὸς τὸν Ε.</s> <s>ἀλλ' ὡς ὁ Α πρὸς τὸν Β, ὁ Γ πρὸς τὸν Δ· <lb n="1799,001,,,,8,2,19"></lb> |
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6922 καὶ ὡς ἄρα ὁ Γ πρὸς τὸν Δ, ὁ Δ πρὸς τὸν Ε.</s> <s>καὶ ἐπεὶ ὁ Α <lb n="1799,001,,,,8,2,20"></lb> |
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6923 τοὺς Γ, Δ πολλαπλασιάσας τοὺς Ζ, Η πεποίηκεν, ἔστιν <lb n="1799,001,,,,8,2,21"></lb> |
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6924 ἄρα ὡς ὁ Γ πρὸς τὸν Δ, [οὕτωσ] ὁ Ζ πρὸς τὸν Η.</s> <s>ὡς δὲ ὁ <lb n="1799,001,,,,8,2,22"></lb> |
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6925 Γ πρὸς τὸν Δ, οὕτως ἦν ὁ Α πρὸς τὸν Β· καὶ ὡς ἄρα ὁ Α <lb n="1799,001,,,,8,2,23"></lb> |
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6926 πρὸς τὸν Β, ὁ Ζ πρὸς τὸν Η.</s> <s>πάλιν, ἐπεὶ ὁ Α τοὺς Δ, Ε <lb n="1799,001,,,,8,2,24"></lb> |
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6927 πολλαπλασιάσας τοὺς Η, Θ πεποίηκεν, ἔστιν ἄρα ὡς ὁ <lb n="1799,001,,,,8,2,25"></lb> |
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6928 Δ πρὸς τὸν Ε, ὁ Η πρὸς τὸν Θ.</s> <s>ἀλλ' ὡς ὁ Δ πρὸς τὸν Ε, <lb n="1799,001,,,,8,2,26"></lb> |
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6929 ὁ Α πρὸς τὸν Β.</s> <s>καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, οὕτως ὁ Η <lb n="1799,001,,,,8,2,27"></lb> |
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6930 πρὸς τὸν Θ.</s> <s>καὶ ἐπεὶ οἱ Α, Β τὸν Ε πολλαπλασιάσαντες <lb n="1799,001,,,,8,2,28"></lb> |
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6931 τοὺς Θ, Κ πεποιήκασιν, ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, <lb n="1799,001,,,,8,2,29"></lb> |
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6932 οὕτως ὁ Θ πρὸς τὸν Κ.</s> <s>ἀλλ' ὡς ὁ Α πρὸς τὸν Β, οὕτως <lb n="1799,001,,,,8,2,30"></lb> |
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6933 ὅ τε Ζ πρὸς τὸν Η καὶ ὁ Η πρὸς τὸν Θ.</s> <s>καὶ ὡς ἄρα ὁ Ζ <lb n="1799,001,,,,8,2,31"></lb> |
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6934 πρὸς τὸν Η, οὕτως ὅ τε Η πρὸς τὸν Θ καὶ ὁ Θ πρὸς τὸν Κ· <pb></pb> <lb n="1799,001,,,,8,2,32"></lb> |
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6935 οἱ Γ, Δ, Ε ἄρα καὶ οἱ Ζ, Η, Θ, Κ ἀνάλογόν εἰσιν ἐν τῷ <lb n="1799,001,,,,8,2,33"></lb> |
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6936 τοῦ Α πρὸς τὸν Β λόγῳ.</s> <s>λέγω δή, ὅτι καὶ ἐλάχιστοι.</s> <lb n="1799,001,,,,8,2,34"></lb> |
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6937 <s>ἐπεὶ γὰρ οἱ Α, Β ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον <lb n="1799,001,,,,8,2,35"></lb> |
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6938 ἐχόντων αὐτοῖς, οἱ δὲ ἐλάχιστοι τῶν τὸν αὐτὸν λόγον <lb n="1799,001,,,,8,2,36"></lb> |
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6939 ἐχόντων πρῶτοι πρὸς ἀλλήλους εἰσίν, οἱ Α, Β ἄρα πρῶτοι <lb n="1799,001,,,,8,2,37"></lb> |
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6940 πρὸς ἀλλήλους εἰσίν.</s> <s>καὶ ἑκάτερος μὲν τῶν Α, Β ἑαυτὸν <lb n="1799,001,,,,8,2,38"></lb> |
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6941 πολλαπλασιάσας ἑκάτερον τῶν Γ, Ε πεποίηκεν, ἑκάτερον <lb n="1799,001,,,,8,2,39"></lb> |
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6942 δὲ τῶν Γ, Ε πολλαπλασιάσας ἑκάτερον τῶν Ζ, Κ πε-<lb n="1799,001,,,,8,2,40"></lb> |
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6943 ποίηκεν· οἱ Γ, Ε ἄρα καὶ οἱ Ζ, Κ πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,8,2,41"></lb> |
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6944 εἰσίν.</s> <s>ἐὰν δὲ ὦσιν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, οἱ <lb n="1799,001,,,,8,2,42"></lb> |
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6945 δὲ ἄκροι αὐτῶν πρῶτοι πρὸς ἀλλήλους ὦσιν, ἐλάχιστοί <lb n="1799,001,,,,8,2,43"></lb> |
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6946 εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.</s> <s>οἱ Γ, Δ, Ε <lb n="1799,001,,,,8,2,44"></lb> |
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6947 ἄρα καὶ οἱ Ζ, Η, Θ, Κ ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον <lb n="1799,001,,,,8,2,45"></lb> |
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6948 ἐχόντων τοῖς Α, Β· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,2,46"></lb> |
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6949 <s>Πόρισμα <lb n="1799,001,,,,8,2,47"></lb> |
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6950 Ἐκ δὴ τούτου φανερόν, ὅτι ἐὰν τρεῖς ἀριθμοὶ ἑξῆς ἀνά-<lb n="1799,001,,,,8,2,48"></lb> |
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6951 λογον ἐλάχιστοι ὦσι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς, <lb n="1799,001,,,,8,2,49"></lb> |
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6952 οἱ ἄκροι αὐτῶν τετράγωνοί εἰσιν, ἐὰν δὲ τέσσαρες, κύβοι.</s> <lb n="1799,001,,,,8,2,50"></lb> |
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6953 <s>Ἐὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον ἐλάχιστοι <lb n="1799,001,,,,8,3,1"></lb> |
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6954 τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς, οἱ ἄκροι αὐτῶν <lb n="1799,001,,,,8,3,2"></lb> |
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6955 πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s><pb></pb> <lb n="1799,001,,,,8,3,3"></lb> |
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6956 Ἔστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον ἐλάχιστοι <lb n="1799,001,,,,8,3,4"></lb> |
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6957 τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς οἱ Α, Β, Γ, Δ· λέγω, <lb n="1799,001,,,,8,3,5"></lb> |
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6958 ὅτι οἱ ἄκροι αὐτῶν οἱ Α, Δ πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,8,3,6"></lb> |
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6959 <s>Εἰλήφθωσαν γὰρ δύο <lb n="1799,001,,,,8,3,7"></lb> |
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6960 μὲν ἀριθμοὶ ἐλάχιστοι ἐν <lb n="1799,001,,,,8,3,8"></lb> |
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6961 τῷ τῶν Α, Β, Γ, Δ λόγῳ <lb n="1799,001,,,,8,3,9"></lb> |
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6962 οἱ Ε, Ζ, τρεῖς δὲ οἱ Η, <lb n="1799,001,,,,8,3,10"></lb> |
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6963 Θ, Κ, καὶ ἑξῆς ἑνὶ πλεί-<lb n="1799,001,,,,8,3,11"></lb> |
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6964 ους, ἕως τὸ λαμβανόμε-<lb n="1799,001,,,,8,3,12"></lb> |
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6965 νον πλῆθος ἴσον γένηται <lb n="1799,001,,,,8,3,13"></lb> |
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6966 τῷ πλήθει τῶν Α, Β, <lb n="1799,001,,,,8,3,14"></lb> |
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6967 Γ, Δ.</s> <s>εἰλήφθωσαν καὶ <lb n="1799,001,,,,8,3,15"></lb> |
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6968 ἔστωσαν οἱ Λ, Μ, Ν, Ξ.</s> <lb n="1799,001,,,,8,3,16"></lb> |
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6969 <s>Καὶ ἐπεὶ οἱ Ε, Ζ ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον <lb n="1799,001,,,,8,3,17"></lb> |
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6970 ἐχόντων αὐτοῖς, πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,8,3,18"></lb> |
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6971 ἑκάτερος τῶν Ε, Ζ ἑαυτὸν μὲν πολλαπλασιάσας ἑκάτερον <lb n="1799,001,,,,8,3,19"></lb> |
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6972 τῶν Η, Κ πεποίηκεν, ἑκάτερον δὲ τῶν Η, Κ πολλα-<lb n="1799,001,,,,8,3,20"></lb> |
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6973 πλασιάσας ἑκάτερον τῶν Λ, *ξ πεποίηκεν, καὶ οἱ Η, Κ <lb n="1799,001,,,,8,3,21"></lb> |
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6974 ἄρα καὶ οἱ Λ, Ξ πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>καὶ ἐπεὶ οἱ <lb n="1799,001,,,,8,3,22"></lb> |
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6975 Α, Β, Γ, Δ ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων <lb n="1799,001,,,,8,3,23"></lb> |
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6976 αὐτοῖς, εἰσὶ δὲ καὶ οἱ Λ, Μ, Ν, Ξ ἐλάχιστοι ἐν τῷ αὐτῷ <lb n="1799,001,,,,8,3,24"></lb> |
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6977 λόγῳ ὄντες τοῖς Α, Β, Γ, Δ, καί ἐστιν ἴσον τὸ πλῆθος τῶν <lb n="1799,001,,,,8,3,25"></lb> |
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6978 Α, Β, Γ, Δ τῷ πλήθει τῶν Λ, Μ, Ν, Ξ, ἕκαστος ἄρα τῶν <lb n="1799,001,,,,8,3,26"></lb> |
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6979 Α, Β, Γ, Δ ἑκάστῳ τῶν Λ, Μ, Ν, Ξ ἴσος ἐστίν· ἴσος ἄρα <pb></pb> <lb n="1799,001,,,,8,3,27"></lb> |
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6980 ἐστὶν ὁ μὲν Α τῷ Λ, ὁ δὲ Δ τῷ Ξ.</s> <s>καί εἰσιν οἱ Λ, Ξ πρῶτοι <lb n="1799,001,,,,8,3,28"></lb> |
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6981 πρὸς ἀλλήλους.</s> <s>καὶ οἱ Α, Δ ἄρα πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,8,3,29"></lb> |
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6982 εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,3,30"></lb> |
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6983 <s>Λόγων δοθέντων ὁποσωνοῦν ἐν ἐλαχίστοις ἀριθμοῖς <lb n="1799,001,,,,8,4,1"></lb> |
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6984 ἀριθμοὺς εὑρεῖν ἑξῆς ἀνάλογον ἐλαχίστους ἐν τοῖς δοθεῖσι <lb n="1799,001,,,,8,4,2"></lb> |
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6985 λόγοις.</s> <lb n="1799,001,,,,8,4,3"></lb> |
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6986 <s>Ἔστωσαν οἱ δοθέντες λόγοι ἐν ἐλαχίστοις ἀριθμοῖς ὅ τε <lb n="1799,001,,,,8,4,4"></lb> |
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6987 τοῦ Α πρὸς τὸν Β καὶ ὁ τοῦ Γ πρὸς τὸν Δ καὶ ἔτι ὁ τοῦ <lb n="1799,001,,,,8,4,5"></lb> |
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6988 Ε πρὸς τὸν Ζ· δεῖ δὴ ἀριθμοὺς εὑρεῖν ἑξῆς ἀνάλογον <lb n="1799,001,,,,8,4,6"></lb> |
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6989 ἐλαχίστους ἔν τε τῷ τοῦ Α πρὸς τὸν Β λόγῳ καὶ ἐν τῷ <lb n="1799,001,,,,8,4,7"></lb> |
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6990 τοῦ Γ πρὸς τὸν Δ καὶ ἔτι ἐν τῷ τοῦ Ε πρὸς τὸν Ζ.</s> <lb n="1799,001,,,,8,4,8"></lb> |
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6991 <s>Εἰλήφθω γὰρ ὁ ὑπὸ τῶν Β, Γ ἐλάχιστος μετρούμενος <lb n="1799,001,,,,8,4,9"></lb> |
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6992 ἀριθμὸς ὁ Η.</s> <s>καὶ ὁσάκις μὲν ὁ Β τὸν Η μετρεῖ, τοσαυτάκις <lb n="1799,001,,,,8,4,10"></lb> |
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6993 καὶ ὁ Α τὸν Θ μετρείτω, ὁσάκις δὲ ὁ Γ τὸν Η μετρεῖ, <lb n="1799,001,,,,8,4,11"></lb> |
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6994 τοσαυτάκις καὶ ὁ Δ τὸν Κ μετρείτω.</s> <s>ὁ δὲ Ε τὸν Κ ἤτοι <lb n="1799,001,,,,8,4,12"></lb> |
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6995 μετρεῖ ἢ οὐ μετρεῖ.</s> <s>μετρείτω πρότερον.</s> <s>καὶ ὁσάκις ὁ Ε <lb n="1799,001,,,,8,4,13"></lb> |
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6996 τὸν Κ μετρεῖ, τοσαυτάκις καὶ ὁ Ζ τὸν Λ μετρείτω.</s> <s>καὶ <lb n="1799,001,,,,8,4,14"></lb> |
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6997 ἐπεὶ ἰσάκις ὁ Α τὸν Θ μετρεῖ καὶ ὁ Β τὸν Η, ἔστιν ἄρα <lb n="1799,001,,,,8,4,15"></lb> |
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6998 ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Θ πρὸς τὸν Η.</s> <s>διὰ τὰ αὐτὰ <lb n="1799,001,,,,8,4,16"></lb> |
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6999 δὴ καὶ ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Η πρὸς τὸν Κ, καὶ ἔτι <lb n="1799,001,,,,8,4,17"></lb> |
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7000 ὡς ὁ Ε πρὸς τὸν Ζ, οὕτως ὁ Κ πρὸς τὸν Λ· οἱ Θ, Η, Κ, Λ <pb></pb> <lb n="1799,001,,,,8,4,18"></lb> |
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7001 ἄρα ἑξῆς ἀνάλογόν εἰσιν ἔν τε τῷ τοῦ Α πρὸς τὸν Β καὶ <lb n="1799,001,,,,8,4,19"></lb> |
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7002 ἐν τῷ τοῦ Γ πρὸς τὸν Δ καὶ ἔτι ἐν τῷ τοῦ Ε πρὸς τὸν Ζ <lb n="1799,001,,,,8,4,20"></lb> |
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7003 λόγῳ.</s> <s>λέγω δή, ὅτι καὶ ἐλάχιστοι.</s> <s>εἰ γὰρ μή εἰσιν οἱ Θ, <lb n="1799,001,,,,8,4,21"></lb> |
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7004 Η, Κ, Λ ἑξῆς ἀνάλογον ἐλάχιστοι ἔν τε τοῖς τοῦ Α πρὸς <lb n="1799,001,,,,8,4,22"></lb> |
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7005 τὸν Β καὶ τοῦ Γ πρὸς τὸν Δ καὶ ἐν τῷ τοῦ Ε πρὸς τὸν <lb n="1799,001,,,,8,4,23"></lb> |
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7006 Ζ λόγοις, ἔστωσαν οἱ Ν, Ξ, Μ, Ο.</s> <s>καὶ ἐπεί ἐστιν ὡς ὁ Α <lb n="1799,001,,,,8,4,24"></lb> |
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7007 πρὸς τὸν Β, οὕτως ὁ Ν πρὸς τὸν Ξ, οἱ δὲ Α, Β ἐλάχιστοι, <lb n="1799,001,,,,8,4,25"></lb> |
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7008 οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας <lb n="1799,001,,,,8,4,26"></lb> |
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7009 ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, <lb n="1799,001,,,,8,4,27"></lb> |
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7010 τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος <lb n="1799,001,,,,8,4,28"></lb> |
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7011 τὸν ἑπόμενον, ὁ Β ἄρα τὸν Ξ μετρεῖ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὁ <lb n="1799,001,,,,8,4,29"></lb> |
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7012 Γ τὸν Ξ μετρεῖ· οἱ Β, Γ ἄρα τὸν Ξ μετροῦσιν· καὶ ὁ <lb n="1799,001,,,,8,4,30"></lb> |
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7013 ἐλάχιστος ἄρα ὑπὸ τῶν Β, Γ μετρούμενος τὸν Ξ μετρήσει.</s> <lb n="1799,001,,,,8,4,31"></lb> |
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7014 <s>ἐλάχιστος δὲ ὑπὸ τῶν Β, Γ μετρεῖται ὁ Η· ὁ Η ἄρα τὸν Ξ <pb></pb> <lb n="1799,001,,,,8,4,32"></lb> |
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7015 μετρεῖ ὁ μείζων τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ <lb n="1799,001,,,,8,4,33"></lb> |
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7016 ἄρα ἔσονταί τινες τῶν Θ, Η, Κ, Λ ἐλάσσονες ἀριθμοὶ ἑξῆς <lb n="1799,001,,,,8,4,34"></lb> |
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7017 ἔν τε τῷ τοῦ Α πρὸς τὸν Β καὶ τῷ τοῦ Γ πρὸς τὸν Δ καὶ <lb n="1799,001,,,,8,4,35"></lb> |
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7018 ἔτι τῷ τοῦ Ε πρὸς τὸν Ζ λόγῳ.</s> <lb n="1799,001,,,,8,4,36"></lb> |
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7019 <s>Μὴ μετρείτω δὴ ὁ Ε τὸν Κ.</s> <s>καὶ εἰλήφθω ὑπὸ τῶν Ε, Κ <lb n="1799,001,,,,8,4,37"></lb> |
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7020 ἐλάχιστος μετρούμενος ἀριθμὸς ὁ Μ.</s> <s>καὶ ὁσάκις μὲν ὁ Κ <lb n="1799,001,,,,8,4,38"></lb> |
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7021 τὸν Μ μετρεῖ, τοσαυτάκις καὶ ἑκάτερος τῶν Θ, Η ἑκάτερον <lb n="1799,001,,,,8,4,39"></lb> |
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7022 τῶν Ν, Ξ μετρείτω, ὁσάκις δὲ ὁ Ε τὸν Μ μετρεῖ, τοσαυ-<lb n="1799,001,,,,8,4,40"></lb> |
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7023 τάκις καὶ ὁ Ζ τὸν Ο μετρείτω.</s> <s>ἐπεὶ ἰσάκις ὁ Θ τὸν Ν <lb n="1799,001,,,,8,4,41"></lb> |
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7024 μετρεῖ καὶ ὁ Η τὸν Ξ, ἔστιν ἄρα ὡς ὁ Θ πρὸς τὸν Η, οὕτως <lb n="1799,001,,,,8,4,42"></lb> |
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7025 ὁ Ν πρὸς τὸν Ξ.</s> <s>ὡς δὲ ὁ Θ πρὸς τὸν Η, οὕτως ὁ Α πρὸς <lb n="1799,001,,,,8,4,43"></lb> |
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7026 τὸν Β· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, οὕτως ὁ Ν πρὸς τὸν <lb n="1799,001,,,,8,4,44"></lb> |
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7027 Ξ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ξ <lb n="1799,001,,,,8,4,45"></lb> |
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7028 πρὸς τὸν Μ.</s> <s>πάλιν, ἐπεὶ ἰσάκις ὁ Ε τὸν Μ μετρεῖ καὶ ὁ Ζ <lb n="1799,001,,,,8,4,46"></lb> |
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7029 τὸν Ο, ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Ζ, οὕτως ὁ Μ πρὸς <lb n="1799,001,,,,8,4,47"></lb> |
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7030 τὸν Ο· οἱ Ν, Ξ, Μ, Ο ἄρα ἑξῆς ἀνάλογόν εἰσιν ἐν τοῖς τοῦ <lb n="1799,001,,,,8,4,48"></lb> |
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7031 τε Α πρὸς τὸν Β καὶ τοῦ Γ πρὸς τὸν Δ καὶ ἔτι τοῦ Ε <pb></pb> <lb n="1799,001,,,,8,4,49"></lb> |
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7032 πρὸς τὸν Ζ λόγοις.</s> <s>λέγω δή, ὅτι καὶ ἐλάχιστοι ἐν τοῖς ΑΒ, <lb n="1799,001,,,,8,4,50"></lb> |
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7033 ΓΔ, ΕΖ λόγοις.</s> <s>εἰ γὰρ μή, ἔσονταί τινες τῶν Ν, Ξ, Μ, Ο <lb n="1799,001,,,,8,4,51"></lb> |
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7034 ἐλάσσονες ἀριθμοὶ ἑξῆς ἀνάλογον ἐν τοῖς ΑΒ, ΓΔ, ΕΖ <lb n="1799,001,,,,8,4,52"></lb> |
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7035 λόγοις.</s> <s>ἔστωσαν οἱ Π, Ρ, Σ, Τ.</s> <s>καὶ ἐπεί ἐστιν ὡς ὁ Π πρὸς <lb n="1799,001,,,,8,4,53"></lb> |
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7036 τὸν Ρ, οὕτως ὁ Α πρὸς τὸν Β, οἱ δὲ Α, Β ἐλάχιστοι, οἱ δὲ <lb n="1799,001,,,,8,4,54"></lb> |
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7037 ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας αὐτοῖς <lb n="1799,001,,,,8,4,55"></lb> |
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7038 ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν <lb n="1799,001,,,,8,4,56"></lb> |
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7039 ἑπόμενον, ὁ Β ἄρα τὸν Ρ μετρεῖ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὁ Γ <lb n="1799,001,,,,8,4,57"></lb> |
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7040 τὸν Ρ μετρεῖ· οἱ Β, Γ ἄρα τὸν Ρ μετροῦσιν.</s> <s>καὶ ὁ ἐλάχιστος <lb n="1799,001,,,,8,4,58"></lb> |
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7041 ἄρα ὑπὸ τῶν Β, Γ μετρούμενος τὸν Ρ μετρήσει.</s> <s>ἐλάχιστος <lb n="1799,001,,,,8,4,59"></lb> |
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7042 δὲ ὑπὸ τῶν Β, Γ μετρούμενός ἐστιν ὁ Η· ὁ Η ἄρα τὸν Ρ <lb n="1799,001,,,,8,4,60"></lb> |
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7043 μετρεῖ.</s> <s>καί ἐστιν ὡς ὁ Η πρὸς τὸν Ρ, οὕτως ὁ Κ πρὸς τὸν <lb n="1799,001,,,,8,4,61"></lb> |
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7044 Σ· καὶ ὁ Κ ἄρα τὸν Σ μετρεῖ.</s> <s>μετρεῖ δὲ καὶ ὁ Ε τὸν Σ· <lb n="1799,001,,,,8,4,62"></lb> |
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7045 οἱ Ε, Κ ἄρα τὸν Σ μετροῦσιν.</s> <s>καὶ ὁ ἐλάχιστος ἄρα ὑπὸ <lb n="1799,001,,,,8,4,63"></lb> |
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7046 τῶν Ε, Κ μετρούμενος τὸν Σ μετρήσει.</s> <s>ἐλάχιστος δὲ ὑπὸ <lb n="1799,001,,,,8,4,64"></lb> |
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7047 τῶν Ε, Κ μετρούμενός ἐστιν ὁ Μ· ὁ Μ ἄρα τὸν Σ μετρεῖ <lb n="1799,001,,,,8,4,65"></lb> |
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7048 ὁ μείζων τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα <lb n="1799,001,,,,8,4,66"></lb> |
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7049 ἔσονταί τινες τῶν Ν, Ξ, Μ, Ο ἐλάσσονες ἀριθμοὶ ἑξῆς <lb n="1799,001,,,,8,4,67"></lb> |
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7050 ἀνάλογον ἔν τε τοῖς τοῦ Α πρὸς τὸν Β καὶ τοῦ Γ πρὸς <lb n="1799,001,,,,8,4,68"></lb> |
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7051 τὸν Δ καὶ ἔτι τοῦ Ε πρὸς τὸν Ζ λόγοις· οἱ Ν, Ξ, Μ, Ο <lb n="1799,001,,,,8,4,69"></lb> |
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7052 ἄρα ἑξῆς ἀνάλογον ἐλάχιστοί εἰσιν ἐν τοῖς ΑΒ, ΓΔ, ΕΖ <lb n="1799,001,,,,8,4,70"></lb> |
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7053 λόγοις· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,8,4,71"></lb> |
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7054 Οἱ ἐπίπεδοι ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχουσι τὸν <lb n="1799,001,,,,8,5,1"></lb> |
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7055 συγκείμενον ἐκ τῶν πλευρῶν.</s> <lb n="1799,001,,,,8,5,2"></lb> |
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7056 <s>Ἔστωσαν ἐπίπεδοι ἀριθμοὶ οἱ Α, Β, καὶ τοῦ μὲν Α <lb n="1799,001,,,,8,5,3"></lb> |
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7057 πλευραὶ ἔστωσαν οἱ Γ, Δ ἀριθμοί, τοῦ δὲ Β οἱ Ε, Ζ· λέγω, <lb n="1799,001,,,,8,5,4"></lb> |
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7058 ὅτι ὁ Α πρὸς τὸν Β λόγον ἔχει <lb n="1799,001,,,,8,5,5"></lb> |
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7059 τὸν συγκείμενον ἐκ τῶν πλευ-<lb n="1799,001,,,,8,5,6"></lb> |
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7060 ρῶν.</s> <lb n="1799,001,,,,8,5,7"></lb> |
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7061 <s>Λόγων γὰρ δοθέντων τοῦ τε <lb n="1799,001,,,,8,5,8"></lb> |
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7062 ὃν ἔχει ὁ Γ πρὸς τὸν Ε καὶ <lb n="1799,001,,,,8,5,9"></lb> |
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7063 ὁ Δ πρὸς τὸν Ζ εἰλήφθωσαν <lb n="1799,001,,,,8,5,10"></lb> |
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7064 ἀριθμοὶ ἑξῆς ἐλάχιστοι ἐν τοῖς <lb n="1799,001,,,,8,5,11"></lb> |
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7065 ΓΕ, ΔΖ λόγοις, οἱ Η, Θ, Κ, <lb n="1799,001,,,,8,5,12"></lb> |
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7066 ὥστε εἶναι ὡς μὲν τὸν Γ πρὸς <lb n="1799,001,,,,8,5,13"></lb> |
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7067 τὸν Ε, οὕτως τὸν Η πρὸς τὸν <lb n="1799,001,,,,8,5,14"></lb> |
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7068 Θ, ὡς δὲ τὸν Δ πρὸς τὸν Ζ, οὕτως τὸν Θ πρὸς τὸν Κ.</s> <lb n="1799,001,,,,8,5,15"></lb> |
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7069 <s>καὶ ὁ Δ τὸν Ε πολλαπλασιάσας τὸν Λ ποιείτω.</s> <lb n="1799,001,,,,8,5,16"></lb> |
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7070 <s>Καὶ ἐπεὶ ὁ Δ τὸν μὲν Γ πολλαπλασιάσας τὸν Α πε-<lb n="1799,001,,,,8,5,17"></lb> |
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7071 ποίηκεν, τὸν δὲ Ε πολλαπλασιάσας τὸν Λ πεποίηκεν, ἔστιν <lb n="1799,001,,,,8,5,18"></lb> |
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7072 ἄρα ὡς ὁ Γ πρὸς τὸν Ε, οὕτως ὁ Α πρὸς τὸν Λ.</s> <s>ὡς <lb n="1799,001,,,,8,5,19"></lb> |
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7073 δὲ ὁ Γ πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν Θ· καὶ ὡς <pb></pb> <lb n="1799,001,,,,8,5,20"></lb> |
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7074 ἄρα ὁ Η πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Λ.</s> <s>πάλιν, ἐπεὶ <lb n="1799,001,,,,8,5,21"></lb> |
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7075 ὁ Ε τὸν Δ πολλαπλασιάσας τὸν Λ πεποίηκεν, ἀλλὰ μὴν καὶ <lb n="1799,001,,,,8,5,22"></lb> |
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7076 τὸν Ζ πολλαπλασιάσας τὸν Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Δ <lb n="1799,001,,,,8,5,23"></lb> |
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7077 πρὸς τὸν Ζ, οὕτως ὁ Λ πρὸς τὸν Β.</s> <s>ἀλλ' ὡς ὁ Δ πρὸς τὸν <lb n="1799,001,,,,8,5,24"></lb> |
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7078 Ζ, οὕτως ὁ Θ πρὸς τὸν Κ· καὶ ὡς ἄρα ὁ Θ πρὸς τὸν Κ, <lb n="1799,001,,,,8,5,25"></lb> |
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7079 οὕτως ὁ Λ πρὸς τὸν Β.</s> <s>ἐδείχθη δὲ καὶ ὡς ὁ Η πρὸς τὸν <lb n="1799,001,,,,8,5,26"></lb> |
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7080 Θ, οὕτως ὁ Α πρὸς τὸν Λ· δι' ἴσου ἄρα ἐστὶν ὡς ὁ Η <lb n="1799,001,,,,8,5,27"></lb> |
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7081 πρὸς τὸν Κ, [οὕτωσ] ὁ Α πρὸς τὸν Β, ὁ δὲ Η πρὸς τὸν Κ <lb n="1799,001,,,,8,5,28"></lb> |
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7082 λόγον ἔχει τὸν συγκείμενον ἐκ τῶν πλευρῶν· καὶ ὁ Α ἄρα <lb n="1799,001,,,,8,5,29"></lb> |
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7083 πρὸς τὸν Β λόγον ἔχει τὸν συγκείμενον ἐκ τῶν πλευρῶν· <lb n="1799,001,,,,8,5,30"></lb> |
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7084 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,5,31"></lb> |
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7085 <s>Ἐὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, ὁ δὲ πρῶτος <lb n="1799,001,,,,8,6,1"></lb> |
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7086 τὸν δεύτερον μὴ μετρῇ, οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει.</s> <lb n="1799,001,,,,8,6,2"></lb> |
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7087 <s>Ἔστωσαν ὁποσοιοῦν ἀριθ- <lb n="1799,001,,,,8,6,3"></lb> |
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7088 μοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, <lb n="1799,001,,,,8,6,4"></lb> |
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7089 Δ, Ε, ὁ δὲ Α τὸν Β μὴ με-<lb n="1799,001,,,,8,6,5"></lb> |
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7090 τρείτω· λέγω, ὅτι οὐδὲ ἄλ-<lb n="1799,001,,,,8,6,6"></lb> |
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7091 λος οὐδεὶς οὐδένα μετρήσει.</s> <lb n="1799,001,,,,8,6,7"></lb> |
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7092 <s>Ὅτι μὲν οὖν οἱ Α, Β, Γ, <lb n="1799,001,,,,8,6,8"></lb> |
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7093 Δ, Ε ἑξῆς ἀλλήλους οὐ με-<lb n="1799,001,,,,8,6,9"></lb> |
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7094 τροῦσιν, φανερόν· οὐδὲ γὰρ <lb n="1799,001,,,,8,6,10"></lb> |
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7095 ὁ Α τὸν Β μετρεῖ.</s> <s>λέγω δή, <lb n="1799,001,,,,8,6,11"></lb> |
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7096 ὅτι οὐδὲ ἄλλος οὐδεὶς οὐδένα <lb n="1799,001,,,,8,6,12"></lb> |
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7097 μετρήσει.</s> <s>εἰ γὰρ δυνατόν, μετρείτω ὁ Α τὸν Γ.</s> <s>καὶ ὅσοι <lb n="1799,001,,,,8,6,13"></lb> |
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7098 εἰσὶν οἱ Α, Β, Γ, τοσοῦτοι εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ <pb></pb> <lb n="1799,001,,,,8,6,14"></lb> |
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7099 τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β, Γ οἱ Ζ, Η, Θ.</s> <lb n="1799,001,,,,8,6,15"></lb> |
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7100 <s>καὶ ἐπεὶ οἱ Ζ, Η, Θ ἐν τῷ αὐτῷ λόγῳ εἰσὶ τοῖς Α, Β, <lb n="1799,001,,,,8,6,16"></lb> |
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7101 Γ, καί ἐστιν ἴσον τὸ πλῆθος τῶν Α, Β, Γ τῷ πλήθει τῶν <lb n="1799,001,,,,8,6,17"></lb> |
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7102 Ζ, Η, Θ, δι' ἴσου ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Ζ <lb n="1799,001,,,,8,6,18"></lb> |
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7103 πρὸς τὸν Θ.</s> <s>καὶ ἐπεί ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Ζ <lb n="1799,001,,,,8,6,19"></lb> |
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7104 πρὸς τὸν Η, οὐ μετρεῖ δὲ ὁ Α τὸν Β, οὐ μετρεῖ ἄρα οὐδὲ ὁ Ζ <lb n="1799,001,,,,8,6,20"></lb> |
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7105 τὸν Η· οὐκ ἄρα μονάς ἐστιν ὁ Ζ· ἡ γὰρ μονὰς πάντα ἀριθ-<lb n="1799,001,,,,8,6,21"></lb> |
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7106 μὸν μετρεῖ.</s> <s>καί εἰσιν οἱ Ζ, Θ πρῶτοι πρὸς ἀλλήλους [οὐδὲ <lb n="1799,001,,,,8,6,22"></lb> |
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7107 ὁ Ζ ἄρα τὸν Θ μετρεῖ].</s> <s>καί ἐστιν ὡς ὁ Ζ πρὸς τὸν Θ, οὕτως <lb n="1799,001,,,,8,6,23"></lb> |
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7108 ὁ Α πρὸς τὸν Γ· οὐδὲ ὁ Α ἄρα τὸν Γ μετρεῖ.</s> <s>ὁμοίως δὴ <lb n="1799,001,,,,8,6,24"></lb> |
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7109 δείξομεν, ὅτι οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει· ὅπερ ἔδει <lb n="1799,001,,,,8,6,25"></lb> |
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7110 δεῖξαι.</s> <lb n="1799,001,,,,8,6,26"></lb> |
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7111 <s>Ἐὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ [ἑξῆσ] ἀνάλογον, ὁ δὲ <lb n="1799,001,,,,8,7,1"></lb> |
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7112 πρῶτος τὸν ἔσχατον μετρῇ, καὶ τὸν δεύτερον μετρήσει.</s> <lb n="1799,001,,,,8,7,2"></lb> |
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7113 <s>Ἔστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, <lb n="1799,001,,,,8,7,3"></lb> |
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7114 Δ, ὁ δὲ Α τὸν Δ μετρείτω· λέγω, ὅτι καὶ ὁ Α τὸν Β μετρεῖ.</s> <lb n="1799,001,,,,8,7,4"></lb> |
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7115 <s>Εἰ γὰρ οὐ μετρεῖ ὁ Α τὸν Β, οὐδὲ <lb n="1799,001,,,,8,7,5"></lb> |
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7116 ἄλλος οὐδεὶς οὐδένα μετρήσει· μετρεῖ δὲ <lb n="1799,001,,,,8,7,6"></lb> |
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7117 ὁ Α τὸν Δ.</s> <s>μετρεῖ ἄρα καὶ ὁ Α τὸν Β· <lb n="1799,001,,,,8,7,7"></lb> |
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7118 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,7,8"></lb> |
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7119 <s>Ἐὰν δύο ἀριθμῶν μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον <lb n="1799,001,,,,8,8,1"></lb> |
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7120 ἐμπίπτωσιν ἀριθμοί, ὅσοι εἰς αὐτοὺς μεταξὺ κατὰ τὸ <pb></pb> <lb n="1799,001,,,,8,8,2"></lb> |
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7121 συνεχὲς ἀνάλογον ἐμπίπτουσιν ἀριθμοί, τοσοῦτοι καὶ <lb n="1799,001,,,,8,8,3"></lb> |
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7122 εἰς τοὺς τὸν αὐτὸν λόγον ἔχοντας [αὐτοῖσ] μεταξὺ κατὰ <lb n="1799,001,,,,8,8,4"></lb> |
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7123 τὸ συνεχὲς ἀνάλογον ἐμπεσοῦνται.</s> <lb n="1799,001,,,,8,8,5"></lb> |
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7124 <s>Δύο γὰρ ἀριθμῶν τῶν Α, Β μεταξὺ κατὰ τὸ συνεχὲς <lb n="1799,001,,,,8,8,6"></lb> |
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7125 ἀνάλογον ἐμπιπτέτωσαν <lb n="1799,001,,,,8,8,7"></lb> |
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7126 ἀριθμοὶ οἱ Γ, Δ, καὶ πε-<lb n="1799,001,,,,8,8,8"></lb> |
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7127 ποιήσθω ὡς ὁ Α πρὸς <lb n="1799,001,,,,8,8,9"></lb> |
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7128 τὸν Β, οὕτως ὁ Ε πρὸς <lb n="1799,001,,,,8,8,10"></lb> |
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7129 τὸν Ζ· λέγω, ὅτι ὅσοι εἰς <lb n="1799,001,,,,8,8,11"></lb> |
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7130 τοὺς Α, Β μεταξὺ κατὰ <lb n="1799,001,,,,8,8,12"></lb> |
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7131 τὸ συνεχὲς ἀνάλογον ἐμ-<lb n="1799,001,,,,8,8,13"></lb> |
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7132 πεπτώκασιν ἀριθμοί, το-<lb n="1799,001,,,,8,8,14"></lb> |
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7133 σοῦτοι καὶ εἰς τοὺς Ε, Ζ <lb n="1799,001,,,,8,8,15"></lb> |
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7134 μεταξὺ κατὰ τὸ συνεχὲς <lb n="1799,001,,,,8,8,16"></lb> |
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7135 ἀνάλογον ἐμπεσοῦνται.</s> <lb n="1799,001,,,,8,8,17"></lb> |
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7136 <s>Ὅσοι γάρ εἰσι τῷ πλήθει οἱ Α, Β, Γ, Δ, τοσοῦτοι <lb n="1799,001,,,,8,8,18"></lb> |
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7137 εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον <lb n="1799,001,,,,8,8,19"></lb> |
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7138 ἐχόντων τοῖς Α, Γ, Δ, Β οἱ Η, Θ, Κ, Λ· οἱ ἄρα ἄκροι <lb n="1799,001,,,,8,8,20"></lb> |
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7139 αὐτῶν οἱ Η, Λ πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>καὶ ἐπεὶ οἱ <lb n="1799,001,,,,8,8,21"></lb> |
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7140 Α, Γ, Δ, Β τοῖς Η, Θ, Κ, Λ ἐν τῷ αὐτῷ λόγῳ εἰσίν, καί <lb n="1799,001,,,,8,8,22"></lb> |
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7141 ἐστιν ἴσον τὸ πλῆθος τῶν Α, Γ, Δ, Β τῷ πλήθει τῶν <lb n="1799,001,,,,8,8,23"></lb> |
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7142 Η, Θ, Κ, Λ, δι' ἴσου ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Β, οὕτως <lb n="1799,001,,,,8,8,24"></lb> |
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7143 ὁ Η πρὸς τὸν Λ.</s> <s>ὡς δὲ ὁ Α πρὸς τὸν Β, οὕτως ὁ Ε πρὸς <lb n="1799,001,,,,8,8,25"></lb> |
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7144 τὸν Ζ· καὶ ὡς ἄρα ὁ Η πρὸς τὸν Λ, οὕτως ὁ Ε πρὸς <lb n="1799,001,,,,8,8,26"></lb> |
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7145 τὸν Ζ.</s> <s>οἱ δὲ Η, Λ πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ <lb n="1799,001,,,,8,8,27"></lb> |
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7146 δὲ ἐλάχιστοι ἀριθμοὶ μετροῦσι τοὺς τὸν αὐτὸν λόγον <lb n="1799,001,,,,8,8,28"></lb> |
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7147 ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων <pb></pb> <lb n="1799,001,,,,8,8,29"></lb> |
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7148 τὸν ἐλάσσονα, τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον <lb n="1799,001,,,,8,8,30"></lb> |
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7149 καὶ ὁ ἑπόμενος τὸν ἑπόμενον.</s> <s>ἰσάκις ἄρα ὁ Η τὸν Ε μετρεῖ <lb n="1799,001,,,,8,8,31"></lb> |
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7150 καὶ ὁ Λ τὸν Ζ.</s> <s>ὁσάκις δὴ ὁ Η τὸν Ε μετρεῖ, τοσαυτάκις <lb n="1799,001,,,,8,8,32"></lb> |
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7151 καὶ ἑκάτερος τῶν Θ, Κ ἑκάτερον τῶν Μ, Ν μετρείτω· <lb n="1799,001,,,,8,8,33"></lb> |
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7152 οἱ Η, Θ, Κ, Λ ἄρα τοὺς Ε, Μ, Ν, Ζ ἰσάκις μετροῦσιν.</s> <lb n="1799,001,,,,8,8,34"></lb> |
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7153 <s>οἱ Η, Θ, Κ, Λ ἄρα τοῖς Ε, Μ, Ν, Ζ ἐν τῷ αὐτῷ λόγῳ <lb n="1799,001,,,,8,8,35"></lb> |
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7154 εἰσίν.</s> <s>ἀλλὰ οἱ Η, Θ, Κ, Λ τοῖς Α, Γ, Δ, Β ἐν τῷ αὐτῷ <lb n="1799,001,,,,8,8,36"></lb> |
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7155 λόγῳ εἰσίν· καὶ οἱ Α, Γ, Δ, Β ἄρα τοῖς Ε, Μ, Ν, Ζ ἐν τῷ <lb n="1799,001,,,,8,8,37"></lb> |
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7156 αὐτῷ λόγῳ εἰσίν.</s> <s>οἱ δὲ Α, Γ, Δ, Β ἑξῆς ἀνάλογόν εἰσιν· <lb n="1799,001,,,,8,8,38"></lb> |
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7157 καὶ οἱ Ε, Μ, Ν, Ζ ἄρα ἑξῆς ἀνάλογόν εἰσιν.</s> <s>ὅσοι ἄρα εἰς <lb n="1799,001,,,,8,8,39"></lb> |
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7158 τοὺς Α, Β μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεπτώ-<lb n="1799,001,,,,8,8,40"></lb> |
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7159 κασιν ἀριθμοί, τοσοῦτοι καὶ εἰς τοὺς Ε, Ζ μεταξὺ κατὰ <lb n="1799,001,,,,8,8,41"></lb> |
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7160 τὸ συνεχὲς ἀνάλογον ἐμπεπτώκασιν ἀριθμοί· ὅπερ ἔδει <lb n="1799,001,,,,8,8,42"></lb> |
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7161 δεῖξαι.</s> <lb n="1799,001,,,,8,8,43"></lb> |
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7162 <s>Ἐὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, καὶ εἰς <lb n="1799,001,,,,8,9,1"></lb> |
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7163 αὐτοὺς μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπίπτωσιν <lb n="1799,001,,,,8,9,2"></lb> |
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7164 ἀριθμοί, ὅσοι εἰς αὐτοὺς μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον <lb n="1799,001,,,,8,9,3"></lb> |
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7165 ἐμπίπτουσιν ἀριθμοί, τοσοῦτοι καὶ ἑκατέρου αὐτῶν καὶ <lb n="1799,001,,,,8,9,4"></lb> |
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7166 μονάδος μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεσοῦνται.</s> <lb n="1799,001,,,,8,9,5"></lb> |
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7167 <s>Ἔστωσαν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους οἱ Α, Β <lb n="1799,001,,,,8,9,6"></lb> |
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7168 καὶ εἰς αὐτοὺς μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπιπτέ-<lb n="1799,001,,,,8,9,7"></lb> |
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7169 τωσαν οἱ Γ, Δ, καὶ ἐκκείσθω ἡ Ε μονάς· λέγω, ὅτι ὅσοι <lb n="1799,001,,,,8,9,8"></lb> |
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7170 εἰς τοὺς Α, Β μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεπτώ-<lb n="1799,001,,,,8,9,9"></lb> |
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7171 κασιν ἀριθμοί, τοσοῦτοι καὶ ἑκατέρου τῶν Α, Β καὶ τῆς <lb n="1799,001,,,,8,9,10"></lb> |
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7172 μονάδος μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεσοῦνται.</s> <s><pb></pb> <lb n="1799,001,,,,8,9,11"></lb> |
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7173 Εἰλήφθωσαν γὰρ δύο μὲν ἀριθμοὶ ἐλάχιστοι ἐν τῷ <lb n="1799,001,,,,8,9,12"></lb> |
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7174 τῶν Α, Γ, Δ, Β λόγῳ ὄντες οἱ Ζ, Η, τρεῖς δὲ οἱ Θ, Κ, Λ, <lb n="1799,001,,,,8,9,13"></lb> |
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7175 καὶ ἀεὶ ἑξῆς ἑνὶ πλείους, ἕως ἂν ἴσον γένηται τὸ πλῆθος <lb n="1799,001,,,,8,9,14"></lb> |
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7176 αὐτῶν τῷ πλήθει τῶν Α, Γ, Δ, Β.</s> <s>εἰλήφθωσαν, καὶ <lb n="1799,001,,,,8,9,15"></lb> |
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7177 ἔστωσαν οἱ Μ, Ν, Ξ, Ο.</s> <s>φανερὸν δή, ὅτι ὁ μὲν Ζ ἑαυτὸν <lb n="1799,001,,,,8,9,16"></lb> |
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7178 πολλαπλασιάσας τὸν Θ πεποίηκεν, τὸν δὲ Θ πολλαπλα-<lb n="1799,001,,,,8,9,17"></lb> |
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7179 σιάσας τὸν Μ πεποίηκεν, καὶ ὁ Η ἑαυτὸν μὲν πολλαπλα-<lb n="1799,001,,,,8,9,18"></lb> |
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7180 σιάσας τὸν Λ πεποίηκεν, τὸν δὲ Λ πολλαπλασιάσας τὸν Ο <lb n="1799,001,,,,8,9,19"></lb> |
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7181 πεποίηκεν.</s> <s>καὶ ἐπεὶ οἱ Μ, Ν, Ξ, Ο ἐλάχιστοί εἰσι τῶν τὸν <lb n="1799,001,,,,8,9,20"></lb> |
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7182 αὐτὸν λόγον ἐχόντων τοῖς Ζ, Η, εἰσὶ δὲ καὶ οἱ Α, Γ, Δ, Β <lb n="1799,001,,,,8,9,21"></lb> |
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7183 ἐλάχιστοι τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Ζ, Η, καί ἐστιν <lb n="1799,001,,,,8,9,22"></lb> |
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7184 ἴσον τὸ πλῆθος τῶν Μ, Ν, Ξ, Ο τῷ πλήθει τῶν Α, Γ, Δ, Β, <lb n="1799,001,,,,8,9,23"></lb> |
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7185 ἕκαστος ἄρα τῶν Μ, Ν, Ξ, Ο ἑκάστῳ τῶν Α, Γ, Δ, Β <lb n="1799,001,,,,8,9,24"></lb> |
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7186 ἴσος ἐστίν· ἴσος ἄρα ἐστὶν ὁ μὲν Μ τῷ Α, ὁ δὲ Ο τῷ Β.</s> <lb n="1799,001,,,,8,9,25"></lb> |
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7187 <s>καὶ ἐπεὶ ὁ Ζ ἑαυτὸν πολλαπλασιάσας τὸν Θ πεποίηκεν, <lb n="1799,001,,,,8,9,26"></lb> |
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7188 ὁ Ζ ἄρα τὸν Θ μετρεῖ κατὰ τὰς ἐν τῷ Ζ μονάδας.</s> <s>μετρεῖ <lb n="1799,001,,,,8,9,27"></lb> |
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7189 δὲ καὶ ἡ Ε μονὰς τὸν Ζ κατὰ τὰς ἐν αὐτῷ μονάδας· <lb n="1799,001,,,,8,9,28"></lb> |
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7190 ἰσάκις ἄρα ἡ Ε μονὰς τὸν Ζ ἀριθμὸν μετρεῖ καὶ ὁ Ζ τὸν Θ.</s> <s><pb></pb> <lb n="1799,001,,,,8,9,29"></lb> |
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7191 ἔστιν ἄρα ὡς ἡ Ε μονὰς πρὸς τὸν Ζ ἀριθμόν, οὕτως ὁ Ζ <lb n="1799,001,,,,8,9,30"></lb> |
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7192 πρὸς τὸν Θ.</s> <s>πάλιν, ἐπεὶ ὁ Ζ τὸν Θ πολλαπλασιάσας τὸν <lb n="1799,001,,,,8,9,31"></lb> |
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7193 Μ πεποίηκεν, ὁ Θ ἄρα τὸν Μ μετρεῖ κατὰ τὰς ἐν τῷ Ζ <lb n="1799,001,,,,8,9,32"></lb> |
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7194 μονάδας.</s> <s>μετρεῖ δὲ καὶ ἡ Ε μονὰς τὸν Ζ ἀριθμὸν κατὰ <lb n="1799,001,,,,8,9,33"></lb> |
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7195 τὰς ἐν αὐτῷ μονάδας· ἰσάκις ἄρα ἡ Ε μονὰς τὸν Ζ <lb n="1799,001,,,,8,9,34"></lb> |
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7196 ἀριθμὸν μετρεῖ καὶ ὁ Θ τὸν Μ.</s> <s>ἔστιν ἄρα ὡς ἡ Ε μονὰς <lb n="1799,001,,,,8,9,35"></lb> |
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7197 πρὸς τὸν Ζ ἀριθμόν, οὕτως ὁ Θ πρὸς τὸν Μ.</s> <s>ἐδείχθη δὲ <lb n="1799,001,,,,8,9,36"></lb> |
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7198 καὶ ὡς ἡ Ε μονὰς πρὸς τὸν Ζ ἀριθμόν, οὕτως ὁ Ζ πρὸς <lb n="1799,001,,,,8,9,37"></lb> |
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7199 τὸν Θ· καὶ ὡς ἄρα ἡ Ε μονὰς πρὸς τὸν Ζ ἀριθμόν, οὕτως <lb n="1799,001,,,,8,9,38"></lb> |
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7200 ὁ Ζ πρὸς τὸν Θ καὶ ὁ Θ πρὸς τὸν Μ.</s> <s>ἴσος δὲ ὁ Μ τῷ Α· <lb n="1799,001,,,,8,9,39"></lb> |
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7201 ἔστιν ἄρα ὡς ἡ Ε μονὰς πρὸς τὸν Ζ ἀριθμόν, οὕτως ὁ Ζ <lb n="1799,001,,,,8,9,40"></lb> |
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7202 πρὸς τὸν Θ καὶ ὁ Θ πρὸς τὸν Α.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὡς <lb n="1799,001,,,,8,9,41"></lb> |
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7203 ἡ Ε μονὰς πρὸς τὸν Η ἀριθμόν, οὕτως ὁ Η πρὸς τὸν Λ <lb n="1799,001,,,,8,9,42"></lb> |
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7204 καὶ ὁ Λ πρὸς τὸν Β.</s> <s>ὅσοι ἄρα εἰς τοὺς Α, Β μεταξὺ κατὰ <lb n="1799,001,,,,8,9,43"></lb> |
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7205 τὸ συνεχὲς ἀνάλογον ἐμπεπτώκασιν ἀριθμοί, τοσοῦτοι καὶ <lb n="1799,001,,,,8,9,44"></lb> |
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7206 ἑκατέρου τῶν Α, Β καὶ μονάδος τῆς Ε μεταξὺ κατὰ τὸ <lb n="1799,001,,,,8,9,45"></lb> |
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7207 συνεχὲς ἀνάλογον ἐμπεπτώκασιν ἀριθμοί· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,9,46"></lb> |
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7208 <s>Ἐὰν δύο ἀριθμῶν ἑκατέρου καὶ μονάδος μεταξὺ κατὰ <lb n="1799,001,,,,8,10,1"></lb> |
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7209 τὸ συνεχὲς ἀνάλογον ἐμπίπτωσιν ἀριθμοί, ὅσοι ἑκατέρου <lb n="1799,001,,,,8,10,2"></lb> |
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7210 αὐτῶν καὶ μονάδος μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον <pb></pb> <lb n="1799,001,,,,8,10,3"></lb> |
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7211 ἐμπίπτουσιν ἀριθμοί, τοσοῦτοι καὶ εἰς αὐτοὺς μεταξὺ <lb n="1799,001,,,,8,10,4"></lb> |
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7212 κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεσοῦνται.</s> <lb n="1799,001,,,,8,10,5"></lb> |
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7213 <s>Δύο γὰρ ἀριθμῶν τῶν Α, Β <lb n="1799,001,,,,8,10,6"></lb> |
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7214 καὶ μονάδος τῆς Γ μεταξὺ κατὰ τὸ <lb n="1799,001,,,,8,10,7"></lb> |
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7215 συνεχὲς ἀνάλογον ἐμπιπτέτωσαν <lb n="1799,001,,,,8,10,8"></lb> |
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7216 ἀριθμοὶ οἵ τε Δ, Ε καὶ οἱ Ζ, <lb n="1799,001,,,,8,10,9"></lb> |
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7217 Η· λέγω, ὅτι ὅσοι ἑκατέρου τῶν <lb n="1799,001,,,,8,10,10"></lb> |
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7218 Α, Β καὶ μονάδος τῆς Γ μεταξὺ <lb n="1799,001,,,,8,10,11"></lb> |
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7219 κατὰ τὸ συνεχὲς ἀνάλογον ἐμ-<lb n="1799,001,,,,8,10,12"></lb> |
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7220 πεπτώκασιν ἀριθμοί, τοσοῦτοι <lb n="1799,001,,,,8,10,13"></lb> |
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7221 καὶ εἰς τοὺς Α, Β μεταξὺ κατὰ τὸ <lb n="1799,001,,,,8,10,14"></lb> |
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7222 συνεχὲς ἀνάλογον ἐμπεσοῦνται.</s> <lb n="1799,001,,,,8,10,15"></lb> |
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7223 <s>Ὁ Δ γὰρ τὸν Ζ πολλαπλασιάσας τὸν Θ ποιείτω, <lb n="1799,001,,,,8,10,16"></lb> |
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7224 ἑκάτερος δὲ τῶν Δ, Ζ τὸν Θ πολλαπλασιάσας ἑκάτερον <lb n="1799,001,,,,8,10,17"></lb> |
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7225 τῶν Κ, Λ ποιείτω.</s> <lb n="1799,001,,,,8,10,18"></lb> |
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7226 <s>Καὶ ἐπεί ἐστιν ὡς ἡ Γ μονὰς πρὸς τὸν Δ ἀριθμόν, <lb n="1799,001,,,,8,10,19"></lb> |
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7227 οὕτως ὁ Δ πρὸς τὸν Ε, ἰσάκις ἄρα ἡ Γ μονὰς τὸν Δ <lb n="1799,001,,,,8,10,20"></lb> |
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7228 ἀριθμὸν μετρεῖ καὶ ὁ Δ τὸν Ε.</s> <s>ἡ δὲ Γ μονὰς τὸν Δ ἀριθμὸν <lb n="1799,001,,,,8,10,21"></lb> |
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7229 μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας· καὶ ὁ Δ ἄρα ἀριθμὸς <lb n="1799,001,,,,8,10,22"></lb> |
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7230 τὸν Ε μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας· ὁ Δ ἄρα ἑαυτὸν <lb n="1799,001,,,,8,10,23"></lb> |
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7231 πολλαπλασιάσας τὸν Ε πεποίηκεν.</s> <s>πάλιν, ἐπεί ἐστιν ὡς <lb n="1799,001,,,,8,10,24"></lb> |
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7232 ἡ Γ [μονὰσ] πρὸς τὸν Δ ἀριθμὸν, οὕτως ὁ Ε πρὸς τὸν Α, <lb n="1799,001,,,,8,10,25"></lb> |
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7233 ἰσάκις ἄρα ἡ Γ μονὰς τὸν Δ ἀριθμὸν μετρεῖ καὶ ὁ Ε τὸν <lb n="1799,001,,,,8,10,26"></lb> |
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7234 Α.</s> <s>ἡ δὲ Γ μονὰς τὸν Δ ἀριθμὸν μετρεῖ κατὰ τὰς ἐν τῷ Δ <lb n="1799,001,,,,8,10,27"></lb> |
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7235 μονάδας· καὶ ὁ Ε ἄρα τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Δ <lb n="1799,001,,,,8,10,28"></lb> |
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7236 μονάδας· ὁ Δ ἄρα τὸν Ε πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <s><pb></pb> <lb n="1799,001,,,,8,10,29"></lb> |
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7237 διὰ τὰ αὐτὰ δὴ καὶ ὁ μὲν Ζ ἑαυτὸν πολλαπλασιάσας τὸν Η <lb n="1799,001,,,,8,10,30"></lb> |
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7238 πεποίηκεν, τὸν δὲ Η πολλαπλασιάσας τὸν Β πεποίηκεν.</s> <lb n="1799,001,,,,8,10,31"></lb> |
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7239 <s>καὶ ἐπεὶ ὁ Δ ἑαυτὸν μὲν πολλαπλασιάσας τὸν Ε πεποίηκεν, <lb n="1799,001,,,,8,10,32"></lb> |
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7240 τὸν δὲ Ζ πολλαπλασιάσας τὸν Θ πεποίηκεν, ἔστιν ἄρα <lb n="1799,001,,,,8,10,33"></lb> |
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7241 ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Ε πρὸς τὸν Θ.</s> <s>διὰ τὰ αὐτὰ δὴ <lb n="1799,001,,,,8,10,34"></lb> |
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7242 καὶ ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Θ πρὸς τὸν Η.</s> <s>καὶ ὡς <lb n="1799,001,,,,8,10,35"></lb> |
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7243 ἄρα ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Θ πρὸς τὸν Η.</s> <s>πάλιν, ἐπεὶ <lb n="1799,001,,,,8,10,36"></lb> |
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7244 ὁ Δ ἑκάτερον τῶν Ε, Θ πολλαπλασιάσας ἑκάτερον τῶν <lb n="1799,001,,,,8,10,37"></lb> |
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7245 Α, Κ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Θ, οὕτως <lb n="1799,001,,,,8,10,38"></lb> |
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7246 ὁ Α πρὸς τὸν Κ.</s> <s>ἀλλ' ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Δ <lb n="1799,001,,,,8,10,39"></lb> |
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7247 πρὸς τὸν Ζ· καὶ ὡς ἄρα ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Α <lb n="1799,001,,,,8,10,40"></lb> |
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7248 πρὸς τὸν Κ.</s> <s>πάλιν, ἐπεὶ ἑκάτερος τῶν Δ, Ζ τὸν Θ <lb n="1799,001,,,,8,10,41"></lb> |
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7249 πολλαπλασιάσας ἑκάτερον τῶν Κ, Λ πεποίηκεν, ἔστιν ἄρα <lb n="1799,001,,,,8,10,42"></lb> |
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7250 ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Κ πρὸς τὸν Λ.</s> <s>ἀλλ' ὡς <lb n="1799,001,,,,8,10,43"></lb> |
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7251 ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Α πρὸς τὸν Κ· καὶ ὡς ἄρα <lb n="1799,001,,,,8,10,44"></lb> |
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7252 ὁ Α πρὸς τὸν Κ, οὕτως ὁ Κ πρὸς τὸν Λ.</s> <s>ἔτι ἐπεὶ ὁ <lb n="1799,001,,,,8,10,45"></lb> |
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7253 Ζ ἑκάτερον τῶν Θ, Η πολλαπλασιάσας ἑκάτερον τῶν <lb n="1799,001,,,,8,10,46"></lb> |
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7254 Λ, Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Θ πρὸς τὸν Η, οὕτως <lb n="1799,001,,,,8,10,47"></lb> |
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7255 ὁ Λ πρὸς τὸν Β.</s> <s>ὡς δὲ ὁ Θ πρὸς τὸν Η, οὕτως ὁ Δ <lb n="1799,001,,,,8,10,48"></lb> |
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7256 πρὸς τὸν Ζ· καὶ ὡς ἄρα ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Λ πρὸς <lb n="1799,001,,,,8,10,49"></lb> |
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7257 τὸν Β.</s> <s>ἐδείχθη δὲ καὶ ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὅ τε Α <lb n="1799,001,,,,8,10,50"></lb> |
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7258 πρὸς τὸν Κ καὶ ὁ Κ πρὸς τὸν Λ· καὶ ὡς ἄρα ὁ Α πρὸς τὸν <lb n="1799,001,,,,8,10,51"></lb> |
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7259 Κ, οὕτως ὁ Κ πρὸς τὸν Λ καὶ ὁ Λ πρὸς τὸν Β.</s> <s>οἱ Α, Κ, <lb n="1799,001,,,,8,10,52"></lb> |
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7260 Λ, Β ἄρα κατὰ τὸ συνεχὲς ἑξῆς εἰσιν ἀνάλογον.</s> <s>ὅσοι <lb n="1799,001,,,,8,10,53"></lb> |
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7261 ἄρα ἑκατέρου τῶν Α, Β καὶ τῆς Γ μονάδος μεταξὺ κατὰ <pb></pb> <lb n="1799,001,,,,8,10,54"></lb> |
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7262 τὸ συνεχὲς ἀνάλογον ἐμπίπτουσιν ἀριθμοί, τοσοῦτοι καὶ <lb n="1799,001,,,,8,10,55"></lb> |
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7263 εἰς τοὺς Α, Β μεταξὺ κατὰ τὸ συνεχὲς ἐμπεσοῦνται· <lb n="1799,001,,,,8,10,56"></lb> |
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7264 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,10,57"></lb> |
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7265 <s>Δύο τετραγώνων ἀριθμῶν εἷς μέσος ἀνάλογόν ἐστιν <lb n="1799,001,,,,8,11,1"></lb> |
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7266 ἀριθμός, καὶ ὁ τετράγωνος πρὸς τὸν τετράγωνον διπλα-<lb n="1799,001,,,,8,11,2"></lb> |
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7267 σίονα λόγον ἔχει ἤπερ ἡ πλευρὰ πρὸς τὴν πλευράν.</s> <lb n="1799,001,,,,8,11,3"></lb> |
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7268 <s>Ἔστωσαν τετράγωνοι ἀριθμοὶ οἱ Α, Β, καὶ τοῦ μὲν Α <lb n="1799,001,,,,8,11,4"></lb> |
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7269 πλευρὰ ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ· <lb n="1799,001,,,,8,11,5"></lb> |
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7270 λέγω, ὅτι τῶν Α, Β εἷς μέσος <lb n="1799,001,,,,8,11,6"></lb> |
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7271 ἀνάλογόν ἐστιν ἀριθμός, καὶ ὁ Α <lb n="1799,001,,,,8,11,7"></lb> |
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7272 πρὸς τὸν Β διπλασίονα λόγον ἔχει <lb n="1799,001,,,,8,11,8"></lb> |
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7273 ἤπερ ὁ Γ πρὸς τὸν Δ.</s> <lb n="1799,001,,,,8,11,9"></lb> |
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7274 <s>Ὁ Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω.</s> <s>καὶ <lb n="1799,001,,,,8,11,10"></lb> |
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7275 ἐπεὶ τετράγωνός ἐστιν ὁ Α, πλευρὰ δὲ αὐτοῦ ἐστιν ὁ Γ, <lb n="1799,001,,,,8,11,11"></lb> |
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7276 ὁ Γ ἄρα ἑαυτὸν πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <s>διὰ <lb n="1799,001,,,,8,11,12"></lb> |
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7277 τὰ αὐτὰ δὴ καὶ ὁ Δ ἑαυτὸν πολλαπλασιάσας τὸν Β <lb n="1799,001,,,,8,11,13"></lb> |
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7278 πεποίηκεν.</s> <s>ἐπεὶ οὖν ὁ Γ ἑκάτερον τῶν Γ, Δ πολλαπλα-<lb n="1799,001,,,,8,11,14"></lb> |
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7279 σιάσας ἑκάτερον τῶν Α, Ε πεποίηκεν, ἔστιν ἄρα ὡς ὁ Γ <lb n="1799,001,,,,8,11,15"></lb> |
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7280 πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν Ε.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,8,11,16"></lb> |
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7281 ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ε πρὸς τὸν Β.</s> <s>καὶ ὡς ἄρα <lb n="1799,001,,,,8,11,17"></lb> |
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7282 ὁ Α πρὸς τὸν Ε, οὕτως ὁ Ε πρὸς τὸν Β.</s> <s>τῶν Α, Β ἄρα <lb n="1799,001,,,,8,11,18"></lb> |
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7283 εἷς μέσος ἀνάλογόν ἐστιν ἀριθμός.</s> <s><pb></pb> <lb n="1799,001,,,,8,11,19"></lb> |
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7284 Λέγω δή, ὅτι καὶ ὁ Α πρὸς τὸν Β διπλασίονα λόγον <lb n="1799,001,,,,8,11,20"></lb> |
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7285 ἔχει ἤπερ ὁ Γ πρὸς τὸν Δ.</s> <s>ἐπεὶ γὰρ τρεῖς ἀριθμοὶ ἀνάλογόν <lb n="1799,001,,,,8,11,21"></lb> |
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7286 εἰσιν οἱ Α, Ε, Β, ὁ Α ἄρα πρὸς τὸν Β διπλασίονα λόγον <lb n="1799,001,,,,8,11,22"></lb> |
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7287 ἔχει ἤπερ ὁ Α πρὸς τὸν Ε.</s> <s>ὡς δὲ ὁ Α πρὸς τὸν Ε, οὕτως <lb n="1799,001,,,,8,11,23"></lb> |
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7288 ὁ Γ πρὸς τὸν Δ.</s> <s>ὁ Α ἄρα πρὸς τὸν Β διπλασίονα λόγον <lb n="1799,001,,,,8,11,24"></lb> |
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7289 ἔχει ἤπερ ἡ Γ πλευρὰ πρὸς τὴν Δ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,11,25"></lb> |
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7290 <s>Δύο κύβων ἀριθμῶν δύο μέσοι ἀνάλογόν εἰσιν ἀριθμοί, <lb n="1799,001,,,,8,12,1"></lb> |
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7291 καὶ ὁ κύβος πρὸς τὸν κύβον τριπλασίονα λόγον ἔχει ἤπερ <lb n="1799,001,,,,8,12,2"></lb> |
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7292 ἡ πλευρὰ πρὸς τὴν πλευράν.</s> <lb n="1799,001,,,,8,12,3"></lb> |
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7293 <s>Ἔστωσαν κύβοι ἀριθμοὶ οἱ Α, Β καὶ τοῦ μὲν Α πλευρὰ <lb n="1799,001,,,,8,12,4"></lb> |
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7294 ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ· λέγω, ὅτι τῶν Α, Β δύο μέσοι <lb n="1799,001,,,,8,12,5"></lb> |
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7295 ἀνάλογόν εἰσιν ἀριθμοί, καὶ ὁ Α πρὸς τὸν Β τριπλασίονα <lb n="1799,001,,,,8,12,6"></lb> |
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7296 λόγον ἔχει ἤπερ ὁ Γ πρὸς τὸν Δ.</s> <lb n="1799,001,,,,8,12,7"></lb> |
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7297 <s>Ὁ γὰρ Γ ἑαυτὸν μὲν πολλαπλασιάσας τὸν Ε ποιείτω, <lb n="1799,001,,,,8,12,8"></lb> |
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7298 τὸν δὲ Δ πολλαπλασιάσας τὸν Ζ ποιείτω, ὁ δὲ Δ ἑαυτὸν <lb n="1799,001,,,,8,12,9"></lb> |
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7299 πολλαπλασιάσας τὸν Η ποιείτω, ἑκάτερος δὲ τῶν Γ, Δ <lb n="1799,001,,,,8,12,10"></lb> |
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7300 τὸν Ζ πολλαπλασιάσας ἑκάτερον τῶν Θ, Κ ποιείτω.</s> <lb n="1799,001,,,,8,12,11"></lb> |
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7301 <s>Καὶ ἐπεὶ κύβος ἐστὶν ὁ Α, πλευρὰ δὲ αὐτοῦ ὁ Γ, καὶ ὁ <lb n="1799,001,,,,8,12,12"></lb> |
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7302 Γ ἑαυτὸν πολλαπλασιάσας τὸν Ε πεποίηκεν, ὁ Γ ἄρα <lb n="1799,001,,,,8,12,13"></lb> |
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7303 ἑαυτὸν μὲν πολλαπλασιάσας τὸν Ε πεποίηκεν, τὸν δὲ Ε <pb></pb> <lb n="1799,001,,,,8,12,14"></lb> |
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7304 πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,8,12,15"></lb> |
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7305 ὁ Δ ἑαυτὸν μὲν πολλαπλασιάσας τὸν Η πεποίηκεν, τὸν δὲ <lb n="1799,001,,,,8,12,16"></lb> |
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7306 Η πολλαπλασιάσας τὸν Β πεποίηκεν.</s> <s>καὶ ἐπεὶ ὁ Γ ἑκάτερον <lb n="1799,001,,,,8,12,17"></lb> |
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7307 τῶν Γ, Δ πολλαπλασιάσας ἑκάτερον τῶν Ε, Ζ πεποίηκεν, <lb n="1799,001,,,,8,12,18"></lb> |
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7308 ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ε πρὸς τὸν Ζ.</s> <lb n="1799,001,,,,8,12,19"></lb> |
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7309 <s>διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ζ πρὸς τὸν <lb n="1799,001,,,,8,12,20"></lb> |
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7310 Η.</s> <s>πάλιν, ἐπεὶ ὁ Γ ἑκάτερον τῶν Ε, Ζ πολλαπλασιάσας <lb n="1799,001,,,,8,12,21"></lb> |
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7311 ἑκάτερον τῶν Α, Θ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Ε πρὸς <lb n="1799,001,,,,8,12,22"></lb> |
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7312 τὸν Ζ, οὕτως ὁ Α πρὸς τὸν Θ.</s> <s>ὡς δὲ ὁ Ε πρὸς τὸν Ζ, <lb n="1799,001,,,,8,12,23"></lb> |
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7313 οὕτως ὁ Γ πρὸς τὸν Δ· καὶ ὡς ἄρα ὁ Γ πρὸς τὸν Δ, οὕτως <lb n="1799,001,,,,8,12,24"></lb> |
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7314 ὁ Α πρὸς τὸν Θ.</s> <s>πάλιν, ἐπεὶ ἑκάτερος τῶν Γ, Δ τὸν Ζ <lb n="1799,001,,,,8,12,25"></lb> |
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7315 πολλαπλασιάσας ἑκάτερον τῶν Θ, Κ πεποίηκεν, ἔστιν ἄρα <lb n="1799,001,,,,8,12,26"></lb> |
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7316 ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Θ πρὸς τὸν Κ.</s> <s>πάλιν, ἐπεὶ <lb n="1799,001,,,,8,12,27"></lb> |
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7317 ὁ Δ ἑκάτερον τῶν Ζ, Η πολλαπλασιάσας ἑκάτερον τῶν <lb n="1799,001,,,,8,12,28"></lb> |
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7318 Κ, Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Ζ πρὸς τὸν Η, οὕτως <lb n="1799,001,,,,8,12,29"></lb> |
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7319 ὁ Κ πρὸς τὸν Β.</s> <s>ὡς δὲ ὁ Ζ πρὸς τὸν Η, οὕτως ὁ Γ πρὸς <lb n="1799,001,,,,8,12,30"></lb> |
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7320 τὸν Δ· καὶ ὡς ἄρα ὁ Γ πρὸς τὸν Δ, οὕτως ὅ τε Α πρὸς <lb n="1799,001,,,,8,12,31"></lb> |
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7321 τὸν Θ καὶ ὁ Θ πρὸς τὸν Κ καὶ ὁ Κ πρὸς τὸν Β.</s> <s>τῶν Α, Β <lb n="1799,001,,,,8,12,32"></lb> |
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7322 ἄρα δύο μέσοι ἀνάλογόν εἰσιν οἱ Θ, Κ.</s> <lb n="1799,001,,,,8,12,33"></lb> |
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7323 <s>Λέγω δή, ὅτι καὶ ὁ Α πρὸς τὸν Β τριπλασίονα λόγον <lb n="1799,001,,,,8,12,34"></lb> |
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7324 ἔχει ἤπερ ὁ Γ πρὸς τὸν Δ.</s> <s>ἐπεὶ γὰρ τέσσαρες ἀριθμοὶ ἀνά-<lb n="1799,001,,,,8,12,35"></lb> |
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7325 λογόν εἰσιν οἱ Α, Θ, Κ, Β, ὁ Α ἄρα πρὸς τὸν Β τριπλα-<lb n="1799,001,,,,8,12,36"></lb> |
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7326 σίονα λόγον ἔχει ἤπερ ὁ Α πρὸς τὸν Θ.</s> <s>ὡς δὲ ὁ Α πρὸς τὸν <pb></pb> <lb n="1799,001,,,,8,12,37"></lb> |
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7327 Θ, οὕτως ὁ Γ πρὸς τὸν Δ· καὶ ὁ Α [ἄρα] πρὸς τὸν Β τρι-<lb n="1799,001,,,,8,12,38"></lb> |
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7328 πλασίονα λόγον ἔχει ἤπερ ὁ Γ πρὸς τὸν Δ.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,12,39"></lb> |
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7329 <s>Ἐὰν ὦσιν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, καὶ <lb n="1799,001,,,,8,13,1"></lb> |
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7330 πολλαπλασιάσας ἕκαστος ἑαυτὸν ποιῇ τινα, οἱ γενόμενοι <lb n="1799,001,,,,8,13,2"></lb> |
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7331 ἐξ αὐτῶν ἀνάλογον ἔσονται· καὶ ἐὰν οἱ ἐξ ἀρχῆς τοὺς <lb n="1799,001,,,,8,13,3"></lb> |
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7332 γενομένους πολλαπλασιάσαντες ποιῶσί τινας, καὶ αὐτοὶ <lb n="1799,001,,,,8,13,4"></lb> |
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7333 ἀνάλογον ἔσονται [καὶ ἀεὶ περὶ τοὺς ἄκρους τοῦτο <lb n="1799,001,,,,8,13,5"></lb> |
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7334 συμβαίνει].</s> <lb n="1799,001,,,,8,13,6"></lb> |
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7335 <s>Ἔστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, οἱ Α, Β, <lb n="1799,001,,,,8,13,7"></lb> |
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7336 Γ, ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Β πρὸς τὸν Γ, καὶ οἱ <lb n="1799,001,,,,8,13,8"></lb> |
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7337 Α, Β, Γ ἑαυτοὺς μὲν πολλαπλασιάσαντες τοὺς Δ, Ε, Ζ <lb n="1799,001,,,,8,13,9"></lb> |
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7338 ποιείτωσαν, τοὺς δὲ Δ, Ε, Ζ πολλαπλασιάσαντες τοὺς <lb n="1799,001,,,,8,13,10"></lb> |
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7339 Η, Θ, Κ ποιείτωσαν· λέγω, ὅτι οἵ τε Δ, Ε, Ζ καὶ οἱ <lb n="1799,001,,,,8,13,11"></lb> |
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7340 Η, Θ, Κ ἑξῆς ἀνάλογόν εἰσιν.</s> <lb n="1799,001,,,,8,13,12"></lb> |
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7341 <s>Ὁ μὲν γὰρ Α τὸν Β πολλαπλασιάσας τὸν Λ ποιείτω, <lb n="1799,001,,,,8,13,13"></lb> |
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7342 ἑκάτερος δὲ τῶν Α, Β τὸν Λ πολλαπλασιάσας ἑκάτερον <lb n="1799,001,,,,8,13,14"></lb> |
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7343 τῶν Μ, Ν ποιείτω.</s> <s>καὶ πάλιν ὁ μὲν Β τὸν Γ πολλαπλα- <pb></pb><lb n="1799,001,,,,8,13,15"></lb> |
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7344 σιάσας τὸν Ξ ποιείτω, ἑκάτερος δὲ τῶν Β, Γ τὸν Ξ <lb n="1799,001,,,,8,13,16"></lb> |
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7345 πολλαπλασιάσας ἑκάτερον τῶν Ο, Π ποιείτω.</s> <lb n="1799,001,,,,8,13,17"></lb> |
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7346 <s>Ὁμοίως δὴ τοῖς ἐπάνω δείξομεν, ὅτι οἱ Δ, Λ, Ε καὶ οἱ <lb n="1799,001,,,,8,13,18"></lb> |
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7347 Η, Μ, Ν, Θ ἑξῆς εἰσιν ἀνάλογον ἐν τῷ τοῦ Α πρὸς τὸν Β <lb n="1799,001,,,,8,13,19"></lb> |
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7348 λόγῳ, καὶ ἔτι οἱ Ε, Ξ, Ζ καὶ οἱ Θ, Ο, Π, Κ ἑξῆς εἰσιν <lb n="1799,001,,,,8,13,20"></lb> |
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7349 ἀνάλογον ἐν τῷ τοῦ Β πρὸς τὸν Γ λόγῳ.</s> <s>καί ἐστιν ὡς ὁ Α <lb n="1799,001,,,,8,13,21"></lb> |
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7350 πρὸς τὸν Β, οὕτως ὁ Β πρὸς τὸν Γ· καὶ οἱ Δ, Λ, Ε ἄρα <lb n="1799,001,,,,8,13,22"></lb> |
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7351 τοῖς Ε, Ξ, Ζ ἐν τῷ αὐτῷ λόγῳ εἰσὶ καὶ ἔτι οἱ Η, Μ, Ν, Θ <lb n="1799,001,,,,8,13,23"></lb> |
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7352 τοῖς Θ, Ο, Π, Κ.</s> <s>καί ἐστιν ἴσον τὸ μὲν τῶν Δ, Λ, Ε <lb n="1799,001,,,,8,13,24"></lb> |
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7353 πλῆθος τῷ τῶν Ε, Ξ, Ζ πλήθει, τὸ δὲ τῶν Η, Μ, Ν, Θ <lb n="1799,001,,,,8,13,25"></lb> |
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7354 τῷ τῶν Θ, Ο, Π, Κ· δι' ἴσου ἄρα ἐστὶν ὡς μὲν ὁ Δ πρὸς <lb n="1799,001,,,,8,13,26"></lb> |
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7355 τὸν Ε, οὕτως ὁ Ε πρὸς τὸν Ζ, ὡς δὲ ὁ Η πρὸς τὸν Θ, <lb n="1799,001,,,,8,13,27"></lb> |
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7356 οὕτως ὁ Θ πρὸς τὸν Κ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,13,28"></lb> |
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7357 <s>Ἐὰν τετράγωνος τετράγωνον μετρῇ, καὶ ἡ πλευρὰ <lb n="1799,001,,,,8,14,1"></lb> |
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7358 τὴν πλευρὰν μετρήσει· καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν <lb n="1799,001,,,,8,14,2"></lb> |
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7359 μετρῇ, καὶ ὁ τετράγωνος τὸν τετράγωνον μετρήσει.</s> <lb n="1799,001,,,,8,14,3"></lb> |
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7360 <s>Ἔστωσαν τετράγωνοι ἀριθμοὶ οἱ <lb n="1799,001,,,,8,14,4"></lb> |
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7361 Α, Β, πλευραὶ δὲ αὐτῶν ἔστωσαν <lb n="1799,001,,,,8,14,5"></lb> |
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7362 οἱ Γ, Δ, ὁ δὲ Α τὸν Β μετρείτω· <lb n="1799,001,,,,8,14,6"></lb> |
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7363 λέγω, ὅτι καὶ ὁ Γ τὸν Δ μετρεῖ.</s> <s><pb></pb> <lb n="1799,001,,,,8,14,7"></lb> |
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7364 Ὁ Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω· οἱ Α, <lb n="1799,001,,,,8,14,8"></lb> |
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7365 Ε, Β ἄρα ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ <lb n="1799,001,,,,8,14,9"></lb> |
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7366 λόγῳ.</s> <s>καὶ ἐπεὶ οἱ Α, Ε, Β ἑξῆς ἀνάλογόν εἰσιν, καὶ μετρεῖ <lb n="1799,001,,,,8,14,10"></lb> |
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7367 ὁ Α τὸν Β, μετρεῖ ἄρα καὶ ὁ Α τὸν Ε.</s> <s>καί ἐστιν ὡς ὁ Α πρὸς <lb n="1799,001,,,,8,14,11"></lb> |
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7368 τὸν Ε, οὕτως ὁ Γ πρὸς τὸν Δ· μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ.</s> <lb n="1799,001,,,,8,14,12"></lb> |
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7369 <s>Πάλιν δὴ ὁ Γ τὸν Δ μετρείτω· λέγω, ὅτι καὶ ὁ Α τὸν Β <lb n="1799,001,,,,8,14,13"></lb> |
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7370 μετρεῖ.</s> <lb n="1799,001,,,,8,14,14"></lb> |
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7371 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, <lb n="1799,001,,,,8,14,15"></lb> |
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7372 ὅτι οἱ Α, Ε, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν <lb n="1799,001,,,,8,14,16"></lb> |
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7373 Δ λόγῳ.</s> <s>καὶ ἐπεί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α <lb n="1799,001,,,,8,14,17"></lb> |
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7374 πρὸς τὸν Ε, μετρεῖ δὲ ὁ Γ τὸν Δ, μετρεῖ ἄρα καὶ ὁ Α <lb n="1799,001,,,,8,14,18"></lb> |
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7375 τὸν Ε.</s> <s>καί εἰσιν οἱ Α, Ε, Β ἑξῆς ἀνάλογον· μετρεῖ ἄρα <lb n="1799,001,,,,8,14,19"></lb> |
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7376 καὶ ὁ Α τὸν Β.</s> <lb n="1799,001,,,,8,14,20"></lb> |
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7377 <s>Ἐὰν ἄρα τετράγωνος τετράγωνον μετρῇ, καὶ ἡ πλευρὰ <lb n="1799,001,,,,8,14,21"></lb> |
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7378 τὴν πλευρὰν μετρήσει· καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν <lb n="1799,001,,,,8,14,22"></lb> |
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7379 μετρῇ, καὶ ὁ τετράγωνος τὸν τετράγωνον μετρήσει· ὅπερ <lb n="1799,001,,,,8,14,23"></lb> |
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7380 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,14,24"></lb> |
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7381 <s>Ἐὰν κύβος ἀριθμὸς κύβον ἀριθμὸν μετρῇ, καὶ ἡ πλευρὰ <lb n="1799,001,,,,8,15,1"></lb> |
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7382 τὴν πλευρὰν μετρήσει· καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν <lb n="1799,001,,,,8,15,2"></lb> |
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7383 μετρῇ, καὶ ὁ κύβος τὸν κύβον μετρήσει.</s> <lb n="1799,001,,,,8,15,3"></lb> |
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7384 <s>Κύβος γὰρ ἀριθμὸς ὁ Α κύβον τὸν Β μετρείτω, καὶ <lb n="1799,001,,,,8,15,4"></lb> |
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7385 τοῦ μὲν Α πλευρὰ ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ· λέγω, ὅτι ὁ <lb n="1799,001,,,,8,15,5"></lb> |
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7386 Γ τὸν Δ μετρεῖ.</s> <s><pb></pb> <lb n="1799,001,,,,8,15,6"></lb> |
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7387 Ὁ Γ γὰρ ἑαυτὸν πολλαπλασιάσας τὸν Ε ποιείτω, ὁ δὲ <lb n="1799,001,,,,8,15,7"></lb> |
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7388 Δ ἑαυτὸν πολλαπλασιάσας τὸν Η ποιείτω, καὶ ἔτι ὁ Γ <lb n="1799,001,,,,8,15,8"></lb> |
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7389 τὸν Δ πολλαπλασιάσας τὸν Ζ [ποιείτω], ἑκάτερος δὲ <lb n="1799,001,,,,8,15,9"></lb> |
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7390 τῶν Γ, Δ τὸν Ζ πολλαπλασιάσας ἑκάτερον τῶν Θ, Κ <lb n="1799,001,,,,8,15,10"></lb> |
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7391 ποιείτω.</s> <s>φανερὸν δή, ὅτι οἱ Ε, Ζ, <lb n="1799,001,,,,8,15,11"></lb> |
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7392 Η καὶ οἱ Α, Θ, Κ, Β ἑξῆς ἀνά-<lb n="1799,001,,,,8,15,12"></lb> |
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7393 λογόν εἰσιν ἐν τῳ̂ τοῦ Γ πρὸς <lb n="1799,001,,,,8,15,13"></lb> |
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7394 τὸν Δ λόγῳ.</s> <s>καὶ ἐπεὶ οἱ Α, Θ, <lb n="1799,001,,,,8,15,14"></lb> |
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7395 Κ, Β ἑξῆς ἀνάλογόν εἰσιν, καὶ <lb n="1799,001,,,,8,15,15"></lb> |
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7396 μετρεῖ ὁ Α τὸν Β, μετρεῖ ἄρα καὶ <lb n="1799,001,,,,8,15,16"></lb> |
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7397 τὸν Θ.</s> <s>καί ἐστιν ὡς ὁ Α πρὸς τὸν <lb n="1799,001,,,,8,15,17"></lb> |
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7398 Θ, οὕτως ὁ Γ πρὸς τὸν Δ· μετρεῖ <lb n="1799,001,,,,8,15,18"></lb> |
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7399 ἄρα καὶ ὁ Γ τὸν Δ.</s> <lb n="1799,001,,,,8,15,19"></lb> |
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7400 <s>Ἀλλὰ δὴ μετρείτω ὁ Γ τὸν Δ· λέγω, ὅτι καὶ ὁ Α τὸν Β <lb n="1799,001,,,,8,15,20"></lb> |
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7401 μετρήσει.</s> <lb n="1799,001,,,,8,15,21"></lb> |
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7402 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δὴ δείξομεν, <lb n="1799,001,,,,8,15,22"></lb> |
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7403 ὅτι οἱ Α, Θ, Κ, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς <lb n="1799,001,,,,8,15,23"></lb> |
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7404 τὸν Δ λόγῳ.</s> <s>καὶ ἐπεὶ ὁ Γ τὸν Δ μετρεῖ, καί ἐστιν ὡς <lb n="1799,001,,,,8,15,24"></lb> |
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7405 ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν Θ, καὶ ὁ Α ἄρα τὸν Θ <lb n="1799,001,,,,8,15,25"></lb> |
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7406 μετρεῖ· ὥστε καὶ τὸν Β μετρεῖ ὁ Α· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,15,26"></lb> |
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7407 <s>Ἐὰν τετράγωνος ἀριθμὸς τετράγωνον ἀριθμὸν μὴ <lb n="1799,001,,,,8,16,1"></lb> |
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7408 μετρῇ, οὐδὲ ἡ πλευρὰ τὴν πλευρὰν μετρήσει· κἂν ἡ <lb n="1799,001,,,,8,16,2"></lb> |
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7409 πλευρὰ τὴν πλευρὰν μὴ μετρῇ, οὐδὲ ὁ τετράγωνος τὸν <lb n="1799,001,,,,8,16,3"></lb> |
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7410 τετράγωνον μετρήσει.</s> <s><pb></pb> <lb n="1799,001,,,,8,16,4"></lb> |
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7411 Ἔστωσαν τετράγωνοι ἀριθμοὶ οἱ Α, <lb n="1799,001,,,,8,16,5"></lb> |
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7412 Β, πλευραὶ δὲ αὐτῶν ἔστωσαν οἱ Γ, <lb n="1799,001,,,,8,16,6"></lb> |
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7413 Δ, καὶ μὴ μετρείτω ὁ Α τὸν Β· λέγω, <lb n="1799,001,,,,8,16,7"></lb> |
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7414 ὅτι οὐδὲ ὁ Γ τὸν Δ μετρεῖ.</s> <lb n="1799,001,,,,8,16,8"></lb> |
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7415 <s>Εἰ γὰρ μετρεῖ ὁ Γ τὸν Δ, μετρήσει <lb n="1799,001,,,,8,16,9"></lb> |
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7416 καὶ ὁ Α τὸν Β.</s> <s>οὐ μετρεῖ δὲ ὁ Α τὸν Β· οὐδὲ ἄρα <lb n="1799,001,,,,8,16,10"></lb> |
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7417 ὁ Γ τὸν Δ μετρήσει.</s> <lb n="1799,001,,,,8,16,11"></lb> |
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7418 <s>Μὴ μετρείτω [δὴ] πάλιν ὁ Γ τὸν Δ· λέγω, ὅτι οὐδὲ ὁ Α <lb n="1799,001,,,,8,16,12"></lb> |
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7419 τὸν Β μετρήσει.</s> <lb n="1799,001,,,,8,16,13"></lb> |
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7420 <s>Εἰ γὰρ μετρεῖ ὁ Α τὸν Β, μετρήσει καὶ ὁ Γ τὸν Δ.</s> <lb n="1799,001,,,,8,16,14"></lb> |
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7421 <s>οὐ μετρεῖ δὲ ὁ Γ τὸν Δ· οὐδ' ἄρα ὁ Α τὸν Β μετρήσει· <lb n="1799,001,,,,8,16,15"></lb> |
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7422 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,16,16"></lb> |
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7423 <s>Ἐὰν κύβος ἀριθμὸς κύβον ἀριθμὸν μὴ μετρῇ, οὐδὲ ἡ <lb n="1799,001,,,,8,17,1"></lb> |
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7424 πλευρὰ τὴν πλευρὰν μετρήσει· κἂν ἡ πλευρὰ τὴν πλευρὰν <lb n="1799,001,,,,8,17,2"></lb> |
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7425 μὴ μετρῇ, οὐδὲ ὁ κύβος τὸν κύβον μετρήσει.</s> <lb n="1799,001,,,,8,17,3"></lb> |
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7426 <s>Κύβος γὰρ ἀριθμὸς ὁ Α κύβον <lb n="1799,001,,,,8,17,4"></lb> |
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7427 ἀριθμὸν τὸν Β μὴ μετρείτω, καὶ <lb n="1799,001,,,,8,17,5"></lb> |
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7428 τοῦ μὲν Α πλευρὰ ἔστω ὁ Γ, τοῦ δὲ <lb n="1799,001,,,,8,17,6"></lb> |
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7429 Β ὁ Δ· λέγω, ὅτι ὁ Γ τὸν Δ οὐ <lb n="1799,001,,,,8,17,7"></lb> |
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7430 μετρήσει.</s> <lb n="1799,001,,,,8,17,8"></lb> |
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7431 <s>Εἰ γὰρ μετρεῖ ὁ Γ τὸν Δ, καὶ ὁ Α τὸν Β μετρήσει.</s> <s>οὐ <lb n="1799,001,,,,8,17,9"></lb> |
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7432 μετρεῖ δὲ ὁ Α τὸν Β· οὐδ' ἄρα ὁ Γ τὸν Δ μετρεῖ.</s> <s><pb></pb> <lb n="1799,001,,,,8,17,10"></lb> |
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7433 Ἀλλὰ δὴ μὴ μετρείτω ὁ Γ τὸν Δ· λέγω, ὅτι οὐδὲ ὁ Α <lb n="1799,001,,,,8,17,11"></lb> |
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7434 τὸν Β μετρήσει.</s> <lb n="1799,001,,,,8,17,12"></lb> |
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7435 <s>Εἰ γὰρ ὁ Α τὸν Β μετρεῖ, καὶ ὁ Γ τὸν Δ μετρήσει.</s> <lb n="1799,001,,,,8,17,13"></lb> |
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7436 <s>οὐ μετρεῖ δὲ ὁ Γ τὸν Δ· οὐδ' ἄρα ὁ Α τὸν Β μετρήσει· <lb n="1799,001,,,,8,17,14"></lb> |
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7437 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,17,15"></lb> |
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7438 <s>Δύο ὁμοίων ἐπιπέδων ἀριθμῶν εἷς μέσος ἀνάλογόν <lb n="1799,001,,,,8,18,1"></lb> |
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7439 ἐστιν ἀριθμός· καὶ ὁ ἐπίπεδος πρὸς τὸν ἐπίπεδον διπλα-<lb n="1799,001,,,,8,18,2"></lb> |
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7440 σίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν <lb n="1799,001,,,,8,18,3"></lb> |
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7441 ὁμόλογον πλευράν.</s> <lb n="1799,001,,,,8,18,4"></lb> |
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7442 <s>Ἔστωσαν δύο ὅμοιοι ἐπίπεδοι ἀριθμοὶ οἱ Α, Β, καὶ <lb n="1799,001,,,,8,18,5"></lb> |
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7443 τοῦ μὲν Α πλευραὶ ἔστωσαν οἱ Γ, Δ ἀριθμοί, τοῦ δὲ Β <lb n="1799,001,,,,8,18,6"></lb> |
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7444 οἱ Ε, Ζ.</s> <s>καὶ ἐπεὶ ὅμοιοι ἐπίπεδοί εἰσιν οἱ ἀνάλογον ἔχοντες <lb n="1799,001,,,,8,18,7"></lb> |
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7445 τὰς πλευράς, ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ε <lb n="1799,001,,,,8,18,8"></lb> |
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7446 πρὸς τὸν Ζ.</s> <s>λέγω οὖν, ὅτι τῶν Α, Β εἷς μέσος ἀνάλογόν <lb n="1799,001,,,,8,18,9"></lb> |
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7447 ἐστιν ἀριθμός, καὶ ὁ Α πρὸς τὸν Β διπλασίονα λόγον ἔχει <lb n="1799,001,,,,8,18,10"></lb> |
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7448 ἤπερ ὁ Γ πρὸς τὸν Ε ἢ ὁ Δ πρὸς τὸν Ζ, τουτέστιν ἤπερ ἡ <lb n="1799,001,,,,8,18,11"></lb> |
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7449 ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον [πλευράν].</s> <lb n="1799,001,,,,8,18,12"></lb> |
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7450 <s>Καὶ ἐπεί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ε πρὸς τὸν <lb n="1799,001,,,,8,18,13"></lb> |
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7451 Ζ, ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Γ πρὸς τὸν Ε, ὁ Δ πρὸς τὸν Ζ.</s> <lb n="1799,001,,,,8,18,14"></lb> |
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7452 <s>καὶ ἐπεὶ ἐπίπεδός ἐστιν ὁ Α, πλευραὶ δὲ αὐτοῦ οἱ Γ, Δ, <lb n="1799,001,,,,8,18,15"></lb> |
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7453 ὁ Δ ἄρα τὸν Γ πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <s>διὰ <pb></pb> <lb n="1799,001,,,,8,18,16"></lb> |
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7454 τὰ αὐτὰ δὴ καὶ ὁ Ε τὸν Ζ πολλαπλασιάσας τὸν Β πε-<lb n="1799,001,,,,8,18,17"></lb> |
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7455 ποίηκεν.</s> <s>ὁ Δ δὴ τὸν Ε πολλαπλασιάσας τὸν Η ποιείτω.</s> <lb n="1799,001,,,,8,18,18"></lb> |
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7456 <s>καὶ ἐπεὶ ὁ Δ τὸν μὲν Γ πολλαπλασιάσας τὸν Α πεποίηκεν, <lb n="1799,001,,,,8,18,19"></lb> |
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7457 τὸν δὲ Ε πολλαπλασιάσας τὸν Η πεποίηκεν, ἔστιν ἄρα <lb n="1799,001,,,,8,18,20"></lb> |
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7458 ὡς ὁ Γ πρὸς τὸν Ε, οὕτως ὁ Α πρὸς τὸν Η.</s> <s>ἀλλ' ὡς ὁ Γ <lb n="1799,001,,,,8,18,21"></lb> |
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7459 πρὸς τὸν Ε, [οὕτωσ] ὁ Δ πρὸς τὸν Ζ· καὶ ὡς ἄρα ὁ Δ <lb n="1799,001,,,,8,18,22"></lb> |
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7460 πρὸς τὸν Ζ, οὕτως ὁ Α πρὸς τὸν Η.</s> <s>πάλιν, ἐπεὶ ὁ Ε τὸν <lb n="1799,001,,,,8,18,23"></lb> |
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7461 μὲν Δ πολλαπλασιάσας τὸν Η πεποίηκεν, τὸν δὲ Ζ <lb n="1799,001,,,,8,18,24"></lb> |
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7462 πολλαπλασιάσας τὸν Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Δ πρὸς <lb n="1799,001,,,,8,18,25"></lb> |
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7463 τὸν Ζ, οὕτως ὁ Η πρὸς τὸν Β.</s> <s>ἐδείχθη δὲ καὶ ὡς ὁ Δ πρὸς <lb n="1799,001,,,,8,18,26"></lb> |
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7464 τὸν Ζ, οὕτως ὁ Α πρὸς τὸν Η· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Η, <lb n="1799,001,,,,8,18,27"></lb> |
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7465 οὕτως ὁ Η πρὸς τὸν Β.</s> <s>οἱ Α, Η, Β ἄρα ἑξῆς ἀνάλογόν <lb n="1799,001,,,,8,18,28"></lb> |
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7466 εἰσιν.</s> <s>τῶν Α, Β ἄρα εἷς μέσος ἀνάλογόν ἐστιν ἀριθμός.</s> <lb n="1799,001,,,,8,18,29"></lb> |
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7467 <s>Λέγω δή, ὅτι καὶ ὁ Α πρὸς τὸν Β διπλασίονα λόγον <lb n="1799,001,,,,8,18,30"></lb> |
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7468 ἔχει ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον πλευράν, <lb n="1799,001,,,,8,18,31"></lb> |
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7469 τουτέστιν ἤπερ ὁ Γ πρὸς τὸν Ε ἢ ὁ Δ πρὸς τὸν Ζ.</s> <s>ἐπεὶ γὰρ <lb n="1799,001,,,,8,18,32"></lb> |
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7470 οἱ Α, Η, Β ἑξῆς ἀνάλογόν εἰσιν, ὁ Α πρὸς τὸν Β διπλα-<lb n="1799,001,,,,8,18,33"></lb> |
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7471 σίονα λόγον ἔχει ἤπερ πρὸς τὸν Η.</s> <s>καί ἐστιν ὡς ὁ Α πρὸς <lb n="1799,001,,,,8,18,34"></lb> |
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7472 τὸν Η, οὕτως ὅ τε Γ πρὸς τὸν Ε καὶ ὁ Δ πρὸς τὸν Ζ.</s> <s>καὶ <lb n="1799,001,,,,8,18,35"></lb> |
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7473 ὁ Α ἄρα πρὸς τὸν Β διπλασίονα λόγον ἔχει ἤπερ ὁ Γ πρὸς <lb n="1799,001,,,,8,18,36"></lb> |
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7474 τὸν Ε ἢ ὁ Δ πρὸς τὸν Ζ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,18,37"></lb> |
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7475 <s>Δύο ὁμοίων στερεῶν ἀριθμῶν δύο μέσοι ἀνάλογον <lb n="1799,001,,,,8,19,1"></lb> |
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7476 ἐμπίπτουσιν ἀριθμοί· καὶ ὁ στερεὸς πρὸς τὸν ὅμοιον <lb n="1799,001,,,,8,19,2"></lb> |
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7477 στερεὸν τριπλασίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος πλευρὰ <lb n="1799,001,,,,8,19,3"></lb> |
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7478 πρὸς τὴν ὁμόλογον πλευράν.</s> <s><pb></pb> <lb n="1799,001,,,,8,19,4"></lb> |
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7479 Ἔστωσαν δύο ὅμοιοι στερεοὶ οἱ Α, Β, καὶ τοῦ μὲν Α <lb n="1799,001,,,,8,19,5"></lb> |
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7480 πλευραὶ ἔστωσαν οἱ Γ, Δ, Ε, τοῦ δὲ Β οἱ Ζ, Η, Θ.</s> <s>καὶ <lb n="1799,001,,,,8,19,6"></lb> |
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7481 ἐπεὶ ὅμοιοι στερεοί εἰσιν οἱ ἀνάλογον ἔχοντες τὰς πλευράς, <lb n="1799,001,,,,8,19,7"></lb> |
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7482 ἔστιν ἄρα ὡς μὲν ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ζ πρὸς τὸν Η, <lb n="1799,001,,,,8,19,8"></lb> |
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7483 ὡς δὲ ὁ Δ πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν Θ.</s> <s>λέγω, ὅτι <lb n="1799,001,,,,8,19,9"></lb> |
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7484 τῶν Α, Β δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί, καὶ <lb n="1799,001,,,,8,19,10"></lb> |
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7485 ὁ Α πρὸς τὸν Β τριπλασίονα λόγον ἔχει ἤπερ ὁ Γ πρὸς <lb n="1799,001,,,,8,19,11"></lb> |
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7486 τὸν Ζ καὶ ὁ Δ πρὸς τὸν Η καὶ ἔτι ὁ Ε πρὸς τὸν Θ.</s> <lb n="1799,001,,,,8,19,12"></lb> |
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7487 <s>Ὁ Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Κ ποιείτω, ὁ δὲ <lb n="1799,001,,,,8,19,13"></lb> |
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7488 Ζ τὸν Η πολλαπλασιάσας τὸν Λ ποιείτω.</s> <s>καὶ ἐπεὶ οἱ <lb n="1799,001,,,,8,19,14"></lb> |
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7489 Γ, Δ τοῖς Ζ, Η ἐν τῷ αὐτῷ λόγῳ εἰσίν, καὶ ἐκ μὲν τῶν <lb n="1799,001,,,,8,19,15"></lb> |
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7490 Γ, Δ ἐστιν ὁ Κ, ἐκ δὲ τῶν Ζ, Η ὁ Λ, οἱ Κ, Λ [ἄρα] ὅμοιοι <lb n="1799,001,,,,8,19,16"></lb> |
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7491 ἐπίπεδοί εἰσιν ἀριθμοί· τῶν Κ, Λ ἄρα εἷς μέσος ἀνάλογόν <lb n="1799,001,,,,8,19,17"></lb> |
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7492 ἐστιν ἀριθμός.</s> <s>ἔστω ὁ Μ.</s> <s>ὁ Μ ἄρα ἐστὶν ὁ ἐκ τῶν Δ, Ζ, <lb n="1799,001,,,,8,19,18"></lb> |
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7493 ὡς ἐν τῷ πρὸ τούτου θεωρήματι ἐδείχθη.</s> <s>καὶ ἐπεὶ ὁ Δ <lb n="1799,001,,,,8,19,19"></lb> |
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7494 τὸν μὲν Γ πολλαπλασιάσας τὸν Κ πεποίηκεν, τὸν δὲ Ζ <pb></pb> <lb n="1799,001,,,,8,19,20"></lb> |
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7495 πολλαπλασιάσας τὸν Μ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Γ <lb n="1799,001,,,,8,19,21"></lb> |
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7496 πρὸς τὸν Ζ, οὕτως ὁ Κ πρὸς τὸν Μ.</s> <s>ἀλλ' ὡς ὁ Κ πρὸς τὸν <lb n="1799,001,,,,8,19,22"></lb> |
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7497 Μ, ὁ Μ πρὸς τὸν Λ.</s> <s>οἱ Κ, Μ, Λ ἄρα ἑξῆς εἰσιν ἀνάλογον <lb n="1799,001,,,,8,19,23"></lb> |
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7498 ἐν τῷ τοῦ Γ πρὸς τὸν Ζ λόγῳ.</s> <s>καὶ ἐπεί ἐστιν ὡς ὁ Γ πρὸς <lb n="1799,001,,,,8,19,24"></lb> |
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7499 τὸν Δ, οὕτως ὁ Ζ πρὸς τὸν Η, ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Γ <lb n="1799,001,,,,8,19,25"></lb> |
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7500 πρὸς τὸν Ζ, οὕτως ὁ Δ πρὸς τὸν Η.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,8,19,26"></lb> |
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7501 ὡς ὁ Δ πρὸς τὸν Η, οὕτως ὁ Ε πρὸς τὸν Θ.</s> <s>οἱ Κ, Μ, Λ <lb n="1799,001,,,,8,19,27"></lb> |
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7502 ἄρα ἑξῆς εἰσιν ἀνάλογον ἔν τε τῷ τοῦ Γ πρὸς τὸν Ζ λόγῳ <lb n="1799,001,,,,8,19,28"></lb> |
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7503 καὶ τῷ τοῦ Δ πρὸς τὸν Η καὶ ἔτι τῷ τοῦ Ε πρὸς τὸν Θ.</s> <lb n="1799,001,,,,8,19,29"></lb> |
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7504 <s>ἑκάτερος δὴ τῶν Ε, Θ τὸν Μ πολλαπλασιάσας ἑκάτερον <lb n="1799,001,,,,8,19,30"></lb> |
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7505 τῶν Ν, Ξ ποιείτω.</s> <s>καὶ ἐπεὶ στερεός ἐστιν ὁ Α, πλευραὶ δὲ <lb n="1799,001,,,,8,19,31"></lb> |
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7506 αὐτοῦ εἰσιν οἱ Γ, Δ, Ε, ὁ Ε ἄρα τὸν ἐκ τῶν Γ, Δ πολλαπλα-<lb n="1799,001,,,,8,19,32"></lb> |
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7507 σιάσας τὸν Α πεποίηκεν.</s> <s>ὁ δὲ ἐκ τῶν Γ, Δ ἐστιν ὁ Κ· <lb n="1799,001,,,,8,19,33"></lb> |
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7508 ὁ Ε ἄρα τὸν Κ πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <s>διὰ τὰ <lb n="1799,001,,,,8,19,34"></lb> |
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7509 αὐτὰ δὴ καὶ ὁ Θ τὸν Λ πολλαπλασιάσας τὸν Β πεποίηκεν.</s> <lb n="1799,001,,,,8,19,35"></lb> |
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7510 <s>καὶ ἐπεὶ ὁ Ε τὸν Κ πολλαπλασιάσας τὸν Α πεποίηκεν, <lb n="1799,001,,,,8,19,36"></lb> |
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7511 ἀλλὰ μὴν καὶ τὸν Μ πολλαπλασιάσας τὸν Ν πεποίηκεν, <lb n="1799,001,,,,8,19,37"></lb> |
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7512 ἔστιν ἄρα ὡς ὁ Κ πρὸς τὸν Μ, οὕτως ὁ Α πρὸς τὸν Ν.</s> <lb n="1799,001,,,,8,19,38"></lb> |
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7513 <s>ὡς δὲ ὁ Κ πρὸς τὸν Μ, οὕτως ὅ τε Γ πρὸς τὸν Ζ καὶ ὁ Δ <lb n="1799,001,,,,8,19,39"></lb> |
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7514 πρὸς τὸν Η καὶ ἔτι ὁ Ε πρὸς τὸν Θ· καὶ ὡς ἄρα ὁ Γ πρὸς <lb n="1799,001,,,,8,19,40"></lb> |
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7515 τὸν Ζ καὶ ὁ Δ πρὸς τὸν Η καὶ ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Α <lb n="1799,001,,,,8,19,41"></lb> |
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7516 πρὸς τὸν Ν.</s> <s>πάλιν, ἐπεὶ ἑκάτερος τῶν Ε, Θ τὸν Μ πολλα-<lb n="1799,001,,,,8,19,42"></lb> |
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7517 πλασιάσας ἑκάτερον τῶν Ν, Ξ πεποίηκεν, ἔστιν ἄρα <lb n="1799,001,,,,8,19,43"></lb> |
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7518 ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Ν πρὸς τὸν Ξ.</s> <s>ἀλλ' ὡς ὁ Ε <pb></pb> <lb n="1799,001,,,,8,19,44"></lb> |
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7519 πρὸς τὸν Θ, οὕτως ὅ τε Γ πρὸς τὸν Ζ καὶ ὁ Δ πρὸς τὸν <lb n="1799,001,,,,8,19,45"></lb> |
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7520 Η· καὶ ὡς ἄρα ὁ Γ πρὸς τὸν Ζ καὶ ὁ Δ πρὸς τὸν Η <lb n="1799,001,,,,8,19,46"></lb> |
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7521 καὶ ὁ Ε πρὸς τὸν Θ, οὕτως ὅ τε Α πρὸς τὸν Ν καὶ ὁ Ν <lb n="1799,001,,,,8,19,47"></lb> |
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7522 πρὸς τὸν Ξ.</s> <s>πάλιν, ἐπεὶ ὁ Θ τὸν Μ πολλαπλασιάσας τὸν <lb n="1799,001,,,,8,19,48"></lb> |
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7523 Ξ πεποίηκεν, ἀλλὰ μὴν καὶ τὸν Λ πολλαπλασιάσας τὸν Β <lb n="1799,001,,,,8,19,49"></lb> |
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7524 πεποίηκεν, ἔστιν ἄρα ὡς ὁ Μ πρὸς τὸν Λ, οὕτως ὁ Ξ <lb n="1799,001,,,,8,19,50"></lb> |
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7525 πρὸς τὸν Β.</s> <s>ἀλλ' ὡς ὁ Μ πρὸς τὸν Λ, οὕτως ὅ τε Γ πρὸς <lb n="1799,001,,,,8,19,51"></lb> |
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7526 τὸν Ζ καὶ ὁ Δ πρὸς τὸν Η καὶ ὁ Ε πρὸς τὸν Θ.</s> <s>καὶ ὡς <lb n="1799,001,,,,8,19,52"></lb> |
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7527 ἄρα ὁ Γ πρὸς τὸν Ζ καὶ ὁ Δ πρὸς τὸν Η καὶ ὁ Ε πρὸς τὸν <lb n="1799,001,,,,8,19,53"></lb> |
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7528 Θ, οὕτως οὐ μόνον ὁ Ξ πρὸς τὸν Β, ἀλλὰ καὶ ὁ Α πρὸς τὸν <lb n="1799,001,,,,8,19,54"></lb> |
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7529 Ν καὶ ὁ Ν πρὸς τὸν Ξ.</s> <s>οἱ Α, Ν, Ξ, Β ἄρα ἑξῆς εἰσιν <lb n="1799,001,,,,8,19,55"></lb> |
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7530 ἀνάλογον ἐν τοῖς εἰρημένοις τῶν πλευρῶν λόγοις.</s> <lb n="1799,001,,,,8,19,56"></lb> |
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7531 <s>Λέγω, ὅτι καὶ ὁ Α πρὸς τὸν Β τριπλασίονα λόγον ἔχει <lb n="1799,001,,,,8,19,57"></lb> |
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7532 ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον πλευράν, <lb n="1799,001,,,,8,19,58"></lb> |
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7533 τουτέστιν ἤπερ ὁ Γ ἀριθμὸς πρὸς τὸν Ζ ἢ ὁ Δ πρὸς τὸν Η <lb n="1799,001,,,,8,19,59"></lb> |
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7534 καὶ ἔτι ὁ Ε πρὸς τὸν Θ.</s> <s>ἐπεὶ γὰρ τέσσαρες ἀριθμοὶ <lb n="1799,001,,,,8,19,60"></lb> |
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7535 ἑξῆς ἀνάλογόν εἰσιν οἱ Α, Ν, Ξ, Β, ὁ Α ἄρα πρὸς τὸν Β <lb n="1799,001,,,,8,19,61"></lb> |
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7536 τριπλασίονα λόγον ἔχει ἤπερ ὁ Α πρὸς τὸν Ν.</s> <s>ἀλλ' ὡς <lb n="1799,001,,,,8,19,62"></lb> |
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7537 ὁ Α πρὸς τὸν Ν, οὕτως ἐδείχθη ὅ τε Γ πρὸς τὸν Ζ καὶ <lb n="1799,001,,,,8,19,63"></lb> |
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7538 ὁ Δ πρὸς τὸν Η καὶ ἔτι ὁ Ε πρὸς τὸν Θ.</s> <s>καὶ ὁ Α ἄρα πρὸς <lb n="1799,001,,,,8,19,64"></lb> |
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7539 τὸν Β τριπλασίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος πλευρὰ <lb n="1799,001,,,,8,19,65"></lb> |
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7540 πρὸς τὴν ὁμόλογον πλευράν, τουτέστιν ἤπερ ὁ Γ ἀριθμὸς <lb n="1799,001,,,,8,19,66"></lb> |
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7541 πρὸς τὸν Ζ καὶ ὁ Δ πρὸς τὸν Η καὶ ἔτι ὁ Ε πρὸς τὸν Θ· <lb n="1799,001,,,,8,19,67"></lb> |
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7542 ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,8,19,68"></lb> |
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7543 Ἐὰν δύο ἀριθμῶν εἷς μέσος ἀνάλογον ἐμπίπτῃ ἀριθμός, <lb n="1799,001,,,,8,20,1"></lb> |
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7544 ὅμοιοι ἐπίπεδοι ἔσονται οἱ ἀριθμοί.</s> <lb n="1799,001,,,,8,20,2"></lb> |
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7545 <s>Δύο γὰρ ἀριθμῶν τῶν Α, Β εἷς μέσος ἀνάλογον <lb n="1799,001,,,,8,20,3"></lb> |
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7546 ἐμπιπτέτω ἀριθμὸς ὁ Γ· λέγω, ὅτι οἱ Α, Β ὅμοιοι <lb n="1799,001,,,,8,20,4"></lb> |
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7547 ἐπίπεδοί εἰσιν ἀριθμοί.</s> <lb n="1799,001,,,,8,20,5"></lb> |
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7548 <s>Εἰλήφθωσαν [γὰρ] ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν <lb n="1799,001,,,,8,20,6"></lb> |
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7549 λόγον ἐχόντων τοῖς Α, Γ οἱ Δ, Ε· ἰσάκις ἄρα ὁ Δ τὸν Α <lb n="1799,001,,,,8,20,7"></lb> |
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7550 μετρεῖ καὶ ὁ Ε τὸν Γ.</s> <s>ὁσάκις δὴ ὁ Δ τὸν Α μετρεῖ, <lb n="1799,001,,,,8,20,8"></lb> |
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7551 τοσαῦται μονάδες ἔστωσαν ἐν <lb n="1799,001,,,,8,20,9"></lb> |
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7552 τῷ Ζ· ὁ Ζ ἄρα τὸν Δ πολλα-<lb n="1799,001,,,,8,20,10"></lb> |
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7553 πλασιάσας τὸν Α πεποίηκεν.</s> <lb n="1799,001,,,,8,20,11"></lb> |
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7554 <s>ὥστε ὁ Α ἐπίπεδός ἐστιν, πλευ-<lb n="1799,001,,,,8,20,12"></lb> |
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7555 ραὶ δὲ αὐτοῦ οἱ Δ, Ζ.</s> <s>πάλιν, <lb n="1799,001,,,,8,20,13"></lb> |
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7556 ἐπεὶ οἱ Δ, Ε ἐλάχιστοί εἰσι τῶν <lb n="1799,001,,,,8,20,14"></lb> |
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7557 τὸν αὐτὸν λόγον ἐχόντων τοῖς <lb n="1799,001,,,,8,20,15"></lb> |
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7558 Γ, Β, ἰσάκις ἄρα ὁ Δ τὸν Γ μετρεῖ καὶ ὁ Ε τὸν Β.</s> <s>ὁσάκις δὴ ὁ <lb n="1799,001,,,,8,20,16"></lb> |
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7559 Ε τὸν Β μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Η.</s> <s>ὁ Ε <lb n="1799,001,,,,8,20,17"></lb> |
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7560 ἄρα τὸν Β μετρεῖ κατὰ τὰς ἐν τῷ Η μονάδας· ὁ Η ἄρα <lb n="1799,001,,,,8,20,18"></lb> |
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7561 τὸν Ε πολλαπλασιάσας τὸν Β πεποίηκεν.</s> <s>ὁ Β ἄρα ἐπίπεδός <pb></pb> <lb n="1799,001,,,,8,20,19"></lb> |
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7562 ἐστι, πλευραὶ δὲ αὐτοῦ εἰσιν οἱ Ε, Η.</s> <s>οἱ Α, Β ἄρα ἐπίπεδοί <lb n="1799,001,,,,8,20,20"></lb> |
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7563 εἰσιν ἀριθμοί.</s> <s>λέγω δή, ὅτι καὶ ὅμοιοι.</s> <s>ἐπεὶ γὰρ ὁ Ζ τὸν <lb n="1799,001,,,,8,20,21"></lb> |
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7564 μὲν Δ πολλαπλασιάσας τὸν Α πεποίηκεν, τὸν δὲ Ε <lb n="1799,001,,,,8,20,22"></lb> |
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7565 πολλαπλασιάσας τὸν Γ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Δ <lb n="1799,001,,,,8,20,23"></lb> |
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7566 πρὸς τὸν Ε, οὕτως ὁ Α πρὸς τὸν Γ, τουτέστιν ὁ Γ πρὸς <lb n="1799,001,,,,8,20,24"></lb> |
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7567 τὸν Β.</s> <s>πάλιν, ἐπεὶ ὁ Ε ἑκάτερον τῶν Ζ, Η πολλαπλασιάσας <lb n="1799,001,,,,8,20,25"></lb> |
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7568 τοὺς Γ, Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Ζ πρὸς τὸν Η, οὕτως <lb n="1799,001,,,,8,20,26"></lb> |
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7569 ὁ Γ πρὸς τὸν Β.</s> <s>ὡς δὲ ὁ Γ πρὸς τὸν Β, οὕτως ὁ Δ πρὸς <lb n="1799,001,,,,8,20,27"></lb> |
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7570 τὸν Ε· καὶ ὡς ἄρα ὁ Δ πρὸς τὸν Ε, οὕτως ὁ Ζ πρὸς τὸν Η.</s> <lb n="1799,001,,,,8,20,28"></lb> |
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7571 <s>καὶ ἐναλλὰξ ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Ε πρὸς τὸν Η.</s> <lb n="1799,001,,,,8,20,29"></lb> |
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7572 <s>οἱ Α, Β ἄρα ὅμοιοι ἐπίπεδοί εἰσιν ἀριθμοί· αἱ γὰρ πλευ-<lb n="1799,001,,,,8,20,30"></lb> |
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7573 ραὶ αὐτῶν ἀνάλογόν εἰσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,20,31"></lb> |
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7574 <s>Ἐὰν δύο ἀριθμῶν δύο μέσοι ἀνάλογον ἐμπίπτωσιν ἀρι-<lb n="1799,001,,,,8,21,1"></lb> |
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7575 θμοί, ὅμοιοι στερεοί εἰσιν οἱ ἀριθμοί.</s> <lb n="1799,001,,,,8,21,2"></lb> |
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7576 <s>Δύο γὰρ ἀριθμῶν τῶν Α, Β δύο μέσοι ἀνάλογον ἐμπιπτέ-<lb n="1799,001,,,,8,21,3"></lb> |
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7577 τωσαν ἀριθμοὶ οἱ Γ, Δ· λέγω, ὅτι οἱ Α, Β ὅμοιοι στερεοί <lb n="1799,001,,,,8,21,4"></lb> |
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7578 εἰσιν.</s> <lb n="1799,001,,,,8,21,5"></lb> |
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7579 <s>Εἰλήφθωσαν γὰρ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν <pb></pb> <lb n="1799,001,,,,8,21,6"></lb> |
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7580 λόγον ἐχόντων τοῖς Α, Γ, Δ τρεῖς οἱ Ε, Ζ, Η· οἱ ἄρα <lb n="1799,001,,,,8,21,7"></lb> |
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7581 ἄκροι αὐτῶν οἱ Ε, Η πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>καὶ <lb n="1799,001,,,,8,21,8"></lb> |
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7582 ἐπεὶ τῶν Ε, Η εἷς μέσος ἀνάλογον ἐμπέπτωκεν ἀριθμὸς <lb n="1799,001,,,,8,21,9"></lb> |
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7583 ὁ Ζ, οἱ Ε, Η ἄρα ἀριθμοὶ ὅμοιοι ἐπίπεδοί εἰσιν.</s> <s>ἔστωσαν <lb n="1799,001,,,,8,21,10"></lb> |
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7584 οὖν τοῦ μὲν Ε πλευραὶ οἱ Θ, Κ, τοῦ δὲ Η οἱ Λ, Μ.</s> <s>φανερὸν <lb n="1799,001,,,,8,21,11"></lb> |
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7585 ἄρα ἐστὶν ἐκ τοῦ πρὸ τούτου, ὅτι οἱ Ε, Ζ, Η ἑξῆς εἰσιν <lb n="1799,001,,,,8,21,12"></lb> |
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7586 ἀνάλογον ἔν τε τῷ τοῦ Θ πρὸς τὸν Λ λόγῳ καὶ τῷ τοῦ Κ <lb n="1799,001,,,,8,21,13"></lb> |
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7587 πρὸς τὸν Μ.</s> <s>καὶ ἐπεὶ οἱ Ε, Ζ, Η ἐλάχιστοί εἰσι τῶν τὸν <lb n="1799,001,,,,8,21,14"></lb> |
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7588 αὐτὸν λόγον ἐχόντων τοῖς Α, Γ, Δ, καί ἐστιν ἴσον τὸ <lb n="1799,001,,,,8,21,15"></lb> |
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7589 πλῆθος τῶν Ε, Ζ, Η τῷ πλήθει τῶν Α, Γ, Δ, δι' ἴσου <lb n="1799,001,,,,8,21,16"></lb> |
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7590 ἄρα ἐστὶν ὡς ὁ Ε πρὸς τὸν Η, οὕτως ὁ Α πρὸς τὸν Δ.</s> <s>οἱ δὲ <lb n="1799,001,,,,8,21,17"></lb> |
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7591 Ε, Η πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι <lb n="1799,001,,,,8,21,18"></lb> |
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7592 μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας αὐτοῖς ἰσάκις ὅ <lb n="1799,001,,,,8,21,19"></lb> |
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7593 τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, <lb n="1799,001,,,,8,21,20"></lb> |
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7594 τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος <lb n="1799,001,,,,8,21,21"></lb> |
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7595 τὸν ἑπόμενον· ἰσάκις ἄρα ὁ Ε τὸν Α μετρεῖ καὶ ὁ Η τὸν <lb n="1799,001,,,,8,21,22"></lb> |
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7596 Δ.</s> <s>ὁσάκις δὴ ὁ Ε τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστω- <pb></pb><lb n="1799,001,,,,8,21,23"></lb> |
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7597 σαν ἐν τῷ Ν.</s> <s>ὁ Ν ἄρα τὸν Ε πολλαπλασιάσας τὸν Α <lb n="1799,001,,,,8,21,24"></lb> |
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7598 πεποίηκεν.</s> <s>ὁ δὲ Ε ἐστιν ὁ ἐκ τῶν Θ, Κ· ὁ Ν ἄρα τὸν <lb n="1799,001,,,,8,21,25"></lb> |
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7599 ἐκ τῶν Θ, Κ πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <s>στερεὸς <lb n="1799,001,,,,8,21,26"></lb> |
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7600 ἄρα ἐστὶν ὁ Α, πλευραὶ δὲ αὐτοῦ εἰσιν οἱ Θ, Κ, Ν.</s> <s>πάλιν, <lb n="1799,001,,,,8,21,27"></lb> |
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7601 ἐπεὶ οἱ Ε, Ζ, Η ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον <lb n="1799,001,,,,8,21,28"></lb> |
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7602 ἐχόντων τοῖς Γ, Δ, Β, ἰσάκις ἄρα ὁ Ε τὸν Γ μετρεῖ καὶ <lb n="1799,001,,,,8,21,29"></lb> |
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7603 ὁ Η τὸν Β.</s> <s>ὁσάκις δὴ ὁ Ε τὸν Γ μετρεῖ, τοσαῦται μονάδες <lb n="1799,001,,,,8,21,30"></lb> |
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7604 ἔστωσαν ἐν τῷ Ξ.</s> <s>ὁ Η ἄρα τὸν Β μετρεῖ κατὰ τὰς ἐν <lb n="1799,001,,,,8,21,31"></lb> |
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7605 τῷ Ξ μονάδας· ὁ Ξ ἄρα τὸν Η πολλαπλασιάσας τὸν Β <lb n="1799,001,,,,8,21,32"></lb> |
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7606 πεποίηκεν.</s> <s>ὁ δὲ Η ἐστιν ὁ ἐκ τῶν Λ, Μ· ὁ Ξ ἄρα τὸν ἐκ <lb n="1799,001,,,,8,21,33"></lb> |
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7607 τῶν Λ, Μ πολλαπλασιάσας τὸν Β πεποίηκεν.</s> <s>στερεὸς <lb n="1799,001,,,,8,21,34"></lb> |
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7608 ἄρα ἐστὶν ὁ Β, πλευραὶ δὲ αὐτοῦ εἰσιν οἱ Λ, Μ, Ξ· οἱ <lb n="1799,001,,,,8,21,35"></lb> |
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7609 Α, Β ἄρα στερεοί εἰσιν.</s> <lb n="1799,001,,,,8,21,36"></lb> |
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7610 <s>Λέγω [δή], ὅτι καὶ ὅμοιοι.</s> <s>ἐπεὶ γὰρ οἱ Ν, Ξ τὸν Ε <lb n="1799,001,,,,8,21,37"></lb> |
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7611 πολλαπλασιάσαντες τοὺς Α, Γ πεποιήκασιν, ἔστιν ἄρα <lb n="1799,001,,,,8,21,38"></lb> |
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7612 ὡς ὁ Ν πρὸς τὸν Ξ, ὁ Α πρὸς τὸν Γ, τουτέστιν ὁ Ε πρὸς <lb n="1799,001,,,,8,21,39"></lb> |
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7613 τὸν Ζ.</s> <s>ἀλλ' ὡς ὁ Ε πρὸς τὸν Ζ, ὁ Θ πρὸς τὸν Λ καὶ ὁ Κ <lb n="1799,001,,,,8,21,40"></lb> |
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7614 πρὸς τὸν Μ· καὶ ὡς ἄρα ὁ Θ πρὸς τὸν Λ, οὕτως ὁ Κ πρὸς <lb n="1799,001,,,,8,21,41"></lb> |
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7615 τὸν Μ καὶ ὁ Ν πρὸς τὸν Ξ.</s> <s>καί εἰσιν οἱ μὲν Θ, Κ, Ν πλευ-<lb n="1799,001,,,,8,21,42"></lb> |
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7616 ραὶ τοῦ Α, οἱ δὲ Ξ, Λ, Μ πλευραὶ τοῦ Β.</s> <s>οἱ Α, Β ἄρα <lb n="1799,001,,,,8,21,43"></lb> |
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7617 ἀριθμοὶ ὅμοιοι στερεοί εἰσιν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,8,21,44"></lb> |
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7618 Ἐὰν τρεῖς ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν, ὁ δὲ πρῶτος <lb n="1799,001,,,,8,22,1"></lb> |
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7619 τετράγωνος ᾖ, καὶ ὁ τρίτος τετράγωνος ἔσται.</s> <lb n="1799,001,,,,8,22,2"></lb> |
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7620 <s>Ἔστωσαν τρεῖς ἀριθμοὶ ἑξῆς ἀνάλογον <lb n="1799,001,,,,8,22,3"></lb> |
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7621 οἱ Α, Β, Γ, ὁ δὲ πρῶτος ὁ Α τετράγωνος <lb n="1799,001,,,,8,22,4"></lb> |
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7622 ἔστω· λέγω, ὅτι καὶ ὁ τρίτος ὁ Γ τετρά-<lb n="1799,001,,,,8,22,5"></lb> |
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7623 γωνός ἐστιν.</s> <lb n="1799,001,,,,8,22,6"></lb> |
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7624 <s>Ἐπεὶ γὰρ τῶν Α, Γ εἷς μέσος ἀνάλογόν ἐστιν ἀριθμὸς <lb n="1799,001,,,,8,22,7"></lb> |
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7625 ὁ Β, οἱ Α, Γ ἄρα ὅμοιοι ἐπίπεδοί εἰσιν.</s> <s>τετράγωνος δὲ ὁ <lb n="1799,001,,,,8,22,8"></lb> |
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7626 Α· τετράγωνος ἄρα καὶ ὁ Γ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,8,22,9"></lb> |
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7627 <s>Ἐὰν τέσσαρες ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν, ὁ δὲ πρῶτος <lb n="1799,001,,,,8,23,1"></lb> |
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7628 κύβος ᾖ, καὶ ὁ τέταρτος κύβος ἔσται.</s> <lb n="1799,001,,,,8,23,2"></lb> |
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7629 <s>Ἔστωσαν τέσσαρες ἀριθμοὶ ἑξῆς ἀνά-<lb n="1799,001,,,,8,23,3"></lb> |
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7630 λογον οἱ Α, Β, Γ, Δ, ὁ δὲ Α κύβος <lb n="1799,001,,,,8,23,4"></lb> |
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7631 ἔστω· λέγω, ὅτι καὶ ὁ Δ κύβος ἐστίν.</s> <lb n="1799,001,,,,8,23,5"></lb> |
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7632 <s>Ἐπεὶ γὰρ τῶν Α, Δ δύο μέσοι ἀνά-<lb n="1799,001,,,,8,23,6"></lb> |
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7633 λογόν εἰσιν ἀριθμοὶ οἱ Β, Γ, οἱ Α, Δ <lb n="1799,001,,,,8,23,7"></lb> |
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7634 ἄρα ὅμοιοί εἰσι στερεοὶ ἀριθμοί.</s> <s>κύβος δὲ ὁ Α· κύβος ἄρα <lb n="1799,001,,,,8,23,8"></lb> |
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7635 καὶ ὁ Δ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,8,23,9"></lb> |
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7636 Ἐὰν δύο ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχωσιν, ὃν <lb n="1799,001,,,,8,24,1"></lb> |
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7637 τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ὁ δὲ <lb n="1799,001,,,,8,24,2"></lb> |
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7638 πρῶτος τετράγωνος ᾖ, καὶ ὁ δεύτερος τετράγωνος ἔσται.</s> <lb n="1799,001,,,,8,24,3"></lb> |
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7639 <s>Δύο γὰρ ἀριθμοὶ οἱ Α, Β πρὸς ἀλλήλους λόγον ἐχέτωσαν, <lb n="1799,001,,,,8,24,4"></lb> |
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7640 ὃν τετράγωνος ἀριθμὸς ὁ Γ πρὸς τετράγωνον ἀριθμὸν <lb n="1799,001,,,,8,24,5"></lb> |
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7641 τὸν Δ, ὁ δὲ Α τετράγωνος ἔστω· λέγω, ὅτι καὶ ὁ Β τετρά-<lb n="1799,001,,,,8,24,6"></lb> |
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7642 γωνός ἐστιν.</s> <lb n="1799,001,,,,8,24,7"></lb> |
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7643 <s>Ἐπεὶ γὰρ οἱ Γ, Δ τετράγωνοί εἰσιν, οἱ Γ, Δ ἄρα ὅμοιοι <lb n="1799,001,,,,8,24,8"></lb> |
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7644 ἐπίπεδοί εἰσιν.</s> <s>τῶν Γ, Δ ἄρα εἷς μέ- <lb n="1799,001,,,,8,24,9"></lb> |
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7645 σος ἀνάλογον ἐμπίπτει ἀριθμός.</s> <s>καί <lb n="1799,001,,,,8,24,10"></lb> |
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7646 ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, ὁ Α πρὸς <lb n="1799,001,,,,8,24,11"></lb> |
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7647 τὸν Β· καὶ τῶν Α, Β ἄρα εἷς μέσος <lb n="1799,001,,,,8,24,12"></lb> |
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7648 ἀνάλογον ἐμπίπτει ἀριθμός.</s> <s>καί ἐστιν <lb n="1799,001,,,,8,24,13"></lb> |
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7649 ὁ Α τετράγωνος· καὶ ὁ Β ἄρα τετράγωνός ἐστιν· ὅπερ ἔδει <lb n="1799,001,,,,8,24,14"></lb> |
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7650 δεῖξαι.</s> <lb n="1799,001,,,,8,24,15"></lb> |
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7651 <s>Ἐὰν δύο ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχωσιν, ὃν <lb n="1799,001,,,,8,25,1"></lb> |
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7652 κύβος ἀριθμὸς πρὸς κύβον ἀριθμόν, ὁ δὲ πρῶτος κύβος ᾖ, <lb n="1799,001,,,,8,25,2"></lb> |
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7653 καὶ ὁ δεύτερος κύβος ἔσται.</s> <lb n="1799,001,,,,8,25,3"></lb> |
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7654 <s>Δύο γὰρ ἀριθμοὶ οἱ Α, Β πρὸς ἀλλήλους λόγον ἐχέτω-<lb n="1799,001,,,,8,25,4"></lb> |
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7655 σαν, ὃν κύβος ἀριθμὸς ὁ Γ πρὸς κύβον ἀριθμὸν τὸν Δ, <lb n="1799,001,,,,8,25,5"></lb> |
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7656 κύβος δὲ ἔστω ὁ Α· λέγω [δή], ὅτι καὶ ὁ Β κύβος ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,8,25,6"></lb> |
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7657 Ἐπεὶ γὰρ οἱ Γ, Δ κύβοι εἰσίν, οἱ Γ, Δ ὅμοιοι στερεοί <lb n="1799,001,,,,8,25,7"></lb> |
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7658 εἰσιν· τῶν Γ, Δ ἄρα δύο μέσοι ἀνάλογον ἐμπίπτουσιν <lb n="1799,001,,,,8,25,8"></lb> |
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7659 ἀριθμοί.</s> <s>ὅσοι δὲ εἰς τοὺς Γ, Δ μεταξὺ κατὰ τὸ συνεχὲς <lb n="1799,001,,,,8,25,9"></lb> |
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7660 ἀνάλογον ἐμπίπτουσιν, τοσοῦτοι καὶ εἰς τοὺς τὸν αὐτὸν <lb n="1799,001,,,,8,25,10"></lb> |
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7661 λόγον ἔχοντας αὐτοῖς· ὥστε καὶ τῶν Α, Β δύο μέσοι <lb n="1799,001,,,,8,25,11"></lb> |
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7662 ἀνάλογον ἐμπίπτουσιν ἀριθμοί.</s> <s>ἐμπιπτέτωσαν οἱ Ε, Ζ.</s> <lb n="1799,001,,,,8,25,12"></lb> |
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7663 <s>ἐπεὶ οὖν τέσσαρες ἀριθμοὶ οἱ Α, Ε, Ζ, Β ἑξῆς ἀνάλογόν <lb n="1799,001,,,,8,25,13"></lb> |
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7664 εἰσιν, καί ἐστι κύβος ὁ Α, κύβος ἄρα καὶ ὁ Β· ὅπερ ἔδει <lb n="1799,001,,,,8,25,14"></lb> |
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7665 δεῖξαι.</s> <lb n="1799,001,,,,8,25,15"></lb> |
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7666 <s>Οἱ ὅμοιοι ἐπίπεδοι ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχουσιν, <lb n="1799,001,,,,8,26,1"></lb> |
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7667 ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.</s> <lb n="1799,001,,,,8,26,2"></lb> |
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7668 <s>Ἔστωσαν ὅμοιοι ἐπίπεδοι ἀριθμοὶ οἱ Α, Β· λέγω, ὅτι <lb n="1799,001,,,,8,26,3"></lb> |
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7669 ὁ Α πρὸς τὸν Β λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,8,26,4"></lb> |
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7670 τετράγωνον ἀριθμόν.</s> <lb n="1799,001,,,,8,26,5"></lb> |
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7671 <s>Ἐπεὶ γὰρ οἱ Α, Β ὅμοιοι ἐπίπεδοί εἰσιν, τῶν Α, Β <lb n="1799,001,,,,8,26,6"></lb> |
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7672 ἄρα εἷς μέσος ἀνάλογον ἐμπίπτει ἀριθμός.</s> <s>ἐμπιπτέτω καὶ <pb></pb> <lb n="1799,001,,,,8,26,7"></lb> |
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7673 ἔστω ὁ Γ, καὶ εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν <lb n="1799,001,,,,8,26,8"></lb> |
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7674 αὐτὸν λόγον ἐχόντων τοῖς Α, Γ, Β οἱ Δ, Ε, Ζ· οἱ ἄρα <lb n="1799,001,,,,8,26,9"></lb> |
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7675 ἄκροι αὐτῶν οἱ Δ, Ζ τετράγωνοί εἰσιν.</s> <s>καὶ ἐπεί ἐστιν ὡς <lb n="1799,001,,,,8,26,10"></lb> |
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7676 ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Α πρὸς τὸν Β, καί εἰσιν οἱ Δ, Ζ <lb n="1799,001,,,,8,26,11"></lb> |
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7677 τετράγωνοι, ὁ Α ἄρα πρὸς τὸν Β λόγον ἔχει, ὃν τετρά-<lb n="1799,001,,,,8,26,12"></lb> |
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7678 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ὅπερ ἔδει <lb n="1799,001,,,,8,26,13"></lb> |
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7679 δεῖξαι.</s> <lb n="1799,001,,,,8,26,14"></lb> |
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7680 <s>Οἱ ὅμοιοι στερεοὶ ἀριθμοὶ πρὸς ἀλλήλους λόγον <lb n="1799,001,,,,8,27,1"></lb> |
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7681 ἔχουσιν, ὃν κύβος ἀριθμὸς πρὸς κύβον ἀριθμόν.</s> <lb n="1799,001,,,,8,27,2"></lb> |
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7682 <s>Ἔστωσαν ὅμοιοι στερεοὶ ἀριθμοὶ οἱ Α, Β· λέγω, ὅτι <lb n="1799,001,,,,8,27,3"></lb> |
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7683 ὁ Α πρὸς τὸν Β λόγον ἔχει, ὃν κύβος ἀριθμὸς πρὸς κύβον <lb n="1799,001,,,,8,27,4"></lb> |
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7684 ἀριθμόν.</s> <lb n="1799,001,,,,8,27,5"></lb> |
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7685 <s>Ἐπεὶ γὰρ οἱ Α, Β ὅμοιοι στερεοί εἰσιν, τῶν Α, Β ἄρα <lb n="1799,001,,,,8,27,6"></lb> |
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7686 δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί.</s> <s>ἐμπιπτέτωσαν <lb n="1799,001,,,,8,27,7"></lb> |
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7687 οἱ Γ, Δ, καὶ εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν <lb n="1799,001,,,,8,27,8"></lb> |
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7688 λόγον ἐχόντων τοῖς Α, Γ, Δ, Β ἴσοι αὐτοῖς τὸ πλῆθος οἱ <lb n="1799,001,,,,8,27,9"></lb> |
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7689 Ε, Ζ, Η, Θ· οἱ ἄρα ἄκροι αὐτῶν οἱ Ε, Θ κύβοι εἰσίν.</s> <s>καί <lb n="1799,001,,,,8,27,10"></lb> |
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7690 ἐστιν ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Β· καὶ ὁ Α <lb n="1799,001,,,,8,27,11"></lb> |
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7691 ἄρα πρὸς τὸν Β λόγον ἔχει, ὃν κύβος ἀριθμὸς πρὸς κύβον <lb n="1799,001,,,,8,27,12"></lb> |
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7692 ἀριθμόν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,8,27,13"></lb> |
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7693 </s></p> |
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7694 </chap> |
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7695 <chap><p type="main"><s>Ἐὰν δύο ὅμοιοι ἐπίπεδοι ἀριθμοὶ πολλαπλασιάσαντες <lb n="1799,001,,,,9,1,1"></lb> |
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7696 |
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7697 ἀλλήλους ποιῶσί τινα, ὁ γενόμενος τετράγωνος ἔσται.</s> <lb n="1799,001,,,,9,1,2"></lb> |
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7698 <s>Ἔστωσαν δύο ὅμοιοι ἐπί-<lb n="1799,001,,,,9,1,3"></lb> |
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7699 πεδοι ἀριθμοὶ οἱ Α, Β, καὶ <lb n="1799,001,,,,9,1,4"></lb> |
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7700 ὁ Α τὸν Β πολλαπλασιάσας <lb n="1799,001,,,,9,1,5"></lb> |
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7701 τὸν Γ ποιείτω· λέγω, ὅτι <lb n="1799,001,,,,9,1,6"></lb> |
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7702 ὁ Γ τετράγωνός ἐστιν.</s> <lb n="1799,001,,,,9,1,7"></lb> |
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7703 <s>Ὁ γὰρ Α ἑαυτὸν πολλα-<lb n="1799,001,,,,9,1,8"></lb> |
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7704 πλασιάσας τὸν Δ ποιείτω.</s> <s>ὁ Δ ἄρα τετράγωνός ἐστιν.</s> <lb n="1799,001,,,,9,1,9"></lb> |
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7705 <s>ἐπεὶ οὖν ὁ Α ἑαυτὸν μὲν πολλαπλασιάσας τὸν Δ πεποίηκεν, <lb n="1799,001,,,,9,1,10"></lb> |
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7706 τὸν δὲ Β πολλαπλασιάσας τὸν Γ πεποίηκεν, ἔστιν ἄρα ὡς <lb n="1799,001,,,,9,1,11"></lb> |
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7707 ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν Γ.</s> <s>καὶ ἐπεὶ οἱ Α, Β <lb n="1799,001,,,,9,1,12"></lb> |
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7708 ὅμοιοι ἐπίπεδοί εἰσιν ἀριθμοί, τῶν Α, Β ἄρα εἷς μέσος <lb n="1799,001,,,,9,1,13"></lb> |
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7709 ἀνάλογον ἐμπίπτει ἀριθμός.</s> <s>ἐὰν δὲ δύο ἀριθμῶν μεταξὺ <lb n="1799,001,,,,9,1,14"></lb> |
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7710 κατὰ τὸ συνεχὲς ἀνάλογον ἐμπίπτωσιν ἀριθμοί, ὅσοι εἰς <lb n="1799,001,,,,9,1,15"></lb> |
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7711 αὐτοὺς ἐμπίπτουσι, τοσοῦτοι καὶ εἰς τοὺς τὸν αὐτὸν λόγον <lb n="1799,001,,,,9,1,16"></lb> |
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7712 ἔχοντας· ὥστε καὶ τῶν Δ, Γ εἷς μέσος ἀνάλογον ἐμπίπτει <lb n="1799,001,,,,9,1,17"></lb> |
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7713 ἀριθμός.</s> <s>καί ἐστι τετράγωνος ὁ Δ· τετράγωνος ἄρα καὶ <lb n="1799,001,,,,9,1,18"></lb> |
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7714 ὁ Γ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,9,1,19"></lb> |
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7715 Ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσι <lb n="1799,001,,,,9,2,1"></lb> |
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7716 τετράγωνον, ὅμοιοι ἐπίπεδοί εἰσιν ἀριθμοί.</s> <lb n="1799,001,,,,9,2,2"></lb> |
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7717 <s>Ἔστωσαν δύο ἀριθμοὶ οἱ Α, Β, καὶ ὁ Α τὸν Β πολλα-<lb n="1799,001,,,,9,2,3"></lb> |
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7718 πλασιάσας τετράγωνον τὸν Γ ποιείτω· λέγω, ὅτι οἱ Α, Β <lb n="1799,001,,,,9,2,4"></lb> |
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7719 ὅμοιοι ἐπίπεδοί εἰσιν ἀριθμοί.</s> <lb n="1799,001,,,,9,2,5"></lb> |
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7720 <s>Ὁ γὰρ Α ἑαυτὸν πολλαπλασιάσας τὸν Δ ποιείτω· ὁ Δ <lb n="1799,001,,,,9,2,6"></lb> |
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7721 ἄρα τετράγωνός ἐστιν.</s> <s>καὶ ἐπεὶ ὁ Α ἑαυτὸν μὲν πολλαπλα-<lb n="1799,001,,,,9,2,7"></lb> |
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7722 σιάσας τὸν Δ πεποίηκεν, τὸν δὲ Β <lb n="1799,001,,,,9,2,8"></lb> |
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7723 πολλαπλασιάσας τὸν Γ πεποίηκεν, ἔστιν <lb n="1799,001,,,,9,2,9"></lb> |
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7724 ἄρα ὡς ὁ Α πρὸς τὸν Β, ὁ Δ πρὸς τὸν <lb n="1799,001,,,,9,2,10"></lb> |
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7725 Γ.</s> <s>καὶ ἐπεὶ ὁ Δ τετράγωνός ἐστιν, ἀλλὰ <lb n="1799,001,,,,9,2,11"></lb> |
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7726 καὶ ὁ Γ, οἱ Δ, Γ ἄρα ὅμοιοι ἐπίπεδοί <lb n="1799,001,,,,9,2,12"></lb> |
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7727 εἰσιν.</s> <s>τῶν Δ, Γ ἄρα εἷς μέσος ἀνάλογον <lb n="1799,001,,,,9,2,13"></lb> |
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7728 ἐμπίπτει.</s> <s>καί ἐστιν ὡς ὁ Δ πρὸς τὸν Γ, οὕτως ὁ Α πρὸς <lb n="1799,001,,,,9,2,14"></lb> |
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7729 τὸν Β· καὶ τῶν Α, Β ἄρα εἷς μέσος ἀνάλογον ἐμπίπτει.</s> <lb n="1799,001,,,,9,2,15"></lb> |
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7730 <s>ἐὰν δὲ δύο ἀριθμῶν εἷς μέσος ἀνάλογον ἐμπίπτῃ, ὅμοιοι <lb n="1799,001,,,,9,2,16"></lb> |
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7731 ἐπίπεδοί εἰσιν [οἱ] ἀριθμοί· οἱ ἄρα Α, Β ὅμοιοί εἰσιν ἐπί-<lb n="1799,001,,,,9,2,17"></lb> |
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7732 πεδοι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,2,18"></lb> |
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7733 <s>Ἐὰν κύβος ἀριθμὸς ἑαυτὸν πολλαπλασιάσας ποιῇ τινα, <lb n="1799,001,,,,9,3,1"></lb> |
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7734 ὁ γενόμενος κύβος ἔσται.</s> <lb n="1799,001,,,,9,3,2"></lb> |
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7735 <s>Κύβος γὰρ ἀριθμὸς ὁ Α ἑαυτὸν πολλαπλασιάσας τὸν Β <lb n="1799,001,,,,9,3,3"></lb> |
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7736 ποιείτω· λέγω, ὅτι ὁ Β κύβος ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,9,3,4"></lb> |
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7737 Εἰλήφθω γὰρ τοῦ Α πλευρὰ ὁ Γ, καὶ ὁ Γ ἑαυτὸν πολλα-<lb n="1799,001,,,,9,3,5"></lb> |
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7738 πλασιάσας τὸν Δ ποιείτω.</s> <s>φανερὸν δή ἐστιν, ὅτι ὁ Γ τὸν Δ <lb n="1799,001,,,,9,3,6"></lb> |
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7739 πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <s>καὶ ἐπεὶ ὁ Γ ἑαυτὸν <lb n="1799,001,,,,9,3,7"></lb> |
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7740 πολλαπλασιάσας τὸν Δ πεποίηκεν, <lb n="1799,001,,,,9,3,8"></lb> |
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7741 ὁ Γ ἄρα τὸν Δ μετρεῖ κατὰ τὰς ἐν <lb n="1799,001,,,,9,3,9"></lb> |
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7742 αὑτῷ μονάδας.</s> <s>ἀλλὰ μὴν καὶ ἡ μο-<lb n="1799,001,,,,9,3,10"></lb> |
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7743 νὰς τὸν Γ μετρεῖ κατὰ τὰς ἐν αὐτῷ <lb n="1799,001,,,,9,3,11"></lb> |
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7744 μονάδας· ἔστιν ἄρα ὡς ἡ μονὰς <lb n="1799,001,,,,9,3,12"></lb> |
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7745 πρὸς τὸν Γ, ὁ Γ πρὸς τὸν Δ.</s> <s>πάλιν, ἐπεὶ ὁ Γ τὸν Δ πολλα-<lb n="1799,001,,,,9,3,13"></lb> |
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7746 πλασιάσας τὸν Α πεποίηκεν, ὁ Δ ἄρα τὸν Α μετρεῖ κατὰ <lb n="1799,001,,,,9,3,14"></lb> |
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7747 τὰς ἐν τῷ Γ μονάδας.</s> <s>μετρεῖ δὲ καὶ ἡ μονὰς τὸν Γ κατὰ <lb n="1799,001,,,,9,3,15"></lb> |
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7748 τὰς ἐν αὐτῷ μονάδας· ἔστιν ἄρα ὡς ἡ μονὰς πρὸς τὸν Γ, <lb n="1799,001,,,,9,3,16"></lb> |
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7749 ὁ Δ πρὸς τὸν Α.</s> <s>ἀλλ' ὡς ἡ μονὰς πρὸς τὸν Γ, ὁ Γ πρὸς <lb n="1799,001,,,,9,3,17"></lb> |
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7750 τὸν Δ· καὶ ὡς ἄρα ἡ μονὰς πρὸς τὸν Γ, οὕτως ὁ Γ πρὸς <lb n="1799,001,,,,9,3,18"></lb> |
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7751 τὸν Δ καὶ ὁ Δ πρὸς τὸν Α.</s> <s>τῆς ἄρα μονάδος καὶ τοῦ Α <lb n="1799,001,,,,9,3,19"></lb> |
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7752 ἀριθμοῦ δύο μέσοι ἀνάλογον κατὰ τὸ συνεχὲς ἐμπεπτώ-<lb n="1799,001,,,,9,3,20"></lb> |
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7753 κασιν ἀριθμοὶ οἱ Γ, Δ.</s> <s>πάλιν, ἐπεὶ ὁ Α ἑαυτὸν πολλαπλα-<lb n="1799,001,,,,9,3,21"></lb> |
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7754 σιάσας τὸν Β πεποίηκεν, ὁ Α ἄρα τὸν Β μετρεῖ κατὰ τὰς <lb n="1799,001,,,,9,3,22"></lb> |
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7755 ἐν αὑτῷ μονάδας.</s> <s>μετρεῖ δὲ καὶ ἡ μονὰς τὸν Α κατὰ τὰς <lb n="1799,001,,,,9,3,23"></lb> |
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7756 ἐν αὐτῷ μονάδας· ἔστιν ἄρα ὡς ἡ μονὰς πρὸς τὸν Α, ὁ Α <lb n="1799,001,,,,9,3,24"></lb> |
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7757 πρὸς τὸν Β.</s> <s>τῆς δὲ μονάδος καὶ τοῦ Α δύο μέσοι ἀνάλογον <lb n="1799,001,,,,9,3,25"></lb> |
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7758 ἐμπεπτώκασιν ἀριθμοί· καὶ τῶν Α, Β ἄρα δύο μέσοι ἀνάλο-<lb n="1799,001,,,,9,3,26"></lb> |
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7759 γον ἐμπεσοῦνται ἀριθμοί.</s> <s>ἐὰν δὲ δύο ἀριθμῶν δύο μέσοι <lb n="1799,001,,,,9,3,27"></lb> |
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7760 ἀνάλογον ἐμπίπτωσιν, ὁ δὲ πρῶτος κύβος ᾖ, καὶ ὁ δεύτερος <lb n="1799,001,,,,9,3,28"></lb> |
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7761 κύβος ἔσται.</s> <s>καί ἐστιν ὁ Α κύβος· καὶ ὁ Β ἄρα κύβος <lb n="1799,001,,,,9,3,29"></lb> |
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7762 ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,9,3,30"></lb> |
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7763 Ἐὰν κύβος ἀριθμὸς κύβον ἀριθμὸν πολλαπλασιάσας <lb n="1799,001,,,,9,4,1"></lb> |
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7764 ποιῇ τινα, ὁ γενόμενος κύβος ἔσται.</s> <lb n="1799,001,,,,9,4,2"></lb> |
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7765 <s>Κύβος γὰρ ἀριθμὸς ὁ Α κύβον ἀριθμὸν τὸν Β πολλα-<lb n="1799,001,,,,9,4,3"></lb> |
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7766 πλασιάσας τὸν Γ ποιείτω· λέγω, ὅτι ὁ Γ κύβος ἐστίν.</s> <lb n="1799,001,,,,9,4,4"></lb> |
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7767 <s>Ὁ γὰρ Α ἑαυτὸν πολλαπλα- <lb n="1799,001,,,,9,4,5"></lb> |
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7768 σιάσας τὸν Δ ποιείτω· ὁ Δ ἄρα <lb n="1799,001,,,,9,4,6"></lb> |
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7769 κύβος ἐστίν.</s> <s>καὶ ἐπεὶ ὁ Α ἑαυτὸν <lb n="1799,001,,,,9,4,7"></lb> |
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7770 μὲν πολλαπλασιάσας τὸν Δ πεποίη-<lb n="1799,001,,,,9,4,8"></lb> |
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7771 κεν, τὸν δὲ Β πολλαπλασιάσας <lb n="1799,001,,,,9,4,9"></lb> |
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7772 τὸν Γ πεποίηκεν, ἔστιν ἄρα ὡς <lb n="1799,001,,,,9,4,10"></lb> |
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7773 ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν Γ.</s> <s>καὶ ἐπεὶ οἱ Α, <lb n="1799,001,,,,9,4,11"></lb> |
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7774 Β κύβοι εἰσίν, ὅμοιοι στερεοί εἰσιν οἱ Α, Β.</s> <s>τῶν Α, Β <lb n="1799,001,,,,9,4,12"></lb> |
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7775 ἄρα δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί· ὥστε καὶ <lb n="1799,001,,,,9,4,13"></lb> |
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7776 τῶν Δ, Γ δύο μέσοι ἀνάλογον ἐμπεσοῦνται ἀριθμοί.</s> <s>καί <lb n="1799,001,,,,9,4,14"></lb> |
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7777 ἐστι κύβος ὁ Δ· κύβος ἄρα καὶ ὁ Γ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,4,15"></lb> |
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7778 <s>Ἐὰν κύβος ἀριθμὸς ἀριθμόν τινα πολλαπλασιάσας <lb n="1799,001,,,,9,5,1"></lb> |
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7779 κύβον ποιῇ, καὶ ὁ πολλαπλασιασθεὶς κύβος ἔσται.</s> <lb n="1799,001,,,,9,5,2"></lb> |
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7780 <s>Κύβος γὰρ ἀριθμὸς ὁ Α ἀριθμόν τινα τὸν Β πολλα-<lb n="1799,001,,,,9,5,3"></lb> |
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7781 πλασιάσας κύβον τὸν Γ ποιείτω· λέγω, ὅτι ὁ Β κύβος <lb n="1799,001,,,,9,5,4"></lb> |
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7782 ἐστίν.</s> <lb n="1799,001,,,,9,5,5"></lb> |
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7783 <s>Ὁ γὰρ Α ἑαυτὸν πολλαπλασιάσας τὸν Δ ποιείτω· κύβος <lb n="1799,001,,,,9,5,6"></lb> |
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7784 ἄρα ἐστίν ὁ Δ.</s> <s>καὶ ἐπεὶ ὁ Α ἑαυτὸν μὲν πολλαπλασιάσας <pb></pb> <lb n="1799,001,,,,9,5,7"></lb> |
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7785 τὸν Δ πεποίηκεν, τὸν δὲ Β πολλα-<lb n="1799,001,,,,9,5,8"></lb> |
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7786 πλασιάσας τὸν Γ πεποίηκεν, ἔστιν <lb n="1799,001,,,,9,5,9"></lb> |
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7787 ἄρα ὡς ὁ Α πρὸς τὸν Β, ὁ Δ πρὸς <lb n="1799,001,,,,9,5,10"></lb> |
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7788 τὸν Γ.</s> <s>καὶ ἐπεὶ οἱ Δ, Γ κύβοι <lb n="1799,001,,,,9,5,11"></lb> |
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7789 εἰσίν, ὅμοιοι στερεοί εἰσιν.</s> <s>τῶν <lb n="1799,001,,,,9,5,12"></lb> |
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7790 Δ, Γ ἄρα δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί.</s> <s>καί <lb n="1799,001,,,,9,5,13"></lb> |
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7791 ἐστιν ὡς ὁ Δ πρὸς τὸν Γ, οὕτως ὁ Α πρὸς τὸν Β· καὶ τῶν <lb n="1799,001,,,,9,5,14"></lb> |
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7792 Α, Β ἄρα δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί.</s> <s>καί <lb n="1799,001,,,,9,5,15"></lb> |
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7793 ἐστι κύβος ὁ Α· κύβος ἄρα ἐστὶ καὶ ὁ Β· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,5,16"></lb> |
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7794 <s>Ἐὰν ἀριθμὸς ἑαυτὸν πολλαπλασιάσας κύβον ποιῇ, καὶ <lb n="1799,001,,,,9,6,1"></lb> |
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7795 αὐτὸς κύβος ἔσται.</s> <lb n="1799,001,,,,9,6,2"></lb> |
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7796 <s>Ἀριθμὸς γὰρ ὁ Α ἑαυτὸν πολλαπλασιάσας κύβον τὸν Β <lb n="1799,001,,,,9,6,3"></lb> |
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7797 ποιείτω· λέγω, ὅτι καὶ ὁ Α κύβος ἐστίν.</s> <lb n="1799,001,,,,9,6,4"></lb> |
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7798 <s>Ὁ γὰρ Α τὸν Β πολλαπλασιάσας τὸν Γ <lb n="1799,001,,,,9,6,5"></lb> |
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7799 ποιείτω.</s> <s>ἐπεὶ οὖν ὁ Α ἑαυτὸν μὲν πολλα-<lb n="1799,001,,,,9,6,6"></lb> |
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7800 πλασιάσας τὸν Β πεποίηκεν, τὸν δὲ Β <lb n="1799,001,,,,9,6,7"></lb> |
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7801 πολλαπλασιάσας τὸν Γ πεποίηκεν, ὁ Γ <lb n="1799,001,,,,9,6,8"></lb> |
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7802 ἄρα κύβος ἐστίν.</s> <s>καὶ ἐπεὶ ὁ Α ἑαυτὸν <lb n="1799,001,,,,9,6,9"></lb> |
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7803 πολλαπλασιάσας τὸν Β πεποίηκεν, ὁ Α ἄρα τὸν Β μετρεῖ <lb n="1799,001,,,,9,6,10"></lb> |
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7804 κατὰ τὰς ἐν αὑτῷ μονάδας.</s> <s>μετρεῖ δὲ καὶ ἡ μονὰς τὸν Α <lb n="1799,001,,,,9,6,11"></lb> |
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7805 κατὰ τὰς ἐν αὐτῷ μονάδας.</s> <s>ἔστιν ἄρα ὡς ἡ μονὰς πρὸς <lb n="1799,001,,,,9,6,12"></lb> |
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7806 τὸν Α, οὕτως ὁ Α πρὸς τὸν Β.</s> <s>καὶ ἐπεὶ ὁ Α τὸν Β <pb></pb> <lb n="1799,001,,,,9,6,13"></lb> |
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7807 πολλαπλασιάσας τὸν Γ πεποίηκεν, ὁ Β ἄρα τὸν Γ μετρεῖ <lb n="1799,001,,,,9,6,14"></lb> |
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7808 κατὰ τὰς ἐν τῷ Α μονάδας.</s> <s>μετρεῖ δὲ καὶ ἡ μονὰς τὸν Α <lb n="1799,001,,,,9,6,15"></lb> |
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7809 κατὰ τὰς ἐν αὐτῷ μονάδας.</s> <s>ἔστιν ἄρα ὡς ἡ μονὰς πρὸς τὸν <lb n="1799,001,,,,9,6,16"></lb> |
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7810 Α, οὕτως ὁ Β πρὸς τὸν Γ.</s> <s>ἀλλ' ὡς ἡ μονὰς πρὸς τὸν Α, <lb n="1799,001,,,,9,6,17"></lb> |
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7811 οὕτως ὁ Α πρὸς τὸν Β· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, ὁ Β <lb n="1799,001,,,,9,6,18"></lb> |
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7812 πρὸς τὸν Γ.</s> <s>καὶ ἐπεὶ οἱ Β, Γ κύβοι εἰσίν, ὅμοιοι στερεοί <lb n="1799,001,,,,9,6,19"></lb> |
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7813 εἰσιν.</s> <s>τῶν Β, Γ ἄρα δύο μέσοι ἀνάλογόν εἰσιν ἀριθμοί.</s> <s>καί <lb n="1799,001,,,,9,6,20"></lb> |
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7814 ἐστιν ὡς ὁ Β πρὸς τὸν Γ, ὁ Α πρὸς τὸν Β.</s> <s>καὶ τῶν <lb n="1799,001,,,,9,6,21"></lb> |
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7815 Α, Β ἄρα δύο μέσοι ἀνάλογόν εἰσιν ἀριθμοί.</s> <s>καί ἐστι <lb n="1799,001,,,,9,6,22"></lb> |
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7816 κύβος ὁ Β· κύβος ἄρα ἐστὶ καὶ ὁ Α· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,6,23"></lb> |
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7817 <s>Ἐὰν σύνθετος ἀριθμὸς ἀριθμόν τινα πολλαπλασιάσας <lb n="1799,001,,,,9,7,1"></lb> |
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7818 ποιῇ τινα, ὁ γενόμενος στερεὸς ἔσται.</s> <lb n="1799,001,,,,9,7,2"></lb> |
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7819 <s>Σύνθετος γὰρ ἀριθμὸς ὁ Α ἀριθμόν τινα τὸν Β πολλα-<lb n="1799,001,,,,9,7,3"></lb> |
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7820 πλασιάσας τὸν Γ ποιείτω· λέγω, ὅτι ὁ Γ στερεός ἐστιν.</s> <lb n="1799,001,,,,9,7,4"></lb> |
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7821 <s>Ἐπεὶ γὰρ ὁ Α σύνθετός ἐστιν, ὑπὸ <lb n="1799,001,,,,9,7,5"></lb> |
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7822 ἀριθμοῦ τινος μετρηθήσεται.</s> <s>μετρεί-<lb n="1799,001,,,,9,7,6"></lb> |
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7823 σθω ὑπὸ τοῦ Δ, καὶ ὁσάκις ὁ Δ τὸν <lb n="1799,001,,,,9,7,7"></lb> |
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7824 Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν <lb n="1799,001,,,,9,7,8"></lb> |
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7825 ἐν τῷ Ε.</s> <s>ἐπεὶ οὖν ὁ Δ τὸν Α μετρεῖ <lb n="1799,001,,,,9,7,9"></lb> |
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7826 κατὰ τὰς ἐν τῷ Ε μονάδας, ὁ Ε ἄρα <lb n="1799,001,,,,9,7,10"></lb> |
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7827 τὸν Δ πολλαπλασιάσας τὸν Α πεποίηκεν.</s> <s>καὶ ἐπεὶ ὁ Α τὸν <lb n="1799,001,,,,9,7,11"></lb> |
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7828 Β πολλαπλασιάσας τὸν Γ πεποίηκεν, ὁ δὲ Α ἐστιν ὁ ἐκ <pb></pb> <lb n="1799,001,,,,9,7,12"></lb> |
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7829 τῶν Δ, Ε, ὁ ἄρα ἐκ τῶν Δ, Ε τὸν Β πολλαπλασιάσας τὸν Γ <lb n="1799,001,,,,9,7,13"></lb> |
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7830 πεποίηκεν.</s> <s>ὁ Γ ἄρα στερεός ἐστιν, πλευραὶ δὲ αὐτοῦ <lb n="1799,001,,,,9,7,14"></lb> |
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7831 εἰσιν οἱ Δ, Ε, Β· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,7,15"></lb> |
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7832 <s>Ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον <lb n="1799,001,,,,9,8,1"></lb> |
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7833 ὦσιν, ὁ μὲν τρίτος ἀπὸ τῆς μονάδος τετράγωνος ἔσται <lb n="1799,001,,,,9,8,2"></lb> |
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7834 καὶ οἱ ἕνα διαλείποντες, ὁ δὲ τέταρτος κύβος καὶ οἱ δύο <lb n="1799,001,,,,9,8,3"></lb> |
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7835 διαλείποντες πάντες, ὁ δὲ ἕβδομος κύβος ἅμα καὶ τετρά-<lb n="1799,001,,,,9,8,4"></lb> |
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7836 γωνος καὶ οἱ πέντε διαλείποντες.</s> <lb n="1799,001,,,,9,8,5"></lb> |
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7837 <s>Ἔστωσαν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον <lb n="1799,001,,,,9,8,6"></lb> |
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7838 οἱ Α, Β, Γ, Δ, Ε, Ζ· λέγω, ὅτι ὁ μὲν τρίτος ἀπὸ τῆς <lb n="1799,001,,,,9,8,7"></lb> |
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7839 μονάδος ὁ Β τετράγωνός ἐστι <lb n="1799,001,,,,9,8,8"></lb> |
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7840 καὶ οἱ ἕνα διαλείποντες πάντες, <lb n="1799,001,,,,9,8,9"></lb> |
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7841 ὁ δὲ τέταρτος ὁ Γ κύβος καὶ <lb n="1799,001,,,,9,8,10"></lb> |
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7842 οἱ δύο διαλείποντες πάντες, ὁ <lb n="1799,001,,,,9,8,11"></lb> |
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7843 δὲ ἕβδομος ὁ Ζ κύβος ἅμα καὶ <lb n="1799,001,,,,9,8,12"></lb> |
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7844 τετράγωνος καὶ οἱ πέντε δια-<lb n="1799,001,,,,9,8,13"></lb> |
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7845 λείποντες πάντες.</s> <lb n="1799,001,,,,9,8,14"></lb> |
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7846 <s>Ἐπεὶ γάρ ἐστιν ὡς ἡ μονὰς <lb n="1799,001,,,,9,8,15"></lb> |
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7847 πρὸς τὸν Α, οὕτως ὁ Α πρὸς <lb n="1799,001,,,,9,8,16"></lb> |
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7848 τὸν Β, ἰσάκις ἄρα ἡ μονὰς τὸν Α ἀριθμὸν μετρεῖ καὶ <lb n="1799,001,,,,9,8,17"></lb> |
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7849 ὁ Α τὸν Β.</s> <s>ἡ δὲ μονὰς τὸν Α ἀριθμὸν μετρεῖ κατὰ τὰς ἐν <pb></pb> <lb n="1799,001,,,,9,8,18"></lb> |
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7850 αὐτῷ μονάδας· καὶ ὁ Α ἄρα τὸν Β μετρεῖ κατὰ τὰς ἐν <lb n="1799,001,,,,9,8,19"></lb> |
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7851 τῷ Α μονάδας.</s> <s>ὁ Α ἄρα ἑαυτὸν πολλαπλασιάσας τὸν Β <lb n="1799,001,,,,9,8,20"></lb> |
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7852 πεποίηκεν· τετράγωνος ἄρα ἐστὶν ὁ Β.</s> <s>καὶ ἐπεὶ οἱ Β, Γ, <lb n="1799,001,,,,9,8,21"></lb> |
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7853 Δ ἑξῆς ἀνάλογόν εἰσιν, ὁ δὲ Β τετράγωνός ἐστιν, καὶ ὁ Δ <lb n="1799,001,,,,9,8,22"></lb> |
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7854 ἄρα τετράγωνός ἐστιν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ὁ Ζ τετράγωνός <lb n="1799,001,,,,9,8,23"></lb> |
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7855 ἐστιν.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ οἱ ἕνα διαλείποντες <lb n="1799,001,,,,9,8,24"></lb> |
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7856 πάντες τετράγωνοί εἰσιν.</s> <s>λέγω δή, ὅτι καὶ ὁ τέταρτος <lb n="1799,001,,,,9,8,25"></lb> |
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7857 ἀπὸ τῆς μονάδος ὁ Γ κύβος ἐστὶ καὶ οἱ δύο διαλείποντες <lb n="1799,001,,,,9,8,26"></lb> |
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7858 πάντες.</s> <s>ἐπεὶ γάρ ἐστιν ὡς ἡ μονὰς πρὸς τὸν Α, οὕτως ὁ Β <lb n="1799,001,,,,9,8,27"></lb> |
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7859 πρὸς τὸν Γ, ἰσάκις ἄρα ἡ μονὰς τὸν Α ἀριθμὸν μετρεῖ καὶ <lb n="1799,001,,,,9,8,28"></lb> |
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7860 ὁ Β τὸν Γ.</s> <s>ἡ δὲ μονὰς τὸν Α ἀριθμὸν μετρεῖ κατὰ τὰς <lb n="1799,001,,,,9,8,29"></lb> |
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7861 ἐν τῷ Α μονάδας· καὶ ὁ Β ἄρα τὸν Γ μετρεῖ κατὰ τὰς ἐν <lb n="1799,001,,,,9,8,30"></lb> |
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7862 τῷ Α μονάδας· ὁ Α ἄρα τὸν Β πολλαπλασιάσας τὸν Γ <lb n="1799,001,,,,9,8,31"></lb> |
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7863 πεποίηκεν.</s> <s>ἐπεὶ οὖν ὁ Α ἑαυτὸν μὲν πολλαπλασιάσας τὸν Β <lb n="1799,001,,,,9,8,32"></lb> |
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7864 πεποίηκεν, τὸν δὲ Β πολλαπλασιάσας τὸν Γ πεποίηκεν, <lb n="1799,001,,,,9,8,33"></lb> |
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7865 κύβος ἄρα ἐστὶν ὁ Γ.</s> <s>καὶ ἐπεὶ οἱ Γ, Δ, Ε, Ζ ἑξῆς ἀνάλογόν <lb n="1799,001,,,,9,8,34"></lb> |
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7866 εἰσιν, ὁ δὲ Γ κύβος ἐστίν, καὶ ὁ Ζ ἄρα κύβος ἐστίν.</s> <lb n="1799,001,,,,9,8,35"></lb> |
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7867 <s>ἐδείχθη δὲ καὶ τετράγωνος· ὁ ἄρα ἕβδομος ἀπὸ τῆς <lb n="1799,001,,,,9,8,36"></lb> |
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7868 μονάδος κύβος τέ ἐστι καὶ τετράγωνος.</s> <s>ὁμοίως δὴ δείξομεν, <lb n="1799,001,,,,9,8,37"></lb> |
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7869 ὅτι καὶ οἱ πέντε διαλείποντες πάντες κύβοι τέ εἰσι καὶ <lb n="1799,001,,,,9,8,38"></lb> |
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7870 τετράγωνοι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,8,39"></lb> |
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7871 <s>Ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἑξῆς κατὰ τὸ συνεχὲς <lb n="1799,001,,,,9,9,1"></lb> |
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7872 ἀριθμοὶ ἀνάλογον ὦσιν, ὁ δὲ μετὰ τὴν μονάδα τετράγωνος <pb></pb> <lb n="1799,001,,,,9,9,2"></lb> |
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7873 ᾖ, καὶ οἱ λοιποὶ πάντες τετράγωνοι ἔσονται.</s> <s>καὶ ἐὰν ὁ <lb n="1799,001,,,,9,9,3"></lb> |
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7874 μετὰ τὴν μονάδα κύβος ᾖ, καὶ οἱ λοιποὶ πάντες κύβοι <lb n="1799,001,,,,9,9,4"></lb> |
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7875 ἔσονται.</s> <lb n="1799,001,,,,9,9,5"></lb> |
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7876 <s>Ἔστωσαν ἀπὸ μονάδος ἑξῆς ἀνάλογον ὁσοιδηποτοῦν <lb n="1799,001,,,,9,9,6"></lb> |
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7877 ἀριθμοὶ οἱ Α, Β, Γ, Δ, Ε, Ζ, ὁ δὲ μετὰ τὴν μονάδα ὁ Α <lb n="1799,001,,,,9,9,7"></lb> |
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7878 τετράγωνος ἔστω· λέγω, ὅτι καὶ οἱ λοιποὶ πάντες τε-<lb n="1799,001,,,,9,9,8"></lb> |
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7879 τράγωνοι ἔσονται. <lb n="1799,001,,,,9,9,9"></lb> |
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7880 Ὅτι μὲν οὖν ὁ τρίτος ἀπὸ τῆς μονάδος ὁ Β τετράγωνός <lb n="1799,001,,,,9,9,10"></lb> |
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7881 ἐστι καὶ οἱ ἕνα διαλείποντες πάντες, δέδεικται· λέγω <lb n="1799,001,,,,9,9,11"></lb> |
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7882 [δή], ὅτι καὶ οἱ λοιποὶ πάντες τετράγωνοί εἰσιν.</s> <s>ἐπεὶ γὰρ <lb n="1799,001,,,,9,9,12"></lb> |
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7883 οἱ Α, Β, Γ ἑξῆς ἀνάλογόν εἰσιν, καί ἐστιν ὁ Α τετράγωνος, <lb n="1799,001,,,,9,9,13"></lb> |
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7884 καὶ ὁ Γ [ἄρα] τετράγωνός ἐστιν.</s> <s>πάλιν, ἐπεὶ [καὶ] οἱ <lb n="1799,001,,,,9,9,14"></lb> |
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7885 Β, Γ, Δ ἑξῆς ἀνάλογόν εἰσιν, καί ἐστιν ὁ Β τετράγωνος, <lb n="1799,001,,,,9,9,15"></lb> |
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7886 καὶ ὁ Δ [ἄρα] τετράγωνός ἐστιν.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι <lb n="1799,001,,,,9,9,16"></lb> |
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7887 καὶ οἱ λοιποὶ πάντες τετράγωνοί εἰσιν.</s> <lb n="1799,001,,,,9,9,17"></lb> |
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7888 <s>Ἀλλὰ δὴ ἔστω ὁ Α κύβος· λέγω, ὅτι καὶ οἱ λοιποὶ <lb n="1799,001,,,,9,9,18"></lb> |
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7889 πάντες κύβοι εἰσίν.</s> <lb n="1799,001,,,,9,9,19"></lb> |
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7890 <s>Ὅτι μὲν οὖν ὁ τέταρτος ἀπὸ τῆς μονάδος ὁ Γ κύβος <lb n="1799,001,,,,9,9,20"></lb> |
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7891 ἐστὶ καὶ οἱ δύο διαλείποντες πάντες, δέδεικται· λέγω <lb n="1799,001,,,,9,9,21"></lb> |
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7892 [δή], ὅτι καὶ οἱ λοιποὶ πάντες κύβοι εἰσίν.</s> <s>ἐπεὶ γάρ ἐστιν <pb></pb> <lb n="1799,001,,,,9,9,22"></lb> |
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7893 ὡς ἡ μονὰς πρὸς τὸν Α, οὕτως ὁ Α πρὸς τὸν Β, ἰσάκις <lb n="1799,001,,,,9,9,23"></lb> |
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7894 ἄρα ἡ μονὰς τὸν Α μετρεῖ καὶ ὁ Α τὸν Β.</s> <s>ἡ δὲ μονὰς τὸν Α <lb n="1799,001,,,,9,9,24"></lb> |
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7895 μετρεῖ κατὰ τὰς ἐν αὐτῷ μονάδας· καὶ ὁ Α ἄρα τὸν Β <lb n="1799,001,,,,9,9,25"></lb> |
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7896 μετρεῖ κατὰ τὰς ἐν αὑτῷ μονάδας· ὁ Α ἄρα ἑαυτὸν <lb n="1799,001,,,,9,9,26"></lb> |
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7897 πολλαπλασιάσας τὸν Β πεποίηκεν.</s> <s>καί ἐστιν ὁ Α κύβος.</s> <lb n="1799,001,,,,9,9,27"></lb> |
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7898 <s>ἐὰν δὲ κύβος ἀριθμὸς ἑαυτὸν πολλαπλασιάσας ποιῇ τινα, <lb n="1799,001,,,,9,9,28"></lb> |
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7899 ὁ γενόμενος κύβος ἐστίν· καὶ ὁ Β ἄρα κύβος ἐστίν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,9,9,29"></lb> |
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7900 τέσσαρες ἀριθμοὶ οἱ Α, Β, Γ, Δ ἑξῆς ἀνάλογόν εἰσιν, <lb n="1799,001,,,,9,9,30"></lb> |
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7901 καί ἐστιν ὁ Α κύβος, καὶ ὁ Δ ἄρα κύβος ἐστίν.</s> <s>διὰ τὰ <lb n="1799,001,,,,9,9,31"></lb> |
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7902 αὐτὰ δὴ καὶ ὁ Ε κύβος ἐστίν, καὶ ὁμοίως οἱ λοιποὶ πάντες <lb n="1799,001,,,,9,9,32"></lb> |
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7903 κύβοι εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,9,33"></lb> |
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7904 <s>Ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ [ἑξῆσ] ἀνάλογον <lb n="1799,001,,,,9,10,1"></lb> |
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7905 ὦσιν, ὁ δὲ μετὰ τὴν μονάδα μὴ ᾖ τετράγωνος, οὐδ' ἄλλος <lb n="1799,001,,,,9,10,2"></lb> |
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7906 οὐδεὶς τετράγωνος ἔσται χωρὶς τοῦ τρίτου ἀπὸ τῆς <lb n="1799,001,,,,9,10,3"></lb> |
|
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7907 μονάδος καὶ τῶν ἕνα διαλειπόντων πάντων.</s> <s>καὶ ἐὰν ὁ μετὰ <lb n="1799,001,,,,9,10,4"></lb> |
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7908 τὴν μονάδα κύβος μὴ ᾖ, οὐδὲ ἄλλος οὐδεὶς κύβος ἔσται <lb n="1799,001,,,,9,10,5"></lb> |
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7909 χωρὶς τοῦ τετάρτου ἀπὸ τῆς μονάδος καὶ τῶν δύο διαλει-<lb n="1799,001,,,,9,10,6"></lb> |
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7910 πόντων πάντων.</s> <lb n="1799,001,,,,9,10,7"></lb> |
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7911 <s>Ἔστωσαν ἀπὸ μονάδος ἑξῆς ἀνάλογον ὁσοιδηποτοῦν <lb n="1799,001,,,,9,10,8"></lb> |
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7912 ἀριθμοὶ οἱ Α, Β, Γ, Δ, Ε, Ζ, ὁ δὲ μετὰ τὴν μονάδα ὁ Α <lb n="1799,001,,,,9,10,9"></lb> |
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7913 μὴ ἔστω τετράγωνος· λέγω, ὅτι οὐδὲ ἄλλος οὐδεὶς τετρά-<lb n="1799,001,,,,9,10,10"></lb> |
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7914 γωνος ἔσται χωρὶς τοῦ τρίτου ἀπὸ τῆς μονάδος [καὶ τῶν <lb n="1799,001,,,,9,10,11"></lb> |
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7915 ἕνα διαλειπόντων].</s> <s><pb></pb> <lb n="1799,001,,,,9,10,12"></lb> |
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7916 Εἰ γὰρ δυνατόν, ἔστω ὁ Γ <lb n="1799,001,,,,9,10,13"></lb> |
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7917 τετράγωνος.</s> <s>ἔστι δὲ καὶ ὁ Β <lb n="1799,001,,,,9,10,14"></lb> |
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7918 τετράγωνος· οἱ Β, Γ ἄρα <lb n="1799,001,,,,9,10,15"></lb> |
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7919 πρὸς ἀλλήλους λόγον ἔχουσιν, <lb n="1799,001,,,,9,10,16"></lb> |
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7920 ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,9,10,17"></lb> |
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7921 τετράγωνον ἀριθμόν.</s> <s>καί ἐστιν <lb n="1799,001,,,,9,10,18"></lb> |
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7922 ὡς ὁ Β πρὸς τὸν Γ, ὁ Α <lb n="1799,001,,,,9,10,19"></lb> |
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7923 πρὸς τὸν Β· οἱ Α, Β ἄρα <lb n="1799,001,,,,9,10,20"></lb> |
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7924 πρὸς ἀλλήλους λόγον ἔχουσιν, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,9,10,21"></lb> |
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7925 πρὸς τετράγωνον ἀριθμόν· ὥστε οἱ Α, Β ὅμοιοι ἐπίπεδοί <lb n="1799,001,,,,9,10,22"></lb> |
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7926 εἰσιν.</s> <s>καί ἐστι τετράγωνος ὁ Β· τετράγωνος ἄρα ἐστὶ καὶ <lb n="1799,001,,,,9,10,23"></lb> |
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7927 ὁ Α· ὅπερ οὐχ ὑπέκειτο.</s> <s>οὐκ ἄρα ὁ Γ τετράγωνός ἐστιν.</s> <lb n="1799,001,,,,9,10,24"></lb> |
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7928 <s>ὁμοίως δὴ δείξομεν, ὅτι οὐδ' ἄλλος οὐδεὶς τετράγωνός ἐστι <lb n="1799,001,,,,9,10,25"></lb> |
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7929 χωρὶς τοῦ τρίτου ἀπὸ τῆς μονάδος καὶ τῶν ἕνα διαλει-<lb n="1799,001,,,,9,10,26"></lb> |
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7930 πόντων.</s> <lb n="1799,001,,,,9,10,27"></lb> |
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7931 <s>Ἀλλὰ δὴ μὴ ἔστω ὁ Α κύβος.</s> <s>λέγω, ὅτι οὐδ' ἄλλος <lb n="1799,001,,,,9,10,28"></lb> |
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7932 οὐδεὶς κύβος ἔσται χωρὶς τοῦ τετάρτου ἀπὸ τῆς μονάδος <lb n="1799,001,,,,9,10,29"></lb> |
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7933 καὶ τῶν δύο διαλειπόντων.</s> <lb n="1799,001,,,,9,10,30"></lb> |
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7934 <s>Εἰ γὰρ δυνατόν, ἔστω ὁ Δ κύβος.</s> <s>ἔστι δὲ καὶ ὁ Γ <lb n="1799,001,,,,9,10,31"></lb> |
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7935 κύβος· τέταρτος γάρ ἐστιν ἀπὸ τῆς μονάδος.</s> <s>καί ἐστιν ὡς <lb n="1799,001,,,,9,10,32"></lb> |
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7936 ὁ Γ πρὸς τὸν Δ, ὁ Β πρὸς τὸν Γ· καὶ ὁ Β ἄρα πρὸς τὸν Γ <lb n="1799,001,,,,9,10,33"></lb> |
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7937 λόγον ἔχει, ὃν κύβος πρὸς κύβον.</s> <s>καί ἐστιν ὁ Γ κύβος· <lb n="1799,001,,,,9,10,34"></lb> |
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7938 καὶ ὁ Β ἄρα κύβος ἐστίν.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ μονὰς πρὸς <lb n="1799,001,,,,9,10,35"></lb> |
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7939 τὸν Α, ὁ Α πρὸς τὸν Β, ἡ δὲ μονὰς τὸν Α μετρεῖ κατὰ τὰς <lb n="1799,001,,,,9,10,36"></lb> |
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7940 ἐν αὐτῷ μονάδας, καὶ ὁ Α ἄρα τὸν Β μετρεῖ κατὰ τὰς ἐν <lb n="1799,001,,,,9,10,37"></lb> |
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7941 αὑτῷ μονάδας· ὁ Α ἄρα ἑαυτὸν πολλαπλασιάσας κύβον <pb></pb> <lb n="1799,001,,,,9,10,38"></lb> |
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7942 τὸν Β πεποίηκεν.</s> <s>ἐὰν δὲ ἀριθμὸς ἑαυτὸν πολλαπλασιάσας <lb n="1799,001,,,,9,10,39"></lb> |
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7943 κύβον ποιῇ, καὶ αὐτὸς κύβος ἔσται.</s> <s>κύβος ἄρα καὶ ὁ Α· <lb n="1799,001,,,,9,10,40"></lb> |
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7944 ὅπερ οὐχ ὑπόκειται.</s> <s>οὐκ ἄρα ὁ Δ κύβος ἐστίν.</s> <s>ὁμοίως <lb n="1799,001,,,,9,10,41"></lb> |
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7945 δὴ δείξομεν, ὅτι οὐδ' ἄλλος οὐδεὶς κύβος ἐστὶ χωρὶς τοῦ <lb n="1799,001,,,,9,10,42"></lb> |
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7946 τετάρτου ἀπὸ τῆς μονάδος καὶ τῶν δύο διαλειπόντων· <lb n="1799,001,,,,9,10,43"></lb> |
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7947 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,10,44"></lb> |
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7948 <s>Ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον <lb n="1799,001,,,,9,11,1"></lb> |
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7949 ὦσιν, ὁ ἐλάττων τὸν μείζονα μετρεῖ κατά τινα τῶν <lb n="1799,001,,,,9,11,2"></lb> |
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7950 ὑπαρχόντων ἐν τοῖς ἀνάλογον ἀριθμοῖς.</s> <lb n="1799,001,,,,9,11,3"></lb> |
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7951 <s>Ἔστωσαν ἀπὸ μονάδος τῆς Α ὁποσοιοῦν ἀριθμοὶ ἑξῆς <lb n="1799,001,,,,9,11,4"></lb> |
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7952 ἀνάλογον οἱ Β, Γ, Δ, Ε· λέγω, ὅτι τῶν Β, Γ, Δ, Ε ὁ <lb n="1799,001,,,,9,11,5"></lb> |
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7953 ἐλάχιστος ὁ Β τὸν Ε μετρεῖ κατά τινα τῶν Γ, Δ.</s> <lb n="1799,001,,,,9,11,6"></lb> |
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7954 <s>Ἐπεὶ γάρ ἐστιν ὡς ἡ Α μονὰς <lb n="1799,001,,,,9,11,7"></lb> |
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7955 πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν Ε, <lb n="1799,001,,,,9,11,8"></lb> |
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7956 ἰσάκις ἄρα ἡ Α μονὰς τὸν Β ἀριθμὸν <lb n="1799,001,,,,9,11,9"></lb> |
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7957 μετρεῖ καὶ ὁ Δ τὸν Ε· ἐναλλὰξ ἄρα <lb n="1799,001,,,,9,11,10"></lb> |
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7958 ἰσάκις ἡ Α μονὰς τὸν Δ μετρεῖ καὶ <lb n="1799,001,,,,9,11,11"></lb> |
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7959 ὁ Β τὸν Ε.</s> <s>ἡ δὲ Α μονὰς τὸν Δ μετρεῖ <lb n="1799,001,,,,9,11,12"></lb> |
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7960 κατὰ τὰς ἐν αὐτῷ μονάδας· καὶ ὁ Β <lb n="1799,001,,,,9,11,13"></lb> |
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7961 ἄρα τὸν Ε μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας· ὥστε ὁ <lb n="1799,001,,,,9,11,14"></lb> |
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7962 ἐλάσσων ὁ Β τὸν μείζονα τὸν Ε μετρεῖ κατά τινα ἀριθ-<lb n="1799,001,,,,9,11,15"></lb> |
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7963 μὸν τῶν ὑπαρχόντων ἐν τοῖς ἀνάλογον ἀριθμοῖς.</s> <lb n="1799,001,,,,9,11,16"></lb> |
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7964 <s>Πόρισμα <lb n="1799,001,,,,9,11,17"></lb> |
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7965 Καὶ φανερόν, ὅτι ἣν ἔχει τάξιν ὁ μετρῶν ἀπὸ μονάδος, <lb n="1799,001,,,,9,11,18"></lb> |
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7966 τὴν αὐτὴν ἔχει καὶ ὁ καθ' ὃν μετρεῖ ἀπὸ τοῦ μετρουμένου <lb n="1799,001,,,,9,11,19"></lb> |
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7967 ἐπὶ τὸ πρὸ αὐτοῦ.</s> <s>̄ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,9,11,20"></lb> |
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7968 Ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον <lb n="1799,001,,,,9,12,1"></lb> |
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7969 ὦσιν, ὑφ' ὅσων ἂν ὁ ἔσχατος πρώτων ἀριθμῶν μετρῆται, <lb n="1799,001,,,,9,12,2"></lb> |
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7970 ὑπὸ τῶν αὐτῶν καὶ ὁ παρὰ τὴν μονάδα μετρηθήσεται.</s> <lb n="1799,001,,,,9,12,3"></lb> |
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7971 <s>Ἔστωσαν ἀπὸ μονάδος ὁποσοιδηποτοῦν ἀριθμοὶ ἀνά-<lb n="1799,001,,,,9,12,4"></lb> |
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7972 λογον οἱ Α, Β, Γ, Δ· λέγω, ὅτι ὑφ' ὅσων ἂν ὁ Δ πρώτων <lb n="1799,001,,,,9,12,5"></lb> |
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7973 ἀριθμῶν μετρῆται, ὑπὸ τῶν αὐτῶν καὶ ὁ Α μετρηθήσεται.</s> <lb n="1799,001,,,,9,12,6"></lb> |
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7974 <s>Μετρείσθω γὰρ ὁ Δ <lb n="1799,001,,,,9,12,7"></lb> |
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7975 ὑπό τινος πρώτου ἀριθ-<lb n="1799,001,,,,9,12,8"></lb> |
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7976 μοῦ τοῦ Ε· λέγω, ὅτι ὁ <lb n="1799,001,,,,9,12,9"></lb> |
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7977 Ε τὸν Α μετρεῖ.</s> <s>μὴ γάρ· <lb n="1799,001,,,,9,12,10"></lb> |
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7978 καί ἐστιν ὁ Ε πρῶτος, <lb n="1799,001,,,,9,12,11"></lb> |
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7979 ἅπας δὲ πρῶτος ἀριθμὸς <lb n="1799,001,,,,9,12,12"></lb> |
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7980 πρὸς ἅπαντα, ὃν μὴ με-<lb n="1799,001,,,,9,12,13"></lb> |
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7981 τρεῖ, πρῶτός ἐστιν· οἱ Ε, Α ἄρα πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,9,12,14"></lb> |
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7982 εἰσίν.</s> <s>καὶ ἐπεὶ ὁ Ε τὸν Δ μετρεῖ, μετρείτω αὐτὸν κατὰ <lb n="1799,001,,,,9,12,15"></lb> |
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7983 τὸν Ζ· ὁ Ε ἄρα τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν.</s> <lb n="1799,001,,,,9,12,16"></lb> |
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7984 <s>πάλιν, ἐπεὶ ὁ Α τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Γ μονάδας, <lb n="1799,001,,,,9,12,17"></lb> |
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7985 ὁ Α ἄρα τὸν Γ πολλαπλασιάσας τὸν Δ πεποίηκεν.</s> <s>ἀλλὰ μὴν <lb n="1799,001,,,,9,12,18"></lb> |
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7986 καὶ ὁ Ε τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν· ὁ ἄρα ἐκ <lb n="1799,001,,,,9,12,19"></lb> |
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7987 τῶν Α, Γ ἴσος ἐστὶ τῷ ἐκ τῶν Ε, Ζ.</s> <s>ἔστιν ἄρα ὡς ὁ Α <lb n="1799,001,,,,9,12,20"></lb> |
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7988 πρὸς τὸν Ε, ὁ Ζ πρὸς τὸν Γ.</s> <s>οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ <lb n="1799,001,,,,9,12,21"></lb> |
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7989 πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν <lb n="1799,001,,,,9,12,22"></lb> |
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7990 αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον <pb></pb> <lb n="1799,001,,,,9,12,23"></lb> |
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7991 καὶ ὁ ἑπόμενος τὸν ἑπόμενον· μετρεῖ ἄρα ὁ Ε τὸν Γ.</s> <lb n="1799,001,,,,9,12,24"></lb> |
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7992 <s>μετρείτω αὐτὸν κατὰ τὸν Η· ὁ Ε ἄρα τὸν Η πολλαπλα-<lb n="1799,001,,,,9,12,25"></lb> |
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7993 σιάσας τὸν Γ πεποίηκεν.</s> <s>ἀλλὰ μὴν διὰ τὸ πρὸ τούτου καὶ ὁ <lb n="1799,001,,,,9,12,26"></lb> |
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7994 Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν.</s> <s>ὁ ἄρα ἐκ τῶν <lb n="1799,001,,,,9,12,27"></lb> |
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7995 Α, Β ἴσος ἐστὶ τῷ ἐκ τῶν Ε, Η.</s> <s>ἔστιν ἄρα ὡς ὁ Α πρὸς <lb n="1799,001,,,,9,12,28"></lb> |
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7996 τὸν Ε, ὁ Η πρὸς τὸν Β.</s> <s>οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι <lb n="1799,001,,,,9,12,29"></lb> |
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7997 καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι ἀριθμοὶ μετροῦσι τοὺς <lb n="1799,001,,,,9,12,30"></lb> |
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7998 τὸν αὐτὸν λόγον ἔχοντας αὐτοῖς ἰσάκις ὅ τε ἡγούμενος <lb n="1799,001,,,,9,12,31"></lb> |
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7999 τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον· μετρεῖ ἄρα <lb n="1799,001,,,,9,12,32"></lb> |
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8000 ὁ Ε τὸν Β.</s> <s>μετρείτω αὐτὸν κατὰ τὸν Θ· ὁ Ε ἄρα τὸν Θ <lb n="1799,001,,,,9,12,33"></lb> |
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8001 πολλαπλασιάσας τὸν Β πεποίηκεν.</s> <s>ἀλλὰ μὴν καὶ ὁ Α ἑαυ-<lb n="1799,001,,,,9,12,34"></lb> |
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8002 τὸν πολλαπλασιάσας τὸν Β πεποίηκεν· ὁ ἄρα ἐκ τῶν <lb n="1799,001,,,,9,12,35"></lb> |
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8003 Ε, Θ ἴσος ἐστὶ τῷ ἀπὸ τοῦ Α.</s> <s>ἔστιν ἄρα ὡσ ὁ Ε πρὸς τὸν <lb n="1799,001,,,,9,12,36"></lb> |
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8004 Α, ὁ Α πρὸς τὸν Θ.</s> <s>οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ <lb n="1799,001,,,,9,12,37"></lb> |
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8005 ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον <lb n="1799,001,,,,9,12,38"></lb> |
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8006 ἔχοντας ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ <lb n="1799,001,,,,9,12,39"></lb> |
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8007 ἑπόμενος τὸν ἑπόμενον· μετρεῖ ἄρα ὁ Ε τὸν Α ὡς ἡγούμενος <lb n="1799,001,,,,9,12,40"></lb> |
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8008 ἡγούμενον.</s> <s>ἀλλὰ μὴν καὶ οὐ μετρεῖ· ὅπερ ἀδύνατον.</s> <s>οὐκ <lb n="1799,001,,,,9,12,41"></lb> |
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8009 ἄρα οἱ Ε, Α πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>σύνθετοι ἄρα.</s> <lb n="1799,001,,,,9,12,42"></lb> |
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8010 <s>οἱ δὲ σύνθετοι ὑπὸ [πρώτου] ἀριθμοῦ τινος μετροῦνται.</s> <lb n="1799,001,,,,9,12,43"></lb> |
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8011 <s>καὶ ἐπεὶ ὁ Ε πρῶτος ὑπόκειται, ὁ δὲ πρῶτος ὑπὸ ἑτέρου <lb n="1799,001,,,,9,12,44"></lb> |
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8012 ἀριθμοῦ οὐ μετρεῖται ἢ ὑφ' ἑαυτοῦ, ὁ Ε ἄρα τοὺς Α, Ε <lb n="1799,001,,,,9,12,45"></lb> |
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8013 μετρεῖ· ὥστε ὁ Ε τὸν Α μετρεῖ.</s> <s>μετρεῖ δὲ καὶ τὸν Δ· ὁ Ε <pb></pb> <lb n="1799,001,,,,9,12,46"></lb> |
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8014 ἄρα τοὺς Α, Δ μετρεῖ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι ὑφ' ὅσων <lb n="1799,001,,,,9,12,47"></lb> |
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8015 ἂν ὁ Δ πρώτων ἀριθμῶν μετρῆται, ὑπὸ τῶν αὐτῶν καὶ <lb n="1799,001,,,,9,12,48"></lb> |
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8016 ὁ Α μετρηθήσεται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,12,49"></lb> |
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8017 <s>Ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον <lb n="1799,001,,,,9,13,1"></lb> |
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8018 ὦσιν, ὁ δὲ μετὰ τὴν μονάδα πρῶτος ᾖ, ὁ μέγιστος ὑπ' <lb n="1799,001,,,,9,13,2"></lb> |
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8019 οὐδενὸς [ἄλλου] μετρηθήσεται παρὲξ τῶν ὑπαρχόντων ἐν <lb n="1799,001,,,,9,13,3"></lb> |
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8020 τοῖς ἀνάλογον ἀριθμοῖς. <lb n="1799,001,,,,9,13,4"></lb> |
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8021 Ἔστωσαν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλο-<lb n="1799,001,,,,9,13,5"></lb> |
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8022 γον οἱ Α, Β, Γ, Δ, ὁ δὲ μετὰ τὴν μονάδα ὁ Α πρῶτος ἔστω· <lb n="1799,001,,,,9,13,6"></lb> |
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8023 λέγω, ὅτι ὁ μέγιστος αὐτῶν ὁ Δ ὑπ' οὐδενὸς ἄλλου <lb n="1799,001,,,,9,13,7"></lb> |
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8024 μετρηθήσεται παρὲξ τῶν Α, Β, Γ.</s> <lb n="1799,001,,,,9,13,8"></lb> |
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8025 <s>Εἰ γὰρ δυνατόν, μετρείσθω ὑπὸ τοῦ Ε, καὶ ὁ Ε μηδενὶ <lb n="1799,001,,,,9,13,9"></lb> |
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8026 τῶν Α, Β, Γ ἔστω ὁ αὐτός.</s> <s>φανερὸν δή, ὅτι ὁ Ε πρῶτος <lb n="1799,001,,,,9,13,10"></lb> |
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8027 οὔκ ἐστιν.</s> <s>εἰ γὰρ ὁ Ε πρῶτός ἐστι καὶ μετρεῖ τὸν Δ, καὶ <lb n="1799,001,,,,9,13,11"></lb> |
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8028 τὸν Α μετρήσει πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ <lb n="1799,001,,,,9,13,12"></lb> |
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8029 ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ὁ Ε πρῶτός ἐστιν.</s> <s>σύνθετος ἄρα.</s> <lb n="1799,001,,,,9,13,13"></lb> |
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8030 <s>πᾶς δὲ σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ <lb n="1799,001,,,,9,13,14"></lb> |
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8031 μετρεῖται· ὁ Ε ἄρα ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται.</s> <lb n="1799,001,,,,9,13,15"></lb> |
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8032 <s>λέγω δή, ὅτι ὑπ' οὐδενὸς ἄλλου πρώτου μετρηθήσεται <lb n="1799,001,,,,9,13,16"></lb> |
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8033 πλὴν τοῦ Α.</s> <s>εἰ γὰρ ὑφ' ἑτέρου μετρεῖται ὁ Ε, ὁ δὲ Ε <lb n="1799,001,,,,9,13,17"></lb> |
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8034 τὸν Δ μετρεῖ, κἀκεῖνος ἄρα τὸν Δ μετρήσει· ὥστε καὶ <pb></pb> <lb n="1799,001,,,,9,13,18"></lb> |
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8035 τὸν Α μετρήσει πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ <lb n="1799,001,,,,9,13,19"></lb> |
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8036 ἐστὶν ἀδύνατον.</s> <s>ὁ Α ἄρα τὸν Ε μετρεῖ.</s> <s>καὶ ἐπεὶ ὁ Ε τὸν <lb n="1799,001,,,,9,13,20"></lb> |
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8037 Δ μετρεῖ, μετρείτω αὐτὸν κατὰ τὸν Ζ.</s> <s>λέγω, ὅτι ὁ Ζ <lb n="1799,001,,,,9,13,21"></lb> |
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8038 οὐδενὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός.</s> <s>εἰ γὰρ ὁ Ζ ἑνὶ τῶν <lb n="1799,001,,,,9,13,22"></lb> |
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8039 Α, Β, Γ ἐστιν ὁ αὐτὸς καὶ μετρεῖ τὸν Δ κατὰ τὸν Ε, καὶ <lb n="1799,001,,,,9,13,23"></lb> |
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8040 εἷς ἄρα τῶν Α, Β, Γ τὸν Δ μετρεῖ κατὰ τὸν Ε.</s> <s>ἀλλὰ εἷς <lb n="1799,001,,,,9,13,24"></lb> |
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8041 τῶν Α, Β, Γ τὸν Δ μετρεῖ κατά τινα τῶν Α, Β, Γ· καὶ ὁ <lb n="1799,001,,,,9,13,25"></lb> |
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8042 Ε ἄρα ἑνὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός· ὅπερ οὐχ ὑπόκειται.</s> <lb n="1799,001,,,,9,13,26"></lb> |
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8043 <s>οὐκ ἄρα ὁ Ζ ἑνὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός.</s> <s>ὁμοίως δὴ <lb n="1799,001,,,,9,13,27"></lb> |
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8044 δείξομεν, ὅτι μετρεῖται ὁ Ζ ὑπὸ τοῦ Α, δεικνύντες πάλιν, <lb n="1799,001,,,,9,13,28"></lb> |
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8045 ὅτι ὁ Ζ οὔκ ἐστι πρῶτος.</s> <s>εἰ γάρ, καὶ μετρεῖ τὸν Δ, καὶ τὸν <lb n="1799,001,,,,9,13,29"></lb> |
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8046 Α μετρήσει πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ ἐστὶν <lb n="1799,001,,,,9,13,30"></lb> |
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8047 ἀδύνατον· οὐκ ἄρα πρῶτός ἐστιν ὁ Ζ· σύνθετος ἄρα.</s> <s>ἅπας <lb n="1799,001,,,,9,13,31"></lb> |
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8048 δὲ σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖ-<lb n="1799,001,,,,9,13,32"></lb> |
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8049 ται· ὁ Ζ ἄρα ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται.</s> <s>λέγω <lb n="1799,001,,,,9,13,33"></lb> |
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8050 δή, ὅτι ὑφ' ἑτέρου πρώτου οὐ μετρηθήσεται πλὴν τοῦ <lb n="1799,001,,,,9,13,34"></lb> |
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8051 Α.</s> <s>εἰ γὰρ ἕτερός τις πρῶτος τὸν Ζ μετρεῖ, ὁ δὲ Ζ τὸν <lb n="1799,001,,,,9,13,35"></lb> |
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8052 Δ μετρεῖ, κἀκεῖνος ἄρα τὸν Δ μετρήσει· ὥστε καὶ τὸν Α <lb n="1799,001,,,,9,13,36"></lb> |
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8053 μετρήσει πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ ἐστὶν <lb n="1799,001,,,,9,13,37"></lb> |
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8054 ἀδύνατον.</s> <s>ὁ Α ἄρα τὸν Ζ μετρεῖ.</s> <s>καὶ ἐπεὶ ὁ Ε τὸν Δ μετρεῖ <lb n="1799,001,,,,9,13,38"></lb> |
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8055 κατὰ τὸν Ζ, ὁ Ε ἄρα τὸν Ζ πολλαπλασιάσας τὸν Δ <lb n="1799,001,,,,9,13,39"></lb> |
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8056 πεποίηκεν.</s> <s>ἀλλὰ μὴν καὶ ὁ Α τὸν Γ πολλαπλασιάσας <lb n="1799,001,,,,9,13,40"></lb> |
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8057 τὸν Δ πεποίηκεν· ὁ ἄρα ἐκ τῶν Α, Γ ἴσος ἐστὶ τῷ ἐκ <lb n="1799,001,,,,9,13,41"></lb> |
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8058 τῶν Ε, Ζ.</s> <s>ἀνάλογον ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Ε, οὕτως <lb n="1799,001,,,,9,13,42"></lb> |
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8059 ὁ Ζ πρὸς τὸν Γ.</s> <s>ὁ δὲ Α τὸν Ε μετρεῖ· καὶ ὁ Ζ ἄρα τὸν Γ <pb></pb> <lb n="1799,001,,,,9,13,43"></lb> |
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8060 μετρεῖ.</s> <s>μετρείτω αὐτὸν κατὰ τὸν Η.</s> <s>ὁμοίως δὴ δείξομεν, <lb n="1799,001,,,,9,13,44"></lb> |
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8061 ὅτι ὁ Η οὐδενὶ τῶν Α, Β ἐστιν ὁ αὐτός, καὶ ὅτι μετρεῖται <lb n="1799,001,,,,9,13,45"></lb> |
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8062 ὑπὸ τοῦ Α.</s> <s>καὶ ἐπεὶ ὁ Ζ τὸν Γ μετρεῖ κατὰ τὸν Η, ὁ Ζ <lb n="1799,001,,,,9,13,46"></lb> |
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8063 ἄρα τὸν Η πολλαπλασιάσας τὸν Γ πεποίηκεν.</s> <s>ἀλλὰ μὴν <lb n="1799,001,,,,9,13,47"></lb> |
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8064 καὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν· ὁ ἄρα <lb n="1799,001,,,,9,13,48"></lb> |
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8065 ἐκ τῶν Α, Β ἴσος ἐστὶ τῷ ἐκ τῶν Ζ, Η.</s> <s>ἀνάλογον ἄρα ὡς <lb n="1799,001,,,,9,13,49"></lb> |
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8066 ὁ Α πρὸς τὸν Ζ, ὁ Η πρὸς τὸν Β.</s> <s>μετρεῖ δὲ ὁ Α τὸν Ζ· με-<lb n="1799,001,,,,9,13,50"></lb> |
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8067 τρεῖ ἄρα καὶ ὁ Η τὸν Β.</s> <s>μετρείτω αὐτὸν κατὰ τὸν Θ.</s> <lb n="1799,001,,,,9,13,51"></lb> |
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8068 <s>ὁμοίως δὴ δείξομεν, ὅτι ὁ Θ τῷ Α οὐκ ἔστιν ὁ αὐτός.</s> <lb n="1799,001,,,,9,13,52"></lb> |
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8069 <s>καὶ ἐπεὶ ὁ Η τὸν Β μετρεῖ κατὰ τὸν Θ, ὁ Η ἄρα τὸν Θ <lb n="1799,001,,,,9,13,53"></lb> |
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8070 πολλαπλασιάσας τὸν Β πεποίηκεν.</s> <s>ἀλλὰ μὴν καὶ ὁ Α <lb n="1799,001,,,,9,13,54"></lb> |
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8071 ἑαυτὸν πολλαπλασιάσας τὸν Β πεποίηκεν· ὁ ἄρα ὑπὸ <lb n="1799,001,,,,9,13,55"></lb> |
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8072 Θ, Η ἴσος ἐστὶ τῷ ἀπὸ τοῦ Α τετραγώνῳ.</s> <s>ἔστιν ἄρα ὡς <lb n="1799,001,,,,9,13,56"></lb> |
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8073 ὁ Θ πρὸς τὸν Α, ὁ Α πρὸς τὸν Η.</s> <s>μετρεῖ δὲ ὁ Α τὸν Η· <lb n="1799,001,,,,9,13,57"></lb> |
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8074 μετρεῖ ἄρα καὶ ὁ Θ τὸν Α πρῶτον ὄντα μὴ ὢν αὐτῷ ὁ <lb n="1799,001,,,,9,13,58"></lb> |
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8075 αὐτός· ὅπερ ἄτοπον.</s> <s>οὐκ ἄρα ὁ μέγιστος ὁ Δ ὑπὸ ἑτέρου <lb n="1799,001,,,,9,13,59"></lb> |
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8076 ἀριθμοῦ μετρηθήσεται παρὲξ τῶν Α, Β, Γ· ὅπερ ἔδει <lb n="1799,001,,,,9,13,60"></lb> |
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8077 δεῖξαι.</s> <lb n="1799,001,,,,9,13,61"></lb> |
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8078 <s>Ἐὰν ἐλάχιστος ἀριθμὸς ὑπὸ πρώτων ἀριθμῶν μετρῆται, <lb n="1799,001,,,,9,14,1"></lb> |
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8079 ὑπ' οὐδενὸς ἄλλου πρώτου ἀριθμοῦ μετρηθήσεται παρὲξ <lb n="1799,001,,,,9,14,2"></lb> |
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8080 τῶν ἐξ ἀρχῆς μετρούντων.</s> <lb n="1799,001,,,,9,14,3"></lb> |
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8081 <s>Ἐλάχιστος γὰρ ἀριθμὸς ὁ Α ὑπὸ πρώτων ἀριθμῶν <lb n="1799,001,,,,9,14,4"></lb> |
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8082 τῶν Β, Γ, Δ μετρείσθω· λέγω, ὅτι ὁ Α ὑπ' οὐδενὸς ἄλλου <lb n="1799,001,,,,9,14,5"></lb> |
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8083 πρώτου ἀριθμοῦ μετρηθήσεται παρὲξ τῶν Β, Γ, Δ.</s> <s><pb></pb> <lb n="1799,001,,,,9,14,6"></lb> |
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8084 Εἰ γὰρ δυνατόν, μετρείσθω ὑπὸ πρώτου τοῦ Ε, καὶ <lb n="1799,001,,,,9,14,7"></lb> |
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8085 ὁ Ε μηδενὶ τῶν Β, Γ, Δ ἔστω ὁ αὐτός.</s> <s>καὶ ἐπεὶ ὁ Ε τὸν Α <lb n="1799,001,,,,9,14,8"></lb> |
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8086 μετρεῖ, μετρείτω αὐτὸν κατὰ τὸν Ζ· ὁ Ε ἄρα τὸν Ζ <lb n="1799,001,,,,9,14,9"></lb> |
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8087 πολλαπλασιάσας τὸν Α πεποίηκεν. <lb n="1799,001,,,,9,14,10"></lb> |
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8088 καὶ μετρεῖται ὁ Α ὑπὸ πρώτων <lb n="1799,001,,,,9,14,11"></lb> |
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8089 ἀριθμῶν τῶν Β, Γ, Δ.</s> <s>ἐὰν δὲ δύο <lb n="1799,001,,,,9,14,12"></lb> |
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8090 ἀριθμοὶ πολλαπλασιάσαντες ἀλλή-<lb n="1799,001,,,,9,14,13"></lb> |
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8091 λους ποιῶσί τινα, τὸν δὲ γενό-<lb n="1799,001,,,,9,14,14"></lb> |
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8092 μενον ἐξ αὐτῶν μετρῇ τις πρῶτος ἀριθμός, καὶ ἕνα τῶν <lb n="1799,001,,,,9,14,15"></lb> |
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8093 ἐξ ἀρχῆς μετρήσει· οἱ Β, Γ, Δ ἄρα ἕνα τῶν Ε, Ζ μετρή-<lb n="1799,001,,,,9,14,16"></lb> |
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8094 σουσιν.</s> <s>τὸν μὲν οὖν Ε οὐ μετρήσουσιν· ὁ γὰρ Ε πρῶτός <lb n="1799,001,,,,9,14,17"></lb> |
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8095 ἐστι καὶ οὐδενὶ τῶν Β, Γ, Δ ὁ αὐτός.</s> <s>τὸν Ζ ἄρα μετροῦ-<lb n="1799,001,,,,9,14,18"></lb> |
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8096 σιν ἐλάσσονα ὄντα τοῦ Α· ὅπερ ἀδύνατον.</s> <s>ὁ γὰρ Α ὑπό-<lb n="1799,001,,,,9,14,19"></lb> |
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8097 κειται ἐλάχιστος ὑπὸ τῶν Β, Γ, Δ μετρούμενος.</s> <s>οὐκ ἄρα <lb n="1799,001,,,,9,14,20"></lb> |
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8098 τὸν Α μετρήσει πρῶτος ἀριθμὸς παρὲξ τῶν Β, Γ, Δ· <lb n="1799,001,,,,9,14,21"></lb> |
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8099 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,14,22"></lb> |
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8100 <s>Ἐὰν τρεῖς ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν ἐλάχιστοι τῶν <lb n="1799,001,,,,9,15,1"></lb> |
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8101 τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς, δύο ὁποιοιοῦν συντε-<lb n="1799,001,,,,9,15,2"></lb> |
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8102 θέντες πρὸς τὸν λοιπὸν πρῶτοί εἰσιν.</s> <lb n="1799,001,,,,9,15,3"></lb> |
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8103 <s>Ἔστωσαν τρεῖς ἀριθμοὶ ἑξῆς ἀνάλογον ἐλάχιστοι τῶν <lb n="1799,001,,,,9,15,4"></lb> |
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8104 τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς οἱ Α, Β, Γ· λέγω, ὅτι <lb n="1799,001,,,,9,15,5"></lb> |
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8105 τῶν Α, Β, Γ δύο ὁποιοιοῦν συντεθέντες πρὸς τὸν λοιπὸν <lb n="1799,001,,,,9,15,6"></lb> |
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8106 πρῶτοί εἰσιν, οἱ μὲν Α, Β πρὸς τὸν Γ, οἱ δὲ Β, Γ πρὸς <lb n="1799,001,,,,9,15,7"></lb> |
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8107 τὸν Α καὶ ἔτι οἱ Α, Γ πρὸς τὸν Β.</s> <s><pb></pb> <lb n="1799,001,,,,9,15,8"></lb> |
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8108 Εἰλήφθωσαν γὰρ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν <lb n="1799,001,,,,9,15,9"></lb> |
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8109 λόγον ἐχόντων τοῖς Α, Β, Γ δύο οἱ ΔΕ, ΕΖ.</s> <s>φανερὸν δή, <lb n="1799,001,,,,9,15,10"></lb> |
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8110 ὅτι ὁ μὲν ΔΕ ἑαυτὸν πολλαπλασιάσας τὸν Α πεποίηκεν, <lb n="1799,001,,,,9,15,11"></lb> |
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8111 τὸν δὲ ΕΖ πολλαπλασιάσας τὸν Β <lb n="1799,001,,,,9,15,12"></lb> |
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8112 πεποίηκεν, καὶ ἔτι ὁ ΕΖ ἑαυτὸν <lb n="1799,001,,,,9,15,13"></lb> |
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8113 πολλαπλασιάσας τὸν Γ πεποίηκεν.</s> <lb n="1799,001,,,,9,15,14"></lb> |
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8114 <s>καὶ ἐπεὶ οἱ ΔΕ, ΕΖ ἐλάχιστοί <lb n="1799,001,,,,9,15,15"></lb> |
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8115 εἰσιν, πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,9,15,16"></lb> |
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8116 <s>ἐὰν δὲ δύο ἀριθμοί πρῶτοι πρὸς ἀλλήλους ὦσιν, καὶ <lb n="1799,001,,,,9,15,17"></lb> |
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8117 συναμφότερος πρὸς ἑκάτερον πρῶτός ἐστιν· καὶ ὁ ΔΖ <lb n="1799,001,,,,9,15,18"></lb> |
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8118 ἄρα πρὸς ἑκάτερον τῶν ΔΕ, ΕΖ πρῶτός ἐστιν.</s> <s>ἀλλὰ μὴν <lb n="1799,001,,,,9,15,19"></lb> |
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8119 καὶ ὁ ΔΕ πρὸς τὸν ΕΖ πρῶτός ἐστιν· οἱ ΔΖ, ΔΕ ἄρα <lb n="1799,001,,,,9,15,20"></lb> |
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8120 πρὸς τὸν ΕΖ πρῶτοί εἰσιν.</s> <s>ἐὰν δὲ δύο ἀριθμοὶ πρός τινα <lb n="1799,001,,,,9,15,21"></lb> |
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8121 ἀριθμὸν πρῶτοι ὦσιν, καὶ ὁ ἐξ αὐτῶν γενόμενος πρὸς τὸν <lb n="1799,001,,,,9,15,22"></lb> |
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8122 λοιπὸν πρῶτός ἐστιν· ὥστε ὁ ἐκ τῶν ΖΔ, ΔΕ πρὸς τὸν <lb n="1799,001,,,,9,15,23"></lb> |
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8123 ΕΖ πρῶτός ἐστιν· ὥστε καὶ ὁ ἐκ τῶν ΖΔ, ΔΕ πρὸς τὸν <lb n="1799,001,,,,9,15,24"></lb> |
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8124 ἀπὸ τοῦ ΕΖ πρῶτός ἐστιν.</s> <s>[ἐὰν γὰρ δύο ἀριθμοὶ πρῶτοι <lb n="1799,001,,,,9,15,25"></lb> |
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8125 πρὸς ἀλλήλους ὦσιν, ὁ ἐκ τοῦ ἑνὸς αὐτῶν γενόμενος πρὸς <lb n="1799,001,,,,9,15,26"></lb> |
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8126 τὸν λοιπὸν πρῶτός ἐστιν].</s> <s>ἀλλ' ὁ ἐκ τῶν ΖΔ, ΔΕ ὁ ἀπὸ <lb n="1799,001,,,,9,15,27"></lb> |
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8127 τοῦ ΔΕ ἐστι μετὰ τοῦ ἐκ τῶν ΔΕ, ΕΖ· ὁ ἄρα ἀπὸ τοῦ ΔΕ <lb n="1799,001,,,,9,15,28"></lb> |
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8128 μετὰ τοῦ ἐκ τῶν ΔΕ, ΕΖ πρὸς τὸν ἀπὸ τοῦ ΕΖ πρῶτός <pb></pb> <lb n="1799,001,,,,9,15,29"></lb> |
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8129 ἐστιν.</s> <s>καί ἐστιν ὁ μὲν ἀπὸ τοῦ ΔΕ ὁ Α, ὁ δὲ ἐκ τῶν ΔΕ, ΕΖ <lb n="1799,001,,,,9,15,30"></lb> |
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8130 ὁ Β, ὁ δὲ ἀπὸ τοῦ ΕΖ ὁ Γ· οἱ Α, Β ἄρα συντεθέντες πρὸς <lb n="1799,001,,,,9,15,31"></lb> |
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8131 τὸν Γ πρῶτοί εἰσιν.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ οἱ Β, Γ <lb n="1799,001,,,,9,15,32"></lb> |
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8132 πρὸς τὸν Α πρῶτοί εἰσιν.</s> <s>λέγω δή, ὅτι καὶ οἱ Α, Γ πρὸς <lb n="1799,001,,,,9,15,33"></lb> |
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8133 τὸν Β πρῶτοί εἰσιν.</s> <s>ἐπεὶ γὰρ ὁ ΔΖ πρὸς ἑκάτερον τῶν <lb n="1799,001,,,,9,15,34"></lb> |
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8134 ΔΕ, ΕΖ πρῶτός ἐστιν, καὶ ὁ ἀπὸ τοῦ ΔΖ πρὸς τὸν ἐκ <lb n="1799,001,,,,9,15,35"></lb> |
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8135 τῶν ΔΕ, ΕΖ πρῶτός ἐστιν.</s> <s>ἀλλὰ τῷ ἀπὸ τοῦ ΔΖ ἴσοι <lb n="1799,001,,,,9,15,36"></lb> |
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8136 εἰσὶν οἱ ἀπὸ τῶν ΔΕ, ΕΖ μετὰ τοῦ δὶς ἐκ τῶν ΔΕ, ΕΖ· <lb n="1799,001,,,,9,15,37"></lb> |
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8137 καὶ οἱ ἀπὸ τῶν ΔΕ, ΕΖ ἄρα μετὰ τοῦ δὶς ὑπὸ τῶν ΔΕ, <lb n="1799,001,,,,9,15,38"></lb> |
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8138 ΕΖ πρὸς τὸν ὑπὸ τῶν ΔΕ, ΕΖ πρῶτοί [εἰσι].</s> <s>διελόντι οἱ <lb n="1799,001,,,,9,15,39"></lb> |
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8139 ἀπὸ τῶν ΔΕ, ΕΖ μετὰ τοῦ ἅπαξ ὑπὸ ΔΕ, ΕΖ πρὸς τὸν <lb n="1799,001,,,,9,15,40"></lb> |
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8140 ὑπὸ ΔΕ, ΕΖ πρῶτοί εἰσιν.</s> <s>ἔτι διελόντι οἱ ἀπὸ τῶν ΔΕ, <lb n="1799,001,,,,9,15,41"></lb> |
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8141 ΕΖ ἄρα πρὸς τὸν ὑπὸ ΔΕ, ΕΖ πρῶτοί εἰσιν.</s> <s>καί ἐστιν <lb n="1799,001,,,,9,15,42"></lb> |
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8142 ὁ μὲν ἀπὸ τοῦ ΔΕ ὁ Α, ὁ δὲ ὑπὸ τῶν ΔΕ, ΕΖ ὁ Β, ὁ δὲ <lb n="1799,001,,,,9,15,43"></lb> |
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8143 ἀπὸ τοῦ ΕΖ ὁ Γ.</s> <s>οἱ Α, Γ ἄρα συντεθέντες πρὸς τὸν Β <lb n="1799,001,,,,9,15,44"></lb> |
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8144 πρῶτοί εἰσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,15,45"></lb> |
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8145 <s>Ἐὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, οὐκ <lb n="1799,001,,,,9,16,1"></lb> |
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8146 ἔσται ὡς ὁ πρῶτος πρὸς τὸν δεύτερον, οὕτως ὁ δεύτερος <lb n="1799,001,,,,9,16,2"></lb> |
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8147 πρὸς ἄλλον τινά.</s> <s><pb></pb> <lb n="1799,001,,,,9,16,3"></lb> |
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8148 Δύο γὰρ ἀριθμοὶ οἱ Α, Β πρῶτοι πρὸς ἀλλήλους <lb n="1799,001,,,,9,16,4"></lb> |
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8149 ἔστωσαν· λέγω, ὅτι οὐκ ἔστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως <lb n="1799,001,,,,9,16,5"></lb> |
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8150 ὁ Β πρὸς ἄλλον τινά.</s> <lb n="1799,001,,,,9,16,6"></lb> |
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8151 <s>Εἰ γὰρ δυνατόν, ἔστω ὡς ὁ Α πρὸς τὸν Β, ὁ Β πρὸς <lb n="1799,001,,,,9,16,7"></lb> |
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8152 τὸν Γ.</s> <s>οἱ δὲ Α, Β πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ <lb n="1799,001,,,,9,16,8"></lb> |
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8153 δὲ ἐλάχιστοι ἀριθμοὶ μετροῦσι τοὺς τὸν <lb n="1799,001,,,,9,16,9"></lb> |
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8154 αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε ἡγού-<lb n="1799,001,,,,9,16,10"></lb> |
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8155 μενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος <lb n="1799,001,,,,9,16,11"></lb> |
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8156 τὸν ἑπόμενον· μετρεῖ ἄρα ὁ Α τὸν Β <lb n="1799,001,,,,9,16,12"></lb> |
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8157 ὡς ἡγούμενος ἡγούμενον.</s> <s>μετρεῖ δὲ καὶ <lb n="1799,001,,,,9,16,13"></lb> |
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8158 ἑαυτόν· ὁ Α ἄρα τοὺς Α, Β μετρεῖ πρώτους ὄντας πρὸς <lb n="1799,001,,,,9,16,14"></lb> |
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8159 ἀλλήλους· ὅπερ ἄτοπον.</s> <s>οὐκ ἄρα ἔσται ὡς ὁ Α πρὸς τὸν Β, <lb n="1799,001,,,,9,16,15"></lb> |
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8160 οὕτως ὁ Β πρὸς τὸν Γ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,16,16"></lb> |
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8161 <s>Ἐὰν ὦσιν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, οἱ <lb n="1799,001,,,,9,17,1"></lb> |
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8162 δὲ ἄκροι αὐτῶν πρῶτοι πρὸς ἀλλήλους ὦσιν, οὐκ ἔσται <lb n="1799,001,,,,9,17,2"></lb> |
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8163 ὡς ὁ πρῶτος πρὸς τὸν δεύτερον, οὕτως ὁ ἔσχατος πρὸς <lb n="1799,001,,,,9,17,3"></lb> |
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8164 ἄλλον τινά.</s> <lb n="1799,001,,,,9,17,4"></lb> |
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8165 <s>Ἔστωσαν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ <lb n="1799,001,,,,9,17,5"></lb> |
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8166 Α, Β, Γ, Δ, οἱ δὲ ἄκροι αὐτῶν οἱ Α, Δ πρῶτοι πρὸς <lb n="1799,001,,,,9,17,6"></lb> |
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8167 ἀλλήλους ἔστωσαν· λέγω, ὅτι οὐκ ἔστιν ὡς ὁ Α πρὸς τὸν <lb n="1799,001,,,,9,17,7"></lb> |
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8168 Β, οὕτως ὁ Δ πρὸς ἄλλον τινά. <pb></pb> <lb n="1799,001,,,,9,17,8"></lb> |
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8169 Εἰ γὰρ δυνατόν, ἔστω ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ <lb n="1799,001,,,,9,17,9"></lb> |
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8170 πρὸς τὸν Ε· ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Δ, ὁ Β <lb n="1799,001,,,,9,17,10"></lb> |
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8171 πρὸς τὸν Ε.</s> <s>οἱ δὲ Α, Δ πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, <lb n="1799,001,,,,9,17,11"></lb> |
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8172 οἱ δὲ ἐλάχιστοι ἀριθμοὶ μετροῦσι τοὺς τὸν αὐτὸν λόγον <lb n="1799,001,,,,9,17,12"></lb> |
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8173 ἔχοντας ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ <lb n="1799,001,,,,9,17,13"></lb> |
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8174 ἑπόμενος τὸν ἑπόμενον.</s> <s>μετρεῖ ἄρα ὁ Α τὸν Β.</s> <s>καί ἐστιν <lb n="1799,001,,,,9,17,14"></lb> |
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8175 ὡς ὁ Α πρὸς τὸν Β, ὁ Β πρὸς τὸν Γ.</s> <s>καὶ ὁ Β ἄρα τὸν Γ <lb n="1799,001,,,,9,17,15"></lb> |
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8176 μετρεῖ· ὥστε καὶ ὁ Α τὸν Γ μετρεῖ.</s> <s>καὶ ἐπεί ἐστιν ὡς <lb n="1799,001,,,,9,17,16"></lb> |
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8177 ὁ Β πρὸς τὸν Γ, ὁ Γ πρὸς τὸν Δ, μετρεῖ δὲ ὁ Β τὸν Γ, <lb n="1799,001,,,,9,17,17"></lb> |
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8178 μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ.</s> <s>ἀλλ' ὁ Α τὸν Γ ἐμέτρει· ὥστε <lb n="1799,001,,,,9,17,18"></lb> |
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8179 ὁ Α καὶ τὸν Δ μετρεῖ.</s> <s>μετρεῖ δὲ καὶ ἑαυτόν.</s> <s>ὁ Α ἄρα τοὺς <lb n="1799,001,,,,9,17,19"></lb> |
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8180 Α, Δ μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν <lb n="1799,001,,,,9,17,20"></lb> |
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8181 ἀδύνατον.</s> <s>οὐκ ἄρα ἔσται ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ <lb n="1799,001,,,,9,17,21"></lb> |
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8182 πρὸς ἄλλον τινά· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,17,22"></lb> |
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8183 <s>Δύο ἀριθμῶν δοθέντων ἐπισκέψασθαι, εἰ δυνατόν ἐστιν <lb n="1799,001,,,,9,18,1"></lb> |
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8184 αὐτοῖς τρίτον ἀνάλογον προσευρεῖν.</s> <lb n="1799,001,,,,9,18,2"></lb> |
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8185 <s>Ἔστωσαν οἱ δοθέντες δύο ἀριθμοὶ οἱ Α, Β, καὶ δέον <lb n="1799,001,,,,9,18,3"></lb> |
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8186 ἔστω ἐπισκέψασθαι, εἰ δυνατόν ἐστιν αὐτοῖς τρίτον <lb n="1799,001,,,,9,18,4"></lb> |
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8187 ἀνάλογον προσευρεῖν. <pb></pb> <lb n="1799,001,,,,9,18,5"></lb> |
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8188 Οἱ δὴ Α, Β ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ.</s> <lb n="1799,001,,,,9,18,6"></lb> |
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8189 <s>καὶ εἰ πρῶτοι πρὸς ἀλλήλους εἰσίν, δέδεικται, ὅτι ἀδύνατόν <lb n="1799,001,,,,9,18,7"></lb> |
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8190 ἐστιν αὐτοῖς τρίτον ἀνάλογον προσευρεῖν.</s> <lb n="1799,001,,,,9,18,8"></lb> |
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8191 <s>Ἀλλὰ δὴ μὴ ἔστωσαν οἱ Α, Β πρῶτοι πρὸς ἀλλήλους, <lb n="1799,001,,,,9,18,9"></lb> |
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8192 καὶ ὁ Β ἑαυτὸν πολλαπλασιάσας τὸν Γ ποιείτω· ὁ Α δὴ <lb n="1799,001,,,,9,18,10"></lb> |
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8193 τὸν Γ ἤτοι μετρεῖ ἢ οὐ μετρεῖ.</s> <s>μετρείτω πρότερον κατὰ <lb n="1799,001,,,,9,18,11"></lb> |
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8194 τὸν Δ· ὁ Α ἄρα τὸν Δ πολλαπλασιάσας τὸν Γ πεποίηκεν.</s> <lb n="1799,001,,,,9,18,12"></lb> |
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8195 <s>ἀλλὰ μὴν καὶ ὁ Β ἑαυτὸν πολλαπλασιάσας τὸν Γ πεποίηκεν· <lb n="1799,001,,,,9,18,13"></lb> |
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8196 ὁ ἄρα ἐκ τῶν Α, Δ ἴσος ἐστὶ τῷ ἀπὸ τοῦ Β.</s> <s>ἔστιν ἄρα ὡς <lb n="1799,001,,,,9,18,14"></lb> |
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8197 ὁ Α πρὸς τὸν Β, ὁ Β πρὸς τὸν Δ· τοῖς Α, Β ἄρα τρίτος <lb n="1799,001,,,,9,18,15"></lb> |
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8198 ἀριθμὸς ἀνάλογον προσηύρηται ὁ Δ.</s> <lb n="1799,001,,,,9,18,16"></lb> |
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8199 <s>Ἀλλὰ δὴ μὴ μετρείτω ὁ Α τὸν Γ· λέγω, ὅτι τοῖς <lb n="1799,001,,,,9,18,17"></lb> |
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8200 Α, Β ἀδύνατόν ἐστι τρίτον ἀνάλογον προσευρεῖν ἀριθμόν.</s> <lb n="1799,001,,,,9,18,18"></lb> |
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8201 <s>εἰ γὰρ δυνατόν, προσηυρήσθω ὁ Δ.</s> <s>ὁ ἄρα ἐκ τῶν Α, Δ <lb n="1799,001,,,,9,18,19"></lb> |
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8202 ἴσος ἐστὶ τῷ ἀπὸ τοῦ Β.</s> <s>ὁ δὲ ἀπὸ τοῦ Β ἐστιν ὁ Γ· ὁ ἄρα <lb n="1799,001,,,,9,18,20"></lb> |
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8203 ἐκ τῶν Α, Δ ἴσος ἐστὶ τῷ Γ.</s> <s>ὥστε ὁ Α τὸν Δ πολλαπλα-<lb n="1799,001,,,,9,18,21"></lb> |
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8204 σιάσας τὸν Γ πεποίηκεν· ὁ Α ἄρα τὸν Γ μετρεῖ κατὰ τὸν Δ.</s> <lb n="1799,001,,,,9,18,22"></lb> |
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8205 <s>ἀλλὰ μὴν ὑπόκειται καὶ μὴ μετρῶν· ὅπερ ἄτοπον.</s> <s>οὐκ <lb n="1799,001,,,,9,18,23"></lb> |
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8206 ἄρα δυνατόν ἐστι τοῖς Α, Β τρίτον ἀνάλογον προσευρεῖν <lb n="1799,001,,,,9,18,24"></lb> |
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8207 ἀριθμόν, ὅταν ὁ Α τὸν Γ μὴ μετρῇ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,18,25"></lb> |
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8208 <s>Τριῶν ἀριθμῶν δοθέντων ἐπισκέψασθαι, πότε δυνατόν <lb n="1799,001,,,,9,19,1"></lb> |
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8209 ἐστιν αὐτοῖς τέταρτον ἀνάλογον προσευρεῖν.</s> <lb n="1799,001,,,,9,19,2"></lb> |
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8210 <s>Ἔστωσαν οἱ δοθέντες τρεῖς ἀριθμοὶ οἱ Α, Β, Γ, καὶ <lb n="1799,001,,,,9,19,3"></lb> |
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8211 δέον ἔστω ἐπισκέψασθαι, πότε δυνατόν ἐστιν αὐτοῖς <lb n="1799,001,,,,9,19,4"></lb> |
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8212 τέταρτον ἀνάλογον προσευρεῖν.</s> <s><pb></pb> <lb n="1799,001,,,,9,19,5"></lb> |
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8213 Ἤτοι οὖν οὔκ εἰσιν ἑξῆς ἀνάλογον, καὶ οἱ ἄκροι αὐτῶν <lb n="1799,001,,,,9,19,6"></lb> |
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8214 πρῶτοι πρὸς ἀλλήλους εἰσίν, ἢ ἑξῆς εἰσιν ἀνάλογον, καὶ <lb n="1799,001,,,,9,19,7"></lb> |
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8215 οἱ ἄκροι αὐτῶν οὔκ εἰσι πρῶτοι πρὸς ἀλλήλους, ἢ οὔτε <lb n="1799,001,,,,9,19,8"></lb> |
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8216 ἑξῆς εἰσιν ἀνάλογον, οὔτε οἱ ἄκροι αὐτῶν πρῶτοι πρὸς <lb n="1799,001,,,,9,19,9"></lb> |
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8217 ἀλλήλους εἰσίν, ἢ καὶ ἑξῆς εἰσιν ἀνάλογον, καὶ οἱ ἄκροι <lb n="1799,001,,,,9,19,10"></lb> |
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8218 αὐτῶν πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <lb n="1799,001,,,,9,19,11"></lb> |
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8219 <s>Εἰ μὲν οὖν οἱ Α, Β, Γ ἑξῆς εἰσιν ἀνάλογον, καὶ οἱ ἄκροι <lb n="1799,001,,,,9,19,12"></lb> |
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8220 αὐτῶν οἱ Α, Γ πρῶτοι πρὸς ἀλλήλους εἰσίν, δέδεικται, <lb n="1799,001,,,,9,19,13"></lb> |
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8221 ὅτι ἀδύνατόν ἐστιν αὐτοῖς τέταρτον ἀνάλογον προσευρεῖν <lb n="1799,001,,,,9,19,14"></lb> |
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8222 ἀριθμόν.</s> <s>μὴ ἔστωσαν δὴ οἱ Α, Β, Γ ἑξῆς ἀνάλογον τῶν <lb n="1799,001,,,,9,19,15"></lb> |
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8223 ἄκρων πάλιν ὄντων πρώτων πρὸς ἀλλήλους.</s> <s>λέγω, ὅτι <lb n="1799,001,,,,9,19,16"></lb> |
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8224 καὶ οὕτως ἀδύνατόν ἐστιν αὐτοῖς τέταρτον ἀνάλογον <lb n="1799,001,,,,9,19,17"></lb> |
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8225 προσευρεῖν.</s> <s>εἰ γὰρ δυνατόν, προσευρήσθω ὁ Δ, ὥστε <lb n="1799,001,,,,9,19,18"></lb> |
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8226 εἶναι ὡς τὸν Α πρὸς τὸν Β, τὸν Γ πρὸς τὸν Δ, καὶ γεγονέτω <pb></pb> <lb n="1799,001,,,,9,19,19"></lb> |
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8227 ὡς ὁ Β πρὸς τὸν Γ, ὁ Δ πρὸς τὸν Ε.</s> <s>καὶ ἐπεί ἐστιν ὡς μὲν <lb n="1799,001,,,,9,19,20"></lb> |
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8228 ὁ Α πρὸς τὸν Β, ὁ Γ πρὸς τὸν Δ, ὡς δὲ ὁ Β πρὸς τὸν Γ, <lb n="1799,001,,,,9,19,21"></lb> |
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8229 ὁ Δ πρὸς τὸν Ε, δι' ἴσου ἄρα ὡς ὁ Α πρὸς τὸν Γ, ὁ Γ <lb n="1799,001,,,,9,19,22"></lb> |
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8230 πρὸς τὸν Ε.</s> <s>οἱ δὲ Α, Γ πρῶτοι, οἱ δὲ <lb n="1799,001,,,,9,19,23"></lb> |
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8231 πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι <lb n="1799,001,,,,9,19,24"></lb> |
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8232 μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχον-<lb n="1799,001,,,,9,19,25"></lb> |
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8233 τας ὅ τε ἡγούμενος τὸν ἡγούμενον <lb n="1799,001,,,,9,19,26"></lb> |
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8234 καὶ ὁ ἑπόμενος τὸν ἑπόμενον.</s> <s>μετρεῖ <lb n="1799,001,,,,9,19,27"></lb> |
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8235 ἄρα ὁ Α τὸν Γ ὡς ἡγούμενος ἡγού-<lb n="1799,001,,,,9,19,28"></lb> |
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8236 μενον.</s> <s>μετρεῖ δὲ καὶ ἑαυτόν· ὁ Α <lb n="1799,001,,,,9,19,29"></lb> |
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8237 ἄρα τοὺς Α, Γ μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ <lb n="1799,001,,,,9,19,30"></lb> |
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8238 ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τοῖς Α, Β, Γ δυνατόν ἐστι <lb n="1799,001,,,,9,19,31"></lb> |
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8239 τέταρτον ἀνάλογον προσευρεῖν.</s> <lb n="1799,001,,,,9,19,32"></lb> |
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8240 <s>Ἀλλὰ δὴ πάλιν ἔστωσαν οἱ Α, Β, Γ ἑξῆς ἀνάλογον, οἱ <lb n="1799,001,,,,9,19,33"></lb> |
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8241 δὲ Α, Γ μὴ ἔστωσαν πρῶτοι πρὸς ἀλλήλους.</s> <s>λέγω, ὅτι <lb n="1799,001,,,,9,19,34"></lb> |
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8242 δυνατόν ἐστιν αὐτοῖς τέταρτον ἀνάλογον προσευρεῖν.</s> <s>ὁ γὰρ <lb n="1799,001,,,,9,19,35"></lb> |
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8243 Β τὸν Γ πολλαπλασιάσας τὸν Δ ποιείτω· ὁ Α ἄρα τὸν Δ <lb n="1799,001,,,,9,19,36"></lb> |
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8244 ἤτοι μετρεῖ ἢ οὐ μετρεῖ.</s> <s>μετρείτω αὐτὸν πρότερον κατὰ <lb n="1799,001,,,,9,19,37"></lb> |
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8245 τὸν Ε· ὁ Α ἄρα τὸν Ε πολλαπλασιάσας τὸν Δ πεποίηκεν.</s> <lb n="1799,001,,,,9,19,38"></lb> |
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8246 <s>ἀλλὰ μὴν καὶ ὁ Β τὸν Γ πολλαπλασιάσας τὸν Δ πεποίηκεν· <lb n="1799,001,,,,9,19,39"></lb> |
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8247 ὁ ἄρα ἐκ τῶν Α, Ε ἴσος ἐστὶ τῷ ἐκ τῶν Β, Γ.</s> <s>ἀνάλογον <lb n="1799,001,,,,9,19,40"></lb> |
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8248 ἄρα [ἐστὶν] ὡς ὁ Α πρὸς τὸν Β, ὁ Γ πρὸς τὸν Ε· τοῖς <lb n="1799,001,,,,9,19,41"></lb> |
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8249 Α, Β, Γ ἄρα τέταρτος ἀνάλογον προσηύρηται ὁ Ε.</s> <lb n="1799,001,,,,9,19,42"></lb> |
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8250 <s>Ἀλλὰ δὴ μὴ μετρείτω ὁ Α τὸν Δ· λέγω, ὅτι ἀδύνατόν <lb n="1799,001,,,,9,19,43"></lb> |
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8251 ἐστι τοῖς Α, Β, Γ τέταρτον ἀνάλογον προσευρεῖν ἀριθμόν.</s> <lb n="1799,001,,,,9,19,44"></lb> |
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8252 <s>εἰ γὰρ δυνατόν, προσευρήσθω ὁ Ε· ὁ ἄρα ἐκ τῶν Α, Ε <lb n="1799,001,,,,9,19,45"></lb> |
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8253 ἴσος ἐστὶ τῷ ἐκ τῶν Β, Γ.</s> <s>ἀλλὰ ὁ ἐκ τῶν Β, Γ ἐστιν ὁ Δ· <lb n="1799,001,,,,9,19,46"></lb> |
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8254 καὶ ὁ ἐκ τῶν Α, Ε ἄρα ἴσος ἐστὶ τῷ Δ.</s> <s>ὁ Α ἄρα τὸν Ε πολ-<lb n="1799,001,,,,9,19,47"></lb> |
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8255 λαπλασιάσας τὸν Δ πεποίηκεν· ὁ Α ἄρα τὸν Δ μετρεῖ κατὰ <pb></pb> <lb n="1799,001,,,,9,19,48"></lb> |
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8256 τὸν Ε· ὥστε μετρεῖ ὁ Α τὸν Δ.</s> <s>ἀλλὰ καὶ οὐ μετρεῖ· ὅπερ <lb n="1799,001,,,,9,19,49"></lb> |
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8257 ἄτοπον.</s> <s>οὐκ ἄρα δυνατόν ἐστι τοῖς Α, Β, Γ τέταρτον <lb n="1799,001,,,,9,19,50"></lb> |
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8258 ἀνάλογον προσευρεῖν ἀριθμόν, ὅταν ὁ Α τὸν Δ μὴ μετρῇ.</s> <lb n="1799,001,,,,9,19,51"></lb> |
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8259 <s>ἀλλὰ δὴ οἱ Α, Β, Γ μήτε ἑξῆς ἔστωσαν ἀνάλογον μήτε <lb n="1799,001,,,,9,19,52"></lb> |
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8260 οἱ ἄκροι πρῶτοι πρὸς ἀλλήλους.</s> <s>καὶ ὁ Β τὸν Γ πολλαπλα-<lb n="1799,001,,,,9,19,53"></lb> |
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8261 σιάσας τὸν Δ ποιείτω.</s> <s>ὁμοίως δὴ δειχθήσεται, ὅτι εἰ μὲν <lb n="1799,001,,,,9,19,54"></lb> |
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8262 μετρεῖ ὁ Α τὸν Δ, δυνατόν ἐστιν αὐτοῖς ἀνάλογον προσευ-<lb n="1799,001,,,,9,19,55"></lb> |
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8263 ρεῖν, εἰ δὲ οὐ μετρεῖ, ἀδύνατον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,19,56"></lb> |
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8264 <s>Οἱ πρῶτοι ἀριθμοὶ πλείους εἰσὶ παντὸς τοῦ προτε-<lb n="1799,001,,,,9,20,1"></lb> |
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8265 θέντος πλήθους πρώτων ἀριθμῶν.</s> <lb n="1799,001,,,,9,20,2"></lb> |
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8266 <s>Ἔστωσαν οἱ προτεθέντες πρῶτοι ἀριθμοὶ οἱ Α, Β, Γ· <lb n="1799,001,,,,9,20,3"></lb> |
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8267 λέγω, ὅτι τῶν Α, Β, Γ πλείους εἰσὶ πρῶτοι ἀριθμοί. <lb n="1799,001,,,,9,20,4"></lb> |
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8268 Εἰλήφθω γὰρ ὁ ὑπὸ τῶν Α, Β, Γ ἐλάχιστος μετρού-<lb n="1799,001,,,,9,20,5"></lb> |
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8269 μενος καὶ ἔστω ὁ ΔΕ, καὶ προσκείσθω τῷ ΔΕ μονὰς ἡ <lb n="1799,001,,,,9,20,6"></lb> |
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8270 ΔΖ.</s> <s>ὁ δὴ ΕΖ ἤτοι πρῶτός ἐστιν ἢ οὔ.</s> <s>ἔστω πρότερον <lb n="1799,001,,,,9,20,7"></lb> |
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8271 πρῶτος· εὑρημένοι ἄρα εἰσὶ πρῶτοι ἀριθμοὶ οἱ Α, Β, Γ, <lb n="1799,001,,,,9,20,8"></lb> |
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8272 ΕΖ πλείους τῶν Α, Β, Γ.</s> <s><pb></pb> <lb n="1799,001,,,,9,20,9"></lb> |
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8273 Ἀλλὰ δὴ μὴ ἔστω ὁ ΕΖ πρῶτος· ὑπὸ πρώτου ἄρα <lb n="1799,001,,,,9,20,10"></lb> |
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8274 τινὸς ἀριθμοῦ μετρεῖται.</s> <s>μετρείσθω ὑπὸ πρώτου τοῦ Η· <lb n="1799,001,,,,9,20,11"></lb> |
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8275 λέγω, ὅτι ὁ Η οὐδενὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός.</s> <s>εἰ γὰρ <lb n="1799,001,,,,9,20,12"></lb> |
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8276 δυνατόν, ἔστω.</s> <s>οἱ δὲ Α, Β, Γ τὸν ΔΕ μετροῦσιν· καὶ ὁ Η <lb n="1799,001,,,,9,20,13"></lb> |
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8277 ἄρα τὸν ΔΕ μετρήσει.</s> <s>μετρεῖ δὲ καὶ τὸν ΕΖ· καὶ λοιπὴν <lb n="1799,001,,,,9,20,14"></lb> |
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8278 τὴν ΔΖ μονάδα μετρήσει ὁ Η ἀριθμὸς ὤν· ὅπερ ἄτοπον.</s> <lb n="1799,001,,,,9,20,15"></lb> |
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8279 <s>οὐκ ἄρα ὁ Η ἑνὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός.</s> <s>καὶ ὑπόκειται <lb n="1799,001,,,,9,20,16"></lb> |
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8280 πρῶτος.</s> <s>εὑρημένοι ἄρα εἰσὶ πρῶτοι ἀριθμοὶ πλείους τοῦ <lb n="1799,001,,,,9,20,17"></lb> |
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8281 προτεθέντος πλήθους τῶν Α, Β, Γ οἱ Α, Β, Γ, Η· ὅπερ <lb n="1799,001,,,,9,20,18"></lb> |
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8282 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,20,19"></lb> |
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8283 <s>Ἐὰν ἄρτιοι ἀριθμοὶ ὁποσοιοῦν συντεθῶσιν, ὁ ὅλος <lb n="1799,001,,,,9,21,1"></lb> |
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8284 ἄρτιός ἐστιν.</s> <lb n="1799,001,,,,9,21,2"></lb> |
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8285 <s>Συγκείσθωσαν γὰρ ἄρτιοι ἀριθμοὶ ὁποσοιοῦν οἱ ΑΒ, <lb n="1799,001,,,,9,21,3"></lb> |
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8286 ΒΓ, ΓΔ, ΔΕ· λέγω, ὅτι ὅλος ὁ ΑΕ ἄρτιός ἐστιν. <lb n="1799,001,,,,9,21,4"></lb> |
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8287 Ἐπεὶ γὰρ ἕκαστος τῶν ΑΒ, ΒΓ, ΓΔ, ΔΕ ἄρτιός ἐστιν, <lb n="1799,001,,,,9,21,5"></lb> |
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8288 ἔχει μέρος ἥμισυ· ὥστε καὶ ὅλος ὁ ΑΕ ἔχει μέρος ἥμισυ.</s> <lb n="1799,001,,,,9,21,6"></lb> |
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8289 <s>ἄρτιος δὲ ἀριθμός ἐστιν ὁ δίχα διαιρούμενος· ἄρτιος ἄρα <lb n="1799,001,,,,9,21,7"></lb> |
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8290 ἐστὶν ὁ ΑΕ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,9,21,8"></lb> |
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8291 Ἐὰν περισσοὶ ἀριθμοὶ ὁποσοιοῦν συντεθῶσιν, τὸ δὲ <lb n="1799,001,,,,9,22,1"></lb> |
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8292 πλῆθος αὐτῶν ἄρτιον ᾖ, ὁ ὅλος ἄρτιος ἔσται.</s> <lb n="1799,001,,,,9,22,2"></lb> |
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8293 <s>Συγκείσθωσαν γὰρ περισσοὶ ἀριθμοὶ ὁσοιδηποτοῦν <lb n="1799,001,,,,9,22,3"></lb> |
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8294 ἄρτιοι τὸ πλῆθος οἱ ΑΒ, ΒΓ, ΓΔ, ΔΕ· λέγω, ὅτι ὅλος ὁ <lb n="1799,001,,,,9,22,4"></lb> |
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8295 ΑΕ ἄρτιός ἐστιν. <lb n="1799,001,,,,9,22,5"></lb> |
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8296 Ἐπεὶ γὰρ ἕκαστος τῶν ΑΒ, ΒΓ, ΓΔ, ΔΕ περιττός <lb n="1799,001,,,,9,22,6"></lb> |
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8297 ἐστιν, ἀφαιρεθείσης μονάδος ἀφ' ἑκάστου ἕκαστος τῶν <lb n="1799,001,,,,9,22,7"></lb> |
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8298 λοιπῶν ἄρτιος ἔσται· ὥστε καὶ ὁ συγκείμενος ἐξ αὐτῶν <lb n="1799,001,,,,9,22,8"></lb> |
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8299 ἄρτιος ἔσται.</s> <s>ἔστι δὲ καὶ τὸ πλῆθος τῶν μονάδων ἄρτιον.</s> <lb n="1799,001,,,,9,22,9"></lb> |
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8300 <s>καὶ ὅλος ἄρα ὁ ΑΕ ἄρτιός ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,22,10"></lb> |
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8301 <s>Ἐὰν περισσοὶ ἀριθμοὶ ὁποσοιοῦν συντεθῶσιν, τὸ δὲ <lb n="1799,001,,,,9,23,1"></lb> |
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8302 πλῆθος αὐτῶν περισσὸν ᾖ, καὶ ὁ ὅλος περισσὸς ἔσται.</s> <lb n="1799,001,,,,9,23,2"></lb> |
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8303 <s>Συγκείσθωσαν γὰρ ὁποσοιοῦν περισσοὶ ἀριθμοί, ὧν <lb n="1799,001,,,,9,23,3"></lb> |
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8304 τὸ πλῆθος περισσὸν ἔστω, οἱ *αΒ, ΒΓ, ΓΔ· λέγω, ὅτι <lb n="1799,001,,,,9,23,4"></lb> |
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8305 καὶ ὅλος ὁ ΑΔ περισσός ἐστιν. <pb></pb> <lb n="1799,001,,,,9,23,5"></lb> |
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8306 Ἀφῃρήσθω ἀπὸ τοῦ ΓΔ μονὰς ἡ ΔΕ· λοιπὸς ἄρα ὁ ΓΕ <lb n="1799,001,,,,9,23,6"></lb> |
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8307 ἄρτιός ἐστιν.</s> <s>ἔστι δὲ καὶ ὁ ΓΑ ἄρτιος· καὶ ὅλος ἄρα ὁ <lb n="1799,001,,,,9,23,7"></lb> |
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8308 ΑΕ ἄρτιός ἐστιν.</s> <s>καί ἐστι μονὰς ἡ ΔΕ.</s> <s>περισσὸς ἄρα <lb n="1799,001,,,,9,23,8"></lb> |
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8309 ἐστὶν ὁ ΑΔ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,23,9"></lb> |
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8310 <s>Ἐὰν ἀπὸ ἀρτίου ἀριθμοῦ ἄρτιος ἀφαιρεθῇ, ὁ λοιπὸς <lb n="1799,001,,,,9,24,1"></lb> |
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8311 ἄρτιος ἔσται.</s> <lb n="1799,001,,,,9,24,2"></lb> |
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8312 <s>Ἀπὸ γὰρ ἀρτίου τοῦ ΑΒ ἄρτιος ἀφῃρήσθω ὁ ΒΓ· λέγω, <lb n="1799,001,,,,9,24,3"></lb> |
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8313 ὅτι ὁ λοιπὸς ὁ ΓΑ ἄρτιός ἐστιν. <lb n="1799,001,,,,9,24,4"></lb> |
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8314 Ἐπεὶ γὰρ ὁ ΑΒ ἄρτιός ἐστιν, ἔχει μέρος ἥμισυ.</s> <s>διὰ <lb n="1799,001,,,,9,24,5"></lb> |
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8315 τὰ αὐτὰ δὴ καὶ ὁ ΒΓ ἔχει μέρος ἥμισυ· ὥστε καὶ λοιπὸς <lb n="1799,001,,,,9,24,6"></lb> |
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8316 [ὁ ΓΑ ἔχει μέρος ἥμισυ] ἄρτιος [ἄρα] ἐστὶν ὁ ΑΓ· ὅπερ <lb n="1799,001,,,,9,24,7"></lb> |
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8317 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,24,8"></lb> |
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8318 <s>Ἐὰν ἀπὸ ἀρτίου ἀριθμοῦ περισσὸς ἀφαιρεθῇ, ὁ λοιπὸς <lb n="1799,001,,,,9,25,1"></lb> |
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8319 περισσὸς ἔσται.</s> <lb n="1799,001,,,,9,25,2"></lb> |
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8320 <s>Ἀπὸ γὰρ ἀρτίου τοῦ ΑΒ περισσὸς ἀφῃρήσθω ὁ ΒΓ· <lb n="1799,001,,,,9,25,3"></lb> |
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8321 λέγω, ὅτι ὁ λοιπὸς ὁ ΓΑ περισσός ἐστιν. <pb></pb> <lb n="1799,001,,,,9,25,4"></lb> |
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8322 Ἀφῃρήσθω γὰρ ἀπὸ τοῦ ΒΓ μονὰς ἡ ΓΔ· ὁ ΔΒ ἄρα <lb n="1799,001,,,,9,25,5"></lb> |
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8323 ἄρτιός ἐστιν.</s> <s>ἔστι δὲ καὶ ὁ ΑΒ ἄρτιος· καὶ λοιπὸς ἄρα <lb n="1799,001,,,,9,25,6"></lb> |
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8324 ὁ ΑΔ ἄρτιός ἐστιν.</s> <s>καί ἐστι μονὰς ἡ ΓΔ· ὁ ΓΑ ἄρα <lb n="1799,001,,,,9,25,7"></lb> |
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8325 περισσός ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,25,8"></lb> |
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8326 <s>Ἐὰν ἀπὸ περισσοῦ ἀριθμοῦ περισσὸς ἀφαιρεθῇ, ὁ <lb n="1799,001,,,,9,26,1"></lb> |
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8327 λοιπὸς ἄρτιος ἔσται.</s> <lb n="1799,001,,,,9,26,2"></lb> |
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8328 <s>Ἀπὸ γὰρ περισσοῦ τοῦ ΑΒ περισσὸς ἀφῃρήσθω ὁ ΒΓ· <lb n="1799,001,,,,9,26,3"></lb> |
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8329 λέγω, ὅτι ὁ λοιπὸς ὁ ΓΑ ἄρτιός ἐστιν. <lb n="1799,001,,,,9,26,4"></lb> |
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8330 Ἐπεὶ γὰρ ὁ ΑΒ περισσός ἐστιν, ἀφῃρήσθω μονὰς ἡ <lb n="1799,001,,,,9,26,5"></lb> |
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8331 ΒΔ· λοιπὸς ἄρα ὁ ΑΔ ἄρτιός ἐστιν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,9,26,6"></lb> |
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8332 ὁ ΓΔ ἄρτιός ἐστιν· ὥστε καὶ λοιπὸς ὁ ΓΑ ἄρτιός ἐστιν· <lb n="1799,001,,,,9,26,7"></lb> |
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8333 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,26,8"></lb> |
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8334 <s>Ἐὰν ἀπὸ περισσοῦ ἀριθμοῦ ἄρτιος ἀφαιρεθῇ, ὁ λοιπὸς <lb n="1799,001,,,,9,27,1"></lb> |
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8335 περισσὸς ἔσται.</s> <lb n="1799,001,,,,9,27,2"></lb> |
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8336 <s>Ἀπὸ γὰρ περισσοῦ τοῦ ΑΒ ἄρτιος ἀφῃρήσθω ὁ ΒΓ· <lb n="1799,001,,,,9,27,3"></lb> |
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8337 λέγω, ὅτι ὁ λοιπὸς ὁ ΓΑ περισσός ἐστιν. <pb></pb> <lb n="1799,001,,,,9,27,4"></lb> |
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8338 Ἀφῃρήσθω [γὰρ] μονὰς ἡ ΑΔ· ὁ ΔΒ ἄρα ἄρτιός ἐστιν.</s> <lb n="1799,001,,,,9,27,5"></lb> |
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8339 <s>ἔστι δὲ καὶ ὁ ΒΓ ἄρτιος· καὶ λοιπὸς ἄρα ὁ ΓΔ ἄρτιός <lb n="1799,001,,,,9,27,6"></lb> |
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8340 ἐστιν.</s> <s>περισσὸς ἄρα ὁ ΓΑ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,27,7"></lb> |
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8341 <s>Ἐὰν περισσὸς ἀριθμὸς ἄρτιον πολλαπλασιάσας ποιῇ <lb n="1799,001,,,,9,28,1"></lb> |
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8342 τινα, ὁ γενόμενος ἄρτιος ἔσται.</s> <lb n="1799,001,,,,9,28,2"></lb> |
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8343 <s>Περισσὸς γὰρ ἀριθμὸς ὁ Α ἄρτιον τὸν Β πολλαπλα-<lb n="1799,001,,,,9,28,3"></lb> |
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8344 σιάσας τὸν Γ ποιείτω· λέγω, ὅτι ὁ Γ ἄρτιός ἐστιν.</s> <lb n="1799,001,,,,9,28,4"></lb> |
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8345 <s>Ἐπεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας <lb n="1799,001,,,,9,28,5"></lb> |
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8346 τὸν Γ πεποίηκεν, ὁ Γ ἄρα σύγκειται ἐκ <lb n="1799,001,,,,9,28,6"></lb> |
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8347 τοσούτων ἴσων τῷ Β, ὅσαι εἰσὶν ἐν τῷ Α <lb n="1799,001,,,,9,28,7"></lb> |
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8348 μονάδες.</s> <s>καί ἐστιν ὁ Β ἄρτιος· ὁ Γ ἄρα <lb n="1799,001,,,,9,28,8"></lb> |
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8349 σύγκειται ἐξ ἀρτίων.</s> <s>ἐὰν δὲ ἄρτιοι ἀριθμοὶ <lb n="1799,001,,,,9,28,9"></lb> |
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8350 ὁποσοιοῦν συντεθῶσιν, ὁ ὅλος ἄρτιός ἐστιν.</s> <s>ἄρτιος ἄρα <lb n="1799,001,,,,9,28,10"></lb> |
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8351 ἐστὶν ὁ Γ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,28,11"></lb> |
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8352 <s>Ἐὰν περισσὸς ἀριθμὸς περισσὸν ἀριθμὸν πολλαπλα-<lb n="1799,001,,,,9,29,1"></lb> |
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8353 σιάσας ποιῇ τινα, ὁ γενόμενος περισσὸς ἔσται.</s> <lb n="1799,001,,,,9,29,2"></lb> |
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8354 <s>Περισσὸς γὰρ ἀριθμὸς ὁ Α <lb n="1799,001,,,,9,29,3"></lb> |
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8355 περισσὸν τὸν Β πολλαπλασιάσας <lb n="1799,001,,,,9,29,4"></lb> |
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8356 τὸν Γ ποιείτω· λέγω, ὅτι ὁ Γ περισ-<lb n="1799,001,,,,9,29,5"></lb> |
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8357 σός ἐστιν.</s> <s><pb></pb> <lb n="1799,001,,,,9,29,6"></lb> |
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8358 Ἐπεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν, <lb n="1799,001,,,,9,29,7"></lb> |
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8359 ὁ Γ ἄρα σύγκειται ἐκ τοσούτων ἴσων τῷ Β, ὅσαι εἰσὶν <lb n="1799,001,,,,9,29,8"></lb> |
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8360 ἐν τῷ Α μονάδες.</s> <s>καί ἐστιν ἑκάτερος τῶν Α, Β περισσός· <lb n="1799,001,,,,9,29,9"></lb> |
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8361 ὁ Γ ἄρα σύγκειται ἐκ περισσῶν ἀριθμῶν, ὧν τὸ πλῆθος <lb n="1799,001,,,,9,29,10"></lb> |
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8362 περισσόν ἐστιν.</s> <s>ὥστε ὁ Γ περισσός ἐστιν· ὅπερ ἔδει <lb n="1799,001,,,,9,29,11"></lb> |
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8363 δεῖξαι.</s> <lb n="1799,001,,,,9,29,12"></lb> |
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8364 <s>Ἐὰν περισσὸς ἀριθμὸς ἄρτιον ἀριθμὸν μετρῇ, καὶ τὸν <lb n="1799,001,,,,9,30,1"></lb> |
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8365 ἥμισυν αὐτοῦ μετρήσει.</s> <lb n="1799,001,,,,9,30,2"></lb> |
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8366 <s>Περισσὸς γὰρ ἀριθμὸς ὁ Α ἄρτιον τὸν Β μετρείτω· <lb n="1799,001,,,,9,30,3"></lb> |
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8367 λέγω, ὅτι καὶ τὸν ἥμισυν αὐτοῦ μετρήσει.</s> <lb n="1799,001,,,,9,30,4"></lb> |
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8368 <s>Ἐπεὶ γὰρ ὁ Α τὸν Β μετρεῖ, μετρείτω αὐτὸν κατὰ τὸν Γ· <lb n="1799,001,,,,9,30,5"></lb> |
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8369 λέγω, ὅτι ὁ Γ οὐκ ἔστι περισσός.</s> <s>εἰ γὰρ δυνατόν, ἔστω.</s> <s>καὶ <lb n="1799,001,,,,9,30,6"></lb> |
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8370 ἐπεὶ ὁ Α τὸν Β μετρεῖ κατὰ τὸν Γ, ὁ Α ἄρα τὸν Γ πολλα-<lb n="1799,001,,,,9,30,7"></lb> |
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8371 πλασιάσας τὸν Β πεποίηκεν.</s> <s>ὁ Β ἄρα σύγκει- <lb n="1799,001,,,,9,30,8"></lb> |
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8372 ται ἐκ περισσῶν ἀριθμῶν, ὧν τὸ πλῆθος περισ-<lb n="1799,001,,,,9,30,9"></lb> |
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8373 σόν ἐστιν.</s> <s>ὁ Β ἄρα περισσός ἐστιν· ὅπερ <lb n="1799,001,,,,9,30,10"></lb> |
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8374 ἄτοπον· ὑπόκειται γὰρ ἄρτιος.</s> <s>οὐκ ἄρα ὁ Γ <lb n="1799,001,,,,9,30,11"></lb> |
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8375 περισσός ἐστιν· ἄρτιος ἄρα ἐστὶν ὁ Γ.</s> <s>ὥστε ὁ <lb n="1799,001,,,,9,30,12"></lb> |
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8376 Α τὸν Β μετρεῖ ἀρτιάκις.</s> <s>διὰ δὴ τοῦτο καὶ <lb n="1799,001,,,,9,30,13"></lb> |
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8377 τὸν ἥμισυν αὐτοῦ μετρήσει· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,30,14"></lb> |
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8378 <s>Ἐὰν περισσὸς ἀριθμὸς πρός τινα ἀριθμὸν πρῶτος ᾖ, καὶ <lb n="1799,001,,,,9,31,1"></lb> |
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8379 πρὸς τὸν διπλασίονα αὐτοῦ πρῶτος ἔσται.</s> <s><pb></pb> <lb n="1799,001,,,,9,31,2"></lb> |
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8380 Περισσὸς γὰρ ἀριθμὸς ὁ Α πρός τινα ἀριθμὸν τὸν Β <lb n="1799,001,,,,9,31,3"></lb> |
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8381 πρῶτος ἔστω, τοῦ δὲ Β διπλασίων ἔστω ὁ Γ· λέγω, ὅτι <lb n="1799,001,,,,9,31,4"></lb> |
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8382 ὁ Α [καὶ] πρὸς τὸν Γ πρῶτός ἐστιν.</s> <lb n="1799,001,,,,9,31,5"></lb> |
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8383 <s>Εἰ γὰρ μή εἰσιν [οἱ Α, Γ] πρῶτοι, <lb n="1799,001,,,,9,31,6"></lb> |
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8384 μετρήσει τις αὐτοὺς ἀριθμός.</s> <s>μετρείτω, <lb n="1799,001,,,,9,31,7"></lb> |
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8385 καὶ ἔστω ὁ Δ.</s> <s>καί ἐστιν ὁ Α περισσός· <lb n="1799,001,,,,9,31,8"></lb> |
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8386 περισσὸς ἄρα καὶ ὁ Δ.</s> <s>καὶ ἐπεὶ ὁ Δ <lb n="1799,001,,,,9,31,9"></lb> |
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8387 περισσὸς ὢν τὸν Γ μετρεῖ, καί ἐστιν <lb n="1799,001,,,,9,31,10"></lb> |
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8388 ὁ Γ ἄρτιος, καὶ τὸν ἥμισυν ἄρα τοῦ Γ <lb n="1799,001,,,,9,31,11"></lb> |
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8389 μετρήσει [ὁ Δ].</s> <s>τοῦ δὲ Γ ἥμισύ ἐστιν ὁ Β· ὁ Δ ἄρα τὸν Β <lb n="1799,001,,,,9,31,12"></lb> |
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8390 μετρεῖ.</s> <s>μετρεῖ δὲ καὶ τὸν Α.</s> <s>ὁ Δ ἄρα τοὺς Α, Β μετρεῖ <lb n="1799,001,,,,9,31,13"></lb> |
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8391 πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ <lb n="1799,001,,,,9,31,14"></lb> |
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8392 ἄρα ὁ Α πρὸς τὸν Γ πρῶτος οὔκ ἐστιν.</s> <s>οἱ Α, Γ ἄρα πρῶτοι <lb n="1799,001,,,,9,31,15"></lb> |
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8393 πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,31,16"></lb> |
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8394 <s>Τῶν ἀπὸ δυάδος διπλασιαζομένων ἀριθμῶν ἕκαστος <lb n="1799,001,,,,9,32,1"></lb> |
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8395 ἀρτιάκις ἄρτιός ἐστι μόνον.</s> <lb n="1799,001,,,,9,32,2"></lb> |
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8396 <s>Ἀπὸ γὰρ δυάδος τῆς Α δεδιπλασιάσθωσαν ὁσοιδηποτοῦν <lb n="1799,001,,,,9,32,3"></lb> |
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8397 ἀριθμοὶ οἱ Β, Γ, Δ· λέγω, ὅτι οἱ Β, Γ, Δ ἀρτιάκις ἄρτιοί <lb n="1799,001,,,,9,32,4"></lb> |
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8398 εἰσι μόνον. <pb></pb> <lb n="1799,001,,,,9,32,5"></lb> |
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8399 Ὅτι μὲν οὖν ἕκαστος [τῶν Β, Γ, Δ] ἀρτιάκις ἄρτιός <lb n="1799,001,,,,9,32,6"></lb> |
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8400 ἐστιν, φανερόν· ἀπὸ γὰρ δυάδος ἐστὶ διπλασιασθείς.</s> <lb n="1799,001,,,,9,32,7"></lb> |
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8401 <s>λέγω, ὅτι καὶ μόνον.</s> <s>ἐκκείσθω γὰρ μονάς.</s> <s>ἐπεὶ οὖν ἀπὸ <lb n="1799,001,,,,9,32,8"></lb> |
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8402 μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογόν εἰσιν, ὁ δὲ μετὰ <lb n="1799,001,,,,9,32,9"></lb> |
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8403 τὴν μονάδα ὁ Α πρῶτός ἐστιν, ὁ μέγιστος τῶν Α, Β, Γ, Δ <lb n="1799,001,,,,9,32,10"></lb> |
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8404 ὁ Δ ὑπ' οὐδενὸς ἄλλου μετρηθήσεται παρὲξ τῶν Α, Β, Γ.</s> <lb n="1799,001,,,,9,32,11"></lb> |
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8405 <s>καί ἐστιν ἕκαστος τῶν Α, Β, Γ ἄρτιος· ὁ Δ ἄρα ἀρτιάκις <lb n="1799,001,,,,9,32,12"></lb> |
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8406 ἄρτιός ἐστι μόνον.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι [καὶ] ἑκάτερος <lb n="1799,001,,,,9,32,13"></lb> |
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8407 τῶν Β, Γ ἀρτιάκις ἄρτιός ἐστι μόνον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,32,14"></lb> |
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8408 <s>Ἐὰν ἀριθμὸς τὸν ἥμισυν ἔχῃ περισσόν, ἀρτιάκις περισ-<lb n="1799,001,,,,9,33,1"></lb> |
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8409 σός ἐστι μόνον.</s> <lb n="1799,001,,,,9,33,2"></lb> |
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8410 <s>Ἀριθμὸς γὰρ ὁ Α τὸν ἥμισυν ἐχέτω περισσόν· λέγω, <lb n="1799,001,,,,9,33,3"></lb> |
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8411 ὅτι ὁ Α ἀρτιάκις περισσός ἐστι μόνον. <lb n="1799,001,,,,9,33,4"></lb> |
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8412 Ὅτι μὲν οὖν ἀρτιάκις περισσός ἐστιν, φανερόν· ὁ γὰρ <lb n="1799,001,,,,9,33,5"></lb> |
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8413 ἥμισυς αὐτοῦ περισσὸς ὢν μετρεῖ αὐτὸν ἀρτιάκις.</s> <s>λέγω <lb n="1799,001,,,,9,33,6"></lb> |
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8414 δή, ὅτι καὶ μόνον.</s> <s>εἰ γὰρ ἔσται ὁ Α καὶ ἀρτιάκις ἄρτιος, <lb n="1799,001,,,,9,33,7"></lb> |
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8415 μετρηθήσεται ὑπὸ ἀρτίου κατὰ ἄρτιον ἀριθμόν· ὥστε <lb n="1799,001,,,,9,33,8"></lb> |
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8416 καὶ ὁ ἥμισυς αὐτοῦ μετρηθήσεται ὑπὸ ἀρτίου ἀριθμοῦ <lb n="1799,001,,,,9,33,9"></lb> |
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8417 περισσὸς ὤν· ὅπερ ἐστὶν ἄτοπον.</s> <s>ὁ Α ἄρα ἀρτιάκις <lb n="1799,001,,,,9,33,10"></lb> |
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8418 περισσός ἐστι μόνον· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,9,33,11"></lb> |
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8419 Ἐὰν ἀριθμὸς μήτε τῶν ἀπὸ δυάδος διπλασιαζομένων <lb n="1799,001,,,,9,34,1"></lb> |
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8420 ᾖ μήτε τὸν ἥμισυν ἔχῃ περισσόν, ἀρτιάκις τε ἄρτιός ἐστι <lb n="1799,001,,,,9,34,2"></lb> |
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8421 καὶ ἀρτιάκις περισσός.</s> <lb n="1799,001,,,,9,34,3"></lb> |
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8422 <s>Ἀριθμὸς γὰρ ὁ Α μήτε τῶν ἀπὸ δυάδος διπλασια-<lb n="1799,001,,,,9,34,4"></lb> |
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8423 ζομένων ἔστω μήτε τὸν ἥμισυν ἐχέτω περισσόν· λέγω, <lb n="1799,001,,,,9,34,5"></lb> |
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8424 ὅτι ὁ Α ἀρτιάκις τέ ἐστιν ἄρτιος καὶ ἀρτιάκις περισσός. <lb n="1799,001,,,,9,34,6"></lb> |
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8425 Ὅτι μὲν οὖν ὁ Α ἀρτιάκις ἐστὶν ἄρτιος, φανερόν· τὸν <lb n="1799,001,,,,9,34,7"></lb> |
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8426 γὰρ ἥμισυν οὐκ ἔχει περισσόν.</s> <s>λέγω δή, ὅτι καὶ ἀρτιάκις <lb n="1799,001,,,,9,34,8"></lb> |
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8427 περισσός ἐστιν.</s> <s>ἐὰν γὰρ τὸν Α τέμνωμεν δίχα καὶ τὸν <lb n="1799,001,,,,9,34,9"></lb> |
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8428 ἥμισυν αὐτοῦ δίχα καὶ τοῦτο ἀεὶ ποιῶμεν, καταντήσομεν <lb n="1799,001,,,,9,34,10"></lb> |
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8429 εἴς τινα ἀριθμὸν περισσόν, ὃς μετρήσει τὸν Α κατὰ <lb n="1799,001,,,,9,34,11"></lb> |
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8430 ἄρτιον ἀριθμόν.</s> <s>εἰ γὰρ οὔ, καταντήσομεν εἰς δυάδα, καὶ <lb n="1799,001,,,,9,34,12"></lb> |
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8431 ἔσται ὁ Α τῶν ἀπὸ δυάδος διπλασιαζομένων· ὅπερ οὐχ <lb n="1799,001,,,,9,34,13"></lb> |
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8432 ὑπόκειται.</s> <s>ὥστε ὁ Α ἀρτιάκις περισσός ἐστιν.</s> <s>ἐδείχθη δὲ <lb n="1799,001,,,,9,34,14"></lb> |
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8433 καὶ ἀρτιάκις ἄρτιος.</s> <s>ὁ Α ἄρα ἀρτιάκις τε ἄρτιός ἐστι καὶ <lb n="1799,001,,,,9,34,15"></lb> |
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8434 ἀρτιάκις περισσός· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,9,34,16"></lb> |
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8435 Ἐὰν ὦσιν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, <lb n="1799,001,,,,9,35,1"></lb> |
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8436 ἀφαιρεθῶσι δὲ ἀπό τε τοῦ δευτέρου καὶ τοῦ ἐσχάτου ἴσοι <lb n="1799,001,,,,9,35,2"></lb> |
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8437 τῷ πρώτῳ, ἔσται ὡς ἡ τοῦ δευτέρου ὑπεροχὴ πρὸς τὸν <lb n="1799,001,,,,9,35,3"></lb> |
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8438 πρῶτον, οὕτως ἡ τοῦ ἐσχάτου ὑπεροχὴ πρὸς τοὺς πρὸ <lb n="1799,001,,,,9,35,4"></lb> |
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8439 ἑαυτοῦ πάντας.</s> <lb n="1799,001,,,,9,35,5"></lb> |
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8440 <s>Ἔστωσαν ὁποσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ <lb n="1799,001,,,,9,35,6"></lb> |
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8441 Α, ΒΓ, Δ, ΕΖ ἀρχόμενοι ἀπὸ ἐλαχίστου τοῦ Α, καὶ <lb n="1799,001,,,,9,35,7"></lb> |
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8442 ἀφῃρήσθω ἀπὸ τοῦ ΒΓ καὶ τοῦ ΕΖ τῷ Α ἴσος ἑκάτερος <lb n="1799,001,,,,9,35,8"></lb> |
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8443 τῶν ΒΗ, ΖΘ· λέγω, ὅτι ἐστὶν ὡς ὁ ΗΓ πρὸς τὸν Α, <lb n="1799,001,,,,9,35,9"></lb> |
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8444 οὕτως ὁ ΕΘ πρὸς τοὺς Α, ΒΓ, Δ.</s> <lb n="1799,001,,,,9,35,10"></lb> |
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8445 <s>Κείσθω γὰρ τῷ μὲν ΒΓ ἴσος ὁ ΖΚ, τῷ δὲ Δ ἴσος ὁ <lb n="1799,001,,,,9,35,11"></lb> |
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8446 ΖΛ.</s> <s>καὶ ἐπεὶ ὁ ΖΚ τῷ ΒΓ ἴσος ἐστίν, ὧν ὁ ΖΘ τῷ ΒΗ <lb n="1799,001,,,,9,35,12"></lb> |
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8447 ἴσος ἐστίν, λοιπὸς ἄρα ὁ ΘΚ λοιπῷ τῷ ΗΓ ἐστιν ἴσος.</s> <lb n="1799,001,,,,9,35,13"></lb> |
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8448 <s>καὶ ἐπεί ἐστιν ὡς ὁ ΕΖ πρὸς τὸν Δ, οὕτως ὁ Δ πρὸς τὸν <lb n="1799,001,,,,9,35,14"></lb> |
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8449 ΒΓ καὶ ὁ ΒΓ πρὸς τὸν Α, ἴσος δὲ ὁ μὲν Δ τῷ ΖΛ, ὁ δὲ <lb n="1799,001,,,,9,35,15"></lb> |
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8450 ΒΓ τῷ ΖΚ, ὁ δὲ Α τῷ ΖΘ, ἔστιν ἄρα ὡς ὁ ΕΖ πρὸς <lb n="1799,001,,,,9,35,16"></lb> |
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8451 τὸν ΖΛ, οὕτως ὁ ΛΖ πρὸς τὸν ΖΚ καὶ ὁ ΖΚ πρὸς τὸν <pb></pb> <lb n="1799,001,,,,9,35,17"></lb> |
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8452 ΖΘ.</s> <s>διελόντι, ὡς ὁ ΕΛ πρὸς τὸν ΛΖ, οὕτως ὁ ΛΚ πρὸς <lb n="1799,001,,,,9,35,18"></lb> |
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8453 τὸν ΖΚ καὶ ὁ ΚΘ πρὸς τὸν ΖΘ.</s> <s>ἔστιν ἄρα καὶ ὡς εἷς <lb n="1799,001,,,,9,35,19"></lb> |
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8454 τῶν ἡγουμένων πρὸς ἕνα τῶν ἑπομένων, οὕτως ἅπαντες <lb n="1799,001,,,,9,35,20"></lb> |
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8455 οἱ ἡγούμενοι πρὸς ἅπαντας τοὺς ἑπομένους· ἔστιν ἄρα <lb n="1799,001,,,,9,35,21"></lb> |
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8456 ὡς ὁ ΚΘ πρὸς τὸν ΖΘ, οὕτως οἱ ΕΛ, ΛΚ, ΚΘ πρὸς τοὺς <lb n="1799,001,,,,9,35,22"></lb> |
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8457 ΛΖ, ΖΚ, ΘΖ.</s> <s>ἴσος δὲ ὁ μὲν ΚΘ τῷ ΓΗ, ὁ δὲ ΖΘ τῷ <lb n="1799,001,,,,9,35,23"></lb> |
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8458 Α, οἱ δὲ ΛΖ, ΖΚ, ΘΖ τοῖς Δ, ΒΓ, Α· ἔστιν ἄρα ὡς <lb n="1799,001,,,,9,35,24"></lb> |
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8459 ὁ ΓΗ πρὸς τὸν Α, οὕτως ὁ ΕΘ πρὸς τοὺς Δ, ΒΓ, Α.</s> <lb n="1799,001,,,,9,35,25"></lb> |
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8460 <s>ἔστιν ἄρα ὡς ἡ τοῦ δευτέρου ὑπεροχὴ πρὸς τὸν πρῶτον, <lb n="1799,001,,,,9,35,26"></lb> |
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8461 οὕτως ἡ τοῦ ἐσχάτου ὑπεροχὴ πρὸς τοὺς πρὸ ἑαυτοῦ <lb n="1799,001,,,,9,35,27"></lb> |
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8462 πάντας· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,9,35,28"></lb> |
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8463 <s>Ἐὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἐκτεθῶσιν <lb n="1799,001,,,,9,36,1"></lb> |
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8464 ἐν τῇ διπλασίονι ἀναλογίᾳ, ἕως οὗ ὁ σύμπας συντεθεὶς <lb n="1799,001,,,,9,36,2"></lb> |
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8465 πρῶτος γένηται, καὶ ὁ σύμπας ἐπὶ τὸν <lb n="1799,001,,,,9,36,3"></lb> |
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8466 ἔσχατον πολλαπλασιασθεὶς ποιῇ τινα, <lb n="1799,001,,,,9,36,4"></lb> |
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8467 ὁ γενόμενος τέλειος ἔσται.</s> <lb n="1799,001,,,,9,36,5"></lb> |
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8468 <s>Ἀπὸ γὰρ μονάδος ἐκκείσθωσαν <lb n="1799,001,,,,9,36,6"></lb> |
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8469 ὁσοιδηποτοῦν ἀριθμοὶ ἐν τῇ διπλα-<lb n="1799,001,,,,9,36,7"></lb> |
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8470 σίονι ἀναλογίᾳ, ἕως οὗ ὁ σύμπας συντεθεὶς πρῶτος <lb n="1799,001,,,,9,36,8"></lb> |
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8471 γένηται, οἱ Α, Β, Γ, Δ, καὶ τῷ σύμπαντι ἴσος ἔστω <lb n="1799,001,,,,9,36,9"></lb> |
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8472 ὁ Ε, καὶ ὁ Ε τὸν Δ πολλαπλασιάσας τὸν ΖΗ ποιείτω.</s> <lb n="1799,001,,,,9,36,10"></lb> |
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8473 <s>λέγω, ὅτι ὁ ΖΗ τέλειός ἐστιν.</s> <s><pb></pb> <lb n="1799,001,,,,9,36,11"></lb> |
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8474 Ὅσοι γάρ εἰσιν οἱ Α, Β, Γ, Δ τῷ πλήθει, τοσοῦτοι <lb n="1799,001,,,,9,36,12"></lb> |
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8475 ἀπὸ τοῦ Ε εἰλήφθωσαν ἐν τῇ διπλασίονι ἀναλογίᾳ οἱ <lb n="1799,001,,,,9,36,13"></lb> |
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8476 Ε, ΘΚ, Λ, Μ· δι' ἴσου ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Δ, <lb n="1799,001,,,,9,36,14"></lb> |
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8477 οὕτως ὁ Ε πρὸς τὸν Μ.</s> <s>ὁ ἄρα ἐκ τῶν Ε, Δ ἴσος ἐστὶ τῷ <lb n="1799,001,,,,9,36,15"></lb> |
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8478 ἐκ τῶν Α, Μ.</s> <s>καί ἐστιν ὁ ἐκ τῶν Ε, Δ ὁ ΖΗ· καὶ ὁ ἐκ <lb n="1799,001,,,,9,36,16"></lb> |
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8479 τῶν Α, Μ ἄρα ἐστὶν ὁ ΖΗ.</s> <s>ὁ Α ἄρα τὸν Μ πολλαπλα-<lb n="1799,001,,,,9,36,17"></lb> |
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8480 σιάσας τὸν ΖΗ πεποίηκεν· ὁ Μ ἄρα τὸν ΖΗ μετρεῖ κατὰ <lb n="1799,001,,,,9,36,18"></lb> |
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8481 τὰς ἐν τῷ Α μονάδας.</s> <s>καί ἐστι δυὰς ὁ Α· διπλάσιος ἄρα <lb n="1799,001,,,,9,36,19"></lb> |
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8482 ἐστὶν ὁ ΖΗ τοῦ Μ.</s> <s>εἰσὶ δὲ καὶ οἱ Μ, Λ, ΘΚ, Ε ἑξῆς <lb n="1799,001,,,,9,36,20"></lb> |
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8483 διπλάσιοι ἀλλήλων· οἱ Ε, ΘΚ, Λ, Μ, ΖΗ ἄρα ἑξῆς <lb n="1799,001,,,,9,36,21"></lb> |
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8484 ἀνάλογόν εἰσιν ἐν τῇ διπλασίονι ἀναλογίᾳ.</s> <s>ἀφῃρήσθω δὴ <lb n="1799,001,,,,9,36,22"></lb> |
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8485 ἀπὸ τοῦ δευτέρου τοῦ ΘΚ καὶ τοῦ ἐσχάτου τοῦ ΖΗ τῷ <lb n="1799,001,,,,9,36,23"></lb> |
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8486 πρώτῳ τῷ Ε ἴσος ἑκάτερος τῶν ΘΝ, ΖΞ· ἔστιν ἄρα ὡς <lb n="1799,001,,,,9,36,24"></lb> |
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8487 ἡ τοῦ δευτέρου ἀριθμοῦ ὑπεροχὴ πρὸς τὸν πρῶτον, οὕτως <lb n="1799,001,,,,9,36,25"></lb> |
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8488 ἡ τοῦ ἐσχάτου ὑπεροχὴ πρὸς τοὺς πρὸ ἑαυτοῦ πάντας.</s> <lb n="1799,001,,,,9,36,26"></lb> |
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8489 <s>ἔστιν ἄρα ὡς ὁ ΝΚ πρὸς τὸν Ε, οὕτως ὁ ΞΗ πρὸς τοὺς <lb n="1799,001,,,,9,36,27"></lb> |
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8490 Μ, Λ, ΚΘ, Ε.</s> <s>καί ἐστιν ὁ ΝΚ ἴσος τῷ Ε· καὶ ὁ ΞΗ <lb n="1799,001,,,,9,36,28"></lb> |
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8491 ἄρα ἴσος ἐστὶ τοῖς Μ, Λ, ΘΚ, Ε.</s> <s>ἔστι δὲ καὶ ὁ ΖΞ τῷ Ε <lb n="1799,001,,,,9,36,29"></lb> |
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8492 ἴσος, ὁ δὲ Ε τοῖς Α, Β, Γ, Δ καὶ τῇ μονάδι.</s> <s>ὅλος ἄρα ὁ <lb n="1799,001,,,,9,36,30"></lb> |
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8493 ΖΗ ἴσος ἐστὶ τοῖς τε Ε, ΘΚ, Λ, Μ καὶ τοῖς Α, Β, Γ, Δ <lb n="1799,001,,,,9,36,31"></lb> |
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8494 καὶ τῇ μονάδι· καὶ μετρεῖται ὑπ' αὐτῶν.</s> <s>λέγω, ὅτι καὶ ὁ <lb n="1799,001,,,,9,36,32"></lb> |
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8495 ΖΗ ὑπ' οὐδενὸς ἄλλου μετρηθήσεται παρὲξ τῶν Α, Β, <pb></pb> <lb n="1799,001,,,,9,36,33"></lb> |
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8496 Γ, Δ, Ε, ΘΚ, Λ, Μ καὶ τῆς μονάδος.</s> <s>εἰ γὰρ δυνατόν, <lb n="1799,001,,,,9,36,34"></lb> |
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8497 μετρείτω τις τὸν ΖΗ ὁ Ο, καὶ ὁ Ο μηδενὶ τῶν Α, Β, Γ, <lb n="1799,001,,,,9,36,35"></lb> |
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8498 Δ, Ε, ΘΚ, Λ, Μ ἔστω ὁ αὐτός.</s> <s>καὶ ὁσάκις ὁ Ο τὸν ΖΗ <lb n="1799,001,,,,9,36,36"></lb> |
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8499 μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Π· ὁ Π ἄρα τὸν <lb n="1799,001,,,,9,36,37"></lb> |
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8500 Ο πολλαπλασιάσας τὸν ΖΗ πεποίηκεν.</s> <s>ἀλλὰ μὴν καὶ ὁ Ε <lb n="1799,001,,,,9,36,38"></lb> |
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8501 τὸν Δ πολλαπλασιάσας τὸν ΖΗ πεποίηκεν· ἔστιν ἄρα ὡς ὁ <lb n="1799,001,,,,9,36,39"></lb> |
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8502 Ε πρὸς τὸν Π, ὁ Ο πρὸς τὸν Δ.</s> <s>καὶ ἐπεὶ ἀπὸ μονάδος ἑξῆς <lb n="1799,001,,,,9,36,40"></lb> |
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8503 ἀνάλογόν εἰσιν οἱ Α, Β, Γ, Δ, ὁ Δ ἄρα ὑπ' οὐδενὸς ἄλλου <lb n="1799,001,,,,9,36,41"></lb> |
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8504 ἀριθμοῦ μετρηθήσεται παρὲξ τῶν Α, Β, Γ.</s> <s>καὶ ὑπόκειται <lb n="1799,001,,,,9,36,42"></lb> |
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8505 ὁ Ο οὐδενὶ τῶν Α, Β, Γ ὁ αὐτός· οὐκ ἄρα μετρήσει ὁ Ο <lb n="1799,001,,,,9,36,43"></lb> |
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8506 τὸν Δ.</s> <s>ἀλλ' ὡς ὁ Ο πρὸς τὸν Δ, ὁ Ε πρὸς τὸν Π· οὐδὲ ὁ Ε <lb n="1799,001,,,,9,36,44"></lb> |
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8507 ἄρα τὸν Π μετρεῖ.</s> <s>καί ἐστιν ὁ Ε πρῶτος· πᾶς δὲ πρῶτος <lb n="1799,001,,,,9,36,45"></lb> |
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8508 ἀριθμὸς πρὸς ἅπαντα, ὃν μὴ μετρεῖ, πρῶτος [ἐστιν].</s> <lb n="1799,001,,,,9,36,46"></lb> |
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8509 <s>οἱ Ε, Π ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν.</s> <s>οἱ δὲ πρῶτοι καὶ <lb n="1799,001,,,,9,36,47"></lb> |
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8510 ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον <lb n="1799,001,,,,9,36,48"></lb> |
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8511 ἔχοντας ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ <lb n="1799,001,,,,9,36,49"></lb> |
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8512 ἑπόμενος τὸν ἑπόμενον· καί ἐστιν ὡς ὁ Ε πρὸς τὸν Π, <lb n="1799,001,,,,9,36,50"></lb> |
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8513 ὁ Ο πρὸς τὸν Δ· ἰσάκις ἄρα ὁ Ε τὸν Ο μετρεῖ καὶ ὁ Π τὸν <lb n="1799,001,,,,9,36,51"></lb> |
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8514 Δ· ἰσάκις ἄρα ὁ Ε τὸν Ο μετρεῖ καὶ ὁ Π τὸν Δ.</s> <s>ὁ δὲ Δ <lb n="1799,001,,,,9,36,52"></lb> |
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8515 ὑπ' οὐδενὸς ἄλλου μετρεῖται παρὲξ τῶν Α, Β, Γ· ὁ Π <lb n="1799,001,,,,9,36,53"></lb> |
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8516 ἄρα ἑνὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός.</s> <s>ἔστω τῷ Β ὁ αὐτός.</s> <lb n="1799,001,,,,9,36,54"></lb> |
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8517 <s>καὶ ὅσοι εἰσὶν οἱ Β, Γ, Δ τῷ πλήθει τοσοῦτοι εἰλήφθωσαν <lb n="1799,001,,,,9,36,55"></lb> |
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8518 ἀπὸ τοῦ Ε οἱ Ε, ΘΚ, Λ.</s> <s>καί εἰσιν οἱ Ε, ΘΚ, Λ τοῖς Β, <lb n="1799,001,,,,9,36,56"></lb> |
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8519 Γ, Δ ἐν τῷ αὐτῷ λόγῳ· δι' ἴσου ἄρα ἐστὶν ὡς ὁ Β πρὸς <pb></pb> <lb n="1799,001,,,,9,36,57"></lb> |
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8520 τὸν Δ, ὁ Ε πρὸς τὸν Λ.</s> <s>ὁ ἄρα ἐκ τῶν Β, Λ ἴσος ἐστὶ τῷ <lb n="1799,001,,,,9,36,58"></lb> |
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8521 ἐκ τῶν Δ, Ε· ἀλλ' ὁ ἐκ τῶν Δ, Ε ἴσος ἐστὶ τῷ ἐκ τῶν <lb n="1799,001,,,,9,36,59"></lb> |
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8522 Π, Ο· καὶ ὁ ἐκ τῶν Π, Ο ἄρα ἴσος ἐστὶ τῷ ἐκ τῶν Β, Λ.</s> <lb n="1799,001,,,,9,36,60"></lb> |
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8523 <s>ἔστιν ἄρα ὡς ὁ Π πρὸς τὸν Β, ὁ Λ πρὸς τὸν Ο.</s> <s>καί ἐστιν ὁ Π <lb n="1799,001,,,,9,36,61"></lb> |
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8524 τῷ Β ὁ αὐτός· καὶ ὁ Λ ἄρα τῷ Ο ἐστιν ὁ αὐτός· ὅπερ <lb n="1799,001,,,,9,36,62"></lb> |
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8525 ἀδύνατον· ὁ γὰρ Ο ὑπόκειται μηδενὶ τῶν ἐκκειμένων ὁ <lb n="1799,001,,,,9,36,63"></lb> |
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8526 αὐτός.</s> <s>οὐκ ἄρα τὸν ΖΗ μετρήσει τις ἀριθμὸς παρὲξ τῶν <lb n="1799,001,,,,9,36,64"></lb> |
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8527 Α, Β, Γ, Δ, Ε, ΘΚ, Λ, Μ καὶ τῆς μονάδος.</s> <s>καὶ ἐδείχθη <lb n="1799,001,,,,9,36,65"></lb> |
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8528 ὁ ΖΗ τοῖς Α, Β, Γ, Δ, Ε, ΘΚ, Λ, Μ καὶ τῇ μονάδι ἴσος.</s> <lb n="1799,001,,,,9,36,66"></lb> |
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8529 <s>τέλειος δὲ ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος ὤν· <lb n="1799,001,,,,9,36,67"></lb> |
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8530 τέλειος ἄρα ἐστὶν ὁ ΖΗ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,9,36,68"></lb> |
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8531 </s></p> |
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8532 </chap> |
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8533 <chap><p type="main"><s>Σύμμετρα μεγέθη λέγεται τὰ τῷ αὐτῷ μέτρῳ <lb n="1799,001,,,,10;HOR,1,1"></lb> |
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8534 |
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8535 μετρούμενα, ἀσύμμετρα δέ, ὧν μηδὲν ἐνδέχεται κοινὸν <lb n="1799,001,,,,10;HOR,1,2"></lb> |
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8536 μέτρον γενέσθαι.</s> <lb n="1799,001,,,,10;HOR,1,3"></lb> |
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8537 <s>Εὐθεῖαι δυνάμει σύμμετροί εἰσιν, ὅταν τὰ ἀπ' <lb n="1799,001,,,,10;HOR,2,1"></lb> |
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8538 αὐτῶν τετράγωνα τῷ αὐτῷ χωρίῳ μετρῆται, ἀσύμμετροι <lb n="1799,001,,,,10;HOR,2,2"></lb> |
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8539 δέ, ὅταν τοῖς ἀπ' αὐτῶν τετραγώνοις μηδὲν ἐνδέχηται <lb n="1799,001,,,,10;HOR,2,3"></lb> |
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8540 χωρίον κοινὸν μέτρον γενέσθαι.</s> <lb n="1799,001,,,,10;HOR,2,4"></lb> |
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8541 <s>Τούτων ὑποκειμένων δείκνυται, ὅτι τῇ προτεθείσῃ <lb n="1799,001,,,,10;HOR,3,1"></lb> |
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8542 εὐθείᾳ ὑπάρχουσιν εὐθεῖαι πλήθει ἄπειροι σύμμετροί τε <lb n="1799,001,,,,10;HOR,3,2"></lb> |
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8543 καὶ ἀσύμμετροι αἱ μὲν μήκει μόνον, αἱ δὲ καὶ δυνάμει.</s> <lb n="1799,001,,,,10;HOR,3,3"></lb> |
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8544 <s>καλείσθω οὖν ἡ μὲν προτεθεῖσα εὐθεῖα ῥητή, καὶ αἱ <lb n="1799,001,,,,10;HOR,3,4"></lb> |
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8545 ταύτῃ σύμμετροι εἴτε μήκει καὶ δυνάμει εἴτε δυνάμει <lb n="1799,001,,,,10;HOR,3,5"></lb> |
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8546 μόνον ῥηταί, αἱ δὲ ταύτῃ ἀσύμμετροι ἄλογοι καλεί-<lb n="1799,001,,,,10;HOR,3,6"></lb> |
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8547 σθωσαν.</s> <lb n="1799,001,,,,10;HOR,3,7"></lb> |
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8548 <s>Καὶ τὸ μὲν ἀπὸ τῆς προτεθείσης εὐθείας τετράγωνον <pb></pb> <lb n="1799,001,,,,10;HOR,4,1"></lb> |
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8549 ῥητόν, καὶ τὰ τούτῳ σύμμετρα ῥητά, τὰ δὲ τούτῳ ἀσύμ-<lb n="1799,001,,,,10;HOR,4,2"></lb> |
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8550 μετρα ἄλογα καλείσθω, καὶ αἱ δυνάμεναι αὐτὰ ἄλογοι, εἰ <lb n="1799,001,,,,10;HOR,4,3"></lb> |
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8551 μὲν τετράγωνα εἴη, αὐταὶ αἱ πλευραί, εἰ δὲ ἕτερά τινα <lb n="1799,001,,,,10;HOR,4,4"></lb> |
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8552 εὐθύγραμμα, αἱ ἴσα αὐτοῖς τετράγωνα ἀναγράφουσαι.</s> <lb n="1799,001,,,,10;HOR,4,5"></lb> |
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8553 <s></s></p> |
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8554 </chap> |
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8555 <chap><p type="main"><s>Δύο μεγεθῶν ἀνίσων ἐκκειμένων, ἐὰν ἀπὸ τοῦ μείζονος <lb n="1799,001,,,,10,1,1"></lb> |
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8556 |
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8557 ἀφαιρεθῇ μεῖζον ἢ τὸ ἥμισυ καὶ τοῦ καταλειπομένου <lb n="1799,001,,,,10,1,2"></lb> |
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8558 μεῖζον ἢ τὸ ἥμισυ, καὶ τοῦτο ἀεὶ γίγνηται, λειφθήσεταί τι <lb n="1799,001,,,,10,1,3"></lb> |
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8559 μέγεθος, ὃ ἔσται ἔλασσον τοῦ ἐκκειμένου ἐλάσσονος με-<lb n="1799,001,,,,10,1,4"></lb> |
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8560 γέθους.</s> <lb n="1799,001,,,,10,1,5"></lb> |
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8561 <s>Ἔστω δύο μεγέθη ἄνισα τὰ ΑΒ, Γ, ὧν μεῖζον τὸ ΑΒ· <lb n="1799,001,,,,10,1,6"></lb> |
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8562 λέγω, ὅτι, ἐὰν ἀπὸ τοῦ ΑΒ ἀφαιρεθῇ μεῖζον ἢ τὸ ἥμισυ <lb n="1799,001,,,,10,1,7"></lb> |
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8563 καὶ τοῦ καταλειπομένου μεῖζον ἢ τὸ ἥμισυ, καὶ τοῦτο <lb n="1799,001,,,,10,1,8"></lb> |
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8564 ἀεὶ γίγνηται, λειφθήσεταί τι μέγεθος, ὃ ἔσται ἔλασσον <lb n="1799,001,,,,10,1,9"></lb> |
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8565 τοῦ Γ μεγέθους.</s> <lb n="1799,001,,,,10,1,10"></lb> |
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8566 <s>Τὸ Γ γὰρ πολλαπλασιαζόμενον ἔσται ποτὲ τοῦ ΑΒ <lb n="1799,001,,,,10,1,11"></lb> |
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8567 μεῖζον.</s> <s>πεπολλαπλασιάσθω, καὶ ἔστω τὸ ΔΕ τοῦ μὲν Γ <lb n="1799,001,,,,10,1,12"></lb> |
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8568 πολλαπλάσιον, τοῦ δὲ ΑΒ μεῖζον, καὶ διῃρήσθω τὸ ΔΕ <lb n="1799,001,,,,10,1,13"></lb> |
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8569 εἰς τὰ τῷ Γ ἴσα τὰ ΔΖ, <lb n="1799,001,,,,10,1,14"></lb> |
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8570 ΖΗ, ΗΕ, καὶ ἀφῃρήσθω <lb n="1799,001,,,,10,1,15"></lb> |
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8571 ἀπὸ μὲν τοῦ ΑΒ μεῖζον ἢ τὸ <lb n="1799,001,,,,10,1,16"></lb> |
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8572 ἥμισυ τὸ ΒΘ, ἀπὸ δὲ τοῦ ΑΘ <pb></pb> <lb n="1799,001,,,,10,1,17"></lb> |
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8573 μεῖζον ἢ τὸ ἥμισυ τὸ ΘΚ, καὶ τοῦτο ἀεὶ γιγνέσθω, ἕως <lb n="1799,001,,,,10,1,18"></lb> |
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8574 ἂν αἱ ἐν τῷ ΑΒ διαιρέσεις ἰσοπληθεῖς γένωνται ταῖς ἐν τῷ <lb n="1799,001,,,,10,1,19"></lb> |
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8575 ΔΕ διαιρέσεσιν.</s> <lb n="1799,001,,,,10,1,20"></lb> |
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8576 <s>Ἔστωσαν οὖν αἱ ΑΚ, ΚΘ, *θΒ διαιρέσεις ἰσοπληθεῖς <lb n="1799,001,,,,10,1,21"></lb> |
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8577 οὖσαι ταῖς ΔΖ, ΖΗ, ΗΕ· καὶ ἐπεὶ μεῖζόν ἐστι τὸ ΔΕ <lb n="1799,001,,,,10,1,22"></lb> |
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8578 τοῦ ΑΒ, καὶ ἀφῄρηται ἀπὸ μὲν τοῦ ΔΕ ἔλασσον τοῦ <lb n="1799,001,,,,10,1,23"></lb> |
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8579 ἡμίσεος τὸ ΕΗ, ἀπὸ δὲ τοῦ ΑΒ μεῖζον ἢ τὸ ἥμισυ τὸ ΒΘ, <lb n="1799,001,,,,10,1,24"></lb> |
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8580 λοιπὸν ἄρα τὸ ΗΔ λοιποῦ τοῦ ΘΑ μεῖζόν ἐστιν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,1,25"></lb> |
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8581 μεῖζόν ἐστι τὸ ΗΔ τοῦ ΘΑ, καὶ ἀφῄρηται τοῦ μὲν ΗΔ <lb n="1799,001,,,,10,1,26"></lb> |
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8582 ἥμισυ τὸ ΗΖ, τοῦ δὲ ΘΑ μεῖζον ἢ τὸ ἥμισυ τὸ ΘΚ, λοιπὸν <lb n="1799,001,,,,10,1,27"></lb> |
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8583 ἄρα τὸ ΔΖ λοιποῦ τοῦ ΑΚ μεῖζόν ἐστιν.</s> <s>ἴσον δὲ τὸ ΔΖ <lb n="1799,001,,,,10,1,28"></lb> |
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8584 τῷ Γ· καὶ τὸ Γ ἄρα τοῦ ΑΚ μεῖζόν ἐστιν.</s> <s>ἔλασσον ἄρα <lb n="1799,001,,,,10,1,29"></lb> |
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8585 τὸ ΑΚ τοῦ Γ.</s> <lb n="1799,001,,,,10,1,30"></lb> |
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8586 <s>Καταλείπεται ἄρα ἀπὸ τοῦ ΑΒ μεγέθους τὸ ΑΚ μέγε-<lb n="1799,001,,,,10,1,31"></lb> |
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8587 θος ἔλασσον ὂν τοῦ ἐκκειμένου ἐλάσσονος μεγέθους τοῦ Γ· <lb n="1799,001,,,,10,1,32"></lb> |
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8588 ὅπερ ἔδει δεῖξαι.</s> <s>̄ὁμοίως δὲ δειχθήσεται, κἂν ἡμίση ᾖ τὰ <lb n="1799,001,,,,10,1,33"></lb> |
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8589 ἀφαιρούμενα.</s> <lb n="1799,001,,,,10,1,34"></lb> |
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8590 <s>Ἐὰν δύο μεγεθῶν [ἐκκειμένων] ἀνίσων ἀνθυφαιρουμένου <lb n="1799,001,,,,10,2,1"></lb> |
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8591 ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος τὸ καταλειπόμενον <lb n="1799,001,,,,10,2,2"></lb> |
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8592 μηδέποτε καταμετρῇ τὸ πρὸ ἑαυτοῦ, ἀσύμμετρα ἔσται τὰ <lb n="1799,001,,,,10,2,3"></lb> |
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8593 μεγέθη.</s> <lb n="1799,001,,,,10,2,4"></lb> |
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8594 <s>Δύο γὰρ μεγεθῶν ὄντων ἀνίσων τῶν ΑΒ, ΓΔ καὶ <lb n="1799,001,,,,10,2,5"></lb> |
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8595 ἐλάσσονος τοῦ ΑΒ ἀνθυφαιρουμένου ἀεὶ τοῦ ἐλάσσονος <lb n="1799,001,,,,10,2,6"></lb> |
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8596 ἀπὸ τοῦ μείζονος τὸ περιλειπόμενον μηδέποτε κατα- <pb></pb><lb n="1799,001,,,,10,2,7"></lb> |
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8597 μετρείτω τὸ πρὸ ἑαυ-<lb n="1799,001,,,,10,2,8"></lb> |
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8598 τοῦ· λέγω, ὅτι ἀσύμ-<lb n="1799,001,,,,10,2,9"></lb> |
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8599 μετρά ἐστι τὰ ΑΒ, <lb n="1799,001,,,,10,2,10"></lb> |
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8600 ΓΔ μεγέθη.</s> <lb n="1799,001,,,,10,2,11"></lb> |
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8601 <s>Εἰ γάρ ἐστι σύμμετρα, μετρήσει τι αὐτὰ μέγεθος.</s> <lb n="1799,001,,,,10,2,12"></lb> |
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8602 <s>μετρείτω, εἰ δυνατόν, καὶ ἔστω τὸ Ε· καὶ τὸ μὲν ΑΒ τὸ <lb n="1799,001,,,,10,2,13"></lb> |
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8603 ΖΔ καταμετροῦν λειπέτω ἑαυτοῦ ἔλασσον τὸ ΓΖ, τὸ δὲ <lb n="1799,001,,,,10,2,14"></lb> |
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8604 ΓΖ τὸ ΒΗ καταμετροῦν λειπέτω ἑαυτοῦ ἔλασσον τὸ ΑΗ, <lb n="1799,001,,,,10,2,15"></lb> |
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8605 καὶ τοῦτο ἀεὶ γινέσθω, ἕως οὗ λειφθῇ τι μέγεθος, ὅ ἐστιν <lb n="1799,001,,,,10,2,16"></lb> |
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8606 ἔλασσον τοῦ Ε.</s> <s>γεγονέτω, καὶ λελείφθω τὸ ΑΗ ἔλασσον <lb n="1799,001,,,,10,2,17"></lb> |
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8607 τοῦ Ε.</s> <s>ἐπεὶ οὖν τὸ Ε τὸ ΑΒ μετρεῖ, ἀλλὰ τὸ ΑΒ τὸ ΔΖ <lb n="1799,001,,,,10,2,18"></lb> |
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8608 μετρεῖ, καὶ τὸ Ε ἄρα τὸ ΖΔ μετρήσει.</s> <s>μετρεῖ δὲ καὶ ὅλον τὸ <lb n="1799,001,,,,10,2,19"></lb> |
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8609 ΓΔ· καὶ λοιπὸν ἄρα τὸ ΓΖ μετρήσει.</s> <s>ἀλλὰ τὸ ΓΖ τὸ ΒΗ <lb n="1799,001,,,,10,2,20"></lb> |
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8610 μετρεῖ· καὶ τὸ Ε ἄρα τὸ ΒΗ μετρεῖ.</s> <s>μετρεῖ δὲ καὶ ὅλον τὸ <lb n="1799,001,,,,10,2,21"></lb> |
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8611 ΑΒ· καὶ λοιπὸν ἄρα τὸ ΑΗ μετρήσει, τὸ μεῖζον τὸ ἔλασσον.</s> <lb n="1799,001,,,,10,2,22"></lb> |
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8612 <s>ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τὰ ΑΒ, ΓΔ μεγέθη <lb n="1799,001,,,,10,2,23"></lb> |
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8613 μετρήσει τι μέγεθος· ἀσύμμετρα ἄρα ἐστὶ τὰ ΑΒ, ΓΔ <lb n="1799,001,,,,10,2,24"></lb> |
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8614 μεγέθη.</s> <lb n="1799,001,,,,10,2,25"></lb> |
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8615 <s>Ἐὰν ἄρα δύο μεγεθῶν ἀνίσων, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,2,26"></lb> |
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8616 <s>Δύο μεγεθῶν συμμέτρων δοθέντων τὸ μέγιστον αὐτῶν <lb n="1799,001,,,,10,3,1"></lb> |
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8617 κοινὸν μέτρον εὑρεῖν.</s> <s><pb></pb> <lb n="1799,001,,,,10,3,2"></lb> |
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8618 Ἔστω τὰ δοθέντα δύο μεγέθη σύμμετρα τὰ ΑΒ, ΓΔ, <lb n="1799,001,,,,10,3,3"></lb> |
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8619 ὧν ἔλασσον τὸ ΑΒ· δεῖ δὴ τῶν ΑΒ, ΓΔ τὸ μέγιστον <lb n="1799,001,,,,10,3,4"></lb> |
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8620 κοινὸν μέτρον εὑρεῖν.</s> <lb n="1799,001,,,,10,3,5"></lb> |
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8621 <s>Τὸ ΑΒ γὰρ μέγεθος ἤτοι μετρεῖ τὸ ΓΔ ἢ οὔ.</s> <s>εἰ μὲν <lb n="1799,001,,,,10,3,6"></lb> |
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8622 οὖν μετρεῖ, μετρεῖ δὲ καὶ ἑαυτό, τὸ ΑΒ ἄρα τῶν ΑΒ, ΓΔ <lb n="1799,001,,,,10,3,7"></lb> |
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8623 κοινὸν μέτρον ἐστίν· καὶ φανερόν, ὅτι καὶ μέγιστον.</s> <s>μεῖζον <lb n="1799,001,,,,10,3,8"></lb> |
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8624 γὰρ τοῦ ΑΒ μεγέθους τὸ ΑΒ οὐ μετρήσει.</s> <lb n="1799,001,,,,10,3,9"></lb> |
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8625 <s>Μὴ μετρείτω δὴ τὸ ΑΒ τὸ ΓΔ.</s> <s>καὶ ἀνθυφαιρουμένου <lb n="1799,001,,,,10,3,10"></lb> |
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8626 ἀεὶ τοῦ ἐλάσσονος ἀπὸ τοῦ μείζονος, τὸ περιλειπόμενον <lb n="1799,001,,,,10,3,11"></lb> |
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8627 μετρήσει ποτὲ τὸ πρὸ ἑαυτοῦ διὰ τὸ μὴ εἶναι ἀσύμμετρα <lb n="1799,001,,,,10,3,12"></lb> |
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8628 τὰ ΑΒ, ΓΔ· καὶ τὸ μὲν ΑΒ τὸ ΕΔ καταμετροῦν λειπέτω <lb n="1799,001,,,,10,3,13"></lb> |
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8629 ἑαυτοῦ ἔλασσον τὸ ΕΓ, τὸ δὲ ΕΓ τὸ ΖΒ καταμετροῦν <lb n="1799,001,,,,10,3,14"></lb> |
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8630 λειπέτω ἑαυτοῦ ἔλασσον τὸ ΑΖ, τὸ δὲ ΑΖ τὸ ΓΕ <lb n="1799,001,,,,10,3,15"></lb> |
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8631 μετρείτω.</s> <lb n="1799,001,,,,10,3,16"></lb> |
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8632 <s>Ἐπεὶ οὖν τὸ ΑΖ τὸ ΓΕ μετρεῖ, ἀλλὰ τὸ ΓΕ τὸ ΖΒ <lb n="1799,001,,,,10,3,17"></lb> |
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8633 μετρεῖ, καὶ τὸ ΑΖ ἄρα τὸ ΖΒ μετρήσει.</s> <s>μετρεῖ δὲ καὶ <lb n="1799,001,,,,10,3,18"></lb> |
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8634 ἑαυτό· καὶ ὅλον ἄρα τὸ ΑΒ μετρήσει τὸ ΑΖ.</s> <s>ἀλλὰ τὸ ΑΒ <lb n="1799,001,,,,10,3,19"></lb> |
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8635 τὸ ΔΕ μετρεῖ· καὶ τὸ ΑΖ ἄρα τὸ ΕΔ μετρήσει.</s> <s>μετρεῖ <lb n="1799,001,,,,10,3,20"></lb> |
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8636 δὲ καὶ τὸ ΓΕ· καὶ ὅλον ἄρα τὸ ΓΔ μετρεῖ· τὸ ΑΖ ἄρα <lb n="1799,001,,,,10,3,21"></lb> |
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8637 τῶν ΑΒ, ΓΔ κοινὸν μέτρον ἐστίν.</s> <s>λέγω δή, ὅτι καὶ <lb n="1799,001,,,,10,3,22"></lb> |
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8638 μέγιστον.</s> <s>εἰ γὰρ μή, ἔσται τι μέγεθος μεῖζον τοῦ ΑΖ, ὃ <pb></pb> <lb n="1799,001,,,,10,3,23"></lb> |
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8639 μετρήσει τὰ ΑΒ, ΓΔ.</s> <s>ἔστω τὸ Η.</s> <s>ἐπεὶ οὖν τὸ Η τὸ ΑΒ <lb n="1799,001,,,,10,3,24"></lb> |
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8640 μετρεῖ, ἀλλὰ τὸ ΑΒ τὸ ΕΔ μετρεῖ, καὶ τὸ Η ἄρα τὸ ΕΔ <lb n="1799,001,,,,10,3,25"></lb> |
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8641 μετρήσει.</s> <s>μετρεῖ δὲ καὶ ὅλον τὸ ΓΔ· καὶ λοιπὸν ἄρα τὸ <lb n="1799,001,,,,10,3,26"></lb> |
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8642 ΓΕ μετρήσει τὸ Η.</s> <s>ἀλλὰ τὸ ΓΕ τὸ ΖΒ μετρεῖ· καὶ τὸ Η <lb n="1799,001,,,,10,3,27"></lb> |
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8643 ἄρα τὸ ΖΒ μετρήσει.</s> <s>μετρεῖ δὲ καὶ ὅλον τὸ ΑΒ, καὶ λοιπὸν <lb n="1799,001,,,,10,3,28"></lb> |
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8644 τὸ ΑΖ μετρήσει, τὸ μεῖζον τὸ ἔλασσον· ὅπερ ἐστὶν <lb n="1799,001,,,,10,3,29"></lb> |
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8645 ἀδύνατον.</s> <s>οὐκ ἄρα μεῖζόν τι μέγεθος τοῦ ΑΖ τὰ ΑΒ, <lb n="1799,001,,,,10,3,30"></lb> |
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8646 ΓΔ μετρήσει· τὸ ΑΖ ἄρα τῶν ΑΒ, ΓΔ τὸ μέγιστον <lb n="1799,001,,,,10,3,31"></lb> |
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8647 κοινὸν μέτρον ἐστίν.</s> <lb n="1799,001,,,,10,3,32"></lb> |
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8648 <s>Δύο ἄρα μεγεθῶν συμμέτρων δοθέντων τῶν ΑΒ, ΓΔ <lb n="1799,001,,,,10,3,33"></lb> |
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8649 τὸ μέγιστον κοινὸν μέτρον ηὕρηται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,3,34"></lb> |
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8650 <s>Πόρισμα <lb n="1799,001,,,,10,3,35"></lb> |
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8651 Ἐκ δὴ τούτου φανερόν, ὅτι, ἐὰν μέγεθος δύο μεγέθη <lb n="1799,001,,,,10,3,36"></lb> |
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8652 μετρῇ, καὶ τὸ μέγιστον αὐτῶν κοινὸν μέτρον μετρήσει.</s> <lb n="1799,001,,,,10,3,37"></lb> |
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8653 <s>Τριῶν μεγεθῶν συμμέτρων δοθέντων τὸ μέγιστον <lb n="1799,001,,,,10,4,1"></lb> |
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8654 αὐτῶν κοινὸν μέτρον εὑρεῖν.</s> <lb n="1799,001,,,,10,4,2"></lb> |
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8655 <s>Ἔστω τὰ δοθέντα τρία μεγέθη σύμμετρα τὰ Α, Β, Γ· <lb n="1799,001,,,,10,4,3"></lb> |
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8656 δεῖ δὴ τῶν Α, Β, Γ τὸ μέγιστον κοινὸν μέτρον εὑρεῖν.</s> <lb n="1799,001,,,,10,4,4"></lb> |
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8657 <s>Εἰλήφθω γὰρ δύο τῶν Α, Β τὸ μέγιστον κοινὸν μέτρον, <lb n="1799,001,,,,10,4,5"></lb> |
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8658 καὶ ἔστω τὸ Δ· τὸ δὴ Δ τὸ Γ ἤτοι μετρεῖ ἢ οὔ [μετρεῖ].</s> <lb n="1799,001,,,,10,4,6"></lb> |
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8659 <s>μετρείτω πρότερον.</s> <s>ἐπεὶ οὖν τὸ Δ τὸ Γ μετρεῖ, μετρεῖ δὲ <pb></pb> <lb n="1799,001,,,,10,4,7"></lb> |
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8660 καὶ τὰ Α, Β, τὸ Δ ἄρα τὰ Α, Β, Γ <lb n="1799,001,,,,10,4,8"></lb> |
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8661 μετρεῖ· τὸ Δ ἄρα τῶν Α, Β, Γ κοινὸν <lb n="1799,001,,,,10,4,9"></lb> |
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8662 μέτρον ἐστίν.</s> <s>καὶ φανερόν, ὅτι καὶ <lb n="1799,001,,,,10,4,10"></lb> |
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8663 μέγιστον· μεῖζον γὰρ τοῦ Δ μεγέθους <lb n="1799,001,,,,10,4,11"></lb> |
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8664 τὰ Α, Β οὐ μετρεῖ.</s> <lb n="1799,001,,,,10,4,12"></lb> |
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8665 <s>Μὴ μετρείτω δὴ τὸ Δ τὸ Γ.</s> <s>λέγω πρῶτον, ὅτι σύμμετρά <lb n="1799,001,,,,10,4,13"></lb> |
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8666 ἐστι τὰ Γ, Δ.</s> <s>ἐπεὶ γὰρ σύμμετρά ἐστι τὰ Α, Β, Γ, μετρήσει <lb n="1799,001,,,,10,4,14"></lb> |
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8667 τι αὐτὰ μέγεθος, ὃ δηλαδὴ καὶ τὰ Α, Β μετρήσει· ὥστε <lb n="1799,001,,,,10,4,15"></lb> |
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8668 καὶ τὸ τῶν Α, Β μέγιστον κοινὸν μέτρον τὸ Δ μετρήσει.</s> <lb n="1799,001,,,,10,4,16"></lb> |
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8669 <s>μετρεῖ δὲ καὶ τὸ Γ· ὥστε τὸ εἰρημένον μέγεθος μετρήσει <lb n="1799,001,,,,10,4,17"></lb> |
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8670 τὰ Γ, Δ· σύμμετρα ἄρα ἐστὶ τὰ Γ, Δ.</s> <s>εἰλήφθω οὖν αὐτῶν <lb n="1799,001,,,,10,4,18"></lb> |
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8671 τὸ μέγιστον κοινὸν μέτρον, καὶ ἔστω τὸ Ε.</s> <s>ἐπεὶ οὖν τὸ Ε <lb n="1799,001,,,,10,4,19"></lb> |
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8672 τὸ Δ μετρεῖ, ἀλλὰ τὸ Δ τὰ Α, Β μετρεῖ, καὶ τὸ Ε ἄρα τὰ <lb n="1799,001,,,,10,4,20"></lb> |
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8673 Α, Β μετρήσει.</s> <s>μετρεῖ δὲ καὶ τὸ Γ.</s> <s>τὸ Ε ἄρα τὰ Α, Β, Γ <lb n="1799,001,,,,10,4,21"></lb> |
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8674 μετρεῖ· τὸ Ε ἄρα τῶν Α, Β, Γ κοινόν ἐστι μέτρον.</s> <s>λέγω <lb n="1799,001,,,,10,4,22"></lb> |
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8675 δή, ὅτι καὶ μέγιστον.</s> <s>εἰ γὰρ δυνατόν, ἔστω τι τοῦ Ε μεῖζον <lb n="1799,001,,,,10,4,23"></lb> |
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8676 μέγεθος τὸ Ζ, καὶ μετρείτω τὰ Α, Β, Γ.</s> <s>καὶ ἐπεὶ τὸ Ζ <lb n="1799,001,,,,10,4,24"></lb> |
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8677 τὰ Α, Β, Γ μετρεῖ, καὶ τὰ Α, Β ἄρα μετρήσει καὶ τὸ τῶν <lb n="1799,001,,,,10,4,25"></lb> |
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8678 Α, Β μέγιστον κοινὸν μέτρον μετρήσει.</s> <s>τὸ δὲ τῶν Α, Β <lb n="1799,001,,,,10,4,26"></lb> |
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8679 μέγιστον κοινὸν μέτρον ἐστὶ τὸ Δ· τὸ Ζ ἄρα τὸ Δ μετρεῖ.</s> <lb n="1799,001,,,,10,4,27"></lb> |
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8680 <s>μετρεῖ δὲ καὶ τὸ Γ· τὸ Ζ ἄρα τὰ Γ, Δ μετρεῖ· καὶ τὸ τῶν <pb></pb> <lb n="1799,001,,,,10,4,28"></lb> |
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8681 Γ, Δ ἄρα μέγιστον κοινὸν μέτρον μετρήσει τὸ Ζ.</s> <s>ἔστι δὲ τὸ <lb n="1799,001,,,,10,4,29"></lb> |
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8682 Ε· τὸ Ζ ἄρα τὸ Ε μετρήσει, τὸ μεῖζον τὸ ἔλασσον· ὅπερ <lb n="1799,001,,,,10,4,30"></lb> |
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8683 ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα μεῖζόν τι τοῦ Ε μεγέθους <lb n="1799,001,,,,10,4,31"></lb> |
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8684 [μέγεθοσ] τὰ Α, Β, Γ μετρεῖ· τὸ Ε ἄρα τῶν Α, Β, Γ τὸ <lb n="1799,001,,,,10,4,32"></lb> |
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8685 μέγιστον κοινὸν μέτρον ἐστίν, ἐὰν μὴ μετρῇ τὸ Δ τὸ Γ, <lb n="1799,001,,,,10,4,33"></lb> |
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8686 ἐὰν δὲ μετρῇ, αὐτὸ τὸ Δ.</s> <lb n="1799,001,,,,10,4,34"></lb> |
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8687 <s>Τριῶν ἄρα μεγεθῶν συμμέτρων δοθέντων τὸ μέγιστον <lb n="1799,001,,,,10,4,35"></lb> |
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8688 κοινὸν μέτρον ηὕρηται [ὅπερ ἔδει δεῖξαι].</s> <lb n="1799,001,,,,10,4,36"></lb> |
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8689 <s>Πόρισμα <lb n="1799,001,,,,10,4,37"></lb> |
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8690 Ἐκ δὴ τούτου φανερόν, ὅτι, ἐὰν μέγεθος τρία μεγέθη <lb n="1799,001,,,,10,4,38"></lb> |
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8691 μετρῇ, καὶ τὸ μέγιστον αὐτῶν κοινὸν μέτρον μετρήσει.</s> <lb n="1799,001,,,,10,4,39"></lb> |
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8692 <s>Ὁμοίως δὴ καὶ ἐπὶ πλειόνων τὸ μέγιστον κοινὸν μέτρον <lb n="1799,001,,,,10,4,40"></lb> |
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8693 ληφθήσεται, καὶ τὸ πόρισμα προχωρήσει.</s> <s>ὅπερ ἔδει <lb n="1799,001,,,,10,4,41"></lb> |
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8694 δεῖξαι.</s> <lb n="1799,001,,,,10,4,42"></lb> |
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8695 <s>Τὰ σύμμετρα μεγέθη πρὸς ἄλληλα λόγον ἔχει, ὃν <lb n="1799,001,,,,10,5,1"></lb> |
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8696 ἀριθμὸς πρὸς ἀριθμόν.</s> <lb n="1799,001,,,,10,5,2"></lb> |
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8697 <s>Ἔστω σύμμετρα μεγέθη τὰ Α, Β· λέγω, ὅτι τὸ Α πρὸς <lb n="1799,001,,,,10,5,3"></lb> |
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8698 τὸ Β λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν.</s> <lb n="1799,001,,,,10,5,4"></lb> |
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8699 <s>Ἐπεὶ γὰρ σύμμετρά ἐστι τὰ Α, Β, μετρήσει τι αὐτὰ <lb n="1799,001,,,,10,5,5"></lb> |
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8700 μέγεθος.</s> <s>μετρείτω, καὶ ἔστω τὸ Γ.</s> <s>καὶ ὁσάκις τὸ Γ τὸ Α <lb n="1799,001,,,,10,5,6"></lb> |
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8701 μετρεῖ τοσαῦται μονάδες ἔστωσαν ἐν τῷ Δ, ὁσάκις δὲ <lb n="1799,001,,,,10,5,7"></lb> |
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8702 τὸ Γ τὸ Β μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Ε.</s> <s><pb></pb> <lb n="1799,001,,,,10,5,8"></lb> |
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8703 Ἐπεὶ οὖν τὸ Γ τὸ Α μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας, <lb n="1799,001,,,,10,5,9"></lb> |
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8704 μετρεῖ δὲ καὶ ἡ μονὰς τὸν Δ κατὰ τὰς ἐν αὐτῷ μονάδας, <lb n="1799,001,,,,10,5,10"></lb> |
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8705 ἰσάκις ἄρα ἡ μονὰς τὸν Δ μετρεῖ ἀριθμὸν καὶ τὸ Γ μέγεθος <lb n="1799,001,,,,10,5,11"></lb> |
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8706 τὸ Α· ἔστιν ἄρα ὡς τὸ Γ πρὸς <lb n="1799,001,,,,10,5,12"></lb> |
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8707 τὸ Α, οὕτως ἡ μονὰς πρὸς <lb n="1799,001,,,,10,5,13"></lb> |
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8708 τὸν Δ· ἀνάπαλιν ἄρα, ὡς τὸ Α <lb n="1799,001,,,,10,5,14"></lb> |
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8709 πρὸς τὸ Γ, οὕτως ὁ Δ πρὸς <lb n="1799,001,,,,10,5,15"></lb> |
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8710 τὴν μονάδα.</s> <s>πάλιν ἐπεὶ τὸ Γ τὸ Β μετρεῖ κατὰ τὰς ἐν τῷ Ε <lb n="1799,001,,,,10,5,16"></lb> |
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8711 μονάδας, μετρεῖ δὲ καὶ ἡ μονὰς τὸν Ε κατὰ τὰς ἐν αὐτῷ <lb n="1799,001,,,,10,5,17"></lb> |
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8712 μονάδας, ἰσάκις ἄρα ἡ μονὰς τὸν Ε μετρεῖ καὶ τὸ Γ τὸ Β· <lb n="1799,001,,,,10,5,18"></lb> |
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8713 ἔστιν ἄρα ὡς τὸ Γ πρὸς τὸ Β, οὕτως ἡ μονὰς πρὸς τὸν Ε.</s> <lb n="1799,001,,,,10,5,19"></lb> |
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8714 <s>ἐδείχθη δὲ καὶ ὡς τὸ Α πρὸς τὸ Γ, ὁ Δ πρὸς τὴν μονάδα· <lb n="1799,001,,,,10,5,20"></lb> |
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8715 δι' ἴσου ἄρα ἐστὶν ὡς τὸ Α πρὸς τὸ Β, οὕτως ὁ Δ ἀριθμὸς <lb n="1799,001,,,,10,5,21"></lb> |
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8716 πρὸς τὸν Ε.</s> <lb n="1799,001,,,,10,5,22"></lb> |
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8717 <s>Τὰ ἄρα σύμμετρα μεγέθη τὰ Α, Β πρὸς ἄλληλα λόγον <lb n="1799,001,,,,10,5,23"></lb> |
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8718 ἔχει, ὃν ἀριθμὸς ὁ Δ πρὸς ἀριθμὸν τὸν Ε· ὅπερ ἔδει <lb n="1799,001,,,,10,5,24"></lb> |
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8719 δεῖξαι.</s> <lb n="1799,001,,,,10,5,25"></lb> |
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8720 <s>Ἐὰν δύο μεγέθη πρὸς ἄλληλα λόγον ἔχῃ, ὃν ἀριθμὸς <lb n="1799,001,,,,10,6,1"></lb> |
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8721 πρὸς ἀριθμόν, σύμμετρα ἔσται τὰ μεγέθη.</s> <lb n="1799,001,,,,10,6,2"></lb> |
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8722 <s>Δύο γὰρ μεγέθη τὰ Α, Β πρὸς ἄλληλα λόγον ἐχέτω, <lb n="1799,001,,,,10,6,3"></lb> |
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8723 ὃν ἀριθμὸς ὁ Δ πρὸς ἀριθμὸν τὸν Ε· λέγω, ὅτι σύμμετρά <lb n="1799,001,,,,10,6,4"></lb> |
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8724 ἐστι τὰ Α, Β μεγέθη.</s> <s><pb></pb> <lb n="1799,001,,,,10,6,5"></lb> |
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8725 Ὅσαι γάρ εἰσιν ἐν τῷ Δ μονάδες, εἰς τοσαῦτα ἴσα <lb n="1799,001,,,,10,6,6"></lb> |
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8726 διῃρήσθω τὸ Α, καὶ ἑνὶ αὐτῶν ἴσον ἔστω τὸ Γ· ὅσαι δέ <lb n="1799,001,,,,10,6,7"></lb> |
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8727 εἰσιν ἐν τῷ Ε μονάδες, ἐκ τοσούτων μεγεθῶν ἴσων τῷ Γ <lb n="1799,001,,,,10,6,8"></lb> |
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8728 συγκείσθω τὸ Ζ. <lb n="1799,001,,,,10,6,9"></lb> |
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8729 Ἐπεὶ οὖν, ὅσαι εἰσὶν ἐν τῷ Δ μονάδες, τοσαῦτά εἰσι καὶ <lb n="1799,001,,,,10,6,10"></lb> |
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8730 ἐν τῷ Α μεγέθη ἴσα τῷ Γ, ὃ ἄρα μέρος ἐστὶν ἡ μονὰς <lb n="1799,001,,,,10,6,11"></lb> |
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8731 τοῦ Δ, τὸ αὐτὸ μέρος ἐστὶ καὶ τὸ Γ τοῦ Α· ἔστιν ἄρα ὡς <lb n="1799,001,,,,10,6,12"></lb> |
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8732 τὸ Γ πρὸς τὸ Α, οὕτως ἡ μονὰς πρὸς τὸν Δ.</s> <s>μετρεῖ δὲ ἡ <lb n="1799,001,,,,10,6,13"></lb> |
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8733 μονὰς τὸν Δ ἀριθμόν· μετρεῖ ἄρα καὶ τὸ Γ τὸ Α.</s> <s>καὶ ἐπεί <lb n="1799,001,,,,10,6,14"></lb> |
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8734 ἐστιν ὡς τὸ Γ πρὸς τὸ Α, οὕτως ἡ μονὰς πρὸς τὸν Δ <lb n="1799,001,,,,10,6,15"></lb> |
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8735 [ἀριθμόν], ἀνάπαλιν ἄρα ὡς τὸ Α πρὸς τὸ Γ, οὕτως ὁ Δ <lb n="1799,001,,,,10,6,16"></lb> |
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8736 ἀριθμὸς πρὸς τὴν μονάδα.</s> <s>πάλιν ἐπεί, ὅσαι εἰσὶν ἐν τῷ <lb n="1799,001,,,,10,6,17"></lb> |
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8737 Ε μονάδες, τοσαῦτά εἰσι καὶ ἐν τῷ Ζ ἴσα τῷ Γ, ἔστιν ἄρα <lb n="1799,001,,,,10,6,18"></lb> |
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8738 ὡς τὸ Γ πρὸς τὸ Ζ, οὕτως ἡ μονὰς πρὸς τὸν Ε [ἀριθμόν].</s> <lb n="1799,001,,,,10,6,19"></lb> |
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8739 <s>ἐδείχθη δὲ καὶ ὡς τὸ Α πρὸς τὸ Γ, οὕτως ὁ Δ πρὸς τὴν <lb n="1799,001,,,,10,6,20"></lb> |
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8740 μονάδα· δι' ἴσου ἄρα ἐστὶν ὡς τὸ Α πρὸς τὸ Ζ, οὕτως ὁ Δ <lb n="1799,001,,,,10,6,21"></lb> |
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8741 πρὸς τὸν Ε.</s> <s>ἀλλ' ὡς ὁ Δ πρὸς τὸν Ε, οὕτως ἐστὶ τὸ Α <lb n="1799,001,,,,10,6,22"></lb> |
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8742 πρὸς τὸ Β· καὶ ὡς ἄρα τὸ Α πρὸς τὸ Β, οὕτως καὶ πρὸς <lb n="1799,001,,,,10,6,23"></lb> |
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8743 τὸ Ζ.</s> <s>τὸ Α ἄρα πρὸς ἑκάτερον τῶν Β, Ζ τὸν αὐτὸν ἔχει <lb n="1799,001,,,,10,6,24"></lb> |
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8744 λόγον· ἴσον ἄρα ἐστὶ τὸ Β τῷ Ζ.</s> <s>μετρεῖ δὲ τὸ Γ τὸ Ζ· <pb></pb> <lb n="1799,001,,,,10,6,25"></lb> |
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8745 μετρεῖ ἄρα καὶ τὸ Β.</s> <s>ἀλλὰ μὴν καὶ τὸ Α· τὸ Γ ἄρα τὰ <lb n="1799,001,,,,10,6,26"></lb> |
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8746 Α, Β μετρεῖ.</s> <s>σύμμετρον ἄρα ἐστὶ τὸ Α τῷ Β.</s> <lb n="1799,001,,,,10,6,27"></lb> |
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8747 <s>Ἐὰν ἄρα δύο μεγέθη πρὸς ἄλληλα, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,6,28"></lb> |
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8748 <s>Πόρισμα <lb n="1799,001,,,,10,6,29"></lb> |
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8749 Ἐκ δὴ τούτου φανερόν, ὅτι, ἐὰν ὦσι δύο ἀριθμοί, ὡς <lb n="1799,001,,,,10,6,30"></lb> |
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8750 οἱ Δ, Ε, καὶ εὐθεῖα, ὡς ἡ Α, δύνατόν ἐστι ποιῆσαι ὡς ὁ Δ <lb n="1799,001,,,,10,6,31"></lb> |
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8751 ἀριθμὸς πρὸς τὸν Ε ἀριθμόν, οὕτως τὴν εὐθεῖαν πρὸς <lb n="1799,001,,,,10,6,32"></lb> |
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8752 εὐθεῖαν.</s> <s>ἐὰν δὲ καὶ τῶν Α, Ζ μέση ἀνάλογον ληφθῇ, ὡς <lb n="1799,001,,,,10,6,33"></lb> |
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8753 ἡ Β, ἔσται ὡς ἡ Α πρὸς τὴν Ζ, οὕτως τὸ ἀπὸ τῆς Α πρὸς <lb n="1799,001,,,,10,6,34"></lb> |
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8754 τὸ ἀπὸ τη̂ς Β, τουτέστιν ὡς ἡ πρώτη πρὸς τὴν τρίτην, <lb n="1799,001,,,,10,6,35"></lb> |
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8755 οὕτως τὸ ἀπὸ τῆς πρώτης πρὸς τὸ ἀπὸ τῆς δευτέρας τὸ <lb n="1799,001,,,,10,6,36"></lb> |
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8756 ὅμοιον καὶ ὁμοίως ἀναγραφόμενον.</s> <s>ἀλλ' ὡς ἡ Α πρὸς τὴν <lb n="1799,001,,,,10,6,37"></lb> |
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8757 Ζ, οὕτως ἐστὶν ὁ Δ ἀριθμὸς πρὸς τὸν Ε ἀριθμόν· γέγονεν <lb n="1799,001,,,,10,6,38"></lb> |
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8758 ἄρα καὶ ὡς ὁ Δ ἀριθμὸς πρὸς τὸν Ε ἀριθμόν, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,6,39"></lb> |
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8759 τῆς Α εὐθείας πρὸς τὸ ἀπὸ τῆς Β εὐθείας· ὅπερ ἔδει <lb n="1799,001,,,,10,6,40"></lb> |
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8760 δεῖξαι.</s> <lb n="1799,001,,,,10,6,41"></lb> |
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8761 <s>Τὰ ἀσύμμετρα μεγέθη πρὸς ἄλληλα λόγον οὐκ ἔχει, <lb n="1799,001,,,,10,7,1"></lb> |
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8762 ὃν ἀριθμὸς πρὸς ἀριθμόν.</s> <s><pb></pb> <lb n="1799,001,,,,10,7,2"></lb> |
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8763 Ἔστω ἀσύμμετρα μεγέθη τὰ Α, Β· λέγω, ὅτι τὸ Α <lb n="1799,001,,,,10,7,3"></lb> |
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8764 πρὸς τὸ Β λόγον οὐκ ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν.</s> <lb n="1799,001,,,,10,7,4"></lb> |
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8765 <s>Εἰ γὰρ ἔχει τὸ Α πρὸς τὸ Β λόγον, ὃν ἀριθμὸς <lb n="1799,001,,,,10,7,5"></lb> |
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8766 πρὸς ἀριθμόν, σύμμετρον ἔσται τὸ Α τῷ Β.</s> <lb n="1799,001,,,,10,7,6"></lb> |
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8767 <s>οὐκ ἔστι δέ· οὐκ ἄρα τὸ Α πρὸς τὸ Β λόγον ἔχει, <lb n="1799,001,,,,10,7,7"></lb> |
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8768 ὃν ἀριθμὸς πρὸς ἀριθμόν.</s> <lb n="1799,001,,,,10,7,8"></lb> |
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8769 <s>Τὰ ἄρα ἀσύμμετρα μεγέθη πρὸς ἄλληλα λόγον οὐκ ἔχει, <lb n="1799,001,,,,10,7,9"></lb> |
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8770 καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,7,10"></lb> |
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8771 <s>Ἐὰν δύο μεγέθη πρὸς ἄλληλα λόγον μὴ ἔχῃ, ὃν ἀριθμὸς <lb n="1799,001,,,,10,8,1"></lb> |
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8772 πρὸς ἀριθμόν, ἀσύμμετρα ἔσται τὰ μεγέθη.</s> <lb n="1799,001,,,,10,8,2"></lb> |
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8773 <s>Δύο γὰρ μεγέθη τὰ Α, Β πρὸς ἄλληλα λόγον <lb n="1799,001,,,,10,8,3"></lb> |
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8774 μὴ ἐχέτω, ὃν ἀριθμὸς πρὸς ἀριθμόν· λέγω, ὅτι <lb n="1799,001,,,,10,8,4"></lb> |
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8775 ἀσύμμετρά ἐστι τὰ Α, Β μεγέθη.</s> <lb n="1799,001,,,,10,8,5"></lb> |
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8776 <s>Εἰ γὰρ ἔσται σύμμετρα, τὸ Α πρὸς τὸ Β λόγον ἕξει, ὃν <lb n="1799,001,,,,10,8,6"></lb> |
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8777 ἀριθμὸς πρὸς ἀριθμόν.</s> <s>οὐκ ἔχει δέ.</s> <s>ἀσύμμετρα ἄρα ἐστὶ <lb n="1799,001,,,,10,8,7"></lb> |
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8778 τὰ Α, Β μεγέθη.</s> <lb n="1799,001,,,,10,8,8"></lb> |
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8779 <s>Ἐὰν ἄρα δύο μεγέθη πρὸς ἄλληλα, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,8,9"></lb> |
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8780 <s>Τὰ ἀπὸ τῶν μήκει συμμέτρων εὐθειῶν τετράγωνα πρὸς <lb n="1799,001,,,,10,9,1"></lb> |
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8781 ἄλληλα λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετρά- <pb></pb><lb n="1799,001,,,,10,9,2"></lb> |
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8782 γωνον ἀριθμόν· καὶ τὰ τετράγωνα τὰ πρὸς ἄλληλα λόγον <lb n="1799,001,,,,10,9,3"></lb> |
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8783 ἔχοντα, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθ-<lb n="1799,001,,,,10,9,4"></lb> |
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8784 μόν, καὶ τὰς πλευρὰς ἕξει μήκει συμμέτρους.</s> <s>τὰ δὲ ἀπὸ <lb n="1799,001,,,,10,9,5"></lb> |
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8785 τῶν μήκει ἀσυμμέτρων εὐθειῶν τετράγωνα πρὸς ἄλληλα <lb n="1799,001,,,,10,9,6"></lb> |
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8786 λόγον οὐκ ἔχει, ὅνπερ τετράγωνος ἀριθμὸς πρὸς τετρά-<lb n="1799,001,,,,10,9,7"></lb> |
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8787 γωνον ἀριθμόν· καὶ τὰ τετράγωνα τὰ πρὸς ἄλληλα λόγον <lb n="1799,001,,,,10,9,8"></lb> |
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8788 μὴ ἔχοντα, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,9,9"></lb> |
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8789 ἀριθμόν, οὐδὲ τὰς πλευρὰς ἕξει μήκει συμμέτρους.</s> <lb n="1799,001,,,,10,9,10"></lb> |
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8790 <s>Ἔστωσαν γὰρ αἱ Α, Β μήκει σύμμετροι· λέγω, ὅτι τὸ <lb n="1799,001,,,,10,9,11"></lb> |
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8791 ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς Β τετράγωνον <lb n="1799,001,,,,10,9,12"></lb> |
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8792 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,9,13"></lb> |
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8793 ἀριθμόν.</s> <lb n="1799,001,,,,10,9,14"></lb> |
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8794 <s>Ἐπεὶ γὰρ σύμμετρός ἐστιν ἡ Α τῇ Β μήκει, ἡ Α ἄρα <lb n="1799,001,,,,10,9,15"></lb> |
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8795 πρὸς τὴν Β λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν.</s> <s>ἐχέτω, <lb n="1799,001,,,,10,9,16"></lb> |
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8796 ὃν ὁ Γ πρὸς τὸν Δ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς ἡ Α πρὸς τὴν Β, <lb n="1799,001,,,,10,9,17"></lb> |
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8797 οὕτως ὁ Γ πρὸς τὸν Δ, ἀλλὰ τοῦ μὲν τῆς Α πρὸς τὴν Β <lb n="1799,001,,,,10,9,18"></lb> |
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8798 λόγου διπλασίων ἐστὶν ὁ τοῦ ἀπὸ τῆς Α τετραγώνου πρὸς <lb n="1799,001,,,,10,9,19"></lb> |
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8799 τὸ ἀπὸ τῆς Β τετράγωνον· τὰ γὰρ ὅμοια σχήματα ἐν <lb n="1799,001,,,,10,9,20"></lb> |
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8800 διπλασίονι λόγῳ ἐστὶ τῶν ὁμολόγων πλευρῶν· τοῦ δὲ <lb n="1799,001,,,,10,9,21"></lb> |
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8801 τοῦ Γ [ἀριθμοῦ] πρὸς τὸν Δ [ἀριθ- <lb n="1799,001,,,,10,9,22"></lb> |
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8802 μὸν] λόγου διπλασίων ἐστὶν ὁ τοῦ <lb n="1799,001,,,,10,9,23"></lb> |
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8803 ἀπὸ τοῦ Γ τετραγώνου πρὸς τὸν <lb n="1799,001,,,,10,9,24"></lb> |
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8804 ἀπὸ τοῦ Δ τετράγωνον· δύο γὰρ <lb n="1799,001,,,,10,9,25"></lb> |
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8805 τετραγώνων ἀριθμῶν εἷς μέσος <pb></pb> <lb n="1799,001,,,,10,9,26"></lb> |
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8806 ἀνάλογόν ἐστιν ἀριθμός, καὶ ὁ τετράγωνος πρὸς τὸν <lb n="1799,001,,,,10,9,27"></lb> |
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8807 τετράγωνον [ἀριθμὸν] διπλασίονα λόγον ἔχει, ἤπερ ἡ <lb n="1799,001,,,,10,9,28"></lb> |
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8808 πλευρὰ πρὸς τὴν πλευράν· ἔστιν ἄρα καὶ ὡς τὸ ἀπὸ τῆς Α <lb n="1799,001,,,,10,9,29"></lb> |
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8809 τετράγωνον πρὸς τὸ ἀπὸ τῆς Β τετράγωνον, οὕτως ὁ ἀπὸ <lb n="1799,001,,,,10,9,30"></lb> |
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8810 τοῦ Γ τετράγωνος [ἀριθμὸσ] πρὸς τὸν ἀπὸ τοῦ Δ [ἀριθ-<lb n="1799,001,,,,10,9,31"></lb> |
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8811 μοῦ] τετράγωνον [ἀριθμόν].</s> <lb n="1799,001,,,,10,9,32"></lb> |
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8812 <s>Ἀλλὰ δὴ ἔστω ὡς τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ <lb n="1799,001,,,,10,9,33"></lb> |
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8813 ἀπὸ τῆς Β, οὕτως ὁ ἀπὸ τοῦ Γ τετράγωνος πρὸς τὸν ἀπὸ <lb n="1799,001,,,,10,9,34"></lb> |
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8814 τοῦ Δ [τετράγωνον]· λέγω, ὅτι σύμμετρός ἐστιν ἡ Α τῇ Β <lb n="1799,001,,,,10,9,35"></lb> |
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8815 μήκει.</s> <lb n="1799,001,,,,10,9,36"></lb> |
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8816 <s>Ἐπεὶ γάρ ἐστιν ὡς τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ <lb n="1799,001,,,,10,9,37"></lb> |
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8817 ἀπὸ τῆς Β [τετράγωνον], οὕτως ὁ ἀπὸ τοῦ Γ τετράγωνος <lb n="1799,001,,,,10,9,38"></lb> |
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8818 πρὸς τὸν ἀπὸ τοῦ Δ [τετράγωνον], ἀλλ' ὁ μὲν τοῦ ἀπὸ <lb n="1799,001,,,,10,9,39"></lb> |
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8819 τῆς Α τετραγώνου πρὸς τὸ ἀπὸ τῆς Β [τετράγωνον] λόγος <lb n="1799,001,,,,10,9,40"></lb> |
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8820 διπλασίων ἐστὶ τοῦ τῆς Α πρὸς τὴν Β λόγου, ὁ δὲ τοῦ <lb n="1799,001,,,,10,9,41"></lb> |
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8821 ἀπὸ τοῦ Γ [ἀριθμοῦ] τετραγώνου [ἀριθμοῦ] πρὸς τὸν ἀπὸ <lb n="1799,001,,,,10,9,42"></lb> |
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8822 τοῦ Δ [ἀριθμοῦ] τετράγωνον [ἀριθμὸν] λόγος διπλασίων <lb n="1799,001,,,,10,9,43"></lb> |
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8823 ἐστὶ τοῦ τοῦ Γ [ἀριθμοῦ] πρὸς τὸν Δ [ἀριθμὸν] λόγου, <lb n="1799,001,,,,10,9,44"></lb> |
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8824 ἔστιν ἄρα καὶ ὡς ἡ Α πρὸς τὴν Β, οὕτως ὁ Γ [ἀριθμὸσ] <lb n="1799,001,,,,10,9,45"></lb> |
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8825 πρὸς τὸν Δ [ἀριθμόν].</s> <s>ἡ Α ἄρα πρὸς τὴν Β, λόγον ἔχει, <lb n="1799,001,,,,10,9,46"></lb> |
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8826 ὃν ἀριθμὸς ὁ Γ πρὸς ἀριθμὸν τὸν Δ· σύμμετρος ἄρα ἐστὶν <lb n="1799,001,,,,10,9,47"></lb> |
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8827 ἡ Α τῇ Β μήκει.</s> <lb n="1799,001,,,,10,9,48"></lb> |
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8828 <s>Ἀλλὰ δὴ ἀσύμμετρος ἔστω ἡ Α τῇ Β μήκει· λέγω, ὅτι <lb n="1799,001,,,,10,9,49"></lb> |
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8829 τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς Β [τετράγωνον] <lb n="1799,001,,,,10,9,50"></lb> |
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8830 λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,9,51"></lb> |
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8831 ἀριθμόν.</s> <s><pb></pb> <lb n="1799,001,,,,10,9,52"></lb> |
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8832 Εἰ γὰρ ἔχει τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,9,53"></lb> |
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8833 τῆς Β [τετράγωνον] λόγον, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,9,54"></lb> |
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8834 τετράγωνον ἀριθμόν, σύμμετρος ἔσται ἡ Α τῇ Β.</s> <s>οὐκ <lb n="1799,001,,,,10,9,55"></lb> |
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8835 ἔστι δέ· οὐκ ἄρα τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,9,56"></lb> |
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8836 τῆς Β [τετράγωνον] λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,10,9,57"></lb> |
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8837 πρὸς τετράγωνον ἀριθμόν.</s> <lb n="1799,001,,,,10,9,58"></lb> |
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8838 <s>Πάλιν δὴ τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς Β <lb n="1799,001,,,,10,9,59"></lb> |
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8839 [τετράγωνον] λόγον μὴ ἐχέτω, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,10,9,60"></lb> |
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8840 πρὸς τετράγωνον ἀριθμόν· λέγω, ὅτι ἀσύμμετρός ἐστιν <lb n="1799,001,,,,10,9,61"></lb> |
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8841 ἡ Α τῇ Β μήκει.</s> <lb n="1799,001,,,,10,9,62"></lb> |
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8842 <s>Εἰ γάρ ἐστι σύμμετρος ἡ Α τῇ Β, ἕξει τὸ ἀπὸ τῆς Α <lb n="1799,001,,,,10,9,63"></lb> |
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8843 πρὸς τὸ ἀπὸ τῆς Β λόγον, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,9,64"></lb> |
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8844 τετράγωνον ἀριθμόν.</s> <s>οὐκ ἔχει δέ· οὐκ ἄρα σύμμετρός <lb n="1799,001,,,,10,9,65"></lb> |
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8845 ἐστιν ἡ Α τῇ Β μήκει.</s> <lb n="1799,001,,,,10,9,66"></lb> |
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8846 <s>Τὰ ἄρα ἀπὸ τῶν μήκει συμμέτρων, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,9,67"></lb> |
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8847 <s>Πόρισμα <lb n="1799,001,,,,10,9,68"></lb> |
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8848 Καὶ φανερὸν ἐκ τῶν δεδειγμένων ἔσται, ὅτι αἱ μήκει <lb n="1799,001,,,,10,9,69"></lb> |
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8849 σύμμετροι πάντως καὶ δυνάμει, αἱ δὲ δυνάμει οὐ πάντως <lb n="1799,001,,,,10,9,70"></lb> |
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8850 καὶ μήκει [εἴπερ τὰ ἀπὸ τῶν μήκει συμμέτρων εὐθειῶν <lb n="1799,001,,,,10,9,71"></lb> |
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8851 τετράγωνα λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,9,72"></lb> |
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8852 τετράγωνον ἀριθμόν, τὰ δὲ λόγον ἔχοντα, ὃν ἀριθμὸς πρὸς <lb n="1799,001,,,,10,9,73"></lb> |
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8853 ἀριθμόν, σύμμετρά ἐστιν.</s> <s>ὥστε αἱ μήκει σύμμετροι <pb></pb> <lb n="1799,001,,,,10,9,74"></lb> |
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8854 εὐθεῖαι οὐ μόνον [εἰσὶ] μήκει σύμμετροι, ἀλλὰ καὶ <lb n="1799,001,,,,10,9,75"></lb> |
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8855 δυνάμει.</s> <lb n="1799,001,,,,10,9,76"></lb> |
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8856 <s>πάλιν ἐπεί, ὅσα τετράγωνα πρὸς ἄλληλα λόγον ἔχει, ὃν <lb n="1799,001,,,,10,9,77"></lb> |
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8857 τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, μήκει <lb n="1799,001,,,,10,9,78"></lb> |
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8858 ἐδείχθη σύμμετρα καὶ δυνάμει ὄντα σύμμετρα τῷ τὰ <lb n="1799,001,,,,10,9,79"></lb> |
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8859 τετράγωνα λόγον ἔχειν, ὃν ἀριθμὸς πρὸς ἀριθμόν, ὅσα ἄρα <lb n="1799,001,,,,10,9,80"></lb> |
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8860 τετράγωνα λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,9,81"></lb> |
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8861 τετράγωνον ἀριθμόν, ἀλλὰ ἁπλῶς, ὃν ἀριθμὸς πρὸς <lb n="1799,001,,,,10,9,82"></lb> |
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8862 ἀριθμόν, σύμμετρα μὲν ἔσται αὐτὰ τὰ τετράγωνα δυνάμει, <lb n="1799,001,,,,10,9,83"></lb> |
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8863 οὐκέτι δὲ καὶ μήκει· ὥστε τὰ μὲν μήκει σύμμετρα πάντως <lb n="1799,001,,,,10,9,84"></lb> |
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8864 καὶ δυνάμει, τὰ δὲ δυνάμει οὐ πάντως καὶ μήκει, εἰ μὴ καὶ <lb n="1799,001,,,,10,9,85"></lb> |
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8865 λόγον ἔχοιεν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,9,86"></lb> |
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8866 ἀριθμόν.</s> <lb n="1799,001,,,,10,9,87"></lb> |
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8867 <s>λέγω δή, ὅτι [καὶ] αἱ μήκει ἀσύμμετροι οὐ πάντως <lb n="1799,001,,,,10,9,88"></lb> |
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8868 καὶ δυνάμει, ἐπειδήπερ αἱ δυνάμει σύμμετροι δύνανται <lb n="1799,001,,,,10,9,89"></lb> |
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8869 λόγον μὴ ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,9,90"></lb> |
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8870 ἀριθμόν, καὶ διὰ τοῦτο δυνάμει οὖσαι σύμμετροι μήκει <lb n="1799,001,,,,10,9,91"></lb> |
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8871 εἰσὶν ἀσύμμετροι.</s> <s>ὥστε οὐχ αἱ τῷ μήκει ἀσύμμετροι <lb n="1799,001,,,,10,9,92"></lb> |
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8872 πάντως καὶ δυνάμει, ἀλλὰ δύνανται μήκει οὖσαι <lb n="1799,001,,,,10,9,93"></lb> |
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8873 ἀσύμμετροι δυνάμει εἶναι καὶ ἀσύμμετροι καὶ σύμμετροι.</s> <lb n="1799,001,,,,10,9,94"></lb> |
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8874 <s>αἱ δὲ δυνάμει ἀσύμμετροι πάντως καὶ μήκει ἀσύμ-<lb n="1799,001,,,,10,9,95"></lb> |
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8875 μετροι· εἰ γὰρ [εἰσι] μήκει σύμμετροι, ἔσονται καὶ δυνάμει <pb></pb> <lb n="1799,001,,,,10,9,96"></lb> |
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8876 σύμμετροι.</s> <s>ὑπόκεινται δὲ καὶ ἀσύμμετροι· ὅπερ ἄτοπον.</s> <lb n="1799,001,,,,10,9,97"></lb> |
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8877 <s>αἱ ἄρα δυνάμει ἀσύμμετροι πάντως καὶ μήκει].</s> <lb n="1799,001,,,,10,9,98"></lb> |
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8878 <s>Λῆμμα <lb n="1799,001,,,,10,9,99"></lb> |
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8879 Δέδεικται ἐν τοῖς ἀριθμητικοῖς, ὅτι οἱ ὅμοιοι ἐπίπεδοι <lb n="1799,001,,,,10,9,100"></lb> |
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8880 ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχουσιν, ὃν τετράγωνος <lb n="1799,001,,,,10,9,101"></lb> |
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8881 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, καὶ ὅτι, ἐὰν δύο <lb n="1799,001,,,,10,9,102"></lb> |
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8882 ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχωσιν, ὃν τετράγωνος <lb n="1799,001,,,,10,9,103"></lb> |
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8883 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ὅμοιοί εἰσιν ἐπίπεδοι.</s> <lb n="1799,001,,,,10,9,104"></lb> |
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8884 <s>καὶ δῆλον ἐκ τούτων, ὅτι οἱ μὴ ὅμοιοι ἐπίπεδοι ἀριθμοί, <lb n="1799,001,,,,10,9,105"></lb> |
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8885 τουτέστιν οἱ μὴ ἀνάλογον ἔχοντες τὰς πλευράς, πρὸς <lb n="1799,001,,,,10,9,106"></lb> |
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8886 ἀλλήλους λόγον οὐκ ἔχουσιν, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,10,9,107"></lb> |
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8887 πρὸς τετράγωνον ἀριθμόν.</s> <s>εἰ γὰρ ἕξουσιν, ὅμοιοι ἐπίπεδοι <lb n="1799,001,,,,10,9,108"></lb> |
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8888 ἔσονται· ὅπερ οὐχ ὑπόκειται.</s> <s>οἱ ἄρα μὴ ὅμοιοι ἐπίπεδοι <lb n="1799,001,,,,10,9,109"></lb> |
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8889 πρὸς ἀλλήλους λόγον οὐκ ἔχουσιν, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,10,9,110"></lb> |
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8890 πρὸς τετράγωνον ἀριθμόν.</s> <lb n="1799,001,,,,10,9,111"></lb> |
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8891 <s>Τῇ προτεθείσῃ εὐθείᾳ προσευρεῖν δύο εὐθείας ἀσυμ-<lb n="1799,001,,,,10,10,1"></lb> |
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8892 μέτρους, τὴν μὲν μήκει μόνον, τὴν δὲ καὶ δυνάμει.</s> <lb n="1799,001,,,,10,10,2"></lb> |
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8893 <s>Ἔστω ἡ προτεθεῖσα εὐθεῖα ἡ Α· δεῖ δὴ τῇ Α προσευρεῖν <lb n="1799,001,,,,10,10,3"></lb> |
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8894 δύο εὐθείας ἀσυμμέτρους, τὴν μὲν μήκει μόνον, τὴν δὲ καὶ <lb n="1799,001,,,,10,10,4"></lb> |
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8895 δυνάμει.</s> <s><pb></pb> <lb n="1799,001,,,,10,10,5"></lb> |
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8896 Ἐκκείσθωσαν γὰρ δύο ἀριθμοὶ οἱ Β, Γ πρὸς ἀλλήλους <lb n="1799,001,,,,10,10,6"></lb> |
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8897 λόγον μὴ ἔχοντες, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,10,7"></lb> |
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8898 ἀριθμόν, τουτέστι μὴ ὅμοιοι ἐπίπεδοι, καὶ γεγονέτω ὡς <lb n="1799,001,,,,10,10,8"></lb> |
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8899 ὁ Β πρὸς τὸν Γ, οὕτως τὸ ἀπὸ τῆς Α <lb n="1799,001,,,,10,10,9"></lb> |
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8900 τετράγωνον πρὸς τὸ ἀπὸ τῆς Δ τε-<lb n="1799,001,,,,10,10,10"></lb> |
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8901 τράγωνον· ἐμάθομεν γάρ· σύμμετρον <lb n="1799,001,,,,10,10,11"></lb> |
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8902 ἄρα τὸ ἀπὸ τῆς Α τῷ ἀπὸ τῆς Δ.</s> <lb n="1799,001,,,,10,10,12"></lb> |
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8903 <s>καὶ ἐπεὶ ὁ Β πρὸς τὸν Γ λόγον οὐκ <lb n="1799,001,,,,10,10,13"></lb> |
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8904 ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,10,14"></lb> |
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8905 τετράγωνον ἀριθμόν, οὐδ' ἄρα τὸ ἀπὸ <lb n="1799,001,,,,10,10,15"></lb> |
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8906 τῆς Α πρὸς τὸ ἀπὸ τῆς Δ λόγον ἔχει, ὃν τετράγωνος ἀριθ-<lb n="1799,001,,,,10,10,16"></lb> |
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8907 μὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ <lb n="1799,001,,,,10,10,17"></lb> |
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8908 Α τῇ Δ μήκει.</s> <s>εἰλήφθω τῶν Α, Δ μέση ἀνάλογον ἡ Ε· <lb n="1799,001,,,,10,10,18"></lb> |
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8909 ἔστιν ἄρα ὡς ἡ Α πρὸς τὴν Δ, οὕτως τὸ ἀπὸ τῆς Α τετρά-<lb n="1799,001,,,,10,10,19"></lb> |
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8910 γωνον πρὸς τὸ ἀπὸ τῆς Ε.</s> <s>ἀσύμμετρος δέ ἐστιν ἡ Α τῇ Δ <lb n="1799,001,,,,10,10,20"></lb> |
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8911 μήκει· ἀσύμμετρον ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς Α τετράγωνον <lb n="1799,001,,,,10,10,21"></lb> |
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8912 τῷ ἀπὸ τῆς Ε τετραγώνῳ· ἀσύμμετρος ἄρα ἐστὶν ἡ Α τῇ <lb n="1799,001,,,,10,10,22"></lb> |
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8913 Ε δυνάμει.</s> <lb n="1799,001,,,,10,10,23"></lb> |
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8914 <s>Τῇ ἄρα προτεθείσῃ εὐθείᾳ τῇ Α προσεύρηνται δύο <lb n="1799,001,,,,10,10,24"></lb> |
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8915 εὐθεῖαι ἀσύμμετροι αἱ Δ, Ε, μήκει μὲν μόνον ἡ Δ, δυνάμει <lb n="1799,001,,,,10,10,25"></lb> |
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8916 δὲ καὶ μήκει δηλαδὴ ἡ Ε [ὅπερ ἔδει δεῖξαι].</s> <s><pb></pb> <lb n="1799,001,,,,10,10,26"></lb> |
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8917 Ἐὰν τέσσαρα μεγέθη ἀνάλογον ᾖ, τὸ δὲ πρῶτον τῷ <lb n="1799,001,,,,10,11,1"></lb> |
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8918 δευτέρῳ σύμμετρον ᾖ, καὶ τὸ τρίτον τῷ τετάρτῳ σύμμετρον <lb n="1799,001,,,,10,11,2"></lb> |
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8919 ἔσται· κἂν τὸ πρῶτον τῷ δευτέρῳ ἀσύμμετρον ᾖ, καὶ τὸ <lb n="1799,001,,,,10,11,3"></lb> |
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8920 τρίτον τῷ τετάρτῳ ἀσύμμετρον ἔσται. <lb n="1799,001,,,,10,11,4"></lb> |
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8921 Ἔστωσαν τέσσαρα μεγέθη ἀνάλογον τὰ Α, Β, Γ, Δ, ὡς <lb n="1799,001,,,,10,11,5"></lb> |
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8922 τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ πρὸς τὸ Δ, τὸ Α δὲ τῷ Β <lb n="1799,001,,,,10,11,6"></lb> |
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8923 σύμμετρον ἔστω· λέγω, ὅτι καὶ τὸ Γ τῷ Δ σύμμετρον <lb n="1799,001,,,,10,11,7"></lb> |
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8924 ἔσται.</s> <lb n="1799,001,,,,10,11,8"></lb> |
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8925 <s>Ἐπεὶ γὰρ σύμμετρόν ἐστι τὸ Α τῷ Β, τὸ Α ἄρα πρὸς <lb n="1799,001,,,,10,11,9"></lb> |
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8926 τὸ Β λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν.</s> <s>καί ἐστιν ὡς <lb n="1799,001,,,,10,11,10"></lb> |
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8927 τὸ Α πρὸς τὸ Β, οὕτως τὸ Γ πρὸς τὸ Δ· καὶ τὸ Γ ἄρα πρὸς <lb n="1799,001,,,,10,11,11"></lb> |
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8928 τὸ Δ λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν· σύμμετρον ἄρα <lb n="1799,001,,,,10,11,12"></lb> |
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8929 ἐστὶ τὸ Γ τῷ Δ.</s> <lb n="1799,001,,,,10,11,13"></lb> |
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8930 <s>Ἀλλὰ δὴ τὸ Α τῷ Β ἀσύμμετρον ἔστω· λέγω, ὅτι καὶ <lb n="1799,001,,,,10,11,14"></lb> |
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8931 τὸ Γ τῷ Δ ἀσύμμετρον ἔσται.</s> <s>ἐπεὶ γὰρ ἀσύμμετρόν ἐστι <lb n="1799,001,,,,10,11,15"></lb> |
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8932 τὸ Α τῷ Β, τὸ Α ἄρα πρὸς τὸ Β λόγον οὐκ ἔχει, ὃν <lb n="1799,001,,,,10,11,16"></lb> |
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8933 ἀριθμὸς πρὸς ἀριθμόν.</s> <s>καί ἐστιν ὡς τὸ Α πρὸς τὸ Β, <lb n="1799,001,,,,10,11,17"></lb> |
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8934 οὕτως τὸ Γ πρὸς τὸ Δ· οὐδὲ τὸ Γ ἄρα πρὸς τὸ Δ λόγον <lb n="1799,001,,,,10,11,18"></lb> |
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8935 ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν· ἀσύμμετρον ἄρα ἐστὶ τὸ Γ <lb n="1799,001,,,,10,11,19"></lb> |
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8936 τῷ Δ.</s> <lb n="1799,001,,,,10,11,20"></lb> |
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8937 <s>Ἐὰν ἄρα τέσσαρα μεγέθη, καὶ τὰ ἑξῆς.</s> <s><pb></pb> <lb n="1799,001,,,,10,11,21"></lb> |
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8938 Τὰ τῷ αὐτῷ μεγέθει σύμμετρα καὶ ἀλλήλοις ἐστὶ <lb n="1799,001,,,,10,12,1"></lb> |
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8939 σύμμετρα.</s> <lb n="1799,001,,,,10,12,2"></lb> |
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8940 <s>Ἑκάτερον γὰρ τῶν Α, Β τῷ Γ ἔστω σύμμετρον.</s> <s>λέγω, <lb n="1799,001,,,,10,12,3"></lb> |
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8941 ὅτι καὶ τὸ Α τῷ Β ἐστι σύμμετρον.</s> <lb n="1799,001,,,,10,12,4"></lb> |
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8942 <s>Ἐπεὶ γὰρ σύμμετρόν ἐστι τὸ Α τῷ Γ, τὸ Α ἄρα πρὸς <lb n="1799,001,,,,10,12,5"></lb> |
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8943 τὸ Γ λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν.</s> <s>ἐχέτω, ὃν ὁ Δ <lb n="1799,001,,,,10,12,6"></lb> |
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8944 πρὸς τὸν Ε.</s> <s>πάλιν, ἐπεὶ σύμμετρόν ἐστι τὸ Γ τῷ Β, τὸ Γ <lb n="1799,001,,,,10,12,7"></lb> |
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8945 ἄρα πρὸς τὸ Β λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν.</s> <lb n="1799,001,,,,10,12,8"></lb> |
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8946 <s>ἐχέτω, ὃν ὁ Ζ πρὸς τὸν Η.</s> <s>καὶ λόγων δοθέντων ὁποσωνοῦν <lb n="1799,001,,,,10,12,9"></lb> |
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8947 τοῦ τε, ὃν ἔχει ὁ Δ πρὸς τὸν Ε, καὶ ὁ Ζ πρὸς τὸν Η <lb n="1799,001,,,,10,12,10"></lb> |
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8948 εἰλήφθωσαν ἀριθμοὶ ἑξῆς ἐν τοῖς δοθεῖσι λόγοις οἱ Θ, Κ, <lb n="1799,001,,,,10,12,11"></lb> |
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8949 Λ· ὥστε εἶναι ὡς μὲν τὸν Δ πρὸς τὸν Ε, οὕτως τὸν Θ <lb n="1799,001,,,,10,12,12"></lb> |
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8950 πρὸς τὸν Κ, ὡς δὲ τὸν Ζ πρὸς τὸν Η, οὕτως τὸν Κ πρὸς <lb n="1799,001,,,,10,12,13"></lb> |
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8951 τὸν Λ.</s> <lb n="1799,001,,,,10,12,14"></lb> |
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8952 <s>Ἐπεὶ οὖν ἐστιν ὡς τὸ Α πρὸς τὸ Γ, οὕτως ὁ Δ πρὸς <lb n="1799,001,,,,10,12,15"></lb> |
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8953 τὸν Ε, ἀλλ' ὡς ὁ Δ πρὸς τὸν Ε, οὕτως ὁ Θ πρὸς τὸν Κ, <lb n="1799,001,,,,10,12,16"></lb> |
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8954 ἔστιν ἄρα καὶ ὡς τὸ Α πρὸς τὸ Γ, οὕτως ὁ Θ πρὸς τὸν Κ.</s> <lb n="1799,001,,,,10,12,17"></lb> |
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8955 <s>πάλιν, ἐπεί ἐστιν ὡς τὸ Γ πρὸς τὸ Β, οὕτως ὁ Ζ πρὸς τὸν <lb n="1799,001,,,,10,12,18"></lb> |
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8956 Η, ἀλλ' ὡς ὁ Ζ πρὸς τὸν Η, [οὕτωσ] ὁ Κ πρὸς τὸν Λ, καὶ <pb></pb> <lb n="1799,001,,,,10,12,19"></lb> |
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8957 ὡς ἄρα τὸ Γ πρὸς τὸ Β, οὕτως ὁ Κ πρὸς τὸν Λ.</s> <s>ἔστι δὲ <lb n="1799,001,,,,10,12,20"></lb> |
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8958 καὶ ὡς τὸ Α πρὸς τὸ Γ, οὕτως ὁ Θ πρὸς τὸν Κ· δι' ἴσου <lb n="1799,001,,,,10,12,21"></lb> |
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8959 ἄρα ἐστὶν ὡς τὸ Α πρὸς τὸ Β, οὕτως ὁ Θ πρὸς τὸν Λ.</s> <lb n="1799,001,,,,10,12,22"></lb> |
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8960 <s>τὸ Α ἄρα πρὸς τὸ Β λόγον ἔχει, ὃν ἀριθμὸς ὁ Θ πρὸς <lb n="1799,001,,,,10,12,23"></lb> |
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8961 ἀριθμὸν τὸν Λ· σύμμετρον ἄρα ἐστὶ τὸ Α τῷ Β.</s> <lb n="1799,001,,,,10,12,24"></lb> |
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8962 <s>Τὰ ἄρα τῷ αὐτῷ μεγέθει σύμμετρα καὶ ἀλλήλοις ἐστὶ <lb n="1799,001,,,,10,12,25"></lb> |
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8963 σύμμετρα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,12,26"></lb> |
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8964 <s>Ἐὰν ᾖ δύο μεγέθη σύμμετρα, τὸ δὲ ἕτερον αὐτῶν <lb n="1799,001,,,,10,13,1"></lb> |
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8965 μεγέθει τινὶ ἀσύμμετρον ᾖ, καὶ τὸ λοιπὸν τῷ αὐτῷ ἀσύμ-<lb n="1799,001,,,,10,13,2"></lb> |
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8966 μετρον ἔσται.</s> <lb n="1799,001,,,,10,13,3"></lb> |
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8967 <s>Ἔστω δύο μεγέθη σύμμετρα τὰ Α, Β, τὸ δὲ ἕτερον <lb n="1799,001,,,,10,13,4"></lb> |
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8968 αὐτῶν τὸ Α ἄλλῳ τινὶ τῷ Γ ἀσύμμετρον ἔστω· λέγω, ὅτι <lb n="1799,001,,,,10,13,5"></lb> |
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8969 καὶ τὸ λοιπὸν τὸ Β τῷ Γ ἀσύμμε- <lb n="1799,001,,,,10,13,6"></lb> |
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8970 τρόν ἐστιν.</s> <lb n="1799,001,,,,10,13,7"></lb> |
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8971 <s>Εἰ γάρ ἐστι σύμμετρον τὸ Β τῷ Γ, <lb n="1799,001,,,,10,13,8"></lb> |
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8972 ἀλλὰ καὶ τὸ Α τῷ Β σύμμετρόν <lb n="1799,001,,,,10,13,9"></lb> |
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8973 ἐστιν, καὶ τὸ Α ἄρα τῷ Γ σύμμετρόν ἐστιν.</s> <s>ἀλλὰ καὶ <lb n="1799,001,,,,10,13,10"></lb> |
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8974 ἀσύμμετρον· ὅπερ ἀδύνατον.</s> <s>οὐκ ἄρα σύμμετρόν ἐστι τὸ <lb n="1799,001,,,,10,13,11"></lb> |
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8975 Β τῷ Γ· ἀσύμμετρον ἄρα.</s> <lb n="1799,001,,,,10,13,12"></lb> |
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8976 <s>Ἐὰν ἄρα ᾖ δύο μεγέθη σύμμετρα, καὶ τὰ ἑξῆς.</s> <s><pb></pb> <lb n="1799,001,,,,10,13,13"></lb> |
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8977 Λῆμμα <lb n="1799,001,,,,10,13,14"></lb> |
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8978 Δύο δοθεισῶν εὐθειῶν ἀνίσων εὑρεῖν, τίνι μεῖζον <lb n="1799,001,,,,10,13,15"></lb> |
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8979 δύναται ἡ μείζων τῆς ἐλάσσονος.</s> <lb n="1799,001,,,,10,13,16"></lb> |
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8980 <s>Ἔστωσαν αἱ δοθεῖσαι δύο ἄνισοι εὐθεῖαι αἱ ΑΒ, Γ, <lb n="1799,001,,,,10,13,17"></lb> |
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8981 ὧν μείζων ἔστω ἡ ΑΒ· δεῖ δὴ εὑρεῖν, τίνι μεῖζον δύναται <lb n="1799,001,,,,10,13,18"></lb> |
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8982 ἡ ΑΒ τῆς Γ.</s> <lb n="1799,001,,,,10,13,19"></lb> |
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8983 <s>Γεγράφθω ἐπὶ τῆς ΑΒ ἡμικύκλιον τὸ ΑΔΒ, καὶ <lb n="1799,001,,,,10,13,20"></lb> |
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8984 εἰς αὐτὸ ἐνηρμόσθω τῇ Γ ἴση ἡ ΑΔ, καὶ ἐπεζεύχθω ἡ ΔΒ.</s> <lb n="1799,001,,,,10,13,21"></lb> |
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8985 <s>φανερὸν δή, ὅτι ὀρθή ἐστιν ἡ ὑπὸ ΑΔΒ γωνία, καὶ ὅτι <lb n="1799,001,,,,10,13,22"></lb> |
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8986 ἡ ΑΒ τῆς ΑΔ, τουτέστι τῆς Γ, <lb n="1799,001,,,,10,13,23"></lb> |
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8987 μεῖζον δύναται τῇ ΔΒ.</s> <lb n="1799,001,,,,10,13,24"></lb> |
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8988 <s>Ὁμοίως δὲ καὶ δύο δοθεισῶν <lb n="1799,001,,,,10,13,25"></lb> |
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8989 εὐθειῶν ἡ δυναμένη αὐτὰς εὑρί-<lb n="1799,001,,,,10,13,26"></lb> |
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8990 σκεται οὕτως.</s> <lb n="1799,001,,,,10,13,27"></lb> |
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8991 <s>Ἔστωσαν αἱ δοθεῖσαι δύο εὐθεῖαι αἱ ΑΔ, ΔΒ, καὶ δέον <lb n="1799,001,,,,10,13,28"></lb> |
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8992 ἔστω εὑρεῖν τὴν δυναμένην αὐτάς.</s> <s>κείσθωσαν γάρ, ὥστε <lb n="1799,001,,,,10,13,29"></lb> |
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8993 ὀρθὴν γωνίαν περιέχειν τὴν ὑπὸ ΑΔ, ΔΒ, καὶ ἐπεζεύχθω <lb n="1799,001,,,,10,13,30"></lb> |
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8994 ἡ ΑΒ· φανερὸν πάλιν, ὅτι ἡ τὰς ΑΔ, ΔΒ δυναμένη ἐστὶν <lb n="1799,001,,,,10,13,31"></lb> |
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8995 ἡ ΑΒ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,13,32"></lb> |
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8996 <s>Ἐὰν τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, δύνηται δὲ ἡ <lb n="1799,001,,,,10,14,1"></lb> |
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8997 πρώτη τῆς δευτέρας μεῖζον τῷ ἀπὸ συμμέτρου ἑαυτῇ <lb n="1799,001,,,,10,14,2"></lb> |
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8998 [μήκει], καὶ ἡ τρίτη τῆς τετάρτης μεῖζον δυνήσεται τῷ <lb n="1799,001,,,,10,14,3"></lb> |
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8999 ἀπὸ συμμέτρου ἑαυτῇ [μήκει].</s> <s>καὶ ἐὰν ἡ πρώτη τῆς <pb></pb> <lb n="1799,001,,,,10,14,4"></lb> |
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9000 δευτέρας μεῖζον δύνηται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ <lb n="1799,001,,,,10,14,5"></lb> |
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9001 [μήκει], καὶ ἡ τρίτη τῆς τετάρτης μεῖζον δυνήσεται τῷ <lb n="1799,001,,,,10,14,6"></lb> |
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9002 ἀπὸ ἀσυμμέτρου ἑαυτῇ [μήκει].</s> <lb n="1799,001,,,,10,14,7"></lb> |
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9003 <s>Ἔστωσαν τέσσαρες εὐθεῖαι ἀνάλογον αἱ Α, Β, Γ, Δ, <lb n="1799,001,,,,10,14,8"></lb> |
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9004 ὡς ἡ Α πρὸς τὴν Β, οὕτως ἡ Γ πρὸς τὴν Δ, καὶ ἡ Α μὲν τῆς <lb n="1799,001,,,,10,14,9"></lb> |
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9005 Β μεῖζον δυνάσθω τῷ ἀπὸ τῆς Ε, ἡ δὲ Γ τῆς Δ μεῖζον <lb n="1799,001,,,,10,14,10"></lb> |
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9006 δυνάσθω τῷ ἀπὸ τῆς Ζ· λέγω, ὅτι, εἴτε σύμμετρόσ ἐστιν <lb n="1799,001,,,,10,14,11"></lb> |
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9007 ἡ Α τῇ Ε, σύμμετρός ἐστι καὶ ἡ Γ τῇ Ζ, εἴτε ἀσύμμετρός <lb n="1799,001,,,,10,14,12"></lb> |
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9008 ἐστιν ἡ Α τῇ Ε, ἀσύμμετρός ἐστι καὶ ἡ Γ τῇ Ζ.</s> <lb n="1799,001,,,,10,14,13"></lb> |
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9009 <s>Ἐπεὶ γάρ ἐστιν ὡς ἡ Α πρὸς τὴν Β, <lb n="1799,001,,,,10,14,14"></lb> |
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9010 οὕτως ἡ Γ πρὸς τὴν Δ, ἔστιν ἄρα καὶ <lb n="1799,001,,,,10,14,15"></lb> |
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9011 ὡς τὸ ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς Β, <lb n="1799,001,,,,10,14,16"></lb> |
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9012 οὕτως τὸ ἀπὸ τῆς Γ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,14,17"></lb> |
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9013 Δ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς Α ἴσα ἐστὶ <lb n="1799,001,,,,10,14,18"></lb> |
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9014 τὰ ἀπὸ τῶν Ε, Β, τῷ δὲ ἀπὸ τῆς Γ <lb n="1799,001,,,,10,14,19"></lb> |
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9015 ἴσα ἐστὶ τὰ ἀπὸ τῶν Δ, Ζ.</s> <s>ἔστιν ἄρα <lb n="1799,001,,,,10,14,20"></lb> |
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9016 ὡς τὰ ἀπὸ τῶν Ε, Β πρὸς τὸ <lb n="1799,001,,,,10,14,21"></lb> |
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9017 ἀπὸ τῆς Β, οὕτως τὰ ἀπὸ τῶν Δ, Ζ πρὸς τὸ ἀπὸ τῆς Δ· <lb n="1799,001,,,,10,14,22"></lb> |
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9018 διελόντι ἄρα ἐστὶν ὡς τὸ ἀπὸ τῆς Ε πρὸς τὸ ἀπὸ τῆς Β, <lb n="1799,001,,,,10,14,23"></lb> |
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9019 οὕτως τὸ ἀπὸ τῆς Ζ πρὸς τὸ ἀπὸ τῆς Δ· ἔστιν ἄρα καὶ ὡς <lb n="1799,001,,,,10,14,24"></lb> |
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9020 ἡ Ε πρὸς τὴν Β, οὕτως ἡ Ζ πρὸς τὴν Δ· ἀνάπαλιν ἄρα <lb n="1799,001,,,,10,14,25"></lb> |
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9021 ἐστὶν ὡς ἡ Β πρὸς τὴν Ε, οὕτως ἡ Δ πρὸς τὴν Ζ.</s> <s>ἔστι δὲ <lb n="1799,001,,,,10,14,26"></lb> |
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9022 καὶ ὡς ἡ Α πρὸς τὴν Β, οὕτως ἡ Γ πρὸς τὴν Δ· δι' ἴσου <pb></pb> <lb n="1799,001,,,,10,14,27"></lb> |
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9023 ἄρα ἐστὶν ὡς ἡ Α πρὸς τὴν Ε, οὕτως ἡ Γ πρὸς τὴν Ζ.</s> <lb n="1799,001,,,,10,14,28"></lb> |
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9024 <s>εἴτε οὖν σύμμετρός ἐστιν ἡ Α τῇ Ε, σύμμετρός ἐστι καὶ <lb n="1799,001,,,,10,14,29"></lb> |
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9025 ἡ Γ τῇ Ζ, εἴτε ἀσύμμετρός ἐστιν ἡ Α τῇ Ε, ἀσύμμετρός <lb n="1799,001,,,,10,14,30"></lb> |
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9026 ἐστι καὶ ἡ Γ τῇ Ζ.</s> <lb n="1799,001,,,,10,14,31"></lb> |
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9027 <s>Ἐὰν ἄρα, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,14,32"></lb> |
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9028 <s>Ἐὰν δύο μεγέθη σύμμετρα συντεθῇ, καὶ τὸ ὅλον <lb n="1799,001,,,,10,15,1"></lb> |
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9029 ἑκατέρῳ αὐτῶν σύμμετρον ἔσται· κἂν τὸ ὅλον ἑνὶ αὐτῶν <lb n="1799,001,,,,10,15,2"></lb> |
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9030 σύμμετρον ᾖ, καὶ τὰ ἐξ ἀρχῆς μεγέθη σύμμετρα ἔσται.</s> <lb n="1799,001,,,,10,15,3"></lb> |
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9031 <s>Συγκείσθω γὰρ δύο μεγέθη σύμμετρα τὰ ΑΒ, ΒΓ· <lb n="1799,001,,,,10,15,4"></lb> |
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9032 λέγω, ὅτι καὶ ὅλον τὸ ΑΓ ἑκατέρῳ τῶν ΑΒ, ΒΓ ἐστι <lb n="1799,001,,,,10,15,5"></lb> |
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9033 σύμμετρον.</s> <lb n="1799,001,,,,10,15,6"></lb> |
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9034 <s>Ἐπεὶ γὰρ σύμμετρά ἐστι τὰ ΑΒ, ΒΓ, μετρήσει τι <lb n="1799,001,,,,10,15,7"></lb> |
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9035 αὐτὰ μέγεθος.</s> <s>μετρείτω, καὶ ἔστω τὸ Δ.</s> <s>ἐπεὶ οὖν τὸ Δ τὰ <lb n="1799,001,,,,10,15,8"></lb> |
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9036 ΑΒ, ΒΓ μετρεῖ, καὶ ὅλον τὸ ΑΓ μετρήσει.</s> <s>μετρεῖ δὲ καὶ <lb n="1799,001,,,,10,15,9"></lb> |
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9037 τὰ ΑΒ, ΒΓ.</s> <s>τὸ Δ ἄρα τὰ <lb n="1799,001,,,,10,15,10"></lb> |
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9038 ΑΒ, ΒΓ, ΑΓ μετρεῖ· σύμ-<lb n="1799,001,,,,10,15,11"></lb> |
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9039 μετρον ἄρα ἐστὶ τὸ ΑΓ <lb n="1799,001,,,,10,15,12"></lb> |
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9040 ἑκατέρῳ τῶν ΑΒ, ΒΓ.</s> <lb n="1799,001,,,,10,15,13"></lb> |
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9041 <s>Ἀλλὰ δὴ τὸ ΑΓ ἔστω σύμμετρον τῷ ΑΒ· λέγω δή, <lb n="1799,001,,,,10,15,14"></lb> |
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9042 ὅτι καὶ τὰ ΑΒ, ΒΓ σύμμετρά ἐστιν.</s> <lb n="1799,001,,,,10,15,15"></lb> |
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9043 <s>Ἐπεὶ γὰρ σύμμετρά ἐστι τὰ ΑΓ, ΑΒ, μετρήσει τι <lb n="1799,001,,,,10,15,16"></lb> |
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9044 αὐτὰ μέγεθος.</s> <s>μετρείτω, καὶ ἔστω τὸ Δ.</s> <s>ἐπεὶ οὖν τὸ Δ <pb></pb> <lb n="1799,001,,,,10,15,17"></lb> |
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9045 τὰ ΓΑ, ΑΒ μετρεῖ, καὶ λοιπὸν ἄρα τὸ ΒΓ μετρήσει.</s> <lb n="1799,001,,,,10,15,18"></lb> |
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9046 <s>μετρεῖ δὲ καὶ τὸ ΑΒ· τὸ Δ ἄρα τὰ ΑΒ, ΒΓ μετρήσει· <lb n="1799,001,,,,10,15,19"></lb> |
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9047 σύμμετρα ἄρα ἐστὶ τὰ ΑΒ, ΒΓ.</s> <lb n="1799,001,,,,10,15,20"></lb> |
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9048 <s>Ἐὰν ἄρα δύο μεγέθη, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,15,21"></lb> |
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9049 <s>Ἐὰν δύο μεγέθη ἀσύμμετρα συντεθῇ, καὶ τὸ ὅλον <lb n="1799,001,,,,10,16,1"></lb> |
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9050 ἑκατέρῳ αὐτῶν ἀσύμμετρον ἔσται· κἂν τὸ ὅλον ἑνὶ αὐτῶν <lb n="1799,001,,,,10,16,2"></lb> |
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9051 ἀσύμμετρον ᾖ, καὶ τὰ ἐξ ἀρχῆς μεγέθη ἀσύμμετρα ἔσται.</s> <lb n="1799,001,,,,10,16,3"></lb> |
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9052 <s>Συγκείσθω γὰρ δύο μεγέθη ἀσύμμετρα τὰ ΑΒ, ΒΓ· <lb n="1799,001,,,,10,16,4"></lb> |
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9053 λέγω, ὅτι καὶ ὅλον τὸ ΑΓ ἑκατέρῳ τῶν ΑΒ, ΒΓ ἀσύμ-<lb n="1799,001,,,,10,16,5"></lb> |
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9054 μετρόν ἐστιν.</s> <lb n="1799,001,,,,10,16,6"></lb> |
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9055 <s>Εἰ γὰρ μή ἐστιν ἀσύμμετρα τὰ ΓΑ, ΑΒ, μετρήσει τι <lb n="1799,001,,,,10,16,7"></lb> |
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9056 [αὐτὰ] μέγεθος.</s> <s>μετρείτω, εἰ δυνατόν, καὶ ἔστω τὸ Δ.</s> <lb n="1799,001,,,,10,16,8"></lb> |
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9057 <s>ἐπεὶ οὖν τὸ Δ τὰ ΓΑ, ΑΒ μετρεῖ, καὶ λοιπὸν ἄρα τὸ ΒΓ <lb n="1799,001,,,,10,16,9"></lb> |
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9058 μετρήσει.</s> <s>μετρεῖ δὲ καὶ τὸ ΑΒ· τὸ Δ ἄρα τὰ ΑΒ, ΒΓ <lb n="1799,001,,,,10,16,10"></lb> |
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9059 μετρεῖ.</s> <s>σύμμετρα ἄρα ἐστὶ τὰ ΑΒ, ΒΓ· ὑπέκειντο δὲ καὶ <lb n="1799,001,,,,10,16,11"></lb> |
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9060 ἀσύμμετρα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τὰ <lb n="1799,001,,,,10,16,12"></lb> |
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9061 ΓΑ, ΑΒ μετρήσει τι μέγεθος· ἀσύμμετρα ἄρα <lb n="1799,001,,,,10,16,13"></lb> |
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9062 ἐστὶ τὰ ΓΑ, ΑΒ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ τὰ <lb n="1799,001,,,,10,16,14"></lb> |
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9063 ΑΓ, ΓΒ ἀσύμμετρά ἐστιν.</s> <s>τὸ ΑΓ ἄρα ἑκατέρῳ <lb n="1799,001,,,,10,16,15"></lb> |
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9064 τῶν ΑΒ, ΒΓ ἀσύμμετρόν ἐστιν.</s> <lb n="1799,001,,,,10,16,16"></lb> |
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9065 <s>Ἀλλὰ δὴ τὸ ΑΓ ἑνὶ τῶν ΑΒ, ΒΓ ἀσύμμετρον <lb n="1799,001,,,,10,16,17"></lb> |
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9066 ἔστω.</s> <s>ἔστω δὴ πρότερον τῷ ΑΒ· λέγω, ὅτι <lb n="1799,001,,,,10,16,18"></lb> |
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9067 καὶ τὰ ΑΒ, ΒΓ ἀσύμμετρά ἐστιν.</s> <s>εἰ γὰρ <pb></pb> <lb n="1799,001,,,,10,16,19"></lb> |
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9068 ἔσται σύμμετρα, μετρήσει τι αὐτὰ μέγεθος.</s> <s>μετρείτω, καὶ <lb n="1799,001,,,,10,16,20"></lb> |
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9069 ἔστω τὸ Δ.</s> <s>ἐπεὶ οὖν τὸ Δ τὰ ΑΒ, ΒΓ μετρεῖ, καὶ ὅλον ἄρα <lb n="1799,001,,,,10,16,21"></lb> |
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9070 τὸ ΑΓ μετρήσει.</s> <s>μετρεῖ δὲ καὶ τὸ ΑΒ· τὸ Δ ἄρα τὰ ΓΑ, <lb n="1799,001,,,,10,16,22"></lb> |
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9071 ΑΒ μετρεῖ.</s> <s>σύμμετρα ἄρα ἐστὶ τὰ ΓΑ, ΑΒ· ὑπέκειτο δὲ <lb n="1799,001,,,,10,16,23"></lb> |
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9072 καὶ ἀσύμμετρα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα τὰ ΑΒ, ΒΓ <lb n="1799,001,,,,10,16,24"></lb> |
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9073 μετρήσει τι μέγεθος· ἀσύμμετρα ἄρα ἐστὶ τὰ ΑΒ, ΒΓ.</s> <lb n="1799,001,,,,10,16,25"></lb> |
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9074 <s>Ἐὰν ἄρα δύο μεγέθη, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,16,26"></lb> |
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9075 <s>Λῆμμα <lb n="1799,001,,,,10,16,27"></lb> |
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9076 Ἐὰν παρά τινα εὐθεῖαν παραβληθῇ παραλληλόγραμμον <lb n="1799,001,,,,10,16,28"></lb> |
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9077 ἐλλεῖπον εἴδει τετραγώνῳ, τὸ παραβληθὲν ἴσον ἐστὶ τῷ <lb n="1799,001,,,,10,16,29"></lb> |
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9078 ὑπὸ τῶν ἐκ τῆς παραβολῆς γενομένων τμημάτων τῆς <lb n="1799,001,,,,10,16,30"></lb> |
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9079 εὐθείας.</s> <lb n="1799,001,,,,10,16,31"></lb> |
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9080 <s>Παρὰ γὰρ εὐθεῖαν τὴν ΑΒ παραβεβλήσθω παραλ-<lb n="1799,001,,,,10,16,32"></lb> |
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9081 ληλόγραμμον τὸ ΑΔ ἐλλεῖπον εἴδει <lb n="1799,001,,,,10,16,33"></lb> |
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9082 τετραγώνῳ τῷ ΔΒ· λέγω, ὅτι ἴσον <lb n="1799,001,,,,10,16,34"></lb> |
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9083 ἐστὶ τὸ ΑΔ τῷ ὑπὸ τῶν ΑΓ, ΓΒ.</s> <lb n="1799,001,,,,10,16,35"></lb> |
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9084 <s>Καί ἐστιν αὐτόθεν φανερόν· ἐπεὶ <lb n="1799,001,,,,10,16,36"></lb> |
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9085 γὰρ τετράγωνόν ἐστι τὸ ΔΒ, ἴση ἐστὶν <lb n="1799,001,,,,10,16,37"></lb> |
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9086 ἡ ΔΓ τῇ ΓΒ, καί ἐστι τὸ ΑΔ τὸ ὑπὸ τῶν ΑΓ, ΓΔ, <lb n="1799,001,,,,10,16,38"></lb> |
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9087 τουτέστι τὸ ὑπὸ τῶν ΑΓ, ΓΒ.</s> <lb n="1799,001,,,,10,16,39"></lb> |
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9088 <s>Ἐὰν ἄρα παρά τινα εὐθεῖαν, καὶ τὰ ἑξῆς.</s> <s><pb></pb> <lb n="1799,001,,,,10,16,40"></lb> |
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9089 Ἐὰν ὦσι δύο εὐθεῖαι ἄνισοι, τῷ δὲ τετάρτῳ μέρει τοῦ <lb n="1799,001,,,,10,17,1"></lb> |
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9090 ἀπὸ τῆς ἐλάσσονος ἴσον παρὰ τὴν μείζονα παραβληθῇ <lb n="1799,001,,,,10,17,2"></lb> |
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9091 ἐλλεῖπον εἴδει τετραγώνῳ καὶ εἰς σύμμετρα αὐτὴν διαιρῇ <lb n="1799,001,,,,10,17,3"></lb> |
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9092 μήκει, ἡ μείζων τῆς ἐλάσσονος μεῖζον δυνήσεται τῷ ἀπὸ <lb n="1799,001,,,,10,17,4"></lb> |
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9093 συμμέτρου ἑαυτῇ [μήκει].</s> <s>καὶ ἐὰν ἡ μείζων τῆς ἐλάσσονος <lb n="1799,001,,,,10,17,5"></lb> |
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9094 μεῖζον δύνηται τῷ ἀπὸ συμμέτρου ἑαυτῇ [μήκει], τῷ δὲ <lb n="1799,001,,,,10,17,6"></lb> |
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9095 τετάρτῳ τοῦ ἀπὸ τῆς ἐλάσσονος ἴσον παρὰ τὴν μείζονα <lb n="1799,001,,,,10,17,7"></lb> |
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9096 παραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ, εἰς σύμμετρα αὐτὴν <lb n="1799,001,,,,10,17,8"></lb> |
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9097 διαιρεῖ μήκει.</s> <lb n="1799,001,,,,10,17,9"></lb> |
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9098 <s>Ἔστωσαν δύο εὐθεῖαι ἄνισοι αἱ Α, ΒΓ, ὧν μείζων ἡ ΒΓ, <lb n="1799,001,,,,10,17,10"></lb> |
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9099 τῷ δὲ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ἐλάσσονος τῆς Α, <lb n="1799,001,,,,10,17,11"></lb> |
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9100 τουτέστι τῷ ἀπὸ τῆς ἡμισείας τῆς Α, ἴσον παρὰ τὴν ΒΓ <lb n="1799,001,,,,10,17,12"></lb> |
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9101 παραβεβλήσθω ἐλλεῖπον εἴδει τετραγώ- <lb n="1799,001,,,,10,17,13"></lb> |
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9102 νῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΒΔ, ΔΓ, σύμ-<lb n="1799,001,,,,10,17,14"></lb> |
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9103 μετρος δὲ ἔστω ἡ ΒΔ τῇ ΔΓ μήκει· λέ-<lb n="1799,001,,,,10,17,15"></lb> |
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9104 γω, ὅτι ἡ ΒΓ τῆς Α μεῖζον δύναται τῷ <lb n="1799,001,,,,10,17,16"></lb> |
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9105 ἀπὸ συμμέτρου ἑαυτῇ.</s> <lb n="1799,001,,,,10,17,17"></lb> |
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9106 <s>Τετμήσθω γὰρ ἡ ΒΓ δίχα κατὰ τὸ Ε σημεῖον, καὶ <lb n="1799,001,,,,10,17,18"></lb> |
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9107 κείσθω τῇ ΔΕ ἴση ἡ ΕΖ.</s> <s>λοιπὴ ἄρα ἡ ΔΓ ἴση ἐστὶ τῇ <lb n="1799,001,,,,10,17,19"></lb> |
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9108 ΒΖ.</s> <s>καὶ ἐπεὶ εὐθεῖα ἡ ΒΓ τέτμηται εἰς μὲν ἴσα κατὰ τὸ Ε, <lb n="1799,001,,,,10,17,20"></lb> |
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9109 εἰς δὲ ἄνισα κατὰ τὸ Δ, τὸ ἄρα ὑπὸ ΒΔ, ΔΓ περιεχόμενον <lb n="1799,001,,,,10,17,21"></lb> |
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9110 ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΕΔ τετραγώνου ἴσον ἐστὶ <pb></pb> <lb n="1799,001,,,,10,17,22"></lb> |
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9111 τῷ ἀπὸ τῆς ΕΓ τετραγώνῳ· καὶ τὰ τετραπλάσια· τὸ ἄρα <lb n="1799,001,,,,10,17,23"></lb> |
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9112 τετράκις ὑπὸ τῶν ΒΔ, ΔΓ μετὰ τοῦ τετραπλασίου τοῦ <lb n="1799,001,,,,10,17,24"></lb> |
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9113 ἀπὸ τῆς ΔΕ ἴσον ἐστὶ τῷ τετράκις ἀπὸ τῆς ΕΓ τετραγώνῳ.</s> <lb n="1799,001,,,,10,17,25"></lb> |
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9114 <s>ἀλλὰ τῷ μέν τετραπλασίῳ τοῦ ὑπὸ τῶν ΒΔ, ΔΓ ἴσον ἐστὶ <lb n="1799,001,,,,10,17,26"></lb> |
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9115 τὸ ἀπὸ τῆς Α τετράγωνον, τῷ δὲ τετραπλασίῳ τοῦ ἀπὸ τῆς <lb n="1799,001,,,,10,17,27"></lb> |
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9116 ΔΕ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΔΖ τετράγωνον· διπλασίων γάρ <lb n="1799,001,,,,10,17,28"></lb> |
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9117 ἐστιν ἡ ΔΖ τῆς ΔΕ.</s> <s>τῷ δὲ τετραπλασίῳ τοῦ ἀπὸ τῆς <lb n="1799,001,,,,10,17,29"></lb> |
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9118 ΕΓ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΒΓ τετράγωνον· διπλασίων γάρ <lb n="1799,001,,,,10,17,30"></lb> |
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9119 ἐστι πάλιν ἡ ΒΓ τῆς ΓΕ.</s> <s>τὰ ἄρα ἀπὸ τῶν Α, ΔΖ τετρά-<lb n="1799,001,,,,10,17,31"></lb> |
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9120 γωνα ἴσα ἐστὶ τῷ ἀπὸ τῆς ΒΓ τετραγώνῳ· ὥστε τὸ ἀπὸ <lb n="1799,001,,,,10,17,32"></lb> |
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9121 τῆς ΒΓ τοῦ ἀπὸ τῆς Α μεῖζόν ἐστι τῷ ἀπὸ τῆς ΔΖ· ἡ ΒΓ <lb n="1799,001,,,,10,17,33"></lb> |
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9122 ἄρα τῆς Α μεῖζον δύναται τῇ ΔΖ.</s> <s>δεικτέον, ὅτι καὶ <lb n="1799,001,,,,10,17,34"></lb> |
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9123 σύμμετρός ἐστιν ἡ ΒΓ τῇ ΔΖ.</s> <s>ἐπεὶ γὰρ σύμμετρός ἐστιν <lb n="1799,001,,,,10,17,35"></lb> |
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9124 ἡ ΒΔ τῇ ΔΓ μήκει, σύμμετρος ἄρα ἐστὶ καὶ ἡ ΒΓ τῇ <lb n="1799,001,,,,10,17,36"></lb> |
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9125 ΓΔ μήκει.</s> <s>ἀλλὰ ἡ ΓΔ ταῖς ΓΔ, ΒΖ ἐστι σύμμετρος <lb n="1799,001,,,,10,17,37"></lb> |
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9126 μήκει· ἴση γάρ ἐστιν ἡ ΓΔ τῇ ΒΖ.</s> <s>καὶ ἡ ΒΓ ἄρα σύμ-<lb n="1799,001,,,,10,17,38"></lb> |
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9127 μετρός ἐστι ταῖς ΒΖ, ΓΔ μήκει· ὥστε καὶ λοιπῇ τῇ ΖΔ <lb n="1799,001,,,,10,17,39"></lb> |
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9128 σύμμετρός ἐστιν ἡ ΒΓ μήκει· ἡ ΒΓ ἄρα τῆς Α μεῖζον <lb n="1799,001,,,,10,17,40"></lb> |
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9129 δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <lb n="1799,001,,,,10,17,41"></lb> |
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9130 <s>Ἀλλὰ δὴ ἡ ΒΓ τῆς Α μεῖζον δυνάσθω τῷ ἀπὸ συμμέτρου <lb n="1799,001,,,,10,17,42"></lb> |
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9131 ἑαυτῇ, τῷ δὲ τετάρτῳ τοῦ ἀπὸ τῆς Α ἴσον παρὰ τὴν ΒΓ <lb n="1799,001,,,,10,17,43"></lb> |
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9132 παραβεβλήσθω ἐλλεῖπον εἴδει τετραγώνῳ, καὶ ἔστω τὸ <lb n="1799,001,,,,10,17,44"></lb> |
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9133 ὑπὸ τῶν ΒΔ, ΔΓ.</s> <s>δεικτέον, ὅτι σύμμετρός ἐστιν ἡ ΒΔ <lb n="1799,001,,,,10,17,45"></lb> |
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9134 τῇ ΔΓ μήκει.</s> <s><pb></pb> <lb n="1799,001,,,,10,17,46"></lb> |
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9135 Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, <lb n="1799,001,,,,10,17,47"></lb> |
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9136 ὅτι ἡ ΒΓ τῆς Α μεῖζον δύναται τῷ ἀπὸ τῆς ΖΔ.</s> <s>δύναται <lb n="1799,001,,,,10,17,48"></lb> |
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9137 δὲ ἡ ΒΓ τῆς Α μεῖζον τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <s>σύμμε-<lb n="1799,001,,,,10,17,49"></lb> |
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9138 τρος ἄρα ἐστὶν ἡ ΒΓ τῇ ΖΔ μήκει· ὥστε καὶ λοιπῇ <lb n="1799,001,,,,10,17,50"></lb> |
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9139 συναμφοτέρῳ τῇ ΒΖ, ΔΓ σύμμετρός ἐστιν ἡ ΒΓ μήκει.</s> <lb n="1799,001,,,,10,17,51"></lb> |
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9140 <s>ἀλλὰ συναμφότερος ἡ ΒΖ, ΔΓ σύμμετρός ἐστι τῇ ΔΓ <lb n="1799,001,,,,10,17,52"></lb> |
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9141 [μήκει].</s> <s>ὥστε καὶ ἡ ΒΓ τῇ ΓΔ σύμμετρός ἐστι μήκει· <lb n="1799,001,,,,10,17,53"></lb> |
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9142 καὶ διελόντι ἄρα ἡ ΒΔ τῇ ΔΓ ἐστι σύμμετρος μήκει.</s> <lb n="1799,001,,,,10,17,54"></lb> |
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9143 <s>Ἐὰν ἄρα ὦσι δύο εὐθεῖαι ἄνισοι, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,17,55"></lb> |
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9144 <s>Ἐὰν ὦσι δύο εὐθεῖαι ἄνισοι, τῷ δὲ τετάρτῳ μέρει τοῦ ἀπὸ <lb n="1799,001,,,,10,18,1"></lb> |
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9145 τῆς ἐλάσσονος ἴσον παρὰ τὴν μείζονα παραβληθῇ ἐλλεῖπον <lb n="1799,001,,,,10,18,2"></lb> |
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9146 εἴδει τετραγώνῳ, καὶ εἰς ἀσύμμετρα αὐτὴν διαιρῇ [μήκει], <lb n="1799,001,,,,10,18,3"></lb> |
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9147 ἡ μείζων τῆς ἐλάσσονος μεῖζον δυνήσεται τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,18,4"></lb> |
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9148 μέτρου ἑαυτῇ.</s> <s>καὶ ἐὰν ἡ μείζων τῆς ἐλάσσονος μεῖζον <lb n="1799,001,,,,10,18,5"></lb> |
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9149 δύνηται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, τῷ δὲ τετάρτῳ τοῦ <lb n="1799,001,,,,10,18,6"></lb> |
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9150 ἀπὸ τῆς ἐλάσσονος ἴσον παρὰ τὴν μείζονα παραβληθῇ <lb n="1799,001,,,,10,18,7"></lb> |
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9151 ἐλλεῖπον εἴδει τετραγώνῳ, εἰς ἀσύμμετρα αὐτὴν διαιρεῖ <lb n="1799,001,,,,10,18,8"></lb> |
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9152 [μήκει].</s> <s><pb></pb> <lb n="1799,001,,,,10,18,9"></lb> |
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9153 Ἔστωσαν δύο εὐθεῖαι ἄνισοι αἱ Α, ΒΓ, ὧν <lb n="1799,001,,,,10,18,10"></lb> |
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9154 μείζων ἡ ΒΓ, τῷ δὲ τετάρτῳ [μέρει] τοῦ ἀπὸ <lb n="1799,001,,,,10,18,11"></lb> |
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9155 τῆς ἐλάσσονος τῆς Α ἴσον παρὰ τὴν ΒΓ <lb n="1799,001,,,,10,18,12"></lb> |
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9156 παραβεβλήσθω ἐλλεῖπον εἴδει τετραγώνῳ, <lb n="1799,001,,,,10,18,13"></lb> |
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9157 καὶ ἔστω τὸ ὑπὸ τῶν ΒΔΓ, ἀσύμμετρος δὲ <lb n="1799,001,,,,10,18,14"></lb> |
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9158 ἔστω ἡ ΒΔ τῇ ΔΓ μήκει· λέγω, ὅτι ἡ ΒΓ <lb n="1799,001,,,,10,18,15"></lb> |
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9159 τῆς Α μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου <lb n="1799,001,,,,10,18,16"></lb> |
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9160 ἑαυτῇ.</s> <lb n="1799,001,,,,10,18,17"></lb> |
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9161 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων τῷ πρότερον <lb n="1799,001,,,,10,18,18"></lb> |
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9162 ὁμοίως δείξομεν, ὅτι ἡ ΒΓ τῆς Α μεῖζον δύναται τῷ <lb n="1799,001,,,,10,18,19"></lb> |
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9163 ἀπὸ τῆς ΖΔ.</s> <s>δεικτέον [οὖν], ὅτι ἀσύμμετρός ἐστιν ἡ ΒΓ <lb n="1799,001,,,,10,18,20"></lb> |
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9164 τῇ ΔΖ μήκει.</s> <s>ἐπεὶ γὰρ ἀσύμμετρός ἐστιν ἡ ΒΔ τῇ ΔΓ <lb n="1799,001,,,,10,18,21"></lb> |
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9165 μήκει, ἀσύμμετρος ἄρα ἐστὶ καὶ ἡ ΒΓ τῇ ΓΔ μήκει.</s> <s>ἀλλὰ <lb n="1799,001,,,,10,18,22"></lb> |
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9166 ἡ ΔΓ σύμμετρός ἐστι συναμφοτέραις ταῖς ΒΖ, ΔΓ· καὶ <lb n="1799,001,,,,10,18,23"></lb> |
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9167 ἡ ΒΓ ἄρα ἀσύμμετρός ἐστι συναμφοτέραις ταῖς ΒΖ, ΔΓ.</s> <lb n="1799,001,,,,10,18,24"></lb> |
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9168 <s>ὥστε καὶ λοιπῇ τῇ ΖΔ ἀσύμμετρός ἐστιν ἡ ΒΓ μήκει.</s> <lb n="1799,001,,,,10,18,25"></lb> |
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9169 <s>καὶ ἡ ΒΓ τῆς Α μεῖζον δύναται τῷ ἀπὸ τῆς ΖΔ· ἡ ΒΓ <lb n="1799,001,,,,10,18,26"></lb> |
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9170 ἄρα τῆς Α μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <lb n="1799,001,,,,10,18,27"></lb> |
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9171 <s>Δυνάσθω δὴ πάλιν ἡ ΒΓ τῆς Α μεῖζον τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,18,28"></lb> |
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9172 μέτρου ἑαυτῇ, τῷ δὲ τετάρτῳ τοῦ ἀπὸ τῆς Α ἴσον παρὰ <lb n="1799,001,,,,10,18,29"></lb> |
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9173 τὴν ΒΓ παραβεβλήσθω ἐλλεῖπον εἴδει τετραγώνῳ, καὶ <lb n="1799,001,,,,10,18,30"></lb> |
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9174 ἔστω τὸ ὑπὸ τῶν ΒΔ, ΔΓ.</s> <s>δεικτέον, ὅτι ἀσύμμετρός <lb n="1799,001,,,,10,18,31"></lb> |
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9175 ἐστιν ἡ ΒΔ τῇ ΔΓ μήκει.</s> <lb n="1799,001,,,,10,18,32"></lb> |
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9176 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, <lb n="1799,001,,,,10,18,33"></lb> |
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9177 ὅτι ἡ ΒΓ τῆς Α μεῖζον δύναται τῷ ἀπὸ τῆς ΖΔ.</s> <s>ἀλλὰ <pb></pb> <lb n="1799,001,,,,10,18,34"></lb> |
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9178 ἡ ΒΓ τῆς Α μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <lb n="1799,001,,,,10,18,35"></lb> |
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9179 <s>ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΓ τῇ ΖΔ μήκει· ὥστε καὶ <lb n="1799,001,,,,10,18,36"></lb> |
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9180 λοιπῇ συναμφοτέρῳ τῇ ΒΖ, ΔΓ ἀσύμμετρός ἐστιν ἡ ΒΓ.</s> <lb n="1799,001,,,,10,18,37"></lb> |
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9181 <s>ἀλλὰ συναμφότερος ἡ ΒΖ, ΔΓ τῇ ΔΓ σύμμετρός ἐστι <lb n="1799,001,,,,10,18,38"></lb> |
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9182 μήκει· καὶ ἡ ΒΓ ἄρα τῇ ΔΓ ἀσύμμετρός ἐστι μήκει· <lb n="1799,001,,,,10,18,39"></lb> |
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9183 ὥστε καὶ διελόντι ἡ ΒΔ τῇ ΔΓ ἀσύμμετρός ἐστι μήκει.</s> <lb n="1799,001,,,,10,18,40"></lb> |
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9184 <s>Ἐὰν ἄρα ὦσι δύο εὐθεῖαι, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,18,41"></lb> |
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9185 <s>Λῆμμα <lb n="1799,001,,,,10,18,42"></lb> |
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9186 Ἐπεὶ δέδεικται, ὅτι αἱ μήκει σύμμετροι πάντως καὶ <lb n="1799,001,,,,10,18,43"></lb> |
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9187 δυνάμει [εἰσὶ σύμμετροι], αἱ δὲ δυνάμει οὐ πάντως καὶ <lb n="1799,001,,,,10,18,44"></lb> |
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9188 μήκει, ἀλλὰ δὴ δύνανται μήκει καὶ σύμμετροι εἶναι καὶ <lb n="1799,001,,,,10,18,45"></lb> |
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9189 ἀσύμμετροι, φανερόν, ὅτι, ἐὰν τῇ ἐκκειμένῃ ῥητῇ σύμ-<lb n="1799,001,,,,10,18,46"></lb> |
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9190 μετρός τις ᾖ μήκει, λέγεται ῥητὴ καὶ σύμμετρος αὐτῇ <lb n="1799,001,,,,10,18,47"></lb> |
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9191 οὐ μόνον μήκει, ἀλλὰ καὶ δυνάμει, ἐπεὶ αἱ μήκει σύμ-<lb n="1799,001,,,,10,18,48"></lb> |
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9192 μετροι πάντως καὶ δυνάμει.</s> <s>ἐὰν δὲ τῇ ἐκκειμένῃ ῥητῇ <lb n="1799,001,,,,10,18,49"></lb> |
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9193 σύμμετρός τις ᾖ δυνάμει, εἰ μὲν καὶ μήκει, λέγεται καὶ <lb n="1799,001,,,,10,18,50"></lb> |
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9194 οὕτως ῥητὴ καὶ σύμμετρος αὐτῇ μήκει καὶ δυνάμει· εἰ δὲ <lb n="1799,001,,,,10,18,51"></lb> |
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9195 τῇ ἐκκειμένῃ πάλιν ῥητῇ σύμμετρός τις οὖσα δυνάμει <lb n="1799,001,,,,10,18,52"></lb> |
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9196 μήκει αὐτῇ ᾖ ἀσύμμετρος, λέγεται καὶ οὕτως ῥητὴ <lb n="1799,001,,,,10,18,53"></lb> |
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9197 δυνάμει μόνον σύμμετρος.</s> <s><pb></pb> <lb n="1799,001,,,,10,18,54"></lb> |
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9198 Τὸ ὑπὸ ῥητῶν μήκει συμμέτρων κατά τινα τῶν προ-<lb n="1799,001,,,,10,19,1"></lb> |
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9199 ειρημένων τρόπων εὐθειῶν περιεχόμενον ὀρθογώνιον <lb n="1799,001,,,,10,19,2"></lb> |
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9200 ῥητόν ἐστιν.</s> <lb n="1799,001,,,,10,19,3"></lb> |
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9201 <s>Ὑπὸ γὰρ ῥητῶν μήκει συμμέτρων εὐθειῶν τῶν <lb n="1799,001,,,,10,19,4"></lb> |
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9202 ΑΒ, ΒΓ ὀρθογώνιον περιεχέσθω τὸ ΑΓ· λέγω, <lb n="1799,001,,,,10,19,5"></lb> |
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9203 ὅτι ῥητόν ἐστι τὸ ΑΓ.</s> <lb n="1799,001,,,,10,19,6"></lb> |
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9204 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τε-<lb n="1799,001,,,,10,19,7"></lb> |
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9205 τράγωνον τὸ ΑΔ· ῥητὸν ἄρα ἐστὶ τὸ ΑΔ.</s> <lb n="1799,001,,,,10,19,8"></lb> |
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9206 <s>καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΑΒ τῇ ΒΓ <lb n="1799,001,,,,10,19,9"></lb> |
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9207 μήκει, ἴση δέ ἐστιν ἡ ΑΒ τῇ ΒΔ, σύμ-<lb n="1799,001,,,,10,19,10"></lb> |
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9208 μετρος ἄρα ἐστὶν ἡ ΒΔ τῇ ΒΓ μήκει.</s> <lb n="1799,001,,,,10,19,11"></lb> |
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9209 <s>καί ἐστιν ὡς ἡ ΒΔ πρὸς τὴν ΒΓ, οὕτως <lb n="1799,001,,,,10,19,12"></lb> |
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9210 τὸ ΔΑ πρὸς τὸ ΑΓ.</s> <s>σύμμετρον ἄρα ἐστὶ τὸ ΔΑ τῷ <lb n="1799,001,,,,10,19,13"></lb> |
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9211 ΑΓ.</s> <s>ῥητὸν δὲ τὸ ΔΑ· ῥητὸν ἄρα ἐστὶ καὶ τὸ ΑΓ.</s> <lb n="1799,001,,,,10,19,14"></lb> |
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9212 <s>Τὸ ἄρα ὑπὸ ῥητῶν μήκει συμμέτρων, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,19,15"></lb> |
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9213 <s>Ἐὰν ῥητὸν παρὰ ῥητὴν παραβληθῇ, πλάτος ποιεῖ <lb n="1799,001,,,,10,20,1"></lb> |
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9214 ῥητὴν καὶ σύμμετρον τῇ, παρ' ἣν παράκειται, μήκει.</s> <s><pb></pb> <lb n="1799,001,,,,10,20,2"></lb> |
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9215 Ῥητὸν γὰρ τὸ ΑΓ παρὰ ῥητὴν κατά τινα πάλιν τῶν <lb n="1799,001,,,,10,20,3"></lb> |
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9216 προειρημένων τρόπων τὴν ΑΒ παραβεβλήσθω πλάτος <lb n="1799,001,,,,10,20,4"></lb> |
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9217 ποιοῦν τὴν ΒΓ· λέγω, ὅτι ῥητή ἐστιν ἡ ΒΓ καὶ σύμμετρος <lb n="1799,001,,,,10,20,5"></lb> |
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9218 τῇ ΒΑ μήκει.</s> <lb n="1799,001,,,,10,20,6"></lb> |
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9219 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετρά- <lb n="1799,001,,,,10,20,7"></lb> |
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9220 γωνον τὸ ΑΔ· ῥητὸν ἄρα ἐστὶ τὸ ΑΔ.</s> <s>ῥητὸν <lb n="1799,001,,,,10,20,8"></lb> |
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9221 δὲ καὶ τὸ ΑΓ· σύμμετρον ἄρα ἐστὶ τὸ ΔΑ <lb n="1799,001,,,,10,20,9"></lb> |
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9222 τῷ ΑΓ.</s> <s>καί ἐστιν ὡς τὸ ΔΑ πρὸς τὸ ΑΓ, <lb n="1799,001,,,,10,20,10"></lb> |
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9223 οὕτως ἡ ΔΒ πρὸς τὴν ΒΓ.</s> <s>σύμμετρος ἄρα <lb n="1799,001,,,,10,20,11"></lb> |
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9224 ἐστὶ καὶ ἡ ΔΒ τῇ ΒΓ· ἴση δὲ ἡ ΔΒ τῇ <lb n="1799,001,,,,10,20,12"></lb> |
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9225 ΒΑ· σύμμετρος ἄρα καὶ ἡ ΑΒ τῇ ΒΓ.</s> <s>ῥητὴ <lb n="1799,001,,,,10,20,13"></lb> |
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9226 δέ ἐστιν ἡ ΑΒ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΒΓ καὶ σύμμετρος <lb n="1799,001,,,,10,20,14"></lb> |
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9227 τῇ ΑΒ μήκει.</s> <lb n="1799,001,,,,10,20,15"></lb> |
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9228 <s>Ἐὰν ἄρα ῥητὸν παρὰ ῥητὴν παραβληθῇ, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,20,16"></lb> |
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9229 <s>Τὸ ὑπὸ ῥητῶν δυνάμει μόνον συμμέτρων εὐθειῶν <lb n="1799,001,,,,10,21,1"></lb> |
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9230 περιεχόμενον ὀρθογώνιον ἄλογόν ἐστιν, καὶ ἡ δυναμένη <lb n="1799,001,,,,10,21,2"></lb> |
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9231 αὐτὸ ἄλογός ἐστιν, καλείσθω δὲ μέση.</s> <lb n="1799,001,,,,10,21,3"></lb> |
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9232 <s>Ὑπὸ γὰρ ῥητῶν δυνάμει μόνον συμμέτρων εὐθειῶν τῶν <lb n="1799,001,,,,10,21,4"></lb> |
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9233 ΑΒ, ΒΓ ὀρθογώνιον περιεχέσθω τὸ ΑΓ· λέγω, ὅτι <lb n="1799,001,,,,10,21,5"></lb> |
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9234 ἄλογόν ἐστι τὸ ΑΓ, καὶ ἡ δυναμένη αὐτὸ ἄλογός ἐστιν, <lb n="1799,001,,,,10,21,6"></lb> |
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9235 καλείσθω δὲ μέση.</s> <s><pb></pb> <lb n="1799,001,,,,10,21,7"></lb> |
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9236 Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔ· <lb n="1799,001,,,,10,21,8"></lb> |
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9237 ῥητὸν ἄρα ἐστὶ τὸ ΑΔ.</s> <s>καὶ ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΑΒ <lb n="1799,001,,,,10,21,9"></lb> |
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9238 τῇ ΒΓ μήκει· δυνάμει γὰρ μόνον ὑπόκεινται σύμμετροι· <lb n="1799,001,,,,10,21,10"></lb> |
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9239 ἴση δὲ ἡ ΑΒ τῇ ΒΔ, ἀσύμμετρος ἄρα ἐστὶ <lb n="1799,001,,,,10,21,11"></lb> |
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9240 καὶ ἡ ΔΒ τῇ ΒΓ μήκει.</s> <s>καί ἐστιν ὡς ἡ ΔΒ <lb n="1799,001,,,,10,21,12"></lb> |
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9241 πρὸς τὴν ΒΓ, οὕτως τὸ ΑΔ πρὸς τὸ ΑΓ· <lb n="1799,001,,,,10,21,13"></lb> |
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9242 ἀσύμμετρον ἄρα [ἐστὶ] τὸ ΔΑ τῷ ΑΓ.</s> <s>ῥητὸν <lb n="1799,001,,,,10,21,14"></lb> |
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9243 δὲ τὸ ΔΑ· ἄλογον ἄρα ἐστὶ τὸ ΑΓ· ὥστε καὶ <lb n="1799,001,,,,10,21,15"></lb> |
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9244 ἡ δυναμένη τὸ ΑΓ [τουτέστιν ἡ ἴσον αὐτῷ <lb n="1799,001,,,,10,21,16"></lb> |
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9245 τετράγωνον δυναμένη] ἄλογός ἐστιν, καλεί-<lb n="1799,001,,,,10,21,17"></lb> |
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9246 σθω δὲ μέση· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,21,18"></lb> |
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9247 <s>Λῆμμα <lb n="1799,001,,,,10,21,19"></lb> |
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9248 Ἐὰν ὦσι δύο εὐθεῖαι, ἔστιν ὡς ἡ πρώτη πρὸς τὴν <lb n="1799,001,,,,10,21,20"></lb> |
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9249 δευτέραν, οὕτως τὸ ἀπὸ τῆς πρώτης πρὸς τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,21,21"></lb> |
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9250 δύο εὐθειῶν.</s> <lb n="1799,001,,,,10,21,22"></lb> |
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9251 <s>Ἔστωσαν δύο εὐθεῖαι αἱ ΖΕ, ΕΗ.</s> <s>λέγω, ὅτι ἐστὶν ὡς <lb n="1799,001,,,,10,21,23"></lb> |
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9252 ἡ ΖΕ πρὸς τὴν ΕΗ, οὕτως τὸ ἀπὸ τῆς ΖΕ πρὸς τὸ ὑπὸ <lb n="1799,001,,,,10,21,24"></lb> |
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9253 τῶν ΖΕ, ΕΗ.</s> <lb n="1799,001,,,,10,21,25"></lb> |
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9254 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΖΕ τετράγωνον τὸ ΔΖ, <lb n="1799,001,,,,10,21,26"></lb> |
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9255 καὶ συμπεπληρώσθω τὸ ΗΔ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς ἡ ΖΕ <pb></pb> <lb n="1799,001,,,,10,21,27"></lb> |
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9256 πρὸς τὴν ΕΗ, οὕτως τὸ ΖΔ πρὸς <lb n="1799,001,,,,10,21,28"></lb> |
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9257 τὸ ΔΗ, καί ἐστι τὸ μὲν ΖΔ τὸ <lb n="1799,001,,,,10,21,29"></lb> |
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9258 ἀπὸ τῆς ΖΕ, τὸ δὲ ΔΗ τὸ ὑπὸ <lb n="1799,001,,,,10,21,30"></lb> |
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9259 τῶν ΔΕ, ΕΗ, τουτέστι τὸ ὑπὸ <lb n="1799,001,,,,10,21,31"></lb> |
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9260 τῶν ΖΕ, ΕΗ, ἔστιν ἄρα ὡς ἡ ΖΕ <lb n="1799,001,,,,10,21,32"></lb> |
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9261 τὴν ΕΗ, οὕτως τὸ ἀπὸ τῆς ΖΕ πρὸς τὸ ὑπὸ τῶν ΖΕ, ΕΗ.</s> <lb n="1799,001,,,,10,21,33"></lb> |
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9262 <s>ὁμοίως δὲ καὶ ὡς τὸ ὑπὸ τῶν ΗΕ, ΕΖ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,21,34"></lb> |
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9263 ΕΖ, τουτέστιν ὡς τὸ ΗΔ πρὸς τὸ ΖΔ, οὕτως ἡ ΗΕ πρὸς <lb n="1799,001,,,,10,21,35"></lb> |
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9264 τὴν ΕΖ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,21,36"></lb> |
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9265 <s>Τὸ ἀπὸ μέσης παρὰ ῥητὴν παραβαλλόμενον πλάτος <lb n="1799,001,,,,10,22,1"></lb> |
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9266 ποιεῖ ῥητὴν καὶ ἀσύμμετρον τῇ, παρ' ἣν παράκειται, <lb n="1799,001,,,,10,22,2"></lb> |
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9267 μήκει.</s> <lb n="1799,001,,,,10,22,3"></lb> |
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9268 <s>Ἔστω μέση μὲν ἡ Α, ῥητὴ δὲ ἡ ΓΒ, καὶ τῷ ἀπὸ τῆς Α <lb n="1799,001,,,,10,22,4"></lb> |
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9269 ἴσον παρὰ τὴν ΒΓ παραβεβλήσθω χωρίον ὀρθογώνιον <lb n="1799,001,,,,10,22,5"></lb> |
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9270 τὸ ΒΔ πλάτος ποιοῦν τὴν ΓΔ· λέγω, ὅτι ῥητή ἐστιν ἡ ΓΔ <lb n="1799,001,,,,10,22,6"></lb> |
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9271 καὶ ἀσύμμετρος τῇ ΓΒ μήκει.</s> <lb n="1799,001,,,,10,22,7"></lb> |
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9272 <s>Ἐπεὶ γὰρ μέση ἐστὶν ἡ Α, δύναται χωρίον περιεχόμενον <lb n="1799,001,,,,10,22,8"></lb> |
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9273 ὑπὸ ῥητῶν δυνάμει μόνον συμμέτρων.</s> <s>δυνάσθω τὸ ΗΖ.</s> <lb n="1799,001,,,,10,22,9"></lb> |
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9274 <s>δύναται δὲ καὶ τὸ ΒΔ· ἴσον ἄρα ἐστὶ τὸ ΒΔ τῷ ΗΖ.</s> <s>ἔστι <lb n="1799,001,,,,10,22,10"></lb> |
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9275 δὲ αὐτῷ καὶ ἰσογώνιον· τῶν δὲ ἴσων τε καὶ ἰσογωνίων <lb n="1799,001,,,,10,22,11"></lb> |
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9276 παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς <lb n="1799,001,,,,10,22,12"></lb> |
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9277 ἴσας γωνίας· ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΒΓ πρὸς τὴν ΕΗ, <lb n="1799,001,,,,10,22,13"></lb> |
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9278 οὕτως ἡ ΕΖ πρὸς τὴν ΓΔ.</s> <s>ἔστιν ἄρα καὶ ὡς τὸ ἀπὸ τῆς <pb></pb> <lb n="1799,001,,,,10,22,14"></lb> |
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9279 ΒΓ πρὸς τὸ ἀπὸ τῆς ΕΗ, οὕτως τὸ <lb n="1799,001,,,,10,22,15"></lb> |
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9280 ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς ΓΔ.</s> <lb n="1799,001,,,,10,22,16"></lb> |
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9281 <s>σύμμετρον δέ ἐστι τὸ ἀπὸ τῆς ΓΒ τῷ <lb n="1799,001,,,,10,22,17"></lb> |
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9282 ἀπὸ τῆς ΕΗ· ῥητὴ γάρ ἐστιν ἑκατέρα <lb n="1799,001,,,,10,22,18"></lb> |
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9283 αὐτῶν· σύμμετρον ἄρα ἐστὶ καὶ τὸ <lb n="1799,001,,,,10,22,19"></lb> |
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9284 ἀπὸ τῆς ΕΖ τῷ ἀπὸ τῆς ΓΔ.</s> <s>ῥητὸν δέ <lb n="1799,001,,,,10,22,20"></lb> |
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9285 ἐστι τὸ ἀπὸ τῆς ΕΖ· ῥητὸν ἄρα ἐστὶ <lb n="1799,001,,,,10,22,21"></lb> |
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9286 καὶ τὸ ἀπὸ τῆς ΓΔ· ῥητὴ ἄρα ἐστὶν ἡ ΓΔ.</s> <s>καὶ ἐπεὶ ἀσύμ-<lb n="1799,001,,,,10,22,22"></lb> |
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9287 μετρός ἐστιν ἡ ΕΖ τῇ ΕΗ μήκει· δυνάμει γὰρ μόνον εἰσὶ <lb n="1799,001,,,,10,22,23"></lb> |
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9288 σύμμετροι· ὡς δὲ ἡ ΕΖ πρὸς τὴν ΕΗ, οὕτως τὸ ἀπὸ τῆς ΕΖ <lb n="1799,001,,,,10,22,24"></lb> |
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9289 πρὸς τὸ ὑπὸ τῶν ΖΕ, ΕΗ, ἀσύμμετρον ἄρα [ἐστὶ] τὸ ἀπὸ <lb n="1799,001,,,,10,22,25"></lb> |
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9290 τῆς ΕΖ τῷ ὑπὸ τῶν ΖΕ, ΕΗ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς ΕΖ <lb n="1799,001,,,,10,22,26"></lb> |
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9291 σύμμετρόν ἐστι τὸ ἀπὸ τῆς ΓΔ· ῥηταὶ γάρ εἰσι δυνάμει· τῷ <lb n="1799,001,,,,10,22,27"></lb> |
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9292 δὲ ὑπὸ τῶν ΖΕ, ΕΗ σύμμετρόν ἐστι τὸ ὑπὸ τῶν ΑΓ, ΓΒ· <lb n="1799,001,,,,10,22,28"></lb> |
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9293 ἴσα γάρ ἐστι τῷ ἀπὸ τῆς Α· ἀσύμμετρον ἄρα ἐστὶ καὶ τὸ ἀπὸ <lb n="1799,001,,,,10,22,29"></lb> |
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9294 τῆς ΓΔ τῷ ὑπὸ τῶν ΔΓ, ΓΒ.</s> <s>ὡς δὲ τὸ ἀπὸ τῆς ΓΔ πρὸς <lb n="1799,001,,,,10,22,30"></lb> |
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9295 τὸ ὑπὸ τῶν ΔΓ, ΓΒ, οὕτως ἐστὶν ἡ ΔΓ πρὸς τὴν ΓΒ· <lb n="1799,001,,,,10,22,31"></lb> |
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9296 ἀσύμμετρος ἄρα ἐστὶν ἡ ΔΓ τῇ ΓΒ μήκει.</s> <s>ῥητὴ ἄρα ἐστὶν <lb n="1799,001,,,,10,22,32"></lb> |
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9297 ἡ ΓΔ καὶ ἀσύμμετρος τῇ ΓΒ μήκει· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,22,33"></lb> |
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9298 <s>Ἡ τῇ μέσῃ σύμμετρος μέση ἐστίν.</s> <lb n="1799,001,,,,10,23,1"></lb> |
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9299 <s>Ἔστω μέση ἡ Α, καὶ τῇ Α σύμμετρος ἔστω ἡ Β· λέγω, <lb n="1799,001,,,,10,23,2"></lb> |
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9300 ὅτι καὶ ἡ Β μέση ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,10,23,3"></lb> |
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9301 Ἐκκείσθω γὰρ ῥητὴ ἡ ΓΔ, καὶ τῷ μὲν <lb n="1799,001,,,,10,23,4"></lb> |
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9302 ἀπὸ τῆς Α ἴσον παρὰ τὴν ΓΔ παραβε-<lb n="1799,001,,,,10,23,5"></lb> |
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9303 βλήσθω χωρίον ὀρθογώνιον τὸ ΓΕ πλά-<lb n="1799,001,,,,10,23,6"></lb> |
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9304 τος ποιοῦν τὴν ΕΔ· ῥητὴ ἄρα ἐστὶν ἡ <lb n="1799,001,,,,10,23,7"></lb> |
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9305 ΕΔ καὶ ἀσύμμετρος τῇ ΓΔ μήκει.</s> <s>τῷ <lb n="1799,001,,,,10,23,8"></lb> |
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9306 δὲ ἀπὸ τῆς Β ἴσον παρὰ τὴν ΓΔ παρα-<lb n="1799,001,,,,10,23,9"></lb> |
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9307 βεβλήσθω χωρίον ὀρθογώνιον τὸ ΓΖ <lb n="1799,001,,,,10,23,10"></lb> |
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9308 πλάτος ποιοῦν τὴν ΔΖ.</s> <s>ἐπεὶ οὖν σύμμετρός ἐστιν ἡ Α τῇ <lb n="1799,001,,,,10,23,11"></lb> |
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9309 Β, σύμμετρόν ἐστι καὶ τὸ ἀπὸ τῆς Α τῷ ἀπὸ τῆς Β.</s> <s>ἀλλὰ τῷ <lb n="1799,001,,,,10,23,12"></lb> |
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9310 μὲν ἀπὸ τῆς Α ἴσον ἐστὶ τὸ ΕΓ, τῷ δὲ ἀπὸ τῆς Β ἴσον ἐστὶ <lb n="1799,001,,,,10,23,13"></lb> |
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9311 τὸ ΓΖ· σύμμετρον ἄρα ἐστὶ τὸ ΕΓ τῷ ΓΖ.</s> <s>καί ἐστιν ὡς <lb n="1799,001,,,,10,23,14"></lb> |
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9312 τὸ ΕΓ πρὸς τὸ ΓΖ, οὕτως ἡ ΕΔ πρὸς τὴν ΔΖ· σύμμετρος <lb n="1799,001,,,,10,23,15"></lb> |
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9313 ἄρα ἐστὶν ἡ ΕΔ τῇ ΔΖ μήκει.</s> <s>ῥητὴ δέ ἐστιν ἡ ΕΔ καὶ <lb n="1799,001,,,,10,23,16"></lb> |
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9314 ἀσύμμετρος τῇ ΔΓ μήκει· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΔΖ καὶ <lb n="1799,001,,,,10,23,17"></lb> |
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9315 ἀσύμμετρος τῇ ΔΓ μήκει· αἱ ΓΔ, ΔΖ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,23,18"></lb> |
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9316 δυνάμει μόνον σύμμετροι.</s> <s>ἡ δὲ τὸ ὑπὸ ῥητῶν δυνάμει <lb n="1799,001,,,,10,23,19"></lb> |
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9317 μόνον συμμέτρων δυναμένη μέση ἐστίν.</s> <s>ἡ ἄρα τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,23,20"></lb> |
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9318 ΓΔ, ΔΖ δυναμένη μέση ἐστίν· καὶ δύναται τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,23,21"></lb> |
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9319 ΓΔ, ΔΖ ἡ Β· μέση ἄρα ἐστὶν ἡ Β.</s> <lb n="1799,001,,,,10,23,22"></lb> |
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9320 <s>Πόρισμα <lb n="1799,001,,,,10,23,23"></lb> |
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9321 Ἐκ δὴ τούτου φανερόν, ὅτι τὸ τῷ μέσῳ χωρίῳ σύμ-<lb n="1799,001,,,,10,23,24"></lb> |
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9322 μετρον μέσον ἐστίν.</s> <s>[δύνανται γὰρ αὐτὰ εὐθεῖαι, αἵ εἰσι <pb></pb> <lb n="1799,001,,,,10,23,25"></lb> |
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9323 δυνάμει σύμμετροι, ὧν ἡ ἑτέρα μέση· ὥστε καὶ ἡ λοιπὴ <lb n="1799,001,,,,10,23,26"></lb> |
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9324 μέση ἐστίν.] <lb n="1799,001,,,,10,23,27"></lb> |
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9325 Ὡσαύτως δὲ τοῖς ἐπὶ τῶν ῥητῶν εἰρημένοις καὶ ἐπὶ <lb n="1799,001,,,,10,23,28"></lb> |
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9326 τῶν μέσων ἐξακολουθεῖ, τὴν τῇ μέσῃ μήκει σύμμετρον <lb n="1799,001,,,,10,23,29"></lb> |
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9327 λέγεσθαι μέσην καὶ σύμμετρον αὐτῇ μὴ μόνον μήκει, <lb n="1799,001,,,,10,23,30"></lb> |
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9328 ἀλλὰ καὶ δυνάμει, ἐπειδήπερ καθόλου αἱ μήκει σύμμετροι <lb n="1799,001,,,,10,23,31"></lb> |
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9329 πάντως καὶ δυνάμει.</s> <s>ἐὰν δὲ τῇ μέσῃ σύμμετρός τις ᾖ <lb n="1799,001,,,,10,23,32"></lb> |
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9330 δυνάμει, εἰ μὲν καὶ μήκει, λέγονται καὶ οὕτως μέσαι καὶ <lb n="1799,001,,,,10,23,33"></lb> |
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9331 σύμμετροι μήκει καὶ δυνάμει, εἰ δὲ δυνάμει μόνον, <lb n="1799,001,,,,10,23,34"></lb> |
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9332 λέγονται μέσαι δυνάμει μόνον σύμμετροι.</s> <lb n="1799,001,,,,10,23,35"></lb> |
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9333 <s>Τὸ ὑπὸ μέσων μήκει συμμέτρων εὐθειῶν κατά τινα τῶν <lb n="1799,001,,,,10,24,1"></lb> |
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9334 εἰρημένων τρόπων περιεχόμενον ὀρθογώνιον μέσον ἐστίν.</s> <lb n="1799,001,,,,10,24,2"></lb> |
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9335 <s>Ὑπὸ γὰρ μέσων μήκει συμμέτρων εὐθειῶν τῶν ΑΒ, <lb n="1799,001,,,,10,24,3"></lb> |
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9336 ΒΓ περιεχέσθω ὀρθογώνιον τὸ ΑΓ· λέγω, ὅτι τὸ <lb n="1799,001,,,,10,24,4"></lb> |
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9337 ΑΓ μέσον ἐστίν.</s> <lb n="1799,001,,,,10,24,5"></lb> |
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9338 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον <lb n="1799,001,,,,10,24,6"></lb> |
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9339 τὸ ΑΔ· μέσον ἄρα ἐστὶ τὸ ΑΔ.</s> <s>καὶ ἐπεὶ σύμ-<lb n="1799,001,,,,10,24,7"></lb> |
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9340 μετρός ἐστιν ἡ ΑΒ τῇ ΒΓ μήκει, ἴση δὲ ἡ ΑΒ <lb n="1799,001,,,,10,24,8"></lb> |
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9341 τῇ ΒΔ, σύμμετρος ἄρα ἐστὶ καὶ ἡ ΔΒ τῇ ΒΓ <lb n="1799,001,,,,10,24,9"></lb> |
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9342 μήκει· ὥστε καὶ τὸ ΔΑ τῷ ΑΓ σύμμετρόν <lb n="1799,001,,,,10,24,10"></lb> |
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9343 ἐστιν.</s> <s>μέσον δὲ τὸ ΔΑ· μέσον ἄρα καὶ τὸ <lb n="1799,001,,,,10,24,11"></lb> |
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9344 ΑΓ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,24,12"></lb> |
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9345 Τὸ ὑπὸ μέσων δυνάμει μόνον συμμέτρων εὐθειῶν <lb n="1799,001,,,,10,25,1"></lb> |
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9346 περιεχόμενον ὀρθογώνιον ἤτοι ῥητὸν ἢ μέσον ἐστίν.</s> <lb n="1799,001,,,,10,25,2"></lb> |
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9347 <s>Ὑπὸ γὰρ μέσων δυνάμει μόνον συμμέτρων εὐθειῶν <lb n="1799,001,,,,10,25,3"></lb> |
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9348 τῶν ΑΒ, ΒΓ ὀρθογώνιον περιεχέσθω τὸ ΑΓ· λέγω, ὅτι <lb n="1799,001,,,,10,25,4"></lb> |
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9349 τὸ ΑΓ ἤτοι ῥητὸν ἢ μέσον ἐστίν.</s> <lb n="1799,001,,,,10,25,5"></lb> |
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9350 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῶν ΑΒ, ΒΓ τετράγωνα τὰ ΑΔ, <lb n="1799,001,,,,10,25,6"></lb> |
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9351 ΒΕ· μέσον ἄρα ἐστὶν ἑκάτερον <lb n="1799,001,,,,10,25,7"></lb> |
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9352 τῶν ΑΔ, ΒΕ.</s> <s>καὶ ἐκκείσθω <lb n="1799,001,,,,10,25,8"></lb> |
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9353 ῥητὴ ἡ ΖΗ, καὶ τῷ μὲν ΑΔ <lb n="1799,001,,,,10,25,9"></lb> |
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9354 ἴσον παρὰ τὴν ΖΗ παραβε-<lb n="1799,001,,,,10,25,10"></lb> |
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9355 βλήσθω ὀρθογώνιον παραλλη-<lb n="1799,001,,,,10,25,11"></lb> |
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9356 λόγραμμον τὸ ΗΘ πλάτος <lb n="1799,001,,,,10,25,12"></lb> |
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9357 ποιοῦν τὴν ΖΘ, τῷ δὲ ΑΓ ἴσον <lb n="1799,001,,,,10,25,13"></lb> |
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9358 παρὰ τὴν ΘΜ παραβεβλήσθω <lb n="1799,001,,,,10,25,14"></lb> |
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9359 ὀρθογώνιον παραλληλόγραμμον τὸ ΜΚ πλάτος ποιοῦν <lb n="1799,001,,,,10,25,15"></lb> |
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9360 τὴν ΘΚ, καὶ ἔτι τῷ ΒΕ ἴσον ὁμοίως παρὰ τὴν ΚΝ <lb n="1799,001,,,,10,25,16"></lb> |
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9361 παραβεβλήσθω τὸ ΝΛ πλάτος ποιοῦν τὴν ΚΛ· ἐπ' <lb n="1799,001,,,,10,25,17"></lb> |
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9362 εὐθείας ἄρα εἰσὶν αἱ ΖΘ, ΘΚ, ΚΛ.</s> <s>ἐπεὶ οὖν μέσον <lb n="1799,001,,,,10,25,18"></lb> |
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9363 ἐστὶν ἑκάτερον τῶν ΑΔ, ΒΕ, καί ἐστιν ἴσον τὸ μὲν <lb n="1799,001,,,,10,25,19"></lb> |
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9364 ΑΔ τῷ ΗΘ, τὸ δὲ ΒΕ τῷ ΝΛ, μέσον ἄρα καὶ ἑκάτερον <lb n="1799,001,,,,10,25,20"></lb> |
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9365 τῶν ΗΘ, ΝΛ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΖΗ παράκειται· ῥητὴ <lb n="1799,001,,,,10,25,21"></lb> |
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9366 ἄρα ἐστὶν ἑκατέρα τῶν ΖΘ, ΚΛ καὶ ἀσύμμετρος τῇ ΖΗ <lb n="1799,001,,,,10,25,22"></lb> |
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9367 μήκει.</s> <s>καὶ ἐπεὶ σύμμετρόν ἐστι τὸ ΑΔ τῷ ΒΕ, σύμμετρον <lb n="1799,001,,,,10,25,23"></lb> |
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9368 ἄρα ἐστὶ καὶ τὸ ΗΘ τῷ ΝΛ.</s> <s>καί ἐστιν ὡς τὸ ΗΘ πρὸς <pb></pb> <lb n="1799,001,,,,10,25,24"></lb> |
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9369 τὸ ΝΛ, οὕτως ἡ ΖΘ πρὸς τὴν ΚΛ· σύμμετρος ἄρα ἐστὶν <lb n="1799,001,,,,10,25,25"></lb> |
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9370 ἡ ΖΘ τῇ ΚΛ μήκει.</s> <s>αἱ ΖΘ, ΚΛ ἄρα ῥηταί εἰσι μήκει <lb n="1799,001,,,,10,25,26"></lb> |
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9371 σύμμετροι· ῥητὸν ἄρα ἐστὶ τὸ ὑπὸ τῶν ΖΘ, ΚΛ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,25,27"></lb> |
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9372 ἴση ἐστὶν ἡ μὲν ΔΒ τῇ ΒΑ, ἡ δὲ ΞΒ τῇ ΒΓ, ἔστιν ἄρα <lb n="1799,001,,,,10,25,28"></lb> |
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9373 ὡς ἡ ΔΒ πρὸς τὴν ΒΓ, οὕτως ἡ ΑΒ πρὸς τὴν ΒΞ.</s> <s>ἀλλ' <lb n="1799,001,,,,10,25,29"></lb> |
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9374 ὡς μὲν ἡ ΔΒ πρὸς τὴν ΒΓ, οὕτως τὸ ΔΑ πρὸς τὸ ΑΓ· <lb n="1799,001,,,,10,25,30"></lb> |
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9375 ὡς δὲ ἡ ΑΒ πρὸς τὴν ΒΞ, οὕτως τὸ ΑΓ πρὸς τὸ ΓΞ· <lb n="1799,001,,,,10,25,31"></lb> |
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9376 ἔστιν ἄρα ὡς τὸ ΔΑ πρὸς τὸ ΑΓ, οὕτως τὸ ΑΓ πρὸς τὸ <lb n="1799,001,,,,10,25,32"></lb> |
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9377 ΓΞ.</s> <s>ἴσον δέ ἐστι τὸ μὲν ΑΔ τῷ ΗΘ, τὸ δὲ ΑΓ τῷ ΜΚ, <lb n="1799,001,,,,10,25,33"></lb> |
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9378 τὸ δὲ ΓΞ τῷ ΝΛ· ἔστιν ἄρα ὡς τὸ ΗΘ πρὸς τὸ ΜΚ, <lb n="1799,001,,,,10,25,34"></lb> |
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9379 οὕτως τὸ ΜΚ πρὸς τὸ ΝΛ· ἔστιν ἄρα καὶ ὡς ἡ ΖΘ πρὸς <lb n="1799,001,,,,10,25,35"></lb> |
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9380 τὴν ΘΚ, οὕτως ἡ ΘΚ πρὸς τὴν ΚΛ· τὸ ἄρα ὑπὸ τῶν <lb n="1799,001,,,,10,25,36"></lb> |
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9381 ΖΘ, ΚΛ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΘΚ.</s> <s>ῥητὸν δὲ τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,25,37"></lb> |
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9382 ΖΘ, ΚΛ· ῥητὸν ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς ΘΚ· ῥητὴ ἄρα <lb n="1799,001,,,,10,25,38"></lb> |
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9383 ἐστὶν ἡ ΘΚ.</s> <s>καὶ εἰ μὲν σύμμετρός ἐστι τῇ ΖΗ μήκει, <lb n="1799,001,,,,10,25,39"></lb> |
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9384 ῥητόν ἐστι τὸ ΘΝ· εἰ δὲ ἀσύμμετρός ἐστι τῇ ΖΗ μήκει, <lb n="1799,001,,,,10,25,40"></lb> |
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9385 αἱ ΚΘ, ΘΜ ῥηταί εἰσι δυνάμει μόνον σύμμετροι· μέσον <lb n="1799,001,,,,10,25,41"></lb> |
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9386 ἄρα τὸ ΘΝ.</s> <s>τὸ ΘΝ ἄρα ἤτοι ῥητὸν ἢ μέσον ἐστίν.</s> <s>ἴσον <lb n="1799,001,,,,10,25,42"></lb> |
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9387 δὲ τὸ ΘΝ τῷ ΑΓ· τὸ ΑΓ ἄρα ἤτοι ῥητὸν ἢ μέσον ἐστίν.</s> <lb n="1799,001,,,,10,25,43"></lb> |
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9388 <s>Τὸ ἄρα ὑπὸ μέσων δυνάμει μόνον συμμέτρων, καὶ τὰ <lb n="1799,001,,,,10,25,44"></lb> |
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9389 ἑξῆς.</s> <s><pb></pb> <lb n="1799,001,,,,10,25,45"></lb> |
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9390 Μέσον μέσου οὐχ ὑπερέχει ῥητῷ.</s> <lb n="1799,001,,,,10,26,1"></lb> |
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9391 <s>Εἰ γὰρ δυνατόν, μέσον τὸ ΑΒ μέσου τοῦ ΑΓ ὑπερεχέτω <lb n="1799,001,,,,10,26,2"></lb> |
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9392 ῥητῷ τῷ ΔΒ, καὶ ἐκκείσθω ῥητὴ ἡ ΕΖ, καὶ τῷ ΑΒ ἴσον <lb n="1799,001,,,,10,26,3"></lb> |
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9393 παρὰ τὴν ΕΖ παραβεβλήσθω παραλληλόγραμμον ὀρθο-<lb n="1799,001,,,,10,26,4"></lb> |
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9394 γώνιον τὸ ΖΘ πλάτος ποιοῦν τὴν ΕΘ, τῷ <lb n="1799,001,,,,10,26,5"></lb> |
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9395 δὲ ΑΓ ἴσον ἀφῃρήσθω τὸ ΖΗ· λοιπὸν ἄρα <lb n="1799,001,,,,10,26,6"></lb> |
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9396 τὸ ΒΔ λοιπῷ τῷ ΚΘ ἐστιν ἴσον.</s> <s>ῥητὸν δέ <lb n="1799,001,,,,10,26,7"></lb> |
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9397 ἐστι τὸ ΔΒ· ῥητὸν ἄρα ἐστὶ καὶ τὸ ΚΘ.</s> <lb n="1799,001,,,,10,26,8"></lb> |
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9398 <s>ἐπεὶ οὖν μέσον ἐστὶν ἑκάτερον τῶν ΑΒ, <lb n="1799,001,,,,10,26,9"></lb> |
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9399 ΑΓ, καί ἐστι τὸ μὲν ΑΒ τῷ ΖΘ ἴσον, τὸ <lb n="1799,001,,,,10,26,10"></lb> |
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9400 δὲ ΑΓ τῷ ΖΗ, μέσον ἄρα καὶ ἑκάτερον <lb n="1799,001,,,,10,26,11"></lb> |
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9401 τῶν ΖΘ, ΖΗ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΕΖ πα-<lb n="1799,001,,,,10,26,12"></lb> |
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9402 ράκειται· ῥητὴ ἄρα ἐστὶν ἑκατέρα τῶν ΘΕ, <lb n="1799,001,,,,10,26,13"></lb> |
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9403 ΕΗ καὶ ἀσύμμετρος τῇ ΕΖ μήκει.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,26,14"></lb> |
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9404 ῥητόν ἐστι τὸ ΔΒ καί ἐστιν ἴσον τῷ ΚΘ, <lb n="1799,001,,,,10,26,15"></lb> |
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9405 ῥητὸν ἄρα ἐστὶ καὶ τὸ ΚΘ.</s> <s>καὶ παρὰ ῥητὴν <lb n="1799,001,,,,10,26,16"></lb> |
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9406 τὴν ΕΖ παράκειται· ῥητὴ ἄρα ἐστὶν ἡ ΗΘ <lb n="1799,001,,,,10,26,17"></lb> |
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9407 καὶ σύμμετρος τῇ ΕΖ μήκει.</s> <s>ἀλλὰ καὶ ἡ ΕΗ ῥητή ἐστι καὶ <lb n="1799,001,,,,10,26,18"></lb> |
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9408 ἀσύμμετρος τῇ ΕΖ μήκει· ἀσύμμετρος ἄρα ἐστὶν ἡ ΕΗ τῇ <lb n="1799,001,,,,10,26,19"></lb> |
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9409 ΗΘ μήκει.</s> <s>καί ἐστιν ὡς ἡ ΕΗ πρὸς τὴν ΗΘ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,26,20"></lb> |
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9410 τῆς ΕΗ πρὸς τὸ ὑπὸ τῶν ΕΗ, ΗΘ· ἀσύμμετρον ἄρα ἐστὶ <lb n="1799,001,,,,10,26,21"></lb> |
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9411 τὸ ἀπὸ τῆς ΕΗ τῷ ὑπὸ τῶν ΕΗ, ΗΘ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς <lb n="1799,001,,,,10,26,22"></lb> |
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9412 ΕΗ σύμμετρά ἐστι τὰ ἀπὸ τῶν ΕΗ, ΗΘ τετράγωνα· ῥητὰ <lb n="1799,001,,,,10,26,23"></lb> |
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9413 γὰρ ἀμφότερα· τῷ δὲ ὑπὸ τῶν ΕΗ, ΗΘ σύμμετρόν ἐστι τὸ <pb></pb> <lb n="1799,001,,,,10,26,24"></lb> |
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9414 δὶς ὑπὸ τῶν ΕΗ, ΗΘ· διπλάσιον γάρ ἐστιν αὐτοῦ· ἀσύμ-<lb n="1799,001,,,,10,26,25"></lb> |
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9415 μετρα ἄρα ἐστὶ τὰ ἀπὸ τῶν ΕΗ, ΗΘ τῷ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,26,26"></lb> |
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9416 ΕΗ, ΗΘ· καὶ συναμφότερα ἄρα τά τε ἀπὸ τῶν ΕΗ, ΗΘ <lb n="1799,001,,,,10,26,27"></lb> |
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9417 καὶ τὸ δὶς ὑπὸ τῶν ΕΗ, ΗΘ, ὅπερ ἐστὶ τὸ ἀπὸ τῆς ΕΘ, <lb n="1799,001,,,,10,26,28"></lb> |
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9418 ἀσύμμετρόν ἐστι τοῖς ἀπὸ τῶν ΕΗ, ΗΘ.</s> <s>ῥητὰ δὲ τὰ <lb n="1799,001,,,,10,26,29"></lb> |
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9419 ἀπὸ τῶν ΕΗ, ΗΘ· ἄλογον ἄρα τὸ ἀπὸ τῆς ΕΘ.</s> <s>ἄλογος <lb n="1799,001,,,,10,26,30"></lb> |
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9420 ἄρα ἐστὶν ἡ ΕΘ.</s> <s>ἀλλὰ καὶ ῥητή· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,10,26,31"></lb> |
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9421 <s>Μέσον ἄρα μέσου οὐχ ὑπερέχει ῥητῷ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,26,32"></lb> |
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9422 <s>Μέσας εὑρεῖν δυνάμει μόνον συμμέτρους ῥητὸν περι-<lb n="1799,001,,,,10,27,1"></lb> |
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9423 εχούσας.</s> <lb n="1799,001,,,,10,27,2"></lb> |
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9424 <s>Ἐκκείσθωσαν δύο ῥηταὶ δυνάμει μόνον σύμμετροι αἱ <lb n="1799,001,,,,10,27,3"></lb> |
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9425 Α, Β, καὶ εἰλήφθω τῶν Α, Β μέση ἀνάλογον ἡ Γ, καὶ <lb n="1799,001,,,,10,27,4"></lb> |
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9426 γεγονέτω ὡς ἡ Α πρὸς τὴν Β, οὕτως ἡ Γ πρὸς τὴν Δ.</s> <lb n="1799,001,,,,10,27,5"></lb> |
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9427 <s>Καὶ ἐπεὶ αἱ Α, Β ῥηταί εἰσι δυνάμει μόνον σύμμετροι, <lb n="1799,001,,,,10,27,6"></lb> |
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9428 τὸ ἄρα ὑπὸ τῶν Α, Β, τουτέστι τὸ ἀπὸ τῆς Γ, μέσον <lb n="1799,001,,,,10,27,7"></lb> |
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9429 ἐστίν.</s> <s>μέση ἄρα ἡ Γ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ Α πρὸς τὴν Β, <lb n="1799,001,,,,10,27,8"></lb> |
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9430 [οὕτωσ] ἡ Γ πρὸς τὴν Δ, αἱ δὲ Α, Β δυνάμει μόνον [εἰσὶ] <lb n="1799,001,,,,10,27,9"></lb> |
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9431 σύμμετροι, καὶ αἱ Γ, Δ ἄρα δυνάμει μόνον εἰσὶ σύμμετροι.</s> <lb n="1799,001,,,,10,27,10"></lb> |
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9432 <s>καί ἐστι μέση ἡ Γ· μέση ἄρα καὶ ἡ Δ.</s> <s>αἱ Γ, Δ ἄρα μέσαι <pb></pb> <lb n="1799,001,,,,10,27,11"></lb> |
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9433 εἰσὶ δυνάμει μόνον σύμμετροι.</s> <s>λέγω, ὅτι καὶ <lb n="1799,001,,,,10,27,12"></lb> |
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9434 ῥητὸν περιέχουσιν.</s> <s>ἐπεὶ γάρ ἐστιν ὡς ἡ Α πρὸς <lb n="1799,001,,,,10,27,13"></lb> |
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9435 τὴν Β, οὕτως ἡ Γ πρὸς τὴν Δ, ἐναλλὰξ ἄρα <lb n="1799,001,,,,10,27,14"></lb> |
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9436 ἐστὶν ὡς ἡ Α πρὸς τὴν Γ, ἡ Β πρὸς τὴν Δ.</s> <s>ἀλλ' <lb n="1799,001,,,,10,27,15"></lb> |
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9437 ὡς ἡ Α πρὸς τὴν Γ, ἡ Γ πρὸς τὴν Β· καὶ ὡς <lb n="1799,001,,,,10,27,16"></lb> |
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9438 ἄρα ἡ Γ πρὸς τὴν Β, οὕτως ἡ Β πρὸς τὴν Δ· <lb n="1799,001,,,,10,27,17"></lb> |
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9439 τὸ ἄρα ὑπὸ τῶν Γ, Δ ἴσον ἐστὶ τῷ ἀπὸ τῆς Β.</s> <lb n="1799,001,,,,10,27,18"></lb> |
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9440 <s>ῥητὸν δὲ τὸ ἀπὸ τῆς Β· ῥητὸν ἄρα [ἐστὶ] καὶ τὸ ὑπὸ <lb n="1799,001,,,,10,27,19"></lb> |
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9441 τῶν Γ, Δ.</s> <lb n="1799,001,,,,10,27,20"></lb> |
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9442 <s>Εὕρηνται ἄρα μέσαι δυνάμει μόνον σύμμετροι ῥητὸν <lb n="1799,001,,,,10,27,21"></lb> |
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9443 περιέχουσαι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,27,22"></lb> |
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9444 <s>Μέσας εὑρεῖν δυνάμει μόνον συμμέτρους μέσον περι-<lb n="1799,001,,,,10,28,1"></lb> |
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9445 εχούσας.</s> <lb n="1799,001,,,,10,28,2"></lb> |
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9446 <s>Ἐκκείσθωσαν [τρεῖσ] ῥηταὶ δυνάμει μόνον σύμμετροι <lb n="1799,001,,,,10,28,3"></lb> |
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9447 αἱ Α, Β, Γ, καὶ εἰλήφθω τῶν Α, Β μέση ἀνάλογον ἡ Δ, <lb n="1799,001,,,,10,28,4"></lb> |
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9448 καὶ γεγονέτω ὡς ἡ Β πρὸς τὴν Γ, ἡ Δ πρὸς τὴν Ε. <lb n="1799,001,,,,10,28,5"></lb> |
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9449 Ἐπεὶ αἱ Α, Β ῥηταί εἰσι δυνάμει μόνον σύμμετροι, τὸ <lb n="1799,001,,,,10,28,6"></lb> |
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9450 ἄρα ὑπὸ τῶν Α, Β, τουτέστι τὸ ἀπὸ τῆς Δ, μέσον ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,10,28,7"></lb> |
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9451 μέση ἄρα ἡ Δ.</s> <s>καὶ ἐπεὶ αἱ Β, Γ δυνάμει μόνον εἰσὶ σύμ-<lb n="1799,001,,,,10,28,8"></lb> |
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9452 μετροι, καί ἐστιν ὡς ἡ Β πρὸς τὴν Γ, ἡ Δ πρὸς τὴν Ε, καὶ <lb n="1799,001,,,,10,28,9"></lb> |
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9453 αἱ Δ, Ε ἄρα δυνάμει μόνον εἰσὶ σύμμετροι.</s> <s>μέση δὲ ἡ Δ· <lb n="1799,001,,,,10,28,10"></lb> |
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9454 μέση ἄρα καὶ ἡ Ε· αἱ Δ, Ε ἄρα μέσαι εἰσὶ δυνάμει μόνον <lb n="1799,001,,,,10,28,11"></lb> |
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9455 σύμμετροι.</s> <s>λέγω δή, ὅτι καὶ μέσον περιέχουσιν.</s> <s>ἐπεὶ γάρ <lb n="1799,001,,,,10,28,12"></lb> |
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9456 ἐστιν ὡς ἡ Β πρὸς τὴν Γ, ἡ Δ πρὸς τὴν Ε, ἐναλλὰξ ἄρα ὡς <lb n="1799,001,,,,10,28,13"></lb> |
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9457 ἡ Β πρὸς τὴν Δ, ἡ Γ πρὸς τὴν Ε.</s> <s>ὡς δὲ ἡ Β πρὸς τὴν Δ, <lb n="1799,001,,,,10,28,14"></lb> |
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9458 ἡ Δ πρὸς τὴν Α· καὶ ὡς ἄρα ἡ Δ πρὸς τὴν Α, ἡ Γ πρὸς <lb n="1799,001,,,,10,28,15"></lb> |
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9459 τὴν Ε· τὸ ἄρα ὑπὸ τῶν Α, Γ ἴσον ἐστὶ τῷ ὑπὸ τῶν Δ, Ε.</s> <lb n="1799,001,,,,10,28,16"></lb> |
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9460 <s>μέσον δὲ τὸ ὑπὸ τῶν Α, Γ· μέσον ἄρα καὶ τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,28,17"></lb> |
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9461 Δ, Ε.</s> <lb n="1799,001,,,,10,28,18"></lb> |
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9462 <s>Εὕρηνται ἄρα μέσαι δυνάμει μόνον σύμμετροι μέσον <lb n="1799,001,,,,10,28,19"></lb> |
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9463 περιέχουσαι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,28,20"></lb> |
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9464 <s>Λῆμμα <lb n="1799,001,,,,10,28,21"></lb> |
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9465 Εὑρεῖν δύο τετραγώνους ἀριθμούς, ὥστε καὶ τὸν <lb n="1799,001,,,,10,28,22"></lb> |
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9466 συγκείμενον ἐξ αὐτῶν εἶναι τετράγωνον.</s> <lb n="1799,001,,,,10,28,23"></lb> |
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9467 <s>Ἐκκείσθωσαν δύο ἀριθμοὶ οἱ ΑΒ, ΒΓ, ἔστωσαν δὲ <lb n="1799,001,,,,10,28,24"></lb> |
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9468 ἤτοι ἄρτιοι ἢ περιττοί.</s> <s>καὶ ἐπεί, ἐάν τε ἀπὸ ἀρτίου ἄρτιος <lb n="1799,001,,,,10,28,25"></lb> |
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9469 ἀφαιρεθῇ, ἐάν τε ἀπὸ περισσοῦ περισσός, ὁ λοιπὸς <lb n="1799,001,,,,10,28,26"></lb> |
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9470 ἄρτιός ἐστιν, ὁ λοιπὸς ἄρα ὁ ΑΓ ἄρτιός ἐστιν.</s> <s>τετμήσθω <lb n="1799,001,,,,10,28,27"></lb> |
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9471 ὁ ΑΓ δίχα κατὰ τὸ Δ.</s> <s>ἔστωσαν δὲ καὶ οἱ ΑΒ, ΒΓ ἤτοι <lb n="1799,001,,,,10,28,28"></lb> |
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9472 ὅμοιοι ἐπίπεδοι ἢ τετράγωνοι, οἳ καὶ αὐτοὶ ὅμοιοί εἰσιν <pb></pb> <lb n="1799,001,,,,10,28,29"></lb> |
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9473 ἐπίπεδοι· ὁ ἄρα ἐκ τῶν ΑΒ, ΒΓ μετὰ τοῦ ἀπὸ [τοῦ] <lb n="1799,001,,,,10,28,30"></lb> |
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9474 ΓΔ τετραγώνου ἴσος ἐστὶ τῷ ἀπὸ τοῦ ΒΔ τετρα-<lb n="1799,001,,,,10,28,31"></lb> |
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9475 γώνῳ.</s> <s>καί ἐστι τετράγωνος ὁ ἐκ τῶν ΑΒ, ΒΓ, <lb n="1799,001,,,,10,28,32"></lb> |
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9476 ἐπειδήπερ ἐδείχθη, ὅτι, ἐὰν δύο ὅμοιοι ἐπίπεδοι <lb n="1799,001,,,,10,28,33"></lb> |
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9477 πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμε-<lb n="1799,001,,,,10,28,34"></lb> |
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9478 νος τετράγωνός ἐστιν.</s> <s>εὕρηνται ἄρα δύο τετράγωνοι <lb n="1799,001,,,,10,28,35"></lb> |
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9479 ἀριθμοὶ ὅ τε ἐκ τῶν ΑΒ, ΒΓ καὶ ὁ ἀπὸ τοῦ ΓΔ, <lb n="1799,001,,,,10,28,36"></lb> |
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9480 οἳ συντεθέντες ποιοῦσι τὸν ἀπὸ τοῦ ΒΔ τετρά-<lb n="1799,001,,,,10,28,37"></lb> |
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9481 γωνον.</s> <lb n="1799,001,,,,10,28,38"></lb> |
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9482 <s>Καὶ φανερόν, ὅτι εὕρηνται πάλιν δύο τετράγωνοι ὅ τε <lb n="1799,001,,,,10,28,39"></lb> |
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9483 ἀπὸ τοῦ ΒΔ καὶ ὁ ἀπὸ τοῦ ΓΔ, ὥστε τὴν ὑπεροχὴν <lb n="1799,001,,,,10,28,40"></lb> |
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9484 αὐτῶν τὸν ὑπὸ ΑΒ, ΒΓ εἶναι τετράγωνον, ὅταν οἱ ΑΒ, <lb n="1799,001,,,,10,28,41"></lb> |
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9485 ΒΓ ὅμοιοι ὦσιν ἐπίπεδοι.</s> <s>ὅταν δὲ μὴ ὦσιν ὅμοιοι ἐπίπεδοι, <lb n="1799,001,,,,10,28,42"></lb> |
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9486 εὕρηνται δύο τετράγωνοι ὅ τε ἀπὸ τοῦ ΒΔ καὶ ὁ ἀπὸ τοῦ <lb n="1799,001,,,,10,28,43"></lb> |
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9487 ΔΓ, ὧν ἡ ὑπεροχὴ ὁ ὑπὸ τῶν ΑΒ, ΒΓ οὐκ ἔστι τετράγω-<lb n="1799,001,,,,10,28,44"></lb> |
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9488 νος· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,28,45"></lb> |
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9489 <s>Λῆμμα <lb n="1799,001,,,,10,28,46"></lb> |
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9490 Εὑρεῖν δύο τετραγώνους ἀριθμούς, ὥστε τὸν ἐξ αὐτῶν <lb n="1799,001,,,,10,28,47"></lb> |
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9491 συγκείμενον μὴ εἶναι τετράγωνον.</s> <lb n="1799,001,,,,10,28,48"></lb> |
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9492 <s>Ἔστω γὰρ ὁ ἐκ τῶν ΑΒ, ΒΓ, ὡς ἔφαμεν, τετράγωνος, <lb n="1799,001,,,,10,28,49"></lb> |
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9493 καὶ ἄρτιος ὁ ΓΑ, καὶ τετμήσθω ὁ ΓΑ δίχα τῷ Δ.</s> <s>φανερὸν <lb n="1799,001,,,,10,28,50"></lb> |
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9494 δή, ὅτι ὁ ἐκ τῶν ΑΒ, ΒΓ τετράγωνος μετὰ τοῦ ἀπὸ [τοῦ] <pb></pb> <lb n="1799,001,,,,10,28,51"></lb> |
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9495 ΓΔ τετραγώνου ἴσος ἐστὶ τῷ ἀπὸ [τοῦ] ΒΔ <lb n="1799,001,,,,10,28,52"></lb> |
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9496 τετραγώνῳ.</s> <s>ἀφῃρήσθω μονὰς ἡ ΔΕ· ὁ ἄρα ἐκ <lb n="1799,001,,,,10,28,53"></lb> |
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9497 τῶν ΑΒ, ΒΓ μετὰ τοῦ ἀπὸ [τοῦ] ΓΕ ἐλάσσων <lb n="1799,001,,,,10,28,54"></lb> |
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9498 ἐστὶ τοῦ ἀπὸ [τοῦ] ΒΔ τετραγώνου.</s> <s>λέγω οὖν, <lb n="1799,001,,,,10,28,55"></lb> |
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9499 ὅτι ὁ ἐκ τῶν ΑΒ, ΒΓ τετράγωνος μετὰ τοῦ <lb n="1799,001,,,,10,28,56"></lb> |
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9500 ἀπὸ [τοῦ] ΓΕ οὐκ ἔσται τετράγωνος.</s> <lb n="1799,001,,,,10,28,57"></lb> |
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9501 <s>Εἰ γὰρ ἔσται τετράγωνος, ἤτοι ἴσος ἐστὶ τῷ <lb n="1799,001,,,,10,28,58"></lb> |
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9502 ἀπὸ [τοῦ] ΒΕ ἢ ἐλάσσων τοῦ ἀπὸ [τοῦ] ΒΕ, <lb n="1799,001,,,,10,28,59"></lb> |
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9503 οὐκέτι δὲ καὶ μείζων, ἵνα μὴ τμηθῇ ἡ μονάς.</s> <lb n="1799,001,,,,10,28,60"></lb> |
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9504 <s>ἔστω, εἰ δυνατόν, πρότερον ὁ ἐκ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,10,28,61"></lb> |
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9505 μετὰ τοῦ ἀπὸ ΓΕ ἴσος τῷ ἀπὸ ΒΕ, καὶ ἔστω τῆς ΔΕ <lb n="1799,001,,,,10,28,62"></lb> |
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9506 μονάδος διπλασίων ὁ ΗΑ.</s> <s>ἐπεὶ οὖν ὅλος ὁ ΑΓ ὅλου <lb n="1799,001,,,,10,28,63"></lb> |
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9507 τοῦ ΓΔ ἐστι διπλασίων, ὧν ὁ ΑΗ τοῦ ΔΕ ἐστι διπλα-<lb n="1799,001,,,,10,28,64"></lb> |
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9508 σίων, καὶ λοιπὸς ἄρα ὁ ΗΓ λοιποῦ τοῦ ΕΓ ἐστι <lb n="1799,001,,,,10,28,65"></lb> |
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9509 διπλασίων· δίχα ἄρα τέτμηται ὁ ΗΓ τῷ Ε.</s> <s>ὁ ἄρα ἐκ τῶν <lb n="1799,001,,,,10,28,66"></lb> |
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9510 ΗΒ, ΒΓ μετὰ τοῦ ἀπὸ ΓΕ ἴσος ἐστὶ τῷ ἀπὸ ΒΕ τετρα-<lb n="1799,001,,,,10,28,67"></lb> |
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9511 γώνῳ.</s> <s>ἀλλὰ καὶ ὁ ἐκ τῶν ΑΒ, ΒΓ μετὰ τοῦ ἀπὸ ΓΕ ἴσος <lb n="1799,001,,,,10,28,68"></lb> |
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9512 ὑπόκειται τῷ ἀπὸ [τοῦ] ΒΕ τετραγώνῳ· ὁ ἄρα ἐκ τῶν <pb></pb> <lb n="1799,001,,,,10,28,69"></lb> |
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9513 ΗΒ, ΒΓ μετὰ τοῦ ἀπὸ ΓΕ ἴσος ἐστὶ τῷ ἐκ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,10,28,70"></lb> |
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9514 μετὰ τοῦ ἀπὸ ΓΕ.</s> <s>καὶ κοινοῦ ἀφαιρεθέντος τοῦ ἀπὸ ΓΕ <lb n="1799,001,,,,10,28,71"></lb> |
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9515 συνάγεται ὁ ΑΒ ἴσος τῷ ΗΒ· ὅπερ ἄτοπον.</s> <s>οὐκ ἄρα ὁ ἐκ <lb n="1799,001,,,,10,28,72"></lb> |
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9516 τῶν ΑΒ, ΒΓ μετὰ τοῦ ἀπὸ [τοῦ] ΓΕ ἴσος ἐστὶ τῷ ἀπὸ ΒΕ.</s> <lb n="1799,001,,,,10,28,73"></lb> |
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9517 <s>λέγω δή, ὅτι οὐδὲ ἐλάσσων τοῦ ἀπὸ ΒΕ.</s> <s>εἰ γὰρ δυνατόν, <lb n="1799,001,,,,10,28,74"></lb> |
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9518 ἔστω τῷ ἀπὸ ΒΖ ἴσος, καὶ τοῦ ΔΖ διπλασίων ὁ ΘΑ.</s> <s>καὶ <lb n="1799,001,,,,10,28,75"></lb> |
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9519 συναχθήσεται πάλιν διπλασίων ὁ ΘΓ τοῦ ΓΖ· ὥστε καὶ <lb n="1799,001,,,,10,28,76"></lb> |
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9520 τὸν ΓΘ δίχα τετμῆσθαι κατὰ τὸ Ζ, καὶ διὰ τοῦτο τὸν <lb n="1799,001,,,,10,28,77"></lb> |
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9521 ἐκ τῶν ΘΒ, ΒΓ μετὰ τοῦ ἀπὸ ΖΓ ἴσον γίνεσθαι τῷ ἀπὸ <lb n="1799,001,,,,10,28,78"></lb> |
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9522 ΒΖ.</s> <s>ὑπόκειται δὲ καὶ ὁ ἐκ τῶν ΑΒ, ΒΓ μετὰ τοῦ ἀπὸ <lb n="1799,001,,,,10,28,79"></lb> |
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9523 ΓΕ ἴσος τῷ ἀπὸ ΒΖ.</s> <s>ὥστε καὶ ὁ ἐκ τῶν ΘΒ, ΒΓ μετὰ <lb n="1799,001,,,,10,28,80"></lb> |
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9524 τοῦ ἀπὸ ΓΖ ἴσος ἔσται τῷ ἐκ τῶν ΑΒ, ΒΓ μετὰ τοῦ <lb n="1799,001,,,,10,28,81"></lb> |
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9525 ἀπὸ ΓΕ· ὅπερ ἄτοπον.</s> <s>οὐκ ἄρα ὁ ἐκ τῶν ΑΒ, ΒΓ μετὰ <lb n="1799,001,,,,10,28,82"></lb> |
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9526 τοῦ ἀπὸ ΓΕ ἴσος ἐστὶ [τῷ] ἐλάσσονι τοῦ ἀπὸ ΒΕ.</s> <lb n="1799,001,,,,10,28,83"></lb> |
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9527 <s>ἐδείχθη δέ, ὅτι οὐδὲ [αὐτῷ] τῷ ἀπὸ ΒΕ.</s> <s>οὐκ ἄρα ὁ ἐκ <lb n="1799,001,,,,10,28,84"></lb> |
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9528 τῶν ΑΒ, ΒΓ μετὰ τοῦ ἀπὸ ΓΕ τετράγωνός ἐστιν.</s> <lb n="1799,001,,,,10,28,85"></lb> |
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9529 <s>[δυνατοῦ δὲ ὄντος καὶ κατὰ πλείονας τρόπους τοὺς <lb n="1799,001,,,,10,28,86"></lb> |
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9530 εἰρημένους ἀριθμοὺς ἐπιδεικνύειν, ἀρκείσθωσαν ἡμῖν οἱ <pb></pb> <lb n="1799,001,,,,10,28,87"></lb> |
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9531 εἰρημένοι, ἵνα μὴ μακροτέρας οὔσης τῆς πραγματείας ἐπὶ <lb n="1799,001,,,,10,28,88"></lb> |
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9532 πλέον αὐτὴν μηκύνωμεν.] ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,28,89"></lb> |
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9533 <s>Εὑρεῖν δύο ῥητὰς δυνάμει μόνον συμμέτρους, ὥστε τὴν <lb n="1799,001,,,,10,29,1"></lb> |
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9534 μείζονα τῆς ἐλάσσονος μεῖζον δύνασθαι τῷ ἀπὸ συμμέ-<lb n="1799,001,,,,10,29,2"></lb> |
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9535 τρου ἑαυτῇ μήκει.</s> <lb n="1799,001,,,,10,29,3"></lb> |
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9536 <s>Ἐκκείσθω γάρ τις ῥητὴ ἡ ΑΒ καὶ δύο τετράγωνοι <lb n="1799,001,,,,10,29,4"></lb> |
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9537 ἀριθμοὶ οἱ ΓΔ, ΔΕ, ὥστε τὴν ὑπεροχὴν αὐτῶν τὸν ΓΕ <lb n="1799,001,,,,10,29,5"></lb> |
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9538 μὴ εἶναι τετράγωνον, καὶ γεγράφθω <lb n="1799,001,,,,10,29,6"></lb> |
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9539 ἐπὶ τῆς ΑΒ ἡμικύκλιον τὸ ΑΖΒ, <lb n="1799,001,,,,10,29,7"></lb> |
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9540 καὶ πεποιήσθω ὡς ὁ ΔΓ πρὸς τὸν <lb n="1799,001,,,,10,29,8"></lb> |
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9541 ΓΕ, οὕτως τὸ ἀπὸ τῆς ΒΑ τετρά-<lb n="1799,001,,,,10,29,9"></lb> |
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9542 γωνον πρὸς τὸ ἀπὸ τῆς ΑΖ τετρά-<lb n="1799,001,,,,10,29,10"></lb> |
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9543 γωνον καὶ ἐπεζεύχθω ἡ ΖΒ.</s> <lb n="1799,001,,,,10,29,11"></lb> |
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9544 <s>Ἐπεὶ [οὖν] ἐστιν ὡς τὸ ἀπὸ τῆς ΒΑ <lb n="1799,001,,,,10,29,12"></lb> |
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9545 πρὸς τὸ ἀπὸ τῆς ΑΖ, οὕτως ὁ ΔΓ πρὸς τὸν ΓΕ, τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,29,13"></lb> |
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9546 ΒΑ ἄρα πρὸς τὸ ἀπὸ τῆς ΑΖ λόγον ἔχει, ὃν ἀριθμὸς ὁ ΔΓ <lb n="1799,001,,,,10,29,14"></lb> |
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9547 πρὸς ἀριθμὸν τὸν ΓΕ· σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΒΑ <lb n="1799,001,,,,10,29,15"></lb> |
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9548 τῷ ἀπὸ τῆς ΑΖ.</s> <s>ῥητὸν δὲ τὸ ἀπὸ τῆς ΑΒ· ῥητὸν ἄρα καὶ <lb n="1799,001,,,,10,29,16"></lb> |
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9549 τὸ ἀπὸ τῆς ΑΖ· ῥητὴ ἄρα καὶ ἡ ΑΖ.</s> <s>καὶ ἐπεὶ ὁ ΔΓ πρὸς <lb n="1799,001,,,,10,29,17"></lb> |
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9550 τὸν ΓΕ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετρά-<lb n="1799,001,,,,10,29,18"></lb> |
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9551 γωνον ἀριθμόν, οὐδὲ τὸ ἀπὸ τῆς ΒΑ ἄρα πρὸς τὸ ἀπὸ τῆς <pb></pb> <lb n="1799,001,,,,10,29,19"></lb> |
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9552 ΑΖ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,29,20"></lb> |
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9553 ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΒ τῇ ΑΖ μήκει· αἱ <lb n="1799,001,,,,10,29,21"></lb> |
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9554 ΒΑ, ΑΖ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι.</s> <s>καὶ ἐπεί <lb n="1799,001,,,,10,29,22"></lb> |
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9555 [ἐστιν] ὡς ὁ ΔΓ πρὸς τὸν ΓΕ, οὕτως τὸ ἀπὸ τῆς ΒΑ πρὸς <lb n="1799,001,,,,10,29,23"></lb> |
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9556 τὸ ἀπὸ τῆς ΑΖ, ἀναστρέψαντι ἄρα ὡς ὁ ΓΔ πρὸς τὸν <lb n="1799,001,,,,10,29,24"></lb> |
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9557 ΔΕ, οὕτως τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΖ.</s> <s>ὁ δὲ <lb n="1799,001,,,,10,29,25"></lb> |
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9558 ΓΔ πρὸς τὸν ΔΕ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,29,26"></lb> |
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9559 τετράγωνον ἀριθμόν· καὶ τὸ ἀπὸ τῆς ΑΒ ἄρα πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,29,27"></lb> |
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9560 τῆς ΒΖ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγω-<lb n="1799,001,,,,10,29,28"></lb> |
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9561 νον ἀριθμόν· σύμμετρος ἄρα ἐστὶν ἡ ΑΒ τῇ ΒΖ μήκει.</s> <s>καί <lb n="1799,001,,,,10,29,29"></lb> |
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9562 ἐστι τὸ ἀπὸ τῆς ΑΒ ἴσον τοῖς ἀπὸ τῶν ΑΖ, ΖΒ· ἡ ΑΒ <lb n="1799,001,,,,10,29,30"></lb> |
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9563 ἄρα τῆς ΑΖ μεῖζον δύναται τῇ ΒΖ συμμέτρῳ ἑαυτῇ.</s> <lb n="1799,001,,,,10,29,31"></lb> |
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9564 <s>Εὕρηνται ἄρα δύο ῥηταὶ δυνάμει μόνον σύμμετροι αἱ <lb n="1799,001,,,,10,29,32"></lb> |
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9565 ΒΑ, ΑΖ, ὥστε τὴν μείζονα τὴν ΑΒ τῆς ἐλάσσονος τῆς <lb n="1799,001,,,,10,29,33"></lb> |
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9566 ΑΖ μεῖζον δύνασθαι τῷ ἀπὸ τῆς ΒΖ συμμέτρου ἑαυτῇ <lb n="1799,001,,,,10,29,34"></lb> |
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9567 μήκει· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,29,35"></lb> |
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9568 <s>Εὑρεῖν δύο ῥητὰς δυνάμει μόνον συμμέτρους, ὥστε τὴν <lb n="1799,001,,,,10,30,1"></lb> |
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9569 μείζονα τῆς ἐλάσσονος μεῖζον δύνασθαι τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,30,2"></lb> |
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9570 μέτρου ἑαυτῇ μήκει.</s> <lb n="1799,001,,,,10,30,3"></lb> |
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9571 <s>Ἐκκείσθω ῥητὴ ἡ ΑΒ καὶ δύο τετράγωνοι ἀριθμοὶ <lb n="1799,001,,,,10,30,4"></lb> |
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9572 οἱ ΓΕ, ΕΔ, ὥστε τὸν συγκείμενον ἐξ αὐτῶν τὸν ΓΔ μὴ <pb></pb> <lb n="1799,001,,,,10,30,5"></lb> |
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9573 εἶναι τετράγωνον, καὶ γεγράφθω <lb n="1799,001,,,,10,30,6"></lb> |
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9574 ἐπὶ τῆς ΑΒ ἡμικύκλιον τὸ ΑΖΒ, <lb n="1799,001,,,,10,30,7"></lb> |
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9575 καὶ πεποιήσθω ὡς ὁ ΔΓ πρὸς τὸν <lb n="1799,001,,,,10,30,8"></lb> |
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9576 ΓΕ, οὕτως τὸ ἀπὸ τῆς ΒΑ πρὸς τὸ <lb n="1799,001,,,,10,30,9"></lb> |
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9577 ἀπὸ τῆς ΑΖ, καὶ ἐπεζεύχθω ἡ <lb n="1799,001,,,,10,30,10"></lb> |
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9578 ΖΒ.</s> <lb n="1799,001,,,,10,30,11"></lb> |
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9579 <s>Ὁμοίως δὴ δείξομεν τῷ πρὸ τούτου, ὅτι αἱ ΒΑ, ΑΖ <lb n="1799,001,,,,10,30,12"></lb> |
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9580 ῥηταί εἰσι δυνάμει μόνον σύμμετροι.</s> <s>καὶ ἐπεί ἐστιν ὡς <lb n="1799,001,,,,10,30,13"></lb> |
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9581 ὁ ΔΓ πρὸς τὸν ΓΕ, οὕτως τὸ ἀπὸ τῆς ΒΑ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,30,14"></lb> |
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9582 τῆς ΑΖ, ἀναστρέψαντι ἄρα ὡς ὁ ΓΔ πρὸς τὸν ΔΕ, οὕτως <lb n="1799,001,,,,10,30,15"></lb> |
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9583 τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΖ.</s> <s>ὁ δὲ ΓΔ πρὸς τὸν <lb n="1799,001,,,,10,30,16"></lb> |
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9584 ΔΕ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετρά-<lb n="1799,001,,,,10,30,17"></lb> |
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9585 γωνον ἀριθμόν· οὐδ' ἄρα τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,30,18"></lb> |
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9586 ΒΖ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,30,19"></lb> |
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9587 ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΒ τῇ ΒΖ μήκει.</s> <s>καὶ <lb n="1799,001,,,,10,30,20"></lb> |
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9588 δύναται ἡ ΑΒ τῆσ ΑΖ μεῖζον τῷ ἀπὸ τῆς ΖΒ ἀσυμμέτρου <lb n="1799,001,,,,10,30,21"></lb> |
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9589 ἑαυτῇ.</s> <lb n="1799,001,,,,10,30,22"></lb> |
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9590 <s>Αἱ ΑΒ, ΑΖ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι, <lb n="1799,001,,,,10,30,23"></lb> |
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9591 καὶ ἡ ΑΒ τῆς ΑΖ μεῖζον δύναται τῷ ἀπὸ τῆς ΖΒ <lb n="1799,001,,,,10,30,24"></lb> |
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9592 ἀσυμμέτρου ἑαυτῇ μήκει· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,30,25"></lb> |
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9593 Εὑρεῖν δύο μέσας δυνάμει μόνον συμμέτρους ῥητὸν <lb n="1799,001,,,,10,31,1"></lb> |
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9594 περιεχούσας, ὥστε τὴν μείζονα τῆς ἐλάσσονος μεῖζον <lb n="1799,001,,,,10,31,2"></lb> |
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9595 δύνασθαι τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει.</s> <lb n="1799,001,,,,10,31,3"></lb> |
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9596 <s>Ἐκκείσθωσαν δύο ῥηταὶ δυνάμει μόνον σύμμετροι αἱ <lb n="1799,001,,,,10,31,4"></lb> |
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9597 Α, Β, ὥστε τὴν Α μείζονα οὖσαν τῆς ἐλάσσονος τῆς Β <lb n="1799,001,,,,10,31,5"></lb> |
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9598 μεῖζον δύνασθαι τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει.</s> <s>καὶ τῷ <lb n="1799,001,,,,10,31,6"></lb> |
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9599 ὑπὸ τῶν Α, Β ἴσον ἔστω τὸ ἀπὸ τῆς Γ.</s> <s>μέσον δὲ τὸ ὑπὸ <lb n="1799,001,,,,10,31,7"></lb> |
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9600 τῶν Α, Β· μέσον ἄρα καὶ τὸ ἀπὸ τῆς Γ· μέση ἄρα καὶ <lb n="1799,001,,,,10,31,8"></lb> |
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9601 ἡ Γ.</s> <s>τῷ δὲ ἀπὸ τῆς Β ἴσον ἔστω τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,31,9"></lb> |
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9602 Γ, Δ.</s> <s>ῥητὸν δὲ τὸ ἀπὸ τῆς Β· ῥητὸν ἄρα καὶ <lb n="1799,001,,,,10,31,10"></lb> |
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9603 τὸ ὑπὸ τῶν Γ, Δ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ Α πρὸς <lb n="1799,001,,,,10,31,11"></lb> |
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9604 τὴν Β, οὕτως τὸ ὑπὸ τῶν Α, Β πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,31,12"></lb> |
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9605 τῆς Β, ἀλλὰ τῷ μὲν ὑπὸ τῶν Α, Β ἴσον ἐστὶ τὸ <lb n="1799,001,,,,10,31,13"></lb> |
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9606 ἀπὸ τῆς Γ, τῷ δὲ ἀπὸ τῆς Β ἴσον τὸ ὑπὸ τῶν Γ, <lb n="1799,001,,,,10,31,14"></lb> |
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9607 Δ, ὡς ἄρα ἡ Α πρὸς τὴν Β, οὕτως τὸ ἀπὸ τῆς Γ πρὸς τὸ ὑπὸ <lb n="1799,001,,,,10,31,15"></lb> |
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9608 τῶν Γ, Δ.</s> <s>ὡς δὲ τὸ ἀπὸ τῆς Γ πρὸς τὸ ὑπὸ τῶν Γ, Δ, οὕτως <lb n="1799,001,,,,10,31,16"></lb> |
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9609 ἡ Γ πρὸς τὴν Δ· καὶ ὡς ἄρα ἡ Α πρὸς τὴν Β, οὕτως ἡ Γ πρὸς <lb n="1799,001,,,,10,31,17"></lb> |
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9610 τὴν Δ.</s> <s>σύμμετρος δὲ ἡ Α τῇ Β δυνάμει μόνον· σύμμετρος <lb n="1799,001,,,,10,31,18"></lb> |
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9611 ἄρα καὶ ἡ Γ τῇ Δ δυνάμει μόνον.</s> <s>καί ἐστι μέση ἡ Γ· μέση <lb n="1799,001,,,,10,31,19"></lb> |
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9612 ἄρα καὶ ἡ Δ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ Α πρὸς τὴν Β, ἡ Γ πρὸς <lb n="1799,001,,,,10,31,20"></lb> |
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9613 τὴν Δ, ἡ δὲ Α τῆς Β μεῖζον δύναται τῷ ἀπὸ συμμέτρου <lb n="1799,001,,,,10,31,21"></lb> |
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9614 ἑαυτῇ, καὶ ἡ Γ ἄρα τῆς Δ μεῖζον δύναται τῷ ἀπὸ συμ-<lb n="1799,001,,,,10,31,22"></lb> |
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9615 μέτρου ἑαυτῇ.</s> <s><pb></pb> <lb n="1799,001,,,,10,31,23"></lb> |
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9616 Εὕρηνται ἄρα δύο μέσαι δυνάμει μόνον σύμμετροι αἱ <lb n="1799,001,,,,10,31,24"></lb> |
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9617 Γ, Δ ῥητὸν περιέχουσαι, καὶ ἡ Γ τῆς Δ μεῖζον δύναται <lb n="1799,001,,,,10,31,25"></lb> |
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9618 τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει.</s> <lb n="1799,001,,,,10,31,26"></lb> |
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9619 <s>Ὁμοίως δὴ δειχθήσεται καὶ τῷ ἀπὸ ἀσυμμέτρου, ὅταν <lb n="1799,001,,,,10,31,27"></lb> |
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9620 ἡ Α τῆς Β μεῖζον δύνηται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <lb n="1799,001,,,,10,31,28"></lb> |
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9621 <s>Εὑρεῖν δύο μέσας δυνάμει μόνον συμμέτρους μέσον περι-<lb n="1799,001,,,,10,32,1"></lb> |
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9622 εχούσας, ὥστε τὴν μείζονα τῆς ἐλάσσονος μεῖζον δύνασθαι <lb n="1799,001,,,,10,32,2"></lb> |
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9623 τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <lb n="1799,001,,,,10,32,3"></lb> |
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9624 <s>Ἐκκείσθωσαν τρεῖς ῥηταὶ δυνάμει μόνον σύμμετροι αἱ <lb n="1799,001,,,,10,32,4"></lb> |
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9625 Α, Β, Γ, ὥστε τὴν Α τῆς Γ μεῖζον δύνασθαι τῷ ἀπὸ <lb n="1799,001,,,,10,32,5"></lb> |
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9626 συμμέτρου ἑαυτῇ, καὶ τῷ μὲν ὑπὸ τῶν Α, Β ἴσον ἔστω τὸ <lb n="1799,001,,,,10,32,6"></lb> |
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9627 ἀπὸ τῆς Δ.</s> <s>μέσον ἄρα τὸ ἀπὸ <lb n="1799,001,,,,10,32,7"></lb> |
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9628 τῆς Δ· καὶ ἡ Δ ἄρα μέση <lb n="1799,001,,,,10,32,8"></lb> |
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9629 ἐστίν.</s> <s>τῷ δὲ ὑπὸ τῶν Β, Γ ἴσον <lb n="1799,001,,,,10,32,9"></lb> |
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9630 ἔστω τὸ ὑπὸ τῶν Δ, Ε.</s> <s>καὶ <lb n="1799,001,,,,10,32,10"></lb> |
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9631 ἐπεί ἐστιν ὡς τὸ ὑπὸ τῶν Α, Β <lb n="1799,001,,,,10,32,11"></lb> |
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9632 πρὸς τὸ ὑπὸ τῶν Β, Γ, οὕτως ἡ Α πρὸς τὴν Γ, ἀλλὰ τῷ μὲν <lb n="1799,001,,,,10,32,12"></lb> |
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9633 ὑπὸ τῶν Α, Β ἴσον ἐστὶ τὸ ἀπὸ τῆς Δ, τῷ δὲ ὑπὸ τῶν Β, <lb n="1799,001,,,,10,32,13"></lb> |
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9634 Γ ἴσον τὸ ὑπὸ τῶν Δ, Ε, ἔστιν ἄρα ὡς ἡ Α πρὸς τὴν Γ, <pb></pb> <lb n="1799,001,,,,10,32,14"></lb> |
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9635 οὕτως τὸ ἀπὸ τῆς Δ πρὸς τὸ ὑπὸ τῶν Δ, Ε.</s> <s>ὡς δὲ τὸ ἀπὸ <lb n="1799,001,,,,10,32,15"></lb> |
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9636 τῆς Δ πρὸς τὸ ὑπὸ τῶν Δ, Ε, οὕτως ἡ Δ πρὸς τὴν Ε· καὶ <lb n="1799,001,,,,10,32,16"></lb> |
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9637 ὡς ἄρα ἡ Α πρὸς τὴν Γ, οὕτως ἡ Δ πρὸς τὴν Ε· σύμμετρος <lb n="1799,001,,,,10,32,17"></lb> |
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9638 δὲ ἡ Α τῇ Γ δυνάμει [μόνον].</s> <s>σύμμετρος ἄρα καὶ ἡ Δ τῇ Ε <lb n="1799,001,,,,10,32,18"></lb> |
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9639 δυνάμει μόνον.</s> <s>μέση δὲ ἡ Δ· μέση ἄρα καὶ ἡ Ε.</s> <s>καὶ ἐπεί <lb n="1799,001,,,,10,32,19"></lb> |
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9640 ἐστιν ὡς ἡ Α πρὸς τὴν Γ, ἡ Δ πρὸς τὴν Ε, ἡ δὲ Α τῆς Γ μεῖ-<lb n="1799,001,,,,10,32,20"></lb> |
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9641 ζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ ἡ Δ ἄρα τῆς Ε <lb n="1799,001,,,,10,32,21"></lb> |
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9642 μεῖζον δυνήσεται τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <s>λέγω δή, ὅτι <lb n="1799,001,,,,10,32,22"></lb> |
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9643 καὶ μέσον ἐστὶ τὸ ὑπὸ τῶν Δ, Ε.</s> <s>ἐπεὶ γὰρ ἴσον ἐστὶ τὸ ὑπὸ <lb n="1799,001,,,,10,32,23"></lb> |
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9644 τῶν Β, Γ τῷ ὑπὸ τῶν Δ, Ε, μέσον δὲ τὸ ὑπὸ τῶν Β, Γ <lb n="1799,001,,,,10,32,24"></lb> |
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9645 [αἱ γὰρ Β, Γ ῥηταί εἰσι δυνάμει μόνον σύμμετροι], μέσον <lb n="1799,001,,,,10,32,25"></lb> |
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9646 ἄρα καὶ τὸ ὑπὸ τῶν Δ, Ε.</s> <lb n="1799,001,,,,10,32,26"></lb> |
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9647 <s>Εὕρηνται ἄρα δύο μέσαι δυνάμει μόνον σύμμετροι αἱ <lb n="1799,001,,,,10,32,27"></lb> |
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9648 Δ, Ε μέσον περιέχουσαι, ὥστε τὴν μείζονα τῆς ἐλάσσονος <lb n="1799,001,,,,10,32,28"></lb> |
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9649 μεῖζον δύνασθαι τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <lb n="1799,001,,,,10,32,29"></lb> |
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9650 <s>Ὁμοίως δὴ πάλιν δειχθήσεται καὶ τῷ ἀπὸ ἀσυμμέτρου, <lb n="1799,001,,,,10,32,30"></lb> |
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9651 ὅταν ἡ Α τῆς Γ μεῖζον δύνηται τῷ ἀπὸ ἀσυμμέτρου <lb n="1799,001,,,,10,32,31"></lb> |
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9652 ἑαυτῇ.</s> <lb n="1799,001,,,,10,32,32"></lb> |
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9653 <s>Λῆμμα <lb n="1799,001,,,,10,32,33"></lb> |
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9654 Ἔστω τρίγωνον ὀρθογώνιον τὸ ΑΒΓ ὀρθὴν ἔχον τὴν <lb n="1799,001,,,,10,32,34"></lb> |
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9655 Α, καὶ ἤχθω κάθετος ἡ ΑΔ· λέγω, ὅτι τὸ μὲν ὑπὸ τῶν <pb></pb> <lb n="1799,001,,,,10,32,35"></lb> |
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9656 ΓΒΔ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΒΑ, τὸ δὲ ὑπὸ τῶν ΒΓΔ <lb n="1799,001,,,,10,32,36"></lb> |
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9657 ἴσον τῷ ἀπὸ τῆς ΓΑ, καὶ τὸ ὑπὸ τῶν ΒΔ, ΔΓ ἴσον τῷ <lb n="1799,001,,,,10,32,37"></lb> |
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9658 ἀπὸ τῆς ΑΔ, καὶ ἔτι τὸ ὑπὸ τῶν ΒΓ, ΑΔ ἴσον [ἐστὶ] τῷ <lb n="1799,001,,,,10,32,38"></lb> |
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9659 ὑπὸ τῶν ΒΑ, ΑΓ.</s> <lb n="1799,001,,,,10,32,39"></lb> |
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9660 <s>Καὶ πρῶτον, ὅτι τὸ ὑπὸ τῶν ΓΒΔ ἴσον [ἐστὶ] τῷ ἀπὸ <lb n="1799,001,,,,10,32,40"></lb> |
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9661 τῆς ΒΑ.</s> <lb n="1799,001,,,,10,32,41"></lb> |
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9662 <s>Ἐπεὶ γὰρ ἐν ὀρθογωνίῳ τριγώνῳ ἀπὸ τῆς ὀρθῆς <lb n="1799,001,,,,10,32,42"></lb> |
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9663 γωνίας ἐπὶ τὴν βάσιν κάθετος ἦκται ἡ ΑΔ, τὰ ΑΒΔ, <lb n="1799,001,,,,10,32,43"></lb> |
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9664 ΑΔΓ ἄρα τρίγωνα ὅμοιά ἐστι τῷ τε ὅλῳ <lb n="1799,001,,,,10,32,44"></lb> |
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9665 τῷ ΑΒΓ καὶ ἀλλήλοις.</s> <s>καὶ ἐπεὶ ὅμοιόν <lb n="1799,001,,,,10,32,45"></lb> |
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9666 ἐστι τὸ ΑΒΓ τρίγωνον τῷ ΑΒΔ τρι-<lb n="1799,001,,,,10,32,46"></lb> |
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9667 γώνῳ, ἔστιν ἄρα ὡς ἡ ΓΒ πρὸς τὴν <lb n="1799,001,,,,10,32,47"></lb> |
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9668 ΒΑ, οὕτως ἡ ΒΑ πρὸς τὴν ΒΔ· <lb n="1799,001,,,,10,32,48"></lb> |
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9669 τὸ ἄρα ὑπὸ τῶν ΓΒΔ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ.</s> <lb n="1799,001,,,,10,32,49"></lb> |
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9670 <s>Διὰ τὰ αὐτὰ δὴ καὶ τὸ ὑπὸ τῶν ΒΓΔ ἴσον ἐστὶ τῷ ἀπὸ <lb n="1799,001,,,,10,32,50"></lb> |
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9671 τῆς ΑΓ.</s> <lb n="1799,001,,,,10,32,51"></lb> |
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9672 <s>Καὶ ἐπεί, ἐὰν ἐν ὀρθογωνίῳ τριγώνῳ ἀπὸ τῆς ὀρθῆς <lb n="1799,001,,,,10,32,52"></lb> |
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9673 γωνίας ἐπὶ τὴν βάσιν κάθετος ἀχθῇ, ἡ ἀχθεῖσα τῶν τῆς <lb n="1799,001,,,,10,32,53"></lb> |
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9674 βάσεως τμημάτων μέση ἀνάλογόν ἐστιν, ἔστιν ἄρα ὡς <lb n="1799,001,,,,10,32,54"></lb> |
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9675 ἡ ΒΔ πρὸς τὴν ΔΑ, οὕτως ἡ ΑΔ πρὸς τὴν ΔΓ· τὸ ἄρα <lb n="1799,001,,,,10,32,55"></lb> |
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9676 ὑπὸ τῶν ΒΔ, ΔΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΔΑ.</s> <lb n="1799,001,,,,10,32,56"></lb> |
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9677 <s>Λέγω, ὅτι καὶ τὸ ὑπὸ τῶν ΒΓ, ΑΔ ἴσον ἐστὶ τῷ ὑπὸ <lb n="1799,001,,,,10,32,57"></lb> |
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9678 τῶν ΒΑ, ΑΓ.</s> <s>ἐπεὶ γάρ, ὡς ἔφαμεν, ὅμοιόν ἐστι τὸ ΑΒΓ <pb></pb> <lb n="1799,001,,,,10,32,58"></lb> |
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9679 τῷ ΑΒΔ, ἔστιν ἄρα ὡς ἡ ΒΓ πρὸς τὴν ΓΑ, οὕτως ἡ ΒΑ <lb n="1799,001,,,,10,32,59"></lb> |
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9680 πρὸς τὴν ΑΔ.</s> <s>[ἐὰν δὲ τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, τὸ <lb n="1799,001,,,,10,32,60"></lb> |
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9681 ὑπὸ τῶν ἄκρων ἴσον ἐστὶ τῷ ὑπὸ τῶν μέσων.] τὸ ἄρα ὑπὸ <lb n="1799,001,,,,10,32,61"></lb> |
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9682 τῶν ΒΓ, ΑΔ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΒΑ, ΑΓ· ὅπερ ἔδει <lb n="1799,001,,,,10,32,62"></lb> |
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9683 δεῖξαι.</s> <lb n="1799,001,,,,10,32,63"></lb> |
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9684 <s>Εὑρεῖν δύο εὐθείας δυνάμει ἀσυμμέτρους ποιούσας τὸ <lb n="1799,001,,,,10,33,1"></lb> |
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9685 μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων ῥητόν, <lb n="1799,001,,,,10,33,2"></lb> |
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9686 τὸ δ' ὑπ' αὐτῶν μέσον.</s> <lb n="1799,001,,,,10,33,3"></lb> |
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9687 <s>Ἐκκείσθωσαν δύο ῥηταὶ δυνάμει μόνον σύμμετροι αἱ <lb n="1799,001,,,,10,33,4"></lb> |
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9688 ΑΒ, ΒΓ, ὥστε τὴν μείζονα τὴν ΑΒ τῆς ἐλάσσονος τῆς <lb n="1799,001,,,,10,33,5"></lb> |
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9689 ΒΓ μεῖζον δύνασθαι τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, καὶ <lb n="1799,001,,,,10,33,6"></lb> |
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9690 τετμήσθω ἡ ΒΓ δίχα κατὰ <lb n="1799,001,,,,10,33,7"></lb> |
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9691 τὸ Δ, καὶ τῷ ἀφ' ὁποτέρας <lb n="1799,001,,,,10,33,8"></lb> |
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9692 τῶν ΒΔ, ΔΓ ἴσον παρὰ τὴν <lb n="1799,001,,,,10,33,9"></lb> |
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9693 ΑΒ παραβεβλήσθω παραλ-<lb n="1799,001,,,,10,33,10"></lb> |
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9694 ληλόγραμμον ἐλλεῖπον εἴδει <lb n="1799,001,,,,10,33,11"></lb> |
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9695 τετραγώνῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΑΕΒ, καὶ γεγράφθω <lb n="1799,001,,,,10,33,12"></lb> |
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9696 ἐπὶ τῆς ΑΒ ἡμικύκλιον τὸ ΑΖΒ, καὶ ἤχθω τῇ ΑΒ <lb n="1799,001,,,,10,33,13"></lb> |
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9697 πρὸς ὀρθὰς ἡ ΕΖ, καὶ ἐπεζεύχθωσαν αἱ ΑΖ, ΖΒ.</s> <lb n="1799,001,,,,10,33,14"></lb> |
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9698 <s>Καὶ ἐπεὶ [δύο] εὐθεῖαι ἄνισοί εἰσιν αἱ ΑΒ, ΒΓ, καὶ ἡ <pb></pb> <lb n="1799,001,,,,10,33,15"></lb> |
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9699 ΑΒ τῆς ΒΓ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, <lb n="1799,001,,,,10,33,16"></lb> |
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9700 τῷ δὲ τετάρτῳ τοῦ ἀπὸ τῆς ΒΓ, τουτέστι τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,33,17"></lb> |
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9701 ἡμισείας αὐτῆς, ἴσον παρὰ τὴν ΑΒ παραβέβληται <lb n="1799,001,,,,10,33,18"></lb> |
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9702 παραλληλόγραμμον ἐλλεῖπον εἴδει τετραγώνῳ καὶ ποιεῖ τὸ <lb n="1799,001,,,,10,33,19"></lb> |
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9703 ὑπὸ τῶν ΑΕΒ, ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΕ τῇ ΕΒ.</s> <s>καί <lb n="1799,001,,,,10,33,20"></lb> |
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9704 ἐστιν ὡς ἡ ΑΕ πρὸς ΕΒ, οὕτως τὸ ὑπὸ τῶν ΒΑ, ΑΕ <lb n="1799,001,,,,10,33,21"></lb> |
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9705 πρὸς τὸ ὑπὸ τῶν ΑΒ, ΒΕ, ἴσον δὲ τὸ μὲν ὑπὸ τῶν ΒΑ, <lb n="1799,001,,,,10,33,22"></lb> |
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9706 ΑΕ τῷ ἀπὸ τῆς ΑΖ, τὸ δὲ ὑπὸ τῶν ΑΒ, ΒΕ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,33,23"></lb> |
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9707 ΒΖ· ἀσύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΑΖ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,33,24"></lb> |
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9708 ΖΒ· αἱ ΑΖ, ΖΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,33,25"></lb> |
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9709 ἡ ΑΒ ῥητή ἐστιν, ῥητὸν ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς ΑΒ· ὥστε <lb n="1799,001,,,,10,33,26"></lb> |
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9710 καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΖ, ΖΒ ῥητόν ἐστιν.</s> <lb n="1799,001,,,,10,33,27"></lb> |
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9711 <s>καὶ ἐπεὶ πάλιν τὸ ὑπὸ τῶν ΑΕ, ΕΒ ἴσον ἐστὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,33,28"></lb> |
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9712 ΕΖ, ὑπόκειται δὲ τὸ ὑπὸ τῶν ΑΕ, ΕΒ καὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,33,29"></lb> |
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9713 ΒΔ ἴσον, ἴση ἄρα ἐστὶν ἡ ΖΕ τῇ ΒΔ· διπλῆ ἄρα ἡ ΒΓ <lb n="1799,001,,,,10,33,30"></lb> |
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9714 τῆς ΖΕ· ὥστε καὶ τὸ ὑπὸ τῶν ΑΒ, ΒΓ σύμμετρόν ἐστι <lb n="1799,001,,,,10,33,31"></lb> |
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9715 τῷ ὑπὸ τῶν ΑΒ, ΕΖ.</s> <s>μέσον δὲ τὸ ὑπὸ τῶν ΑΒ, ΒΓ· <lb n="1799,001,,,,10,33,32"></lb> |
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9716 μέσον ἄρα καὶ τὸ ὑπὸ τῶν ΑΒ, ΕΖ.</s> <s>ἴσον δὲ τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,33,33"></lb> |
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9717 ΑΒ, ΕΖ τῷ ὑπὸ τῶν ΑΖ, ΖΒ· μέσον ἄρα καὶ τὸ ὑπὸ <pb></pb> <lb n="1799,001,,,,10,33,34"></lb> |
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9718 τῶν ΑΖ, ΖΒ.</s> <s>ἐδείχθη δὲ καὶ ῥητὸν τὸ συγκείμενον ἐκ τῶν <lb n="1799,001,,,,10,33,35"></lb> |
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9719 ἀπ' αὐτῶν τετραγώνων.</s> <lb n="1799,001,,,,10,33,36"></lb> |
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9720 <s>Εὕρηνται ἄρα δύο εὐθεῖαι δυνάμει ἀσύμμετροι αἱ ΑΖ, <lb n="1799,001,,,,10,33,37"></lb> |
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9721 ΖΒ ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν <lb n="1799,001,,,,10,33,38"></lb> |
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9722 τετραγώνων ῥητόν, τὸ δὲ ὑπ' αὐτῶν μέσον· ὅπερ ἔδει <lb n="1799,001,,,,10,33,39"></lb> |
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9723 δεῖξαι.</s> <lb n="1799,001,,,,10,33,40"></lb> |
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9724 <s>Εὑρεῖν δύο εὐθείας δυνάμει ἀσυμμέτρους ποιούσας τὸ <lb n="1799,001,,,,10,34,1"></lb> |
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9725 μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων μέσον, <lb n="1799,001,,,,10,34,2"></lb> |
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9726 τὸ δ' ὑπ' αὐτῶν ῥητόν.</s> <lb n="1799,001,,,,10,34,3"></lb> |
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9727 <s>Ἐκκείσθωσαν δύο μέσαι δυνάμει μόνον σύμμετροι αἱ <lb n="1799,001,,,,10,34,4"></lb> |
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9728 ΑΒ, ΒΓ ῥητὸν περιέχουσαι τὸ ὑπ' αὐτῶν, ὥστε τὴν <lb n="1799,001,,,,10,34,5"></lb> |
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9729 ΑΒ τῆς ΒΓ μεῖζον δύνασθαι τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, <lb n="1799,001,,,,10,34,6"></lb> |
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9730 καὶ γεγράφθω ἐπὶ τῆς ΑΒ τὸ ΑΔΒ ἡμικύκλιον, καὶ <lb n="1799,001,,,,10,34,7"></lb> |
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9731 τετμήσθω ἡ ΒΓ δίχα κατὰ τὸ Ε, καὶ παραβεβλήσθω <lb n="1799,001,,,,10,34,8"></lb> |
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9732 παρὰ τὴν ΑΒ τῷ ἀπὸ τῆς ΒΕ ἴσον παραλληλόγραμμον <lb n="1799,001,,,,10,34,9"></lb> |
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9733 ἐλλεῖπον εἴδει τετραγώνῳ τὸ ὑπὸ τῶν ΑΖΒ· ἀσύμμετρος <lb n="1799,001,,,,10,34,10"></lb> |
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9734 ἄρα [ἐστὶν] ἡ ΑΖ τῇ ΖΒ μήκει.</s> <s>καὶ ἤχθω ἀπὸ τοῦ Ζ <lb n="1799,001,,,,10,34,11"></lb> |
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9735 τῇ ΑΒ πρὸς ὀρθὰς ἡ ΖΔ, καὶ ἐπεζεύχθωσαν αἱ ΑΔ, ΔΒ.</s> <s><pb></pb> <lb n="1799,001,,,,10,34,12"></lb> |
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9736 Ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΑΖ τῇ ΖΒ, ἀσύμμετρον <lb n="1799,001,,,,10,34,13"></lb> |
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9737 ἄρα ἐστὶ καὶ τὸ ὑπὸ τῶν ΒΑ, ΑΖ τῷ ὑπὸ τῶν ΑΒ, ΒΖ.</s> <lb n="1799,001,,,,10,34,14"></lb> |
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9738 <s>ἴσον δὲ τὸ μὲν ὑπὸ τῶν ΒΑ, ΑΖ τῷ ἀπὸ τῆς ΑΔ, τὸ δὲ <lb n="1799,001,,,,10,34,15"></lb> |
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9739 ὑπὸ τῶν ΑΒ, ΒΖ τῷ ἀπὸ τῆς ΔΒ· ἀσύμμετρον ἄρα ἐστὶ <lb n="1799,001,,,,10,34,16"></lb> |
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9740 καὶ τὸ ἀπὸ τῆς ΑΔ τῷ ἀπὸ τῆς ΔΒ.</s> <s>καὶ ἐπεὶ μέσον ἐστὶ <lb n="1799,001,,,,10,34,17"></lb> |
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9741 τὸ ἀπὸ τῆς ΑΒ, μέσον ἄρα καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,34,18"></lb> |
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9742 τῶν ΑΔ, ΔΒ.</s> <s>καὶ ἐπεὶ διπλῆ ἐστιν ἡ ΒΓ τῆς ΔΖ, <lb n="1799,001,,,,10,34,19"></lb> |
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9743 διπλάσιον ἄρα καὶ τὸ ὑπὸ τῶν ΑΒ, ΒΓ τοῦ ὑπὸ τῶν <lb n="1799,001,,,,10,34,20"></lb> |
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9744 ΑΒ, ΖΔ.</s> <s>ῥητὸν δὲ τὸ ὑπὸ τῶν ΑΒ, ΒΓ· ῥητὸν ἄρα καὶ <lb n="1799,001,,,,10,34,21"></lb> |
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9745 τὸ ὑπὸ τῶν ΑΒ, ΖΔ.</s> <s>τὸ δὲ ὑπὸ τῶν ΑΒ, ΖΔ ἴσον τῷ ὑπὸ <lb n="1799,001,,,,10,34,22"></lb> |
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9746 τῶν ΑΔ, ΔΒ· ὥστε καὶ τὸ ὑπὸ τῶν ΑΔ, ΔΒ ῥητόν <lb n="1799,001,,,,10,34,23"></lb> |
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9747 ἐστιν.</s> <lb n="1799,001,,,,10,34,24"></lb> |
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9748 <s>Εὕρηνται ἄρα δύο εὐθεῖαι δυνάμει ἀσύμμετροι αἱ ΑΔ, <lb n="1799,001,,,,10,34,25"></lb> |
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9749 ΔΒ ποιοῦσαι τὸ [μὲν] συγκείμενον ἐκ τῶν ἀπ' αὐτῶν <lb n="1799,001,,,,10,34,26"></lb> |
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9750 τετραγώνων μέσον, τὸ δ' ὑπ' αὐτῶν ῥητόν· ὅπερ ἔδει <lb n="1799,001,,,,10,34,27"></lb> |
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9751 δεῖξαι.</s> <lb n="1799,001,,,,10,34,28"></lb> |
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9752 <s>Εὑρεῖν δύο εὐθείας δυνάμει ἀσυμμέτρους ποιούσας τό <lb n="1799,001,,,,10,35,1"></lb> |
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9753 τε συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων μέσον καὶ <lb n="1799,001,,,,10,35,2"></lb> |
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9754 τὸ ὑπ' αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τῷ συγκειμένῳ <lb n="1799,001,,,,10,35,3"></lb> |
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9755 ἐκ τῶν ἀπ' αὐτῶν τετραγώνῳ.</s> <s><pb></pb> <lb n="1799,001,,,,10,35,4"></lb> |
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9756 Ἐκκείσθωσαν δύο μέσαι δυνάμει μόνον σύμμετροι αἱ <lb n="1799,001,,,,10,35,5"></lb> |
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9757 ΑΒ, ΒΓ μέσον περιέχουσαι, ὥστε τὴν ΑΒ τῆς ΒΓ μεῖζον <lb n="1799,001,,,,10,35,6"></lb> |
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9758 δύνασθαι τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, καὶ γεγράφθω ἐπὶ <lb n="1799,001,,,,10,35,7"></lb> |
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9759 τῆς ΑΒ ἡμικύκλιον τὸ ΑΔΒ, καὶ τὰ λοιπὰ γεγονέτω <lb n="1799,001,,,,10,35,8"></lb> |
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9760 τοῖς ἐπάνω ὁμοίως.</s> <lb n="1799,001,,,,10,35,9"></lb> |
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9761 <s>Καὶ ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΑΖ τῇ ΖΒ μήκει, <lb n="1799,001,,,,10,35,10"></lb> |
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9762 ἀσύμμετρός ἐστι καὶ ἡ ΑΔ τῇ ΔΒ δυνάμει.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,35,11"></lb> |
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9763 μέσον ἐστὶ τὸ ἀπὸ τῆς ΑΒ, <lb n="1799,001,,,,10,35,12"></lb> |
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9764 μέσον ἄρα καὶ τὸ συγκεί-<lb n="1799,001,,,,10,35,13"></lb> |
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9765 μενον ἐκ τῶν ἀπὸ τῶν ΑΔ, <lb n="1799,001,,,,10,35,14"></lb> |
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9766 ΔΒ.</s> <s>καὶ ἐπεὶ τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,35,15"></lb> |
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9767 ΑΖ, ΖΒ ἴσον ἐστὶ τῷ ἀφ' <lb n="1799,001,,,,10,35,16"></lb> |
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9768 ἑκατέρας τῶν ΒΕ, ΔΖ, ἴση ἄρα ἐστὶν ἡ ΒΕ τῇ ΔΖ· <lb n="1799,001,,,,10,35,17"></lb> |
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9769 διπλῆ ἄρα ἡ ΒΓ τῆς ΖΔ· ὥστε καὶ τὸ ὑπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,35,18"></lb> |
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9770 ΒΓ διπλάσιόν ἐστι τοῦ ὑπὸ τῶν ΑΒ, ΖΔ.</s> <s>μέσον δὲ τὸ <lb n="1799,001,,,,10,35,19"></lb> |
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9771 ὑπὸ τῶν ΑΒ, ΒΓ· μέσον ἄρα καὶ τὸ ὑπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,35,20"></lb> |
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9772 ΖΔ.</s> <s>καί ἐστιν ἴσον τῷ ὑπὸ τῶν ΑΔ, ΔΒ· μέσον ἄρα καὶ <lb n="1799,001,,,,10,35,21"></lb> |
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9773 τὸ ὑπὸ τῶν ΑΔ, ΔΒ.</s> <s>καὶ ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΑΒ τῇ <lb n="1799,001,,,,10,35,22"></lb> |
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9774 ΒΓ μήκει, σύμμετρος δὲ ἡ ΓΒ τῇ ΒΕ, ἀσύμμετρος ἄρα <lb n="1799,001,,,,10,35,23"></lb> |
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9775 καὶ ἡ ΑΒ τῇ ΒΕ μήκει· ὥστε καὶ τὸ ἀπὸ τῆς ΑΒ τῷ ὑπὸ <lb n="1799,001,,,,10,35,24"></lb> |
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9776 τῶν ΑΒ, ΒΕ ἀσύμμετρόν ἐστιν.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς <lb n="1799,001,,,,10,35,25"></lb> |
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9777 ΑΒ ἴσα ἐστὶ τὰ ἀπὸ τῶν ΑΔ, ΔΒ, τῷ δὲ ὑπὸ τῶν ΑΒ, <pb></pb> <lb n="1799,001,,,,10,35,26"></lb> |
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9778 ΒΕ ἴσον ἐστὶ τὸ ὑπὸ τῶν ΑΒ, ΖΔ, τουτέστι τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,35,27"></lb> |
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9779 ΑΔ, ΔΒ· ἀσύμμετρον ἄρα ἐστὶ τὸ συγκείμενον ἐκ τῶν <lb n="1799,001,,,,10,35,28"></lb> |
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9780 ἀπὸ τῶν ΑΔ, ΔΒ τῷ ὑπὸ τῶν ΑΔ, ΔΒ.</s> <lb n="1799,001,,,,10,35,29"></lb> |
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9781 <s>Εὕρηνται ἄρα δύο εὐθεῖαι αἱ ΑΔ, ΔΒ δυνάμει ἀσύμ-<lb n="1799,001,,,,10,35,30"></lb> |
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9782 μετροι ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπ' αὐτῶν <lb n="1799,001,,,,10,35,31"></lb> |
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9783 μέσον καὶ τὸ ὑπ' αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τῷ <lb n="1799,001,,,,10,35,32"></lb> |
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9784 συγκειμένῳ ἐκ τῶν ἀπ' αὐτῶν τετραγώνων· ὅπερ ἔδει <lb n="1799,001,,,,10,35,33"></lb> |
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9785 δεῖξαι.</s> <lb n="1799,001,,,,10,35,34"></lb> |
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9786 <s>Ἐὰν δύο ῥηταὶ δυνάμει μόνον σύμμετροι συντεθῶσιν, <lb n="1799,001,,,,10,36,1"></lb> |
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9787 ἡ ὅλη ἄλογός ἐστιν, καλείσθω δὲ ἐκ δύο ὀνομάτων.</s> <lb n="1799,001,,,,10,36,2"></lb> |
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9788 <s>Συγκείσθωσαν γὰρ δύο ῥηταὶ δυνάμει μόνον σύμμετροι <lb n="1799,001,,,,10,36,3"></lb> |
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9789 αἱ ΑΒ, ΒΓ· λέγω, ὅτι ὅλη ἡ ΑΓ ἄλογός ἐστιν. <lb n="1799,001,,,,10,36,4"></lb> |
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9790 Ἐπεὶ γὰρ ἀσύμμετρός ἐστιν ἡ ΑΒ τῇ ΒΓ μήκει· <lb n="1799,001,,,,10,36,5"></lb> |
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9791 δυνάμει γὰρ μόνον εἰσὶ σύμμετροι· ὡς δὲ ἡ ΑΒ πρὸς τὴν <lb n="1799,001,,,,10,36,6"></lb> |
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9792 ΒΓ, οὕτως τὸ ὑπὸ τῶν ΑΒΓ πρὸς τὸ ἀπὸ τῆς ΒΓ, <lb n="1799,001,,,,10,36,7"></lb> |
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9793 ἀσύμμετρον ἄρα ἐστὶ τὸ ὑπὸ τῶν ΑΒ, ΒΓ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,36,8"></lb> |
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9794 ΒΓ.</s> <s>ἀλλὰ τῷ μὲν ὑπὸ τῶν ΑΒ, ΒΓ σύμμετρόν ἐστι τὸ <pb></pb> <lb n="1799,001,,,,10,36,9"></lb> |
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9795 δὶς ὑπὸ τῶν ΑΒ, ΒΓ, τῷ δὲ ἀπὸ τῆς ΒΓ σύμμετρά ἐστι <lb n="1799,001,,,,10,36,10"></lb> |
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9796 τὰ ἀπὸ τῶν ΑΒ, ΒΓ· αἱ γὰρ ΑΒ, ΒΓ ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,36,11"></lb> |
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9797 μόνον σύμμετροι· ἀσύμμετρον ἄρα ἐστὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,36,12"></lb> |
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9798 ΑΒ, ΒΓ τοῖς ἀπὸ τῶν ΑΒ, ΒΓ.</s> <s>καὶ συνθέντι τὸ δὶς ὑπὸ <lb n="1799,001,,,,10,36,13"></lb> |
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9799 τῶν ΑΒ, ΒΓ μετὰ τῶν ἀπὸ τῶν ΑΒ, ΒΓ, τουτέστι τὸ <lb n="1799,001,,,,10,36,14"></lb> |
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9800 ἀπὸ τῆς ΑΓ, ἀσύμμετρόν ἐστι τῷ συγκειμένῳ ἐκ τῶν <lb n="1799,001,,,,10,36,15"></lb> |
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9801 ἀπὸ τῶν ΑΒ, ΒΓ.</s> <s>ῥητὸν δὲ τὸ συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,36,16"></lb> |
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9802 τῶν ΑΒ, ΒΓ· ἄλογον ἄρα [ἐστὶ] τὸ ἀπὸ τῆς ΑΓ· ὥστε <lb n="1799,001,,,,10,36,17"></lb> |
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9803 καὶ ἡ ΑΓ ἄλογός ἐστιν, καλείσθω δὲ ἐκ δύο ὀνομάτων· <lb n="1799,001,,,,10,36,18"></lb> |
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9804 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,36,19"></lb> |
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9805 <s>Ἐὰν δύο μέσαι δυνάμει μόνον σύμμετροι συντεθῶσι <lb n="1799,001,,,,10,37,1"></lb> |
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9806 ῥητὸν περιέχουσαι, ἡ ὅλη ἄλογός ἐστιν, καλείσθω δὲ ἐκ <lb n="1799,001,,,,10,37,2"></lb> |
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9807 δύο μέσων πρώτη.</s> <lb n="1799,001,,,,10,37,3"></lb> |
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9808 <s>Συγκείσθωσαν γὰρ δύο μέσαι δυνάμει μόνον σύμμετροι <lb n="1799,001,,,,10,37,4"></lb> |
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9809 αἱ ΑΒ, ΒΓ ῥητὸν περιέχουσαι· λέγω, ὅτι ὅλη ἡ ΑΓ <lb n="1799,001,,,,10,37,5"></lb> |
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9810 ἄλογός ἐστιν.</s> <lb n="1799,001,,,,10,37,6"></lb> |
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9811 <s>Ἐπεὶ γὰρ ἀσύμμετρόσ ἐστιν ἡ ΑΒ τῇ ΒΓ μήκει, καὶ <lb n="1799,001,,,,10,37,7"></lb> |
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9812 τὰ ἀπὸ τῶν ΑΒ, ΒΓ ἄρα ἀσύμμετρά ἐστι τῷ δὶς ὑπὸ τῶν <pb></pb> <lb n="1799,001,,,,10,37,8"></lb> |
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9813 ΑΒ, ΒΓ· καὶ συνθέντι τὰ ἀπὸ τῶν ΑΒ, ΒΓ μετὰ τοῦ <lb n="1799,001,,,,10,37,9"></lb> |
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9814 δὶς ὑπὸ τῶν ΑΒ, ΒΓ, ὅπερ ἐστὶ τὸ ἀπὸ τῆς ΑΓ, ἀσύμ-<lb n="1799,001,,,,10,37,10"></lb> |
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9815 μετρόν ἐστι τῷ ὑπὸ τῶν ΑΒ, ΒΓ.</s> <s>ῥητὸν δὲ τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,37,11"></lb> |
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9816 ΑΒ, ΒΓ· ὑπόκεινται γὰρ αἱ ΑΒ, ΒΓ ῥητὸν περιέχουσαι· <lb n="1799,001,,,,10,37,12"></lb> |
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9817 ἄλογον ἄρα τὸ ἀπὸ τῆς ΑΓ· ἄλογος ἄρα ἡ ΑΓ, καλείσθω <lb n="1799,001,,,,10,37,13"></lb> |
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9818 δὲ ἐκ δύο μέσων πρώτη· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,37,14"></lb> |
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9819 <s>Ἐὰν δύο μέσαι δυνάμει μόνον σύμμετροι συντεθῶσι <lb n="1799,001,,,,10,38,1"></lb> |
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9820 μέσον περιέχουσαι, ἡ ὅλη ἄλογός ἐστιν, καλείσθω δὲ ἐκ <lb n="1799,001,,,,10,38,2"></lb> |
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9821 δύο μέσων δευτέρα.</s> <lb n="1799,001,,,,10,38,3"></lb> |
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9822 <s>Συγκείσθωσαν γὰρ δύο μέσαι δυνάμει μόνον σύμμετροι <lb n="1799,001,,,,10,38,4"></lb> |
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9823 αἱ ΑΒ, ΒΓ μέσον περιέχουσαι· λέγω, ὅτι ἄλογός ἐστιν <lb n="1799,001,,,,10,38,5"></lb> |
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9824 ἡ ΑΓ.</s> <lb n="1799,001,,,,10,38,6"></lb> |
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9825 <s>Ἐκκείσθω γὰρ ῥητὴ ἡ ΔΕ, καὶ τῷ ἀπὸ τῆς ΑΓ ἴσον <lb n="1799,001,,,,10,38,7"></lb> |
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9826 παρὰ τὴν ΔΕ παραβεβλήσθω τὸ ΔΖ πλάτος ποιοῦν <lb n="1799,001,,,,10,38,8"></lb> |
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9827 τὴν ΔΗ.</s> <s>καὶ ἐπεὶ τὸ ἀπὸ τῆς ΑΓ ἴσον ἐστὶ τοῖς τε ἀπὸ <lb n="1799,001,,,,10,38,9"></lb> |
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9828 τῶν ΑΒ, ΒΓ καὶ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ, παραβεβλήσθω <pb></pb> <lb n="1799,001,,,,10,38,10"></lb> |
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9829 δὴ τοῖς ἀπὸ τῶν ΑΒ, ΒΓ παρὰ τὴν ΔΕ ἴσον τὸ ΕΘ· <lb n="1799,001,,,,10,38,11"></lb> |
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9830 λοιπὸν ἄρα τὸ ΘΖ ἴσον ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ.</s> <s>καὶ <lb n="1799,001,,,,10,38,12"></lb> |
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9831 ἐπεὶ μέση ἐστὶν ἑκατέρα τῶν ΑΒ, ΒΓ, μέσα ἄρα ἐστὶ καὶ <lb n="1799,001,,,,10,38,13"></lb> |
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9832 τὰ ἀπὸ τῶν ΑΒ, ΒΓ.</s> <s>μέσον δὲ ὑπόκειται καὶ τὸ δὶς ὑπὸ <lb n="1799,001,,,,10,38,14"></lb> |
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9833 τῶν ΑΒ, ΒΓ.</s> <s>καί ἐστι τοῖς μὲν ἀπὸ τῶν ΑΒ, ΒΓ ἴσον <lb n="1799,001,,,,10,38,15"></lb> |
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9834 τὸ ΕΘ, τῷ δὲ δὶς ὑπὸ τῶν ΑΒ, ΒΓ ἴσον τὸ ΖΘ· μέσον ἄρα <lb n="1799,001,,,,10,38,16"></lb> |
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9835 ἑκάτερον τῶν ΕΘ, ΘΖ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΔΕ παρά-<lb n="1799,001,,,,10,38,17"></lb> |
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9836 κειται· ῥητὴ ἄρα ἐστὶν ἑκατέρα τῶν ΔΘ, ΘΗ καὶ ἀσύμ-<lb n="1799,001,,,,10,38,18"></lb> |
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9837 μετρος τῇ ΔΕ μήκει.</s> <s>ἐπεὶ οὖν ἀσύμμετρός ἐστιν ἡ ΑΒ τῇ <lb n="1799,001,,,,10,38,19"></lb> |
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9838 ΒΓ μήκει, καί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ <lb n="1799,001,,,,10,38,20"></lb> |
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9839 ἀπὸ τῆς ΑΒ πρὸς τὸ ὑπὸ τῶν ΑΒ, ΒΓ, ἀσύμμετρον ἄρα <lb n="1799,001,,,,10,38,21"></lb> |
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9840 ἐστὶ τὸ ἀπὸ τῆς ΑΒ τῷ ὑπὸ τῶν ΑΒ, ΒΓ.</s> <s>ἀλλὰ τῷ μὲν <lb n="1799,001,,,,10,38,22"></lb> |
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9841 ἀπὸ τῆς ΑΒ σύμμετρόν ἐστι τὸ συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,38,23"></lb> |
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9842 τῶν ΑΒ, ΒΓ τετραγώνων, τῷ δὲ ὑπὸ τῶν ΑΒ, ΒΓ σύμ-<lb n="1799,001,,,,10,38,24"></lb> |
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9843 μετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ.</s> <s>ἀσύμμετρον ἄρα <pb></pb> <lb n="1799,001,,,,10,38,25"></lb> |
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9844 ἐστὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΒ, ΒΓ τῷ δὶς <lb n="1799,001,,,,10,38,26"></lb> |
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9845 ὑπὸ τῶν ΑΒ, ΒΓ.</s> <s>ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΑΒ, ΒΓ ἴσον <lb n="1799,001,,,,10,38,27"></lb> |
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9846 ἐστὶ τὸ ΕΘ, τῷ δὲ δὶς ὑπὸ τῶν ΑΒ, ΒΓ ἴσον ἐστὶ τὸ ΘΖ.</s> <lb n="1799,001,,,,10,38,28"></lb> |
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9847 <s>ἀσύμμετρον ἄρα ἐστὶ τὸ ΕΘ τῷ ΘΖ· ὥστε καὶ ἡ ΔΘ <lb n="1799,001,,,,10,38,29"></lb> |
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9848 τῇ ΘΗ ἐστιν ἀσύμμετρος μήκει.</s> <s>αἱ ΔΘ, ΘΗ ἄρα ῥηταί <lb n="1799,001,,,,10,38,30"></lb> |
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9849 εἰσι δυνάμει μόνον σύμμετροι.</s> <s>ὥστε ἡ ΔΗ ἄλογός ἐστιν.</s> <lb n="1799,001,,,,10,38,31"></lb> |
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9850 <s>ῥητὴ δὲ ἡ ΔΕ· τὸ δὲ ὑπὸ ἀλόγου καὶ ῥητῆς περιεχόμενον <lb n="1799,001,,,,10,38,32"></lb> |
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9851 ὀρθογώνιον ἄλογόν ἐστιν· ἄλογον ἄρα ἐστὶ τὸ ΔΖ χωρίον, <lb n="1799,001,,,,10,38,33"></lb> |
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9852 καὶ ἡ δυναμένη [αὐτὸ] ἄλογός ἐστιν.</s> <s>δύναται δὲ τὸ ΔΖ <lb n="1799,001,,,,10,38,34"></lb> |
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9853 ἡ ΑΓ· ἄλογος ἄρα ἐστὶν ἡ ΑΓ, καλείσθω δὲ ἐκ δύο μέσων <lb n="1799,001,,,,10,38,35"></lb> |
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9854 δευτέρα.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,38,36"></lb> |
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9855 <s>Ἐὰν δύο εὐθεῖαι δυνάμει ἀσύμμετροι συντεθῶσι ποιοῦσαι <lb n="1799,001,,,,10,39,1"></lb> |
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9856 τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων <lb n="1799,001,,,,10,39,2"></lb> |
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9857 ῥητόν, τὸ δ' ὑπ' αὐτῶν μέσον, ἡ ὅλη εὐθεῖα ἄλογός ἐστιν, <lb n="1799,001,,,,10,39,3"></lb> |
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9858 καλείσθω δὲ μείζων.</s> <lb n="1799,001,,,,10,39,4"></lb> |
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9859 <s>Συγκείσθωσαν γὰρ δύο εὐθεῖαι δυνάμει ἀσύμμετροι <lb n="1799,001,,,,10,39,5"></lb> |
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9860 αἱ ΑΒ, ΒΓ ποιοῦσαι τὰ προκείμενα· λέγω, ὅτι ἄλογός <lb n="1799,001,,,,10,39,6"></lb> |
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9861 ἐστιν ἡ ΑΓ.</s> <lb n="1799,001,,,,10,39,7"></lb> |
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9862 <s>Ἐπεὶ γὰρ τὸ ὑπὸ τῶν ΑΒ, ΒΓ μέσον ἐστίν, καὶ τὸ δὶς <lb n="1799,001,,,,10,39,8"></lb> |
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9863 [ἄρα] ὑπὸ τῶν ΑΒ, ΒΓ μέσον ἐστίν.</s> <s>τὸ δὲ συγκείμενον ἐκ <pb></pb> <lb n="1799,001,,,,10,39,9"></lb> |
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9864 τῶν ἀπὸ τῶν ΑΒ, ΒΓ ῥητόν· ἀσύμμετρον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,10,39,10"></lb> |
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9865 δὶς ὑπὸ τῶν ΑΒ, ΒΓ τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν <lb n="1799,001,,,,10,39,11"></lb> |
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9866 ΑΒ, ΒΓ· ὥστε καὶ τὰ ἀπὸ τῶν ΑΒ, ΒΓ μετὰ τοῦ δὶς <lb n="1799,001,,,,10,39,12"></lb> |
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9867 ὑπὸ τῶν ΑΒ, ΒΓ, ὅπερ ἐστὶ τὸ ἀπὸ τῆς ΑΓ, ἀσύμ-<lb n="1799,001,,,,10,39,13"></lb> |
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9868 μετρόν ἐστι τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,10,39,14"></lb> |
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9869 [ῥητὸν δὲ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΒ, ΒΓ]· <lb n="1799,001,,,,10,39,15"></lb> |
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9870 ἄλογον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΑΓ.</s> <s>ὥστε καὶ ἡ ΑΓ ἄλογός <lb n="1799,001,,,,10,39,16"></lb> |
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9871 ἐστιν, καλείσθω δὲ μείζων.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,39,17"></lb> |
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9872 <s>Ἐὰν δύο εὐθεῖαι δυνάμει ἀσύμμετροι συντεθῶσι ποιοῦσαι <lb n="1799,001,,,,10,40,1"></lb> |
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9873 τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων μέσον, <lb n="1799,001,,,,10,40,2"></lb> |
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9874 τὸ δ' ὑπ' αὐτῶν ῥητόν, ἡ ὅλη εὐθεῖα ἄλογός ἐστιν, <lb n="1799,001,,,,10,40,3"></lb> |
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9875 καλείσθω δὲ ῥητὸν καὶ μέσον δυναμένη.</s> <lb n="1799,001,,,,10,40,4"></lb> |
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9876 <s>Συγκείσθωσαν γὰρ δύο εὐθεῖαι δυνάμει ἀσύμ- <lb n="1799,001,,,,10,40,5"></lb> |
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9877 μετροι αἱ ΑΒ, ΒΓ ποιοῦσαι τὰ προκείμενα· λέγω, <lb n="1799,001,,,,10,40,6"></lb> |
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9878 ὅτι ἄλογός ἐστιν ἡ ΑΓ.</s> <lb n="1799,001,,,,10,40,7"></lb> |
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9879 <s>Ἐπεὶ γὰρ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,40,8"></lb> |
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9880 ΒΓ μέσον ἐστίν, τὸ δὲ δὶς ὑπὸ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,10,40,9"></lb> |
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9881 ῥητόν, ἀσύμμετρον ἄρα ἐστὶ τὸ συγκείμενον ἐκ τῶν <pb></pb> <lb n="1799,001,,,,10,40,10"></lb> |
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9882 ἀπὸ τῶν ΑΒ, ΒΓ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ· ὥστε καὶ <lb n="1799,001,,,,10,40,11"></lb> |
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9883 τὸ ἀπὸ τῆς ΑΓ ἀσύμμετρόν ἐστι τῷ δὶς ὑπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,40,12"></lb> |
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9884 ΒΓ.</s> <s>ῥητὸν δὲ τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ· ἄλογον ἄρα <lb n="1799,001,,,,10,40,13"></lb> |
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9885 τὸ ἀπὸ τῆς ΑΓ.</s> <s>ἄλογος ἄρα ἡ ΑΓ, καλείσθω δὲ ῥητὸν <lb n="1799,001,,,,10,40,14"></lb> |
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9886 καὶ μέσον δυναμένη.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,40,15"></lb> |
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9887 <s>Ἐὰν δύο εὐθεῖαι δυνάμει ἀσύμμετροι συντεθῶσι <lb n="1799,001,,,,10,41,1"></lb> |
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9888 ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων <lb n="1799,001,,,,10,41,2"></lb> |
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9889 μέσον καὶ τὸ ὑπ' αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τῷ <lb n="1799,001,,,,10,41,3"></lb> |
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9890 συγκειμένῳ ἐκ τῶν ἀπ' αὐτῶν τετραγώνων, ἡ ὅλη εὐθεῖα <lb n="1799,001,,,,10,41,4"></lb> |
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9891 ἄλογός ἐστιν, καλείσθω δὲ δύο μέσα δυναμένη.</s> <lb n="1799,001,,,,10,41,5"></lb> |
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9892 <s>Συγκείσθωσαν γὰρ δύο εὐθεῖαι δυνάμει ἀσύμμετροι αἱ <lb n="1799,001,,,,10,41,6"></lb> |
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9893 ΑΒ, ΒΓ ποιοῦσαι τὰ προκείμενα· λέγω, <lb n="1799,001,,,,10,41,7"></lb> |
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9894 ὅτι ἡ ΑΓ ἄλογός ἐστιν.</s> <lb n="1799,001,,,,10,41,8"></lb> |
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9895 <s>Ἐκκείσθω ῥητὴ ἡ ΔΕ, καὶ παραβεβλή-<lb n="1799,001,,,,10,41,9"></lb> |
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9896 σθω παρὰ τὴν ΔΕ τοῖς μὲν ἀπὸ τῶν <lb n="1799,001,,,,10,41,10"></lb> |
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9897 ΑΒ, ΒΓ ἴσον τὸ ΔΖ, τῷ δὲ δὶς ὑπὸ <lb n="1799,001,,,,10,41,11"></lb> |
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9898 τῶν ΑΒ, ΒΓ ἴσον τὸ ΗΘ· ὅλον ἄρα τὸ <lb n="1799,001,,,,10,41,12"></lb> |
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9899 ΔΘ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΓ τετραγώνῳ.</s> <lb n="1799,001,,,,10,41,13"></lb> |
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9900 <s>καὶ ἐπεὶ μέσον ἐστὶ τὸ συγκείμενον ἐκ <lb n="1799,001,,,,10,41,14"></lb> |
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9901 τῶν ἀπὸ τῶν ΑΒ, ΒΓ, καί ἐστιν ἴσον <pb></pb> <lb n="1799,001,,,,10,41,15"></lb> |
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9902 τῷ ΔΖ, μέσον ἄρα ἐστὶ καὶ τὸ ΔΖ.</s> <s>καὶ παρὰ ῥητὴν τὴν <lb n="1799,001,,,,10,41,16"></lb> |
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9903 ΔΕ παράκειται· ῥητὴ ἄρα ἐστὶν ἡ ΔΗ καὶ ἀσύμμετρος <lb n="1799,001,,,,10,41,17"></lb> |
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9904 τῇ ΔΕ μήκει.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΗΚ ῥητή ἐστι καὶ <lb n="1799,001,,,,10,41,18"></lb> |
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9905 ἀσύμμετρος τῇ ΗΖ, τουτέστι τῇ ΔΕ, μήκει.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,41,19"></lb> |
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9906 ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ τῷ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,41,20"></lb> |
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9907 ΑΒ, ΒΓ, ἀσύμμετρόν ἐστι τὸ ΔΖ τῷ ΗΘ· ὥστε καὶ <lb n="1799,001,,,,10,41,21"></lb> |
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9908 ἡ ΔΗ τῇ ΗΚ ἀσύμμετρός ἐστιν.</s> <s>καί εἰσι ῥηταί· αἱ ΔΗ, <lb n="1799,001,,,,10,41,22"></lb> |
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9909 ΗΚ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἄλογος ἄρα <lb n="1799,001,,,,10,41,23"></lb> |
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9910 ἐστὶν ἡ ΔΚ ἡ καλουμένη ἐκ δύο ὀνομάτων.</s> <s>ῥητὴ δὲ ἡ ΔΕ· <lb n="1799,001,,,,10,41,24"></lb> |
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9911 ἄλογον ἄρα ἐστὶ τὸ ΔΘ καὶ ἡ δυναμένη αὐτὸ ἄλογός ἐστιν.</s> <lb n="1799,001,,,,10,41,25"></lb> |
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9912 <s>δύναται δὲ τὸ ΘΔ ἡ ΑΓ· ἄλογος ἄρα ἐστὶν ἡ ΑΓ, καλείσθω <lb n="1799,001,,,,10,41,26"></lb> |
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9913 δὲ δύο μέσα δυναμένη.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,41,27"></lb> |
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9914 <s>Λῆμμα <lb n="1799,001,,,,10,41,28"></lb> |
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9915 Ὅτι δὲ αἱ εἰρημέναι ἄλογοι μοναχῶς διαιροῦνται εἰς <lb n="1799,001,,,,10,41,29"></lb> |
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9916 τὰς εὐθείας, ἐξ ὧν σύγκεινται ποιουσῶν τὰ προκείμενα <lb n="1799,001,,,,10,41,30"></lb> |
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9917 εἴδη, δείξομεν ἤδη προεκθέμενοι λημμάτιον τοιοῦτον· <lb n="1799,001,,,,10,41,31"></lb> |
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9918 Ἐκκείσθω εὐθεῖα ἡ ΑΒ καὶ τετμήσθω ἡ ὅλη εἰς ἄνισα <lb n="1799,001,,,,10,41,32"></lb> |
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9919 καθ' ἑκάτερον τῶν Γ, Δ, ὑποκείσθω δὲ μείζων ἡ ΑΓ τῆς <lb n="1799,001,,,,10,41,33"></lb> |
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9920 ΔΒ· λέγω, ὅτι τὰ ἀπὸ τῶν ΑΓ, ΓΒ μείζονά ἐστι τῶν ἀπὸ <lb n="1799,001,,,,10,41,34"></lb> |
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9921 τῶν ΑΔ, ΔΒ. <pb></pb> <lb n="1799,001,,,,10,41,35"></lb> |
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9922 Τετμήσθω γὰρ ἡ ΑΒ δίχα κατὰ τὸ Ε.</s> <s>καὶ ἐπεὶ μείζων <lb n="1799,001,,,,10,41,36"></lb> |
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9923 ἐστὶν ἡ ΑΓ τῆς ΔΒ, κοινὴ ἀφῃρήσθω ἡ ΔΓ· λοιπὴ ἄρα <lb n="1799,001,,,,10,41,37"></lb> |
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9924 ἡ ΑΔ λοιπῆς τῆς ΓΒ μείζων ἐστίν.</s> <s>ἴση δὲ ἡ ΑΕ τῇ ΕΒ· <lb n="1799,001,,,,10,41,38"></lb> |
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9925 ἐλάττων ἄρα ἡ ΔΕ τῆς ΕΓ· τὰ Γ, Δ ἄρα σημεῖα οὐκ <lb n="1799,001,,,,10,41,39"></lb> |
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9926 ἴσον ἀπέχουσι τῆς διχοτομίας.</s> <s>καὶ ἐπεὶ τὸ ὑπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,41,40"></lb> |
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9927 ΓΒ μετὰ τοῦ ἀπὸ τῆς ΕΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΕΒ, <lb n="1799,001,,,,10,41,41"></lb> |
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9928 ἀλλὰ μὴν καὶ τὸ ὑπὸ τῶν ΑΔ, ΔΒ μετὰ τοῦ ἀπὸ ΔΕ ἴσον <lb n="1799,001,,,,10,41,42"></lb> |
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9929 ἐστὶ τῷ ἀπὸ τῆς ΕΒ, τὸ ἄρα ὑπὸ τῶν ΑΓ, ΓΒ μετὰ τοῦ <lb n="1799,001,,,,10,41,43"></lb> |
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9930 ἀπὸ τῆς ΕΓ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΑΔ, ΔΒ μετὰ τοῦ ἀπὸ <lb n="1799,001,,,,10,41,44"></lb> |
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9931 τῆς ΔΕ· ὧν τὸ ἀπὸ τῆς ΔΕ ἔλασσόν ἐστι τοῦ ἀπὸ τῆς <lb n="1799,001,,,,10,41,45"></lb> |
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9932 ΕΓ· καὶ λοιπὸν ἄρα τὸ ὑπὸ τῶν ΑΓ, ΓΒ ἔλασσόν ἐστι τοῦ <lb n="1799,001,,,,10,41,46"></lb> |
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9933 ὑπὸ τῶν ΑΔ, ΔΒ.</s> <s>ὥστε καὶ τὸ δὶς ὑπὸ τῶν ΑΓ, ΓΒ <lb n="1799,001,,,,10,41,47"></lb> |
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9934 ἔλασσόν ἐστι τοῦ δὶς ὑπὸ ΑΔ, ΔΒ.</s> <s>καὶ λοιπὸν ἄρα τὸ <lb n="1799,001,,,,10,41,48"></lb> |
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9935 συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, ΓΒ μεῖζόν ἐστι τοῦ <lb n="1799,001,,,,10,41,49"></lb> |
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9936 συγκειμένου ἐκ τῶν ἀπὸ τῶν ΑΔ, ΔΒ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,41,50"></lb> |
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9937 <s>Ἡ ἐκ δύο ὀνομάτων κατὰ ἓν μόνον σημεῖον διαιρεῖται <lb n="1799,001,,,,10,42,1"></lb> |
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9938 εἰς τὰ ὀνόματα.</s> <lb n="1799,001,,,,10,42,2"></lb> |
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9939 <s>Ἔστω ἐκ δύο ὀνομάτων ἡ ΑΒ διῃρημένη εἰς τὰ ὀνόματα <lb n="1799,001,,,,10,42,3"></lb> |
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9940 κατὰ τὸ Γ· αἱ ΑΓ, ΓΒ ἄρα ῥηταί εἰσι δυνάμει μόνον <lb n="1799,001,,,,10,42,4"></lb> |
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9941 σύμμετροι.</s> <s>λέγω, ὅτι ἡ ΑΒ κατ' ἄλλο σημεῖον οὐ διαιρεῖται <lb n="1799,001,,,,10,42,5"></lb> |
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9942 εἰς δύο ῥητὰς δυνάμει μόνον συμμέτρους.</s> <s><pb></pb> <lb n="1799,001,,,,10,42,6"></lb> |
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9943 Εἰ γὰρ δυνατόν, διῃρήσθω καὶ κατὰ τὸ Δ, <lb n="1799,001,,,,10,42,7"></lb> |
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9944 ὥστε καὶ τὰς ΑΔ, ΔΒ ῥητὰς εἶναι δυνάμει <lb n="1799,001,,,,10,42,8"></lb> |
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9945 μόνον συμμέτρους.</s> <s>φανερὸν δή, ὅτι ἡ ΑΓ τῇ ΔΒ <lb n="1799,001,,,,10,42,9"></lb> |
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9946 οὐκ ἔστιν ἡ αὐτή.</s> <s>εἰ γὰρ δυνατόν, ἔστω.</s> <s>ἔσται <lb n="1799,001,,,,10,42,10"></lb> |
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9947 δὴ καὶ ἡ ΑΔ τῇ ΓΒ ἡ αὐτή· καὶ ἔσται ὡς ἡ <lb n="1799,001,,,,10,42,11"></lb> |
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9948 ΑΓ πρὸς τὴν ΓΒ, οὕτως ἡ ΒΔ πρὸς τὴν ΔΑ, <lb n="1799,001,,,,10,42,12"></lb> |
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9949 καὶ ἔσται ἡ ΑΒ κατὰ τὸ αὐτὸ τῇ κατὰ τὸ Γ <lb n="1799,001,,,,10,42,13"></lb> |
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9950 διαιρέσει διαιρεθεῖσα καὶ κατὰ τὸ Δ· ὅπερ οὐχ <lb n="1799,001,,,,10,42,14"></lb> |
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9951 ὑπόκειται.</s> <s>οὐκ ἄρα ἡ ΑΓ τῇ ΔΒ ἐστιν ἡ αὐτή.</s> <lb n="1799,001,,,,10,42,15"></lb> |
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9952 <s>διὰ δὴ τοῦτο καὶ τὰ Γ, Δ σημεῖα οὐκ ἴσον ἀπέχουσι τῆς <lb n="1799,001,,,,10,42,16"></lb> |
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9953 διχοτομίας.</s> <s>ᾧ ἄρα διαφέρει τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν <lb n="1799,001,,,,10,42,17"></lb> |
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9954 ἀπὸ τῶν ΑΔ, ΔΒ, τούτῳ διαφέρει καὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,42,18"></lb> |
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9955 ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ διὰ τὸ καὶ τὰ ἀπὸ <lb n="1799,001,,,,10,42,19"></lb> |
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9956 τῶν ΑΓ, ΓΒ μετὰ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ καὶ τὰ <lb n="1799,001,,,,10,42,20"></lb> |
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9957 ἀπὸ τῶν ΑΔ, ΔΒ μετὰ τοῦ δὶς ὑπὸ τῶν ΑΔ, ΔΒ ἴσα <lb n="1799,001,,,,10,42,21"></lb> |
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9958 εἶναι τῷ ἀπὸ τῆς ΑΒ.</s> <s>ἀλλὰ τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν <lb n="1799,001,,,,10,42,22"></lb> |
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9959 ἀπὸ τῶν ΑΔ, ΔΒ διαφέρει ῥητῷ· ῥητὰ γὰρ ἀμφότερα· <lb n="1799,001,,,,10,42,23"></lb> |
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9960 καὶ τὸ δὶς ἄρα ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,42,24"></lb> |
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9961 ΓΒ διαφέρει ῥητῷ μέσα ὄντα· ὅπερ ἄτοπον· μέσον γὰρ <lb n="1799,001,,,,10,42,25"></lb> |
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9962 μέσου οὐχ ὑπερέχει ῥητῷ.</s> <lb n="1799,001,,,,10,42,26"></lb> |
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9963 <s>Οὐκ ἄρα ἡ ἐκ δύο ὀνομάτων κατ' ἄλλο καὶ ἄλλο σημεῖον <lb n="1799,001,,,,10,42,27"></lb> |
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9964 διαιρεῖται· καθ' ἓν ἄρα μόνον· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,42,28"></lb> |
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9965 Ἡ ἐκ δύο μέσων πρώτη καθ' ἓν μόνον σημεῖον διαιρεῖται.</s> <lb n="1799,001,,,,10,43,1"></lb> |
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9966 <s>Ἔστω ἐκ δύο μέσων πρώτη ἡ ΑΒ διῃρημένη κατὰ <lb n="1799,001,,,,10,43,2"></lb> |
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9967 τὸ Γ, ὥστε τὰς ΑΓ, ΓΒ μέσας εἶναι δυνάμει μόνον <lb n="1799,001,,,,10,43,3"></lb> |
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9968 συμμέτρους ῥητὸν περιεχούσας· λέγω, ὅτι ἡ ΑΒ κατ' <lb n="1799,001,,,,10,43,4"></lb> |
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9969 ἄλλο σημεῖον οὐ διαιρεῖται. <lb n="1799,001,,,,10,43,5"></lb> |
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9970 Εἰ γὰρ δυνατόν, διῃρήσθω καὶ κατὰ τὸ Δ, ὥστε καὶ <lb n="1799,001,,,,10,43,6"></lb> |
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9971 τὰς ΑΔ, ΔΒ μέσας εἶναι δυνάμει μόνον συμμέτρους <lb n="1799,001,,,,10,43,7"></lb> |
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9972 ῥητὸν περιεχούσας.</s> <s>ἐπεὶ οὖν, ᾧ διαφέρει τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,43,8"></lb> |
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9973 ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ, τούτῳ διαφέρει τὰ ἀπὸ <lb n="1799,001,,,,10,43,9"></lb> |
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9974 τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ, ῥητῷ δὲ διαφέρει <lb n="1799,001,,,,10,43,10"></lb> |
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9975 τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ· ῥητὰ <lb n="1799,001,,,,10,43,11"></lb> |
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9976 γὰρ ἀμφότερα· ῥητῷ ἄρα διαφέρει καὶ τὰ ἀπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,43,12"></lb> |
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9977 ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ μέσα ὄντα· ὅπερ ἄτοπον.</s> <lb n="1799,001,,,,10,43,13"></lb> |
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9978 <s>Οὐκ ἄρα ἡ ἐκ δύο μέσων πρώτη κατ' ἄλλο καὶ ἄλλο <lb n="1799,001,,,,10,43,14"></lb> |
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9979 σημεῖον διαιρεῖται εἰς τὰ ὀνόματα· καθ' ἓν ἄρα μόνον· <lb n="1799,001,,,,10,43,15"></lb> |
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9980 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,43,16"></lb> |
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9981 <s>Ἡ ἐκ μέσων δευτέρα καθ' ἓν μόνον σημεῖον διαιρεῖται.</s> <lb n="1799,001,,,,10,44,1"></lb> |
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9982 <s>Ἔστω ἐκ δύο μέσων δευτέρα ἡ ΑΒ διῃρημένη κατὰ <lb n="1799,001,,,,10,44,2"></lb> |
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9983 τὸ Γ, ὥστε τὰς ΑΓ, ΓΒ μέσας εἶναι δυνάμει μόνον <pb></pb> <lb n="1799,001,,,,10,44,3"></lb> |
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9984 συμμέτρους μέσον περιεχούσας· φανερὸν δή, ὅτι τὸ Γ <lb n="1799,001,,,,10,44,4"></lb> |
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9985 οὐκ ἔστι κατὰ τῆς διχοτομίας, ὅτι οὐκ εἰσὶ μήκει σύμ-<lb n="1799,001,,,,10,44,5"></lb> |
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9986 μετροι.</s> <s>λέγω, ὅτι ἡ ΑΒ κατ' ἄλλο σημεῖον οὐ διαιρεῖται.</s> <lb n="1799,001,,,,10,44,6"></lb> |
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9987 <s>Εἰ γὰρ δυνατόν, διῃρήσθω καὶ κατὰ τὸ Δ, ὥστε τὴν ΑΓ <lb n="1799,001,,,,10,44,7"></lb> |
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9988 τῇ ΔΒ μὴ εἶναι τὴν αὐτήν, ἀλλὰ μείζονα καθ' ὑπόθεσιν <lb n="1799,001,,,,10,44,8"></lb> |
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9989 τὴν ΑΓ· δῆλον δή, ὅτι καὶ τὰ ἀπὸ τῶν ΑΔ, ΔΒ, ὡς <lb n="1799,001,,,,10,44,9"></lb> |
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9990 ἐπάνω ἐδείξαμεν, ἐλάσσονα τῶν ἀπὸ τῶν ΑΓ, ΓΒ· καὶ τὰς <lb n="1799,001,,,,10,44,10"></lb> |
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9991 ΑΔ, ΔΒ μέσας εἶναι δυνάμει μόνον συμμέτρους μέσον <lb n="1799,001,,,,10,44,11"></lb> |
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9992 περιεχούσας.</s> <s>καὶ ἐκκείσθω ῥητὴ ἡ ΕΖ, καὶ τῷ μὲν ἀπὸ <lb n="1799,001,,,,10,44,12"></lb> |
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9993 τῆς ΑΒ ἴσον παρὰ τὴν ΕΖ παραλληλόγραμμον ὀρθογώνιον <lb n="1799,001,,,,10,44,13"></lb> |
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9994 παραβεβλήσθω τὸ ΕΚ, τοῖς δὲ ἀπὸ τῶν ΑΓ, ΓΒ ἴσον <lb n="1799,001,,,,10,44,14"></lb> |
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9995 ἀφῃρήσθω τὸ ΕΗ· λοιπὸν ἄρα τὸ ΘΚ ἴσον ἐστὶ τῷ δὶς <lb n="1799,001,,,,10,44,15"></lb> |
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9996 ὑπὸ τῶν ΑΓ, ΓΒ.</s> <s>πάλιν δὴ τοῖς ἀπὸ τῶν ΑΔ, ΔΒ, ἅπερ <lb n="1799,001,,,,10,44,16"></lb> |
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9997 ἐλάσσονα ἐδείχθη τῶν ἀπὸ τῶν ΑΓ, ΓΒ, ἴσον ἀφῃρή-<lb n="1799,001,,,,10,44,17"></lb> |
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9998 σθω τὸ ΕΛ· καὶ λοιπὸν ἄρα τὸ ΜΚ ἴσον τῷ δὶς ὑπὸ <lb n="1799,001,,,,10,44,18"></lb> |
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9999 τῶν ΑΔ, ΔΒ.</s> <s>καὶ ἐπεὶ μέσα ἐστὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ, <lb n="1799,001,,,,10,44,19"></lb> |
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10000 μέσον ἄρα [καὶ] τὸ ΕΗ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΕΖ παρά- <pb></pb><lb n="1799,001,,,,10,44,20"></lb> |
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10001 κειται· ῥητὴ ἄρα ἐστὶν ἡ ΕΘ καὶ ἀσύμμετρος τῇ ΕΖ <lb n="1799,001,,,,10,44,21"></lb> |
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10002 μήκει.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΘΝ ῥητή ἐστι καὶ ἀσύμμετρος <lb n="1799,001,,,,10,44,22"></lb> |
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10003 τῇ ΕΖ μήκει.</s> <s>καὶ ἐπεὶ αἱ ΑΓ, ΓΒ μέσαι εἰσὶ δυνάμει <lb n="1799,001,,,,10,44,23"></lb> |
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10004 μόνον σύμμετροι, ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΓ τῇ ΓΒ <lb n="1799,001,,,,10,44,24"></lb> |
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10005 μήκει.</s> <s>ὡς δὲ ἡ ΑΓ πρὸς τὴν ΓΒ, οὕτως τὸ ἀπὸ τῆς ΑΓ <lb n="1799,001,,,,10,44,25"></lb> |
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10006 πρὸς τὸ ὑπὸ τῶν ΑΓ, ΓΒ· ἀσύμμετρον ἄρα ἐστὶ τὸ ἀπὸ <lb n="1799,001,,,,10,44,26"></lb> |
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10007 τῆς ΑΓ τῷ ὑπὸ τῶν ΑΓ, ΓΒ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΓ <lb n="1799,001,,,,10,44,27"></lb> |
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10008 σύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΓ, ΓΒ· δυνάμει γάρ εἰσι <lb n="1799,001,,,,10,44,28"></lb> |
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10009 σύμμετροι αἱ ΑΓ, ΓΒ.</s> <s>τῷ δὲ ὑπὸ τῶν ΑΓ, ΓΒ σύμ-<lb n="1799,001,,,,10,44,29"></lb> |
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10010 μετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΓ, ΓΒ.</s> <s>καὶ τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,44,30"></lb> |
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10011 ΑΓ, ΓΒ ἄρα ἀσύμμετρά ἐστι τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ.</s> <lb n="1799,001,,,,10,44,31"></lb> |
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10012 <s>ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΑΓ, ΓΒ ἴσον ἐστὶ τὸ ΕΗ, τῷ <lb n="1799,001,,,,10,44,32"></lb> |
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10013 δὲ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ἴσον τὸ ΘΚ· ἀσύμμετρον ἄρα <lb n="1799,001,,,,10,44,33"></lb> |
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10014 ἐστὶ τὸ ΕΗ τῷ ΘΚ· ὥστε καὶ ἡ ΕΘ τῇ ΘΝ ἀσύμμετρός <lb n="1799,001,,,,10,44,34"></lb> |
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10015 ἐστι μήκει.</s> <s>καί εἰσι ῥηταί· αἱ ΕΘ, ΘΝ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,44,35"></lb> |
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10016 δυνάμει μόνον σύμμετροι.</s> <s>ἐὰν δὲ δύο ῥηταὶ δυνάμει μόνον <lb n="1799,001,,,,10,44,36"></lb> |
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10017 σύμμετροι συντεθῶσιν, ἡ ὅλη ἄλογός ἐστιν ἡ καλουμένη <lb n="1799,001,,,,10,44,37"></lb> |
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10018 ἐκ δύο ὀνομάτων· ἡ ΕΝ ἄρα ἐκ δύο ὀνομάτων ἐστὶ <lb n="1799,001,,,,10,44,38"></lb> |
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10019 διῃρημένη κατὰ τὸ Θ.</s> <s>κατὰ τὰ αὐτὰ δὴ δειχθήσονται καὶ <lb n="1799,001,,,,10,44,39"></lb> |
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10020 αἱ ΕΜ, ΜΝ ῥηταὶ δυνάμει μόνον σύμμετροι· καὶ ἔσται <lb n="1799,001,,,,10,44,40"></lb> |
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10021 ἡ ΕΝ ἐκ δύο ὀνομάτων κατ' ἄλλο καὶ ἄλλο διῃρημένη τό <lb n="1799,001,,,,10,44,41"></lb> |
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10022 τε Θ καὶ τὸ Μ, καὶ οὐκ ἔστιν ἡ ΕΘ τῇ ΜΝ ἡ αὐτή, ὅτι <pb></pb> <lb n="1799,001,,,,10,44,42"></lb> |
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10023 τὰ ἀπὸ τῶν ΑΓ, ΓΒ μείζονά ἐστι τῶν ἀπὸ τῶν ΑΔ, ΔΒ.</s> <lb n="1799,001,,,,10,44,43"></lb> |
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10024 <s>ἀλλὰ τὰ ἀπὸ τῶν ΑΔ, ΔΒ μείζονά ἐστι τοῦ δὶς ὑπὸ <lb n="1799,001,,,,10,44,44"></lb> |
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10025 ΑΔ, ΔΒ· πολλῷ ἄρα καὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ, τουτέστι <lb n="1799,001,,,,10,44,45"></lb> |
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10026 τὸ ΕΗ, μεῖζόν ἐστι τοῦ δὶς ὑπὸ τῶν ΑΔ, ΔΒ, τουτέστι <lb n="1799,001,,,,10,44,46"></lb> |
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10027 τοῦ ΜΚ· ὥστε καὶ ἡ ΕΘ τῆς ΜΝ μείζων ἐστίν.</s> <s>ἡ ἄρα <lb n="1799,001,,,,10,44,47"></lb> |
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10028 ΕΘ τῇ ΜΝ οὐκ ἔστιν ἡ αὐτή· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,44,48"></lb> |
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10029 <s>Ἡ μείζων κατὰ τὸ αὐτὸ μόνον σημεῖον διαιρεῖται.</s> <lb n="1799,001,,,,10,45,1"></lb> |
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10030 <s>Ἔστω μείζων ἡ ΑΒ διῃρημένη κατὰ τὸ Γ, ὥστε τὰς <lb n="1799,001,,,,10,45,2"></lb> |
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10031 ΑΓ, ΓΒ δυνάμει ἀσυμμέτρους εἶναι ποιούσας τὸ μὲν <lb n="1799,001,,,,10,45,3"></lb> |
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10032 συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνων ῥητόν, <lb n="1799,001,,,,10,45,4"></lb> |
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10033 τὸ δ' ὑπὸ τῶν ΑΓ, ΓΒ μέσον· λέγω, ὅτι ἡ ΑΒ κατ' <lb n="1799,001,,,,10,45,5"></lb> |
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10034 ἄλλο σημεῖον οὐ διαιρεῖται.</s> <lb n="1799,001,,,,10,45,6"></lb> |
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10035 <s>Εἰ γὰρ δυνατόν, διῃρήσθω καὶ κατὰ τὸ Δ, ὥστε <lb n="1799,001,,,,10,45,7"></lb> |
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10036 καὶ τὰς ΑΔ, ΔΒ δυνάμει ἀσυμμέτρους εἶναι ποιού-<lb n="1799,001,,,,10,45,8"></lb> |
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10037 σας τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΔ, ΔΒ <lb n="1799,001,,,,10,45,9"></lb> |
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10038 ῥητόν, τὸ δ' ὑπ' αὐτῶν μέσον.</s> <s>καὶ ἐπεί, ᾧ διαφέρει <lb n="1799,001,,,,10,45,10"></lb> |
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10039 τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ, τούτῳ <pb></pb> <lb n="1799,001,,,,10,45,11"></lb> |
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10040 διαφέρει καὶ τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,45,12"></lb> |
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10041 ΓΒ, ἀλλὰ τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ <lb n="1799,001,,,,10,45,13"></lb> |
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10042 ὑπερέχει ῥητῷ· ῥητὰ γὰρ ἀμφότερα· καὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,45,14"></lb> |
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10043 ΑΔ, ΔΒ ἄρα τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ὑπερέχει ῥητῷ <lb n="1799,001,,,,10,45,15"></lb> |
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10044 μέσα ὄντα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ἡ μείζων κατ' <lb n="1799,001,,,,10,45,16"></lb> |
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10045 ἄλλο καὶ ἄλλο σημεῖον διαιρεῖται· κατὰ τὸ αὐτὸ ἄρα <lb n="1799,001,,,,10,45,17"></lb> |
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10046 μόνον διαιρεῖται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,45,18"></lb> |
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10047 <s>Ἡ ῥητὸν καὶ μέσον δυναμένη καθ' ἓν μόνον σημεῖον <lb n="1799,001,,,,10,46,1"></lb> |
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10048 διαιρεῖται.</s> <lb n="1799,001,,,,10,46,2"></lb> |
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10049 <s>Ἔστω ῥητὸν καὶ μέσον δυναμένη ἡ ΑΒ διῃρημένη κατὰ <lb n="1799,001,,,,10,46,3"></lb> |
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10050 τὸ Γ, ὥστε τὰς ΑΓ, ΓΒ δυνάμει ἀσυμμέτρους εἶναι <lb n="1799,001,,,,10,46,4"></lb> |
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10051 ποιούσας τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, ΓΒ <lb n="1799,001,,,,10,46,5"></lb> |
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10052 μέσον, τὸ δὲ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ῥητόν· λέγω, ὅτι ἡ ΑΒ <lb n="1799,001,,,,10,46,6"></lb> |
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10053 κατ' ἄλλο σημεῖον οὐ διαιρεῖται.</s> <lb n="1799,001,,,,10,46,7"></lb> |
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10054 <s>Εἰ γὰρ δυνατόν, διῃρήσθω καὶ κατὰ τὸ Δ, ὥστε <lb n="1799,001,,,,10,46,8"></lb> |
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10055 καὶ τὰς ΑΔ, ΔΒ δυνάμει ἀσυμμέτρους εἶναι ποι-<lb n="1799,001,,,,10,46,9"></lb> |
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10056 ούσας τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΔ, <lb n="1799,001,,,,10,46,10"></lb> |
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10057 ΔΒ μέσον, τὸ δὲ δὶς ὑπὸ τῶν ΑΔ, ΔΒ ῥητόν.</s> <s>ἐπεὶ <lb n="1799,001,,,,10,46,11"></lb> |
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10058 οὖν, ᾧ διαφέρει τὸ δὶς ὑπὸ τῶν ΑΓ, ΓΒ τοῦ δὶς <lb n="1799,001,,,,10,46,12"></lb> |
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10059 ὑπὸ τῶν ΑΔ, ΔΒ, τούτῳ διαφέρει καὶ τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,46,13"></lb> |
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10060 ΑΔ, ΔΒ τῶν ἀπὸ τῶν ΑΓ, ΓΒ, τὸ δὲ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,46,14"></lb> |
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10061 ΑΓ, ΓΒ τοῦ δὶς ὑπὸ τῶν ΑΔ, ΔΒ ὑπερέχει ῥητῷ, <pb></pb> <lb n="1799,001,,,,10,46,15"></lb> |
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10062 καὶ τὰ ἀπὸ τῶν ΑΔ, ΔΒ ἄρα τῶν ἀπὸ τῶν ΑΓ, ΓΒ ὑπερ-<lb n="1799,001,,,,10,46,16"></lb> |
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10063 έχει ῥητῷ μέσα ὄντα· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ἡ <lb n="1799,001,,,,10,46,17"></lb> |
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10064 ῥητὸν καὶ μέσον δυναμένη κατ' ἄλλο καὶ ἄλλο σημεῖον <lb n="1799,001,,,,10,46,18"></lb> |
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10065 διαιρεῖται.</s> <s>κατὰ ἓν ἄρα σημεῖον διαιρεῖται· ὅπερ ἔδει <lb n="1799,001,,,,10,46,19"></lb> |
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10066 δεῖξαι.</s> <lb n="1799,001,,,,10,46,20"></lb> |
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10067 <s>Ἡ δύο μέσα δυναμένη καθ' ἓν μόνον σημεῖον διαιρεῖται.</s> <lb n="1799,001,,,,10,47,1"></lb> |
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10068 <s>Ἔστω [δύο μέσα δυναμένη] ἡ ΑΒ διῃρημένη κατὰ τὸ Γ, <lb n="1799,001,,,,10,47,2"></lb> |
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10069 ὥστε τὰς ΑΓ, ΓΒ δυνάμει ἀσυμμέτρους εἰ̂ναι ποιούσας <lb n="1799,001,,,,10,47,3"></lb> |
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10070 τό τε συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, ΓΒ μέσον καὶ τὸ <lb n="1799,001,,,,10,47,4"></lb> |
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10071 ὑπὸ τῶν ΑΓ, ΓΒ μέσον καὶ ἔτι ἀσύμμετρον τῷ συγκει-<lb n="1799,001,,,,10,47,5"></lb> |
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10072 μένῳ ἐκ τῶν ἀπ' αὐτῶν.</s> <s>λέγω, ὅτι ἡ ΑΒ κατ' ἄλλο <lb n="1799,001,,,,10,47,6"></lb> |
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10073 σημεῖον οὐ διαιρεῖται ποιοῦσα τὰ προκείμενα.</s> <lb n="1799,001,,,,10,47,7"></lb> |
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10074 <s>Εἰ γὰρ δυνατόν, διῃρήσθω κατὰ τὸ Δ, ὥστε πάλιν <lb n="1799,001,,,,10,47,8"></lb> |
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10075 δηλονότι τὴν ΑΓ τῇ ΔΒ μὴ εἶναι τὴν αὐτήν, ἀλλὰ <lb n="1799,001,,,,10,47,9"></lb> |
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10076 μείζονα καθ' ὑπόθεσιν τὴν ΑΓ, <lb n="1799,001,,,,10,47,10"></lb> |
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10077 καὶ ἐκκείσθω ῥητὴ ἡ ΕΖ, καὶ <lb n="1799,001,,,,10,47,11"></lb> |
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10078 παραβεβλήσθω παρὰ τὴν ΕΖ <lb n="1799,001,,,,10,47,12"></lb> |
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10079 τοῖς μὲν ἀπὸ τῶν ΑΓ, ΓΒ ἴσον <lb n="1799,001,,,,10,47,13"></lb> |
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10080 τὸ ΕΗ, τῷ δὲ δὶς ὑπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,47,14"></lb> |
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10081 ΓΒ ἴσον τὸ ΘΚ· ὅλον ἄρα τὸ <lb n="1799,001,,,,10,47,15"></lb> |
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10082 ΕΚ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ <lb n="1799,001,,,,10,47,16"></lb> |
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10083 τετραγώνῳ.</s> <s>πάλιν δὴ παρα-<lb n="1799,001,,,,10,47,17"></lb> |
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10084 βεβλήσθω παρὰ τὴν ΕΖ τοῖς <pb></pb> <lb n="1799,001,,,,10,47,18"></lb> |
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10085 ἀπὸ τῶν ΑΔ, ΔΒ ἴσον τὸ ΕΛ· λοιπὸν ἄρα τὸ δὶς ὑπὸ <lb n="1799,001,,,,10,47,19"></lb> |
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10086 τῶν ΑΔ, ΔΒ λοιπῷ τῷ ΜΚ ἴσον ἐστίν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,47,20"></lb> |
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10087 μέσον ὑπόκειται τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,47,21"></lb> |
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10088 ΓΒ, μέσον ἄρα ἐστὶ καὶ τὸ ΕΗ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΕΖ <lb n="1799,001,,,,10,47,22"></lb> |
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10089 παράκειται· ῥητὴ ἄρα ἐστὶν ἡ ΘΕ καὶ ἀσύμμετρος τῇ <lb n="1799,001,,,,10,47,23"></lb> |
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10090 ΕΖ μήκει.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΘΝ ῥητή ἐστι καὶ <lb n="1799,001,,,,10,47,24"></lb> |
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10091 ἀσύμμετρος τῇ ΕΖ μήκει.</s> <s>καὶ ἐπεὶ ἀσύμμετρόν ἐστι τὸ <lb n="1799,001,,,,10,47,25"></lb> |
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10092 συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, ΓΒ τῷ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,47,26"></lb> |
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10093 ΑΓ, ΓΒ, καὶ τὸ ΕΗ ἄρα τῷ ΗΝ ἀσύμμετρόν ἐστιν· <lb n="1799,001,,,,10,47,27"></lb> |
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10094 ὥστε καὶ ἡ ΕΘ τῇ ΘΝ ἀσύμμετρός ἐστιν.</s> <s>καί εἰσι ῥηταί· <lb n="1799,001,,,,10,47,28"></lb> |
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10095 αἱ ΕΘ, ΘΝ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· <lb n="1799,001,,,,10,47,29"></lb> |
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10096 ἡ ΕΝ ἄρα ἐκ δύο ὀνομάτων ἐστὶ διῃρημένη κατὰ τὸ Θ.</s> <lb n="1799,001,,,,10,47,30"></lb> |
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10097 <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ κατὰ τὸ Μ διῄρηται.</s> <s>καὶ οὐκ <lb n="1799,001,,,,10,47,31"></lb> |
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10098 ἔστιν ἡ ΕΘ τῇ ΜΝ ἡ αὐτή· ἡ ἄρα ἐκ δύο ὀνομάτων κατ' <lb n="1799,001,,,,10,47,32"></lb> |
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10099 ἄλλο καὶ ἄλλο σημεῖον διῄρηται· ὅπερ ἐστὶν ἄτοπον.</s> <s>οὐκ <lb n="1799,001,,,,10,47,33"></lb> |
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10100 ἄρα ἡ δύο μέσα δυναμένη κατ' ἄλλο καὶ ἄλλο σημεῖον <lb n="1799,001,,,,10,47,34"></lb> |
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10101 διαιρεῖται· καθ' ἓν ἄρα μόνον [σημεῖον] διαιρεῖται.</s> <lb n="1799,001,,,,10,47,35"></lb> |
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10102 <s></s></p> |
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10103 </chap> |
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10104 <chap><p type="main"><s>Ὑποκειμένης ῥητῆς καὶ τῆς ἐκ δύο ὀνομάτων <lb n="1799,001,,,,10;HOR2,1,1"></lb> |
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10105 |
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10106 διῃρημένης εἰς τὰ ὀνόματα, ἧς τὸ μεῖζον ὄνομα τοῦ <lb n="1799,001,,,,10;HOR2,1,2"></lb> |
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10107 ἐλάσσονος μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει, <lb n="1799,001,,,,10;HOR2,1,3"></lb> |
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10108 ἐὰν μὲν τὸ μεῖζον ὄνομα σύμμετρον ᾖ μήκει τῇ ἐκκειμένῃ <lb n="1799,001,,,,10;HOR2,1,4"></lb> |
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10109 ῥητῇ, καλείσθω [ἡ ὅλη] ἐκ δύο ὀνομάτων πρώτη.</s> <s><pb></pb> <lb n="1799,001,,,,10;HOR2,1,5"></lb> |
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10110 Ἐὰν δὲ τὸ ἔλασσον ὄνομα σύμμετρον ᾖ μήκει τῇ <lb n="1799,001,,,,10;HOR2,2,1"></lb> |
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10111 ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ δύο ὀνομάτων δευτέρα.</s> <lb n="1799,001,,,,10;HOR2,2,2"></lb> |
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10112 <s>Ἐὰν δὲ μηδέτερον τῶν ὀνομάτων σύμμετρον ᾖ <lb n="1799,001,,,,10;HOR2,3,1"></lb> |
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10113 μήκει τῇ ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ δύο ὀνομάτων <lb n="1799,001,,,,10;HOR2,3,2"></lb> |
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10114 τρίτη.</s> <lb n="1799,001,,,,10;HOR2,3,3"></lb> |
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10115 <s>Πάλιν δὴ ἐὰν τὸ μεῖζον ὄνομα [τοῦ ἐλάσσονοσ] μεῖζον <lb n="1799,001,,,,10;HOR2,4,1"></lb> |
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10116 δύνηται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει, ἐὰν μὲν τὸ <lb n="1799,001,,,,10;HOR2,4,2"></lb> |
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10117 μεῖζον ὄνομα σύμμετρον ᾖ μήκει τῇ ἐκκειμένῃ ῥητῇ, <lb n="1799,001,,,,10;HOR2,4,3"></lb> |
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10118 καλείσθω ἐκ δύο ὀνομάτων τετάρτη.</s> <lb n="1799,001,,,,10;HOR2,4,4"></lb> |
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10119 <s>Ἐὰν δὲ τὸ ἔλασσον, πέμπτη.</s> <lb n="1799,001,,,,10;HOR2,5,1"></lb> |
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10120 <s>Ἐὰν δὲ μηδέτερον, ἕκτη.</s> <lb n="1799,001,,,,10;HOR2,6,1"></lb> |
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10121 <s></s></p> |
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10122 </chap> |
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10123 <chap><p type="main"><s>Εὑρεῖν τὴν ἐκ δύο ὀνομάτων πρώτην.</s> <lb n="1799,001,,,,10,48,1"></lb> |
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10124 <s> |
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10125 Ἐκκείσθωσαν δύο ἀριθμοὶ οἱ ΑΓ, ΓΒ, ὥστε τὸν <lb n="1799,001,,,,10,48,2"></lb> |
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10126 συγκείμενον ἐξ αὐτῶν τὸν ΑΒ πρὸς μὲν τὸν ΒΓ λόγον <lb n="1799,001,,,,10,48,3"></lb> |
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10127 ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, <lb n="1799,001,,,,10,48,4"></lb> |
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10128 πρὸς δὲ τὸν ΓΑ λόγον μὴ ἔχειν, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,10,48,5"></lb> |
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10129 πρὸς τετράγωνον ἀριθμόν, καὶ ἐκκείσθω τις ῥητὴ ἡ Δ, <lb n="1799,001,,,,10,48,6"></lb> |
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10130 καὶ τῇ Δ σύμμετρος ἔστω μήκει ἡ ΕΖ.</s> <s>ῥητὴ ἄρα ἐστὶ <pb></pb> <lb n="1799,001,,,,10,48,7"></lb> |
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10131 καὶ ἡ ΕΖ.</s> <s>καὶ γεγονέτω ὡς ὁ ΒΑ ἀριθμὸς πρὸς τὸν <lb n="1799,001,,,,10,48,8"></lb> |
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10132 ΑΓ, οὕτως τὸ ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς ΖΗ.</s> <s>ὁ δὲ <lb n="1799,001,,,,10,48,9"></lb> |
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10133 ΑΒ πρὸς τὸν ΑΓ λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν· <lb n="1799,001,,,,10,48,10"></lb> |
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10134 καὶ τὸ ἀπὸ τῆς ΕΖ ἄρα πρὸς τὸ ἀπὸ τῆς ΖΗ λόγον ἔχει, <lb n="1799,001,,,,10,48,11"></lb> |
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10135 ὃν ἀριθμὸς πρὸς ἀριθμόν· ὥστε σύμμετρόν ἐστι τὸ ἀπὸ <lb n="1799,001,,,,10,48,12"></lb> |
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10136 τῆς ΕΖ τῷ ἀπὸ τῆς ΖΗ.</s> <s>καί ἐστι ῥητὴ ἡ ΕΖ· ῥητὴ ἄρα <lb n="1799,001,,,,10,48,13"></lb> |
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10137 καὶ ἡ ΖΗ.</s> <s>καὶ ἐπεὶ ὁ ΒΑ πρὸς τὸν ΑΓ λόγον οὐκ ἔχει, <lb n="1799,001,,,,10,48,14"></lb> |
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10138 ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδὲ <lb n="1799,001,,,,10,48,15"></lb> |
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10139 τὸ ἀπὸ τῆς ΕΖ ἄρα πρὸς τὸ ἀπὸ τῆς ΖΗ λόγον ἔχει, ὃν <lb n="1799,001,,,,10,48,16"></lb> |
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10140 τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμ-<lb n="1799,001,,,,10,48,17"></lb> |
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10141 μετρος ἄρα ἐστὶν ἡ ΕΖ τῇ ΖΗ μήκει.</s> <s>αἱ ΕΖ, ΖΗ <lb n="1799,001,,,,10,48,18"></lb> |
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10142 ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἐκ δύο ἄρα <lb n="1799,001,,,,10,48,19"></lb> |
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10143 ὀνομάτων ἐστὶν ἡ ΕΗ.</s> <lb n="1799,001,,,,10,48,20"></lb> |
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10144 <s>Λέγω, ὅτι καὶ πρώτη.</s> <lb n="1799,001,,,,10,48,21"></lb> |
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10145 <s>Ἐπεὶ γάρ ἐστιν ὡς ὁ ΒΑ ἀριθμὸς πρὸς τὸν ΑΓ, οὕτως <lb n="1799,001,,,,10,48,22"></lb> |
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10146 τὸ ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς ΖΗ, μείζων δὲ ὁ ΒΑ <lb n="1799,001,,,,10,48,23"></lb> |
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10147 τοῦ ΑΓ, μεῖζον ἄρα καὶ τὸ ἀπὸ τῆς ΕΖ τοῦ ἀπὸ τῆς <lb n="1799,001,,,,10,48,24"></lb> |
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10148 ΖΗ.</s> <s>ἔστω οὖν τῷ ἀπὸ τῆς ΕΖ ἴσα τὰ ἀπὸ τῶν ΖΗ, Θ.</s> <lb n="1799,001,,,,10,48,25"></lb> |
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10149 <s>καὶ ἐπεί ἐστιν ὡς ὁ ΒΑ πρὸς τὸν ΑΓ, οὕτως τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,48,26"></lb> |
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10150 ΕΖ πρὸς τὸ ἀπὸ τῆς ΖΗ, ἀναστρέψαντι ἄρα ἐστὶν ὡς <lb n="1799,001,,,,10,48,27"></lb> |
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10151 ὁ ΑΒ πρὸς τὸν ΒΓ, οὕτως τὸ ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,48,28"></lb> |
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10152 τῆς Θ.</s> <s>ὁ δὲ ΑΒ πρὸς τὸν ΒΓ λόγον ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,48,29"></lb> |
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10153 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· καὶ τὸ ἀπὸ τῆς ΕΖ <lb n="1799,001,,,,10,48,30"></lb> |
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10154 ἄρα πρὸς τὸ ἀπὸ τῆς Θ λόγον ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,48,31"></lb> |
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10155 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.</s> <s>σύμμετρος ἄρα ἐστὶν <pb></pb> <lb n="1799,001,,,,10,48,32"></lb> |
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10156 ἡ ΕΖ τῇ Θ μήκει· ἡ ΕΖ ἄρα τῆς ΖΗ μεῖζον δύναται <lb n="1799,001,,,,10,48,33"></lb> |
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10157 τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <s>καί εἰσι ῥηταὶ αἱ ΕΖ, ΖΗ, καὶ <lb n="1799,001,,,,10,48,34"></lb> |
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10158 σύμμετρος ἡ ΕΖ τῇ Δ μήκει.</s> <lb n="1799,001,,,,10,48,35"></lb> |
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10159 <s>Ἡ ΕΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ πρώτη· ὅπερ ἔδει <lb n="1799,001,,,,10,48,36"></lb> |
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10160 δεῖξαι.</s> <lb n="1799,001,,,,10,48,37"></lb> |
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10161 <s>Εὑρεῖν τὴν ἐκ δύο ὀνομάτων δευτέραν.</s> <lb n="1799,001,,,,10,49,1"></lb> |
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10162 <s>Ἐκκείσθωσαν δύο ἀριθμοὶ οἱ ΑΓ, ΓΒ, ὥστε τὸν <lb n="1799,001,,,,10,49,2"></lb> |
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10163 συγκείμενον ἐξ αὐτῶν τὸν ΑΒ πρὸς μὲν τὸν ΒΓ λόγον <lb n="1799,001,,,,10,49,3"></lb> |
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10164 ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, <lb n="1799,001,,,,10,49,4"></lb> |
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10165 πρὸς δὲ τὸν ΑΓ λόγον μὴ ἔχειν, ὃν <lb n="1799,001,,,,10,49,5"></lb> |
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10166 τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,49,6"></lb> |
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10167 ἀριθμόν, καὶ ἐκκείσθω ῥητὴ ἡ Δ, καὶ <lb n="1799,001,,,,10,49,7"></lb> |
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10168 τῇ Δ σύμμετρος ἔστω ἡ ΕΖ μήκει· <lb n="1799,001,,,,10,49,8"></lb> |
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10169 ῥητὴ ἄρα ἐστὶν ἡ ΕΖ.</s> <s>γεγονέτω δὴ <lb n="1799,001,,,,10,49,9"></lb> |
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10170 καὶ ὡς ὁ ΓΑ ἀριθμὸς πρὸς τὸν ΑΒ, <lb n="1799,001,,,,10,49,10"></lb> |
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10171 οὕτως τὸ ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,49,11"></lb> |
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10172 ΖΗ· σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,49,12"></lb> |
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10173 ΕΖ τῷ ἀπὸ τῆς ΖΗ.</s> <s>ῥητὴ ἄρα ἐστὶ καὶ ἡ ΖΗ.</s> <s>καὶ <lb n="1799,001,,,,10,49,13"></lb> |
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10174 ἐπεὶ ὁ ΓΑ ἀριθμὸς πρὸς τὸν ΑΒ λόγον οὐκ ἔχει, ὃν <lb n="1799,001,,,,10,49,14"></lb> |
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10175 τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδὲ τὸ <pb></pb> <lb n="1799,001,,,,10,49,15"></lb> |
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10176 ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς ΖΗ λόγον ἔχει, ὃν τετρά-<lb n="1799,001,,,,10,49,16"></lb> |
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10177 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.</s> <s>ἀσύμμετρος <lb n="1799,001,,,,10,49,17"></lb> |
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10178 ἄρα ἐστὶν ἡ ΕΖ τῇ ΖΗ μήκει· αἱ ΕΖ, ΖΗ ἄρα ῥηταί <lb n="1799,001,,,,10,49,18"></lb> |
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10179 εἰσι δυνάμει μόνον σύμμετροι· ἐκ δύο ἄρα ὀνομάτων <lb n="1799,001,,,,10,49,19"></lb> |
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10180 ἐστὶν ἡ ΕΗ.</s> <lb n="1799,001,,,,10,49,20"></lb> |
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10181 <s>Δεικτέον δή, ὅτι καὶ δευτέρα.</s> <lb n="1799,001,,,,10,49,21"></lb> |
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10182 <s>Ἐπεὶ γὰρ ἀνάπαλίν ἐστιν ὡς ὁ ΒΑ ἀριθμὸς πρὸς τὸν <lb n="1799,001,,,,10,49,22"></lb> |
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10183 ΑΓ, οὕτως τὸ ἀπὸ τῆς ΗΖ πρὸς τὸ ἀπὸ τῆς ΖΕ, μείζων <lb n="1799,001,,,,10,49,23"></lb> |
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10184 δὲ ὁ ΒΑ τοῦ ΑΓ, μεῖζον ἄρα [καὶ] τὸ ἀπὸ τῆς ΗΖ τοῦ <lb n="1799,001,,,,10,49,24"></lb> |
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10185 ἀπὸ τῆς ΖΕ.</s> <s>ἔστω τῷ ἀπὸ τῆς ΗΖ ἴσα τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,49,25"></lb> |
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10186 ΕΖ, Θ· ἀναστρέψαντι ἄρα ἐστὶν ὡς ὁ ΑΒ πρὸς τὸν ΒΓ, <lb n="1799,001,,,,10,49,26"></lb> |
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10187 οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς Θ.</s> <s>ἀλλ' ὁ ΑΒ <lb n="1799,001,,,,10,49,27"></lb> |
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10188 πρὸς τὸν ΒΓ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,49,28"></lb> |
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10189 τετράγωνον ἀριθμόν· καὶ τὸ ἀπὸ τῆς ΖΗ ἄρα πρὸς τὸ <lb n="1799,001,,,,10,49,29"></lb> |
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10190 ἀπὸ τῆς Θ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,49,30"></lb> |
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10191 τετράγωνον ἀριθμόν.</s> <s>σύμμετρος ἄρα ἐστὶν ἡ ΖΗ τῇ Θ <lb n="1799,001,,,,10,49,31"></lb> |
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10192 μήκει· ὥστε ἡ ΖΗ τῆς ΖΕ μεῖζον δύναται τῷ ἀπὸ <lb n="1799,001,,,,10,49,32"></lb> |
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10193 συμμέτρου ἑαυτῇ.</s> <s>καί εἰσι ῥηταὶ αἱ ΖΗ, ΖΕ δυνάμει <lb n="1799,001,,,,10,49,33"></lb> |
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10194 μόνον σύμμετροι, καὶ τὸ ΕΖ ἔλασσον ὄνομα τῇ ἐκκειμένῃ <lb n="1799,001,,,,10,49,34"></lb> |
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10195 ῥητῇ σύμμετρόν ἐστι τῇ Δ μήκει.</s> <lb n="1799,001,,,,10,49,35"></lb> |
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10196 <s>Ἡ ΕΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ δευτέρα· ὅπερ ἔδει <lb n="1799,001,,,,10,49,36"></lb> |
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10197 δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,49,37"></lb> |
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10198 Εὑρεῖν τὴν ἐκ δύο ὀνομάτων τρίτην.</s> <lb n="1799,001,,,,10,50,1"></lb> |
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10199 <s>Ἐκκείσθωσαν δύο ἀριθμοὶ οἱ ΑΓ, ΓΒ, ὥστε τὸν <lb n="1799,001,,,,10,50,2"></lb> |
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10200 συγκείμενον ἐξ αὐτῶν τὸν ΑΒ πρὸς μὲν τὸν ΒΓ λόγον <lb n="1799,001,,,,10,50,3"></lb> |
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10201 ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, <lb n="1799,001,,,,10,50,4"></lb> |
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10202 πρὸς δὲ τὸν ΑΓ λόγον μὴ ἔχειν, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,10,50,5"></lb> |
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10203 πρὸς τετράγωνον ἀριθμόν.</s> <s>ἐκκείσθω δέ τις καὶ ἄλλος <lb n="1799,001,,,,10,50,6"></lb> |
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10204 μὴ τετράγωνος ἀριθμὸς ὁ Δ, καὶ πρὸς ἑκάτερον τῶν <lb n="1799,001,,,,10,50,7"></lb> |
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10205 ΒΑ, ΑΓ λόγον μὴ ἐχέτω, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,50,8"></lb> |
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10206 τετράγωνον ἀριθμόν· καὶ ἐκκείσθω τις ῥητὴ εὐθεῖα ἡ Ε, <lb n="1799,001,,,,10,50,9"></lb> |
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10207 καὶ γεγονέτω ὡς ὁ Δ πρὸς τὸν ΑΒ, οὕτως τὸ ἀπὸ τῆς Ε <lb n="1799,001,,,,10,50,10"></lb> |
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10208 πρὸς τὸ ἀπὸ τῆς ΖΗ· σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τη̂ς <lb n="1799,001,,,,10,50,11"></lb> |
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10209 Ε τῷ ἀπὸ τῆς ΖΗ.</s> <s>καί ἐστι ῥητὴ ἡ Ε· ῥητὴ ἄρα ἐστὶ καὶ <lb n="1799,001,,,,10,50,12"></lb> |
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10210 ἡ ΖΗ.</s> <s>καὶ ἐπεὶ ὁ Δ πρὸς τὸν ΑΒ λόγον οὐκ ἔχει, ὃν <lb n="1799,001,,,,10,50,13"></lb> |
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10211 τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδὲ τὸ <lb n="1799,001,,,,10,50,14"></lb> |
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10212 ἀπὸ τῆς Ε πρὸς τὸ ἀπὸ τῆς ΖΗ λόγον ἔχει, ὃν τετρά-<lb n="1799,001,,,,10,50,15"></lb> |
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10213 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος <lb n="1799,001,,,,10,50,16"></lb> |
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10214 ἄρα ἐστὶν ἡ Ε τῇ ΖΗ μήκει.</s> <s>γεγονέτω δὴ πάλιν ὡς ὁ <lb n="1799,001,,,,10,50,17"></lb> |
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10215 ΒΑ ἀριθμὸς πρὸς τὸν ΑΓ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς <lb n="1799,001,,,,10,50,18"></lb> |
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10216 τὸ ἀπὸ τῆς ΗΘ· σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΖΗ <lb n="1799,001,,,,10,50,19"></lb> |
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10217 τῷ ἀπὸ τῆς ΗΘ.</s> <s>ῥητὴ δὲ ἡ ΖΗ· ῥητὴ ἄρα καὶ ἡ ΗΘ.</s> <lb n="1799,001,,,,10,50,20"></lb> |
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10218 <s>καὶ ἐπεὶ ὁ ΒΑ πρὸς τὸν ΑΓ λόγον οὐκ ἔχει, ὅν τετρά- <pb></pb><lb n="1799,001,,,,10,50,21"></lb> |
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10219 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδὲ τὸ ἀπὸ <lb n="1799,001,,,,10,50,22"></lb> |
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10220 τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΘΗ λόγον ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,50,23"></lb> |
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10221 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν <lb n="1799,001,,,,10,50,24"></lb> |
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10222 ἡ ΖΗ τῇ ΗΘ μήκει.</s> <s>αἱ ΖΗ, ΗΘ ἄρα ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,50,25"></lb> |
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10223 μόνον σύμμετροι· ἡ ΖΘ ἄρα ἐκ δύο ὀνομάτων ἐστίν.</s> <lb n="1799,001,,,,10,50,26"></lb> |
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10224 <s>Λέγω δή, ὅτι καὶ τρίτη.</s> <lb n="1799,001,,,,10,50,27"></lb> |
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10225 <s>Ἐπεὶ γάρ ἐστιν ὡς ὁ Δ πρὸς τὸν ΑΒ, οὕτως τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,50,28"></lb> |
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10226 Ε πρὸς τὸ ἀπὸ τῆς ΖΗ, ὡς δὲ ὁ ΒΑ πρὸς τὸν ΑΓ, οὕτως <lb n="1799,001,,,,10,50,29"></lb> |
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10227 τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ, δι' ἴσου ἄρα ἐστὶν <lb n="1799,001,,,,10,50,30"></lb> |
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10228 ὡς ὁ Δ πρὸς τὸν ΑΓ, οὕτως τὸ ἀπὸ τῆς Ε πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,50,31"></lb> |
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10229 τῆς ΗΘ.</s> <s>ὁ δὲ Δ πρὸς τὸν ΑΓ λόγον οὐκ ἔχει, ὃν τετρά-<lb n="1799,001,,,,10,50,32"></lb> |
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10230 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· οὐδὲ τὸ ἀπὸ <lb n="1799,001,,,,10,50,33"></lb> |
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10231 τῆς Ε ἄρα πρὸς τὸ ἀπὸ τῆς ΗΘ λόγον ἔχει, ὃν τετρά-<lb n="1799,001,,,,10,50,34"></lb> |
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10232 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος <lb n="1799,001,,,,10,50,35"></lb> |
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10233 ἄρα ἐστὶν ἡ Ε τῇ ΗΘ μήκει.</s> <s>καὶ ἐπεί ἐστιν ὡς ὁ ΒΑ <lb n="1799,001,,,,10,50,36"></lb> |
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10234 πρὸς τὸν ΑΓ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,50,37"></lb> |
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10235 ΗΘ, μεῖζον ἄρα τὸ ἀπὸ τῆς ΖΗ τοῦ ἀπὸ τῆς ΗΘ.</s> <s>ἔστω <lb n="1799,001,,,,10,50,38"></lb> |
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10236 οὖν τῷ ἀπὸ τῆς ΖΗ ἴσα τὰ ἀπὸ τῶν ΗΘ, Κ· ἀνα-<lb n="1799,001,,,,10,50,39"></lb> |
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10237 στρέψαντι ἄρα [ἐστὶν] ὡς ὁ ΑΒ πρὸς τὸν ΒΓ, οὕτως τὸ <lb n="1799,001,,,,10,50,40"></lb> |
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10238 ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς Κ.</s> <s>ὁ δὲ ΑΒ πρὸς τὸν ΒΓ <lb n="1799,001,,,,10,50,41"></lb> |
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10239 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,50,42"></lb> |
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10240 ἀριθμόν· καὶ τὸ ἀπὸ τῆς ΖΗ ἄρα πρὸς τὸ ἀπὸ τῆς Κ <lb n="1799,001,,,,10,50,43"></lb> |
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10241 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,50,44"></lb> |
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10242 ἀριθμόν· σύμμετρος ἄρα [ἐστὶν] ἡ ΖΗ τῇ Κ μήκει.</s> <s>ἡ ΖΗ <pb></pb> <lb n="1799,001,,,,10,50,45"></lb> |
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10243 ἄρα τῆς ΗΘ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <lb n="1799,001,,,,10,50,46"></lb> |
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10244 <s>καί εἰσιν αἱ ΖΗ, ΗΘ ῥηταὶ δυνάμει μόνον σύμμετροι, <lb n="1799,001,,,,10,50,47"></lb> |
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10245 καὶ οὐδετέρα αὐτῶν σύμμετρός ἐστι τῇ Ε μήκει.</s> <lb n="1799,001,,,,10,50,48"></lb> |
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10246 <s>Ἡ ΖΘ ἄρα ἐκ δύο ὀνομάτων ἐστὶ τρίτη· ὅπερ ἔδει <lb n="1799,001,,,,10,50,49"></lb> |
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10247 δεῖξαι.</s> <lb n="1799,001,,,,10,50,50"></lb> |
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10248 <s>Εὑρεῖν τὴν ἐκ δύο ὀνομάτων τετάρτην.</s> <lb n="1799,001,,,,10,51,1"></lb> |
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10249 <s>Ἐκκείσθωσαν δύο ἀριθμοὶ οἱ ΑΓ, ΓΒ, ὥστε τὸν ΑΒ <lb n="1799,001,,,,10,51,2"></lb> |
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10250 πρὸς τὸν ΒΓ λόγον μὴ ἔχειν μήτε μὴν πρὸς τὸν ΑΓ, <lb n="1799,001,,,,10,51,3"></lb> |
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10251 ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.</s> <s>καὶ <lb n="1799,001,,,,10,51,4"></lb> |
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10252 ἐκκείσθω ῥητὴ ἡ Δ, καὶ τῇ Δ σύμμετρος <lb n="1799,001,,,,10,51,5"></lb> |
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10253 ἔστω μήκει ἡ ΕΖ· ῥητὴ ἄρα ἐστὶ καὶ ἡ <lb n="1799,001,,,,10,51,6"></lb> |
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10254 ΕΖ.</s> <s>καὶ γεγονέτω ὡς ὁ ΒΑ ἀριθμὸς πρὸς <lb n="1799,001,,,,10,51,7"></lb> |
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10255 τὸν ΑΓ, οὕτως τὸ ἀπὸ τῆς ΕΖ πρὸς τὸ <lb n="1799,001,,,,10,51,8"></lb> |
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10256 ἀπὸ τῆς ΖΗ· σύμμετρον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,10,51,9"></lb> |
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10257 ἀπὸ τῆς ΕΖ τῷ ἀπὸ τῆς ΖΗ· ῥητὴ ἄρα <lb n="1799,001,,,,10,51,10"></lb> |
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10258 ἐστὶ καὶ ἡ ΖΗ.</s> <s>καὶ ἐπεὶ ὁ ΒΑ πρὸς <lb n="1799,001,,,,10,51,11"></lb> |
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10259 τὸν ΑΓ λόγον οὐκ ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,51,12"></lb> |
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10260 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδὲ τὸ <lb n="1799,001,,,,10,51,13"></lb> |
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10261 ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς ΖΗ λόγον ἔχει, ὃν τετράγω-<lb n="1799,001,,,,10,51,14"></lb> |
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10262 νος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα <pb></pb> <lb n="1799,001,,,,10,51,15"></lb> |
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10263 ἐστὶν ἡ ΕΖ τῇ ΖΗ μήκει.</s> <s>αἱ ΕΖ, ΖΗ ἄρα ῥηταί εἰσι δυνά-<lb n="1799,001,,,,10,51,16"></lb> |
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10264 μει μόνον σύμμετροι· ὥστε ἡ ΕΗ ἐκ δύο ὀνομάτων ἐστίν.</s> <lb n="1799,001,,,,10,51,17"></lb> |
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10265 <s>Λέγω δή, ὅτι καὶ τετάρτη.</s> <lb n="1799,001,,,,10,51,18"></lb> |
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10266 <s>Ἐπεὶ γάρ ἐστιν ὡς ὁ ΒΑ πρὸς τὸν ΑΓ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,51,19"></lb> |
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10267 τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς ΖΗ [μείζων δὲ ὁ ΒΑ τοῦ ΑΓ], <lb n="1799,001,,,,10,51,20"></lb> |
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10268 μεῖζον ἄρα τὸ ἀπὸ τῆς ΕΖ τοῦ ἀπὸ τῆς ΖΗ.</s> <s>ἔστω οὖν <lb n="1799,001,,,,10,51,21"></lb> |
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10269 τῷ ἀπὸ τῆς ΕΖ ἴσα τὰ ἀπὸ τῶν ΖΗ, Θ· ἀναστρέψαντι <lb n="1799,001,,,,10,51,22"></lb> |
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10270 ἄρα ὡς ὁ ΑΒ ἀριθμὸς πρὸς τὸν ΒΓ, οὕτως τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,51,23"></lb> |
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10271 ΕΖ πρὸς τὸ ἀπὸ τῆς Θ.</s> <s>ὁ δὲ ΑΒ πρὸς τὸν ΒΓ λόγον <lb n="1799,001,,,,10,51,24"></lb> |
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10272 οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,51,25"></lb> |
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10273 ἀριθμόν· οὐδ' ἄρα τὸ ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς Θ <lb n="1799,001,,,,10,51,26"></lb> |
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10274 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,51,27"></lb> |
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10275 ἀριθμόν.</s> <s>ἀσύμμετρος ἄρα ἐστὶν ἡ ΕΖ τῇ Θ μήκει· ἡ <lb n="1799,001,,,,10,51,28"></lb> |
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10276 ΕΖ ἄρα τῆς ΗΖ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου <lb n="1799,001,,,,10,51,29"></lb> |
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10277 ἑαυτῇ.</s> <s>καί εἰσιν αἱ ΕΖ, ΖΗ ῥηταὶ δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,51,30"></lb> |
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10278 μετροι, καὶ ἡ ΕΖ τῇ Δ σύμμετρός ἐστι μήκει.</s> <lb n="1799,001,,,,10,51,31"></lb> |
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10279 <s>Ἡ ΕΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ τετάρτη· ὅπερ ἔδει <lb n="1799,001,,,,10,51,32"></lb> |
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10280 δεῖξαι.</s> <lb n="1799,001,,,,10,51,33"></lb> |
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10281 <s>Εὑρεῖν τὴν ἐκ δύο ὀνομάτων πέμπτην.</s> <lb n="1799,001,,,,10,52,1"></lb> |
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10282 <s>Ἐκκείσθωσαν δύο ἀριθμοὶ οἱ ΑΓ, ΓΒ, ὥστε τὸν ΑΒ <lb n="1799,001,,,,10,52,2"></lb> |
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10283 πρὸς ἑκάτερον αὐτῶν λόγον μὴ ἔχειν, ὃν τετράγωνος <pb></pb> <lb n="1799,001,,,,10,52,3"></lb> |
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10284 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, καὶ <lb n="1799,001,,,,10,52,4"></lb> |
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10285 ἐκκείσθω ῥητή τις εὐθεῖα ἡ Δ, καὶ τῇ Δ <lb n="1799,001,,,,10,52,5"></lb> |
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10286 σύμμετρος ἔστω [μήκει] ἡ ΕΖ· ῥητὴ ἄρα <lb n="1799,001,,,,10,52,6"></lb> |
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10287 ἡ ΕΖ.</s> <s>καὶ γεγονέτω ὡς ὁ ΓΑ πρὸς τὸν <lb n="1799,001,,,,10,52,7"></lb> |
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10288 ΑΒ, οὕτως τὸ ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,52,8"></lb> |
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10289 τῆς ΖΗ.</s> <s>ὁ δὲ ΓΑ πρὸς τὸν ΑΒ λόγον οὐκ <lb n="1799,001,,,,10,52,9"></lb> |
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10290 ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετρά-<lb n="1799,001,,,,10,52,10"></lb> |
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10291 γωνον ἀριθμόν· οὐδὲ τὸ ἀπὸ τῆς ΕΖ ἄρα <lb n="1799,001,,,,10,52,11"></lb> |
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10292 πρὸς τὸ ἀπὸ τῆς ΖΗ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,10,52,12"></lb> |
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10293 πρὸς τετράγωνον ἀριθμόν.</s> <s>αἱ ΕΖ, ΖΗ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,52,13"></lb> |
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10294 δυνάμει μόνον σύμμετροι· ἐκ δύο ἄρα ὀνομάτων ἐστὶν <lb n="1799,001,,,,10,52,14"></lb> |
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10295 ἡ ΕΗ.</s> <lb n="1799,001,,,,10,52,15"></lb> |
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10296 <s>Λέγω δή, ὅτι καὶ πέμπτη.</s> <lb n="1799,001,,,,10,52,16"></lb> |
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10297 <s>Ἐπεὶ γάρ ἐστιν ὡς ὁ ΓΑ πρὸς τὸν ΑΒ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,52,17"></lb> |
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10298 τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς ΖΗ, ἀνάπαλιν ὡς ὁ ΒΑ πρὸς <lb n="1799,001,,,,10,52,18"></lb> |
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10299 τὸν ΑΓ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΖΕ· <lb n="1799,001,,,,10,52,19"></lb> |
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10300 μεῖζον ἄρα τὸ ἀπὸ τῆς ΗΖ τοῦ ἀπὸ τῆς ΖΕ.</s> <s>ἔστω οὖν <pb></pb> <lb n="1799,001,,,,10,52,20"></lb> |
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10301 τῷ ἀπὸ τῆς ΗΖ ἴσα τὰ ἀπὸ τῶν ΕΖ, Θ· ἀναστρέψαντι <lb n="1799,001,,,,10,52,21"></lb> |
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10302 ἄρα ἐστὶν ὡς ὁ ΑΒ ἀριθμὸς πρὸς τὸν ΒΓ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,52,22"></lb> |
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10303 τῆς ΗΖ πρὸς τὸ ἀπὸ τῆς Θ.</s> <s>ὁ δὲ ΑΒ πρὸς τὸν ΒΓ λόγον <lb n="1799,001,,,,10,52,23"></lb> |
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10304 οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,52,24"></lb> |
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10305 ἀριθμόν· οὐδ' ἄρα τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς Θ <lb n="1799,001,,,,10,52,25"></lb> |
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10306 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,52,26"></lb> |
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10307 ἀριθμόν.</s> <s>ἀσύμμετρος ἄρα ἐστὶν ἡ ΖΗ τῇ Θ μήκει· ὥστε <lb n="1799,001,,,,10,52,27"></lb> |
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10308 ἡ ΖΗ τῆς ΖΕ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυ-<lb n="1799,001,,,,10,52,28"></lb> |
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10309 τῇ.</s> <s>καί εἰσιν αἱ ΗΖ, ΖΕ ῥηταὶ δυνάμει μόνον σύμμετροι <lb n="1799,001,,,,10,52,29"></lb> |
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10310 καὶ τὸ ΕΖ ἔλαττον ὄνομα σύμμετρόν ἐστι τῇ ἐκκειμένῃ <lb n="1799,001,,,,10,52,30"></lb> |
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10311 ῥητῇ τῇ Δ μήκει.</s> <lb n="1799,001,,,,10,52,31"></lb> |
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10312 <s>Ἡ ΕΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ πέμπτη· ὅπερ <lb n="1799,001,,,,10,52,32"></lb> |
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10313 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,52,33"></lb> |
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10314 <s>Εὑρεῖν τὴν ἐκ δύο ὀνομάτων ἕκτην.</s> <lb n="1799,001,,,,10,53,1"></lb> |
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10315 <s>Ἐκκείσθωσαν δύο ἀριθμοὶ οἱ ΑΓ, ΓΒ, ὥστε τὸν ΑΒ <lb n="1799,001,,,,10,53,2"></lb> |
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10316 πρὸς ἑκάτερον αὐτῶν λόγον μὴ ἔχειν, ὃν τετράγωνος <lb n="1799,001,,,,10,53,3"></lb> |
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10317 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἔστω δὲ καὶ ἕτερος <lb n="1799,001,,,,10,53,4"></lb> |
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10318 ἀριθμὸς ὁ Δ μὴ τετράγωνος ὢν μηδὲ πρὸς ἑκάτερον <lb n="1799,001,,,,10,53,5"></lb> |
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10319 τῶν ΒΑ, ΑΓ λόγον ἔχων, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,53,6"></lb> |
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10320 τετράγωνον ἀριθμόν· καὶ ἐκκείσθω τις ῥητὴ εὐθεῖα <pb></pb> <lb n="1799,001,,,,10,53,7"></lb> |
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10321 ἡ Ε, καὶ γεγονέτω ὡς ὁ Δ πρὸς τὸν ΑΒ, <lb n="1799,001,,,,10,53,8"></lb> |
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10322 οὕτως τὸ ἀπὸ τῆς Ε πρὸς τὸ ἀπὸ τῆς ΖΗ· <lb n="1799,001,,,,10,53,9"></lb> |
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10323 σύμμετρον ἄρα τὸ ἀπὸ τῆς Ε τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,53,10"></lb> |
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10324 ΖΗ.</s> <s>καί ἐστι ῥητὴ ἡ Ε· ῥητὴ ἄρα καὶ ἡ <lb n="1799,001,,,,10,53,11"></lb> |
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10325 ΖΗ.</s> <s>καὶ ἐπεὶ οὐκ ἔχει ὁ Δ πρὸς τὸν ΑΒ <lb n="1799,001,,,,10,53,12"></lb> |
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10326 λόγον, ὃν τετράγωνος ἀριθμὸς πρὸς τετρά-<lb n="1799,001,,,,10,53,13"></lb> |
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10327 γωνον ἀριθμόν, οὐδὲ τὸ ἀπὸ τῆς Ε ἄρα <lb n="1799,001,,,,10,53,14"></lb> |
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10328 πρὸς τὸ ἀπὸ τῆς ΖΗ λόγον ἔχει, ὃν τετρά-<lb n="1799,001,,,,10,53,15"></lb> |
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10329 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· <lb n="1799,001,,,,10,53,16"></lb> |
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10330 ἀσύμμετρος ἄρα ἡ Ε τῇ ΖΗ μήκει.</s> <s>γεγο-<lb n="1799,001,,,,10,53,17"></lb> |
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10331 νέτω δὴ πάλιν ὡς ὁ ΒΑ πρὸς τὸν ΑΓ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,53,18"></lb> |
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10332 τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ.</s> <s>σύμμετρον ἄρα τὸ ἀπὸ <lb n="1799,001,,,,10,53,19"></lb> |
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10333 τῆς ΖΗ τῷ ἀπὸ τῆς ΘΗ.</s> <s>ῥητὸν ἄρα τὸ ἀπὸ τῆς ΘΗ· <lb n="1799,001,,,,10,53,20"></lb> |
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10334 ῥητὴ ἄρα ἡ ΘΗ.</s> <s>καὶ ἐπεὶ ὁ ΒΑ πρὸς τὸν ΑΓ λόγον <lb n="1799,001,,,,10,53,21"></lb> |
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10335 οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθ-<lb n="1799,001,,,,10,53,22"></lb> |
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10336 μόν, οὐδὲ τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ λόγον <lb n="1799,001,,,,10,53,23"></lb> |
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10337 ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· <lb n="1799,001,,,,10,53,24"></lb> |
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10338 ἀσύμμετρος ἄρα ἐστὶν ἡ ΖΗ τῇ ΗΘ μήκει.</s> <s>αἱ ΖΗ, ΗΘ <lb n="1799,001,,,,10,53,25"></lb> |
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10339 ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἐκ δύο ἄρα <lb n="1799,001,,,,10,53,26"></lb> |
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10340 ὀνομάτων ἐστὶν ἡ ΖΘ.</s> <lb n="1799,001,,,,10,53,27"></lb> |
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10341 <s>Δεικτέον δή, ὅτι καὶ ἕκτη.</s> <lb n="1799,001,,,,10,53,28"></lb> |
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10342 <s>Ἐπεὶ γάρ ἐστιν ὡς ὁ Δ πρὸς τὸν ΑΒ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,53,29"></lb> |
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10343 τῆς Ε πρὸς τὸ ἀπὸ τῆς ΖΗ, ἔστι δὲ καὶ ὡς ὁ ΒΑ πρὸς <lb n="1799,001,,,,10,53,30"></lb> |
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10344 τὸν ΑΓ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ, <lb n="1799,001,,,,10,53,31"></lb> |
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10345 δι' ἴσου ἄρα ἐστὶν ὡς ὁ Δ πρὸς τὸν ΑΓ, οὕτως τὸ ἀπὸ <pb></pb> <lb n="1799,001,,,,10,53,32"></lb> |
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10346 τῆς Ε πρὸς τὸ ἀπὸ τῆς ΗΘ.</s> <s>ὁ δὲ Δ πρὸς τὸν ΑΓ λόγον <lb n="1799,001,,,,10,53,33"></lb> |
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10347 οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,53,34"></lb> |
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10348 ἀριθμόν· οὐδὲ τὸ ἀπὸ τῆς Ε ἄρα πρὸς τὸ ἀπὸ τῆς ΗΘ <lb n="1799,001,,,,10,53,35"></lb> |
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10349 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,53,36"></lb> |
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10350 ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ Ε τῇ ΗΘ μήκει.</s> <lb n="1799,001,,,,10,53,37"></lb> |
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10351 <s>ἐδείχθη δὲ καὶ τῇ ΖΗ ἀσύμμετρος· ἑκατέρα ἄρα τῶν <lb n="1799,001,,,,10,53,38"></lb> |
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10352 ΖΗ, ΗΘ ἀσύμμετρός ἐστι τῇ Ε μήκει.</s> <s>καὶ ἐπεί ἐστιν ὡς <lb n="1799,001,,,,10,53,39"></lb> |
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10353 ὁ ΒΑ πρὸς τὸν ΑΓ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,53,40"></lb> |
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10354 τῆς ΗΘ, μεῖζον ἄρα τὸ ἀπὸ τῆς ΖΗ τοῦ ἀπὸ τῆς ΗΘ.</s> <lb n="1799,001,,,,10,53,41"></lb> |
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10355 <s>ἔστω οὖν τῷ ἀπὸ [τῆσ] ΖΗ ἴσα τὰ ἀπὸ τῶν ΗΘ, Κ· <lb n="1799,001,,,,10,53,42"></lb> |
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10356 ἀναστρέψαντι ἄρα ὡς ὁ ΑΒ πρὸς ΒΓ, οὕτως τὸ ἀπὸ ΖΗ <lb n="1799,001,,,,10,53,43"></lb> |
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10357 πρὸς τὸ ἀπὸ τῆς Κ.</s> <s>ὁ δὲ ΑΒ πρὸς τὸν ΒΓ λόγον οὐκ ἔχει, <lb n="1799,001,,,,10,53,44"></lb> |
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10358 ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ὥστε <lb n="1799,001,,,,10,53,45"></lb> |
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10359 οὐδὲ τὸ ἀπὸ ΖΗ πρὸς τὸ ἀπὸ τῆς Κ λόγον ἔχει, ὃν <lb n="1799,001,,,,10,53,46"></lb> |
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10360 τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.</s> <s>ἀσύμ-<lb n="1799,001,,,,10,53,47"></lb> |
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10361 μετρος ἄρα ἐστὶν ἡ ΖΗ τῇ Κ μήκει· ἡ ΖΗ ἄρα τῆς ΗΘ <lb n="1799,001,,,,10,53,48"></lb> |
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10362 μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <s>καί εἰσιν <lb n="1799,001,,,,10,53,49"></lb> |
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10363 αἱ ΖΗ, ΗΘ ῥηταὶ δυνάμει μόνον σύμμετροι, καὶ οὐδετέρα <lb n="1799,001,,,,10,53,50"></lb> |
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10364 αὐτῶν σύμμετρός ἐστι μήκει τῇ ἐκκειμένῃ ῥητῇ τῇ Ε.</s> <lb n="1799,001,,,,10,53,51"></lb> |
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10365 <s>Ἡ ΖΘ ἄρα ἐκ δύο ὀνομάτων ἐστὶν ἕκτη· ὅπερ ἔδει <lb n="1799,001,,,,10,53,52"></lb> |
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10366 δεῖξαι.</s> <lb n="1799,001,,,,10,53,53"></lb> |
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10367 <s>Λῆμμα <lb n="1799,001,,,,10,53,54"></lb> |
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10368 Ἔστω δύο τετράγωνα τὰ ΑΒ, ΒΓ καὶ κείσθωσαν ὥστε <lb n="1799,001,,,,10,53,55"></lb> |
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10369 ἐπ' εὐθείας εἶναι τὴν ΔΒ τῇ ΒΕ· ἐπ' εὐθείας ἄρα ἐστὶ <pb></pb> <lb n="1799,001,,,,10,53,56"></lb> |
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10370 καὶ ἡ ΖΒ τῇ ΒΗ.</s> <s>καὶ συμπεπληρώσθω <lb n="1799,001,,,,10,53,57"></lb> |
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10371 τὸ ΑΓ παραλληλόγραμμον· λέγω, ὅτι <lb n="1799,001,,,,10,53,58"></lb> |
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10372 τετράγωνόν ἐστι τὸ ΑΓ, καὶ ὅτι τῶν <lb n="1799,001,,,,10,53,59"></lb> |
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10373 ΑΒ, ΒΓ μέσον ἀνάλογόν ἐστι τὸ ΔΗ, <lb n="1799,001,,,,10,53,60"></lb> |
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10374 καὶ ἔτι τῶν ΑΓ, ΓΒ μέσον ἀνάλογόν <lb n="1799,001,,,,10,53,61"></lb> |
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10375 ἐστι τὸ ΔΓ.</s> <lb n="1799,001,,,,10,53,62"></lb> |
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10376 <s>Ἐπεὶ γὰρ ἴση ἐστὶν ἡ μὲν ΔΒ τῇ ΒΖ, ἡ δὲ ΒΕ τῇ <lb n="1799,001,,,,10,53,63"></lb> |
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10377 ΒΗ, ὅλη ἄρα ἡ ΔΕ ὅλῃ τῇ ΖΗ ἐστιν ἴση.</s> <s>ἀλλ' ἡ μὲν <lb n="1799,001,,,,10,53,64"></lb> |
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10378 ΔΕ ἑκατέρᾳ τῶν ΑΘ, ΚΓ ἐστιν ἴση, ἡ δὲ ΖΗ ἑκατέρᾳ <lb n="1799,001,,,,10,53,65"></lb> |
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10379 τῶν ΑΚ, ΘΓ ἐστιν ἴση· καὶ ἑκατέρα ἄρα τῶν ΑΘ, ΚΓ <lb n="1799,001,,,,10,53,66"></lb> |
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10380 ἑκατέρᾳ τῶν ΑΚ, ΘΓ ἐστιν ἴση.</s> <s>ἰσόπλευρον ἄρα ἐστὶ <lb n="1799,001,,,,10,53,67"></lb> |
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10381 τὸ ΑΓ παραλληλόγραμμον· ἔστι δὲ καὶ ὀρθογώνιον· <lb n="1799,001,,,,10,53,68"></lb> |
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10382 τετράγωνον ἄρα ἐστὶ τὸ ΑΓ.</s> <lb n="1799,001,,,,10,53,69"></lb> |
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10383 <s>Καὶ ἐπεί ἐστιν ὡς ἡ ΖΒ πρὸς τὴν ΒΗ, οὕτως ἡ ΔΒ <lb n="1799,001,,,,10,53,70"></lb> |
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10384 πρὸς τὴν ΒΕ, ἀλλ' ὡς μὲν ἡ ΖΒ πρὸς τὴν ΒΗ, οὕτως <lb n="1799,001,,,,10,53,71"></lb> |
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10385 τὸ ΑΒ πρὸς τὸ ΔΗ, ὡς δὲ ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως <lb n="1799,001,,,,10,53,72"></lb> |
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10386 τὸ ΔΗ πρὸς τὸ ΒΓ, καὶ ὡς ἄρα τὸ ΑΒ πρὸς τὸ ΔΗ, <lb n="1799,001,,,,10,53,73"></lb> |
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10387 οὕτως τὸ ΔΗ πρὸς τὸ ΒΓ.</s> <s>τῶν ΑΒ, ΒΓ ἄρα μέσον <lb n="1799,001,,,,10,53,74"></lb> |
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10388 ἀνάλογόν ἐστι τὸ ΔΗ.</s> <lb n="1799,001,,,,10,53,75"></lb> |
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10389 <s>Λέγω δή, ὅτι καὶ τῶν ΑΓ, ΓΒ μέσον ἀνάλογόν [ἐστι] <lb n="1799,001,,,,10,53,76"></lb> |
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10390 τὸ ΔΓ.</s> <lb n="1799,001,,,,10,53,77"></lb> |
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10391 <s>Ἐπεὶ γάρ ἐστιν ὡς ἡ ΑΔ πρὸς τὴν ΔΚ, οὕτως ἡ ΚΗ <lb n="1799,001,,,,10,53,78"></lb> |
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10392 πρὸς τὴν ΗΓ· ἴση γάρ [ἐστιν] ἑκατέρα ἑκατέρᾳ· καὶ <lb n="1799,001,,,,10,53,79"></lb> |
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10393 συνθέντι ὡς ἡ ΑΚ πρὸς ΚΔ, οὕτως ἡ ΚΓ πρὸς ΓΗ, <lb n="1799,001,,,,10,53,80"></lb> |
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10394 ἀλλ' ὡς μὲν ἡ ΑΚ πρὸς ΚΔ, οὕτως τὸ ΑΓ πρὸς τὸ <pb></pb> <lb n="1799,001,,,,10,53,81"></lb> |
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10395 ΓΔ, ὡς δὲ ἡ ΚΓ πρὸς ΓΗ, οὕτως τὸ ΔΓ πρὸς ΓΒ, <lb n="1799,001,,,,10,53,82"></lb> |
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10396 καὶ ὡς ἄρα τὸ ΑΓ πρὸς ΔΓ, οὕτως τὸ ΔΓ πρὸς τὸ ΒΓ.</s> <lb n="1799,001,,,,10,53,83"></lb> |
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10397 <s>τῶν ΑΓ, ΓΒ ἄρα μέσον ἀνάλογόν ἐστι τὸ ΔΓ· ἃ προ-<lb n="1799,001,,,,10,53,84"></lb> |
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10398 έκειτο δεῖξαι.</s> <lb n="1799,001,,,,10,53,85"></lb> |
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10399 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο <lb n="1799,001,,,,10,54,1"></lb> |
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10400 ὀνομάτων πρώτης, ἡ τὸ χωρίον δυναμένη ἄλογός ἐστιν <lb n="1799,001,,,,10,54,2"></lb> |
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10401 ἡ καλουμένη ἐκ δύο ὀνομάτων. <lb n="1799,001,,,,10,54,3"></lb> |
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10402 Χωρίον γὰρ τὸ ΑΓ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΒ <lb n="1799,001,,,,10,54,4"></lb> |
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10403 καὶ τῆς ἐκ δύο ὀνομάτων πρώτης τῆς ΑΔ· λέγω, ὅτι <lb n="1799,001,,,,10,54,5"></lb> |
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10404 ἡ τὸ ΑΓ χωρίον δυναμένη ἄλογός ἐστιν ἡ καλουμένη ἐκ <lb n="1799,001,,,,10,54,6"></lb> |
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10405 δύο ὀνομάτων.</s> <lb n="1799,001,,,,10,54,7"></lb> |
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10406 <s>Ἐπεὶ γὰρ ἐκ δύο ὀνομάτων ἐστὶ πρώτη ἡ ΑΔ, διῃ-<lb n="1799,001,,,,10,54,8"></lb> |
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10407 ρήσθω εἰς τὰ ὀνόματα κατὰ τὸ Ε, καὶ ἔστω τὸ μεῖζον <lb n="1799,001,,,,10,54,9"></lb> |
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10408 ὄνομα τὸ ΑΕ.</s> <s>φανερὸν δή, ὅτι αἱ ΑΕ, ΕΔ ῥηταί εἰσι <lb n="1799,001,,,,10,54,10"></lb> |
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10409 δυνάμει μόνον σύμμετροι, καὶ ἡ ΑΕ τῆς ΕΔ μεῖζον <pb></pb> <lb n="1799,001,,,,10,54,11"></lb> |
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10410 δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ ἡ ΑΕ σύμμετρός <lb n="1799,001,,,,10,54,12"></lb> |
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10411 ἐστι τῇ ἐκκειμένῃ ῥητῇ τῇ ΑΒ μήκει.</s> <s>τετμήσθω δὴ ἡ <lb n="1799,001,,,,10,54,13"></lb> |
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10412 ΕΔ δίχα κατὰ τὸ Ζ σημεῖον.</s> <s>καὶ ἐπεὶ ἡ ΑΕ τῆς ΕΔ <lb n="1799,001,,,,10,54,14"></lb> |
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10413 μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, ἐὰν ἄρα τῷ <lb n="1799,001,,,,10,54,15"></lb> |
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10414 τετάρτῳ μέρει τοῦ ἀπὸ τῆς ἐλάσσονος, τουτέστι τῷ ἀπὸ <lb n="1799,001,,,,10,54,16"></lb> |
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10415 τῆς ΕΖ, ἴσον παρὰ τὴν μείζονα τὴν ΑΕ παραβληθῇ <lb n="1799,001,,,,10,54,17"></lb> |
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10416 ἐλλεῖπον εἴδει τετραγώνῳ, εἰς σύμμετρα αὐτὴν διαιρεῖ.</s> <lb n="1799,001,,,,10,54,18"></lb> |
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10417 <s>παραβεβλήσθω οὖν παρὰ τὴν ΑΕ τῷ ἀπὸ τῆς ΕΖ ἴσον <lb n="1799,001,,,,10,54,19"></lb> |
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10418 τὸ ὑπὸ ΑΗ, ΗΕ· σύμμετρος ἄρα ἐστὶν ἡ ΑΗ τῇ ΕΗ <lb n="1799,001,,,,10,54,20"></lb> |
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10419 μήκει.</s> <s>καὶ ἤχθωσαν ἀπὸ τῶν Η, Ε, Ζ ὁποτέρᾳ τῶν <lb n="1799,001,,,,10,54,21"></lb> |
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10420 ΑΒ, ΓΔ παράλληλοι αἱ ΗΘ, ΕΚ, ΖΛ· καὶ τῷ μὲν ΑΘ <lb n="1799,001,,,,10,54,22"></lb> |
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10421 παραλληλογράμμῳ ἴσον τετράγωνον συνεστάτω τὸ ΣΝ, <lb n="1799,001,,,,10,54,23"></lb> |
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10422 τῷ δὲ ΗΚ ἴσον τὸ ΝΠ, καὶ κείσθω ὥστε ἐπ' εὐθείας <lb n="1799,001,,,,10,54,24"></lb> |
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10423 εἶναι τὴν ΜΝ τῇ ΝΞ· ἐπ' εὐθείας ἄρα ἐστὶ καὶ ἡ ΡΝ τῇ <lb n="1799,001,,,,10,54,25"></lb> |
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10424 ΝΟ.</s> <s>καὶ συμπεπληρώσθω τὸ ΣΠ παραλληλόγραμμον· <lb n="1799,001,,,,10,54,26"></lb> |
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10425 τετράγωνον ἄρα ἐστὶ τὸ ΣΠ.</s> <s>καὶ ἐπεὶ τὸ ὑπὸ τῶν ΑΗ, <lb n="1799,001,,,,10,54,27"></lb> |
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10426 ΗΕ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΕΖ, ἔστιν ἄρα ὡς ἡ ΑΗ πρὸς <lb n="1799,001,,,,10,54,28"></lb> |
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10427 ΕΖ, οὕτως ἡ ΖΕ πρὸς ΕΗ· καὶ ὡς ἄρα τὸ ΑΘ πρὸς <lb n="1799,001,,,,10,54,29"></lb> |
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10428 ΕΛ, τὸ ΕΛ πρὸς ΚΗ· τῶν ΑΘ, ΗΚ ἄρα μέσον ἀνά-<lb n="1799,001,,,,10,54,30"></lb> |
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10429 λογόν ἐστι τὸ ΕΛ.</s> <s>ἀλλὰ τὸ μὲν ΑΘ ἴσον ἐστὶ τῷ ΣΝ, <lb n="1799,001,,,,10,54,31"></lb> |
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10430 τὸ δὲ ΗΚ ἴσον τῷ ΝΠ· τῶν ΣΝ, ΝΠ ἄρα μέσον ἀνά- <pb></pb><lb n="1799,001,,,,10,54,32"></lb> |
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10431 λογόν ἐστι τὸ ΕΛ.</s> <s>ἔστι δὲ τῶν αὐτῶν τῶν ΣΝ, ΝΠ <lb n="1799,001,,,,10,54,33"></lb> |
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10432 μέσον ἀνάλογον καὶ τὸ ΜΡ· ἴσον ἄρα ἐστὶ τὸ ΕΛ τῷ <lb n="1799,001,,,,10,54,34"></lb> |
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10433 ΜΡ· ὥστε καὶ τῷ ΟΞ ἴσον ἐστίν.</s> <s>ἔστι δὲ καὶ τὰ ΑΘ, <lb n="1799,001,,,,10,54,35"></lb> |
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10434 ΗΚ τοῖς ΣΝ, ΝΠ ἴσα· ὅλον ἄρα τὸ ΑΓ ἴσον ἐστὶν ὅλῳ <lb n="1799,001,,,,10,54,36"></lb> |
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10435 τῷ ΣΠ, τουτέστι τῷ ἀπὸ τῆς ΜΞ τετραγώνῳ· τὸ ΑΓ <lb n="1799,001,,,,10,54,37"></lb> |
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10436 ἄρα δύναται ἡ ΜΞ.</s> <lb n="1799,001,,,,10,54,38"></lb> |
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10437 <s>Λέγω, ὅτι ἡ ΜΞ ἐκ δύο ὀνομάτων ἐστίν.</s> <lb n="1799,001,,,,10,54,39"></lb> |
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10438 <s>Ἐπεὶ γὰρ σύμμετρός ἐστιν ἡ ΑΗ τῇ ΗΕ, σύμμετρός <lb n="1799,001,,,,10,54,40"></lb> |
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10439 ἐστι καὶ ἡ ΑΕ ἑκατέρᾳ τῶν ΑΗ, ΗΕ.</s> <s>ὑπόκειται δὲ <lb n="1799,001,,,,10,54,41"></lb> |
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10440 καὶ ἡ ΑΕ τῇ ΑΒ σύμμετρος· καὶ αἱ ΑΗ, ΗΕ ἄρα τῇ <lb n="1799,001,,,,10,54,42"></lb> |
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10441 ΑΒ σύμμετροί εἰσιν.</s> <s>καί ἐστι ῥητὴ ἡ ΑΒ· ῥητὴ ἄρα <lb n="1799,001,,,,10,54,43"></lb> |
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10442 ἐστὶ καὶ ἑκατέρα τῶν ΑΗ, ΗΕ· ῥητὸν ἄρα ἐστὶν ἑκά-<lb n="1799,001,,,,10,54,44"></lb> |
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10443 τερον τῶν ΑΘ, ΗΚ, καί ἐστι σύμμετρον τὸ ΑΘ τῷ <lb n="1799,001,,,,10,54,45"></lb> |
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10444 ΗΚ.</s> <s>ἀλλὰ τὸ μὲν ΑΘ τῷ ΣΝ ἴσον ἐστίν, τὸ δὲ ΗΚ <lb n="1799,001,,,,10,54,46"></lb> |
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10445 τῷ ΝΠ· καὶ τὰ ΣΝ, ΝΠ ἄρα, τουτέστι τὰ ἀπὸ τῶν ΜΝ, <lb n="1799,001,,,,10,54,47"></lb> |
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10446 ΝΞ, ῥητά ἐστι καὶ σύμμετρα.</s> <s>καὶ ἐπεὶ ἀσύμμετρός <lb n="1799,001,,,,10,54,48"></lb> |
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10447 ἐστιν ἡ ΑΕ τῇ ΕΔ μήκει, ἀλλ' ἡ μὲν ΑΕ τῇ ΑΗ ἐστι <lb n="1799,001,,,,10,54,49"></lb> |
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10448 σύμμετρος, ἡ δὲ ΔΕ τῇ ΕΖ σύμμετρος, ἀσύμμετρος <lb n="1799,001,,,,10,54,50"></lb> |
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10449 ἄρα καὶ ἡ ΑΗ τῇ ΕΖ· ὥστε καὶ τὸ ΑΘ τῷ ΕΛ ἀσύμ-<lb n="1799,001,,,,10,54,51"></lb> |
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10450 μετρόν ἐστιν.</s> <s>ἀλλὰ τὸ μὲν ΑΘ τῷ ΣΝ ἐστιν ἴσον, τὸ δὲ <lb n="1799,001,,,,10,54,52"></lb> |
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10451 ΕΛ τῷ ΜΡ· καὶ τὸ ΣΝ ἄρα τῷ ΜΡ ἀσύμμετρόν ἐστιν.</s> <s><pb></pb> <lb n="1799,001,,,,10,54,53"></lb> |
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10452 ἀλλ' ὡς τὸ ΣΝ πρὸς ΜΡ, ἡ ΟΝ πρὸς τὴν ΝΡ· ἀσύμ-<lb n="1799,001,,,,10,54,54"></lb> |
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10453 μετρος ἄρα ἐστὶν ἡ ΟΝ τῇ ΝΡ.</s> <s>ἴση δὲ ἡ μὲν ΟΝ τῇ <lb n="1799,001,,,,10,54,55"></lb> |
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10454 ΜΝ, ἡ δὲ ΝΡ τῇ ΝΞ· ἀσύμμετρος ἄρα ἐστὶν ἡ ΜΝ τῇ <lb n="1799,001,,,,10,54,56"></lb> |
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10455 ΝΞ.</s> <s>καί ἐστι τὸ ἀπὸ τῆς ΜΝ σύμμετρον τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,54,57"></lb> |
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10456 ΝΞ, καὶ ῥητὸν ἑκάτερον· αἱ ΜΝ, ΝΞ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,54,58"></lb> |
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10457 δυνάμει μόνον σύμμετροι.</s> <lb n="1799,001,,,,10,54,59"></lb> |
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10458 <s>Ἡ ΜΞ ἄρα ἐκ δύο ὀνομάτων ἐστὶ καὶ δύναται τὸ ΑΓ· <lb n="1799,001,,,,10,54,60"></lb> |
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10459 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,54,61"></lb> |
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10460 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο <lb n="1799,001,,,,10,55,1"></lb> |
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10461 ὀνομάτων δευτέρας, ἡ τὸ χωρίον δυναμένη ἄλογός ἐστιν <lb n="1799,001,,,,10,55,2"></lb> |
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10462 ἡ καλουμένη ἐκ δύο μέσων πρώτη.</s> <lb n="1799,001,,,,10,55,3"></lb> |
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10463 <s>Περιεχέσθω γὰρ χωρίον τὸ ΑΒΓΔ ὑπὸ ῥητῆς τῆς <lb n="1799,001,,,,10,55,4"></lb> |
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10464 ΑΒ καὶ τῆς ἐκ δύο ὀνομάτων δευτέρας τῆς ΑΔ· λέγω, <lb n="1799,001,,,,10,55,5"></lb> |
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10465 ὅτι ἡ τὸ ΑΓ χωρίον δυναμένη ἐκ δύο μέσων πρώτη <lb n="1799,001,,,,10,55,6"></lb> |
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10466 ἐστίν.</s> <lb n="1799,001,,,,10,55,7"></lb> |
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10467 <s>Ἐπεὶ γὰρ ἐκ δύο ὀνομάτων δευτέρα ἐστὶν ἡ ΑΔ, <lb n="1799,001,,,,10,55,8"></lb> |
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10468 διῃρήσθω εἰς τὰ ὀνόματα κατὰ τὸ Ε, ὥστε τὸ μεῖζον <lb n="1799,001,,,,10,55,9"></lb> |
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10469 ὄνομα εἶναι τὸ ΑΕ· αἱ ΑΕ, ΕΔ ἄρα ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,55,10"></lb> |
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10470 μόνον σύμμετροι, καὶ ἡ ΑΕ τῆς ΕΔ μεῖζον δύναται <lb n="1799,001,,,,10,55,11"></lb> |
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10471 τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ τὸ ἔλαττον ὄνομα ἡ ΕΔ <lb n="1799,001,,,,10,55,12"></lb> |
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10472 σύμμετρόν ἐστι τῇ ΑΒ μήκει.</s> <s>τετμήσθω ἡ ΕΔ δίχα κατὰ <lb n="1799,001,,,,10,55,13"></lb> |
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10473 τὸ Ζ, καὶ τῷ ἀπὸ τῆς ΕΖ ἴσον παρὰ τὴν ΑΕ παρα-<lb n="1799,001,,,,10,55,14"></lb> |
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10474 βεβλήσθω ἐλλεῖπον εἴδει τετραγώνῳ τὸ ὑπὸ τῶν ΑΗΕ· <pb></pb> <lb n="1799,001,,,,10,55,15"></lb> |
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10475 σύμμετρος ἄρα ἡ ΑΗ τῇ ΗΕ μήκει.</s> <s>καὶ διὰ τῶν Η, <lb n="1799,001,,,,10,55,16"></lb> |
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10476 Ε, Ζ παράλληλοι ἤχθωσαν ταῖς ΑΒ, ΓΔ αἱ ΗΘ, ΕΚ, <lb n="1799,001,,,,10,55,17"></lb> |
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10477 ΖΛ, καὶ τῷ μὲν ΑΘ παραλληλογράμμῳ ἴσον τετρά-<lb n="1799,001,,,,10,55,18"></lb> |
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10478 γωνον συνεστάτω τὸ ΣΝ, τῷ δὲ ΗΚ ἴσον τετράγωνον <lb n="1799,001,,,,10,55,19"></lb> |
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10479 τὸ ΝΠ, καὶ κείσθω ὥστε ἐπ' εὐθείας εἶναι τὴν ΜΝ τῇ <lb n="1799,001,,,,10,55,20"></lb> |
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10480 ΝΞ· ἐπ' εὐθείας ἄρα [ἐστὶ] καὶ ἡ ΡΝ τῇ ΝΟ.</s> <s>καὶ <lb n="1799,001,,,,10,55,21"></lb> |
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10481 συμπεπληρώσθω τὸ ΣΠ τετράγωνον· φανερὸν δὴ ἐκ <lb n="1799,001,,,,10,55,22"></lb> |
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10482 τοῦ προδεδειγμένου, ὅτι τὸ ΜΡ μέσον ἀνάλογόν ἐστι <lb n="1799,001,,,,10,55,23"></lb> |
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10483 τῶν ΣΝ, ΝΠ, καὶ ἴσον τῷ ΕΛ, καὶ ὅτι τὸ ΑΓ χωρίον <lb n="1799,001,,,,10,55,24"></lb> |
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10484 δύναται ἡ ΜΞ.</s> <s>δεικτέον δή, ὅτι ἡ ΜΞ ἐκ δύο μέσων ἐστὶ <lb n="1799,001,,,,10,55,25"></lb> |
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10485 πρώτη.</s> <s>ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΑΕ τῇ ΕΔ μήκει, <lb n="1799,001,,,,10,55,26"></lb> |
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10486 σύμμετρος δὲ ἡ ΕΔ τῇ ΑΒ, ἀσύμμετρος ἄρα ἡ ΑΕ τῇ <lb n="1799,001,,,,10,55,27"></lb> |
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10487 ΑΒ.</s> <s>καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΑΗ τῇ ΕΗ, σύμμετρός <lb n="1799,001,,,,10,55,28"></lb> |
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10488 ἐστι καὶ ἡ ΑΕ ἑκατέρᾳ τῶν ΑΗ, ΗΕ.</s> <s>ἀλλὰ ἡ ΑΕ <pb></pb> <lb n="1799,001,,,,10,55,29"></lb> |
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10489 ἀσύμμετρος τῇ ΑΒ μήκει· καὶ αἱ ΑΗ, ΗΕ ἄρα ἀσύμ-<lb n="1799,001,,,,10,55,30"></lb> |
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10490 μετροί εἰσι τῇ ΑΒ.</s> <s>αἱ ΒΑ, ΑΗ, ΗΕ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,55,31"></lb> |
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10491 δυνάμει μόνον σύμμετροι· ὥστε μέσον ἐστὶν ἑκάτερον <lb n="1799,001,,,,10,55,32"></lb> |
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10492 τῶν ΑΘ, ΗΚ.</s> <s>ὥστε καὶ ἑκάτερον τῶν ΣΝ, ΝΠ μέσον <lb n="1799,001,,,,10,55,33"></lb> |
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10493 ἐστίν.</s> <s>καὶ αἱ ΜΝ, ΝΞ ἄρα μέσαι εἰσίν.</s> <s>καὶ ἐπεὶ σύμ-<lb n="1799,001,,,,10,55,34"></lb> |
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10494 μετρος ἡ ΑΗ τῇ ΗΕ μήκει, σύμμετρόν ἐστι καὶ τὸ ΑΘ <lb n="1799,001,,,,10,55,35"></lb> |
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10495 τῷ ΗΚ, τουτέστι τὸ ΣΝ τῷ ΝΠ, τουτέστι τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,55,36"></lb> |
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10496 ΜΝ τῷ ἀπὸ τῆς ΝΞ [ὥστε δυνάμει εἰσὶ σύμμετροι αἱ ΜΝ, <lb n="1799,001,,,,10,55,37"></lb> |
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10497 ΝΞ].</s> <s>καὶ ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΑΕ τῇ ΕΔ μήκει, <lb n="1799,001,,,,10,55,38"></lb> |
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10498 ἀλλ' ἡ μὲν ΑΕ σύμμετρός ἐστι τῇ ΑΗ, ἡ δὲ ΕΔ τῇ ΕΖ <lb n="1799,001,,,,10,55,39"></lb> |
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10499 σύμμετρος, ἀσύμμετρος ἄρα ἡ ΑΗ τῇ ΕΖ· ὥστε καὶ <lb n="1799,001,,,,10,55,40"></lb> |
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10500 τὸ ΑΘ τῷ ΕΛ ἀσύμμετρόν ἐστιν, τουτέστι τὸ ΣΝ τῷ <lb n="1799,001,,,,10,55,41"></lb> |
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10501 ΜΡ, τουτέστιν ἡ ΟΝ τῇ ΝΡ, τουτέστιν ἡ ΜΝ τῇ ΝΞ <lb n="1799,001,,,,10,55,42"></lb> |
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10502 ἀσύμμετρός ἐστι μήκει.</s> <s>ἐδείχθησαν δὲ αἱ ΜΝ, ΝΞ <lb n="1799,001,,,,10,55,43"></lb> |
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10503 καὶ μέσαι οὖσαι καὶ δυνάμει σύμμετροι· αἱ ΜΝ, ΝΞ <lb n="1799,001,,,,10,55,44"></lb> |
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10504 ἄρα μέσαι εἰσὶ δυνάμει μόνον σύμμετροι.</s> <s>λέγω δή, ὅτι <lb n="1799,001,,,,10,55,45"></lb> |
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10505 καὶ ῥητὸν περιέχουσιν.</s> <s>ἐπεὶ γὰρ ἡ ΔΕ ὑπόκειται ἑκατέρᾳ <lb n="1799,001,,,,10,55,46"></lb> |
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10506 τῶν ΑΒ, ΕΖ σύμμετρος, σύμμετρος ἄρα καὶ ἡ ΕΖ <lb n="1799,001,,,,10,55,47"></lb> |
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10507 τῇ ΕΚ.</s> <s>καὶ ῥητὴ ἑκατέρα αὐτῶν· ῥητὸν ἄρα τὸ ΕΛ, <lb n="1799,001,,,,10,55,48"></lb> |
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10508 τουτέστι τὸ ΜΡ· τὸ δὲ ΜΡ ἐστι τὸ ὑπὸ τῶν ΜΝΞ.</s> <lb n="1799,001,,,,10,55,49"></lb> |
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10509 <s>ἐὰν δὲ δύο μέσαι δυνάμει μόνον σύμμετροι συντεθῶσι <pb></pb> <lb n="1799,001,,,,10,55,50"></lb> |
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10510 ῥητὸν περιέχουσαι, ἡ ὅλη ἄλογός ἐστιν, καλεῖται δὲ ἐκ <lb n="1799,001,,,,10,55,51"></lb> |
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10511 δύο μέσων πρώτη.</s> <lb n="1799,001,,,,10,55,52"></lb> |
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10512 <s>Ἡ ἄρα ΜΞ ἐκ δύο μέσων ἐστὶ πρώτη· ὅπερ ἔδει <lb n="1799,001,,,,10,55,53"></lb> |
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10513 δεῖξαι.</s> <lb n="1799,001,,,,10,55,54"></lb> |
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10514 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο <lb n="1799,001,,,,10,56,1"></lb> |
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10515 ὀνομάτων τρίτης, ἡ τὸ χωρίον δυναμένη ἄλογός ἐστιν <lb n="1799,001,,,,10,56,2"></lb> |
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10516 ἡ καλουμένη ἐκ δύο μέσων δευτέρα.</s> <lb n="1799,001,,,,10,56,3"></lb> |
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10517 <s>Χωρίον γὰρ τὸ ΑΒΓΔ περιεχέσθω ὑπὸ ῥητῆς τῆς <lb n="1799,001,,,,10,56,4"></lb> |
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10518 ΑΒ καὶ τῆς ἐκ δύο ὀνομάτων τρίτης τῆς ΑΔ διῃρημένης <lb n="1799,001,,,,10,56,5"></lb> |
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10519 εἰς τὰ ὀνόματα κατὰ τὸ Ε, ὧν μεῖζόν ἐστι τὸ ΑΕ· λέγω, <lb n="1799,001,,,,10,56,6"></lb> |
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10520 ὅτι ἡ τὸ ΑΓ χωρίον δυναμένη ἄλογός ἐστιν ἡ καλουμένη <lb n="1799,001,,,,10,56,7"></lb> |
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10521 ἐκ δύο μέσων δευτέρα.</s> <lb n="1799,001,,,,10,56,8"></lb> |
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10522 <s>Κατεσκευάσθω γὰρ τὰ αὐτὰ τοῖς πρότερον.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,56,9"></lb> |
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10523 ἐκ δύο ὀνομάτων ἐστὶ τρίτη ἡ ΑΔ, αἱ ΑΕ, ΕΔ ἄρα <lb n="1799,001,,,,10,56,10"></lb> |
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10524 ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ ἡ ΑΕ τῆς ΕΔ <lb n="1799,001,,,,10,56,11"></lb> |
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10525 μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ οὐδετέρα <pb></pb> <lb n="1799,001,,,,10,56,12"></lb> |
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10526 τῶν ΑΕ, ΕΔ σύμμετρός [ἐστι] τῇ ΑΒ μήκει.</s> <s>ὁμοίως <lb n="1799,001,,,,10,56,13"></lb> |
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10527 δὴ τοῖς προδεδειγμένοις δείξομεν, ὅτι ἡ ΜΞ ἐστιν ἡ τὸ <lb n="1799,001,,,,10,56,14"></lb> |
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10528 ΑΓ χωρίον δυναμένη, καὶ αἱ ΜΝ, ΝΞ μέσαι εἰσὶ δυνάμει <lb n="1799,001,,,,10,56,15"></lb> |
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10529 μόνον σύμμετροι· ὥστε ἡ ΜΞ ἐκ δύο μέσων ἐστίν.</s> <lb n="1799,001,,,,10,56,16"></lb> |
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10530 <s>Δεικτέον δή, ὅτι καὶ δευτέρα.</s> <lb n="1799,001,,,,10,56,17"></lb> |
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10531 <s>[Καὶ] ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΔΕ τῇ ΑΒ μήκει, <lb n="1799,001,,,,10,56,18"></lb> |
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10532 τουτέστι τῇ ΕΚ, σύμμετρος δὲ ἡ ΔΕ τῇ ΕΖ, ἀσύμ-<lb n="1799,001,,,,10,56,19"></lb> |
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10533 μετρος ἄρα ἐστὶν ἡ ΕΖ τῇ ΕΚ μήκει.</s> <s>καί εἰσι ῥηταί· <lb n="1799,001,,,,10,56,20"></lb> |
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10534 αἱ ΖΕ, ΕΚ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι.</s> <lb n="1799,001,,,,10,56,21"></lb> |
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10535 <s>μέσον ἄρα [ἐστὶ] τὸ ΕΛ, τουτέστι τὸ ΜΡ· καὶ περι-<lb n="1799,001,,,,10,56,22"></lb> |
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10536 έχεται ὑπὸ τῶν ΜΝΞ· μέσον ἄρα ἐστὶ τὸ ὑπὸ τῶν ΜΝΞ.</s> <lb n="1799,001,,,,10,56,23"></lb> |
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10537 <s>Ἡ ΜΞ ἄρα ἐκ δύο μέσων ἐστὶ δευτέρα· ὅπερ ἔδει <lb n="1799,001,,,,10,56,24"></lb> |
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10538 δεῖξαι.</s> <lb n="1799,001,,,,10,56,25"></lb> |
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10539 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο <lb n="1799,001,,,,10,57,1"></lb> |
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10540 ὀνομάτων τετάρτης, ἡ τὸ χωρίον δυναμένη ἄλογός ἐστιν <lb n="1799,001,,,,10,57,2"></lb> |
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10541 ἡ καλουμένη μείζων.</s> <lb n="1799,001,,,,10,57,3"></lb> |
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10542 <s>Χωρίον γὰρ τὸ ΑΓ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΒ <lb n="1799,001,,,,10,57,4"></lb> |
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10543 καὶ τῆς ἐκ δύο ὀνομάτων τετάρτης τῆς ΑΔ διῃρημένης <lb n="1799,001,,,,10,57,5"></lb> |
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10544 εἰς τὰ ὀνόματα κατὰ τὸ Ε, ὧν μεῖζον ἔστω τὸ ΑΕ· <lb n="1799,001,,,,10,57,6"></lb> |
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10545 λέγω, ὅτι ἡ τὸ ΑΓ χωρίον δυναμένη ἄλογός ἐστιν ἡ <lb n="1799,001,,,,10,57,7"></lb> |
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10546 καλουμένη μείζων.</s> <s><pb></pb> <lb n="1799,001,,,,10,57,8"></lb> |
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10547 Ἐπεὶ γὰρ ἡ ΑΔ ἐκ δύο ὀνομάτων ἐστὶ τετάρτη, αἱ <lb n="1799,001,,,,10,57,9"></lb> |
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10548 ΑΕ, ΕΔ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ <lb n="1799,001,,,,10,57,10"></lb> |
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10549 ἡ ΑΕ τῆς ΕΔ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου <lb n="1799,001,,,,10,57,11"></lb> |
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10550 ἑαυτῇ, καὶ ἡ ΑΕ τῇ ΑΒ σύμμετρός [ἐστι] μήκει.</s> <s>τε-<lb n="1799,001,,,,10,57,12"></lb> |
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10551 τμήσθω ἡ ΔΕ δίχα κατὰ τὸ Ζ, καὶ τῷ ἀπὸ τῆς ΕΖ <lb n="1799,001,,,,10,57,13"></lb> |
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10552 ἴσον παρὰ τὴν ΑΕ παραβεβλήσθω παραλληλόγραμμον <lb n="1799,001,,,,10,57,14"></lb> |
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10553 τὸ ὑπὸ ΑΗ, ΗΕ· ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΗ τῇ ΗΕ <lb n="1799,001,,,,10,57,15"></lb> |
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10554 μήκει.</s> <s>ἤχθωσαν παράλληλοι τῇ ΑΒ αἱ ΗΘ, ΕΚ, ΖΛ, <lb n="1799,001,,,,10,57,16"></lb> |
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10555 καὶ τὰ λοιπὰ τὰ αὐτὰ τοῖς πρὸ τούτου γεγονέτω· φανερὸν <lb n="1799,001,,,,10,57,17"></lb> |
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10556 δή, ὅτι ἡ τὸ ΑΓ χωρίον δυναμένη ἐστὶν ἡ ΜΞ.</s> <s>δεικτέον <lb n="1799,001,,,,10,57,18"></lb> |
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10557 δή, ὅτι ἡ ΜΞ ἄλογός ἐστιν ἡ καλουμένη μείζων.</s> <s>ἐπεὶ <lb n="1799,001,,,,10,57,19"></lb> |
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10558 ἀσύμμετρός ἐστιν ἡ ΑΗ τῇ ΕΗ μήκει, ἀσύμμετρόν <lb n="1799,001,,,,10,57,20"></lb> |
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10559 ἐστι καὶ τὸ ΑΘ τῷ ΗΚ, τουτέστι τὸ ΣΝ τῷ ΝΠ· αἱ <lb n="1799,001,,,,10,57,21"></lb> |
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10560 ΜΝ, ΝΞ ἄρα δυνάμει εἰσὶν ἀσύμμετροι.</s> <s>καὶ ἐπεὶ σύμ- <pb></pb><lb n="1799,001,,,,10,57,22"></lb> |
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10561 μετρός ἐστιν ἡ ΑΕ τῇ ΑΒ μήκει, ῥητόν ἐστι τὸ ΑΚ· <lb n="1799,001,,,,10,57,23"></lb> |
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10562 καί ἐστιν ἴσον τοῖς ἀπὸ τῶν ΜΝ, ΝΞ· ῥητὸν ἄρα [ἐστὶ] <lb n="1799,001,,,,10,57,24"></lb> |
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10563 καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΜΝ, ΝΞ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,57,25"></lb> |
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10564 ἀσύμμετρός [ἐστιν] ἡ ΔΕ τῇ ΑΒ μήκει, τουτέστι τῇ ΕΚ, <lb n="1799,001,,,,10,57,26"></lb> |
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10565 ἀλλὰ ἡ ΔΕ σύμμετρός ἐστι τῇ ΕΖ, ἀσύμμετρος ἄρα <lb n="1799,001,,,,10,57,27"></lb> |
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10566 ἡ ΕΖ τῇ ΕΚ μήκει.</s> <s>αἱ ΕΚ, ΕΖ ἄρα ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,57,28"></lb> |
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10567 μόνον σύμμετροι· μέσον ἄρα τὸ ΛΕ, τουτέστι τὸ ΜΡ.</s> <lb n="1799,001,,,,10,57,29"></lb> |
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10568 <s>καὶ περιέχεται ὑπὸ τῶν ΜΝ, ΝΞ· μέσον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,10,57,30"></lb> |
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10569 ὑπὸ τῶν ΜΝ, ΝΞ.</s> <s>καὶ ῥητὸν τὸ [συγκείμενον] ἐκ τῶν <lb n="1799,001,,,,10,57,31"></lb> |
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10570 ἀπὸ τῶν ΜΝ, ΝΞ, καί εἰσιν ἀσύμμετροι αἱ ΜΝ, ΝΞ <lb n="1799,001,,,,10,57,32"></lb> |
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10571 δυνάμει.</s> <s>ἐὰν δὲ δύο εὐθεῖαι δυνάμει ἀσύμμετροι συν-<lb n="1799,001,,,,10,57,33"></lb> |
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10572 τεθῶσι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν <lb n="1799,001,,,,10,57,34"></lb> |
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10573 τετραγώνων ῥητόν, τὸ δ' ὑπ' αὐτῶν μέσον, ἡ ὅλη ἄλογός <lb n="1799,001,,,,10,57,35"></lb> |
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10574 ἐστιν, καλεῖται δὲ μείζων.</s> <lb n="1799,001,,,,10,57,36"></lb> |
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10575 <s>Ἡ ΜΞ ἄρα ἄλογός ἐστιν ἡ καλουμένη μείζων, καὶ <lb n="1799,001,,,,10,57,37"></lb> |
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10576 δύναται τὸ ΑΓ χωρίον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,57,38"></lb> |
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10577 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο <lb n="1799,001,,,,10,58,1"></lb> |
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10578 ὀνομάτων πέμπτης, ἡ τὸ χωρίον δυναμένη ἄλογός ἐστιν <lb n="1799,001,,,,10,58,2"></lb> |
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10579 ἡ καλουμένη ῥητὸν καὶ μέσον δυναμένη.</s> <lb n="1799,001,,,,10,58,3"></lb> |
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10580 <s>Χωρίον γὰρ τὸ ΑΓ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΒ <lb n="1799,001,,,,10,58,4"></lb> |
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10581 καὶ τῆς ἐκ δύο ὀνομάτων πέμπτης τῆς ΑΔ διῃρημένης <lb n="1799,001,,,,10,58,5"></lb> |
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10582 εἰς τὰ ὀνόματα κατὰ τὸ Ε, ὥστε τὸ μεῖζον ὄνομα εἶναι <lb n="1799,001,,,,10,58,6"></lb> |
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10583 τὸ ΑΕ· λέγω [δή], ὅτι ἡ τὸ ΑΓ χωρίον δυναμένη ἄλογός <lb n="1799,001,,,,10,58,7"></lb> |
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10584 ἐστιν ἡ καλουμένη ῥητὸν καὶ μέσον δυναμένη.</s> <s><pb></pb> <lb n="1799,001,,,,10,58,8"></lb> |
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10585 Κατεσκευάσθω γὰρ τὰ αὐτὰ τοῖς πρότερον δεδειγμέ-<lb n="1799,001,,,,10,58,9"></lb> |
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10586 νοις· φανερὸν δή, ὅτι ἡ τὸ ΑΓ χωρίον δυναμένη ἐστὶν <lb n="1799,001,,,,10,58,10"></lb> |
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10587 ἡ ΜΞ.</s> <s>δεικτέον δή, ὅτι ἡ ΜΞ ἐστιν ἡ ῥητὸν καὶ μέσον <lb n="1799,001,,,,10,58,11"></lb> |
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10588 δυναμένη.</s> <s>ἐπεὶ γὰρ ἀσύμμετρός ἐστιν ἡ ΑΗ τῇ ΗΕ, <lb n="1799,001,,,,10,58,12"></lb> |
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10589 ἀσύμμετρον ἄρα ἐστὶ καὶ τὸ ΑΘ τῷ ΘΕ, τουτέστι τὸ <lb n="1799,001,,,,10,58,13"></lb> |
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10590 ἀπὸ τῆς ΜΝ τῷ ἀπὸ τῆς ΝΞ· αἱ ΜΝ, ΝΞ ἄρα δυνάμει <lb n="1799,001,,,,10,58,14"></lb> |
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10591 εἰσὶν ἀσύμμετροι.</s> <s>καὶ ἐπεὶ ἡ ΑΔ ἐκ δύο ὀνομάτων ἐστὶ <lb n="1799,001,,,,10,58,15"></lb> |
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10592 πέμπτη, καί [ἐστιν] ἔλασσον αὐτῆς τμῆμα τὸ ΕΔ, <lb n="1799,001,,,,10,58,16"></lb> |
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10593 σύμμετρος ἄρα ἡ ΕΔ τῇ ΑΒ μήκει.</s> <s>ἀλλὰ ἡ ΑΕ τῇ ΕΔ <lb n="1799,001,,,,10,58,17"></lb> |
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10594 ἐστιν ἀσύμμετρος· καὶ ἡ ΑΒ ἄρα τῇ ΑΕ ἐστιν ἀσύμ-<lb n="1799,001,,,,10,58,18"></lb> |
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10595 μετρος μήκει.</s> <s>[αἱ ΒΑ, ΑΕ ῥηταί εἰσι δυνάμει μόνον <lb n="1799,001,,,,10,58,19"></lb> |
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10596 σύμμετροι.] μέσον ἄρα ἐστὶ τὸ ΑΚ, τουτέστι τὸ συγκεί-<lb n="1799,001,,,,10,58,20"></lb> |
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10597 μενον ἐκ τῶν ἀπὸ τῶν ΜΝ, ΝΞ.</s> <s>καὶ ἐπεὶ σύμμετρός <lb n="1799,001,,,,10,58,21"></lb> |
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10598 ἐστιν ἡ ΔΕ τῇ ΑΒ μήκει, τουτέστι τῇ ΕΚ, ἀλλὰ ἡ ΔΕ <pb></pb> <lb n="1799,001,,,,10,58,22"></lb> |
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10599 τῇ ΕΖ σύμμετρός ἐστιν, καὶ ἡ ΕΖ ἄρα τῇ ΕΚ σύμ-<lb n="1799,001,,,,10,58,23"></lb> |
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10600 μετρός ἐστιν.</s> <s>καὶ ῥητὴ ἡ ΕΚ· ῥητὸν ἄρα καὶ τὸ ΕΛ, <lb n="1799,001,,,,10,58,24"></lb> |
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10601 τουτέστι τὸ ΜΡ, τουτέστι τὸ ὑπὸ ΜΝΞ· αἱ ΜΝ, ΝΞ <lb n="1799,001,,,,10,58,25"></lb> |
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10602 ἄρα δυνάμει ἀσύμμετροί εἰσι ποιοῦσαι τὸ μὲν συγκεί-<lb n="1799,001,,,,10,58,26"></lb> |
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10603 μενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων μέσον, τὸ δ' ὑπ' <lb n="1799,001,,,,10,58,27"></lb> |
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10604 αὐτω̂ν ῥητόν.</s> <lb n="1799,001,,,,10,58,28"></lb> |
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10605 <s>Ἡ ΜΞ ἄρα ῥητὸν καὶ μέσον δυναμένη ἐστὶ καὶ δύναται <lb n="1799,001,,,,10,58,29"></lb> |
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10606 τὸ ΑΓ χωρίον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,58,30"></lb> |
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10607 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο <lb n="1799,001,,,,10,59,1"></lb> |
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10608 ὀνομάτων ἕκτης, ἡ τὸ χωρίον δυναμένη ἄλογός ἐστιν <lb n="1799,001,,,,10,59,2"></lb> |
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10609 ἡ καλουμένη δύο μέσα δυναμένη.</s> <lb n="1799,001,,,,10,59,3"></lb> |
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10610 <s>Χωρίον γὰρ τὸ ΑΒΓΔ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΒ <lb n="1799,001,,,,10,59,4"></lb> |
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10611 καὶ τῆς ἐκ δύο ὀνομάτων ἕκτης τῆς ΑΔ διῃρημένης <lb n="1799,001,,,,10,59,5"></lb> |
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10612 εἰς τὰ ὀνόματα κατὰ τὸ Ε, ὥστε τὸ μεῖζον ὄνομα εἶναι <lb n="1799,001,,,,10,59,6"></lb> |
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10613 τὸ ΑΕ· λέγω, ὅτι ἡ τὸ ΑΓ δυναμένη ἡ δύο μέσα δυναμένη <lb n="1799,001,,,,10,59,7"></lb> |
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10614 ἐστίν.</s> <lb n="1799,001,,,,10,59,8"></lb> |
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10615 <s>Κατεσκευάσθω [γὰρ] τὰ αὐτὰ τοῖς προδεδειγμένοις.</s> <lb n="1799,001,,,,10,59,9"></lb> |
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10616 <s>φανερὸν δή, ὅτι [ἡ] τὸ ΑΓ δυναμένη ἐστὶν ἡ ΜΞ, καὶ <lb n="1799,001,,,,10,59,10"></lb> |
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10617 ὅτι ἀσύμμετρός ἐστι ἡ ΜΝ τῇ ΝΞ δυνάμει.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,59,11"></lb> |
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10618 ἀσύμμετρός ἐστιν ἡ ΕΑ τῇ ΑΒ μήκει, αἱ ΕΑ, ΑΒ ἄρα <lb n="1799,001,,,,10,59,12"></lb> |
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10619 ῥηταί εἰσι δυνάμει μόνον σύμμετροι· μέσον ἄρα ἐστὶ <pb></pb> <lb n="1799,001,,,,10,59,13"></lb> |
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10620 τὸ ΑΚ, τουτέστι τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΜΝ, <lb n="1799,001,,,,10,59,14"></lb> |
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10621 ΝΞ.</s> <s>πάλιν, ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΕΔ τῇ ΑΒ μήκει, <lb n="1799,001,,,,10,59,15"></lb> |
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10622 ἀσύμμετρος ἄρα ἐστὶ καὶ ἡ ΖΕ τῇ ΕΚ· αἱ ΖΕ, ΕΚ <lb n="1799,001,,,,10,59,16"></lb> |
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10623 ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· μέσον ἄρα <lb n="1799,001,,,,10,59,17"></lb> |
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10624 ἐστὶ τὸ ΕΛ, τουτέστι τὸ ΜΡ, τουτέστι τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,59,18"></lb> |
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10625 ΜΝΞ.</s> <s>καὶ ἐπεὶ ἀσύμμετρος ἡ ΑΕ τῇ ΕΖ, καὶ τὸ ΑΚ <lb n="1799,001,,,,10,59,19"></lb> |
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10626 τῷ ΕΛ ἀσύμμετρόν ἐστιν.</s> <s>ἀλλὰ τὸ μὲν ΑΚ ἐστι τὸ <lb n="1799,001,,,,10,59,20"></lb> |
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10627 συγκείμενον ἐκ τῶν ἀπὸ τῶν ΜΝ, ΝΞ, τὸ δὲ ΕΛ ἐστι <lb n="1799,001,,,,10,59,21"></lb> |
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10628 τὸ ὑπὸ τῶν ΜΝΞ· ἀσύμμετρον ἄρα ἐστὶ τὸ συγκείμενον <lb n="1799,001,,,,10,59,22"></lb> |
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10629 ἐκ τῶν ἀπὸ τῶν ΜΝΞ τῷ ὑπὸ τῶν ΜΝΞ.</s> <s>καί ἐστι μέσον <lb n="1799,001,,,,10,59,23"></lb> |
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10630 ἑκάτερον αὐτῶν, καὶ αἱ ΜΝ, ΝΞ δυνάμει εἰσὶν ἀσύμ-<lb n="1799,001,,,,10,59,24"></lb> |
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10631 μετροι.</s> <lb n="1799,001,,,,10,59,25"></lb> |
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10632 <s>Ἡ ΜΞ ἄρα δύο μέσα δυναμένη ἐστὶ καὶ δύναται τὸ <lb n="1799,001,,,,10,59,26"></lb> |
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10633 ΑΓ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,59,27"></lb> |
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10634 [Λῆμμα <lb n="1799,001,,,,10,59,28"></lb> |
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10635 Ἐὰν εὐθεῖα γραμμὴ τμηθῇ εἰς ἄνισα, τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,59,29"></lb> |
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10636 ἀνίσων τετράγωνα μείζονά ἐστι τοῦ δὶς ὑπὸ τῶν ἀνίσων <lb n="1799,001,,,,10,59,30"></lb> |
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10637 περιεχομένου ὀρθογωνίου.</s> <lb n="1799,001,,,,10,59,31"></lb> |
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10638 <s>Ἔστω εὐθεῖα ἡ ΑΒ καὶ τετμήσθω εἰς ἄνισα κατὰ <lb n="1799,001,,,,10,59,32"></lb> |
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10639 τὸ Γ, καὶ ἔστω μείζων ἡ ΑΓ· λέγω, ὅτι τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,59,33"></lb> |
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10640 ΑΓ, ΓΒ μείζονά ἐστι τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ.</s> <lb n="1799,001,,,,10,59,34"></lb> |
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10641 <s>Τετμήσθω γὰρ ἡ ΑΒ δίχα κατὰ τὸ Δ.</s> <s>ἐπεὶ <lb n="1799,001,,,,10,59,35"></lb> |
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10642 οὖν εὐθεῖα γραμμὴ τέτμηται εἰς μὲν ἴσα κατὰ <lb n="1799,001,,,,10,59,36"></lb> |
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10643 τὸ Δ, εἰς δὲ ἄνισα κατὰ τὸ Γ, τὸ ἄρα ὑπὸ τῶν <lb n="1799,001,,,,10,59,37"></lb> |
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10644 ΑΓ, ΓΒ μετὰ τοῦ ἀπὸ ΓΔ ἴσον ἐστὶ τῷ ἀπὸ ΑΔ· <lb n="1799,001,,,,10,59,38"></lb> |
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10645 ὥστε τὸ ὑπὸ τῶν ΑΓ, ΓΒ ἔλαττόν ἐστι τοῦ ἀπὸ <lb n="1799,001,,,,10,59,39"></lb> |
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10646 ΑΔ· τὸ ἄρα δὶς ὑπὸ τῶν ΑΓ, ΓΒ ἔλαττον ἢ δι-<lb n="1799,001,,,,10,59,40"></lb> |
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10647 πλάσιόν ἐστι τοῦ ἀπὸ ΑΔ.</s> <s>ἀλλὰ τὰ ἀπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,59,41"></lb> |
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10648 ΓΒ διπλάσιά [ἐστι] τῶν ἀπὸ τῶν ΑΔ, ΔΓ· τὰ ἄρα <lb n="1799,001,,,,10,59,42"></lb> |
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10649 ἀπὸ τῶν ΑΓ, ΓΒ μείζονά ἐστι τοῦ δὶς ὑπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,59,43"></lb> |
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10650 ΓΒ· ὅπερ ἔδει δεῖξαι.] <lb n="1799,001,,,,10,59,44"></lb> |
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10651 Τὸ ἀπὸ τῆς ἐκ δύο ὀνομάτων παρὰ ῥητὴν παραβαλλό-<lb n="1799,001,,,,10,60,1"></lb> |
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10652 μενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων πρώτην.</s> <lb n="1799,001,,,,10,60,2"></lb> |
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10653 <s>Ἔστω ἐκ δύο ὀνομάτων ἡ ΑΒ διῃρημένη εἰς τὰ ὀνόματα <lb n="1799,001,,,,10,60,3"></lb> |
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10654 κατὰ τὸ Γ, ὥστε τὸ μεῖζον ὄνομα εἶναι τὸ ΑΓ, καὶ <pb></pb> <lb n="1799,001,,,,10,60,4"></lb> |
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10655 ἐκκείσθω ῥητὴ ἡ ΔΕ, καὶ τῷ <lb n="1799,001,,,,10,60,5"></lb> |
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10656 ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν <lb n="1799,001,,,,10,60,6"></lb> |
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10657 ΔΕ παραβεβλήσθω τὸ ΔΕΖΗ <lb n="1799,001,,,,10,60,7"></lb> |
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10658 πλάτος ποιοῦν τὴν ΔΗ· λέγω, <lb n="1799,001,,,,10,60,8"></lb> |
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10659 ὅτι ἡ ΔΗ ἐκ δύο ὀνομάτων ἐστὶ <lb n="1799,001,,,,10,60,9"></lb> |
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10660 πρώτη.</s> <lb n="1799,001,,,,10,60,10"></lb> |
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10661 <s>Παραβεβλήσθω γὰρ παρὰ τὴν ΔΕ τῷ μὲν ἀπὸ τῆς ΑΓ <lb n="1799,001,,,,10,60,11"></lb> |
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10662 ἴσον τὸ ΔΘ, τῷ δὲ ἀπὸ τῆς ΒΓ ἴσον τὸ ΚΛ· λοιπὸν <lb n="1799,001,,,,10,60,12"></lb> |
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10663 ἄρα τὸ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ἴσον ἐστὶ τῷ ΜΖ.</s> <s>τετμήσθω <lb n="1799,001,,,,10,60,13"></lb> |
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10664 ἡ ΜΗ δίχα κατὰ τὸ Ν, καὶ παράλληλος ἤχθω ἡ ΝΞ <lb n="1799,001,,,,10,60,14"></lb> |
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10665 [ἑκατέρᾳ τῶν ΜΛ, ΗΖ].</s> <s>ἑκάτερον ἄρα τῶν ΜΞ, ΝΖ <lb n="1799,001,,,,10,60,15"></lb> |
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10666 ἴσον ἐστὶ τῷ ἅπαξ ὑπὸ τῶν ΑΓΒ.</s> <s>καὶ ἐπεὶ ἐκ δύο ὀνο-<lb n="1799,001,,,,10,60,16"></lb> |
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10667 μάτων ἐστὶν ἡ ΑΒ διῃρημένη εἰς τὰ ὀνόματα κατὰ τὸ <lb n="1799,001,,,,10,60,17"></lb> |
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10668 Γ, αἱ ΑΓ, ΓΒ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· <lb n="1799,001,,,,10,60,18"></lb> |
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10669 τὰ ἄρα ἀπὸ τῶν ΑΓ, ΓΒ ῥητά ἐστι καὶ σύμμετρα <lb n="1799,001,,,,10,60,19"></lb> |
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10670 ἀλλήλοις· ὥστε καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν <lb n="1799,001,,,,10,60,20"></lb> |
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10671 ΑΓ, ΓΒ [σύμμετρόν ἐστι τοῖς ἀπὸ τῶν ΑΓ, ΓΒ· <lb n="1799,001,,,,10,60,21"></lb> |
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10672 ῥητὸν ἄρα ἐστὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,60,22"></lb> |
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10673 ΓΒ].</s> <s>καί ἐστιν ἴσον τῷ ΔΛ· ῥητὸν ἄρα ἐστὶ τὸ ΔΛ.</s> <s>καὶ <lb n="1799,001,,,,10,60,23"></lb> |
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10674 παρὰ ῥητὴν τὴν ΔΕ παράκειται· ῥητὴ ἄρα ἐστὶν ἡ ΔΜ <lb n="1799,001,,,,10,60,24"></lb> |
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10675 καὶ σύμμετρος τῇ ΔΕ μήκει.</s> <s>πάλιν, ἐπεὶ αἱ ΑΓ, ΓΒ <lb n="1799,001,,,,10,60,25"></lb> |
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10676 ῥηταί εἰσι δυνάμει μόνον σύμμετροι, μέσον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,10,60,26"></lb> |
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10677 δὶς ὑπὸ τῶν ΑΓ, ΓΒ, τουτέστι τὸ ΜΖ.</s> <s>καὶ παρὰ ῥητὴν <lb n="1799,001,,,,10,60,27"></lb> |
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10678 τὴν ΜΛ παράκειται· ῥητὴ ἄρα καὶ ἡ ΜΗ ἐστι καὶ ἀσύμ- <pb></pb><lb n="1799,001,,,,10,60,28"></lb> |
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10679 μετρος τῇ ΜΛ, τουτέστι τῇ ΔΕ, μήκει.</s> <s>ἔστι δὲ καὶ <lb n="1799,001,,,,10,60,29"></lb> |
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10680 ἡ ΜΔ ῥητὴ καὶ τῇ ΔΕ μήκει σύμμετρος· ἀσύμμετρος <lb n="1799,001,,,,10,60,30"></lb> |
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10681 ἄρα ἐστὶν ἡ ΔΜ τῇ ΜΗ μήκει.</s> <s>καί εἰσι ῥηταί· αἱ ΔΜ, <lb n="1799,001,,,,10,60,31"></lb> |
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10682 ΜΗ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἐκ δύο <lb n="1799,001,,,,10,60,32"></lb> |
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10683 ἄρα ὀνομάτων ἐστὶν ἡ ΔΗ.</s> <lb n="1799,001,,,,10,60,33"></lb> |
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10684 <s>Δεικτέον δή, ὅτι καὶ πρώτη.</s> <lb n="1799,001,,,,10,60,34"></lb> |
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10685 <s>Ἐπεὶ τῶν ἀπὸ τῶν ΑΓ, ΓΒ μέσον ἀνάλογόν ἐστι τὸ <lb n="1799,001,,,,10,60,35"></lb> |
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10686 ὑπὸ τῶν ΑΓΒ, καὶ τῶν ΔΘ, ΚΛ ἄρα μέσον ἀνάλογόν <lb n="1799,001,,,,10,60,36"></lb> |
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10687 ἐστι τὸ ΜΞ.</s> <s>ἔστιν ἄρα ὡς τὸ ΔΘ πρὸς τὸ ΜΞ, οὕτως <lb n="1799,001,,,,10,60,37"></lb> |
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10688 τὸ ΜΞ πρὸς τὸ ΚΛ, τουτέστιν ὡς ἡ ΔΚ πρὸς τὴν ΜΝ, <lb n="1799,001,,,,10,60,38"></lb> |
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10689 ἡ ΜΝ πρὸς τὴν ΜΚ· τὸ ἄρα ὑπὸ τῶν ΔΚ, ΚΜ ἴσον <lb n="1799,001,,,,10,60,39"></lb> |
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10690 ἐστὶ τῷ ἀπὸ τῆς ΜΝ.</s> <s>καὶ ἐπεὶ σύμμετρόν ἐστι τὸ ἀπὸ <lb n="1799,001,,,,10,60,40"></lb> |
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10691 τῆς ΑΓ τῷ ἀπὸ τῆς ΓΒ, σύμμετρόν ἐστι καὶ τὸ ΔΘ <lb n="1799,001,,,,10,60,41"></lb> |
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10692 τῷ ΚΛ· ὥστε καὶ ἡ ΔΚ τῇ ΚΜ σύμμετρός ἐστιν.</s> <s>καὶ <lb n="1799,001,,,,10,60,42"></lb> |
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10693 ἐπεὶ μείζονά ἐστι τὰ ἀπὸ τῶν ΑΓ, ΓΒ τοῦ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,60,43"></lb> |
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10694 ΑΓ, ΓΒ, μεῖζον ἄρα καὶ τὸ ΔΛ τοῦ ΜΖ· ὥστε καὶ <lb n="1799,001,,,,10,60,44"></lb> |
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10695 ἡ ΔΜ τῆς ΜΗ μείζων ἐστίν.</s> <s>καί ἐστιν ἴσον τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,60,45"></lb> |
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10696 ΔΚ, ΚΜ τῷ ἀπὸ τῆς ΜΝ, τουτέστι τῷ τετάρτῳ τοῦ <lb n="1799,001,,,,10,60,46"></lb> |
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10697 ἀπὸ τῆς ΜΗ, καὶ σύμμετρος ἡ ΔΚ τῇ ΚΜ.</s> <s>ἐὰν δὲ ὦσι <lb n="1799,001,,,,10,60,47"></lb> |
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10698 δύο εὐθεῖαι ἄνισοι, τῷ δὲ τετάρτῳ μέρει τοῦ ἀπὸ τῆς <lb n="1799,001,,,,10,60,48"></lb> |
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10699 ἐλάσσονος ἴσον παρὰ τὴν μείζονα παραβληθῇ ἐλλεῖπον <lb n="1799,001,,,,10,60,49"></lb> |
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10700 εἴδει τετραγώνῳ καὶ εἰς σύμμετρα αὐτὴν διαιρῇ, ἡ <lb n="1799,001,,,,10,60,50"></lb> |
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10701 μείζων τῆς ἐλάσσονος μεῖζον δύναται τῷ ἀπὸ συμ-<lb n="1799,001,,,,10,60,51"></lb> |
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10702 μέτρου ἑαυτῇ· ἡ ΔΜ ἄρα τῆς ΜΗ μεῖζον δύναται τῷ <pb></pb> <lb n="1799,001,,,,10,60,52"></lb> |
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10703 ἀπὸ συμμέτρου ἑαυτῇ.</s> <s>καί εἰσι ῥηταὶ αἱ ΔΜ, ΜΗ, καὶ <lb n="1799,001,,,,10,60,53"></lb> |
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10704 ἡ ΔΜ μεῖζον ὄνομα σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ <lb n="1799,001,,,,10,60,54"></lb> |
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10705 τῇ ΔΕ μήκει.</s> <lb n="1799,001,,,,10,60,55"></lb> |
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10706 <s>Ἡ ΔΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ πρώτη· ὅπερ ἔδει <lb n="1799,001,,,,10,60,56"></lb> |
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10707 δεῖξαι.</s> <lb n="1799,001,,,,10,60,57"></lb> |
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10708 <s>Τὸ ἀπὸ τῆς ἐκ δύο μέσων πρώτης παρὰ ῥητὴν παρα-<lb n="1799,001,,,,10,61,1"></lb> |
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10709 βαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων δευτέραν.</s> <lb n="1799,001,,,,10,61,2"></lb> |
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10710 <s>Ἔστω ἐκ δύο μέσων πρώτη ἡ ΑΒ διῃρημένη εἰς τὰς <lb n="1799,001,,,,10,61,3"></lb> |
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10711 μέσας κατὰ τὸ Γ, ὧν μείζων ἡ <lb n="1799,001,,,,10,61,4"></lb> |
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10712 ΑΓ, καὶ ἐκκείσθω ῥητὴ ἡ ΔΕ, <lb n="1799,001,,,,10,61,5"></lb> |
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10713 καὶ παραβεβλήσθω παρὰ τὴν ΔΕ <lb n="1799,001,,,,10,61,6"></lb> |
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10714 τῷ ἀπὸ τῆς ΑΒ ἴσον παραλλη-<lb n="1799,001,,,,10,61,7"></lb> |
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10715 λόγραμμον τὸ ΔΖ πλάτος ποιοῦν <lb n="1799,001,,,,10,61,8"></lb> |
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10716 τὴν ΔΗ· λέγω, ὅτι ἡ ΔΗ ἐκ δύο <lb n="1799,001,,,,10,61,9"></lb> |
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10717 ὀνομάτων ἐστὶ δευτέρα.</s> <lb n="1799,001,,,,10,61,10"></lb> |
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10718 <s>Κατεσκευάσθω γὰρ τὰ αὐτὰ τοῖς πρὸ τούτου.</s> <s>καὶ <lb n="1799,001,,,,10,61,11"></lb> |
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10719 ἐπεὶ ἡ ΑΒ ἐκ δύο μέσων ἐστὶ πρώτη διῃρημένη κατὰ <lb n="1799,001,,,,10,61,12"></lb> |
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10720 τὸ Γ, αἱ ΑΓ, ΓΒ ἄρα μέσαι εἰσὶ δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,61,13"></lb> |
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10721 μετροι ῥητὸν περιέχουσαι· ὥστε καὶ τὰ ἀπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,61,14"></lb> |
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10722 ΓΒ μέσα ἐστίν.</s> <s>μέσον ἄρα ἐστὶ τὸ ΔΛ.</s> <s>καὶ παρὰ ῥητὴν <lb n="1799,001,,,,10,61,15"></lb> |
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10723 τὴν ΔΕ παραβέβληται· ῥητὴ ἄρα ἐστίν ἡ ΜΔ καὶ <lb n="1799,001,,,,10,61,16"></lb> |
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10724 ἀσύμμετρος τῇ ΔΕ μήκει.</s> <s>πάλιν, ἐπεὶ ῥητόν ἐστι τὸ <lb n="1799,001,,,,10,61,17"></lb> |
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10725 δὶς ὑπὸ τῶν ΑΓ, ΓΒ, ῥητόν ἐστι καὶ τὸ ΜΖ.</s> <s>καὶ παρὰ <lb n="1799,001,,,,10,61,18"></lb> |
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10726 ῥητὴν τὴν ΜΛ παράκειται· ῥητὴ ἄρα [ἐστὶ] καὶ ἡ ΜΗ <pb></pb> <lb n="1799,001,,,,10,61,19"></lb> |
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10727 καὶ μήκει σύμμετρος τῇ ΜΛ, τουτέστι τῇ ΔΕ· ἀσύμ-<lb n="1799,001,,,,10,61,20"></lb> |
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10728 μετρος ἄρα ἐστὶν ἡ ΔΜ τῇ ΜΗ μήκει.</s> <s>καί εἰσι ῥηταί· <lb n="1799,001,,,,10,61,21"></lb> |
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10729 αἱ ΔΜ, ΜΗ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· <lb n="1799,001,,,,10,61,22"></lb> |
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10730 ἐκ δύο ἄρα ὀνομάτων ἐστὶν ἡ ΔΗ.</s> <lb n="1799,001,,,,10,61,23"></lb> |
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10731 <s>Δεικτέον δή, ὅτι καὶ δευτέρα.</s> <lb n="1799,001,,,,10,61,24"></lb> |
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10732 <s>Ἐπεὶ γὰρ τὰ ἀπὸ τῶν ΑΓ, ΓΒ μείζονά ἐστι τοῦ δὶς <lb n="1799,001,,,,10,61,25"></lb> |
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10733 ὑπὸ τῶν ΑΓ, ΓΒ, μεῖζον ἄρα καὶ τὸ ΔΛ τοῦ ΜΖ· <lb n="1799,001,,,,10,61,26"></lb> |
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10734 ὥστε καὶ ἡ ΔΜ τῆς ΜΗ.</s> <s>καὶ ἐπεὶ σύμμετρόν ἐστι τὸ <lb n="1799,001,,,,10,61,27"></lb> |
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10735 ἀπὸ τῆς ΑΓ τῷ ἀπὸ τῆς ΓΒ, σύμμετρόν ἐστι καὶ τὸ <lb n="1799,001,,,,10,61,28"></lb> |
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10736 ΔΘ τῷ ΚΛ· ὥστε καὶ ἡ ΔΚ τῇ ΚΜ σύμμετρός ἐστιν.</s> <lb n="1799,001,,,,10,61,29"></lb> |
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10737 <s>καί ἐστι τὸ ὑπὸ τῶν ΔΚΜ ἴσον τῷ ἀπὸ τῆς ΜΝ· ἡ ΔΜ <lb n="1799,001,,,,10,61,30"></lb> |
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10738 ἄρα τῆς ΜΗ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <lb n="1799,001,,,,10,61,31"></lb> |
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10739 <s>καί ἐστιν ἡ ΜΗ σύμμετρος τῇ ΔΕ μήκει.</s> <lb n="1799,001,,,,10,61,32"></lb> |
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10740 <s>Ἡ ΔΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ δευτέρα.</s> <lb n="1799,001,,,,10,61,33"></lb> |
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10741 <s>Τὸ ἀπὸ τῆς ἐκ δύο μέσων δευτέρας παρὰ ῥητὴν παρα-<lb n="1799,001,,,,10,62,1"></lb> |
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10742 βαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων τρίτην.</s> <lb n="1799,001,,,,10,62,2"></lb> |
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10743 <s>Ἔστω ἐκ δύο μέσων δευτέρα <lb n="1799,001,,,,10,62,3"></lb> |
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10744 ἡ ΑΒ διῃρημένη εἰς τὰς μέσας <lb n="1799,001,,,,10,62,4"></lb> |
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10745 κατὰ τὸ Γ, ὥστε τὸ μεῖζον τμῆμα <lb n="1799,001,,,,10,62,5"></lb> |
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10746 εἶναι τὸ ΑΓ, ῥητὴ δέ τις ἔστω <lb n="1799,001,,,,10,62,6"></lb> |
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10747 ἡ ΔΕ, καὶ παρὰ τὴν ΔΕ τῷ ἀπὸ <pb></pb> <lb n="1799,001,,,,10,62,7"></lb> |
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10748 τῆς ΑΒ ἴσον παραλληλόγραμμον παραβεβλήσθω τὸ ΔΖ <lb n="1799,001,,,,10,62,8"></lb> |
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10749 πλάτος ποιοῦν τὴν ΔΗ· λέγω, ὅτι ἡ ΔΗ ἐκ δύο ὀνομάτων <lb n="1799,001,,,,10,62,9"></lb> |
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10750 ἐστὶ τρίτη.</s> <lb n="1799,001,,,,10,62,10"></lb> |
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10751 <s>Κατεσκευάσθω τὰ αὐτὰ τοῖς προδεδειγμένοις.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,62,11"></lb> |
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10752 ἐκ δύο μέσων δευτέρα ἐστὶν ἡ ΑΒ διῃρημένη κατὰ τὸ Γ, <lb n="1799,001,,,,10,62,12"></lb> |
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10753 αἱ ΑΓ, ΓΒ ἄρα μέσαι εἰσὶ δυνάμει μόνον σύμμετροι <lb n="1799,001,,,,10,62,13"></lb> |
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10754 μέσον περιέχουσαι· ὥστε καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,62,14"></lb> |
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10755 τῶν ΑΓ, ΓΒ μέσον ἐστίν.</s> <s>καί ἐστιν ἴσον τῷ ΔΛ· μέσον <lb n="1799,001,,,,10,62,15"></lb> |
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10756 ἄρα καὶ τὸ ΔΛ.</s> <s>καὶ παράκειται παρὰ ῥητὴν τὴν ΔΕ· <lb n="1799,001,,,,10,62,16"></lb> |
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10757 ῥητὴ ἄρα ἐστὶ καὶ ἡ ΜΔ καὶ ἀσύμμετρος τῇ ΔΕ μήκει.</s> <lb n="1799,001,,,,10,62,17"></lb> |
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10758 <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΜΗ ῥητή ἐστι καὶ ἀσύμμετρος τῇ <lb n="1799,001,,,,10,62,18"></lb> |
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10759 ΜΛ, τουτέστι τῇ ΔΕ, μήκει· ῥητὴ ἄρα ἐστὶν ἑκατέρα <lb n="1799,001,,,,10,62,19"></lb> |
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10760 τῶν ΔΜ, ΜΗ καὶ ἀσύμμετρος τῇ ΔΕ μήκει.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,62,20"></lb> |
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10761 ἀσύμμετρός ἐστιν ἡ ΑΓ τῇ ΓΒ μήκει, ὡς δὲ ἡ ΑΓ πρὸς <lb n="1799,001,,,,10,62,21"></lb> |
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10762 τὴν ΓΒ, οὕτως τὸ ἀπὸ τῆς ΑΓ πρὸς τὸ ὑπὸ τῶν ΑΓΒ, <lb n="1799,001,,,,10,62,22"></lb> |
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10763 ἀσύμμετρον ἄρα καὶ τὸ ἀπὸ τῆς ΑΓ τῷ ὑπὸ τῶν ΑΓΒ.</s> <lb n="1799,001,,,,10,62,23"></lb> |
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10764 <s>ὥστε καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, ΓΒ τῷ <lb n="1799,001,,,,10,62,24"></lb> |
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10765 δὶς ὑπὸ τῶν ΑΓΒ ἀσύμμετρόν ἐστιν, τουτέστι τὸ ΔΛ <lb n="1799,001,,,,10,62,25"></lb> |
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10766 τῷ ΜΖ· ὥστε καὶ ἡ ΔΜ τῇ ΜΗ ἀσύμμετρός ἐστιν.</s> <s>καί <lb n="1799,001,,,,10,62,26"></lb> |
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10767 εἰσι ῥηταί· ἐκ δύο ἄρα ὀνομάτων ἐστὶν ἡ ΔΗ.</s> <lb n="1799,001,,,,10,62,27"></lb> |
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10768 <s>Δεικτέον [δή], ὅτι καὶ τρίτη.</s> <lb n="1799,001,,,,10,62,28"></lb> |
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10769 <s>Ὁμοίως δὴ τοῖς προτέροις ἐπιλογιούμεθα, ὅτι μείζων <pb></pb> <lb n="1799,001,,,,10,62,29"></lb> |
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10770 ἐστὶν ἡ ΔΜ τῆς ΜΗ, καὶ σύμμετρος ἡ ΔΚ τῇ ΚΜ.</s> <lb n="1799,001,,,,10,62,30"></lb> |
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10771 <s>καί ἐστι τὸ ὑπὸ τῶν ΔΚΜ ἴσον τῷ ἀπὸ τῆς ΜΝ· ἡ ΔΜ <lb n="1799,001,,,,10,62,31"></lb> |
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10772 ἄρα τῆς ΜΗ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <s>καὶ <lb n="1799,001,,,,10,62,32"></lb> |
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10773 οὐδετέρα τῶν ΔΜ, ΜΗ σύμμετρός ἐστι τῇ ΔΕ μήκει.</s> <lb n="1799,001,,,,10,62,33"></lb> |
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10774 <s>Ἡ ΔΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ τρίτη· ὅπερ ἔδει <lb n="1799,001,,,,10,62,34"></lb> |
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10775 δεῖξαι.</s> <lb n="1799,001,,,,10,62,35"></lb> |
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10776 <s>Τὸ ἀπὸ τῆς μείζονος παρὰ ῥητὴν παραβαλλόμενον <lb n="1799,001,,,,10,63,1"></lb> |
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10777 πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων τετάρτην.</s> <lb n="1799,001,,,,10,63,2"></lb> |
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10778 <s>Ἔστω μείζων ἡ ΑΒ διῃρημένη κατὰ τὸ Γ, ὥστε μείζονα <lb n="1799,001,,,,10,63,3"></lb> |
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10779 εἶναι τὴν ΑΓ τῆς ΓΒ, ῥητὴ δὲ ἡ ΔΕ, καὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,63,4"></lb> |
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10780 ΑΒ ἴσον παρὰ τὴν ΔΕ παραβε- <lb n="1799,001,,,,10,63,5"></lb> |
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10781 βλήσθω τὸ ΔΖ παραλληλόγραμ-<lb n="1799,001,,,,10,63,6"></lb> |
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10782 μον πλάτος ποιοῦν τὴν ΔΗ· λέγω, <lb n="1799,001,,,,10,63,7"></lb> |
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10783 ὅτι ἡ ΔΗ ἐκ δύο ὀνομάτων ἐστὶ <lb n="1799,001,,,,10,63,8"></lb> |
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10784 τετάρτη.</s> <lb n="1799,001,,,,10,63,9"></lb> |
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10785 <s>Κατεσκευάσθω τὰ αὐτὰ τοῖς <lb n="1799,001,,,,10,63,10"></lb> |
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10786 προδεδειγμένοις.</s> <s>καὶ ἐπεὶ μείζων <lb n="1799,001,,,,10,63,11"></lb> |
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10787 ἐστὶν ἡ ΑΒ διῃρημένη κατὰ τὸ Γ, αἱ ΑΓ, ΓΒ δυνά-<lb n="1799,001,,,,10,63,12"></lb> |
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10788 μει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ <lb n="1799,001,,,,10,63,13"></lb> |
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10789 τῶν ἀπ' αὐτῶν τετραγώνων ῥητόν, τὸ δὲ ὑπ' αὐτῶν <lb n="1799,001,,,,10,63,14"></lb> |
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10790 μέσον.</s> <s>ἐπεὶ οὖν ῥητόν ἐστι τὸ συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,63,15"></lb> |
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10791 τῶν ΑΓ, ΓΒ, ῥητὸν ἄρα ἐστὶ τὸ ΔΛ· ῥητὴ ἄρα καὶ ἡ ΔΜ <pb></pb> <lb n="1799,001,,,,10,63,16"></lb> |
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10792 καὶ σύμμετρος τῇ ΔΕ μήκει.</s> <s>πάλιν, ἐπεὶ μέσον ἐστὶ τὸ <lb n="1799,001,,,,10,63,17"></lb> |
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10793 δὶς ὑπὸ τῶν ΑΓ, ΓΒ, τουτέστι τὸ ΜΖ, καὶ παρὰ ῥητήν <lb n="1799,001,,,,10,63,18"></lb> |
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10794 ἐστι τὴν ΜΛ, ῥητὴ ἄρα ἐστὶ καὶ ἡ ΜΗ καὶ ἀσύμμετρος <lb n="1799,001,,,,10,63,19"></lb> |
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10795 τῇ ΔΕ μήκει· ἀσύμμετρος ἄρα ἐστὶ καὶ ἡ ΔΜ τῇ ΜΗ <lb n="1799,001,,,,10,63,20"></lb> |
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10796 μήκει.</s> <s>αἱ ΔΜ, ΜΗ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,63,21"></lb> |
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10797 μετροι· ἐκ δύο ἄρα ὀνομάτων ἐστὶν ἡ ΔΗ.</s> <lb n="1799,001,,,,10,63,22"></lb> |
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10798 <s>Δεικτέον [δή], ὅτι καὶ τετάρτη.</s> <lb n="1799,001,,,,10,63,23"></lb> |
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10799 <s>Ὁμοίως δὴ δείξομεν τοῖς πρότερον, ὅτι μείζων ἐστὶν <lb n="1799,001,,,,10,63,24"></lb> |
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10800 ἡ ΔΜ τῆς ΜΗ, καὶ ὅτι τὸ ὑπὸ ΔΚΜ ἴσον ἐστὶ τῷ ἀπὸ <lb n="1799,001,,,,10,63,25"></lb> |
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10801 τῆς ΜΝ.</s> <s>ἐπεὶ οὖν ἀσύμμετρόν ἐστι τὸ ἀπὸ τῆς ΑΓ τῷ <lb n="1799,001,,,,10,63,26"></lb> |
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10802 ἀπὸ τῆς ΓΒ, ἀσύμμετρον ἄρα ἐστὶ καὶ τὸ ΔΘ τῷ ΚΛ· <lb n="1799,001,,,,10,63,27"></lb> |
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10803 ὥστε ἀσύμμετρος καὶ ἡ ΔΚ τῇ ΚΜ ἐστιν.</s> <s>ἐὰν δὲ ὦσι <lb n="1799,001,,,,10,63,28"></lb> |
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10804 δύο εὐθεῖαι ἄνισοι, τῷ δὲ τετάρτῳ μέρει τοῦ ἀπὸ τῆς <lb n="1799,001,,,,10,63,29"></lb> |
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10805 ἐλάσσονος ἴσον παραλληλόγραμμον παρὰ τὴν μείζονα <lb n="1799,001,,,,10,63,30"></lb> |
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10806 παραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ καὶ εἰς ἀσύμμετρα <lb n="1799,001,,,,10,63,31"></lb> |
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10807 αὐτὴν διαιρῇ, ἡ μείζων τῆς ἐλάσσονος μεῖζον δυνήσεται <lb n="1799,001,,,,10,63,32"></lb> |
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10808 τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει· ἡ ΔΜ ἄρα τῆς ΜΗ <lb n="1799,001,,,,10,63,33"></lb> |
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10809 μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <s>καί εἰσιν αἱ <lb n="1799,001,,,,10,63,34"></lb> |
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10810 ΔΜ, ΜΗ ῥηταὶ δυνάμει μόνον σύμμετροι, καὶ ἡ ΔΜ <lb n="1799,001,,,,10,63,35"></lb> |
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10811 σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ τῇ ΔΕ.</s> <lb n="1799,001,,,,10,63,36"></lb> |
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10812 <s>Ἡ ΔΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ τετάρτη· ὅπερ ἔδει <lb n="1799,001,,,,10,63,37"></lb> |
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10813 δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,63,38"></lb> |
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10814 Τὸ ἀπὸ τῆς ῥητὸν καὶ μέσον δυναμένης παρὰ ῥητὴν <lb n="1799,001,,,,10,64,1"></lb> |
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10815 παραβαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων <lb n="1799,001,,,,10,64,2"></lb> |
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10816 πέμπτην.</s> <lb n="1799,001,,,,10,64,3"></lb> |
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10817 <s>Ἔστω ῥητὸν καὶ μέσον δυναμένη ἡ ΑΒ διῃρημένη εἰς <lb n="1799,001,,,,10,64,4"></lb> |
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10818 τὰς εὐθείας κατὰ τὸ Γ, ὥστε μείζονα εἶναι τὴν ΑΓ, καὶ <lb n="1799,001,,,,10,64,5"></lb> |
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10819 ἐκκείσθω ῥητὴ ἡ ΔΕ, καὶ τῷ ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν <lb n="1799,001,,,,10,64,6"></lb> |
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10820 ΔΕ παραβεβλήσθω τὸ ΔΖ πλά- <lb n="1799,001,,,,10,64,7"></lb> |
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10821 τος ποιοῦν τὴν ΔΗ· λέγω, ὅτι <lb n="1799,001,,,,10,64,8"></lb> |
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10822 ἡ ΔΗ ἐκ δύο ὀνομάτων ἐστὶ <lb n="1799,001,,,,10,64,9"></lb> |
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10823 πέμπτη.</s> <lb n="1799,001,,,,10,64,10"></lb> |
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10824 <s>Κατεσκευάσθω τὰ αὐτὰ τοῖς <lb n="1799,001,,,,10,64,11"></lb> |
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10825 πρὸ τούτου.</s> <s>ἐπεὶ οὖν ῥητὸν καὶ <lb n="1799,001,,,,10,64,12"></lb> |
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10826 μέσον δυναμένη ἐστὶν ἡ ΑΒ δι-<lb n="1799,001,,,,10,64,13"></lb> |
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10827 ῃρημένη κατὰ τὸ Γ, αἱ ΑΓ, ΓΒ ἄρα δυνάμει εἰσὶν ἀσύμ-<lb n="1799,001,,,,10,64,14"></lb> |
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10828 μετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν <lb n="1799,001,,,,10,64,15"></lb> |
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10829 τετραγώνων μέσον, τὸ δ' ὑπ' αὐτῶν ῥητόν.</s> <s>ἐπεὶ οὖν μέσον <lb n="1799,001,,,,10,64,16"></lb> |
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10830 ἐστὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, ΓΒ, μέσον ἄρα <lb n="1799,001,,,,10,64,17"></lb> |
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10831 ἐστὶ τὸ ΔΛ· ὥστε ῥητή ἐστιν ἡ ΔΜ καὶ μήκει ἀσύμ-<lb n="1799,001,,,,10,64,18"></lb> |
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10832 μετρος τῇ ΔΕ.</s> <s>πάλιν, ἐπεὶ ῥητόν ἐστι τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,64,19"></lb> |
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10833 ΑΓΒ, τουτέστι τὸ ΜΖ, ῥητὴ ἄρα ἡ ΜΗ καὶ σύμμετρος <lb n="1799,001,,,,10,64,20"></lb> |
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10834 τῇ ΔΕ.</s> <s>ἀσύμμετρος ἄρα ἡ ΔΜ τῇ ΜΗ· αἱ ΔΜ, ΜΗ <lb n="1799,001,,,,10,64,21"></lb> |
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10835 ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἐκ δύο ἄρα <lb n="1799,001,,,,10,64,22"></lb> |
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10836 ὀνομάτων ἐστὶν ἡ ΔΗ.</s> <lb n="1799,001,,,,10,64,23"></lb> |
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10837 <s>Λέγω δή, ὅτι καὶ πέμπτη.</s> <lb n="1799,001,,,,10,64,24"></lb> |
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10838 <s>Ὁμοίως γὰρ δειχθήσεται, ὅτι τὸ ὑπὸ τῶν ΔΚΜ ἴσον <pb></pb> <lb n="1799,001,,,,10,64,25"></lb> |
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10839 ἐστὶ τῷ ἀπὸ τῆς ΜΝ, καὶ ἀσύμμετρος ἡ ΔΚ τῇ ΚΜ <lb n="1799,001,,,,10,64,26"></lb> |
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10840 μήκει· ἡ ΔΜ ἄρα τῆς ΜΗ μεῖζον δύναται τῷ ἀπὸ <lb n="1799,001,,,,10,64,27"></lb> |
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10841 ἀσυμμέτρου ἑαυτῇ.</s> <s>καί εἰσιν αἱ ΔΜ, ΜΗ [ῥηταὶ] δυνάμει <lb n="1799,001,,,,10,64,28"></lb> |
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10842 μόνον σύμμετροι, καὶ ἡ ἐλάσσων ἡ ΜΗ σύμμετρος τῇ <lb n="1799,001,,,,10,64,29"></lb> |
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10843 ΔΕ μήκει.</s> <lb n="1799,001,,,,10,64,30"></lb> |
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10844 <s>Ἡ ΔΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ πέμπτη· ὅπερ ἔδει <lb n="1799,001,,,,10,64,31"></lb> |
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10845 δεῖξαι.</s> <lb n="1799,001,,,,10,64,32"></lb> |
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10846 <s>Τὸ ἀπὸ τῆς δύο μέσα δυναμένης παρὰ ῥητὴν παρα-<lb n="1799,001,,,,10,65,1"></lb> |
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10847 βαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων ἕκτην.</s> <lb n="1799,001,,,,10,65,2"></lb> |
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10848 <s>Ἔστω δύο μέσα δυναμένη ἡ ΑΒ διῃρημένη κατὰ τὸ Γ, <lb n="1799,001,,,,10,65,3"></lb> |
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10849 ῥητὴ δὲ ἔστω ἡ ΔΕ.</s> <s>καὶ παρὰ τὴν ΔΕ τῷ ἀπὸ τῆς ΑΒ <lb n="1799,001,,,,10,65,4"></lb> |
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10850 ἴσον παραβεβλήσθω τὸ ΔΖ πλά-<lb n="1799,001,,,,10,65,5"></lb> |
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10851 τος ποιοῦν τὴν ΔΗ· λέγω, ὅτι <lb n="1799,001,,,,10,65,6"></lb> |
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10852 ἡ ΔΗ ἐκ δύο ὀνομάτων ἐστὶν <lb n="1799,001,,,,10,65,7"></lb> |
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10853 ἕκτη.</s> <lb n="1799,001,,,,10,65,8"></lb> |
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10854 <s>Κατεσκευάσθω γὰρ τὰ αὐτὰ <lb n="1799,001,,,,10,65,9"></lb> |
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10855 τοῖς πρότερον.</s> <s>καὶ ἐπεὶ ἡ ΑΒ δύο <lb n="1799,001,,,,10,65,10"></lb> |
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10856 μέσα δυναμένη ἐστὶ διῃρημένη <lb n="1799,001,,,,10,65,11"></lb> |
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10857 κατὰ τὸ Γ, αἱ ΑΓ, ΓΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι <lb n="1799,001,,,,10,65,12"></lb> |
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10858 ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων <lb n="1799,001,,,,10,65,13"></lb> |
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10859 μέσον καὶ τὸ ὑπ' αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τὸ ἐκ <lb n="1799,001,,,,10,65,14"></lb> |
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10860 τῶν ἀπ' αὐτῶν τετραγώνων συγκείμενον τῷ ὑπ' αὐτῶν· <lb n="1799,001,,,,10,65,15"></lb> |
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10861 ὥστε κατὰ τὰ προδεδειγμένα μέσον ἐστὶν ἑκάτερον τῶν <pb></pb> <lb n="1799,001,,,,10,65,16"></lb> |
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10862 ΔΛ, ΜΖ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΔΕ παράκειται· ῥητὴ ἄρα <lb n="1799,001,,,,10,65,17"></lb> |
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10863 ἐστὶν ἑκατέρα τῶν ΔΜ, ΜΗ καὶ ἀσύμμετρος τῇ ΔΕ μήκει.</s> <lb n="1799,001,,,,10,65,18"></lb> |
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10864 <s>καὶ ἐπεὶ ἀσύμμετρόν ἐστι τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν <lb n="1799,001,,,,10,65,19"></lb> |
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10865 ΑΓ, ΓΒ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ, ἀσύμμετρον ἄρα ἐστὶ <lb n="1799,001,,,,10,65,20"></lb> |
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10866 τὸ ΔΛ τῷ ΜΖ.</s> <s>ἀσύμμετρος ἄρα καὶ ἡ ΔΜ τῇ ΜΗ· <lb n="1799,001,,,,10,65,21"></lb> |
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10867 αἱ ΔΜ, ΜΗ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἐκ <lb n="1799,001,,,,10,65,22"></lb> |
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10868 δύο ἄρα ὀνομάτων ἐστὶν ἡ ΔΗ.</s> <lb n="1799,001,,,,10,65,23"></lb> |
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10869 <s>Λέγω δή, ὅτι καὶ ἕκτη.</s> <lb n="1799,001,,,,10,65,24"></lb> |
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10870 <s>Ὁμοίως δὴ πάλιν δείξομεν, ὅτι τὸ ὑπὸ τῶν ΔΚΜ ἴσον <lb n="1799,001,,,,10,65,25"></lb> |
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10871 ἐστὶ τῷ ἀπὸ τῆς ΜΝ, καὶ ὅτι ἡ ΔΚ τῇ ΚΜ μήκει ἐστὶν <lb n="1799,001,,,,10,65,26"></lb> |
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10872 ἀσύμμετρος· καὶ διὰ τὰ αὐτὰ δὴ ἡ ΔΜ τῆς ΜΗ μεῖζον <lb n="1799,001,,,,10,65,27"></lb> |
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10873 δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει.</s> <s>καὶ οὐδετέρα <lb n="1799,001,,,,10,65,28"></lb> |
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10874 τῶν ΔΜ, ΜΗ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ τῇ <lb n="1799,001,,,,10,65,29"></lb> |
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10875 ΔΕ μήκει.</s> <lb n="1799,001,,,,10,65,30"></lb> |
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10876 <s>Ἡ ΔΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶν ἕκτη· ὅπερ ἔδει <lb n="1799,001,,,,10,65,31"></lb> |
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10877 δεῖξαι.</s> <lb n="1799,001,,,,10,65,32"></lb> |
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10878 <s>Ἡ τῇ ἐκ δύο ὀνομάτων μήκει σύμμετρος καὶ αὐτὴ ἐκ <lb n="1799,001,,,,10,66,1"></lb> |
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10879 δύο ὀνομάτων ἐστὶ καὶ τῇ τάξει ἡ αὐτή.</s> <lb n="1799,001,,,,10,66,2"></lb> |
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10880 <s>Ἔστω ἐκ δύο ὀνομάτων ἡ ΑΒ, καὶ τῇ ΑΒ μήκει <lb n="1799,001,,,,10,66,3"></lb> |
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10881 σύμμετρος ἔστω ἡ ΓΔ· λέγω, ὅτι ἡ ΓΔ ἐκ δύο ὀνομάτων <lb n="1799,001,,,,10,66,4"></lb> |
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10882 ἐστὶ καὶ τῇ τάξει ἡ αὐτὴ τῇ ΑΒ. <pb></pb> <lb n="1799,001,,,,10,66,5"></lb> |
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10883 Ἐπεὶ γὰρ ἐκ δύο ὀνομάτων ἐστὶν ἡ ΑΒ, διῃρήσθω εἰς <lb n="1799,001,,,,10,66,6"></lb> |
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10884 τὰ ὀνόματα κατὰ τὸ Ε, καὶ ἔστω μεῖζον ὄνομα τὸ ΑΕ· <lb n="1799,001,,,,10,66,7"></lb> |
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10885 αἱ ΑΕ, ΕΒ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι.</s> <lb n="1799,001,,,,10,66,8"></lb> |
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10886 <s>γεγονέτω ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΑΕ πρὸς τὴν <lb n="1799,001,,,,10,66,9"></lb> |
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10887 ΓΖ· καὶ λοιπὴ ἄρα ἡ ΕΒ πρὸς λοιπὴν τὴν ΖΔ ἐστιν, ὡς <lb n="1799,001,,,,10,66,10"></lb> |
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10888 ἡ ΑΒ πρὸς τὴν ΓΔ.</s> <s>σύμμετρος δὲ ἡ ΑΒ τῇ ΓΔ μήκει.</s> <lb n="1799,001,,,,10,66,11"></lb> |
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10889 <s>σύμμετρος ἄρα ἐστὶ καὶ ἡ μὲν ΑΕ τῇ ΓΖ, ἡ δὲ ΕΒ τῇ <lb n="1799,001,,,,10,66,12"></lb> |
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10890 ΖΔ.</s> <s>καί εἰσι ῥηταὶ αἱ ΑΕ, ΕΒ· ῥηταὶ ἄρα εἰσὶ καὶ αἱ <lb n="1799,001,,,,10,66,13"></lb> |
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10891 ΓΖ, ΖΔ.</s> <s>καὶ [ἐπεί] ἐστιν ὡς ἡ ΑΕ πρὸς ΓΖ, ἡ ΕΒ <lb n="1799,001,,,,10,66,14"></lb> |
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10892 πρὸς ΖΔ.</s> <s>ἐναλλὰξ ἄρα ἐστὶν ὡς ἡ ΑΕ πρὸς ΕΒ, ἡ ΓΖ <lb n="1799,001,,,,10,66,15"></lb> |
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10893 πρὸς ΖΔ.</s> <s>αἱ δὲ ΑΕ, ΕΒ δυνάμει μόνον [εἰσὶ] σύμμετροι· <lb n="1799,001,,,,10,66,16"></lb> |
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10894 καὶ αἱ ΓΖ, ΖΔ ἄρα δυνάμει μόνον εἰσὶ σύμμετροι.</s> <s>καί εἰσι <lb n="1799,001,,,,10,66,17"></lb> |
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10895 ῥηταί· ἐκ δύο ἄρα ὀνομάτων ἐστὶν ἡ ΓΔ.</s> <lb n="1799,001,,,,10,66,18"></lb> |
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10896 <s>Λέγω δή, ὅτι τῇ τάξει ἐστὶν ἡ αὐτὴ τῇ ΑΒ.</s> <lb n="1799,001,,,,10,66,19"></lb> |
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10897 <s>Ἡ γὰρ ΑΕ τῆς ΕΒ μεῖζον δύναται ἤτοι τῷ ἀπὸ <lb n="1799,001,,,,10,66,20"></lb> |
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10898 συμμέτρου ἑαυτῇ ἢ τῷ ἀπὸ ἀσυμμέτρου.</s> <s>εἰ μὲν οὖν ἡ <lb n="1799,001,,,,10,66,21"></lb> |
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10899 ΑΕ τῆς ΕΒ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, <lb n="1799,001,,,,10,66,22"></lb> |
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10900 καὶ ἡ ΓΖ τῆς ΖΔ μεῖζον δυνήσεται τῷ ἀπὸ συμμέτρου <lb n="1799,001,,,,10,66,23"></lb> |
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10901 ἑαυτῇ.</s> <s>καὶ εἰ μὲν σύμμετρός ἐστιν ἡ ΑΕ τῇ ἐκκειμένῃ <lb n="1799,001,,,,10,66,24"></lb> |
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10902 ῥητῇ, καὶ ἡ ΓΖ σύμμετρος αὐτῇ ἔσται, καὶ διὰ τοῦτο <lb n="1799,001,,,,10,66,25"></lb> |
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10903 ἑκατέρα τῶν ΑΒ, ΓΔ ἐκ δύο ὀνομάτων ἐστὶ πρώτη, <pb></pb> <lb n="1799,001,,,,10,66,26"></lb> |
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10904 τουτέστι τῇ τάξει ἡ αὐτή.</s> <s>εἰ δὲ ἡ ΕΒ σύμμετρός ἐστι τῇ <lb n="1799,001,,,,10,66,27"></lb> |
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10905 ἐκκειμένῃ ῥητῇ, καὶ ἡ ΖΔ σύμμετρός ἐστιν αὐτῇ, καὶ <lb n="1799,001,,,,10,66,28"></lb> |
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10906 διὰ τοῦτο πάλιν τῇ τάξει ἡ αὐτὴ ἔσται τῇ ΑΒ· ἑκατέρα <lb n="1799,001,,,,10,66,29"></lb> |
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10907 γὰρ αὐτῶν ἔσται ἐκ δύο ὀνομάτων δευτέρα.</s> <s>εἰ δὲ οὐδετέρα <lb n="1799,001,,,,10,66,30"></lb> |
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10908 τῶν ΑΕ, ΕΒ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ, οὐδετέρα <lb n="1799,001,,,,10,66,31"></lb> |
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10909 τῶν ΓΖ, ΖΔ σύμμετρος αὐτῇ ἔσται, καί ἐστιν ἑκατέρα <lb n="1799,001,,,,10,66,32"></lb> |
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10910 τρίτη.</s> <s>εἰ δὲ ἡ ΑΕ τῆς ΕΒ μεῖζον δύναται τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,66,33"></lb> |
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10911 μέτρου ἑαυτῇ, καὶ ἡ ΓΖ τῆς ΖΔ μεῖζον δύναται τῷ ἀπὸ <lb n="1799,001,,,,10,66,34"></lb> |
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10912 ἀσυμμέτρου ἑαυτῇ.</s> <s>καὶ εἰ μὲν ἡ ΑΕ σύμμετρός ἐστι τῇ <lb n="1799,001,,,,10,66,35"></lb> |
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10913 ἐκκειμένῃ ῥητῇ, καὶ ἡ ΓΖ σύμμετρός ἐστιν αὐτῇ, καί <lb n="1799,001,,,,10,66,36"></lb> |
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10914 ἐστιν ἑκατέρα τετάρτη.</s> <s>εἰ δὲ ἡ ΕΒ, καὶ ἡ ΖΔ, καὶ ἔσται <lb n="1799,001,,,,10,66,37"></lb> |
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10915 ἑκατέρα πέμπτη.</s> <s>εἰ δὲ οὐδετέρα τῶν ΑΕ, ΕΒ, καὶ τῶν <lb n="1799,001,,,,10,66,38"></lb> |
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10916 ΓΖ, ΖΔ οὐδετέρα σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ, <lb n="1799,001,,,,10,66,39"></lb> |
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10917 καὶ ἔσται ἑκατέρα ἕκτη.</s> <lb n="1799,001,,,,10,66,40"></lb> |
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10918 <s>Ὥστε ἡ τῇ ἐκ δύο ὀνομάτων μήκει σύμμετρος ἐκ <lb n="1799,001,,,,10,66,41"></lb> |
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10919 δύο ὀνομάτων ἐστὶ καὶ τῇ τάξει ἡ αὐτή· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,66,42"></lb> |
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10920 <s>Ἡ τῇ ἐκ δύο μέσων μήκει σύμμετρος καὶ αὐτὴ ἐκ δύο <lb n="1799,001,,,,10,67,1"></lb> |
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10921 μέσων ἐστὶ καὶ τῇ τάξει ἡ αὐτή.</s> <s><pb></pb> <lb n="1799,001,,,,10,67,2"></lb> |
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10922 Ἔστω ἐκ δύο μέσων ἡ ΑΒ, καὶ τῇ ΑΒ σύμμετρος ἔστω <lb n="1799,001,,,,10,67,3"></lb> |
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10923 μήκει ἡ ΓΔ· λέγω, ὅτι ἡ ΓΔ ἐκ δύο μέσων ἐστὶ καὶ τῇ <lb n="1799,001,,,,10,67,4"></lb> |
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10924 τάξει ἡ αὐτὴ τῇ ΑΒ.</s> <lb n="1799,001,,,,10,67,5"></lb> |
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10925 <s>Ἐπεὶ γὰρ ἐκ δύο μέσων ἐστὶν ἡ ΑΒ, διῃρήσθω εἰς <lb n="1799,001,,,,10,67,6"></lb> |
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10926 τὰς μέσας κατὰ τὸ Ε· αἱ ΑΕ, ΕΒ ἄρα μέσαι εἰσὶ δυνάμει <lb n="1799,001,,,,10,67,7"></lb> |
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10927 μόνον σύμμετροι.</s> <s>καὶ γεγονέτω ὡς ἡ ΑΒ πρὸς ΓΔ, ἡ <lb n="1799,001,,,,10,67,8"></lb> |
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10928 ΑΕ πρὸς ΓΖ· καὶ λοιπὴ ἄρα ἡ ΕΒ πρὸς <lb n="1799,001,,,,10,67,9"></lb> |
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10929 λοιπὴν τὴν ΖΔ ἐστιν, ὡς ἡ ΑΒ πρὸς <lb n="1799,001,,,,10,67,10"></lb> |
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10930 ΓΔ.</s> <s>σύμμετρος δὲ ἡ ΑΒ τῇ ΓΔ μήκει· <lb n="1799,001,,,,10,67,11"></lb> |
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10931 σύμμετρος ἄρα καὶ ἑκατέρα τῶν ΑΕ, ΕΒ <lb n="1799,001,,,,10,67,12"></lb> |
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10932 ἑκατέρᾳ τῶν ΓΖ, ΖΔ.</s> <s>μέσαι δὲ αἱ ΑΕ, ΕΒ· <lb n="1799,001,,,,10,67,13"></lb> |
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10933 μέσαι ἄρα καὶ αἱ ΓΖ, ΖΔ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ ΑΕ πρὸς ΕΒ, <lb n="1799,001,,,,10,67,14"></lb> |
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10934 ἡ ΓΖ πρὸς ΖΔ, αἱ δὲ ΑΕ, ΕΒ δυνάμει μόνον σύμμετροί <lb n="1799,001,,,,10,67,15"></lb> |
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10935 εἰσιν, καὶ αἱ ΓΖ, ΖΔ [ἄρα] δυνάμει μόνον σύμμετροί εἰσιν.</s> <lb n="1799,001,,,,10,67,16"></lb> |
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10936 <s>ἐδείχθησαν δὲ καὶ μέσαι· ἡ ΓΔ ἄρα ἐκ δύο μέσων ἐστίν.</s> <lb n="1799,001,,,,10,67,17"></lb> |
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10937 <s>Λέγω δή, ὅτι καὶ τῇ τάξει ἡ αὐτή ἐστι τῇ ΑΒ.</s> <lb n="1799,001,,,,10,67,18"></lb> |
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10938 <s>Ἐπεὶ γάρ ἐστιν ὡς ἡ ΑΕ πρὸς ΕΒ, ἡ ΓΖ πρὸς ΖΔ, <lb n="1799,001,,,,10,67,19"></lb> |
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10939 καὶ ὡς ἄρα τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ὑπὸ τῶν ΑΕΒ, <lb n="1799,001,,,,10,67,20"></lb> |
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10940 οὕτως τὸ ἀπὸ τῆς ΓΖ πρὸς τὸ ὑπὸ τῶν ΓΖΔ· ἐναλλὰξ <lb n="1799,001,,,,10,67,21"></lb> |
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10941 ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ τῆς ΓΖ, οὕτως τὸ ὑπὸ <lb n="1799,001,,,,10,67,22"></lb> |
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10942 τῶν ΑΕΒ πρὸς τὸ ὑπὸ τῶν ΓΖΔ.</s> <s>σύμμετρον δὲ τὸ ἀπὸ <pb></pb> <lb n="1799,001,,,,10,67,23"></lb> |
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10943 τῆς ΑΕ τῷ ἀπὸ τῆς ΓΖ· σύμμετρον ἄρα καὶ τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,67,24"></lb> |
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10944 ΑΕΒ τῷ ὑπὸ τῶν ΓΖΔ.</s> <s>εἴτε οὖν ῥητόν ἐστι τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,67,25"></lb> |
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10945 ΑΕΒ, καὶ τὸ ὑπὸ τῶν ΓΖΔ ῥητόν ἐστιν [καὶ διὰ τοῦτό <lb n="1799,001,,,,10,67,26"></lb> |
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10946 ἐστιν ἐκ δύο μέσων πρώτη].</s> <s>εἴτε μέσον, μέσον, καί <lb n="1799,001,,,,10,67,27"></lb> |
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10947 ἐστιν ἑκατέρα δευτέρα.</s> <lb n="1799,001,,,,10,67,28"></lb> |
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10948 <s>Καὶ διὰ τοῦτο ἔσται ἡ ΓΔ τῇ ΑΒ τῇ τάξει ἡ αὐτή· <lb n="1799,001,,,,10,67,29"></lb> |
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10949 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,67,30"></lb> |
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10950 <s>Ἡ τῇ μείζονι σύμμετρος καὶ αὐτὴ μείζων ἐστίν.</s> <lb n="1799,001,,,,10,68,1"></lb> |
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10951 <s>Ἔστω μείζων ἡ ΑΒ, καὶ τῇ ΑΒ σύμμετρος ἔστω ἡ <lb n="1799,001,,,,10,68,2"></lb> |
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10952 ΓΔ· λέγω, ὅτι ἡ ΓΔ μείζων ἐστίν.</s> <lb n="1799,001,,,,10,68,3"></lb> |
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10953 <s>Διῃρήσθω ἡ ΑΒ κατὰ τὸ Ε· αἱ ΑΕ, ΕΒ ἄρα δυνά-<lb n="1799,001,,,,10,68,4"></lb> |
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10954 μει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ <lb n="1799,001,,,,10,68,5"></lb> |
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10955 τῶν ἀπ' αὐτῶν τετραγώνων ῥητόν, τὸ δ' ὑπ' αὐτῶν <lb n="1799,001,,,,10,68,6"></lb> |
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10956 μέσον· καὶ γεγονέτω τὰ αὐτὰ τοῖς πρότε- <lb n="1799,001,,,,10,68,7"></lb> |
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10957 ρον.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, <lb n="1799,001,,,,10,68,8"></lb> |
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10958 οὕτως ἥ τε ΑΕ πρὸς τὴν ΓΖ καὶ ἡ ΕΒ <lb n="1799,001,,,,10,68,9"></lb> |
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10959 πρὸς τὴν ΖΔ, καὶ ὡς ἄρα ἡ ΑΕ πρὸς τὴν <lb n="1799,001,,,,10,68,10"></lb> |
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10960 ΓΖ, οὕτως ἡ ΕΒ πρὸς τὴν ΖΔ.</s> <s>σύμμετρος <lb n="1799,001,,,,10,68,11"></lb> |
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10961 δὲ ἡ ΑΒ τῇ ΓΔ.</s> <s>σύμμετρος ἄρα καὶ ἑκα-<lb n="1799,001,,,,10,68,12"></lb> |
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10962 τέρα τῶν ΑΕ, ΕΒ ἑκατέρᾳ τῶν ΓΖ, ΖΔ.</s> <s><pb></pb> <lb n="1799,001,,,,10,68,13"></lb> |
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10963 καὶ ἐπεί ἐστιν ὡς ἡ ΑΕ πρὸς τὴν ΓΖ, οὕτως ἡ ΕΒ <lb n="1799,001,,,,10,68,14"></lb> |
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10964 πρὸς τὴν ΖΔ, καὶ ἐναλλὰξ ὡς ἡ ΑΕ πρὸς ΕΒ, οὕτως <lb n="1799,001,,,,10,68,15"></lb> |
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10965 ἡ ΓΖ πρὸς ΖΔ, καὶ συνθέντι ἄρα ἐστὶν ὡς ἡ ΑΒ πρὸς <lb n="1799,001,,,,10,68,16"></lb> |
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10966 τὴν ΒΕ, οὕτως ἡ ΓΔ πρὸς τὴν ΔΖ· καὶ ὡς ἄρα τὸ <lb n="1799,001,,,,10,68,17"></lb> |
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10967 ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΕ, οὕτως τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,68,18"></lb> |
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10968 ΓΔ πρὸς τὸ ἀπὸ τῆς ΔΖ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ <lb n="1799,001,,,,10,68,19"></lb> |
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10969 ὡς τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΑΕ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,68,20"></lb> |
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10970 τῆς ΓΔ πρὸς τὸ ἀπὸ τῆς ΓΖ.</s> <s>καὶ ὡς ἄρα τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,68,21"></lb> |
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10971 ΑΒ πρὸς τὰ ἀπὸ τῶν ΑΕ, ΕΒ, οὕτως τὸ ἀπὸ τῆς ΓΔ <lb n="1799,001,,,,10,68,22"></lb> |
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10972 πρὸς τὰ ἀπὸ τῶν ΓΖ, ΖΔ· καὶ ἐναλλὰξ ἄρα ἐστὶν ὡς <lb n="1799,001,,,,10,68,23"></lb> |
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10973 τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΓΔ, οὕτως τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,68,24"></lb> |
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10974 ΑΕ, ΕΒ πρὸς τὰ ἀπὸ τῶν ΓΖ, ΖΔ.</s> <s>σύμμετρον δὲ τὸ <lb n="1799,001,,,,10,68,25"></lb> |
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10975 ἀπὸ τῆς ΑΒ τῷ ἀπὸ τῆς ΓΔ· σύμμετρα ἄρα καὶ τὰ ἀπὸ <lb n="1799,001,,,,10,68,26"></lb> |
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10976 τῶν ΑΕ, ΕΒ τοῖς ἀπὸ τῶν ΓΖ, ΖΔ.</s> <s>καί ἐστι τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,68,27"></lb> |
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10977 ΑΕ, ΕΒ ἅμα ῥητόν, καὶ τὰ ἀπὸ τῶν ΓΖ, ΖΔ ἅμα ῥητόν <lb n="1799,001,,,,10,68,28"></lb> |
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10978 ἐστιν.</s> <s>ὁμοίως δὲ καὶ τὸ δὶς ὑπὸ τῶν ΑΕ, ΕΒ σύμμετρόν <lb n="1799,001,,,,10,68,29"></lb> |
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10979 ἐστι τῷ δὶς ὑπὸ τῶν ΓΖ, ΖΔ.</s> <s>καί ἐστι μέσον τὸ δὶς ὑπὸ <lb n="1799,001,,,,10,68,30"></lb> |
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10980 τῶν ΑΕ, ΕΒ· μέσον ἄρα καὶ τὸ δὶς ὑπὸ τῶν ΓΖ, ΖΔ.</s> <lb n="1799,001,,,,10,68,31"></lb> |
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10981 <s>αἱ ΓΖ, ΖΔ ἄρα δυνάμει ἀσύμμετροί εἰσι ποιοῦσαι τὸ <lb n="1799,001,,,,10,68,32"></lb> |
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10982 μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων ἅμα <pb></pb> <lb n="1799,001,,,,10,68,33"></lb> |
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10983 ῥητόν, τὸ δὲ δὶς ὑπ' αὐτῶν μέσον· ὅλη ἄρα ἡ ΓΔ ἄλογός <lb n="1799,001,,,,10,68,34"></lb> |
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10984 ἐστιν ἡ καλουμένη μείζων.</s> <lb n="1799,001,,,,10,68,35"></lb> |
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10985 <s>Ἡ ἄρα τῇ μείζονι σύμμετρος μείζων ἐστίν· ὅπερ ἔδει <lb n="1799,001,,,,10,68,36"></lb> |
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10986 δεῖξαι.</s> <lb n="1799,001,,,,10,68,37"></lb> |
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10987 <s>Ἡ τῇ ῥητὸν καὶ μέσον δυναμένῃ σύμμετρος [καὶ αὐτὴ] <lb n="1799,001,,,,10,69,1"></lb> |
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10988 ῥητὸν καὶ μέσον δυναμένη ἐστίν.</s> <lb n="1799,001,,,,10,69,2"></lb> |
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10989 <s>Ἔστω ῥητὸν καὶ μέσον δυναμένη ἡ ΑΒ, καὶ τῇ ΑΒ <lb n="1799,001,,,,10,69,3"></lb> |
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10990 σύμμετρος ἔστω ἡ ΓΔ· δεικτέον, ὅτι καὶ ἡ ΓΔ ῥητὸν καὶ <lb n="1799,001,,,,10,69,4"></lb> |
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10991 μέσον δυναμένη ἐστίν.</s> <lb n="1799,001,,,,10,69,5"></lb> |
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10992 <s>Διῃρήσθω ἡ ΑΒ εἰς τὰς εὐθείας κατὰ τὸ <lb n="1799,001,,,,10,69,6"></lb> |
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10993 Ε· αἱ ΑΕ, ΕΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι <lb n="1799,001,,,,10,69,7"></lb> |
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10994 ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν <lb n="1799,001,,,,10,69,8"></lb> |
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10995 τετραγώνων μέσον, τὸ δ' ὑπ' αὐτῶν ῥητόν· <lb n="1799,001,,,,10,69,9"></lb> |
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10996 καὶ τὰ αὐτὰ κατεσκευάσθω τοῖς πρότερον.</s> <lb n="1799,001,,,,10,69,10"></lb> |
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10997 <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ ΓΖ, ΖΔ δυνά-<lb n="1799,001,,,,10,69,11"></lb> |
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10998 μει εἰσὶν ἀσύμμετροι, καὶ σύμμετρον τὸ μὲν <lb n="1799,001,,,,10,69,12"></lb> |
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10999 συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τῷ συγκειμένῳ <lb n="1799,001,,,,10,69,13"></lb> |
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11000 ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ, τὸ δὲ ὑπὸ ΑΕ, ΕΒ τῷ ὑπὸ <lb n="1799,001,,,,10,69,14"></lb> |
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11001 ΓΖ, ΖΔ· ὥστε καὶ τὸ [μὲν] συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,69,15"></lb> |
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11002 τῶν ΓΖ, ΖΔ τετραγώνων ἐστὶ μέσον, τὸ δ' ὑπὸ τῶν <lb n="1799,001,,,,10,69,16"></lb> |
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11003 ΓΖ, ΖΔ ῥητόν.</s> <lb n="1799,001,,,,10,69,17"></lb> |
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11004 <s>Ῥητὸν ἄρα καὶ μέσον δυναμένη ἐστὶν ἡ ΓΔ· ὅπερ ἔδει <lb n="1799,001,,,,10,69,18"></lb> |
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11005 δεῖξαι.</s> <lb n="1799,001,,,,10,69,19"></lb> |
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11006 <s>Ἡ τῇ δύο μέσα δυναμένῃ σύμμετρος δύο μέσα δυνα-<lb n="1799,001,,,,10,70,1"></lb> |
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11007 μένη ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,10,70,2"></lb> |
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11008 Ἔστω δύο μέσα δυναμένη ἡ ΑΒ, καὶ τῇ ΑΒ σύμμετρος <lb n="1799,001,,,,10,70,3"></lb> |
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11009 ἡ ΓΔ· δεικτέον, ὅτι καὶ ἡ ΓΔ δύο μέσα δυναμένη ἐστίν.</s> <lb n="1799,001,,,,10,70,4"></lb> |
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11010 <s>Ἐπεὶ γὰρ δύο μέσα δυναμένη ἐστὶν ἡ ΑΒ, <lb n="1799,001,,,,10,70,5"></lb> |
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11011 διῃρήσθω εἰς τὰς εὐθείας κατὰ τὸ Ε· αἱ ΑΕ, <lb n="1799,001,,,,10,70,6"></lb> |
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11012 ΕΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι <lb n="1799,001,,,,10,70,7"></lb> |
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11013 τό τε συγκείμενον ἐκ τῶν ἀπ' αὐτῶν [τετρα-<lb n="1799,001,,,,10,70,8"></lb> |
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11014 γώνων] μέσον καὶ τὸ ὑπ' αὐτῶν μέσον καὶ <lb n="1799,001,,,,10,70,9"></lb> |
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11015 ἔτι ἀσύμμετρον τὸ συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,70,10"></lb> |
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11016 τῶν ΑΕ, ΕΒ τετραγώνων τῷ ὑπὸ τῶν ΑΕ, <lb n="1799,001,,,,10,70,11"></lb> |
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11017 ΕΒ· καὶ κατεσκευάσθω τὰ αὐτὰ τοῖς πρότερον.</s> <s>ὁμοίως <lb n="1799,001,,,,10,70,12"></lb> |
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11018 δὴ δείξομεν, ὅτι καὶ αἱ ΓΖ, ΖΔ δυνάμει εἰσὶν ἀσύμμε-<lb n="1799,001,,,,10,70,13"></lb> |
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11019 τροι καὶ σύμμετρον τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν <lb n="1799,001,,,,10,70,14"></lb> |
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11020 ΑΕ, ΕΒ τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ, τὸ <lb n="1799,001,,,,10,70,15"></lb> |
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11021 δὲ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, ΖΔ· ὥστε καὶ τὸ <lb n="1799,001,,,,10,70,16"></lb> |
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11022 συγκείμενον ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων μέσον <lb n="1799,001,,,,10,70,17"></lb> |
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11023 ἐστὶ καὶ τὸ ὑπὸ τῶν ΓΖ, ΖΔ μέσον καὶ ἔτι ἀσύμμετρον <lb n="1799,001,,,,10,70,18"></lb> |
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11024 τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων <lb n="1799,001,,,,10,70,19"></lb> |
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11025 τῷ ὑπὸ τῶν ΓΖ, ΖΔ.</s> <lb n="1799,001,,,,10,70,20"></lb> |
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11026 <s>Ἡ ἄρα ΓΔ δύο μέσα δυναμένη ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,70,21"></lb> |
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11027 <s>Ῥητοῦ καὶ μέσου συντιθεμένου τέσσαρες ἄλογοι γί-<lb n="1799,001,,,,10,71,1"></lb> |
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11028 γνονται ἤτοι ἐκ δύο ὀνομάτων ἢ ἐκ δύο μέσων πρώτη ἢ <lb n="1799,001,,,,10,71,2"></lb> |
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11029 μείζων ἢ ῥητὸν καὶ μέσον δυναμένη.</s> <lb n="1799,001,,,,10,71,3"></lb> |
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11030 <s>Ἔστω ῥητὸν μὲν τὸ ΑΒ, μέσον δὲ τὸ ΓΔ· λέγω, ὅτι ἡ <lb n="1799,001,,,,10,71,4"></lb> |
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11031 τὸ ΑΔ χωρίον δυναμένη ἤτοι ἐκ δύο ὀνομάτων ἐστὶν ἢ ἐκ <lb n="1799,001,,,,10,71,5"></lb> |
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11032 δύο μέσων πρώτη ἢ μείζων ἢ ῥητὸν καὶ μέσον δυναμένη.</s> <s><pb></pb> <lb n="1799,001,,,,10,71,6"></lb> |
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11033 Τὸ γὰρ ΑΒ τοῦ ΓΔ ἤτοι μεῖζόν ἐστιν ἢ ἔλασσον.</s> <lb n="1799,001,,,,10,71,7"></lb> |
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11034 <s>ἔστω πρότερον μεῖζον· καὶ ἐκκείσθω ῥητὴ ἡ ΕΖ, καὶ <lb n="1799,001,,,,10,71,8"></lb> |
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11035 παραβεβλήσθω παρὰ τὴν ΕΖ <lb n="1799,001,,,,10,71,9"></lb> |
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11036 τῷ ΑΒ ἴσον τὸ ΕΗ πλάτος <lb n="1799,001,,,,10,71,10"></lb> |
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11037 ποιοῦν τὴν ΕΘ· τῷ δὲ ΔΓ <lb n="1799,001,,,,10,71,11"></lb> |
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11038 ἴσον παρὰ τὴν ΕΖ παραβε-<lb n="1799,001,,,,10,71,12"></lb> |
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11039 βλήσθω τὸ ΘΙ πλάτος ποιοῦν <lb n="1799,001,,,,10,71,13"></lb> |
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11040 τὴν ΘΚ.</s> <s>καὶ ἐπεὶ ῥητόν ἐστι <lb n="1799,001,,,,10,71,14"></lb> |
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11041 τὸ ΑΒ καί ἐστιν ῥητόν ἐστι <lb n="1799,001,,,,10,71,15"></lb> |
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11042 τὸ ΑΒ καί ἐστιν ἴσον τῷ <lb n="1799,001,,,,10,71,16"></lb> |
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11043 ΕΗ, ῥητὸν ἄρα καὶ τὸ ΕΗ.</s> <s>καὶ παρὰ [ῥητὴν] τὴν <lb n="1799,001,,,,10,71,17"></lb> |
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11044 ΕΖ παραβέβληται πλάτος ποιοῦν τὴν ΕΘ· ἡ ΕΘ ἄρα <lb n="1799,001,,,,10,71,18"></lb> |
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11045 ῥητή ἐστι καὶ σύμμετρος τῇ ΕΖ μήκει.</s> <s>πάλιν, ἐπεὶ μέσον <lb n="1799,001,,,,10,71,19"></lb> |
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11046 ἐστὶ τὸ ΓΔ καί ἐστιν ἴσον τῷ ΘΙ, μέσον ἄρα ἐστὶ καὶ <lb n="1799,001,,,,10,71,20"></lb> |
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11047 τὸ ΘΙ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΕΖ παράκειται πλάτος <lb n="1799,001,,,,10,71,21"></lb> |
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11048 ποιοῦν τὴν ΘΚ· ῥητὴ ἄρα ἐστὶν ἡ ΘΚ καὶ ἀσύμμετρος <lb n="1799,001,,,,10,71,22"></lb> |
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11049 τῇ ΕΖ μήκει.</s> <s>καὶ ἐπεὶ μέσον ἐστὶ τὸ ΓΔ, ῥητὸν δὲ τὸ <lb n="1799,001,,,,10,71,23"></lb> |
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11050 ΑΒ, ἀσύμμετρον ἄρα ἐστὶ τὸ ΑΒ τῷ ΓΔ· ὥστε καὶ τὸ <lb n="1799,001,,,,10,71,24"></lb> |
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11051 ΕΗ ἀσύμμετρόν ἐστι τῷ ΘΙ.</s> <s>ὡς δὲ τὸ ΕΗ πρὸς τὸ ΘΙ, <lb n="1799,001,,,,10,71,25"></lb> |
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11052 οὕτως ἐστὶν ἡ ΕΘ πρὸς τὴν ΘΚ· ἀσύμμετρος ἄρα ἐστὶ <lb n="1799,001,,,,10,71,26"></lb> |
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11053 καὶ ἡ ΕΘ τῇ ΘΚ μήκει.</s> <s>καί εἰσιν ἀμφότεραι ῥηταί· αἱ <lb n="1799,001,,,,10,71,27"></lb> |
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11054 ΕΘ, ΘΚ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἐκ δύο <lb n="1799,001,,,,10,71,28"></lb> |
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11055 ἄρα ὀνομάτων ἐστὶν ἡ ΕΚ διῃρημένη κατὰ τὸ Θ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,71,29"></lb> |
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11056 μεῖζόν ἐστι τὸ ΑΒ τοῦ ΓΔ, ἴσον δὲ τὸ μὲν ΑΒ τῷ ΕΗ, <pb></pb> <lb n="1799,001,,,,10,71,30"></lb> |
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11057 τὸ δὲ ΓΔ τῷ ΘΙ, μεῖζον ἄρα καὶ τὸ ΕΗ τοῦ ΘΙ· καὶ ἡ <lb n="1799,001,,,,10,71,31"></lb> |
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11058 ΕΘ ἄρα μείζων ἐστὶ τῆς ΘΚ.</s> <s>ἤτοι οὖν ἡ ΕΘ τῆς ΘΚ <lb n="1799,001,,,,10,71,32"></lb> |
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11059 μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει ἢ τῷ <lb n="1799,001,,,,10,71,33"></lb> |
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11060 ἀπὸ ἀσυμμέτρου.</s> <s>δυνάσθω πρότερον τῷ ἀπὸ συμμέτρου <lb n="1799,001,,,,10,71,34"></lb> |
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11061 ἑαυτῇ.</s> <s>καί ἐστιν ἡ μείζων ἡ ΘΕ σύμμετρος τῇ ἐκκειμένῃ <lb n="1799,001,,,,10,71,35"></lb> |
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11062 ῥητῇ τῇ ΕΖ· ἡ ἄρα ΕΚ ἐκ δύο ὀνομάτων ἐστὶ πρώτη.</s> <lb n="1799,001,,,,10,71,36"></lb> |
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11063 <s>ῥητὴ δὲ ἡ ΕΖ· ἐὰν δὲ χωρίον περιέχηται ὑπὸ ῥητῆς καὶ <lb n="1799,001,,,,10,71,37"></lb> |
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11064 τῆς ἐκ δύο ὀνομάτων πρώτης, ἡ τὸ χωρίον δυναμένη ἐκ <lb n="1799,001,,,,10,71,38"></lb> |
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11065 δύο ὀνομάτων ἐστίν.</s> <s>ἡ ἄρα τὸ ΕΙ δυναμένη ἐκ δύο <lb n="1799,001,,,,10,71,39"></lb> |
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11066 ὀνομάτων ἐστίν· ὥστε καὶ ἡ τὸ ΑΔ δυναμένη ἐκ δύο <lb n="1799,001,,,,10,71,40"></lb> |
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11067 ὀνομάτων ἐστίν.</s> <s>ἀλλὰ δὴ δυνάσθω ἡ ΕΘ τῆς ΘΚ μεῖζον <lb n="1799,001,,,,10,71,41"></lb> |
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11068 τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ· καί ἐστιν ἡ μείζων ἡ ΕΘ <lb n="1799,001,,,,10,71,42"></lb> |
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11069 σύμμετρος τῇ ἐκκειμένῃ ῥητῇ τῇ ΕΖ μήκει· ἡ ἄρα ΕΚ <lb n="1799,001,,,,10,71,43"></lb> |
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11070 ἐκ δύο ὀνομάτων ἐστὶ τετάρτη.</s> <s>ῥητὴ δὲ ἡ ΕΖ· ἐὰν δὲ <lb n="1799,001,,,,10,71,44"></lb> |
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11071 χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο ὀνομάτων <lb n="1799,001,,,,10,71,45"></lb> |
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11072 τετάρτης, ἡ τὸ χωρίον δυναμένη ἄλογός ἐστιν ἡ καλουμένη <lb n="1799,001,,,,10,71,46"></lb> |
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11073 μείζων.</s> <s>ἡ ἄρα τὸ ΕΙ χωρίον <lb n="1799,001,,,,10,71,47"></lb> |
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11074 δυναμένη μείζων ἐστίν· ὥστε <lb n="1799,001,,,,10,71,48"></lb> |
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11075 καὶ ἡ τὸ ΑΔ δυναμένη μείζων <lb n="1799,001,,,,10,71,49"></lb> |
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11076 ἐστίν.</s> <lb n="1799,001,,,,10,71,50"></lb> |
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11077 <s>Ἀλλὰ δὴ ἔστω ἔλασσον τὸ <lb n="1799,001,,,,10,71,51"></lb> |
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11078 ΑΒ τοῦ ΓΔ· καὶ τὸ ΕΗ ἄρα <lb n="1799,001,,,,10,71,52"></lb> |
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11079 ἔλασσόν ἐστι τοῦ ΘΙ· ὥστε καὶ <lb n="1799,001,,,,10,71,53"></lb> |
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11080 ἡ ΕΘ ἐλάσσων ἐστὶ τῆς ΘΚ.</s> <s><pb></pb> <lb n="1799,001,,,,10,71,54"></lb> |
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11081 ἤτοι δὲ ἡ ΘΚ τῆς ΕΘ μεῖζον δύναται τῷ ἀπὸ συμ-<lb n="1799,001,,,,10,71,55"></lb> |
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11082 μέτρου ἑαυτῇ ἢ τῷ ἀπὸ ἀσυμμέτρου.</s> <s>δυνάσθω πρό-<lb n="1799,001,,,,10,71,56"></lb> |
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11083 τερον τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει· καί ἐστιν ἡ <lb n="1799,001,,,,10,71,57"></lb> |
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11084 ἐλάσσων ἡ ΕΘ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ τῇ ΕΖ <lb n="1799,001,,,,10,71,58"></lb> |
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11085 μήκει· ἡ ἄρα ΕΚ ἐκ δύο ὀνομάτων ἐστὶ δευτέρα.</s> <s>ῥητὴ δὲ <lb n="1799,001,,,,10,71,59"></lb> |
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11086 ἡ ΕΖ· ἐὰν δὲ χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς <lb n="1799,001,,,,10,71,60"></lb> |
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11087 ἐκ δύο ὀνομάτων δευτέρας, ἡ τὸ χωρίον δυναμένη ἐκ <lb n="1799,001,,,,10,71,61"></lb> |
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11088 δύο μέσων ἐστὶ πρώτη.</s> <s>ἡ ἄρα τὸ ΕΙ χωρίον δυναμένη ἐκ <lb n="1799,001,,,,10,71,62"></lb> |
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11089 δύο μέσων ἐστὶ πρώτη· ὥστε καὶ ἡ τὸ ΑΔ δυναμένη <lb n="1799,001,,,,10,71,63"></lb> |
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11090 ἐκ δύο μέσων ἐστὶ πρώτη.</s> <s>ἀλλὰ δὴ ἡ ΘΚ τῆς ΘΕ μεῖζον <lb n="1799,001,,,,10,71,64"></lb> |
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11091 δυνάσθω τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <s>καί ἐστιν ἡ ἐλάσσων <lb n="1799,001,,,,10,71,65"></lb> |
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11092 ἡ ΕΘ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ τῇ ΕΖ· ἡ ἄρα ΕΚ ἐκ <lb n="1799,001,,,,10,71,66"></lb> |
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11093 δύο ὀνομάτων ἐστὶ πέμπτη.</s> <s>ῥητὴ δὲ ἡ ΕΖ· ἐὰν δὲ χωρίον <lb n="1799,001,,,,10,71,67"></lb> |
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11094 περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο ὀνομάτων πέμπτης, <lb n="1799,001,,,,10,71,68"></lb> |
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11095 ἡ τὸ χωρίον δυναμένη ῥητὸν καὶ μέσον δυναμένη ἐστίν.</s> <lb n="1799,001,,,,10,71,69"></lb> |
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11096 <s>ἡ ἄρα τὸ ΕΙ χωρίον δυναμένη ῥητὸν καὶ μέσον δυναμένη <lb n="1799,001,,,,10,71,70"></lb> |
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11097 ἐστίν· ὥστε καὶ ἡ τὸ ΑΔ χωρίον δυναμένη ῥητὸν καὶ <lb n="1799,001,,,,10,71,71"></lb> |
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11098 μέσον δυναμένη ἐστίν.</s> <lb n="1799,001,,,,10,71,72"></lb> |
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11099 <s>Ῥητοῦ ἄρα καὶ μέσου συντιθεμένου τέσσαρες ἄλογοι <lb n="1799,001,,,,10,71,73"></lb> |
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11100 γίγνονται ἤτοι ἐκ δύο ὀνομάτων ἢ ἐκ δύο μέσων πρώτη <lb n="1799,001,,,,10,71,74"></lb> |
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11101 ἢ μείζων ἢ ῥητὸν καὶ μέσον δυναμένη· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,71,75"></lb> |
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11102 Δύο μέσων ἀσυμμέτρων ἀλλήλοις συντιθεμένων αἱ <lb n="1799,001,,,,10,72,1"></lb> |
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11103 λοιπαὶ δύο ἄλογοι γίγνονται ἤτοι ἐκ δύο μέσων δευτέρα <lb n="1799,001,,,,10,72,2"></lb> |
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11104 ἢ [ἡ] δύο μέσα δυναμένη.</s> <lb n="1799,001,,,,10,72,3"></lb> |
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11105 <s>Συγκείσθω γὰρ δύο μέσα ἀσύμμετρα ἀλλήλοις τὰ <lb n="1799,001,,,,10,72,4"></lb> |
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11106 ΑΒ, ΓΔ· λέγω, ὅτι ἡ τὸ ΑΔ χωρίον δυναμένη ἤτοι ἐκ <lb n="1799,001,,,,10,72,5"></lb> |
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11107 δύο μέσων ἐστὶ δευτέρα ἢ δύο μέσα δυναμένη.</s> <lb n="1799,001,,,,10,72,6"></lb> |
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11108 <s>Τὸ γὰρ ΑΒ τοῦ ΓΔ ἤτοι <lb n="1799,001,,,,10,72,7"></lb> |
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11109 μεῖζόν ἐστιν ἢ ἔλασσον.</s> <s>ἔστω, <lb n="1799,001,,,,10,72,8"></lb> |
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11110 εἰ τύχοι, πρότερον μεῖζον τὸ <lb n="1799,001,,,,10,72,9"></lb> |
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11111 ΑΒ τοῦ ΓΔ· καὶ ἐκκείσθω <lb n="1799,001,,,,10,72,10"></lb> |
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11112 ῥητὴ ἡ ΕΖ, καὶ τῷ μὲν ΑΒ <lb n="1799,001,,,,10,72,11"></lb> |
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11113 ἴσον παρὰ τὴν ΕΖ παραβεβλή-<lb n="1799,001,,,,10,72,12"></lb> |
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11114 σθω τὸ ΕΗ πλάτος ποιοῦν τὴν <lb n="1799,001,,,,10,72,13"></lb> |
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11115 ΕΘ, τῷ δὲ ΓΔ ἴσον τὸ ΘΙ πλά-<lb n="1799,001,,,,10,72,14"></lb> |
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11116 τος ποιοῦν τὴν ΘΚ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,72,15"></lb> |
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11117 μέσον ἐστὶν ἑκάτερον τῶν ΑΒ, ΓΔ, μέσον ἄρα καὶ <lb n="1799,001,,,,10,72,16"></lb> |
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11118 ἑκάτερον τῶν ΕΗ, ΘΙ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΖΕ παρά-<lb n="1799,001,,,,10,72,17"></lb> |
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11119 κειται πλάτος ποιοῦν τὰς ΕΘ, ΘΚ· ἑκατέρα ἄρα τῶν <lb n="1799,001,,,,10,72,18"></lb> |
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11120 ΕΘ, ΘΚ ῥητή ἐστι καὶ ἀσύμμετρος τῇ ΕΖ μήκει.</s> <s>καὶ <lb n="1799,001,,,,10,72,19"></lb> |
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11121 ἐπεὶ ἀσύμμετρόν ἐστι τὸ ΑΒ τῷ ΓΔ, καί ἐστιν ἴσον τὸ <lb n="1799,001,,,,10,72,20"></lb> |
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11122 μὲν ΑΒ τῷ ΕΗ, τὸ δὲ ΓΔ τῷ ΘΙ, ἀσύμμετρον ἄρα ἐστὶ <lb n="1799,001,,,,10,72,21"></lb> |
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11123 καὶ τὸ ΕΗ τῷ ΘΙ.</s> <s>ὡς δὲ τὸ ΕΗ πρὸς τὸ ΘΙ, οὕτως ἐστὶν <pb></pb> <lb n="1799,001,,,,10,72,22"></lb> |
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11124 ἡ ΕΘ πρὸς ΘΚ· ἀσύμμετρος ἄρα ἐστὶν ἡ ΕΘ τῇ ΘΚ <lb n="1799,001,,,,10,72,23"></lb> |
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11125 μήκει.</s> <s>αἱ ΕΘ, ΘΚ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,72,24"></lb> |
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11126 μετροι· ἐκ δύο ἄρα ὀνομάτων ἐστὶν ἡ ΕΚ.</s> <s>ἤτοι δὲ ἡ ΕΘ <lb n="1799,001,,,,10,72,25"></lb> |
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11127 τῆς ΘΚ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ ἢ τῷ <lb n="1799,001,,,,10,72,26"></lb> |
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11128 ἀπὸ ἀσυμμέτρου.</s> <s>δυνάσθω πρότερον τῷ ἀπὸ συμμέτρου <lb n="1799,001,,,,10,72,27"></lb> |
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11129 ἑαυτῇ μήκει· καὶ οὐδετέρα τῶν ΕΘ, ΘΚ σύμμετρός <lb n="1799,001,,,,10,72,28"></lb> |
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11130 ἐστι τῇ ἐκκειμένῃ ῥητῇ τῇ ΕΖ μήκει· ἡ ΕΚ ἄρα ἐκ δύο <lb n="1799,001,,,,10,72,29"></lb> |
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11131 ὀνομάτων ἐστὶ τρίτη.</s> <s>ῥητὴ δὲ ἡ ΕΖ· ἐὰν δὲ χωρίον <lb n="1799,001,,,,10,72,30"></lb> |
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11132 περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο ὀνομάτων τρίτης, <lb n="1799,001,,,,10,72,31"></lb> |
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11133 ἡ τὸ χωρίον δυναμένη ἐκ δύο μέσων ἐστὶ δευτέρα· ἡ ἄρα <lb n="1799,001,,,,10,72,32"></lb> |
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11134 τὸ ΕΙ, τουτέστι τὸ ΑΔ, δυναμένη ἐκ δύο μέσων ἐστὶ <lb n="1799,001,,,,10,72,33"></lb> |
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11135 δευτέρα.</s> <s>ἀλλὰ δὴ ἡ ΕΘ τῆς ΘΚ μεῖζον δυνάσθω τῷ ἀπὸ <lb n="1799,001,,,,10,72,34"></lb> |
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11136 ἀσυμμέτρου ἑαυτῇ μήκει· καὶ ἀσύμμετρός ἐστιν ἑκατέρα <lb n="1799,001,,,,10,72,35"></lb> |
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11137 τῶν ΕΘ, ΘΚ τῇ ΕΖ μήκει· ἡ ἄρα ΕΚ ἐκ δύο ὀνομάτων <lb n="1799,001,,,,10,72,36"></lb> |
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11138 ἐστὶν ἕκτη.</s> <s>ἐὰν δὲ χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς <lb n="1799,001,,,,10,72,37"></lb> |
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11139 ἐκ δύο ὀνομάτων ἕκτης, ἡ τὸ χωρίον δυναμένη ἡ δύο μέσα <lb n="1799,001,,,,10,72,38"></lb> |
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11140 δυναμένη ἐστίν· ὥστε καὶ ἡ τὸ ΑΔ χωρίον δυναμένη <lb n="1799,001,,,,10,72,39"></lb> |
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11141 ἡ δύο μέσα δυναμένη ἐστίν.</s> <lb n="1799,001,,,,10,72,40"></lb> |
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11142 <s>[Ὁμοίως δὴ δείξομεν, ὅτι κἂν ἔλαττον ᾖ τὸ ΑΒ τοῦ <lb n="1799,001,,,,10,72,41"></lb> |
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11143 ΓΔ, ἡ τὸ ΑΔ χωρίον δυναμένη ἢ ἐκ δύο μέσων δευτέρα <lb n="1799,001,,,,10,72,42"></lb> |
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11144 ἐστὶν ἤτοι δύο μέσα δυναμένη].</s> <lb n="1799,001,,,,10,72,43"></lb> |
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11145 <s>Δύο ἄρα μέσων ἀσυμμέτρων ἀλλήλοις συντιθεμένων <lb n="1799,001,,,,10,72,44"></lb> |
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11146 αἱ λοιπαὶ δύο ἄλογοι γίγνονται ἤτοι ἐκ δύο μέσων δευτέρα <lb n="1799,001,,,,10,72,45"></lb> |
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11147 ἢ δύο μέσα δυναμένη.</s> <s><pb></pb> <lb n="1799,001,,,,10,72,46"></lb> |
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11148 Ἡ ἐκ δύο ὀνομάτων καὶ αἱ μετ' αὐτὴν ἄλογοι οὔτε τῇ <lb n="1799,001,,,,10,72,47"></lb> |
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11149 μέσῃ οὔτε ἀλλήλαις εἰσὶν αἱ αὐταί.</s> <s>τὸ μὲν γὰρ ἀπὸ μέσης <lb n="1799,001,,,,10,72,48"></lb> |
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11150 παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ῥητὴν καὶ <lb n="1799,001,,,,10,72,49"></lb> |
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11151 ἀσύμμετρον τῇ παρ' ἣν παράκειται μήκει.</s> <s>τὸ δὲ ἀπὸ τῆς <lb n="1799,001,,,,10,72,50"></lb> |
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11152 ἐκ δύο ὀνομάτων παρὰ ῥητὴν παραβαλλόμενον πλάτος <lb n="1799,001,,,,10,72,51"></lb> |
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11153 ποιεῖ τὴν ἐκ δύο ὀνομάτων πρώτην.</s> <s>τὸ δὲ ἀπὸ τῆς ἐκ <lb n="1799,001,,,,10,72,52"></lb> |
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11154 δύο μέσων πρώτης παρὰ ῥητὴν παραβαλλόμενον πλάτος <lb n="1799,001,,,,10,72,53"></lb> |
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11155 ποιεῖ τὴν ἐκ δύο ὀνομάτων δευτέραν.</s> <s>τὸ δὲ ἀπὸ τῆς ἐκ <lb n="1799,001,,,,10,72,54"></lb> |
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11156 δύο μέσων δευτέρας παρὰ ῥητὴν παραβαλλόμενον πλάτος <lb n="1799,001,,,,10,72,55"></lb> |
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11157 ποιεῖ τὴν ἐκ δύο ὀνομάτων τρίτην.</s> <s>τὸ δὲ ἀπὸ τῆς μείζονος <lb n="1799,001,,,,10,72,56"></lb> |
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11158 παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο <lb n="1799,001,,,,10,72,57"></lb> |
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11159 ὀνομάτων τετάρτην.</s> <s>τὸ δὲ ἀπὸ τῆς ῥητὸν καὶ μέσον <lb n="1799,001,,,,10,72,58"></lb> |
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11160 δυναμένης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ τὴν <lb n="1799,001,,,,10,72,59"></lb> |
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11161 ἐκ δύο ὀνομάτων πέμπτην.</s> <s>τὸ δὲ ἀπὸ τῆς δύο μέσα δυνα-<lb n="1799,001,,,,10,72,60"></lb> |
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11162 μένης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ τὴν <lb n="1799,001,,,,10,72,61"></lb> |
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11163 ἐκ δύο ὀνομάτων ἕκτην.</s> <s>τὰ δ' εἰρημένα πλάτη δια-<lb n="1799,001,,,,10,72,62"></lb> |
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11164 φέρει τοῦ τε πρώτου καὶ ἀλλήλων, τοῦ μὲν πρώτου, ὅτι <lb n="1799,001,,,,10,72,63"></lb> |
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11165 ῥητή ἐστιν, ἀλλήλων δέ, ὅτι τῇ τάξει οὐκ εἰσὶν αἱ αὐταί· <lb n="1799,001,,,,10,72,64"></lb> |
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11166 ὥστε καὶ αὐταὶ αἱ ἄλογοι διαφέρουσιν ἀλλήλων.</s> <lb n="1799,001,,,,10,72,65"></lb> |
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11167 <s>Ἐὰν ἀπὸ ῥητῆς ῥητὴ ἀφαιρεθῇ δυνάμει μόνον σύμμετρος <lb n="1799,001,,,,10,73,1"></lb> |
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11168 οὖσα τῇ ὅλῃ, ἡ λοιπὴ ἄλογός ἐστιν· καλείσθω δὲ ἀποτομή.</s> <s><pb></pb> <lb n="1799,001,,,,10,73,2"></lb> |
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11169 Ἀπὸ γὰρ ῥητῆς τῆς ΑΒ ῥητὴ ἀφῃρήσθω ἡ ΒΓ δυνάμει <lb n="1799,001,,,,10,73,3"></lb> |
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11170 μόνον σύμμετρος οὖσα τῇ ὅλῃ· λέγω, ὅτι ἡ λοιπὴ ἡ ΑΓ <lb n="1799,001,,,,10,73,4"></lb> |
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11171 ἄλογός ἐστιν ἡ καλουμένη ἀποτομή.</s> <lb n="1799,001,,,,10,73,5"></lb> |
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11172 <s>Ἐπεὶ γὰρ ἀσύμμετρός ἐστιν ἡ ΑΒ τῇ ΒΓ μήκει, καί <lb n="1799,001,,,,10,73,6"></lb> |
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11173 ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ τῆς ΑΒ <lb n="1799,001,,,,10,73,7"></lb> |
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11174 πρὸς τὸ ὑπὸ τῶν ΑΒ, ΒΓ, ἀσύμμετρον ἄρα ἐστὶ τὸ ἀπὸ <lb n="1799,001,,,,10,73,8"></lb> |
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11175 τῆς ΑΒ τῷ ὑπὸ τῶν ΑΒ, ΒΓ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΒ <lb n="1799,001,,,,10,73,9"></lb> |
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11176 σύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ τετράγωνα, τῷ δὲ <lb n="1799,001,,,,10,73,10"></lb> |
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11177 ὑπὸ τῶν ΑΒ, ΒΓ σύμμετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,73,11"></lb> |
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11178 ΒΓ.</s> <s>καὶ ἐπειδήπερ τὰ ἀπὸ τῶν ΑΒ, ΒΓ ἴσα ἐστὶ τῷ <lb n="1799,001,,,,10,73,12"></lb> |
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11179 δὶς ὑπὸ τῶν ΑΒ, ΒΓ μετὰ τοῦ ἀπὸ ΓΑ, καὶ λοιπῷ ἄρα <lb n="1799,001,,,,10,73,13"></lb> |
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11180 τῷ ἀπὸ τῆς ΑΓ ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ.</s> <lb n="1799,001,,,,10,73,14"></lb> |
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11181 <s>ῥητὰ δὲ τὰ ἀπὸ τῶν ΑΒ, ΒΓ· ἄλογος ἄρα ἐστὶν ἡ ΑΓ· <lb n="1799,001,,,,10,73,15"></lb> |
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11182 καλείσθω δὲ ἀποτομή.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,73,16"></lb> |
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11183 Ἐὰν ἀπὸ μέσης μέση ἀφαιρεθῇ δυνάμει μόνον σύμμετρος <lb n="1799,001,,,,10,74,1"></lb> |
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11184 οὖσα τῇ ὅλῃ, μετὰ δὲ τῆς ὅλης ῥητὸν περιέχουσα, ἡ <lb n="1799,001,,,,10,74,2"></lb> |
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11185 λοιπὴ ἄλογός ἐστιν· καλείσθω δὲ μέσης ἀποτομὴ πρώτη.</s> <lb n="1799,001,,,,10,74,3"></lb> |
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11186 <s>Ἀπὸ γὰρ μέσης τῆς ΑΒ μέση ἀφῃρήσθω ἡ ΒΓ δυνάμει <lb n="1799,001,,,,10,74,4"></lb> |
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11187 μόνον σύμμετρος οὖσα τῇ ΑΒ, μετὰ δὲ τῆς ΑΒ ῥητὸν <lb n="1799,001,,,,10,74,5"></lb> |
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11188 ποιοῦσα τὸ ὑπὸ τῶν ΑΒ, ΒΓ· λέγω, ὅτι ἡ λοιπὴ ἡ ΑΓ <lb n="1799,001,,,,10,74,6"></lb> |
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11189 ἄλογός ἐστιν· καλείσθω δὲ μέσης ἀποτομὴ πρώτη.</s> <lb n="1799,001,,,,10,74,7"></lb> |
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11190 <s>Ἐπεὶ γὰρ αἱ ΑΒ, ΒΓ μέσαι εἰσίν, μέσα ἐστὶ καὶ τὰ <lb n="1799,001,,,,10,74,8"></lb> |
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11191 ἀπὸ τῶν ΑΒ, ΒΓ.</s> <s>ῥητὸν δὲ τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ· <lb n="1799,001,,,,10,74,9"></lb> |
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11192 ἀσύμμετρα ἄρα τὰ ἀπὸ τῶν ΑΒ, ΒΓ τῷ δὶς ὑπὸ <lb n="1799,001,,,,10,74,10"></lb> |
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11193 τῶν ΑΒ, ΒΓ· καὶ λοιπῷ ἄρα τῷ ἀπὸ τῆς ΑΓ <lb n="1799,001,,,,10,74,11"></lb> |
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11194 ἀσύμμετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ, ἐπεὶ <lb n="1799,001,,,,10,74,12"></lb> |
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11195 κἂν τὸ ὅλον ἑνὶ αὐτῶν ἀσύμμετρον ᾖ, καὶ τὰ ἐξ <lb n="1799,001,,,,10,74,13"></lb> |
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11196 ἀρχῆς μεγέθη ἀσύμμετρα ἔσται.</s> <s>ῥητὸν δὲ τὸ δὶς <lb n="1799,001,,,,10,74,14"></lb> |
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11197 ὑπὸ τῶν ΑΒ, ΒΓ· ἄλογον ἄρα τὸ ἀπὸ τῆς ΑΓ· <lb n="1799,001,,,,10,74,15"></lb> |
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11198 ἄλογος ἄρα ἐστὶν ἡ ΑΓ· καλείσθω δὲ μέσης ἀπο-<lb n="1799,001,,,,10,74,16"></lb> |
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11199 τομὴ πρώτη.</s> <lb n="1799,001,,,,10,74,17"></lb> |
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11200 <s>Ἐὰν ἀπὸ μέσης μέση ἀφαιρεθῇ δυνάμει μόνον σύμμετρος <lb n="1799,001,,,,10,75,1"></lb> |
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11201 οὖσα τῇ ὅλη, μετὰ δὲ τῆς ὅλης μέσον περιέχουσα, ἡ λοιπὴ <lb n="1799,001,,,,10,75,2"></lb> |
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11202 ἄλογός ἐστιν· καλείσθω δὲ μέσης ἀποτομὴ δευτέρα.</s> <s><pb></pb> <lb n="1799,001,,,,10,75,3"></lb> |
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11203 Ἀπὸ γὰρ μέσης τῆς ΑΒ μέση ἀφῃρήσθω ἡ ΓΒ δυνάμει <lb n="1799,001,,,,10,75,4"></lb> |
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11204 μόνον σύμμετρος οὖσα τῇ ὅλῃ τῇ ΑΒ, μετὰ δὲ τῆς ὅλης <lb n="1799,001,,,,10,75,5"></lb> |
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11205 τῆς ΑΒ μέσον περιέχουσα τὸ ὑπὸ τῶν ΑΒ, ΒΓ· λέγω, <lb n="1799,001,,,,10,75,6"></lb> |
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11206 ὅτι ἡ λοιπὴ ἡ ΑΓ ἄλογός ἐστιν· καλείσθω δὲ μέσης <lb n="1799,001,,,,10,75,7"></lb> |
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11207 ἀποτομὴ δευτέρα.</s> <lb n="1799,001,,,,10,75,8"></lb> |
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11208 <s>Ἐκκείσθω γὰρ ῥητὴ ἡ ΔΙ, <lb n="1799,001,,,,10,75,9"></lb> |
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11209 καὶ τοῖς μὲν ἀπὸ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,10,75,10"></lb> |
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11210 ἴσον παρὰ τὴν ΔΙ παραβε-<lb n="1799,001,,,,10,75,11"></lb> |
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11211 βλήσθω τὸ ΔΕ πλάτος ποιοῦν <lb n="1799,001,,,,10,75,12"></lb> |
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11212 τὴν ΔΗ, τῷ δὲ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,75,13"></lb> |
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11213 ΑΒ, ΒΓ ἴσον παρὰ τὴν ΔΙ <lb n="1799,001,,,,10,75,14"></lb> |
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11214 παραβεβλήσθω τὸ ΔΘ πλάτος <lb n="1799,001,,,,10,75,15"></lb> |
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11215 ποιοῦν τὴν ΔΖ· λοιπὸν ἄρα <lb n="1799,001,,,,10,75,16"></lb> |
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11216 τὸ ΖΕ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΓ.</s> <s>καὶ ἐπεὶ μέσα καὶ <lb n="1799,001,,,,10,75,17"></lb> |
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11217 σύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ, μέσον ἄρα καὶ τὸ <lb n="1799,001,,,,10,75,18"></lb> |
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11218 ΔΕ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΔΙ παράκειται πλάτος ποι-<lb n="1799,001,,,,10,75,19"></lb> |
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11219 οῦν τὴν ΔΗ· ῥητὴ ἄρα ἐστὶν ἡ ΔΗ καὶ ἀσύμμετρος <lb n="1799,001,,,,10,75,20"></lb> |
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11220 τῇ ΔΙ μήκει.</s> <s>πάλιν, ἐπεὶ μέσον ἐστὶ τὸ ὑπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,75,21"></lb> |
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11221 ΒΓ, καὶ τὸ δὶς ἄρα ὑπὸ τῶν ΑΒ, ΒΓ μέσον ἐστίν.</s> <s>καί <lb n="1799,001,,,,10,75,22"></lb> |
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11222 ἐστιν ἴσον τῷ ΔΘ· καὶ τὸ ΔΘ ἄρα μέσον ἐστίν.</s> <s>καὶ παρὰ <lb n="1799,001,,,,10,75,23"></lb> |
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11223 ῥητὴν τὴν ΔΙ παραβέβληται πλάτος ποιοῦν τὴν ΔΖ· <pb></pb> <lb n="1799,001,,,,10,75,24"></lb> |
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11224 ῥητὴ ἄρα ἐστὶν ἡ ΔΖ καὶ ἀσύμμετρος τῇ ΔΙ μήκει.</s> <s>καὶ <lb n="1799,001,,,,10,75,25"></lb> |
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11225 ἐπεὶ αἱ ΑΒ, ΒΓ δυνάμει μόνον σύμμετροί εἰσιν, ἀσύμ-<lb n="1799,001,,,,10,75,26"></lb> |
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11226 μετρος ἄρα ἐστὶν ἡ ΑΒ τῇ ΒΓ μήκει· ἀσύμμετρον ἄρα <lb n="1799,001,,,,10,75,27"></lb> |
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11227 καὶ τὸ ἀπὸ τῆς ΑΒ τετράγωνον τῷ ὑπὸ τῶν ΑΒ, ΒΓ.</s> <lb n="1799,001,,,,10,75,28"></lb> |
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11228 <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΒ σύμμετρά ἐστι τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,75,29"></lb> |
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11229 ΑΒ, ΒΓ, τῷ δὲ ὑπὸ τῶν ΑΒ, ΒΓ σύμμετρόν ἐστι τὸ <lb n="1799,001,,,,10,75,30"></lb> |
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11230 δὶς ὑπὸ τῶν ΑΒ, ΒΓ· ἀσύμμετρον ἄρα ἐστὶ τὸ δὶς ὑπὸ <lb n="1799,001,,,,10,75,31"></lb> |
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11231 τῶν ΑΒ, ΒΓ τοῖς ἀπὸ τῶν ΑΒ, ΒΓ.</s> <s>ἴσον δὲ τοῖς μὲν <lb n="1799,001,,,,10,75,32"></lb> |
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11232 ἀπὸ τῶν ΑΒ, ΒΓ τὸ ΔΕ, τῷ δὲ δὶς ὑπὸ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,10,75,33"></lb> |
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11233 τὸ ΔΘ· ἀσύμμετρον ἄρα [ἐστὶ] τὸ ΔΕ τῷ ΔΘ.</s> <s>ὡσ δὲ <lb n="1799,001,,,,10,75,34"></lb> |
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11234 τὸ ΔΕ πρὸς τὸ ΔΘ, οὕτως ἡ ΗΔ πρὸς τὴν ΔΖ· ἀσύμ-<lb n="1799,001,,,,10,75,35"></lb> |
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11235 μετρος ἄρα ἐστὶν ἡ ΗΔ τῇ ΔΖ.</s> <s>καί εἰσιν ἀμφότεραι <lb n="1799,001,,,,10,75,36"></lb> |
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11236 ῥηταί· αἱ ἄρα ΗΔ, ΔΖ ῥηταί εἰσι δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,75,37"></lb> |
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11237 μετροι· ἡ ΖΗ ἄρα ἀποτομή ἐστιν.</s> <s>ῥητὴ δὲ ἡ ΔΙ· τὸ <lb n="1799,001,,,,10,75,38"></lb> |
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11238 δὲ ὑπὸ ῥητῆς καὶ ἀλόγου περιεχόμενον ἄλογόν ἐστιν, καὶ <lb n="1799,001,,,,10,75,39"></lb> |
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11239 ἡ δυναμένη αὐτὸ ἄλογός ἐστιν.</s> <s>καὶ δύναται τὸ ΖΕ ἡ <lb n="1799,001,,,,10,75,40"></lb> |
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11240 ΑΓ· ἡ ΑΓ ἄρα ἄλογός ἐστιν· καλείσθω δὲ μέσης ἀπο-<lb n="1799,001,,,,10,75,41"></lb> |
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11241 τομὴ δευτέρα.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,75,42"></lb> |
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11242 Ἐὰν ἀπὸ εὐθείας εὐθεῖα ἀφαιρεθῇ δυνάμει ἀσύμμετρος <lb n="1799,001,,,,10,76,1"></lb> |
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11243 οὖσα τῇ ὅλῃ, μετὰ δὲ τῆς ὅλης ποιοῦσα τὰ μὲν ἀπ' αὐτῶν <lb n="1799,001,,,,10,76,2"></lb> |
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11244 ἅμα ῥητόν, τὸ δ' ὑπ' αὐτῶν μέσον, ἡ λοιπὴ ἄλογός ἐστιν· <lb n="1799,001,,,,10,76,3"></lb> |
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11245 καλείσθω δὲ ἐλάσσων.</s> <lb n="1799,001,,,,10,76,4"></lb> |
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11246 <s>Ἀπὸ γὰρ εὐθείας τῆς ΑΒ εὐθεῖα ἀφῃρήσθω ἡ ΒΓ <lb n="1799,001,,,,10,76,5"></lb> |
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11247 δυνάμει ἀσύμμετρος οὖσα τῇ ὅλῃ ποιοῦσα τὰ προκείμενα. <lb n="1799,001,,,,10,76,6"></lb> |
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11248 λέγω, ὅτι ἡ λοιπὴ ἡ ΑΓ ἄλογός ἐστιν ἡ καλουμένη <lb n="1799,001,,,,10,76,7"></lb> |
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11249 ἐλάσσων.</s> <lb n="1799,001,,,,10,76,8"></lb> |
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11250 <s>Ἐπεὶ γὰρ τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,76,9"></lb> |
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11251 ΒΓ τετραγώνων ῥητόν ἐστιν, τὸ δὲ δὶς ὑπὸ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,10,76,10"></lb> |
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11252 μέσον, ἀσύμμετρα ἄρα ἐστὶ τὰ ἀπὸ τῶν ΑΒ, ΒΓ τῷ <lb n="1799,001,,,,10,76,11"></lb> |
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11253 δὶς ὑπὸ τῶν ΑΒ, ΒΓ· καὶ ἀναστρέψαντι λοιπῷ τῷ ἀπὸ <lb n="1799,001,,,,10,76,12"></lb> |
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11254 τῆς ΑΓ ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ.</s> <s>ῥητὰ <lb n="1799,001,,,,10,76,13"></lb> |
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11255 δὲ τὰ ἀπὸ τῶν ΑΒ, ΒΓ.</s> <s>ἄλογον ἄρα τὸ ἀπὸ τῆς ΑΓ· <lb n="1799,001,,,,10,76,14"></lb> |
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11256 ἄλογος ἄρα ἡ ΑΓ· καλείσθω δὲ ἐλάσσων.</s> <s>ὅπερ ἔδει <lb n="1799,001,,,,10,76,15"></lb> |
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11257 δεῖξαι.</s> <lb n="1799,001,,,,10,76,16"></lb> |
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11258 <s>Ἐὰν ἀπὸ εὐθείας εὐθεῖα ἀφαιρεθῇ δυνάμει ἀσύμμετρος <lb n="1799,001,,,,10,77,1"></lb> |
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11259 οὖσα τῇ ὅλῃ, μετὰ δὲ τῆς ὅλης ποιοῦσα τὸ μὲν συγκεί- <pb></pb><lb n="1799,001,,,,10,77,2"></lb> |
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11260 μενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων μέσον, τὸ δὲ δὶς <lb n="1799,001,,,,10,77,3"></lb> |
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11261 ὑπ' αὐτῶν ῥητόν, ἡ λοιπὴ ἄλογός ἐστιν· καλείσθω δὲ ἡ <lb n="1799,001,,,,10,77,4"></lb> |
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11262 μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα.</s> <lb n="1799,001,,,,10,77,5"></lb> |
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11263 <s>Ἀπὸ γὰρ εὐθείας τῆς ΑΒ εὐθεῖα ἀφῃρήσθω ἡ ΒΓ <lb n="1799,001,,,,10,77,6"></lb> |
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11264 δυνάμει ἀσύμμετρος οὖσα τῇ ΑΒ ποιοῦσα τὰ προ-<lb n="1799,001,,,,10,77,7"></lb> |
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11265 κείμενα· λέγω, ὅτι ἡ λοιπὴ ἡ ΑΓ ἄλογός ἐστιν ἡ προ-<lb n="1799,001,,,,10,77,8"></lb> |
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11266 ειρημένη.</s> <lb n="1799,001,,,,10,77,9"></lb> |
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11267 <s>Ἐπεὶ γὰρ τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,77,10"></lb> |
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11268 ΒΓ τετραγώνων μέσον ἐστίν, τὸ δὲ δὶς ὑπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,77,11"></lb> |
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11269 ΒΓ ῥητόν, ἀσύμμετρα ἄρα ἐστὶ τὰ ἀπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,77,12"></lb> |
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11270 ΒΓ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ· καὶ λοιπὸν ἄρα τὸ <lb n="1799,001,,,,10,77,13"></lb> |
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11271 ἀπὸ τῆς ΑΓ ἀσύμμετρόν ἐστι τῷ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,77,14"></lb> |
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11272 ΑΒ, ΒΓ.</s> <s>καί ἐστι τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ ῥη-<lb n="1799,001,,,,10,77,15"></lb> |
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11273 τόν· τὸ ἄρα ἀπὸ τῆς ΑΓ ἄλογόν ἐστιν· ἄλογος <lb n="1799,001,,,,10,77,16"></lb> |
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11274 ἄρα ἐστὶν ἡ ΑΓ· καλείσθω δὲ ἡ μετὰ ῥητοῦ μέσον <lb n="1799,001,,,,10,77,17"></lb> |
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11275 τὸ ὅλον ποιοῦσα.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,77,18"></lb> |
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11276 <s>Ἐὰν ἀπὸ εὐθείας εὐθεῖα ἀφαιρεθῇ δυνάμει ἀσύμμετρος <lb n="1799,001,,,,10,78,1"></lb> |
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11277 οὖσα τῇ ὅλῃ, μετὰ δὲ τῆς ὅλης ποιοῦσα τό τε συγκείμενον <lb n="1799,001,,,,10,78,2"></lb> |
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11278 ἐκ τῶν ἀπ' αὐτῶν τετραγώνων μέσον τό τε δὶς ὑπ' <pb></pb> <lb n="1799,001,,,,10,78,3"></lb> |
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11279 αὐτῶν μέσον καὶ ἔτι τὰ ἀπ' αὐτῶν τετράγωνα ἀσύμμετρα <lb n="1799,001,,,,10,78,4"></lb> |
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11280 τῷ δὶς ὑπ' αὐτῶν, ἡ λοιπὴ ἄλογός ἐστιν· καλείσθω δὲ ἡ <lb n="1799,001,,,,10,78,5"></lb> |
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11281 μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα.</s> <lb n="1799,001,,,,10,78,6"></lb> |
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11282 <s>Ἀπὸ γὰρ εὐθείας τῆς ΑΒ εὐθεῖα ἀφῃρήσθω ἡ ΒΓ <lb n="1799,001,,,,10,78,7"></lb> |
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11283 δυνάμει ἀσύμμετρος οὖσα τῇ ΑΒ ποιοῦσα τὰ προκείμενα· <lb n="1799,001,,,,10,78,8"></lb> |
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11284 λέγω, ὅτι ἡ λοιπὴ ἡ ΑΓ ἄλογός ἐστιν ἡ καλουμένη ἡ μετὰ <lb n="1799,001,,,,10,78,9"></lb> |
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11285 μέσου μέσον τὸ ὅλον ποιοῦσα.</s> <lb n="1799,001,,,,10,78,10"></lb> |
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11286 <s>Ἐκκείσθω γὰρ ῥητὴ ἡ ΔΙ, καὶ τοῖς μὲν ἀπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,78,11"></lb> |
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11287 ΒΓ ἴσον παρὰ τὴν ΔΙ παραβεβλήσθω τὸ ΔΕ πλάτος <lb n="1799,001,,,,10,78,12"></lb> |
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11288 ποιοῦν τὴν ΔΗ, τῷ δὲ δὶς <lb n="1799,001,,,,10,78,13"></lb> |
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11289 ὑπὸ τῶν ΑΒ, ΒΓ ἴσον ἀφῃρή-<lb n="1799,001,,,,10,78,14"></lb> |
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11290 σθω τὸ ΔΘ [πλάτος ποιοῦν <lb n="1799,001,,,,10,78,15"></lb> |
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11291 τὴν ΔΖ].</s> <s>λοιπὸν ἄρα τὸ ΖΕ <lb n="1799,001,,,,10,78,16"></lb> |
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11292 ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΓ· <lb n="1799,001,,,,10,78,17"></lb> |
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11293 ὥστε ἡ ΑΓ δύναται τὸ ΖΕ.</s> <lb n="1799,001,,,,10,78,18"></lb> |
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11294 <s>καὶ ἐπεὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,10,78,19"></lb> |
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11295 τετραγώνων μέσον ἐστὶ καί ἐστιν ἴσον τῷ ΔΕ, μέσον <lb n="1799,001,,,,10,78,20"></lb> |
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11296 ἄρα [ἐστὶ] τὸ ΔΕ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΔΙ παράκειται <lb n="1799,001,,,,10,78,21"></lb> |
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11297 πλάτος ποιοῦν τὴν ΔΗ· ῥητὴ ἄρα ἐστὶν ἡ ΔΗ καὶ <lb n="1799,001,,,,10,78,22"></lb> |
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11298 ἀσύμμετρος τῇ ΔΙ μήκει.</s> <s>πάλιν, ἐπεὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,78,23"></lb> |
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11299 ΑΒ, ΒΓ μέσον ἐστὶ καί ἐστιν ἴσον τῷ ΔΘ, τὸ ἄρα ΔΘ <lb n="1799,001,,,,10,78,24"></lb> |
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11300 μέσον ἐστίν.</s> <s>καὶ παρὰ ῥητὴν τὴν ΔΙ παράκειται πλάτος <lb n="1799,001,,,,10,78,25"></lb> |
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11301 ποιοῦν τὴν ΔΖ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΔΖ καὶ ἀσύμμετρος <pb></pb> <lb n="1799,001,,,,10,78,26"></lb> |
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11302 τῇ ΔΙ μήκει.</s> <s>καὶ ἐπεὶ ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,78,27"></lb> |
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11303 ΒΓ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ, ἀσύμμετρον ἄρα καὶ τὸ ΔΕ <lb n="1799,001,,,,10,78,28"></lb> |
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11304 τῷ ΔΘ.</s> <s>ὡς δὲ τὸ ΔΕ πρὸς τὸ ΔΘ, οὕτως ἐστὶ καὶ ἡ <lb n="1799,001,,,,10,78,29"></lb> |
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11305 ΔΗ πρὸς τὴν ΔΖ· ἀσύμμετρος ἄρα ἡ ΔΗ τῇ ΔΖ.</s> <s>καί <lb n="1799,001,,,,10,78,30"></lb> |
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11306 εἰσιν ἀμφότεραι ῥηταί· αἱ ΗΔ, ΔΖ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,78,31"></lb> |
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11307 δυνάμει μόνον σύμμετροι.</s> <s>ἀποτομὴ ἄρα ἐστὶν ἡ ΖΗ· <lb n="1799,001,,,,10,78,32"></lb> |
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11308 ῥητὴ δὲ ἡ ΖΘ.</s> <s>τὸ δὲ ὑπὸ ῥητῆς καὶ ἀποτομῆς περιεχό-<lb n="1799,001,,,,10,78,33"></lb> |
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11309 μενον [ὀρθογώνιον] ἄλογόν ἐστιν, καὶ ἡ δυναμένη αὐτὸ <lb n="1799,001,,,,10,78,34"></lb> |
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11310 ἄλογός ἐστιν.</s> <s>καὶ δύναται τὸ ΖΕ ἡ ΑΓ· ἡ ΑΓ ἄρα <lb n="1799,001,,,,10,78,35"></lb> |
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11311 ἄλογός ἐστιν· καλείσθω δὲ ἡ μετὰ μέσου μέσον τὸ ὅλον <lb n="1799,001,,,,10,78,36"></lb> |
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11312 ποιοῦσα.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,78,37"></lb> |
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11313 <s>Τῇ ἀποτομῇ μία [μόνον] προσαρμόζει εὐθεῖα ῥητὴ <lb n="1799,001,,,,10,79,1"></lb> |
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11314 δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ.</s> <lb n="1799,001,,,,10,79,2"></lb> |
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11315 <s>Ἔστω ἀποτομὴ ἡ ΑΒ, προσαρμόζουσα δὲ αὐτῇ <lb n="1799,001,,,,10,79,3"></lb> |
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11316 ἡ ΒΓ· αἱ ΑΓ, ΓΒ ἄρα ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,79,4"></lb> |
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11317 μόνον σύμμετροι· λέγω, ὅτι τῇ ΑΒ ἑτέρα οὐ <lb n="1799,001,,,,10,79,5"></lb> |
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11318 προσαρμόζει ῥητὴ δυνάμει μόνον σύμμετρος οὖσα <lb n="1799,001,,,,10,79,6"></lb> |
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11319 τῇ ὅλῃ.</s> <lb n="1799,001,,,,10,79,7"></lb> |
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11320 <s>Εἰ γὰρ δυνατόν, προσαρμοζέτω ἡ ΒΔ· καὶ αἱ ΑΔ, <lb n="1799,001,,,,10,79,8"></lb> |
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11321 ΔΒ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι.</s> <s>καὶ <lb n="1799,001,,,,10,79,9"></lb> |
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11322 ἐπεί, ᾧ ὑπερέχει τὰ ἀπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ <pb></pb> <lb n="1799,001,,,,10,79,10"></lb> |
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11323 τῶν ΑΔ, ΔΒ, τούτῳ ὑπερέχει καὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ <lb n="1799,001,,,,10,79,11"></lb> |
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11324 τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ· τῷ γὰρ αὐτῷ τῷ ἀπὸ τῆς ΑΒ <lb n="1799,001,,,,10,79,12"></lb> |
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11325 ἀμφότερα ὑπερέχει· ἐναλλὰξ ἄρα, ᾧ ὑπερέχει τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,79,13"></lb> |
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11326 ΑΔ, ΔΒ τῶν ἀπὸ τῶν ΑΓ, ΓΒ, τούτῳ ὑπερέχει [καὶ] τὸ <lb n="1799,001,,,,10,79,14"></lb> |
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11327 δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ.</s> <s>τὰ δὲ <lb n="1799,001,,,,10,79,15"></lb> |
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11328 ἀπὸ τῶν ΑΔ, ΔΒ τῶν ἀπὸ τῶν ΑΓ, ΓΒ ὑπερέχει ῥητῷ· <lb n="1799,001,,,,10,79,16"></lb> |
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11329 ῥητὰ γὰρ ἀμφότερα.</s> <s>καὶ τὸ δὶς ἄρα ὑπὸ τῶν ΑΔ, ΔΒ <lb n="1799,001,,,,10,79,17"></lb> |
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11330 τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ὑπερέχει ῥητῷ· ὅπερ ἐστὶν <lb n="1799,001,,,,10,79,18"></lb> |
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11331 ἀδύνατον· μέσα γὰρ ἀμφότερα, μέσον δὲ μέσου οὐχ <lb n="1799,001,,,,10,79,19"></lb> |
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11332 ὑπερέχει ῥητῷ.</s> <s>τῇ ἄρα ΑΒ ἑτέρα οὐ προσαρμόζει ῥητὴ <lb n="1799,001,,,,10,79,20"></lb> |
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11333 δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ.</s> <lb n="1799,001,,,,10,79,21"></lb> |
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11334 <s>Μία ἄρα μόνη τῇ ἀποτομῇ προσαρμόζει ῥητὴ δυνάμει <lb n="1799,001,,,,10,79,22"></lb> |
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11335 μόνον σύμμετρος οὖσα τῇ ὅλῃ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,79,23"></lb> |
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11336 <s>Τῇ μέσης ἀποτομῇ πρώτῃ μία μόνον προσαρμόζει <lb n="1799,001,,,,10,80,1"></lb> |
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11337 εὐθεῖα μέση δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ, μετὰ <lb n="1799,001,,,,10,80,2"></lb> |
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11338 δὲ τῆς ὅλης ῥητὸν περιέχουσα.</s> <lb n="1799,001,,,,10,80,3"></lb> |
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11339 <s>Ἔστω γὰρ μέσης ἀποτομὴ πρώτη ἡ ΑΒ, καὶ τῇ ΑΒ <lb n="1799,001,,,,10,80,4"></lb> |
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11340 προσαρμοζέτω ἡ ΒΓ· αἱ ΑΓ, ΓΒ ἄρα μέσαι εἰσὶ δυνάμει <pb></pb> <lb n="1799,001,,,,10,80,5"></lb> |
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11341 μόνον σύμμετροι ῥητὸν περιέχουσαι τὸ ὑπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,80,6"></lb> |
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11342 ΓΒ· λέγω, ὅτι τῇ ΑΒ ἑτέρα οὐ προσαρμόζει μέση <lb n="1799,001,,,,10,80,7"></lb> |
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11343 δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ, μετὰ δὲ <lb n="1799,001,,,,10,80,8"></lb> |
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11344 τῆς ὅλης ῥητὸν περιέχουσα.</s> <lb n="1799,001,,,,10,80,9"></lb> |
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11345 <s>Εἰ γὰρ δυνατόν, προσαρμοζέτω καὶ ἡ ΔΒ.</s> <s>αἱ <lb n="1799,001,,,,10,80,10"></lb> |
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11346 ἄρα ΑΔ, ΔΒ μέσαι εἰσὶ δυνάμει μόνον σύμμε-<lb n="1799,001,,,,10,80,11"></lb> |
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11347 τροι ῥητὸν περιέχουσαι τὸ ὑπὸ τῶν ΑΔ, ΔΒ.</s> <lb n="1799,001,,,,10,80,12"></lb> |
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11348 <s>καὶ ἐπεί, ᾧ ὑπερέχει τὰ ἀπὸ τῶν ΑΔ, ΔΒ τοῦ <lb n="1799,001,,,,10,80,13"></lb> |
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11349 δὶς ὑπὸ τῶν ΑΔ, ΔΒ, τούτῳ ὑπερέχει καὶ τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,80,14"></lb> |
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11350 ΑΓ, ΓΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ· τῷ γὰρ αὐτῷ <lb n="1799,001,,,,10,80,15"></lb> |
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11351 [πάλιν] ὑπερέχουσι τῷ ἀπὸ τῆς ΑΒ· ἐναλλὰξ ἄρα, ᾧ <lb n="1799,001,,,,10,80,16"></lb> |
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11352 ὑπερέχει τὰ ἀπὸ τῶν ΑΔ, ΔΒ τῶν ἀπὸ τῶν ΑΓ, ΓΒ, <lb n="1799,001,,,,10,80,17"></lb> |
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11353 τούτῳ ὑπερέχει καὶ τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς <lb n="1799,001,,,,10,80,18"></lb> |
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11354 ὑπὸ τῶν ΑΓ, ΓΒ.</s> <s>τὸ δὲ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς <lb n="1799,001,,,,10,80,19"></lb> |
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11355 ὑπὸ τῶν ΑΓ, ΓΒ ὑπερέχει ῥητῷ· ῥητὰ γὰρ ἀμφότερα.</s> <lb n="1799,001,,,,10,80,20"></lb> |
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11356 <s>καὶ τὰ ἀπὸ τῶν ΑΔ, ΔΒ ἄρα τῶν ἀπὸ τῶν ΑΓ, ΓΒ <lb n="1799,001,,,,10,80,21"></lb> |
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11357 [τετραγώνων] ὑπερέχει ῥητῷ· ὅπερ ἐστὶν ἀδύνατον· μέσα <lb n="1799,001,,,,10,80,22"></lb> |
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11358 γάρ ἐστιν ἀμφότερα, μέσον δὲ μέσου οὐχ ὑπερέχει ῥητῷ.</s> <lb n="1799,001,,,,10,80,23"></lb> |
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11359 <s>Τῇ ἄρα μέσης ἀποτομῇ πρώτῃ μία μόνον προσαρμόζει <lb n="1799,001,,,,10,80,24"></lb> |
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11360 εὐθεῖα μέση δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ, μετὰ <lb n="1799,001,,,,10,80,25"></lb> |
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11361 δὲ τῆς ὅλης ῥητὸν περιέχουσα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,80,26"></lb> |
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11362 <s>Τῇ μέσης ἀποτομῇ δευτέρᾳ μία μόνον προσαρμόζει <lb n="1799,001,,,,10,81,1"></lb> |
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11363 εὐθεῖα μέση δυνάμει μόνον σύμμετρος τῇ ὅλῃ, μετὰ δὲ τῆς <lb n="1799,001,,,,10,81,2"></lb> |
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11364 ὅλης μέσον περιέχουσα.</s> <s><pb></pb> <lb n="1799,001,,,,10,81,3"></lb> |
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11365 Ἔστω μέσης ἀποτομὴ δευτέ- <lb n="1799,001,,,,10,81,4"></lb> |
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11366 ρα ἡ ΑΒ καὶ τῇ ΑΒ προσαρμό-<lb n="1799,001,,,,10,81,5"></lb> |
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11367 ζουσα ἡ ΒΓ· αἱ ἄρα ΑΓ, ΓΒ <lb n="1799,001,,,,10,81,6"></lb> |
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11368 μέσαι εἰσὶ δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,81,7"></lb> |
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11369 μετροι μέσον περιέχουσαι τὸ <lb n="1799,001,,,,10,81,8"></lb> |
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11370 ὑπὸ τῶν ΑΓ, ΓΒ· λέγω, ὅτι <lb n="1799,001,,,,10,81,9"></lb> |
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11371 τῇ ΑΒ ἑτέρα οὐ προσαρμόσει <lb n="1799,001,,,,10,81,10"></lb> |
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11372 εὐθεῖα μέση δυνάμει μόνον <lb n="1799,001,,,,10,81,11"></lb> |
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11373 σύμμετρος οὖσα τῇ ὅλῃ, μετὰ <lb n="1799,001,,,,10,81,12"></lb> |
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11374 δὲ τῆς ὅλης μέσον περιέχουσα.</s> <lb n="1799,001,,,,10,81,13"></lb> |
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11375 <s>Εἰ γὰρ δυνατόν, προσαρμοζέτω ἡ ΒΔ· καὶ αἱ ΑΔ, ΔΒ <lb n="1799,001,,,,10,81,14"></lb> |
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11376 ἄρα μέσαι εἰσὶ δυνάμει μόνον σύμμετροι μέσον περιέχουσαι <lb n="1799,001,,,,10,81,15"></lb> |
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11377 τὸ ὑπὸ τῶν ΑΔ, ΔΒ.</s> <s>καὶ ἐκκείσθω ῥητὴ ἡ ΕΖ, καὶ τοῖς <lb n="1799,001,,,,10,81,16"></lb> |
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11378 μὲν ἀπὸ τῶν ΑΓ, ΓΒ ἴσον παρὰ τὴν ΕΖ παραβεβλήσθω <lb n="1799,001,,,,10,81,17"></lb> |
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11379 τὸ ΕΗ πλάτος ποιοῦν τὴν ΕΜ· τῷ δὲ δὶς ὑπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,81,18"></lb> |
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11380 ΓΒ ἴσον ἀφῃρήσθω τὸ ΘΗ πλάτος ποιοῦν τὴν ΘΜ· <lb n="1799,001,,,,10,81,19"></lb> |
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11381 λοιπὸν ἄρα τὸ ΕΛ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ· ὥστε ἡ ΑΒ <lb n="1799,001,,,,10,81,20"></lb> |
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11382 δύναται τὸ ΕΛ.</s> <s>πάλιν δὴ τοῖς ἀπὸ τῶν ΑΔ, ΔΒ ἴσον <lb n="1799,001,,,,10,81,21"></lb> |
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11383 παρὰ τὴν ΕΖ παραβεβλήσθω τὸ ΕΙ πλάτος ποιοῦν τὴν <lb n="1799,001,,,,10,81,22"></lb> |
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11384 ΕΝ· ἔστι δὲ καὶ τὸ ΕΛ ἴσον τῷ ἀπὸ τῆς ΑΒ τετραγώνῳ· <lb n="1799,001,,,,10,81,23"></lb> |
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11385 λοιπὸν ἄρα τὸ ΘΙ ἴσον ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΔ, ΔΒ.</s> <s>καὶ <lb n="1799,001,,,,10,81,24"></lb> |
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11386 ἐπεὶ μέσαι εἰσὶν αἱ ΑΓ, ΓΒ, μέσα ἄρα ἐστὶ καὶ τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,81,25"></lb> |
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11387 ΑΓ, ΓΒ.</s> <s>καί ἐστιν ἴσα τῷ ΕΗ· μέσον ἄρα καὶ τὸ ΕΗ.</s> <s>καὶ <lb n="1799,001,,,,10,81,26"></lb> |
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11388 παρὰ ῥητὴν τὴν ΕΖ παράκειται πλάτος ποιοῦν τὴν ΕΜ· <lb n="1799,001,,,,10,81,27"></lb> |
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11389 ῥητὴ ἄρα ἐστὶν ἡ ΕΜ καὶ ἀσύμμετρος τῇ ΕΖ μήκει.</s> <s>πάλιν, <pb></pb> <lb n="1799,001,,,,10,81,28"></lb> |
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11390 ἐπεὶ μέσον ἐστὶ τὸ ὑπὸ τῶν ΑΓ, ΓΒ, καὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,81,29"></lb> |
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11391 ΑΓ, ΓΒ μέσον ἐστίν.</s> <s>καί ἐστιν ἴσον τῷ ΘΗ· καὶ τὸ ΘΗ <lb n="1799,001,,,,10,81,30"></lb> |
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11392 ἄρα μέσον ἐστίν.</s> <s>καὶ παρὰ ῥητὴν τὴν ΕΖ παράκειται πλάτος <lb n="1799,001,,,,10,81,31"></lb> |
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11393 ποιοῦν τὴν ΘΜ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΘΜ καὶ ἀσύμμετρος <lb n="1799,001,,,,10,81,32"></lb> |
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11394 τῇ ΕΖ μήκει.</s> <s>καὶ ἐπεὶ αἱ ΑΓ, ΓΒ δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,81,33"></lb> |
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11395 μετροί εἰσιν, ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΓ τῇ ΓΒ μήκει.</s> <s>ὡς <lb n="1799,001,,,,10,81,34"></lb> |
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11396 δὲ ἡ ΑΓ πρὸς τὴν ΓΒ, οὕτως ἐστὶ τὸ ἀπὸ τῆς ΑΓ πρὸς <lb n="1799,001,,,,10,81,35"></lb> |
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11397 τὸ ὑπὸ τῶν ΑΓ, ΓΒ· ἀσύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,81,36"></lb> |
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11398 ΑΓ τῷ ὑπὸ τῶν ΑΓ, ΓΒ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΓ <lb n="1799,001,,,,10,81,37"></lb> |
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11399 σύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΓ, ΓΒ, τῷ δὲ ὑπὸ τῶν <lb n="1799,001,,,,10,81,38"></lb> |
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11400 ΑΓ, ΓΒ σύμμετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΓ, ΓΒ· <lb n="1799,001,,,,10,81,39"></lb> |
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11401 ἀσύμμετρα ἄρα ἐστὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῷ δὶς ὑπὸ <lb n="1799,001,,,,10,81,40"></lb> |
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11402 τῶν ΑΓ, ΓΒ.</s> <s>καί ἐστι τοῖς μὲν ἀπὸ τῶν ΑΓ, ΓΒ ἴσον <lb n="1799,001,,,,10,81,41"></lb> |
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11403 τὸ ΕΗ, τῷ δὲ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ἴσον τὸ ΗΘ· ἀσύμ-<lb n="1799,001,,,,10,81,42"></lb> |
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11404 μετρον ἄρα ἐστὶ τὸ ΕΗ τῷ ΘΗ.</s> <s>ὡς δὲ τὸ ΕΗ πρὸς <lb n="1799,001,,,,10,81,43"></lb> |
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11405 τὸ ΘΗ, οὕτως ἐστὶν ἡ ΕΜ πρὸς τὴν ΘΜ· ἀσύμμετρος <lb n="1799,001,,,,10,81,44"></lb> |
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11406 ἄρα ἐστὶν ἡ ΕΜ τῇ ΜΘ μήκει.</s> <s>καί εἰσιν ἀμφότεραι <lb n="1799,001,,,,10,81,45"></lb> |
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11407 ῥηταί· αἱ ΕΜ, ΜΘ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,81,46"></lb> |
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11408 μετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΕΘ, προσαρμόζουσα δὲ <lb n="1799,001,,,,10,81,47"></lb> |
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11409 αὐτῇ ἡ ΘΜ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ ΘΝ αὐτῇ <lb n="1799,001,,,,10,81,48"></lb> |
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11410 προσαρμόζει· τῇ ἄρα ἀποτομῇ ἄλλη καὶ ἄλλη προσαρ-<lb n="1799,001,,,,10,81,49"></lb> |
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11411 μόζει εὐθεῖα δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ· <lb n="1799,001,,,,10,81,50"></lb> |
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11412 ὅπερ ἐστὶν ἀδύνατον.</s> <s><pb></pb> <lb n="1799,001,,,,10,81,51"></lb> |
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11413 Τῇ ἄρα μέσης ἀποτομῇ δευτέρᾳ μία μόνον προσ-<lb n="1799,001,,,,10,81,52"></lb> |
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11414 αρμόζει εὐθεῖα μέση δυνάμει μόνον σύμμετρος οὖσα τῇ <lb n="1799,001,,,,10,81,53"></lb> |
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11415 ὅλῃ, μετὰ δὲ τῆς ὅλης μέσον περιέχουσα· ὅπερ ἔδει <lb n="1799,001,,,,10,81,54"></lb> |
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11416 δεῖξαι.</s> <lb n="1799,001,,,,10,81,55"></lb> |
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11417 <s>Τῇ ἐλάσσονι μία μόνον προσαρμόζει εὐθεῖα δυνάμει <lb n="1799,001,,,,10,82,1"></lb> |
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11418 ἀσύμμετρος οὖσα τῇ ὅλῃ ποιοῦσα μετὰ τῆς ὅλης τὸ μὲν <lb n="1799,001,,,,10,82,2"></lb> |
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11419 ἐκ τῶν ἀπ' αὐτῶν τετραγώνων ῥητόν, τὸ δὲ δὶς ὑπ' <lb n="1799,001,,,,10,82,3"></lb> |
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11420 αὐτῶν μέσον.</s> <lb n="1799,001,,,,10,82,4"></lb> |
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11421 <s>Ἔστω ἡ ἐλάσσων ἡ ΑΒ, καὶ τῇ ΑΒ προσαρμόζουσα <lb n="1799,001,,,,10,82,5"></lb> |
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11422 ἔστω ἡ ΒΓ· αἱ ἄρα ΑΓ, ΓΒ δυνάμει εἰσὶν ἀσύμμετροι <lb n="1799,001,,,,10,82,6"></lb> |
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11423 ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετρα-<lb n="1799,001,,,,10,82,7"></lb> |
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11424 γώνων ῥητόν, τὸ δὲ δὶς ὑπ' αὐτῶν μέσον· λέγω, ὅτι <lb n="1799,001,,,,10,82,8"></lb> |
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11425 τῇ ΑΒ ἑτέρα εὐθεῖα οὐ προσαρμόσει τὰ αὐτὰ ποιοῦσα.</s> <lb n="1799,001,,,,10,82,9"></lb> |
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11426 <s>Εἰ γὰρ δυνατόν, προσαρμοζέτω ἡ ΒΔ· καὶ αἱ ΑΔ, <lb n="1799,001,,,,10,82,10"></lb> |
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11427 ΔΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὰ προ-<lb n="1799,001,,,,10,82,11"></lb> |
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11428 ειρημένα.</s> <s>καὶ ἐπεί, ᾧ ὑπερέχει τὰ ἀπὸ τῶν ΑΔ, ΔΒ τῶν <lb n="1799,001,,,,10,82,12"></lb> |
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11429 ἀπὸ τῶν ΑΓ, ΓΒ, τούτῳ ὑπερέχει καὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,82,13"></lb> |
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11430 ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ, τὰ δὲ ἀπὸ τῶν ΑΔ, <pb></pb> <lb n="1799,001,,,,10,82,14"></lb> |
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11431 ΔΒ τετράγωνα τῶν ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνων <lb n="1799,001,,,,10,82,15"></lb> |
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11432 ὑπερέχει ῥητῷ· ῥητὰ γάρ ἐστιν ἀμφότερα· καὶ τὸ δὶς <lb n="1799,001,,,,10,82,16"></lb> |
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11433 ὑπὸ τῶν ΑΔ, ΔΒ ἄρα τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ὑπερέχει <lb n="1799,001,,,,10,82,17"></lb> |
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11434 ῥητῷ· ὅπερ ἐστὶν ἀδύνατον· μέσα γάρ ἐστιν ἀμφότερα.</s> <lb n="1799,001,,,,10,82,18"></lb> |
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11435 <s>Τῇ ἄρα ἐλάσσονι μία μόνον προσαρμόζει εὐθεῖα δυνάμει <lb n="1799,001,,,,10,82,19"></lb> |
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11436 ἀσύμμετρος οὖσα τῇ ὅλῃ καὶ ποιοῦσα τὰ μὲν ἀπ' αὐτῶν <lb n="1799,001,,,,10,82,20"></lb> |
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11437 τετράγωνα ἅμα ῥητόν, τὸ δὲ δὶς ὑπ' αὐτῶν μέσον· ὅπερ <lb n="1799,001,,,,10,82,21"></lb> |
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11438 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,82,22"></lb> |
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11439 <s>Τῇ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιούσῃ μία μόνον <lb n="1799,001,,,,10,83,1"></lb> |
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11440 προσαρμόζει εὐθεῖα δυνάμει ἀσύμμετρος οὖσα τῇ ὅλῃ, <lb n="1799,001,,,,10,83,2"></lb> |
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11441 μετὰ δὲ τῆς ὅλης ποιοῦσα τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' <lb n="1799,001,,,,10,83,3"></lb> |
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11442 αὐτῶν τετραγώνων μέσον, τὸ δὲ δὶς ὑπ' αὐτῶν ῥητόν.</s> <lb n="1799,001,,,,10,83,4"></lb> |
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11443 <s>Ἔστω ἡ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα ἡ ΑΒ, καὶ <lb n="1799,001,,,,10,83,5"></lb> |
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11444 τῇ ΑΒ προσαρμοζέτω ἡ ΒΓ· αἱ ἄρα ΑΓ, ΓΒ δυνάμει <lb n="1799,001,,,,10,83,6"></lb> |
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11445 εἰσὶν ἀσύμμετροι ποιοῦσαι τὰ προκείμενα· λέγω, ὅτι τῇ <lb n="1799,001,,,,10,83,7"></lb> |
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11446 ΑΒ ἑτέρα οὐ προσαρμόσει τὰ αὐτὰ ποιοῦσα.</s> <lb n="1799,001,,,,10,83,8"></lb> |
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11447 <s>Εἰ γὰρ δυνατόν, προσαρμοζέτω ἡ ΒΔ· καὶ αἱ ΑΔ, ΔΒ <lb n="1799,001,,,,10,83,9"></lb> |
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11448 ἄρα εὐθεῖαι δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὰ προ- <pb></pb><lb n="1799,001,,,,10,83,10"></lb> |
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11449 κείμενα.</s> <s>ἐπεὶ οὖν, ᾧ ὑπερέχει τὰ ἀπὸ τῶν ΑΔ, ΔΒ τῶν <lb n="1799,001,,,,10,83,11"></lb> |
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11450 ἀπὸ τῶν ΑΓ, ΓΒ, τούτῳ ὑπερέχει καὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,83,12"></lb> |
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11451 ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ἀκολούθως τοῖς πρὸ <lb n="1799,001,,,,10,83,13"></lb> |
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11452 αὐτοῦ, τὸ δὲ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, <lb n="1799,001,,,,10,83,14"></lb> |
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11453 ΓΒ ὑπερέχει ῥητῷ· ῥητὰ γάρ ἐστιν ἀμφότερα· καὶ τὰ <lb n="1799,001,,,,10,83,15"></lb> |
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11454 ἀπὸ τῶν ΑΔ, ΔΒ ἄρα τῶν ἀπὸ τῶν ΑΓ, ΓΒ ὑπερέχει <lb n="1799,001,,,,10,83,16"></lb> |
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11455 ῥητῷ· ὅπερ ἐστὶν ἀδύνατον· μέσα γάρ ἐστιν ἀμφότερα.</s> <lb n="1799,001,,,,10,83,17"></lb> |
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11456 <s>οὐκ ἄρα τῇ ΑΒ ἑτέρα προσαρμόσει εὐθεῖα δυνάμει ἀσύμ-<lb n="1799,001,,,,10,83,18"></lb> |
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11457 μετρος οὖσα τῇ ὅλῃ, μετὰ δὲ τῆς ὅλης ποιοῦσα τὰ προειρη-<lb n="1799,001,,,,10,83,19"></lb> |
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11458 μένα· μία ἄρα μόνον προσαρμόσει· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,83,20"></lb> |
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11459 <s>Τῇ μετὰ μέσου μέσον τὸ ὅλον ποιούσῃ μία μόνη <lb n="1799,001,,,,10,84,1"></lb> |
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11460 προσαρμόζει εὐθεῖα δυνάμει ἀσύμμετρος οὖσα τῇ ὅλῃ, <lb n="1799,001,,,,10,84,2"></lb> |
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11461 μετὰ δὲ τῆς ὅλης ποιοῦσα τό τε συγκείμενον ἐκ τῶν ἀπ' <lb n="1799,001,,,,10,84,3"></lb> |
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11462 αὐτῶν τετραγώνων μέσον τό τε δὶς ὑπ' αὐτῶν μέσον καὶ <lb n="1799,001,,,,10,84,4"></lb> |
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11463 ἔτι ἀσύμμετρον τῷ συγκειμένῳ ἐκ τῶν ἀπ' αὐτῶν.</s> <lb n="1799,001,,,,10,84,5"></lb> |
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11464 <s>Ἔστω ἡ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα ἡ ΑΒ, <lb n="1799,001,,,,10,84,6"></lb> |
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11465 προσαρμόζουσα δὲ αὐτῇ ἡ ΒΓ· αἱ ἄρα ΑΓ, ΓΒ δυνάμει <lb n="1799,001,,,,10,84,7"></lb> |
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11466 εἰσὶν ἀσύμμετροι ποιοῦσαι τὰ προειρημένα.</s> <s>λέγω, ὅτι τῇ <pb></pb> <lb n="1799,001,,,,10,84,8"></lb> |
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11467 ΑΒ ἑτέρα οὐ προσαρμόσει <lb n="1799,001,,,,10,84,9"></lb> |
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11468 ποιοῦσα τὰ προειρημένα.</s> <lb n="1799,001,,,,10,84,10"></lb> |
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11469 <s>Εἰ γὰρ δυνατόν, προσαρ-<lb n="1799,001,,,,10,84,11"></lb> |
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11470 μοζέτω ἡ ΒΔ, ὥστε καὶ <lb n="1799,001,,,,10,84,12"></lb> |
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11471 τὰς ΑΔ, ΔΒ δυνάμει ἀσυμ-<lb n="1799,001,,,,10,84,13"></lb> |
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11472 μέτρους εἶναι ποιούσας τά <lb n="1799,001,,,,10,84,14"></lb> |
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11473 τε ἀπὸ τῶν ΑΔ, ΔΒ τε-<lb n="1799,001,,,,10,84,15"></lb> |
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11474 τράγωνα ἅμα μέσον καὶ τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ μέ-<lb n="1799,001,,,,10,84,16"></lb> |
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11475 σον καὶ ἔτι τὰ ἀπὸ τῶν ΑΔ, ΔΒ ἀσύμμετρα τῷ δὶς <lb n="1799,001,,,,10,84,17"></lb> |
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11476 ὑπὸ τῶν ΑΔ, ΔΒ· καὶ ἐκκείσθω ῥητὴ ἡ ΕΖ, καὶ τοῖς <lb n="1799,001,,,,10,84,18"></lb> |
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11477 μὲν ἀπὸ τῶν ΑΓ, ΓΒ ἴσον παρὰ τὴν ΕΖ παραβεβλή-<lb n="1799,001,,,,10,84,19"></lb> |
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11478 σθω τὸ ΕΗ πλάτος ποιοῦν τὴν ΕΜ, τῷ δὲ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,84,20"></lb> |
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11479 ΑΓ, ΓΒ ἴσον παρὰ τὴν ΕΖ παραβεβλήσθω τὸ ΘΗ <lb n="1799,001,,,,10,84,21"></lb> |
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11480 πλάτος ποιοῦν τὴν ΘΜ· λοιπὸν ἄρα τὸ ἀπὸ τῆς ΑΒ ἴσον <lb n="1799,001,,,,10,84,22"></lb> |
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11481 ἐστὶ τῷ ΕΛ· ἡ ἄρα ΑΒ δύναται τὸ ΕΛ.</s> <s>πάλιν τοῖς ἀπὸ <lb n="1799,001,,,,10,84,23"></lb> |
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11482 τῶν ΑΔ, ΔΒ ἴσον παρὰ τὴν ΕΖ παραβεβλήσθω τὸ ΕΙ <lb n="1799,001,,,,10,84,24"></lb> |
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11483 πλάτος ποιοῦν τὴν ΕΝ.</s> <s>ἔστι δὲ καὶ τὸ ἀπὸ τῆς ΑΒ ἴσον <lb n="1799,001,,,,10,84,25"></lb> |
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11484 τῷ ΕΛ· λοιπὸν ἄρα τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ ἴσον [ἐστὶ] <lb n="1799,001,,,,10,84,26"></lb> |
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11485 τῷ ΘΙ.</s> <s>καὶ ἐπεὶ μέσον ἐστὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,84,27"></lb> |
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11486 τῶν ΑΓ, ΓΒ καί ἐστιν ἴσον τῷ ΕΗ, μέσον ἄρα ἐστὶ καὶ <lb n="1799,001,,,,10,84,28"></lb> |
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11487 τὸ ΕΗ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΕΖ παράκειται πλάτος <lb n="1799,001,,,,10,84,29"></lb> |
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11488 ποιοῦν τὴν ΕΜ· ῥητὴ ἄρα ἐστὶν ἡ ΕΜ καὶ ἀσύμμετρος <lb n="1799,001,,,,10,84,30"></lb> |
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11489 τῇ ΕΖ μήκει.</s> <s>πάλιν, ἐπεὶ μέσον ἐστὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,84,31"></lb> |
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11490 ΑΓ, ΓΒ καί ἐστιν ἴσον τῷ ΘΗ, μέσον ἄρα καὶ τὸ ΘΗ.</s> <s><pb></pb> <lb n="1799,001,,,,10,84,32"></lb> |
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11491 καὶ παρὰ ῥητὴν τὴν ΕΖ παράκειται πλάτος ποιοῦν τὴν <lb n="1799,001,,,,10,84,33"></lb> |
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11492 ΘΜ· ῥητὴ ἄρα ἐστὶν ἡ ΘΜ καὶ ἀσύμμετρος τῇ ΕΖ μήκει.</s> <lb n="1799,001,,,,10,84,34"></lb> |
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11493 <s>καὶ ἐπεὶ ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῷ δὶς <lb n="1799,001,,,,10,84,35"></lb> |
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11494 ὑπὸ τῶν ΑΓ, ΓΒ, ἀσύμμετρόν ἐστι καὶ τὸ ΕΗ τῷ ΘΗ· <lb n="1799,001,,,,10,84,36"></lb> |
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11495 ἀσύμμετρος ἄρα ἐστὶ καὶ ἡ ΕΜ τῇ ΜΘ μήκει.</s> <s>καί εἰσιν <lb n="1799,001,,,,10,84,37"></lb> |
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11496 ἀμφότεραι ῥηταί· αἱ ἄρα ΕΜ, ΜΘ ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,84,38"></lb> |
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11497 μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΕΘ, προσαρ-<lb n="1799,001,,,,10,84,39"></lb> |
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11498 μόζουσα δὲ αὐτῇ ἡ ΘΜ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι ἡ ΕΘ <lb n="1799,001,,,,10,84,40"></lb> |
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11499 πάλιν ἀποτομή ἐστιν, προσαρμόζουσα δὲ αὐτῇ ἡ ΘΝ.</s> <lb n="1799,001,,,,10,84,41"></lb> |
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11500 <s>τῇ ἄρα ἀποτομῇ ἄλλη καὶ ἄλλη προσαρμόζει ῥητὴ δυνάμει <lb n="1799,001,,,,10,84,42"></lb> |
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11501 μόνον σύμμετρος οὖσα τῇ ὅλῃ· ὅπερ ἐδείχθη ἀδύνατον.</s> <lb n="1799,001,,,,10,84,43"></lb> |
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11502 <s>οὐκ ἄρα τῇ ΑΒ ἑτέρα προσαρμόσει εὐθεῖα.</s> <lb n="1799,001,,,,10,84,44"></lb> |
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11503 <s>Τῇ ἄρα ΑΒ μία μόνον προσαρμόζει εὐθεῖα δυνάμει <lb n="1799,001,,,,10,84,45"></lb> |
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11504 ἀσύμμετρος οὖσα τῇ ὅλῃ, μετὰ δὲ τῆς ὅλης ποιοῦσα τά τε <lb n="1799,001,,,,10,84,46"></lb> |
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11505 ἀπ' αὐτῶν τετράγωνα ἅμα μέσον καὶ τὸ δὶς ὑπ' αὐτῶν <lb n="1799,001,,,,10,84,47"></lb> |
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11506 μέσον καὶ ἔτι τὰ ἀπ' αὐτῶν τετράγωνα ἀσύμμετρα τῷ <lb n="1799,001,,,,10,84,48"></lb> |
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11507 δὶς ὑπ' αὐτῶν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,84,49"></lb> |
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11508 <s></s></p> |
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11509 </chap> |
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11510 <chap><p type="main"><s>Ὑποκειμένης ῥητῆς καὶ ἀποτομῆς, ἐὰν μὲν ἡ ὅλη <lb n="1799,001,,,,10;HOR3,1,1"></lb> |
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11511 |
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11512 τῆς προσαρμοζούσης μεῖζον δύνηται τῷ ἀπὸ συμμέτρου <lb n="1799,001,,,,10;HOR3,1,2"></lb> |
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11513 ἑαυτῇ μήκει, καὶ ἡ ὅλη σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ <lb n="1799,001,,,,10;HOR3,1,3"></lb> |
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11514 μήκει, καλείσθω ἀποτομὴ πρώτη.</s> <lb n="1799,001,,,,10;HOR3,1,4"></lb> |
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11515 <s>Ἐὰν δὲ ἡ προσαρμόζουσα σύμμετρος ᾖ τῇ ἐκ-<lb n="1799,001,,,,10;HOR3,2,1"></lb> |
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11516 κειμένῃ ῥητῇ μήκει, καὶ ἡ ὅλη τῆς προσαρμοζούσης <pb></pb> <lb n="1799,001,,,,10;HOR3,2,2"></lb> |
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11517 μεῖζον δύνηται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καλείσθω <lb n="1799,001,,,,10;HOR3,2,3"></lb> |
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11518 ἀποτομὴ δευτέρα.</s> <lb n="1799,001,,,,10;HOR3,2,4"></lb> |
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11519 <s>Ἐὰν δὲ μηδετέρα σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ <lb n="1799,001,,,,10;HOR3,3,1"></lb> |
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11520 μήκει, ἡ δὲ ὅλη τῆς προσαρμοζούσης μεῖζον δύνηται τῷ <lb n="1799,001,,,,10;HOR3,3,2"></lb> |
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11521 ἀπὸ συμμέτρου ἑαυτῇ, καλείσθω ἀποτομὴ τρίτη.</s> <lb n="1799,001,,,,10;HOR3,3,3"></lb> |
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11522 <s>Πάλιν, ἐὰν ἡ ὅλη τῆς προσαρμοζούσης μεῖζον <lb n="1799,001,,,,10;HOR3,4,1"></lb> |
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11523 δύνηται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ [μήκει], ἐὰν μὲν ἡ ὅλη <lb n="1799,001,,,,10;HOR3,4,2"></lb> |
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11524 σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ μήκει, καλείσθω ἀπο-<lb n="1799,001,,,,10;HOR3,4,3"></lb> |
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11525 τομὴ τετάρτη.</s> <lb n="1799,001,,,,10;HOR3,4,4"></lb> |
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11526 <s>Ἐὰν δὲ ἡ προσαρμόζουσα, πέμπτη.</s> <lb n="1799,001,,,,10;HOR3,5,1"></lb> |
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11527 <s>Ἐὰν δὲ μηδετέρα, ἕκτη.</s> <lb n="1799,001,,,,10;HOR3,6,1"></lb> |
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11528 <s></s></p> |
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11529 </chap> |
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11530 <chap><p type="main"><s>Εὑρεῖν τὴν πρώτην ἀποτομήν.</s> <lb n="1799,001,,,,10,85,1"></lb> |
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11531 <s> |
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11532 Ἐκκείσθω ῥητὴ ἡ Α, καὶ τῇ Α μήκει σύμμετρος ἔστω <lb n="1799,001,,,,10,85,2"></lb> |
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11533 ἡ ΒΗ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΒΗ.</s> <s>καὶ ἐκκείσθωσαν δύο <lb n="1799,001,,,,10,85,3"></lb> |
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11534 τετράγωνοι ἀριθμοὶ οἱ ΔΕ, ΕΖ, ὧν ἡ ὑπεροχὴ ὁ ΖΔ <lb n="1799,001,,,,10,85,4"></lb> |
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11535 μὴ ἔστω τετράγωνος· οὐδ' ἄρα ὁ ΕΔ πρὸς τὸν ΔΖ λόγον <lb n="1799,001,,,,10,85,5"></lb> |
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11536 ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.</s> <lb n="1799,001,,,,10,85,6"></lb> |
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11537 <s>καὶ πεποιήσθω ὡς ὁ ΕΔ πρὸς τὸν ΔΖ, οὕτως τὸ ἀπὸ <pb></pb> <lb n="1799,001,,,,10,85,7"></lb> |
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11538 τῆς ΒΗ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΗΓ τετράγωνον· <lb n="1799,001,,,,10,85,8"></lb> |
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11539 σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΒΗ τῷ ἀπὸ τῆς ΗΓ.</s> <lb n="1799,001,,,,10,85,9"></lb> |
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11540 <s>ῥητὸν δὲ τὸ ἀπὸ τῆς ΒΗ· ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς ΗΓ· <lb n="1799,001,,,,10,85,10"></lb> |
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11541 ῥητὴ ἄρα ἐστὶ καὶ ἡ ΗΓ.</s> <s>καὶ ἐπεὶ ὁ ΕΔ πρὸς τὸν ΔΖ <lb n="1799,001,,,,10,85,11"></lb> |
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11542 λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,85,12"></lb> |
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11543 ἀριθμόν, οὐδ' ἄρα τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ <lb n="1799,001,,,,10,85,13"></lb> |
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11544 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,85,14"></lb> |
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11545 ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ ΗΓ μήκει.</s> <s>καί <lb n="1799,001,,,,10,85,15"></lb> |
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11546 εἰσιν ἀμφότεραι ῥηταί· αἱ ΒΗ, ΗΓ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,85,16"></lb> |
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11547 δυνάμει μόνον σύμμετροι· ἡ ἄρα ΒΓ ἀποτομή ἐστιν.</s> <lb n="1799,001,,,,10,85,17"></lb> |
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11548 <s>Λέγω δή, ὅτι καὶ πρώτη.</s> <lb n="1799,001,,,,10,85,18"></lb> |
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11549 <s>Ὧι γὰρ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΗ τοῦ ἀπὸ τῆς ΗΓ, <lb n="1799,001,,,,10,85,19"></lb> |
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11550 ἔστω τὸ ἀπὸ τῆς Θ.</s> <s>καὶ ἐπεί ἐστιν ὡς ὁ ΕΔ πρὸς τὸν <lb n="1799,001,,,,10,85,20"></lb> |
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11551 ΖΔ, οὕτως τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ, καὶ <lb n="1799,001,,,,10,85,21"></lb> |
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11552 ἀναστρέψαντι ἄρα ἐστὶν ὡς ὁ ΔΕ πρὸς τὸν ΕΖ, οὕτως τὸ <lb n="1799,001,,,,10,85,22"></lb> |
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11553 ἀπὸ τῆς ΗΒ πρὸς τὸ ἀπὸ τῆς Θ.</s> <s>ὁ δὲ ΔΕ πρὸς τὸν ΕΖ <lb n="1799,001,,,,10,85,23"></lb> |
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11554 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,85,24"></lb> |
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11555 ἀριθμόν· ἑκάτερος γὰρ τετράγωνός ἐστιν· καὶ τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,85,25"></lb> |
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11556 ΗΒ ἄρα πρὸς τὸ ἀπὸ τῆς Θ λόγον ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,85,26"></lb> |
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11557 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· σύμμετρος ἄρα ἐστὶν <lb n="1799,001,,,,10,85,27"></lb> |
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11558 ἡ ΒΗ τῇ Θ μήκει.</s> <s>καὶ δύναται ἡ ΒΗ τῆς ΗΓ μεῖζον <lb n="1799,001,,,,10,85,28"></lb> |
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11559 τῷ ἀπὸ τῆς Θ· ἡ ΒΗ ἄρα τῆς ΗΓ μεῖζον δύναται τῷ <lb n="1799,001,,,,10,85,29"></lb> |
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11560 ἀπὸ συμμέτρου ἑαυτῇ μήκει.</s> <s>καί ἐστιν ἡ ὅλη ἡ ΒΗ σύμ- <pb></pb><lb n="1799,001,,,,10,85,30"></lb> |
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11561 μετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ Α.</s> <s>ἡ ΒΓ ἄρα ἀποτομή <lb n="1799,001,,,,10,85,31"></lb> |
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11562 ἐστι πρώτη.</s> <lb n="1799,001,,,,10,85,32"></lb> |
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11563 <s>Εὕρηται ἄρα ἡ πρώτη ἀποτομὴ ἡ ΒΓ· ὅπερ ἔδει <lb n="1799,001,,,,10,85,33"></lb> |
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11564 εὑρεῖν.</s> <lb n="1799,001,,,,10,85,34"></lb> |
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11565 <s>Εὑρεῖν τὴν δευτέραν ἀποτομήν.</s> <lb n="1799,001,,,,10,86,1"></lb> |
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11566 <s>Ἐκκείσθω ῥητὴ ἡ Α καὶ τῇ Α σύμμετρος μήκει ἡ ΗΓ.</s> <lb n="1799,001,,,,10,86,2"></lb> |
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11567 <s>ῥητὴ ἄρα ἐστὶν ἡ ΗΓ.</s> <s>καὶ ἐκκείσθωσαν δύο τετράγωνοι <lb n="1799,001,,,,10,86,3"></lb> |
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11568 ἀριθμοὶ οἱ ΔΕ, ΕΖ, ὧν ἡ ὑπεροχὴ ὁ ΔΖ μὴ ἔστω τετρά-<lb n="1799,001,,,,10,86,4"></lb> |
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11569 γωνος.</s> <s>καὶ πεποιήσθω ὡς ὁ <lb n="1799,001,,,,10,86,5"></lb> |
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11570 ΖΔ πρὸς τὸν ΔΕ, οὕτως τὸ <lb n="1799,001,,,,10,86,6"></lb> |
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11571 ἀπὸ τῆς ΓΗ τετράγωνον πρὸς <lb n="1799,001,,,,10,86,7"></lb> |
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11572 τὸ ἀπὸ τῆς ΗΒ τετράγωνον.</s> <lb n="1799,001,,,,10,86,8"></lb> |
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11573 <s>σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ <lb n="1799,001,,,,10,86,9"></lb> |
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11574 τῆς ΓΗ τετράγωνον τῷ ἀπὸ <lb n="1799,001,,,,10,86,10"></lb> |
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11575 τῆς ΗΒ τετραγώνῳ.</s> <s>ῥητὸν δὲ <lb n="1799,001,,,,10,86,11"></lb> |
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11576 τὸ ἀπὸ τῆς ΗΓ.</s> <s>ῥητὸν ἄρα [ἐστὶ] καὶ τὸ ἀπὸ τῆς ΗΒ· <lb n="1799,001,,,,10,86,12"></lb> |
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11577 ῥητὴ ἄρα ἐστὶν ἡ ΒΗ.</s> <s>καὶ ἐπεὶ τὸ ἀπὸ τῆς ΗΓ τετρά-<lb n="1799,001,,,,10,86,13"></lb> |
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11578 γωνον πρὸς τὸ ἀπὸ τῆς ΗΒ λόγον οὐκ ἔχει, ὃν τετρά-<lb n="1799,001,,,,10,86,14"></lb> |
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11579 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ἀσύμμετρός <lb n="1799,001,,,,10,86,15"></lb> |
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11580 ἐστιν ἡ ΓΗ τῇ ΗΒ μήκει.</s> <s>καί εἰσιν ἀμφότεραι ῥηταί· <lb n="1799,001,,,,10,86,16"></lb> |
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11581 αἱ ΓΗ, ΗΒ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἡ <lb n="1799,001,,,,10,86,17"></lb> |
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11582 ΒΓ ἄρα ἀποτομή ἐστιν.</s> <s><pb></pb> <lb n="1799,001,,,,10,86,18"></lb> |
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11583 Λέγω δή, ὅτι καὶ δευτέρα.</s> <lb n="1799,001,,,,10,86,19"></lb> |
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11584 <s>Ὧι γὰρ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΗ τοῦ ἀπὸ τῆς ΗΓ, <lb n="1799,001,,,,10,86,20"></lb> |
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11585 ἔστω τὸ ἀπὸ τῆς Θ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς τὸ ἀπὸ τῆς ΒΗ <lb n="1799,001,,,,10,86,21"></lb> |
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11586 πρὸς τὸ ἀπὸ τῆς ΗΓ, οὕτως ὁ ΕΔ ἀριθμὸς πρὸς τὸν <lb n="1799,001,,,,10,86,22"></lb> |
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11587 ΔΖ ἀριθμόν, ἀναστρέψαντι ἄρα ἐστὶν ὡς τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,86,23"></lb> |
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11588 ΒΗ πρὸς τὸ ἀπὸ τῆς Θ, οὕτως ὁ ΔΕ πρὸς τὸν ΕΖ.</s> <s>καί <lb n="1799,001,,,,10,86,24"></lb> |
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11589 ἐστιν ἑκάτερος τῶν ΔΕ, ΕΖ τετράγωνος· τὸ ἄρα ἀπὸ <lb n="1799,001,,,,10,86,25"></lb> |
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11590 τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς Θ λόγον ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,86,26"></lb> |
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11591 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· σύμμετρος ἄρα ἐστὶν <lb n="1799,001,,,,10,86,27"></lb> |
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11592 ἡ ΒΗ τῇ Θ μήκει.</s> <s>καὶ δύναται ἡ ΒΗ τῆς ΗΓ μεῖζον <lb n="1799,001,,,,10,86,28"></lb> |
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11593 τῷ ἀπὸ τῆς Θ· ἡ ΒΗ ἄρα τῆς ΗΓ μεῖζον δύναται τῷ <lb n="1799,001,,,,10,86,29"></lb> |
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11594 ἀπὸ συμμέτρου ἑαυτῇ μήκει.</s> <s>καί ἐστιν ἡ προσαρμόζουσα <lb n="1799,001,,,,10,86,30"></lb> |
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11595 ἡ ΓΗ τῇ ἐκκειμένῃ ῥητῇ σύμμετρος τῇ Α.</s> <s>ἡ ΒΓ ἄρα <lb n="1799,001,,,,10,86,31"></lb> |
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11596 ἀποτομή ἐστι δευτέρα.</s> <lb n="1799,001,,,,10,86,32"></lb> |
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11597 <s>Εὕρηται ἄρα δευτέρα ἀποτομὴ ἡ ΒΓ· ὅπερ ἔδει <lb n="1799,001,,,,10,86,33"></lb> |
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11598 δεῖξαι.</s> <lb n="1799,001,,,,10,86,34"></lb> |
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11599 <s>Εὑρεῖν τὴν τρίτην ἀποτομήν.</s> <lb n="1799,001,,,,10,87,1"></lb> |
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11600 <s>Ἐκκείσθω ῥητὴ ἡ Α, καὶ ἐκκείσθωσαν τρεῖς ἀριθμοὶ <lb n="1799,001,,,,10,87,2"></lb> |
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11601 οἱ Ε, ΒΓ, ΓΔ λόγον μὴ ἔχοντες πρὸς ἀλλήλους, ὃν <lb n="1799,001,,,,10,87,3"></lb> |
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11602 τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ὁ δὲ <pb></pb> <lb n="1799,001,,,,10,87,4"></lb> |
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11603 ΓΒ πρὸς τὸν ΒΔ λόγον ἐχέτω, ὃν <lb n="1799,001,,,,10,87,5"></lb> |
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11604 τετράγωνος ἀριθμὸς πρὸς τετρά-<lb n="1799,001,,,,10,87,6"></lb> |
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11605 γωνον ἀριθμόν, καὶ πεποιήσθω <lb n="1799,001,,,,10,87,7"></lb> |
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11606 ὡς μὲν ὁ Ε πρὸς τὸν ΒΓ, οὕτως <lb n="1799,001,,,,10,87,8"></lb> |
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11607 τὸ ἀπὸ τῆς Α τετράγωνον πρὸς <lb n="1799,001,,,,10,87,9"></lb> |
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11608 τὸ ἀπὸ τῆς ΖΗ τετράγωνον, ὡς <lb n="1799,001,,,,10,87,10"></lb> |
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11609 δὲ ὁ ΒΓ πρὸς τὸν ΓΔ, οὕτως <lb n="1799,001,,,,10,87,11"></lb> |
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11610 τὸ ἀπὸ τῆς ΖΗ τετράγωνον πρὸς <lb n="1799,001,,,,10,87,12"></lb> |
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11611 τὸ ἀπὸ τῆς ΗΘ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς ὁ Ε πρὸς τὸν ΒΓ, οὕ-<lb n="1799,001,,,,10,87,13"></lb> |
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11612 τως τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΗ τετρά-<lb n="1799,001,,,,10,87,14"></lb> |
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11613 γωνον, σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς Α τετράγωνον τῷ <lb n="1799,001,,,,10,87,15"></lb> |
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11614 ἀπὸ τῆς ΖΗ τετραγώνῳ.</s> <s>ῥητὸν δὲ τὸ ἀπὸ τῆς Α τετράγω-<lb n="1799,001,,,,10,87,16"></lb> |
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11615 νον.</s> <s>ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς ΖΗ· ῥητὴ ἄρα ἐστὶν ἡ ΖΗ.</s> <lb n="1799,001,,,,10,87,17"></lb> |
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11616 <s>καὶ ἐπεὶ ὁ Ε πρὸς τὸν ΒΓ λόγον οὐκ ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,87,18"></lb> |
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11617 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδ' ἄρα τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,87,19"></lb> |
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11618 Α τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΗ [τετράγωνον] λόγον <lb n="1799,001,,,,10,87,20"></lb> |
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11619 ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· <lb n="1799,001,,,,10,87,21"></lb> |
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11620 ἀσύμμετρος ἄρα ἐστὶν ἡ Α τῇ ΖΗ μήκει.</s> <s>πάλιν, ἐπεί ἐστιν <lb n="1799,001,,,,10,87,22"></lb> |
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11621 ὡς ὁ ΒΓ πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ τετρά-<lb n="1799,001,,,,10,87,23"></lb> |
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11622 γωνον πρὸς τὸ ἀπὸ τῆς ΗΘ, σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ <lb n="1799,001,,,,10,87,24"></lb> |
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11623 τῆς ΖΗ τῷ ἀπὸ τῆς ΗΘ.</s> <s>ῥητὸν δὲ τὸ ἀπὸ τῆς ΖΗ· ῥητὸν <lb n="1799,001,,,,10,87,25"></lb> |
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11624 ἄρα καὶ τὸ ἀπὸ τῆς ΗΘ· ῥητὴ ἄρα ἐστὶν ἡ ΗΘ.</s> <s>καὶ <lb n="1799,001,,,,10,87,26"></lb> |
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11625 ἐπεὶ ὁ ΒΓ πρὸς τὸν ΓΔ λόγον οὐκ ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,87,27"></lb> |
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11626 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδ' ἄρα τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,87,28"></lb> |
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11627 ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ λόγον ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,87,29"></lb> |
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11628 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν <pb></pb> <lb n="1799,001,,,,10,87,30"></lb> |
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11629 ἡ ΖΗ τῇ ΗΘ μήκει.</s> <s>καί εἰσιν ἀμφότεραι ῥηταί· αἱ ΖΗ, <lb n="1799,001,,,,10,87,31"></lb> |
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11630 ΗΘ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἀποτομὴ <lb n="1799,001,,,,10,87,32"></lb> |
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11631 ἄρα ἐστὶν ἡ ΖΘ.</s> <lb n="1799,001,,,,10,87,33"></lb> |
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11632 <s>Λέγω δή, ὅτι καὶ τρίτη.</s> <lb n="1799,001,,,,10,87,34"></lb> |
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11633 <s>Ἐπεὶ γάρ ἐστιν ὡς μὲν ὁ Ε πρὸς τὸν ΒΓ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,87,35"></lb> |
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11634 τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΗ, ὡς δὲ ὁ ΒΓ <lb n="1799,001,,,,10,87,36"></lb> |
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11635 πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,87,37"></lb> |
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11636 ΘΗ, δι' ἴσου ἄρα ἐστὶν ὡς ὁ Ε πρὸς τὸν ΓΔ, οὕτως τὸ <lb n="1799,001,,,,10,87,38"></lb> |
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11637 ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς ΘΗ.</s> <s>ὁ δὲ Ε πρὸς τὸν ΓΔ <lb n="1799,001,,,,10,87,39"></lb> |
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11638 λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,87,40"></lb> |
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11639 ἀριθμόν· οὐδ' ἄρα τὸ ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς ΗΘ <lb n="1799,001,,,,10,87,41"></lb> |
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11640 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,87,42"></lb> |
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11641 ἀριθμόν· ἀσύμμετρος ἄρα ἡ Α τῇ ΗΘ μήκει.</s> <s>οὐδετέρα <lb n="1799,001,,,,10,87,43"></lb> |
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11642 ἄρα τῶν ΖΗ, ΗΘ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ <lb n="1799,001,,,,10,87,44"></lb> |
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11643 τῇ Α μήκει.</s> <s>ᾧ οὖν μεῖζόν ἐστι τὸ ἀπὸ τῆς ΖΗ τοῦ ἀπὸ <lb n="1799,001,,,,10,87,45"></lb> |
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11644 τῆς ΗΘ, ἔστω τὸ ἀπὸ τῆς Κ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς ὁ ΒΓ <lb n="1799,001,,,,10,87,46"></lb> |
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11645 πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,87,47"></lb> |
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11646 ΗΘ, ἀναστρέψαντι ἄρα ἐστὶν ὡς ὁ ΒΓ πρὸς τὸν ΒΔ, <lb n="1799,001,,,,10,87,48"></lb> |
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11647 οὕτως τὸ ἀπὸ τῆς ΖΗ τετράγωνον πρὸς τὸ ἀπὸ τῆς Κ.</s> <lb n="1799,001,,,,10,87,49"></lb> |
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11648 <s>ὁ δὲ ΒΓ πρὸς τὸν ΒΔ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,10,87,50"></lb> |
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11649 πρὸς τετράγωνον ἀριθμόν· καὶ τὸ ἀπὸ τῆς ΖΗ ἄρα πρὸς <lb n="1799,001,,,,10,87,51"></lb> |
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11650 τὸ ἀπὸ τῆς Κ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,87,52"></lb> |
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11651 τετράγωνον ἀριθμόν.</s> <s>σύμμετρος ἄρα ἐστὶν ἡ ΖΗ τῇ Κ <lb n="1799,001,,,,10,87,53"></lb> |
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11652 μήκει, καὶ δύναται ἡ ΖΗ τῆς ΗΘ μεῖζον τῷ ἀπὸ συμ- <pb></pb><lb n="1799,001,,,,10,87,54"></lb> |
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11653 μέτρου ἑαυτῇ.</s> <s>καὶ οὐδετέρα τῶν ΖΗ, ΗΘ σύμμετρός <lb n="1799,001,,,,10,87,55"></lb> |
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11654 ἐστι τῇ ἐκκειμένῃ ῥητῇ τῇ Α μήκει· ἡ ΖΘ ἄρα ἀποτομή <lb n="1799,001,,,,10,87,56"></lb> |
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11655 ἐστι τρίτη.</s> <lb n="1799,001,,,,10,87,57"></lb> |
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11656 <s>Εὕρηται ἄρα ἡ τρίτη ἀποτομὴ ἡ ΖΘ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,87,58"></lb> |
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11657 <s>Εὑρεῖν τὴν τετάρτην ἀποτομήν.</s> <lb n="1799,001,,,,10,88,1"></lb> |
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11658 <s>Ἐκκείσθω ῥητὴ ἡ Α καὶ τῇ Α μήκει σύμμετρος ἡ ΒΗ· <lb n="1799,001,,,,10,88,2"></lb> |
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11659 ῥητὴ ἄρα ἐστὶ καὶ ἡ ΒΗ.</s> <s>καὶ ἐκκείσθωσαν δύο ἀριθμοὶ <lb n="1799,001,,,,10,88,3"></lb> |
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11660 οἱ ΔΖ, ΖΕ, ὥστε τὸν ΔΕ ὅλον πρὸς ἑκάτερον τῶν ΔΖ, <lb n="1799,001,,,,10,88,4"></lb> |
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11661 ΕΖ λόγον μὴ ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγω- <lb n="1799,001,,,,10,88,5"></lb> |
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11662 νον ἀριθμόν.</s> <s>καὶ πεποιήσθω ὡς ὁ ΔΕ πρὸς τὸν ΕΖ, οὕ-<lb n="1799,001,,,,10,88,6"></lb> |
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11663 τως τὸ ἀπὸ τῆς ΒΗ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΗΓ.</s> <lb n="1799,001,,,,10,88,7"></lb> |
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11664 <s>σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΒΗ τῷ ἀπὸ τῆς ΗΓ.</s> <lb n="1799,001,,,,10,88,8"></lb> |
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11665 <s>ῥητὸν δὲ τὸ ἀπὸ τῆς ΒΗ· ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς ΗΓ· <lb n="1799,001,,,,10,88,9"></lb> |
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11666 ῥητὴ ἄρα ἐστὶν ἡ ΗΓ.</s> <s>καὶ ἐπεὶ ὁ ΔΕ πρὸς τὸν ΕΖ λόγον <lb n="1799,001,,,,10,88,10"></lb> |
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11667 οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,88,11"></lb> |
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11668 ἀριθμόν, οὐδ' ἄρα τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ <lb n="1799,001,,,,10,88,12"></lb> |
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11669 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,88,13"></lb> |
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11670 ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ ΗΓ μήκει.</s> <s>καί <pb></pb> <lb n="1799,001,,,,10,88,14"></lb> |
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11671 εἰσιν ἀμφότεραι ῥηταί· αἱ ΒΗ, ΗΓ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,88,15"></lb> |
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11672 δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΒΓ.</s> <lb n="1799,001,,,,10,88,16"></lb> |
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11673 <s>[Λέγω δή, ὅτι καὶ τετάρτη].</s> <lb n="1799,001,,,,10,88,17"></lb> |
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11674 <s>Ὧι οὖν μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΗ τοῦ ἀπὸ τῆς ΗΓ, <lb n="1799,001,,,,10,88,18"></lb> |
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11675 ἔστω τὸ ἀπὸ τῆς Θ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς ὁ ΔΕ πρὸς τὸν <lb n="1799,001,,,,10,88,19"></lb> |
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11676 ΕΖ, οὕτως τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ, καὶ <lb n="1799,001,,,,10,88,20"></lb> |
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11677 ἀναστρέψαντι ἄρα ἐστὶν ὡς ὁ ΕΔ πρὸς τὸν ΔΖ, οὕτως <lb n="1799,001,,,,10,88,21"></lb> |
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11678 τὸ ἀπὸ τῆς ΗΒ πρὸς τὸ ἀπὸ τῆς Θ.</s> <s>ὁ δὲ ΕΔ πρὸς τὸν <lb n="1799,001,,,,10,88,22"></lb> |
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11679 ΔΖ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετρά-<lb n="1799,001,,,,10,88,23"></lb> |
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11680 γωνον ἀριθμόν· οὐδ' ἄρα τὸ ἀπὸ τῆς ΗΒ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,88,24"></lb> |
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11681 τῆς Θ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,88,25"></lb> |
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11682 ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ Θ μήκει.</s> <s>καὶ <lb n="1799,001,,,,10,88,26"></lb> |
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11683 δύναται ἡ ΒΗ τῆς ΗΓ μεῖζον τῷ ἀπὸ τῆς Θ· ἡ ἄρα ΒΗ <lb n="1799,001,,,,10,88,27"></lb> |
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11684 τῆς ΗΓ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <s>καί <lb n="1799,001,,,,10,88,28"></lb> |
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11685 ἐστιν ὅλη ἡ ΒΗ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ <lb n="1799,001,,,,10,88,29"></lb> |
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11686 Α.</s> <s>ἡ ἄρα ΒΓ ἀποτομή ἐστι τετάρτη.</s> <lb n="1799,001,,,,10,88,30"></lb> |
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11687 <s>Εὕρηται ἄρα ἡ τετάρτη ἀποτομή· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,88,31"></lb> |
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11688 <s>Εὑρεῖν τὴν πέμπτην ἀποτομήν.</s> <lb n="1799,001,,,,10,89,1"></lb> |
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11689 <s>Ἐκκείσθω ῥητὴ ἡ Α, καὶ τῇ Α μήκει σύμμετρος ἔστω <lb n="1799,001,,,,10,89,2"></lb> |
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11690 ἡ ΓΗ· ῥητὴ ἄρα [ἐστὶν] ἡ ΓΗ.</s> <s>καὶ ἐκκείσθωσαν δύο <pb></pb> <lb n="1799,001,,,,10,89,3"></lb> |
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11691 ἀριθμοὶ οἱ ΔΖ, ΖΕ, ὥστε τὸν ΔΕ πρὸς <lb n="1799,001,,,,10,89,4"></lb> |
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11692 ἑκάτερον τῶν ΔΖ, ΖΕ λόγον πάλιν μὴ <lb n="1799,001,,,,10,89,5"></lb> |
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11693 ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετρά-<lb n="1799,001,,,,10,89,6"></lb> |
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11694 γωνον ἀριθμόν· καὶ πεποιήσθω ὡς ὁ ΖΕ <lb n="1799,001,,,,10,89,7"></lb> |
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11695 πρὸς τὸν ΕΔ, οὕτως τὸ ἀπὸ τῆς ΓΗ <lb n="1799,001,,,,10,89,8"></lb> |
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11696 πρὸς τὸ ἀπὸ τῆς ΗΒ.</s> <s>ῥητὸν ἄρα καὶ τὸ <lb n="1799,001,,,,10,89,9"></lb> |
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11697 ἀπὸ τῆς ΗΒ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΒΗ.</s> <lb n="1799,001,,,,10,89,10"></lb> |
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11698 <s>καὶ ἐπεί ἐστιν ὡς ὁ ΔΕ πρὸς τὸν ΕΖ, <lb n="1799,001,,,,10,89,11"></lb> |
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11699 οὕτως τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,89,12"></lb> |
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11700 ΗΓ, ὁ δὲ ΔΕ πρὸς τὸν ΕΖ λόγον οὐκ ἔχει, ὃν τετρά-<lb n="1799,001,,,,10,89,13"></lb> |
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11701 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδ' ἄρα τὸ <lb n="1799,001,,,,10,89,14"></lb> |
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11702 ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ λόγον ἔχει, ὃν τετρά-<lb n="1799,001,,,,10,89,15"></lb> |
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11703 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα <lb n="1799,001,,,,10,89,16"></lb> |
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11704 ἐστὶν ἡ ΒΗ τῇ ΗΓ μήκει.</s> <s>καί εἰσιν ἀμφότεραι ῥηταί· <lb n="1799,001,,,,10,89,17"></lb> |
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11705 αἱ ΒΗ, ΗΓ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἡ <lb n="1799,001,,,,10,89,18"></lb> |
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11706 ΒΓ ἄρα ἀποτομή ἐστιν.</s> <lb n="1799,001,,,,10,89,19"></lb> |
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11707 <s>Λέγω δή, ὅτι καὶ πέμπτη.</s> <lb n="1799,001,,,,10,89,20"></lb> |
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11708 <s>Ὧι γὰρ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΗ τοῦ ἀπὸ τῆς ΗΓ, <lb n="1799,001,,,,10,89,21"></lb> |
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11709 ἔστω τὸ ἀπὸ τῆς Θ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς τὸ ἀπὸ τῆς ΒΗ <lb n="1799,001,,,,10,89,22"></lb> |
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11710 πρὸς τὸ ἀπὸ τῆς ΗΓ, οὕτως ὁ ΔΕ πρὸς τὸν ΕΖ, ἀνα-<lb n="1799,001,,,,10,89,23"></lb> |
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11711 στρέψαντι ἄρα ἐστὶν ὡς ὁ ΕΔ πρὸς τὸν ΔΖ, οὕτως τὸ <lb n="1799,001,,,,10,89,24"></lb> |
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11712 ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς Θ.</s> <s>ὁ δὲ ΕΔ πρὸς τὸν ΔΖ <lb n="1799,001,,,,10,89,25"></lb> |
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11713 λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,89,26"></lb> |
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11714 ἀριθμόν· οὐδ' ἄρα τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς Θ <lb n="1799,001,,,,10,89,27"></lb> |
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11715 λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον <pb></pb> <lb n="1799,001,,,,10,89,28"></lb> |
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11716 ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ Θ μήκει.</s> <s>καὶ <lb n="1799,001,,,,10,89,29"></lb> |
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11717 δύναται ἡ ΒΗ τῆς ΗΓ μεῖζον τῷ ἀπὸ τῆς Θ· ἡ ΗΒ <lb n="1799,001,,,,10,89,30"></lb> |
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11718 ἄρα τῆς ΗΓ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ <lb n="1799,001,,,,10,89,31"></lb> |
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11719 μήκει.</s> <s>καί ἐστιν ἡ προσαρμόζουσα ἡ ΓΗ σύμμετρος τῇ <lb n="1799,001,,,,10,89,32"></lb> |
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11720 ἐκκειμένῃ ῥητῇ τῇ Α μήκει· ἡ ἄρα ΒΓ ἀποτομή ἐστι <lb n="1799,001,,,,10,89,33"></lb> |
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11721 πέμπτη.</s> <lb n="1799,001,,,,10,89,34"></lb> |
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11722 <s>Εὕρηται ἄρα ἡ πέμπτη ἀποτομὴ ἡ ΒΓ· ὅπερ ἔδει <lb n="1799,001,,,,10,89,35"></lb> |
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11723 δεῖξαι.</s> <lb n="1799,001,,,,10,89,36"></lb> |
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11724 <s>Εὑρεῖν τὴν ἕκτην ἀποτομήν.</s> <lb n="1799,001,,,,10,90,1"></lb> |
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11725 <s>Ἐκκείσθω ῥητὴ ἡ Α καὶ τρεῖς ἀριθμοὶ οἱ Ε, ΒΓ, ΓΔ <lb n="1799,001,,,,10,90,2"></lb> |
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11726 λόγον μὴ ἔχοντες πρὸς ἀλλήλους, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,10,90,3"></lb> |
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11727 πρὸς τετράγωνον ἀριθμόν· ἔτι δὲ καὶ ὁ ΓΒ πρὸς τὸν <lb n="1799,001,,,,10,90,4"></lb> |
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11728 ΒΔ λόγον μὴ ἐχέτω, ὃν τετρά- <lb n="1799,001,,,,10,90,5"></lb> |
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11729 γωνος ἀριθμὸς πρὸς τετράγωνον <lb n="1799,001,,,,10,90,6"></lb> |
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11730 ἀριθμόν· καὶ πεποιήσθω ὡς μὲν <lb n="1799,001,,,,10,90,7"></lb> |
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11731 ὁ Ε πρὸς τὸν ΒΓ, οὕτως τὸ <lb n="1799,001,,,,10,90,8"></lb> |
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11732 ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς ΖΗ, <lb n="1799,001,,,,10,90,9"></lb> |
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11733 ὡς δὲ ὁ ΒΓ πρὸς τὸν ΓΔ, οὕ-<lb n="1799,001,,,,10,90,10"></lb> |
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11734 τως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,90,11"></lb> |
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11735 τῆς ΗΘ.</s> <lb n="1799,001,,,,10,90,12"></lb> |
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11736 <s>Ἐπεὶ οὖν ἐστιν ὡς ὁ Ε πρὸς τὸν ΒΓ, οὕτως τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,90,13"></lb> |
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11737 Α πρὸς τὸ ἀπὸ τῆς ΖΗ, σύμμετρον ἄρα τὸ ἀπὸ τῆς Α <lb n="1799,001,,,,10,90,14"></lb> |
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11738 τῷ ἀπὸ τῆς ΖΗ.</s> <s>ῥητὸν δὲ τὸ ἀπὸ τῆς Α· ῥητὸν ἄρα καὶ <pb></pb> <lb n="1799,001,,,,10,90,15"></lb> |
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11739 τὸ ἀπὸ τῆς ΖΗ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΖΗ.</s> <s>καὶ ἐπεὶ ὁ Ε <lb n="1799,001,,,,10,90,16"></lb> |
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11740 πρὸς τὸν ΒΓ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς <lb n="1799,001,,,,10,90,17"></lb> |
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11741 τετράγωνον ἀριθμόν, οὐδ' ἄρα τὸ ἀπὸ τῆς Α πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,90,18"></lb> |
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11742 τῆς ΖΗ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετρά-<lb n="1799,001,,,,10,90,19"></lb> |
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11743 γωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ Α τῇ ΖΗ μήκει.</s> <lb n="1799,001,,,,10,90,20"></lb> |
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11744 <s>πάλιν, ἐπεί ἐστιν ὡς ὁ ΒΓ πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,90,21"></lb> |
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11745 τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ, σύμμετρον ἄρα τὸ ἀπὸ <lb n="1799,001,,,,10,90,22"></lb> |
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11746 τῆς ΖΗ τῷ ἀπὸ τῆς ΗΘ.</s> <s>ῥητὸν δὲ τὸ ἀπὸ τῆς ΖΗ· ῥητὸν <lb n="1799,001,,,,10,90,23"></lb> |
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11747 ἄρα καὶ τὸ ἀπὸ τῆς ΗΘ· ῥητὴ ἄρα καὶ ἡ ΗΘ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,90,24"></lb> |
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11748 ὁ ΒΓ πρὸς τὸν ΓΔ λόγον οὐκ ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,10,90,25"></lb> |
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11749 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδ' ἄρα τὸ ἀπὸ <lb n="1799,001,,,,10,90,26"></lb> |
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11750 τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ λόγον ἔχει, ὃν τετρά-<lb n="1799,001,,,,10,90,27"></lb> |
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11751 γωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος <lb n="1799,001,,,,10,90,28"></lb> |
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11752 ἄρα ἐστὶν ἡ ΖΗ τῇ ΗΘ μήκει.</s> <s>καί εἰσιν ἀμφότεραι ῥηταί· <lb n="1799,001,,,,10,90,29"></lb> |
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11753 αἱ ΖΗ, ΗΘ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· <lb n="1799,001,,,,10,90,30"></lb> |
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11754 ἡ ἄρα ΖΘ ἀποτομή ἐστιν.</s> <lb n="1799,001,,,,10,90,31"></lb> |
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11755 <s>Λέγω δή, ὅτι καὶ ἕκτη.</s> <lb n="1799,001,,,,10,90,32"></lb> |
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11756 <s>Ἐπεὶ γάρ ἐστιν ὡς μὲν ὁ Ε πρὸς τὸν ΒΓ, οὕτως τὸ <lb n="1799,001,,,,10,90,33"></lb> |
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11757 ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς ΖΗ, ὡς δὲ ὁ ΒΓ πρὸς τὸν <lb n="1799,001,,,,10,90,34"></lb> |
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11758 ΓΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ, δι' <lb n="1799,001,,,,10,90,35"></lb> |
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11759 ἴσου ἄρα ἐστὶν ὡς ὁ Ε πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,90,36"></lb> |
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11760 Α πρὸς τὸ ἀπὸ τῆς ΗΘ.</s> <s>ὁ δὲ Ε πρὸς τὸν ΓΔ λόγον οὐκ <lb n="1799,001,,,,10,90,37"></lb> |
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11761 ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· <lb n="1799,001,,,,10,90,38"></lb> |
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11762 οὐδ' ἄρα τὸ ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς ΗΘ λόγον ἔχει, <lb n="1799,001,,,,10,90,39"></lb> |
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11763 ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμ-<lb n="1799,001,,,,10,90,40"></lb> |
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11764 μετρος ἄρα ἐστὶν ἡ Α τῇ ΗΘ μήκει· οὐδετέρα ἄρα τῶν <pb></pb> <lb n="1799,001,,,,10,90,41"></lb> |
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11765 ΖΗ, ΗΘ σύμμετρός ἐστι τῇ Α ῥητῇ μήκει.</s> <s>ᾧ οὖν μεῖζόν <lb n="1799,001,,,,10,90,42"></lb> |
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11766 ἐστι τὸ ἀπὸ τῆς ΖΗ τοῦ ἀπὸ τῆς ΗΘ, ἔστω τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,90,43"></lb> |
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11767 Κ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς ὁ ΒΓ πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,10,90,44"></lb> |
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11768 τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ, ἀναστρέψαντι ἄρα ἐστὶν <lb n="1799,001,,,,10,90,45"></lb> |
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11769 ὡς ὁ ΓΒ πρὸς τὸν ΒΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς <lb n="1799,001,,,,10,90,46"></lb> |
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11770 τὸ ἀπὸ τῆς Κ.</s> <s>ὁ δὲ ΓΒ πρὸς τὸν ΒΔ λόγον οὐκ ἔχει, ὃν <lb n="1799,001,,,,10,90,47"></lb> |
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11771 τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· οὐδ' <lb n="1799,001,,,,10,90,48"></lb> |
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11772 ἄρα τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς Κ λόγον ἔχει, ὃν <lb n="1799,001,,,,10,90,49"></lb> |
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11773 τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμ-<lb n="1799,001,,,,10,90,50"></lb> |
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11774 μετρος ἄρα ἐστὶν ἡ ΖΗ τῇ Κ μήκει.</s> <s>καὶ δύναται ἡ ΖΗ <lb n="1799,001,,,,10,90,51"></lb> |
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11775 τῆς ΗΘ μεῖζον τῷ ἀπὸ τῆς Κ· ἡ ΖΗ ἄρα τῆς ΗΘ μεῖζον <lb n="1799,001,,,,10,90,52"></lb> |
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11776 δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει.</s> <s>καὶ οὐδετέρα <lb n="1799,001,,,,10,90,53"></lb> |
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11777 τῶν ΖΗ, ΗΘ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ μήκει <lb n="1799,001,,,,10,90,54"></lb> |
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11778 τῇ Α.</s> <s>ἡ ἄρα ΖΘ ἀποτομή ἐστιν ἕκτη.</s> <lb n="1799,001,,,,10,90,55"></lb> |
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11779 <s>Εὕρηται ἄρα ἡ ἕκτη ἀποτομὴ ἡ ΖΘ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,90,56"></lb> |
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11780 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς <lb n="1799,001,,,,10,91,1"></lb> |
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11781 πρώτης, ἡ τὸ χωρίον δυναμένη ἀποτομή ἐστιν.</s> <lb n="1799,001,,,,10,91,2"></lb> |
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11782 <s>Περιεχέσθω γὰρ χωρίον τὸ ΑΒ ὑπὸ ῥητῆς τῆς ΑΓ <lb n="1799,001,,,,10,91,3"></lb> |
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11783 καὶ ἀποτομῆς πρώτης τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ <lb n="1799,001,,,,10,91,4"></lb> |
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11784 χωρίον δυναμένη ἀποτομή ἐστιν.</s> <s><pb></pb> <lb n="1799,001,,,,10,91,5"></lb> |
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11785 Ἐπεὶ γὰρ ἀποτομή ἐστι πρώ-<lb n="1799,001,,,,10,91,6"></lb> |
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11786 τη ἡ ΑΔ, ἔστω αὐτῇ προσαρ-<lb n="1799,001,,,,10,91,7"></lb> |
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11787 μόζουσα ἡ ΔΗ· αἱ ΑΗ, ΗΔ <lb n="1799,001,,,,10,91,8"></lb> |
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11788 ἄρα ῥηταί εἰσι δυνάμει μόνον <lb n="1799,001,,,,10,91,9"></lb> |
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11789 σύμμετροι.</s> <s>καὶ ὅλη ἡ ΑΗ σύμ-<lb n="1799,001,,,,10,91,10"></lb> |
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11790 μετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ <lb n="1799,001,,,,10,91,11"></lb> |
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11791 τῇ ΑΓ, καὶ ἡ ΑΗ τῆς ΗΔ <lb n="1799,001,,,,10,91,12"></lb> |
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11792 μεῖζον δύναται τῷ ἀπὸ συμ-<lb n="1799,001,,,,10,91,13"></lb> |
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11793 μέτρου ἑαυτῇ μήκει· ἐὰν ἄρα <lb n="1799,001,,,,10,91,14"></lb> |
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11794 τῷ τετάρτῳ μέρει τοῦ ἀπὸ <lb n="1799,001,,,,10,91,15"></lb> |
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11795 τῆς ΔΗ ἴσον παρὰ τὴν ΑΗ <lb n="1799,001,,,,10,91,16"></lb> |
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11796 παραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ, εἰς σύμμετρα <lb n="1799,001,,,,10,91,17"></lb> |
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11797 αὐτὴν διαιρεῖ.</s> <s>τετμήσθω ἡ ΔΗ δίχα κατὰ τὸ Ε, καὶ τῷ <lb n="1799,001,,,,10,91,18"></lb> |
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11798 ἀπὸ τῆς ΕΗ ἴσον παρὰ τὴν ΑΗ παραβεβλήσθω ἐλλεῖπον <lb n="1799,001,,,,10,91,19"></lb> |
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11799 εἴδει τετραγώνῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΑΖ, ΖΗ· σύμ-<lb n="1799,001,,,,10,91,20"></lb> |
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11800 μετρος ἄρα ἐστὶν ἡ ΑΖ τῇ ΖΗ.</s> <s>καὶ διὰ τῶν Ε, Ζ, Η ση-<lb n="1799,001,,,,10,91,21"></lb> |
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11801 μείων τῇ ΑΓ παράλληλοι ἤχθωσαν αἱ ΕΘ, ΖΙ, ΗΚ.</s> <lb n="1799,001,,,,10,91,22"></lb> |
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11802 <s>Καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΑΖ τῇ ΖΗ μήκει, καὶ <lb n="1799,001,,,,10,91,23"></lb> |
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11803 ἡ ΑΗ ἄρα ἑκατέρᾳ τῶν ΑΖ, ΖΗ σύμμετρός ἐστι μήκει.</s> <lb n="1799,001,,,,10,91,24"></lb> |
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11804 <s>ἀλλὰ ἡ ΑΗ σύμμετρός ἐστι τῇ ΑΓ· καὶ ἑκατέρα ἄρα <lb n="1799,001,,,,10,91,25"></lb> |
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11805 τῶν ΑΖ, ΖΗ σύμμετρός ἐστι τῇ ΑΓ μήκει.</s> <s>καί ἐστι <lb n="1799,001,,,,10,91,26"></lb> |
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11806 ῥητὴ ἡ ΑΓ· ῥητὴ ἄρα καὶ ἑκατέρα τῶν ΑΖ, ΖΗ· ὥστε <lb n="1799,001,,,,10,91,27"></lb> |
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11807 καὶ ἑκάτερον τῶν ΑΙ, ΖΚ ῥητόν ἐστιν.</s> <s>καὶ ἐπεὶ σύμμετρός <lb n="1799,001,,,,10,91,28"></lb> |
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11808 ἐστιν ἡ ΔΕ τῇ ΕΗ μήκει, καὶ ἡ ΔΗ ἄρα ἑκατέρᾳ τῶν <pb></pb> <lb n="1799,001,,,,10,91,29"></lb> |
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11809 ΔΕ, ΕΗ σύμμετρός ἐστι μήκει.</s> <s>ῥητὴ δὲ ἡ ΔΗ καὶ <lb n="1799,001,,,,10,91,30"></lb> |
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11810 ἀσύμμετρος τῇ ΑΓ μήκει· ῥητὴ ἄρα καὶ ἑκατέρα τῶν ΔΕ, <lb n="1799,001,,,,10,91,31"></lb> |
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11811 ΕΗ καὶ ἀσύμμετρος τῇ ΑΓ μήκει· ἑκάτερον ἄρα τῶν <lb n="1799,001,,,,10,91,32"></lb> |
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11812 ΔΘ, ΕΚ μέσον ἐστίν.</s> <lb n="1799,001,,,,10,91,33"></lb> |
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11813 <s>Κείσθω δὴ τῷ μὲν ΑΙ ἴσον τετράγωνον τὸ ΛΜ, τῷ <lb n="1799,001,,,,10,91,34"></lb> |
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11814 δὲ ΖΚ ἴσον τετράγωνον ἀφῃρήσθω κοινὴν γωνίαν ἔχον <lb n="1799,001,,,,10,91,35"></lb> |
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11815 αὐτῷ τὴν ὑπὸ ΛΟΜ τὸ ΝΞ· περὶ τὴν αὐτὴν ἄρα διάμετρόν <lb n="1799,001,,,,10,91,36"></lb> |
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11816 ἐστι τὰ ΛΜ, ΝΞ τετράγωνα.</s> <s>ἔστω αὐτῶν διάμετρος <lb n="1799,001,,,,10,91,37"></lb> |
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11817 ἡ ΟΡ, καὶ καταγεγράφθω τὸ σχῆμα.</s> <s>ἐπεὶ οὖν ἴσον ἐστὶ <lb n="1799,001,,,,10,91,38"></lb> |
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11818 τὸ ὑπὸ τῶν ΑΖ, ΖΗ περιεχόμενον ὀρθογώνιον τῷ ἀπὸ <lb n="1799,001,,,,10,91,39"></lb> |
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11819 τῆς ΕΗ τετραγώνῳ, ἔστιν ἄρα ὡς ἡ ΑΖ πρὸς τὴν ΕΗ, <lb n="1799,001,,,,10,91,40"></lb> |
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11820 οὕτως ἡ ΕΗ πρὸς τὴν ΖΗ.</s> <s>ἀλλ' ὡς μὲν ἡ ΑΖ πρὸς τὴν <lb n="1799,001,,,,10,91,41"></lb> |
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11821 ΕΗ, οὕτως τὸ ΑΙ πρὸς τὸ ΕΚ, ὡς δὲ ἡ ΕΗ πρὸς τὴν <lb n="1799,001,,,,10,91,42"></lb> |
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11822 ΖΗ, οὕτως ἐστὶ τὸ ΕΚ πρὸς τὸ ΚΖ· τῶν ἄρα ΑΙ, ΚΖ <lb n="1799,001,,,,10,91,43"></lb> |
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11823 μέσον ἀνάλογόν ἐστι τὸ ΕΚ.</s> <s>ἔστι δὲ καὶ τῶν ΛΜ, ΝΞ <lb n="1799,001,,,,10,91,44"></lb> |
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11824 μέσον ἀνάλογον τὸ ΜΝ, ὡς ἐν τοῖς ἔμπροσθεν ἐδείχθη, καί <lb n="1799,001,,,,10,91,45"></lb> |
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11825 ἐστι τὸ [μὲν] ΑΙ τῷ ΛΜ τετραγώνῳ ἴσον, τὸ δὲ ΚΖ τῷ <lb n="1799,001,,,,10,91,46"></lb> |
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11826 ΝΞ· καὶ τὸ ΜΝ ἄρα τῷ ΕΚ ἴσον ἐστίν.</s> <s>ἀλλὰ τὸ μὲν <lb n="1799,001,,,,10,91,47"></lb> |
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11827 ΕΚ τῷ ΔΘ ἐστιν ἴσον, τὸ δὲ ΜΝ τῷ ΛΞ· τὸ ἄρα ΔΚ <lb n="1799,001,,,,10,91,48"></lb> |
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11828 ἴσον ἐστὶ τῷ ΥΦΧ γνώμονι καὶ τῷ ΝΞ.</s> <s>ἔστι δὲ καὶ τὸ <lb n="1799,001,,,,10,91,49"></lb> |
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11829 ΑΚ ἴσον τοῖς ΛΜ, ΝΞ τετραγώνοις· λοιπὸν ἄρα τὸ ΑΒ <lb n="1799,001,,,,10,91,50"></lb> |
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11830 ἴσον ἐστὶ τῷ ΣΤ.</s> <s>τὸ δὲ ΣΤ τὸ ἀπὸ τῆς ΛΝ ἐστι τετρά- <pb></pb><lb n="1799,001,,,,10,91,51"></lb> |
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11831 γωνον· τὸ ἄρα ἀπὸ τῆς ΛΝ τετράγωνον ἴσον ἐστὶ τῷ <lb n="1799,001,,,,10,91,52"></lb> |
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11832 ΑΒ· ἡ ΛΝ ἄρα δύναται τὸ ΑΒ.</s> <lb n="1799,001,,,,10,91,53"></lb> |
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11833 <s>Λέγω δή, ὅτι ἡ ΛΝ ἀποτομή ἐστιν.</s> <lb n="1799,001,,,,10,91,54"></lb> |
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11834 <s>Ἐπεὶ γὰρ ῥητόν ἐστιν ἑκάτερον τῶν ΑΙ, ΖΚ, καί ἐστιν <lb n="1799,001,,,,10,91,55"></lb> |
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11835 ἴσον τοῖς ΛΜ, ΝΞ, καὶ ἑκάτερον ἄρα τῶν ΛΜ, ΝΞ <lb n="1799,001,,,,10,91,56"></lb> |
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11836 ῥητόν ἐστιν, τουτέστι τὸ ἀπὸ ἑκατέρας τῶν ΛΟ, ΟΝ· <lb n="1799,001,,,,10,91,57"></lb> |
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11837 καὶ ἑκατέρα ἄρα τῶν ΛΟ, Ο*ν ῥητή ἐστιν.</s> <s>πάλιν, ἐπεὶ <lb n="1799,001,,,,10,91,58"></lb> |
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11838 μέσον ἐστὶ τὸ ΔΘ καί ἐστιν ἴσον τῷ ΛΞ, μέσον ἄρα ἐστὶ <lb n="1799,001,,,,10,91,59"></lb> |
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11839 καὶ τὸ ΛΞ.</s> <s>ἐπεὶ οὖν τὸ μὲν ΛΞ μέσον ἐστίν, τὸ δὲ ΝΞ <lb n="1799,001,,,,10,91,60"></lb> |
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11840 ῥητόν, ἀσύμμετρον ἄρα ἐστὶ τὸ ΛΞ τῷ ΝΞ· ὡς δὲ τὸ <lb n="1799,001,,,,10,91,61"></lb> |
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11841 ΛΞ πρὸς τὸ ΝΞ, οὕτως ἐστὶν ἡ ΛΟ πρὸς τὴν ΟΝ· <lb n="1799,001,,,,10,91,62"></lb> |
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11842 ἀσύμμετρος ἄρα ἐστὶν ἡ ΛΟ τῇ ΟΝ μήκει.</s> <s>καί εἰσιν ἀμ-<lb n="1799,001,,,,10,91,63"></lb> |
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11843 φότεραι ῥηταί· αἱ ΛΟ, ΟΝ ἄρα ῥηταί εἰσι δυνάμει μόνον <lb n="1799,001,,,,10,91,64"></lb> |
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11844 σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΛΝ.</s> <s>καὶ δύναται τὸ ΑΒ <lb n="1799,001,,,,10,91,65"></lb> |
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11845 χωρίον· ἡ ἄρα τὸ ΑΒ χωρίον δυναμένη ἀποτομή ἐστιν.</s> <lb n="1799,001,,,,10,91,66"></lb> |
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11846 <s>Ἐὰν ἄρα χωρίον περιέχηται ὑπὸ ῥητῆς, καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,91,67"></lb> |
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11847 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς <lb n="1799,001,,,,10,92,1"></lb> |
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11848 δευτέρας, ἡ τὸ χωρίον δυναμένη μέσης ἀποτομή ἐστι <lb n="1799,001,,,,10,92,2"></lb> |
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11849 πρώτη.</s> <lb n="1799,001,,,,10,92,3"></lb> |
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11850 <s>Χωρίον γὰρ τὸ ΑΒ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΓ καὶ <lb n="1799,001,,,,10,92,4"></lb> |
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11851 ἀποτομῆς δευτέρας τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ χωρίον <lb n="1799,001,,,,10,92,5"></lb> |
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11852 δυναμένη μέσης ἀποτομή ἐστι πρώτη.</s> <s><pb></pb> <lb n="1799,001,,,,10,92,6"></lb> |
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11853 Ἔστω γὰρ τῇ ΑΔ προσαρ- <lb n="1799,001,,,,10,92,7"></lb> |
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11854 μόζουσα ἡ ΔΗ· αἱ ἄρα ΑΗ, <lb n="1799,001,,,,10,92,8"></lb> |
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11855 ΗΔ ῥηταί εἰσι δυνάμει μόνον <lb n="1799,001,,,,10,92,9"></lb> |
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11856 σύμμετροι, καὶ ἡ προσαρμό-<lb n="1799,001,,,,10,92,10"></lb> |
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11857 ζουσα ἡ ΔΗ σύμμετρός ἐστι <lb n="1799,001,,,,10,92,11"></lb> |
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11858 τῇ ἐκκειμένῃ ῥητῇ τῇ ΑΓ, ἡ <lb n="1799,001,,,,10,92,12"></lb> |
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11859 δὲ ὅλη ἡ ΑΗ τῆς προσαρμο-<lb n="1799,001,,,,10,92,13"></lb> |
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11860 ζούσης τῆς ΗΔ μεῖζον δύνα-<lb n="1799,001,,,,10,92,14"></lb> |
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11861 ται τῷ ἀπὸ συμμέτρου ἑαυτῇ <lb n="1799,001,,,,10,92,15"></lb> |
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11862 μήκει.</s> <s>ἐπεὶ οὖν ἡ ΑΗ τῆς <lb n="1799,001,,,,10,92,16"></lb> |
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11863 ΗΔ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, ἐὰν ἄρα <lb n="1799,001,,,,10,92,17"></lb> |
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11864 τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΗΔ ἴσον παρὰ τὴν <lb n="1799,001,,,,10,92,18"></lb> |
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11865 ΑΗ παραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ, εἰς σύμμετρα <lb n="1799,001,,,,10,92,19"></lb> |
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11866 αὐτὴν διαιρεῖ.</s> <s>τετμήσθω οὖν ἡ ΔΗ δίχα κατὰ τὸ Ε· καὶ <lb n="1799,001,,,,10,92,20"></lb> |
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11867 τῷ ἀπὸ τῆς ΕΗ ἴσον παρὰ τὴν ΑΗ παραβεβλήσθω <lb n="1799,001,,,,10,92,21"></lb> |
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11868 ἐλλεῖπον εἴδει τετραγώνῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΑΖ, ΖΗ· <lb n="1799,001,,,,10,92,22"></lb> |
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11869 σύμμετρος ἄρα ἐστὶν ἡ ΑΖ τῇ ΖΗ μήκει.</s> <s>καὶ ἡ ΑΗ ἄρα <lb n="1799,001,,,,10,92,23"></lb> |
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11870 ἑκατέρᾳ τῶν ΑΖ, ΖΗ σύμμετρός ἐστι μήκει.</s> <s>ῥητὴ δὲ <lb n="1799,001,,,,10,92,24"></lb> |
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11871 ἡ ΑΗ καὶ ἀσύμμετρος τῇ ΑΓ μήκει· καὶ ἑκατέρα ἄρα <lb n="1799,001,,,,10,92,25"></lb> |
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11872 τῶν ΑΖ, ΖΗ ῥητή ἐστι καὶ ἀσύμμετρος τῇ ΑΓ μήκει· <lb n="1799,001,,,,10,92,26"></lb> |
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11873 ἑκάτερον ἄρα τῶν ΑΙ, ΖΚ μέσον ἐστίν.</s> <s>πάλιν, ἐπεὶ <lb n="1799,001,,,,10,92,27"></lb> |
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11874 σύμμετρός ἐστιν ἡ ΔΕ τῇ ΕΗ, καὶ ἡ ΔΗ ἄρα ἑκατέρᾳ <lb n="1799,001,,,,10,92,28"></lb> |
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11875 τῶν ΔΕ, ΕΗ σύμμετρός ἐστιν.</s> <s>ἀλλ' ἡ ΔΗ σύμμετρός <pb></pb> <lb n="1799,001,,,,10,92,29"></lb> |
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11876 ἐστι τῇ ΑΓ μήκει.</s> <s>[ῥητὴ ἄρα καὶ ἑκατέρα τῶν ΔΕ, ΕΗ <lb n="1799,001,,,,10,92,30"></lb> |
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11877 καὶ σύμμετρος τῇ ΑΓ μήκει.] ἑκάτερον ἄρα τῶν ΔΘ, ΕΚ <lb n="1799,001,,,,10,92,31"></lb> |
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11878 ῥητόν ἐστιν.</s> <lb n="1799,001,,,,10,92,32"></lb> |
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11879 <s>Συνεστάτω οὖν τῷ μὲν ΑΙ ἴσον τετράγωνον τὸ ΛΜ, <lb n="1799,001,,,,10,92,33"></lb> |
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11880 τῷ δὲ ΖΚ ἴσον ἀφῃρήσθω τὸ ΝΞ περὶ τὴν αὐτὴν γωνίαν <lb n="1799,001,,,,10,92,34"></lb> |
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11881 ὂν τῷ ΛΜ τὴν ὑπὸ τῶν ΛΟΜ· περὶ τὴν αὐτὴν ἄρα ἐστὶ <lb n="1799,001,,,,10,92,35"></lb> |
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11882 διάμετρον τὰ ΛΜ, ΝΞ τετράγωνα.</s> <s>ἔστω αὐτῶν διάμετρος <lb n="1799,001,,,,10,92,36"></lb> |
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11883 ἡ ΟΡ, καὶ καταγεγράφθω τὸ σχῆμα.</s> <s>ἐπεὶ οὖν τὰ ΑΙ, <lb n="1799,001,,,,10,92,37"></lb> |
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11884 ΖΚ μέσα ἐστὶ καί ἐστιν ἴσα τοῖς ἀπὸ τῶν ΛΟ, ΟΝ, καὶ <lb n="1799,001,,,,10,92,38"></lb> |
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11885 τὰ ἀπὸ τῶν ΛΟ, ΟΝ [ἄρα] μέσα ἐστίν· καὶ αἱ ΛΟ, ΟΝ <lb n="1799,001,,,,10,92,39"></lb> |
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11886 ἄρα μέσαι εἰσὶ δυνάμει μόνον σύμμετροι.</s> <s>καὶ ἐπεὶ τὸ ὑπὸ <lb n="1799,001,,,,10,92,40"></lb> |
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11887 τῶν ΑΖ, ΖΗ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΕΗ, ἔστιν ἄρα ὡς <lb n="1799,001,,,,10,92,41"></lb> |
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11888 ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως ἡ ΕΗ πρὸς τὴν ΖΗ· ἀλλ' ὡς <lb n="1799,001,,,,10,92,42"></lb> |
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11889 μὲν ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως τὸ ΑΙ πρὸς τὸ ΕΚ· ὡς <lb n="1799,001,,,,10,92,43"></lb> |
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11890 δὲ ἡ ΕΗ πρὸς τὴν ΖΗ, οὕτως [ἐστὶ] τὸ ΕΚ πρὸς τὸ ΖΚ· <lb n="1799,001,,,,10,92,44"></lb> |
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11891 τῶν ἄρα ΑΙ, ΖΚ μέσον ἀνάλογόν ἐστι τὸ ΕΚ.</s> <s>ἔστι δὲ <lb n="1799,001,,,,10,92,45"></lb> |
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11892 καὶ τῶν ΛΜ, ΝΞ τετραγώνων μέσον ἀνάλογον τὸ ΜΝ· <lb n="1799,001,,,,10,92,46"></lb> |
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11893 καί ἐστιν ἴσον τὸ μὲν ΑΙ τῷ ΛΜ, τὸ δὲ ΖΚ τῷ ΝΞ· <lb n="1799,001,,,,10,92,47"></lb> |
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11894 καὶ τὸ ΜΝ ἄρα ἴσον ἐστὶ τῷ ΕΚ.</s> <s>ἀλλὰ τῷ μὲν ΕΚ ἴσον <lb n="1799,001,,,,10,92,48"></lb> |
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11895 [ἐστὶ] τὸ ΔΘ, τῷ δὲ ΜΝ ἴσον τὸ ΛΞ· ὅλον ἄρα τὸ ΔΚ <lb n="1799,001,,,,10,92,49"></lb> |
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11896 ἴσον ἐστὶ τῷ ΥΦΧ γνώμονι καὶ τῷ ΝΞ.</s> <s>ἐπεὶ οὖν ὅλον <pb></pb> <lb n="1799,001,,,,10,92,50"></lb> |
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11897 τὸ ΑΚ ἴσον ἐστὶ τοῖς ΛΜ, ΝΞ, ὧν τὸ ΔΚ ἴσον ἐστὶ τῷ <lb n="1799,001,,,,10,92,51"></lb> |
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11898 ΥΦΧ γνώμονι καὶ τῷ ΝΞ, λοιπὸν ἄρα τὸ ΑΒ ἴσον ἐστὶ <lb n="1799,001,,,,10,92,52"></lb> |
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11899 τῷ ΤΣ.</s> <s>τὸ δὲ ΤΣ ἐστι τὸ ἀπὸ τῆς ΛΝ· τὸ ἀπὸ τῆς ΛΝ <lb n="1799,001,,,,10,92,53"></lb> |
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11900 ἄρα ἴσον ἐστὶ τῷ ΑΒ χωρίῳ· ἡ ΛΝ ἄρα δύναται τὸ ΑΒ <lb n="1799,001,,,,10,92,54"></lb> |
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11901 χωρίον.</s> <lb n="1799,001,,,,10,92,55"></lb> |
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11902 <s>Λέγω [δή], ὅτι ἡ ΛΝ μέσης ἀποτομή ἐστι πρώτη.</s> <lb n="1799,001,,,,10,92,56"></lb> |
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11903 <s>Ἐπεὶ γὰρ ῥητόν ἐστι τὸ ΕΚ καί ἐστιν ἴσον τῷ ΛΞ, <lb n="1799,001,,,,10,92,57"></lb> |
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11904 ῥητὸν ἄρα ἐστὶ τὸ ΛΞ, τουτέστι τὸ ὑπὸ τῶν ΛΟ, ΟΝ.</s> <lb n="1799,001,,,,10,92,58"></lb> |
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11905 <s>μέσον δὲ ἐδείχθη τὸ ΝΞ· ἀσύμμετρον ἄρα ἐστὶ τὸ ΛΞ <lb n="1799,001,,,,10,92,59"></lb> |
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11906 τῷ ΝΞ· ὡς δὲ τὸ ΛΞ πρὸς τὸ ΝΞ, οὕτως ἐστὶν ἡ ΛΟ <lb n="1799,001,,,,10,92,60"></lb> |
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11907 πρὸς ΟΝ· αἱ ΛΟ, ΟΝ ἄρα ἀσύμμετροί εἰσι μήκει.</s> <s>αἱ <lb n="1799,001,,,,10,92,61"></lb> |
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11908 ἄρα ΛΟ, ΟΝ μέσαι εἰσὶ δυνάμει μόνον σύμμετροι ῥητὸν <lb n="1799,001,,,,10,92,62"></lb> |
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11909 περιέχουσαι· ἡ ΛΝ ἄρα μέσης ἀποτομή ἐστι πρώτη· <lb n="1799,001,,,,10,92,63"></lb> |
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11910 καὶ δύναται τὸ ΑΒ χωρίον.</s> <lb n="1799,001,,,,10,92,64"></lb> |
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11911 <s>Ἡ ἄρα τὸ ΑΒ χωρίον δυναμένη μέσης ἀποτομή ἐστι <lb n="1799,001,,,,10,92,65"></lb> |
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11912 πρώτη· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,92,66"></lb> |
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11913 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς <lb n="1799,001,,,,10,93,1"></lb> |
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11914 τρίτης, ἡ τὸ χωρίον δυναμένη μέσης ἀποτομή ἐστι <lb n="1799,001,,,,10,93,2"></lb> |
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11915 δευτέρα.</s> <s><pb></pb> <lb n="1799,001,,,,10,93,3"></lb> |
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11916 Χωρίον γὰρ τὸ ΑΒ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΓ <lb n="1799,001,,,,10,93,4"></lb> |
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11917 καὶ ἀποτομῆς τρίτης τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ χωρίον <lb n="1799,001,,,,10,93,5"></lb> |
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11918 δυναμένη μέσης ἀποτομή ἐστι δευτέρα.</s> <lb n="1799,001,,,,10,93,6"></lb> |
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11919 <s>Ἔστω γὰρ τῇ ΑΔ προσαρ-<lb n="1799,001,,,,10,93,7"></lb> |
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11920 μόζουσα ἡ ΔΗ· αἱ ΑΗ, ΗΔ <lb n="1799,001,,,,10,93,8"></lb> |
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11921 ἄρα ῥηταί εἰσι δυνάμει μό-<lb n="1799,001,,,,10,93,9"></lb> |
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11922 νον σύμμετροι, καὶ οὐδετέρα <lb n="1799,001,,,,10,93,10"></lb> |
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11923 τῶν ΑΗ, ΗΔ σύμμετρός <lb n="1799,001,,,,10,93,11"></lb> |
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11924 ἐστι μήκει τῇ ἐκκειμένῃ ῥητῇ <lb n="1799,001,,,,10,93,12"></lb> |
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11925 τῇ ΑΓ, ἡ δὲ ὅλη ἡ ΑΗ <lb n="1799,001,,,,10,93,13"></lb> |
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11926 τῆς προσαρμοζούσης τῆς ΔΗ <lb n="1799,001,,,,10,93,14"></lb> |
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11927 μεῖζον δύναται τῷ ἀπὸ συμ-<lb n="1799,001,,,,10,93,15"></lb> |
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11928 μέτρου ἑαυτῇ.</s> <s>ἐπεὶ οὖν ἡ ΑΗ <lb n="1799,001,,,,10,93,16"></lb> |
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11929 τῆς ΗΔ μεῖζον δύναται τῷ <lb n="1799,001,,,,10,93,17"></lb> |
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11930 ἀπὸ συμμέτρου ἑαυτῇ, ἐὰν <lb n="1799,001,,,,10,93,18"></lb> |
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11931 ἄρα τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΔΗ ἴσον παρὰ τὴν <lb n="1799,001,,,,10,93,19"></lb> |
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11932 ΑΗ παραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ, εἰς σύμμετρα <lb n="1799,001,,,,10,93,20"></lb> |
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11933 αὐτὴν διελεῖ.</s> <s>τετμήσθω οὖν ἡ ΔΗ δίχα κατὰ τὸ Ε, καὶ <lb n="1799,001,,,,10,93,21"></lb> |
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11934 τῷ ἀπὸ τῆς ΕΗ ἴσον παρὰ τὴν ΑΗ παραβεβλήσθω <lb n="1799,001,,,,10,93,22"></lb> |
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11935 ἐλλεῖπον εἴδει τετραγώνῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΑΖ, <lb n="1799,001,,,,10,93,23"></lb> |
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11936 ΖΗ.</s> <s>καὶ ἤχθωσαν διὰ τῶν Ε, Ζ, Η σημείων τῇ ΑΓ <lb n="1799,001,,,,10,93,24"></lb> |
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11937 παράλληλοι αἱ ΕΘ, ΖΙ, ΗΚ· σύμμετροι ἄρα εἰσὶν αἱ <lb n="1799,001,,,,10,93,25"></lb> |
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11938 ΑΖ, ΖΗ· σύμμετρον ἄρα καὶ τὸ ΑΙ τῷ ΖΚ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,93,26"></lb> |
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11939 αἱ ΑΖ, ΖΗ σύμμετροί εἰσι μήκει, καὶ ἡ ΑΗ ἄρα ἑκατέρᾳ <lb n="1799,001,,,,10,93,27"></lb> |
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11940 τῶν ΑΖ, ΖΗ σύμμετρός ἐστι μήκει.</s> <s>ῥητὴ δὲ ἡ ΑΗ καὶ <lb n="1799,001,,,,10,93,28"></lb> |
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11941 ἀσύμμετρος τῇ ΑΓ μήκει· ὥστε καὶ αἱ ΑΖ, ΖΗ.</s> <s><pb></pb> <lb n="1799,001,,,,10,93,29"></lb> |
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11942 ἑκάτερον ἄρα τῶν ΑΙ, ΖΚ μέσον ἐστίν.</s> <s>πάλιν, ἐπεὶ <lb n="1799,001,,,,10,93,30"></lb> |
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11943 σύμμετρός ἐστιν ἡ ΔΕ τῇ ΕΗ μήκει, καὶ ἡ ΔΗ ἄρα <lb n="1799,001,,,,10,93,31"></lb> |
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11944 ἑκατέρᾳ τῶν ΔΕ, ΕΗ σύμμετρός ἐστι μήκει.</s> <s>ῥητὴ δὲ <lb n="1799,001,,,,10,93,32"></lb> |
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11945 ἡ ΗΔ καὶ ἀσύμμετρος τῇ ΑΓ μήκει· ῥητὴ ἄρα καὶ <lb n="1799,001,,,,10,93,33"></lb> |
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11946 ἑκατέρα τῶν ΔΕ, ΕΗ καὶ ἀσύμμετρος τῇ ΑΓ μήκει· <lb n="1799,001,,,,10,93,34"></lb> |
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11947 ἑκάτερον ἄρα τῶν ΔΘ, ΕΚ μέσον ἐστίν.</s> <s>καὶ ἐπεὶ αἱ <lb n="1799,001,,,,10,93,35"></lb> |
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11948 ΑΗ, ΗΔ δυνάμει μόνον σύμμετροί εἰσιν, ἀσύμμετρος ἄρα <lb n="1799,001,,,,10,93,36"></lb> |
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11949 ἐστὶ μήκει ἡ ΑΗ τῇ ΗΔ.</s> <s>ἀλλ' ἡ μὲν ΑΗ τῇ ΑΖ σύμ-<lb n="1799,001,,,,10,93,37"></lb> |
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11950 μετρός ἐστι μήκει, ἡ δὲ ΔΗ τῇ ΕΗ· ἀσύμμετρος ἄρα <lb n="1799,001,,,,10,93,38"></lb> |
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11951 ἐστὶν ἡ ΑΖ τῇ ΕΗ μήκει.</s> <s>ὡς δὲ ἡ ΑΖ πρὸς τὴν ΕΗ, <lb n="1799,001,,,,10,93,39"></lb> |
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11952 οὕτως ἐστὶ τὸ ΑΙ πρὸς τὸ ΕΚ· ἀσύμμετρον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,10,93,40"></lb> |
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11953 ΑΙ τῷ ΕΚ.</s> <lb n="1799,001,,,,10,93,41"></lb> |
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11954 <s>Συνεστάτω οὖν τῷ μὲν ΑΙ ἴσον τετράγωνον τὸ ΛΜ, τῷ <lb n="1799,001,,,,10,93,42"></lb> |
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11955 δὲ ΖΚ ἴσον ἀφῃρήσθω τὸ ΝΞ περὶ τὴν αὐτὴν γωνίαν ὂν <lb n="1799,001,,,,10,93,43"></lb> |
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11956 τῷ ΛΜ· περὶ τὴν αὐτὴν ἄρα διάμετρόν ἐστι τὰ ΛΜ, ΝΞ.</s> <lb n="1799,001,,,,10,93,44"></lb> |
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11957 <s>ἔστω αὐτῶν διάμετρος ἡ ΟΡ, καὶ καταγεγράφθω τὸ <lb n="1799,001,,,,10,93,45"></lb> |
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11958 σχῆμα.</s> <s>ἐπεὶ οὖν τὸ ὑπὸ τῶν ΑΖ, ΖΗ ἴσον ἐστὶ τῷ ἀπὸ <lb n="1799,001,,,,10,93,46"></lb> |
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11959 τῆς ΕΗ, ἔστιν ἄρα ὡς ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως ἡ ΕΗ <lb n="1799,001,,,,10,93,47"></lb> |
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11960 πρὸς τὴν ΖΗ.</s> <s>ἀλλ' ὡς μὲν ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως <lb n="1799,001,,,,10,93,48"></lb> |
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11961 ἐστὶ τὸ ΑΙ πρὸς τὸ ΕΚ· ὡς δὲ ἡ ΕΗ πρὸς τὴν ΖΗ, <pb></pb> <lb n="1799,001,,,,10,93,49"></lb> |
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11962 οὕτως ἐστὶ τὸ ΕΚ πρὸς τὸ ΖΚ· καὶ ὡς ἄρα τὸ ΑΙ πρὸς <lb n="1799,001,,,,10,93,50"></lb> |
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11963 τὸ ΕΚ, οὕτως τὸ ΕΚ πρὸς τὸ ΖΚ· τῶν ἄρα ΑΙ, ΖΚ <lb n="1799,001,,,,10,93,51"></lb> |
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11964 μέσον ἀνάλογόν ἐστι τὸ ΕΚ.</s> <s>ἔστι δὲ καὶ τῶν ΛΜ, ΝΞ <lb n="1799,001,,,,10,93,52"></lb> |
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11965 τετραγώνων μέσον ἀνάλογον τὸ ΜΝ· καί ἐστιν ἴσον τὸ <lb n="1799,001,,,,10,93,53"></lb> |
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11966 μὲν ΑΙ τῷ ΛΜ, τὸ δὲ ΖΚ τῷ ΝΞ· καὶ τὸ ΕΚ ἄρα <lb n="1799,001,,,,10,93,54"></lb> |
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11967 ἴσον ἐστὶ τῷ ΜΝ.</s> <s>ἀλλὰ τὸ μὲν ΜΝ ἴσον ἐστὶ τῷ ΛΞ, <lb n="1799,001,,,,10,93,55"></lb> |
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11968 τὸ δὲ ΕΚ ἴσον [ἐστὶ] τῷ ΔΘ· καὶ ὅλον ἄρα τὸ ΔΚ ἴσον <lb n="1799,001,,,,10,93,56"></lb> |
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11969 ἐστὶ τῷ ΥΦΧ γνώμονι καὶ τῷ ΝΞ.</s> <s>ἔστι δὲ καὶ τὸ ΑΚ <lb n="1799,001,,,,10,93,57"></lb> |
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11970 ἴσον τοῖς ΛΜ, ΝΞ· λοιπὸν ἄρα τὸ ΑΒ ἴσον ἐστὶ τῷ ΣΤ, <lb n="1799,001,,,,10,93,58"></lb> |
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11971 τουτέστι τῷ ἀπὸ τῆς ΛΝ τετραγώνῳ· ἡ ΛΝ ἄρα δύναται <lb n="1799,001,,,,10,93,59"></lb> |
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11972 τὸ ΑΒ χωρίον.</s> <lb n="1799,001,,,,10,93,60"></lb> |
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11973 <s>Λέγω, ὅτι ἡ ΛΝ μέσης ἀποτομή ἐστι δευτέρα.</s> <lb n="1799,001,,,,10,93,61"></lb> |
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11974 <s>Ἐπεὶ γὰρ μέσα ἐδείχθη τὰ ΑΙ, ΖΚ καί ἐστιν ἴσα τοῖς <lb n="1799,001,,,,10,93,62"></lb> |
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11975 ἀπὸ τῶν ΛΟ, ΟΝ, μέσον ἄρα καὶ ἑκάτερον τῶν ἀπὸ <lb n="1799,001,,,,10,93,63"></lb> |
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11976 τῶν ΛΟ, ΟΝ· μέση ἄρα ἑκατέρα τῶν ΛΟ, ΟΝ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,93,64"></lb> |
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11977 σύμμετρόν ἐστι τὸ ΑΙ τῷ ΖΚ, σύμμετρον ἄρα καὶ τὸ <lb n="1799,001,,,,10,93,65"></lb> |
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11978 ἀπὸ τῆς ΛΟ τῷ ἀπὸ τῆς ΟΝ.</s> <s>πάλιν, ἐπεὶ ἀσύμμετρον <lb n="1799,001,,,,10,93,66"></lb> |
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11979 ἐδείχθη τὸ ΑΙ τῷ ΕΚ, ἀσύμμετρον ἄρα ἐστὶ καὶ τὸ ΛΜ <lb n="1799,001,,,,10,93,67"></lb> |
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11980 τῷ ΜΝ, τουτέστι τὸ ἀπὸ τῆς ΛΟ τῷ ὑπὸ τῶν ΛΟ, ΟΝ· <lb n="1799,001,,,,10,93,68"></lb> |
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11981 ὥστε καὶ ἡ ΛΟ ἀσύμμετρός ἐστι τῇ ΟΝ· αἱ ΛΟ, ΟΝ <lb n="1799,001,,,,10,93,69"></lb> |
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11982 ἄρα μέσαι εἰσὶ δυνάμει μόνον σύμμετροι.</s> <lb n="1799,001,,,,10,93,70"></lb> |
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11983 <s>Λέγω δή, ὅτι καὶ μέσον περιέχουσιν.</s> <lb n="1799,001,,,,10,93,71"></lb> |
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11984 <s>Ἐπεὶ γὰρ μέσον ἐδείχθη τὸ ΕΚ καί ἐστιν ἴσον τῷ <lb n="1799,001,,,,10,93,72"></lb> |
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11985 ὑπὸ τῶν ΛΟ, ΟΝ, μέσον ἄρα ἐστὶ καὶ τὸ ὑπὸ τῶν ΛΟ, <lb n="1799,001,,,,10,93,73"></lb> |
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11986 ΟΝ· ὥστε αἱ ΛΟ, ΟΝ μέσαι εἰσὶ δυνάμει μόνον σύμμετροι <pb></pb> <lb n="1799,001,,,,10,93,74"></lb> |
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11987 μέσον περιέχουσαι.</s> <s>ἡ ΛΝ ἄρα μέσης ἀποτομή ἐστι <lb n="1799,001,,,,10,93,75"></lb> |
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11988 δευτέρα· καὶ δύναται τὸ ΑΒ χωρίον.</s> <lb n="1799,001,,,,10,93,76"></lb> |
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11989 <s>Ἡ ἄρα τὸ ΑΒ χωρίον δυναμένη μέσης ἀποτομή ἐστι <lb n="1799,001,,,,10,93,77"></lb> |
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11990 δευτέρα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,93,78"></lb> |
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11991 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς <lb n="1799,001,,,,10,94,1"></lb> |
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11992 τετάρτης, ἡ τὸ χωρίον δυναμένη ἐλάσσων ἐστίν.</s> <lb n="1799,001,,,,10,94,2"></lb> |
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11993 <s>Χωρίον γὰρ τὸ ΑΒ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΓ <lb n="1799,001,,,,10,94,3"></lb> |
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11994 καὶ ἀποτομῆς τετάρτης τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ <lb n="1799,001,,,,10,94,4"></lb> |
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11995 χωρίον δυναμένη ἐλάσσων ἐστίν.</s> <lb n="1799,001,,,,10,94,5"></lb> |
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11996 <s>Ἔστω γὰρ τῇ ΑΔ προσαρμόζουσα ἡ ΔΗ· αἱ ἄρα ΑΗ, <lb n="1799,001,,,,10,94,6"></lb> |
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11997 ΗΔ ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ ἡ ΑΗ <lb n="1799,001,,,,10,94,7"></lb> |
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11998 σύμμετρός ἐστι τῇ ἐκκειμένῃ <lb n="1799,001,,,,10,94,8"></lb> |
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11999 ῥητῇ τῇ ΑΓ μήκει, ἡ δὲ ὅλη <lb n="1799,001,,,,10,94,9"></lb> |
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12000 ἡ ΑΗ τῆς προσαρμοζούσης <lb n="1799,001,,,,10,94,10"></lb> |
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12001 τῆς ΔΗ μεῖζον δύναται τῷ <lb n="1799,001,,,,10,94,11"></lb> |
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12002 ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει.</s> <lb n="1799,001,,,,10,94,12"></lb> |
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12003 <s>ἐπεὶ οὖν ἡ ΑΗ τῆς ΗΔ μεῖ-<lb n="1799,001,,,,10,94,13"></lb> |
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12004 ζον δύναται τῷ ἀπὸ ἀσυμμέ-<lb n="1799,001,,,,10,94,14"></lb> |
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12005 τρου ἑαυτῇ μήκει, ἐὰν ἄρα <lb n="1799,001,,,,10,94,15"></lb> |
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12006 τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς <lb n="1799,001,,,,10,94,16"></lb> |
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12007 ΔΗ ἴσον παρὰ τὴν ΑΗ παρα-<lb n="1799,001,,,,10,94,17"></lb> |
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12008 βληθῇ ἐλλεῖπον εἴδει τετρα-<lb n="1799,001,,,,10,94,18"></lb> |
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12009 γώνῳ, εἰς ἀσύμμετρα αὐτὴν <pb></pb> <lb n="1799,001,,,,10,94,19"></lb> |
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12010 διελεῖ.</s> <s>τετμήσθω οὖν ἡ ΔΗ δίχα κατὰ τὸ Ε, καὶ τῷ <lb n="1799,001,,,,10,94,20"></lb> |
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12011 ἀπὸ τῆς ΕΗ ἴσον παρὰ τὴν ΑΗ παραβεβλήσθω ἐλλεῖ-<lb n="1799,001,,,,10,94,21"></lb> |
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12012 πον εἴδει τετραγώνῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΑΖ, ΖΗ· <lb n="1799,001,,,,10,94,22"></lb> |
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12013 ἀσύμμετρος ἄρα ἐστὶ μήκει ἡ ΑΖ τῇ ΖΗ.</s> <s>ἤχθωσαν οὖν <lb n="1799,001,,,,10,94,23"></lb> |
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12014 διὰ τῶν Ε, Ζ, Η παράλληλοι ταῖς ΑΓ, ΒΔ αἱ ΕΘ, <lb n="1799,001,,,,10,94,24"></lb> |
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12015 ΖΙ, ΗΚ.</s> <s>ἐπεὶ οὖν ῥητή ἐστιν ἡ ΑΗ καὶ σύμμετρος τῇ <lb n="1799,001,,,,10,94,25"></lb> |
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12016 ΑΓ μήκει, ῥητὸν ἄρα ἐστὶν ὅλον τὸ ΑΚ.</s> <s>πάλιν, ἐπεὶ <lb n="1799,001,,,,10,94,26"></lb> |
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12017 ἀσύμμετρός ἐστιν ἡ ΔΗ τῇ ΑΓ μήκει, καί εἰσιν ἀμφό-<lb n="1799,001,,,,10,94,27"></lb> |
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12018 τεραι ῥηταί, μέσον ἄρα ἐστὶ τὸ ΔΚ.</s> <s>πάλιν, ἐπεὶ ἀσύμ-<lb n="1799,001,,,,10,94,28"></lb> |
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12019 μετρός ἐστιν ἡ ΑΖ τῇ ΖΗ μήκει, ἀσύμμετρον ἄρα καὶ <lb n="1799,001,,,,10,94,29"></lb> |
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12020 τὸ ΑΙ τῷ ΖΚ.</s> <s>συνεστάτω οὖν τῷ μὲν ΑΙ ἴσον τετράγωνον <lb n="1799,001,,,,10,94,30"></lb> |
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12021 τὸ ΛΜ, τῷ δὲ ΖΚ ἴσον ἀφῃρήσθω περὶ τὴν αὐτὴν γωνίαν <lb n="1799,001,,,,10,94,31"></lb> |
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12022 τὴν ὑπὸ τῶν ΛΟΜ τὸ ΝΞ.</s> <s>περὶ τὴν αὐτὴν ἄρα διάμετρόν <lb n="1799,001,,,,10,94,32"></lb> |
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12023 ἐστι τὰ ΛΜ, ΝΞ τετράγωνα.</s> <s>ἔστω αὐτῶν διάμετρος ἡ <lb n="1799,001,,,,10,94,33"></lb> |
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12024 ΟΡ, καὶ καταγεγράφθω τὸ σχῆμα.</s> <s>ἐπεὶ οὖν τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,94,34"></lb> |
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12025 ΑΖ, ΖΗ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΕΗ, ἀνάλογον ἄρα ἐστὶν <lb n="1799,001,,,,10,94,35"></lb> |
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12026 ὡς ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως ἡ ΕΗ πρὸς τὴν ΖΗ.</s> <s>ἀλλ' <lb n="1799,001,,,,10,94,36"></lb> |
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12027 ὡς μὲν ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως ἐστὶ τὸ ΑΙ πρὸς τὸ <lb n="1799,001,,,,10,94,37"></lb> |
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12028 ΕΚ, ὡς δὲ ἡ ΕΗ πρὸς τὴν ΖΗ, οὕτως ἐστὶ τὸ ΕΚ πρὸς <lb n="1799,001,,,,10,94,38"></lb> |
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12029 τὸ ΖΚ· τῶν ἄρα ΑΙ, ΖΚ μέσον ἀνάλογόν ἐστι τὸ ΕΚ.</s> <lb n="1799,001,,,,10,94,39"></lb> |
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12030 <s>ἔστι δὲ καὶ τῶν ΛΜ, ΝΞ τετραγώνων μέσον ἀνάλογον τὸ <lb n="1799,001,,,,10,94,40"></lb> |
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12031 ΜΝ, καί ἐστιν ἴσον τὸ μὲν ΑΙ τῷ ΛΜ, τὸ δὲ ΖΚ τῷ <pb></pb> <lb n="1799,001,,,,10,94,41"></lb> |
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12032 ΝΞ· καὶ τὸ ΕΚ ἄρα ἴσον ἐστὶ τῷ ΜΝ.</s> <s>ἀλλὰ τῷ μὲν ΕΚ <lb n="1799,001,,,,10,94,42"></lb> |
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12033 ἴσον ἐστὶ τὸ ΔΘ, τῷ δὲ ΜΝ ἴσον ἐστὶ τὸ ΛΞ· ὅλον ἄρα <lb n="1799,001,,,,10,94,43"></lb> |
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12034 τὸ ΔΚ ἴσον ἐστὶ τῷ ΥΦΧ γνώμονι καὶ τῷ ΝΞ.</s> <s>ἐπεὶ οὖν <lb n="1799,001,,,,10,94,44"></lb> |
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12035 ὅλον τὸ ΑΚ ἴσον ἐστὶ τοῖς ΛΜ, ΝΞ τετραγώνοις, ὧν <lb n="1799,001,,,,10,94,45"></lb> |
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12036 τὸ ΔΚ ἴσον ἐστὶ τῷ ΥΦΧ γνώμονι καὶ τῷ ΝΞ τετρα-<lb n="1799,001,,,,10,94,46"></lb> |
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12037 γώνῳ, λοιπὸν ἄρα τὸ ΑΒ ἴσον ἐστὶ τῷ ΣΤ, τουτέστι <lb n="1799,001,,,,10,94,47"></lb> |
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12038 τῷ ἀπὸ τῆς ΛΝ τετραγώνῳ· ἡ ΛΝ ἄρα δύναται τὸ <lb n="1799,001,,,,10,94,48"></lb> |
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12039 ΑΒ χωρίον.</s> <lb n="1799,001,,,,10,94,49"></lb> |
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12040 <s>Λέγω, ὅτι ἡ ΛΝ ἄλογός ἐστιν ἡ καλουμένη ἐλάσσων.</s> <lb n="1799,001,,,,10,94,50"></lb> |
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12041 <s>Ἐπεὶ γὰρ ῥητόν ἐστι τὸ ΑΚ καί ἐστιν ἴσον τοῖς ἀπὸ <lb n="1799,001,,,,10,94,51"></lb> |
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12042 τῶν ΛΟ, ΟΝ τετραγώνοις, τὸ ἄρα συγκείμενον ἐκ τῶν <lb n="1799,001,,,,10,94,52"></lb> |
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12043 ἀπὸ τῶν ΛΟ, ΟΝ ῥητόν ἐστιν.</s> <s>πάλιν, ἐπεὶ τὸ ΔΚ μέσον <lb n="1799,001,,,,10,94,53"></lb> |
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12044 ἐστίν, καί ἐστιν ἴσον τὸ ΔΚ τῷ δὶς ὑπὸ τῶν ΛΟ, ΟΝ, <lb n="1799,001,,,,10,94,54"></lb> |
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12045 τὸ ἄρα δὶς ὑπὸ τῶν ΛΟ, ΟΝ μέσον ἐστίν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,94,55"></lb> |
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12046 ἀσύμμετρον ἐδείχθη τὸ ΑΙ τῷ ΖΚ, ἀσύμμετρον ἄρα <lb n="1799,001,,,,10,94,56"></lb> |
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12047 καὶ τὸ ἀπὸ τῆς ΛΟ τετράγωνον τῷ ἀπὸ τῆς ΟΝ τετρα-<lb n="1799,001,,,,10,94,57"></lb> |
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12048 γώνῳ.</s> <s>αἱ ΛΟ, ΟΝ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι <lb n="1799,001,,,,10,94,58"></lb> |
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12049 τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων ῥητόν, <lb n="1799,001,,,,10,94,59"></lb> |
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12050 τὸ δὲ δὶς ὑπ' αὐτῶν μέσον.</s> <s>ἡ ΛΝ ἄρα ἄλογός ἐστιν ἡ <lb n="1799,001,,,,10,94,60"></lb> |
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12051 καλουμένη ἐλάσσων· καὶ δύναται τὸ ΑΒ χωρίον.</s> <lb n="1799,001,,,,10,94,61"></lb> |
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12052 <s>Ἡ ἄρα τὸ ΑΒ χωρίον δυναμένη ἐλάσσων ἐστίν· ὅπερ <lb n="1799,001,,,,10,94,62"></lb> |
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12053 ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,94,63"></lb> |
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12054 Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς <lb n="1799,001,,,,10,95,1"></lb> |
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12055 πέμπτης, ἡ τὸ χωρίον δυναμένη [ἡ] μετὰ ῥητοῦ μέσον <lb n="1799,001,,,,10,95,2"></lb> |
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12056 τὸ ὅλον ποιοῦσά ἐστιν.</s> <lb n="1799,001,,,,10,95,3"></lb> |
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12057 <s>Χωρίον γὰρ τὸ ΑΒ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΓ <lb n="1799,001,,,,10,95,4"></lb> |
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12058 καὶ ἀποτομῆς πέμπτης τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ <lb n="1799,001,,,,10,95,5"></lb> |
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12059 χωρίον δυναμένη [ἡ] μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσά <lb n="1799,001,,,,10,95,6"></lb> |
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12060 ἐστιν.</s> <lb n="1799,001,,,,10,95,7"></lb> |
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12061 <s>Ἔστω γὰρ τῇ ΑΔ προσαρμόζουσα ἡ ΔΗ· αἱ ἄρα ΑΗ, <lb n="1799,001,,,,10,95,8"></lb> |
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12062 ΗΔ ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ ἡ προσ-<lb n="1799,001,,,,10,95,9"></lb> |
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12063 αρμόζουσα ἡ ΗΔ σύμμετρός <lb n="1799,001,,,,10,95,10"></lb> |
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12064 ἐστι μήκει τῇ ἐκκειμένῃ ῥητῇ <lb n="1799,001,,,,10,95,11"></lb> |
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12065 τῇ ΑΓ, ἡ δὲ ὅλη ἡ ΑΗ τῆς <lb n="1799,001,,,,10,95,12"></lb> |
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12066 προσαρμοζούσης τῆς ΔΗ μεῖ-<lb n="1799,001,,,,10,95,13"></lb> |
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12067 ζον δύναται τῷ ἀπὸ ἀσυμμέ-<lb n="1799,001,,,,10,95,14"></lb> |
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12068 τρου ἑαυτῇ.</s> <s>ἐὰν ἄρα τῷ τετάρ-<lb n="1799,001,,,,10,95,15"></lb> |
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12069 τῳ μέρει τοῦ ἀπὸ τῆς ΔΗ ἴσον <lb n="1799,001,,,,10,95,16"></lb> |
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12070 παρὰ τὴν ΑΗ παραβληθῇ <lb n="1799,001,,,,10,95,17"></lb> |
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12071 ἐλλεῖπον εἴδει τετραγώνῳ, εἰς <lb n="1799,001,,,,10,95,18"></lb> |
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12072 ἀσύμμετρα αὐτὴν διελεῖ.</s> <s>τε-<lb n="1799,001,,,,10,95,19"></lb> |
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12073 τμήσθω οὖν ἡ ΔΗ δίχα κατὰ <lb n="1799,001,,,,10,95,20"></lb> |
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12074 τὸ Ε σημεῖον, καὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,95,21"></lb> |
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12075 ΕΗ ἴσον παρὰ τὴν ΑΗ παρα-<lb n="1799,001,,,,10,95,22"></lb> |
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12076 βεβλήσθω ἐλλεῖπον εἴδει τετραγώνῳ καὶ ἔστω τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,95,23"></lb> |
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12077 ΑΖ, ΖΗ· ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΖ τῇ ΖΗ μήκει.</s> <s>καὶ <lb n="1799,001,,,,10,95,24"></lb> |
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12078 ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΑΗ τῇ ΓΑ μήκει, καί εἰσιν <pb></pb> <lb n="1799,001,,,,10,95,25"></lb> |
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12079 ἀμφότεραι ῥηταί, μέσον ἄρα ἐστὶ τὸ ΑΚ.</s> <s>πάλιν, ἐπεὶ ῥητή <lb n="1799,001,,,,10,95,26"></lb> |
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12080 ἐστιν ἡ ΔΗ καὶ σύμμετρος τῇ ΑΓ μήκει, ῥητόν ἐστι τὸ <lb n="1799,001,,,,10,95,27"></lb> |
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12081 ΔΚ.</s> <s>συνεστάτω οὖν τῷ μὲν ΑΙ ἴσον τετράγωνον τὸ ΛΜ, <lb n="1799,001,,,,10,95,28"></lb> |
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12082 τῷ δὲ ΖΚ ἴσον τετράγωνον ἀφῃρήσθω τὸ ΝΞ περὶ τὴν <lb n="1799,001,,,,10,95,29"></lb> |
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12083 αὐτὴν γωνίαν τὴν ὑπὸ ΛΟΜ· περὶ τὴν αὐτὴν ἄρα διά-<lb n="1799,001,,,,10,95,30"></lb> |
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12084 μετρόν ἐστι τὰ ΛΜ, ΝΞ τετράγωνα.</s> <s>ἔστω αὐτῶν <lb n="1799,001,,,,10,95,31"></lb> |
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12085 διάμετρος ἡ ΟΡ, καὶ καταγεγράφθω τὸ σχῆμα.</s> <s>ὁμοίως <lb n="1799,001,,,,10,95,32"></lb> |
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12086 δὴ δείξομεν, ὅτι ἡ ΛΝ δύναται τὸ ΑΒ χωρίον.</s> <lb n="1799,001,,,,10,95,33"></lb> |
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12087 <s>Λέγω, ὅτι ἡ ΛΝ ἡ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσά <lb n="1799,001,,,,10,95,34"></lb> |
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12088 ἐστιν.</s> <lb n="1799,001,,,,10,95,35"></lb> |
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12089 <s>Ἐπεὶ γὰρ μέσον ἐδείχθη τὸ ΑΚ καί ἐστιν ἴσον τοῖς <lb n="1799,001,,,,10,95,36"></lb> |
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12090 ἀπὸ τῶν ΛΟ, ΟΝ, τὸ ἄρα συγκείμενον ἐκ τῶν ἀπὸ τῶν <lb n="1799,001,,,,10,95,37"></lb> |
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12091 ΛΟ, ΟΝ μέσον ἐστίν.</s> <s>πάλιν, ἐπεὶ ῥητόν ἐστι τὸ ΔΚ <lb n="1799,001,,,,10,95,38"></lb> |
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12092 καί ἐστιν ἴσον τῷ δὶς ὑπὸ τῶν ΛΟ, ΟΝ, καὶ αὐτὸ ῥητόν <lb n="1799,001,,,,10,95,39"></lb> |
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12093 ἐστιν.</s> <s>καὶ ἐπεὶ ἀσύμμετρόν ἐστι τὸ ΑΙ τῷ ΖΚ, ἀσύμ-<lb n="1799,001,,,,10,95,40"></lb> |
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12094 μετρον ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς ΛΟ τῷ ἀπὸ τῆς ΟΝ· <lb n="1799,001,,,,10,95,41"></lb> |
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12095 αἱ ΛΟ, ΟΝ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ <lb n="1799,001,,,,10,95,42"></lb> |
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12096 μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων μέσον, <lb n="1799,001,,,,10,95,43"></lb> |
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12097 τὸ δὲ δὶς ὑπ' αὐτῶν ῥητόν.</s> <s>ἡ λοιπὴ ἄρα ἡ ΛΝ ἄλογός <lb n="1799,001,,,,10,95,44"></lb> |
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12098 ἐστιν ἡ καλουμένη μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα· <lb n="1799,001,,,,10,95,45"></lb> |
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12099 καὶ δύναται τὸ ΑΒ χωρίον.</s> <lb n="1799,001,,,,10,95,46"></lb> |
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12100 <s>Ἡ τὸ ΑΒ ἄρα χωρίον δυναμένη μετὰ ῥητοῦ μέσον τὸ <lb n="1799,001,,,,10,95,47"></lb> |
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12101 ὅλον ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,95,48"></lb> |
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12102 Ἐὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς <lb n="1799,001,,,,10,96,1"></lb> |
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12103 ἕκτης, ἡ τὸ χωρίον δυναμένη μετὰ μέσου μέσον τὸ ὅλον <lb n="1799,001,,,,10,96,2"></lb> |
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12104 ποιοῦσά ἐστιν.</s> <lb n="1799,001,,,,10,96,3"></lb> |
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12105 <s>Χωρίον γὰρ τὸ ΑΒ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΓ καὶ <lb n="1799,001,,,,10,96,4"></lb> |
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12106 ἀποτομῆς ἕκτης τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ χωρίον <lb n="1799,001,,,,10,96,5"></lb> |
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12107 δυναμένη [ἡ] μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν.</s> <lb n="1799,001,,,,10,96,6"></lb> |
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12108 <s>Ἔστω γὰρ τῇ ΑΔ προσαρ-<lb n="1799,001,,,,10,96,7"></lb> |
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12109 μόζουσα ἡ ΔΗ· αἱ ἄρα ΑΗ, <lb n="1799,001,,,,10,96,8"></lb> |
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12110 ΗΔ ῥηταί εἰσι δυνάμει μό-<lb n="1799,001,,,,10,96,9"></lb> |
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12111 νον σύμμετροι, καὶ οὐδετέρα <lb n="1799,001,,,,10,96,10"></lb> |
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12112 αὐτῶν σύμμετρός ἐστι τῇ ἐκ-<lb n="1799,001,,,,10,96,11"></lb> |
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12113 κειμένῃ ῥητῇ τῇ ΑΓ μήκει, <lb n="1799,001,,,,10,96,12"></lb> |
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12114 ἡ δὲ ὅλη ἡ ΑΗ τῆς προσαρ-<lb n="1799,001,,,,10,96,13"></lb> |
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12115 μοζούσης τῆς ΔΗ μεῖζον δύ-<lb n="1799,001,,,,10,96,14"></lb> |
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12116 ναται τῷ ἀπὸ ἀσυμμέτρου <lb n="1799,001,,,,10,96,15"></lb> |
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12117 ἑαυτῇ μήκει.</s> <s>ἐπεὶ οὖν ἡ ΑΗ <lb n="1799,001,,,,10,96,16"></lb> |
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12118 τῆς ΗΔ μεῖζον δύναται τῷ <lb n="1799,001,,,,10,96,17"></lb> |
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12119 ἀπὸ ἀσυμμέτρου ἑαυτῇ μή-<lb n="1799,001,,,,10,96,18"></lb> |
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12120 κει, ἐὰν ἄρα τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΔΗ ἴσον <lb n="1799,001,,,,10,96,19"></lb> |
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12121 παρὰ τὴν ΑΗ παραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ, <lb n="1799,001,,,,10,96,20"></lb> |
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12122 εἰς ἀσύμμετρα αὐτὴν διελεῖ.</s> <s>τετμήσθω οὖν ἡ ΔΗ δίχα <lb n="1799,001,,,,10,96,21"></lb> |
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12123 κατὰ τὸ Ε [σημεῖον], καὶ τῷ ἀπὸ τῆς ΕΗ ἴσον παρὰ <lb n="1799,001,,,,10,96,22"></lb> |
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12124 τὴν ΑΗ παραβεβλήσθω ἐλλεῖπον εἴδει τετραγώνῳ, καὶ <lb n="1799,001,,,,10,96,23"></lb> |
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12125 ἔστω τὸ ὑπὸ τῶν ΑΖ, ΖΗ· ἀσύμμετρος ἄρα ἐστὶν ἡ <pb></pb> <lb n="1799,001,,,,10,96,24"></lb> |
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12126 ΑΖ τῇ ΖΗ μήκει.</s> <s>ὡς δὲ ἡ ΑΖ πρὸς τὴν ΖΗ, οὕτως <lb n="1799,001,,,,10,96,25"></lb> |
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12127 ἐστὶ τὸ ΑΙ πρὸς τὸ ΖΚ· ἀσύμμετρον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,10,96,26"></lb> |
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12128 ΑΙ τῷ ΖΚ.</s> <s>καὶ ἐπεὶ αἱ ΑΗ, ΑΓ ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,96,27"></lb> |
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12129 μόνον σύμμετροι, μέσον ἐστὶ τὸ ΑΚ.</s> <s>πάλιν, ἐπεὶ αἱ ΑΓ, <lb n="1799,001,,,,10,96,28"></lb> |
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12130 ΔΗ ῥηταί εἰσι καὶ ἀσύμμετροι μήκει, μέσον ἐστὶ καὶ <lb n="1799,001,,,,10,96,29"></lb> |
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12131 τὸ ΔΚ.</s> <s>ἐπεὶ οὖν αἱ ΑΗ, ΗΔ δυνάμει μόνον σύμμετροί <lb n="1799,001,,,,10,96,30"></lb> |
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12132 εἰσιν, ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΗ τῇ ΗΔ μήκει.</s> <s>ὡς <lb n="1799,001,,,,10,96,31"></lb> |
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12133 δὲ ἡ ΑΗ πρὸς τὴν ΗΔ, οὕτως ἐστὶ τὸ ΑΚ πρὸς τὸ ΚΔ· <lb n="1799,001,,,,10,96,32"></lb> |
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12134 ἀσύμμετρον ἄρα ἐστὶ τὸ ΑΚ τῷ ΚΔ.</s> <s>συνεστάτω οὖν τῷ <lb n="1799,001,,,,10,96,33"></lb> |
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12135 μὲν ΑΙ ἴσον τετράγωνον τὸ ΛΜ, τῷ δὲ ΖΚ ἴσον ἀφῃρή-<lb n="1799,001,,,,10,96,34"></lb> |
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12136 σθω περὶ τὴν αὐτὴν γωνίαν τὸ ΝΞ· περὶ τὴν αὐτὴν ἄρα <lb n="1799,001,,,,10,96,35"></lb> |
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12137 διάμετρόν ἐστι τὰ ΛΜ, ΝΞ τετράγωνα.</s> <s>ἔστω αὐτῶν <lb n="1799,001,,,,10,96,36"></lb> |
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12138 διάμετρος ἡ ΟΡ, καὶ καταγεγράφθω τὸ σχῆμα.</s> <s>ὁμοίως <lb n="1799,001,,,,10,96,37"></lb> |
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12139 δὴ τοῖς ἐπάνω δείξομεν, ὅτι ἡ ΛΝ δύναται τὸ ΑΒ <lb n="1799,001,,,,10,96,38"></lb> |
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12140 χωρίον.</s> <lb n="1799,001,,,,10,96,39"></lb> |
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12141 <s>Λέγω, ὅτι ἡ ΛΝ [ἡ] μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά <lb n="1799,001,,,,10,96,40"></lb> |
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12142 ἐστιν.</s> <lb n="1799,001,,,,10,96,41"></lb> |
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12143 <s>Ἐπεὶ γὰρ μέσον ἐδείχθη τὸ ΑΚ καί ἐστιν ἴσον τοῖς <lb n="1799,001,,,,10,96,42"></lb> |
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12144 ἀπὸ τῶν ΛΟ, ΟΝ, τὸ ἄρα συγκείμενον ἐκ τῶν ἀπὸ τῶν <lb n="1799,001,,,,10,96,43"></lb> |
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12145 ΛΟ, ΟΝ μέσον ἐστίν.</s> <s>πάλιν, ἐπεὶ μέσον ἐδείχθη τὸ <lb n="1799,001,,,,10,96,44"></lb> |
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12146 ΔΚ καί ἐστιν ἴσον τῷ δὶς ὑπὸ τῶν ΛΟ, ΟΝ, καὶ τὸ δὶς <lb n="1799,001,,,,10,96,45"></lb> |
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12147 ὑπὸ τῶν ΛΟ, ΟΝ μέσον ἐστίν.</s> <s>καὶ ἐπεὶ ἀσύμμετρον <lb n="1799,001,,,,10,96,46"></lb> |
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12148 ἐδείχθη τὸ ΑΚ τῷ ΔΚ, ἀσύμμετρα [ἄρα] ἐστὶ καὶ τὰ <lb n="1799,001,,,,10,96,47"></lb> |
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12149 ἀπὸ τῶν ΛΟ, ΟΝ τετράγωνα τῷ δὶς ὑπὸ τῶν ΛΟ, ΟΝ.</s> <s><pb></pb> <lb n="1799,001,,,,10,96,48"></lb> |
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12150 καὶ ἐπεὶ ἀσύμμετρόν ἐστι τὸ ΑΙ τῷ ΖΚ, ἀσύμμετρον ἄρα <lb n="1799,001,,,,10,96,49"></lb> |
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12151 καὶ τὸ ἀπὸ τῆς ΛΟ τῷ ἀπὸ τῆς ΟΝ· αἱ ΛΟ, ΟΝ ἄρα <lb n="1799,001,,,,10,96,50"></lb> |
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12152 δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τό τε συγκείμενον <lb n="1799,001,,,,10,96,51"></lb> |
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12153 ἐκ τῶν ἀπ' αὐτῶν τετραγώνων μέσον καὶ τὸ δὶς ὑπ' <lb n="1799,001,,,,10,96,52"></lb> |
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12154 αὐτῶν μέσον ἔτι τε τὰ ἀπ' αὐτῶν τετράγωνα ἀσύμμετρα <lb n="1799,001,,,,10,96,53"></lb> |
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12155 τῷ δὶς ὑπ' αὐτῶν.</s> <s>ἡ ἄρα ΛΝ ἄλογός ἐστιν ἡ καλουμένη <lb n="1799,001,,,,10,96,54"></lb> |
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12156 μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα· καὶ δύναται τὸ ΑΒ <lb n="1799,001,,,,10,96,55"></lb> |
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12157 χωρίον.</s> <lb n="1799,001,,,,10,96,56"></lb> |
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12158 <s>Ἡ ἄρα τὸ χωρίον δυναμένη μετὰ μέσου μέσον τὸ ὅλον <lb n="1799,001,,,,10,96,57"></lb> |
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12159 ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,96,58"></lb> |
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12160 <s>Τὸ ἀπὸ ἀποτομῆς παρὰ ῥητὴν παραβαλλόμενον πλάτος <lb n="1799,001,,,,10,97,1"></lb> |
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12161 ποιεῖ ἀποτομὴν πρώτην.</s> <lb n="1799,001,,,,10,97,2"></lb> |
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12162 <s>Ἔστω ἀποτομὴ ἡ ΑΒ, ῥητὴ δὲ ἡ ΓΔ, καὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,97,3"></lb> |
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12163 ΑΒ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω τὸ ΓΕ πλάτος <lb n="1799,001,,,,10,97,4"></lb> |
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12164 ποιοῦν τὴν ΓΖ· λέγω, ὅτι ἡ ΓΖ ἀποτομή ἐστι πρώτη.</s> <lb n="1799,001,,,,10,97,5"></lb> |
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12165 <s>Ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΗ· αἱ ἄρα ΑΗ, <lb n="1799,001,,,,10,97,6"></lb> |
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12166 ΗΒ ῥηταί εἰσι δυνάμει μόνον σύμμετροι.</s> <s>καὶ τῷ μὲν <lb n="1799,001,,,,10,97,7"></lb> |
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12167 ἀπὸ τῆς ΑΗ ἴσον παρὰ τὴν <lb n="1799,001,,,,10,97,8"></lb> |
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12168 ΓΔ παραβεβλήσθω τὸ ΓΘ, τῷ <lb n="1799,001,,,,10,97,9"></lb> |
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12169 δὲ ἀπὸ τῆς ΒΗ τὸ ΚΛ.</s> <s>ὅλον <lb n="1799,001,,,,10,97,10"></lb> |
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12170 ἄρα τὸ ΓΛ ἴσον ἐστὶ τοῖς ἀπὸ <lb n="1799,001,,,,10,97,11"></lb> |
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12171 τῶν ΑΗ, ΗΒ· ὧν τὸ ΓΕ ἴσον <lb n="1799,001,,,,10,97,12"></lb> |
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12172 ἐστὶ τῷ ἀπὸ τῆς ΑΒ· λοιπὸν <lb n="1799,001,,,,10,97,13"></lb> |
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12173 ἄρα τὸ ΖΛ ἴσον ἐστὶ τῷ δὶς ὑπὸ <pb></pb> <lb n="1799,001,,,,10,97,14"></lb> |
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12174 τῶν ΑΗ, ΗΒ.</s> <s>τετμήσθω ἡ ΖΜ δίχα κατὰ τὸ Ν σημεῖον, <lb n="1799,001,,,,10,97,15"></lb> |
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12175 καὶ ἤχθω διὰ τοῦ Ν τῇ ΓΔ παράλληλος ἡ ΝΞ· ἑκάτερον <lb n="1799,001,,,,10,97,16"></lb> |
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12176 ἄρα τῶν ΖΞ, ΛΝ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>καὶ <lb n="1799,001,,,,10,97,17"></lb> |
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12177 ἐπεὶ τὰ ἀπὸ τῶν ΑΗ, ΗΒ ῥητά ἐστιν, καί ἐστι τοῖς ἀπὸ <lb n="1799,001,,,,10,97,18"></lb> |
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12178 τῶν ΑΗ, ΗΒ ἴσον τὸ ΔΜ, ῥητὸν ἄρα ἐστὶ τὸ ΔΜ.</s> <s>καὶ <lb n="1799,001,,,,10,97,19"></lb> |
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12179 παρὰ ῥητὴν τὴν ΓΔ παραβέβληται πλάτος ποιοῦν τὴν <lb n="1799,001,,,,10,97,20"></lb> |
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12180 ΓΜ· ῥητὴ ἄρα ἐστὶν ἡ ΓΜ καὶ σύμμετρος τῇ ΓΔ <lb n="1799,001,,,,10,97,21"></lb> |
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12181 μήκει.</s> <s>πάλιν, ἐπεὶ μέσον ἐστὶ τὸ δὶς ὑπὸ τῶν ΑΗ, ΗΒ, <lb n="1799,001,,,,10,97,22"></lb> |
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12182 καὶ τῷ δὶς ὑπὸ τῶν ΑΗ, ΗΒ ἴσον τὸ ΖΛ, μέσον ἄρα τὸ <lb n="1799,001,,,,10,97,23"></lb> |
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12183 ΖΛ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΓΔ παράκειται πλάτος ποιοῦν <lb n="1799,001,,,,10,97,24"></lb> |
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12184 τὴν ΖΜ· ῥητὴ ἄρα ἐστὶν ἡ ΖΜ καὶ ἀσύμμετρος τῇ <lb n="1799,001,,,,10,97,25"></lb> |
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12185 ΓΔ μήκει.</s> <s>καὶ ἐπεὶ τὰ μὲν ἀπὸ τῶν ΑΗ, ΗΒ ῥητά <lb n="1799,001,,,,10,97,26"></lb> |
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12186 ἐστιν, τὸ δὲ δὶς ὑπὸ τῶν ΑΗ, ΗΒ μέσον, ἀσύμμετρα <lb n="1799,001,,,,10,97,27"></lb> |
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12187 ἄρα ἐστὶ τὰ ἀπὸ τῶν ΑΗ, ΗΒ τῷ δὶς ὑπὸ τῶν ΑΗ, <lb n="1799,001,,,,10,97,28"></lb> |
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12188 ΗΒ.</s> <s>καὶ τοῖς μὲν ἀπὸ τῶν ΑΗ, ΗΒ ἴσον ἐστὶ τὸ ΓΛ, <lb n="1799,001,,,,10,97,29"></lb> |
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12189 τῷ δὲ δὶς ὑπὸ τῶν ΑΗ, ΗΒ τὸ ΖΛ· ἀσύμμετρον ἄρα <lb n="1799,001,,,,10,97,30"></lb> |
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12190 ἐστὶ τὸ ΔΜ τῷ ΖΛ.</s> <s>ὡς δὲ τὸ ΔΜ πρὸς τὸ ΖΛ, οὕτως <lb n="1799,001,,,,10,97,31"></lb> |
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12191 ἐστὶν ἡ ΓΜ πρὸς τὴν ΖΜ.</s> <s>ἀσύμμετρος ἄρα ἐστὶν ἡ <lb n="1799,001,,,,10,97,32"></lb> |
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12192 ΓΜ τῇ ΖΜ μήκει.</s> <s>καί εἰσιν ἀμφότεραι ῥηταί· αἱ ἄρα <lb n="1799,001,,,,10,97,33"></lb> |
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12193 ΓΜ, ΜΖ ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἡ ΓΖ <lb n="1799,001,,,,10,97,34"></lb> |
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12194 ἄρα ἀποτομή ἐστιν.</s> <lb n="1799,001,,,,10,97,35"></lb> |
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12195 <s>Λέγω δή, ὅτι καὶ πρώτη.</s> <lb n="1799,001,,,,10,97,36"></lb> |
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12196 <s>Ἐπεὶ γὰρ τῶν ἀπὸ τῶν ΑΗ, ΗΒ μέσον ἀνάλογόν ἐστι <lb n="1799,001,,,,10,97,37"></lb> |
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12197 τὸ ὑπὸ τῶν ΑΗ, ΗΒ, καί ἐστι τῷ μὲν ἀπὸ τῆς ΑΗ ἴσον <pb></pb> <lb n="1799,001,,,,10,97,38"></lb> |
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12198 τὸ ΓΘ, τῷ δὲ ἀπὸ τῆς ΒΗ ἴσον τὸ ΚΛ, τῷ δὲ ὑπὸ τῶν <lb n="1799,001,,,,10,97,39"></lb> |
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12199 ΑΗ, ΗΒ τὸ ΝΛ, καὶ τῶν ΓΘ, ΚΛ ἄρα μέσον ἀνάλογόν <lb n="1799,001,,,,10,97,40"></lb> |
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12200 ἐστι τὸ ΝΛ· ἔστιν ἄρα ὡς τὸ ΓΘ πρὸς τὸ ΝΛ, οὕτως <lb n="1799,001,,,,10,97,41"></lb> |
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12201 τὸ ΝΛ πρὸς τὸ ΚΛ.</s> <s>ἀλλ' ὡς μὲν τὸ ΓΘ πρὸς τὸ ΝΛ, <lb n="1799,001,,,,10,97,42"></lb> |
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12202 οὕτως ἐστὶν ἡ ΓΚ πρὸς τὴν ΝΜ· ὡς δὲ τὸ ΝΛ πρὸς τὸ <lb n="1799,001,,,,10,97,43"></lb> |
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12203 ΚΛ, οὕτως ἐστὶν ἡ ΝΜ πρὸς τὴν ΚΜ· τὸ ἄρα ὑπὸ τῶν <lb n="1799,001,,,,10,97,44"></lb> |
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12204 ΓΚ, ΚΜ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΝΜ, τουτέστι τῷ τετάρτῳ <lb n="1799,001,,,,10,97,45"></lb> |
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12205 μέρει τοῦ ἀπὸ τῆς ΖΜ.</s> <s>καὶ ἐπεὶ σύμμετρόν ἐστι τὸ ἀπὸ <lb n="1799,001,,,,10,97,46"></lb> |
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12206 τῆς ΑΗ τῷ ἀπὸ τῆς ΗΒ, σύμμετρόν [ἐστι] καὶ τὸ ΓΘ <lb n="1799,001,,,,10,97,47"></lb> |
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12207 τῷ ΚΛ.</s> <s>ὡς δὲ τὸ ΓΘ πρὸς τὸ ΚΛ, οὕτως ἡ ΓΚ πρὸς <lb n="1799,001,,,,10,97,48"></lb> |
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12208 τὴν ΚΜ· σύμμετρος ἄρα ἐστὶν ἡ ΓΚ τῇ ΚΜ.</s> <s>ἐπεὶ οὖν <lb n="1799,001,,,,10,97,49"></lb> |
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12209 δύο εὐθεῖαι ἄνισοί εἰσιν αἱ ΓΜ, ΜΖ, καὶ τῷ τετάρτῳ <lb n="1799,001,,,,10,97,50"></lb> |
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12210 μέρει τοῦ ἀπὸ τῆς ΖΜ ἴσον παρὰ τὴν ΓΜ παραβέβληται <lb n="1799,001,,,,10,97,51"></lb> |
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12211 ἐλλεῖπον εἴδει τετραγώνῳ τὸ ὑπὸ τῶν ΓΚ, ΚΜ, καί ἐστι <lb n="1799,001,,,,10,97,52"></lb> |
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12212 σύμμετρος ἡ ΓΚ τῇ ΚΜ, ἡ ἄρα ΓΜ τῆς ΜΖ μεῖζον <lb n="1799,001,,,,10,97,53"></lb> |
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12213 δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει.</s> <s>καί ἐστιν ἡ ΓΜ <lb n="1799,001,,,,10,97,54"></lb> |
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12214 σύμμετρος τῇ ἐκκειμένῃ ῥητῇ τῇ ΓΔ μήκει· ἡ ἄρα ΓΖ <lb n="1799,001,,,,10,97,55"></lb> |
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12215 ἀποτομή ἐστι πρώτη.</s> <lb n="1799,001,,,,10,97,56"></lb> |
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12216 <s>Τὸ ἄρα ἀπὸ ἀποτομῆς παρὰ ῥητὴν παραβαλλόμενον <lb n="1799,001,,,,10,97,57"></lb> |
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12217 πλάτος ποιεῖ ἀποτομὴν πρώτην· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,97,58"></lb> |
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12218 Τὸ ἀπὸ μέσης ἀποτομῆς πρώτης παρὰ ῥητὴν παρα-<lb n="1799,001,,,,10,98,1"></lb> |
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12219 βαλλόμενον πλάτος ποιεῖ ἀποτομὴν δευτέραν.</s> <lb n="1799,001,,,,10,98,2"></lb> |
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12220 <s>Ἔστω μέσης ἀποτομὴ πρώτη ἡ ΑΒ, ῥητὴ δὲ ἡ ΓΔ, <lb n="1799,001,,,,10,98,3"></lb> |
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12221 καὶ τῷ ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω <lb n="1799,001,,,,10,98,4"></lb> |
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12222 τὸ ΓΕ πλάτος ποιοῦν τὴν ΓΖ· λέγω, ὅτι ἡ ΓΖ ἀποτομή <lb n="1799,001,,,,10,98,5"></lb> |
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12223 ἐστι δευτέρα.</s> <lb n="1799,001,,,,10,98,6"></lb> |
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12224 <s>Ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΗ· αἱ ἄρα ΑΗ, <lb n="1799,001,,,,10,98,7"></lb> |
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12225 ΗΒ μέσαι εἰσὶ δυνάμει μόνον σύμμετροι ῥητὸν περιέχου-<lb n="1799,001,,,,10,98,8"></lb> |
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12226 σαι.</s> <s>καὶ τῷ μὲν ἀπὸ τῆς ΑΗ <lb n="1799,001,,,,10,98,9"></lb> |
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12227 ἴσον παρὰ τὴν ΓΔ παραβεβλή-<lb n="1799,001,,,,10,98,10"></lb> |
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12228 σθω τὸ ΓΘ πλάτος ποιοῦν τὴν <lb n="1799,001,,,,10,98,11"></lb> |
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12229 ΓΚ, τῷ δὲ ἀπὸ τῆς ΗΒ ἴσον τὸ <lb n="1799,001,,,,10,98,12"></lb> |
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12230 ΚΛ πλάτος ποιοῦν τὴν ΚΜ· ὅλον <lb n="1799,001,,,,10,98,13"></lb> |
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12231 ἄρα τὸ ΓΛ ἴσον ἐστὶ τοῖς ἀπὸ <lb n="1799,001,,,,10,98,14"></lb> |
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12232 τῶν ΑΗ, ΗΒ· μέσον ἄρα καὶ τὸ <lb n="1799,001,,,,10,98,15"></lb> |
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12233 ΓΛ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΓΔ παράκειται πλάτος ποιοῦν <lb n="1799,001,,,,10,98,16"></lb> |
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12234 τὴν ΓΜ· ῥητὴ ἄρα ἐστὶν ἡ ΓΜ καὶ ἀσύμμετρος τῇ ΓΔ <lb n="1799,001,,,,10,98,17"></lb> |
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12235 μήκει.</s> <s>καὶ ἐπεὶ τὸ ΓΛ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΑΗ, ΗΒ, <lb n="1799,001,,,,10,98,18"></lb> |
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12236 ὧν τὸ ἀπὸ τῆς ΑΒ ἴσον ἐστὶ τῷ ΓΕ, λοιπὸν ἄρα τὸ δὶς <lb n="1799,001,,,,10,98,19"></lb> |
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12237 ὑπὸ τῶν ΑΗ, ΗΒ ἴσον ἐστὶ τῷ ΖΛ.</s> <s>ῥητὸν δὲ [ἐστι] τὸ <lb n="1799,001,,,,10,98,20"></lb> |
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12238 δὶς ὑπὸ τῶν ΑΗ, ΗΒ· ῥητὸν ἄρα τὸ ΖΛ.</s> <s>καὶ παρὰ <lb n="1799,001,,,,10,98,21"></lb> |
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12239 ῥητὴν τὴν ΖΕ παράκειται πλάτος ποιοῦν τὴν ΖΜ· <lb n="1799,001,,,,10,98,22"></lb> |
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12240 ῥητὴ ἄρα ἐστὶ καὶ ἡ ΖΜ καὶ σύμμετρος τῇ ΓΔ μήκει.</s> <lb n="1799,001,,,,10,98,23"></lb> |
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12241 <s>ἐπεὶ οὖν τὰ μὲν ἀπὸ τῶν ΑΗ, ΗΒ, τουτέστι τὸ ΓΛ, <pb></pb> <lb n="1799,001,,,,10,98,24"></lb> |
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12242 μέσον ἐστίν, τὸ δὲ δὶς ὑπὸ τῶν ΑΗ, ΗΒ, τουτέστι τὸ <lb n="1799,001,,,,10,98,25"></lb> |
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12243 ΖΛ, ῥητόν, ἀσύμμετρον ἄρα ἐστὶ τὸ ΓΛ τῷ ΖΛ.</s> <s>ὡς δὲ τὸ <lb n="1799,001,,,,10,98,26"></lb> |
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12244 ΓΛ πρὸς τὸ ΖΛ, οὕτως ἐστὶν ἡ ΓΜ πρὸς τὴν ΖΜ· <lb n="1799,001,,,,10,98,27"></lb> |
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12245 ἀσύμμετρος ἄρα ἡ ΓΜ τῇ ΖΜ μήκει.</s> <s>καί εἰσιν ἀμφότεραι <lb n="1799,001,,,,10,98,28"></lb> |
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12246 ῥηταί· αἱ ἄρα ΓΜ, ΜΖ ῥηταί εἰσι δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,98,29"></lb> |
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12247 μετροι· ἡ ΓΖ ἄρα ἀποτομή ἐστιν.</s> <lb n="1799,001,,,,10,98,30"></lb> |
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12248 <s>Λέγω δή, ὅτι καὶ δευτέρα.</s> <lb n="1799,001,,,,10,98,31"></lb> |
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12249 <s>Τετμήσθω γὰρ ἡ ΖΜ δίχα κατὰ τὸ Ν, καὶ ἤχθω διὰ <lb n="1799,001,,,,10,98,32"></lb> |
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12250 τοῦ Ν τῇ ΓΔ παράλληλος ἡ ΝΞ· ἑκάτερον ἄρα τῶν ΖΞ, <lb n="1799,001,,,,10,98,33"></lb> |
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12251 ΝΛ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>καὶ ἐπεὶ τῶν ἀπὸ <lb n="1799,001,,,,10,98,34"></lb> |
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12252 τῶν ΑΗ, ΗΒ τετραγώνων μέσον ἀνάλογόν ἐστι τὸ ὑπὸ <lb n="1799,001,,,,10,98,35"></lb> |
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12253 τῶν ΑΗ, ΗΒ, καί ἐστιν ἴσον τὸ μὲν ἀπὸ τῆς ΑΗ τῷ <lb n="1799,001,,,,10,98,36"></lb> |
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12254 ΓΘ, τὸ δὲ ὑπὸ τῶν ΑΗ, ΗΒ τῷ ΝΛ, τὸ δὲ ἀπὸ τῆς ΒΗ <lb n="1799,001,,,,10,98,37"></lb> |
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12255 τῷ ΚΛ, καὶ τῶν ΓΘ, ΚΛ ἄρα μέσον ἀνάλογόν ἐστι τὸ <lb n="1799,001,,,,10,98,38"></lb> |
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12256 ΝΛ· ἔστιν ἄρα ὡς τὸ ΓΘ πρὸς τὸ ΝΛ, οὕτως τὸ ΝΛ <lb n="1799,001,,,,10,98,39"></lb> |
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12257 πρὸς τὸ ΚΛ.</s> <s>ἀλλ' ὡς μὲν τὸ ΓΘ πρὸς τὸ ΝΛ, οὕτως <lb n="1799,001,,,,10,98,40"></lb> |
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12258 ἐστὶν ἡ ΓΚ πρὸς τὴν ΝΜ, ὡς δὲ τὸ ΝΛ πρὸς τὸ ΚΛ, <lb n="1799,001,,,,10,98,41"></lb> |
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12259 οὕτως ἐστὶν ἡ ΝΜ πρὸς τὴν ΜΚ· ὡς ἄρα ἡ ΓΚ πρὸς τὴν <lb n="1799,001,,,,10,98,42"></lb> |
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12260 ΝΜ, οὕτως ἐστὶν ἡ ΝΜ πρὸς τὴν ΚΜ· τὸ ἄρα ὑπὸ <lb n="1799,001,,,,10,98,43"></lb> |
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12261 τῶν ΓΚ, ΚΜ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΝΜ, τουτέστι τῷ <lb n="1799,001,,,,10,98,44"></lb> |
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12262 τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΖΜ.</s> <s>[καὶ ἐπεὶ σύμμετρόν <lb n="1799,001,,,,10,98,45"></lb> |
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12263 ἐστι τὸ ἀπὸ τῆς ΑΗ τῷ ἀπὸ τῆς ΒΗ, σύμμετρόν ἐστι <pb></pb> <lb n="1799,001,,,,10,98,46"></lb> |
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12264 καὶ τὸ ΓΘ τῷ ΚΛ, τουτέστιν ἡ ΓΚ τῇ ΚΜ.] ἐπεὶ οὖν <lb n="1799,001,,,,10,98,47"></lb> |
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12265 δύο εὐθεῖαι ἄνισοί εἰσιν αἱ ΓΜ, ΜΖ, καὶ τῷ τετάρτῳ <lb n="1799,001,,,,10,98,48"></lb> |
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12266 μέρει τοῦ ἀπὸ τῆς ΜΖ ἴσον παρὰ τὴν μείζονα τὴν ΓΜ <lb n="1799,001,,,,10,98,49"></lb> |
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12267 παραβέβληται ἐλλεῖπον εἴδει τετραγώνῳ τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,98,50"></lb> |
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12268 ΓΚ, ΚΜ καὶ εἰς σύμμετρα αὐτὴν διαιρεῖ, ἡ ἄρα ΓΜ <lb n="1799,001,,,,10,98,51"></lb> |
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12269 τῆς ΜΖ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ μή-<lb n="1799,001,,,,10,98,52"></lb> |
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12270 κει.</s> <s>καί ἐστιν ἡ προσαρμόζουσα ἡ ΖΜ σύμμετρος <lb n="1799,001,,,,10,98,53"></lb> |
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12271 μήκει τῇ ἐκκειμένῃ ῥητῇ τῇ ΓΔ· ἡ ἄρα ΓΖ ἀποτομή <lb n="1799,001,,,,10,98,54"></lb> |
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12272 ἐστι δευτέρα.</s> <lb n="1799,001,,,,10,98,55"></lb> |
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12273 <s>Τὸ ἄρα ἀπὸ μέσης ἀποτομῆς πρώτης παρὰ ῥητὴν <lb n="1799,001,,,,10,98,56"></lb> |
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12274 παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν δευτέραν· ὅπερ <lb n="1799,001,,,,10,98,57"></lb> |
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12275 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,98,58"></lb> |
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12276 <s>Τὸ ἀπὸ μέσης ἀποτομῆς δευτέρας παρὰ ῥητὴν παρα-<lb n="1799,001,,,,10,99,1"></lb> |
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12277 βαλλόμενον πλάτος ποιεῖ ἀποτομὴν τρίτην.</s> <lb n="1799,001,,,,10,99,2"></lb> |
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12278 <s>Ἔστω μέσης ἀποτομὴ δευτέρα ἡ ΑΒ, ῥητὴ δὲ ἡ ΓΔ, <lb n="1799,001,,,,10,99,3"></lb> |
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12279 καὶ τῷ ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω <lb n="1799,001,,,,10,99,4"></lb> |
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12280 τὸ ΓΕ πλάτος ποιοῦν τὴν ΓΖ· <lb n="1799,001,,,,10,99,5"></lb> |
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12281 λέγω, ὅτι ἡ ΓΖ ἀποτομή ἐστι <lb n="1799,001,,,,10,99,6"></lb> |
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12282 τρίτη.</s> <lb n="1799,001,,,,10,99,7"></lb> |
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12283 <s>Ἔστω γὰρ τῇ ΑΒ προσαρμό-<lb n="1799,001,,,,10,99,8"></lb> |
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12284 ζουσα ἡ ΒΗ· αἱ ἄρα ΑΗ, ΗΒ <lb n="1799,001,,,,10,99,9"></lb> |
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12285 μέσαι εἰσὶ δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,99,10"></lb> |
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12286 μετροι μέσον περιέχουσαι.</s> <s>καὶ <pb></pb> <lb n="1799,001,,,,10,99,11"></lb> |
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12287 τῷ μὲν ἀπὸ τῆς ΑΗ ἴσον παρὰ τὴν ΓΔ παραβεβλή-<lb n="1799,001,,,,10,99,12"></lb> |
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12288 σθω τὸ ΓΘ πλάτος ποιοῦν τὴν ΓΚ, τῷ δὲ ἀπὸ τῆς <lb n="1799,001,,,,10,99,13"></lb> |
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12289 ΒΗ ἴσον παρὰ τὴν ΚΘ παραβεβλήσθω τὸ ΚΛ πλάτος <lb n="1799,001,,,,10,99,14"></lb> |
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12290 ποιοῦν τὴν ΚΜ· ὅλον ἄρα τὸ ΓΛ ἴσον ἐστὶ τοῖς <lb n="1799,001,,,,10,99,15"></lb> |
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12291 ἀπὸ τῶν ΑΗ, ΗΒ [καί ἐστι μέσα τὰ ἀπὸ τῶν ΑΗ, <lb n="1799,001,,,,10,99,16"></lb> |
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12292 ΗΒ]· μέσον ἄρα καὶ τὸ ΓΛ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΓΔ <lb n="1799,001,,,,10,99,17"></lb> |
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12293 παραβέβληται πλάτος ποιοῦν τὴν ΓΜ· ῥητὴ ἄρα ἐστὶν <lb n="1799,001,,,,10,99,18"></lb> |
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12294 ἡ ΓΜ καὶ ἀσύμμετρος τῇ ΓΔ μήκει.</s> <s>καὶ ἐπεὶ ὅλον τὸ <lb n="1799,001,,,,10,99,19"></lb> |
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12295 ΓΛ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΑΗ, ΗΒ, ὧν τὸ ΓΕ ἴσον <lb n="1799,001,,,,10,99,20"></lb> |
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12296 ἐστὶ τῷ ἀπὸ τῆς ΑΒ, λοιπὸν ἄρα τὸ ΛΖ ἴσον ἐστὶ τῷ <lb n="1799,001,,,,10,99,21"></lb> |
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12297 δὶς ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>τετμήσθω οὖν ἡ ΖΜ δίχα κατὰ <lb n="1799,001,,,,10,99,22"></lb> |
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12298 τὸ Ν σημεῖον, καὶ τῇ ΓΔ παράλληλος ἤχθω ἡ ΝΞ· <lb n="1799,001,,,,10,99,23"></lb> |
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12299 ἑκάτερον ἄρα τῶν ΖΞ, ΝΛ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΑΗ, <lb n="1799,001,,,,10,99,24"></lb> |
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12300 ΗΒ.</s> <s>μέσον δὲ τὸ ὑπὸ τῶν ΑΗ, ΗΒ· μέσον ἄρα ἐστὶ καὶ <lb n="1799,001,,,,10,99,25"></lb> |
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12301 τὸ ΖΛ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΕΖ παράκειται πλάτος <lb n="1799,001,,,,10,99,26"></lb> |
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12302 ποιοῦν τὴν ΖΜ· ῥητὴ ἄρα καὶ ἡ ΖΜ καὶ ἀσύμμετρος τῇ <lb n="1799,001,,,,10,99,27"></lb> |
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12303 ΓΔ μήκει.</s> <s>καὶ ἐπεὶ αἱ ΑΗ, ΗΒ δυνάμει μόνον εἰσὶ σύμ-<lb n="1799,001,,,,10,99,28"></lb> |
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12304 μετροι, ἀσύμμετρος ἄρα [ἐστὶ] μήκει ἡ ΑΗ τῇ ΗΒ· <lb n="1799,001,,,,10,99,29"></lb> |
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12305 ἀσύμμετρον ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς ΑΗ τῷ ὑπὸ τῶν <lb n="1799,001,,,,10,99,30"></lb> |
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12306 ΑΗ, ΗΒ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΗ σύμμετρά ἐστι τὰ <lb n="1799,001,,,,10,99,31"></lb> |
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12307 ἀπὸ τῶν ΑΗ, ΗΒ, τῷ δὲ ὑπὸ τῶν ΑΗ, ΗΒ τὸ δὶς ὑπὸ <lb n="1799,001,,,,10,99,32"></lb> |
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12308 τῶν ΑΗ, ΗΒ· ἀσύμμετρα ἄρα ἐστὶ τὰ ἀπὸ τῶν ΑΗ, <lb n="1799,001,,,,10,99,33"></lb> |
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12309 ΗΒ τῷ δὶς ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>ἀλλὰ τοῖς μὲν ἀπὸ τῶν <lb n="1799,001,,,,10,99,34"></lb> |
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12310 ΑΗ, ΗΒ ἴσον ἐστὶ τὸ ΓΛ, τῷ δὲ δὶς ὑπὸ τῶν ΑΗ, ΗΒ <lb n="1799,001,,,,10,99,35"></lb> |
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12311 ἴσον ἐστὶ τὸ ΖΛ· ἀσύμμετρον ἄρα ἐστὶ τὸ ΓΛ τῷ ΖΛ.</s> <s><pb></pb> <lb n="1799,001,,,,10,99,36"></lb> |
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12312 ὡς δὲ τὸ ΓΛ πρὸς τὸ ΖΛ, οὕτως ἐστὶν ἡ ΓΜ πρὸς <lb n="1799,001,,,,10,99,37"></lb> |
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12313 τὴν ΖΜ· ἀσύμμετρος ἄρα ἐστὶν ἡ ΓΜ τῇ ΖΜ μήκει.</s> <lb n="1799,001,,,,10,99,38"></lb> |
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12314 <s>καί εἰσιν ἀμφότεραι ῥηταί· αἱ ἄρα ΓΜ, ΜΖ ῥηταί εἰσι <lb n="1799,001,,,,10,99,39"></lb> |
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12315 δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΓΖ.</s> <lb n="1799,001,,,,10,99,40"></lb> |
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12316 <s>Λέγω δή, ὅτι καὶ τρίτη.</s> <lb n="1799,001,,,,10,99,41"></lb> |
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12317 <s>Ἐπεὶ γὰρ σύμμετρόν ἐστι τὸ ἀπὸ τῆς ΑΗ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,99,42"></lb> |
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12318 ΗΒ, σύμμετρον ἄρα καὶ τὸ ΓΘ τῷ ΚΛ· ὥστε καὶ ἡ ΓΚ <lb n="1799,001,,,,10,99,43"></lb> |
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12319 τῇ ΚΜ.</s> <s>καὶ ἐπεὶ τῶν ἀπὸ τῶν ΑΗ, ΗΒ μέσον ἀνάλογόν <lb n="1799,001,,,,10,99,44"></lb> |
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12320 ἐστι τὸ ὑπὸ τῶν ΑΗ, ΗΒ, καί ἐστι τῷ μὲν ἀπὸ τῆς ΑΗ <lb n="1799,001,,,,10,99,45"></lb> |
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12321 ἴσον τὸ ΓΘ, τῷ δὲ ἀπὸ τῆς ΗΒ ἴσον τὸ ΚΛ, τῷ δὲ ὑπὸ <lb n="1799,001,,,,10,99,46"></lb> |
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12322 τῶν ΑΗ, ΗΒ ἴσον τὸ ΝΛ, καὶ τῶν ΓΘ, ΚΛ ἄρα μέσον <lb n="1799,001,,,,10,99,47"></lb> |
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12323 ἀνάλογόν ἐστι τὸ ΝΛ· ἔστιν ἄρα ὡς τὸ ΓΘ πρὸς τὸ ΝΛ, <lb n="1799,001,,,,10,99,48"></lb> |
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12324 οὕτως τὸ ΝΛ πρὸς τὸ ΚΛ.</s> <s>ἀλλ' ὡς μὲν τὸ ΓΘ πρὸς τὸ <lb n="1799,001,,,,10,99,49"></lb> |
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12325 ΝΛ, οὕτως ἐστὶν ἡ ΓΚ πρὸς τὴν ΝΜ, ὡς δὲ τὸ ΝΛ <lb n="1799,001,,,,10,99,50"></lb> |
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12326 πρὸς τὸ ΚΛ, οὕτως ἐστὶν ἡ ΝΜ πρὸς τὴν ΚΜ· ὡς ἄρα <lb n="1799,001,,,,10,99,51"></lb> |
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12327 ἡ ΓΚ πρὸς τὴν ΜΝ, οὕτως ἐστὶν ἡ ΜΝ πρὸς τὴν ΚΜ· <lb n="1799,001,,,,10,99,52"></lb> |
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12328 τὸ ἄρα ὑπὸ τῶν ΓΚ, ΚΜ ἴσον ἐστὶ τῷ [ἀπὸ τῆς ΜΝ, <lb n="1799,001,,,,10,99,53"></lb> |
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12329 τουτέστι τῷ] τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΖΜ.</s> <s>ἐπεὶ οὖν <lb n="1799,001,,,,10,99,54"></lb> |
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12330 δύο εὐθεῖαι ἄνισοί εἰσιν αἱ ΓΜ, ΜΖ, καὶ τῷ τετάρτῳ <lb n="1799,001,,,,10,99,55"></lb> |
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12331 μέρει τοῦ ἀπὸ τῆς ΖΜ ἴσον παρὰ τὴν ΓΜ παραβέβληται <lb n="1799,001,,,,10,99,56"></lb> |
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12332 ἐλλεῖπον εἴδει τετραγώνῳ καὶ εἰς σύμμετρα αὐτὴν <lb n="1799,001,,,,10,99,57"></lb> |
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12333 διαιρεῖ, ἡ ΓΜ ἄρα τῆς ΜΖ μεῖζον δύναται τῷ ἀπὸ <lb n="1799,001,,,,10,99,58"></lb> |
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12334 συμμέτρου ἑαυτῇ.</s> <s>καὶ οὐδετέρα τῶν ΓΜ, ΜΖ σύμμετρός <lb n="1799,001,,,,10,99,59"></lb> |
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12335 ἐστι μήκει τῇ ἐκκειμένῃ ῥητῇ τῇ ΓΔ· ἡ ἄρα ΓΖ ἀποτομή <lb n="1799,001,,,,10,99,60"></lb> |
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12336 ἐστι τρίτη.</s> <s><pb></pb> <lb n="1799,001,,,,10,99,61"></lb> |
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12337 Τὸ ἄρα ἀπὸ μέσης ἀποτομῆς δευτέρας παρὰ ῥητὴν <lb n="1799,001,,,,10,99,62"></lb> |
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12338 παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν τρίτην· ὅπερ <lb n="1799,001,,,,10,99,63"></lb> |
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12339 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,99,64"></lb> |
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12340 <s>Τὸ ἀπὸ ἐλάσσονος παρὰ ῥητὴν παραβαλλόμενον πλάτος <lb n="1799,001,,,,10,100,1"></lb> |
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12341 ποιεῖ ἀποτομὴν τετάρτην.</s> <lb n="1799,001,,,,10,100,2"></lb> |
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12342 <s>Ἔστω ἐλάσσων ἡ ΑΒ, ῥητὴ δὲ ἡ ΓΔ, καὶ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,100,3"></lb> |
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12343 ΑΒ ἴσον παρὰ ῥητὴν τὴν ΓΔ παραβεβλήσθω τὸ ΓΕ <lb n="1799,001,,,,10,100,4"></lb> |
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12344 πλάτος ποιοῦν τὴν ΓΖ· λέγω, ὅτι ἡ ΓΖ ἀποτομή ἐστι <lb n="1799,001,,,,10,100,5"></lb> |
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12345 τετάρτη.</s> <lb n="1799,001,,,,10,100,6"></lb> |
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12346 <s>Ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΗ· αἱ ἄρα <lb n="1799,001,,,,10,100,7"></lb> |
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12347 ΑΗ, ΗΒ δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ μὲν <lb n="1799,001,,,,10,100,8"></lb> |
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12348 συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΗ, <lb n="1799,001,,,,10,100,9"></lb> |
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12349 ΗΒ τετραγώνων ῥητόν, τὸ δὲ <lb n="1799,001,,,,10,100,10"></lb> |
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12350 δὶς ὑπὸ τῶν ΑΗ, ΗΒ μέσον.</s> <lb n="1799,001,,,,10,100,11"></lb> |
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12351 <s>καὶ τῷ μὲν ἀπὸ τῆς ΑΗ ἴσον <lb n="1799,001,,,,10,100,12"></lb> |
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12352 παρὰ τὴν ΓΔ παραβεβλήσθω τὸ <lb n="1799,001,,,,10,100,13"></lb> |
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12353 ΓΘ πλάτος ποιοῦν τὴν ΓΚ, τῷ <lb n="1799,001,,,,10,100,14"></lb> |
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12354 δὲ ἀπὸ τῆς ΒΗ ἴσον τὸ ΚΛ <lb n="1799,001,,,,10,100,15"></lb> |
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12355 πλάτος ποιοῦν τὴν ΚΜ· ὅλον ἄρα τὸ ΓΛ ἴσον ἐστὶ τοῖς <lb n="1799,001,,,,10,100,16"></lb> |
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12356 ἀπὸ τῶν ΑΗ, ΗΒ.</s> <s>καί ἐστι τὸ συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,100,17"></lb> |
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12357 τῶν ΑΗ, ΗΒ ῥητόν· ῥητὸν ἄρα ἐστὶ καὶ τὸ ΓΛ.</s> <s>καὶ <lb n="1799,001,,,,10,100,18"></lb> |
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12358 παρὰ ῥητὴν τὴν ΓΔ παράκειται πλάτος ποιοῦν τὴν ΓΜ· <lb n="1799,001,,,,10,100,19"></lb> |
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12359 ῥητὴ ἄρα καὶ ἡ ΓΜ καὶ σύμμετρος τῇ ΓΔ μήκει.</s> <s>καὶ <lb n="1799,001,,,,10,100,20"></lb> |
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12360 ἐπεὶ ὅλον τὸ ΓΛ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΑΗ, ΗΒ, ὧν <lb n="1799,001,,,,10,100,21"></lb> |
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12361 τὸ ΓΕ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ, λοιπὸν ἄρα τὸ ΖΛ ἴσον <pb></pb> <lb n="1799,001,,,,10,100,22"></lb> |
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12362 ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>τετμήσθω οὖν ἡ ΖΜ δίχα <lb n="1799,001,,,,10,100,23"></lb> |
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12363 κατὰ τὸ Ν σημεῖον, καὶ ἤχθω διὰ τοῦ Ν ὁποτέρᾳ τῶν ΓΔ, <lb n="1799,001,,,,10,100,24"></lb> |
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12364 ΜΛ παράλληλος ἡ ΝΞ· ἑκάτερον ἄρα τῶν ΖΞ, ΝΛ <lb n="1799,001,,,,10,100,25"></lb> |
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12365 ἴσον ἐστὶ τῷ ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>καὶ ἐπεὶ τὸ δὶς ὑπὸ <lb n="1799,001,,,,10,100,26"></lb> |
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12366 τῶν ΑΗ, ΗΒ μέσον ἐστὶ καί ἐστιν ἴσον τῷ ΖΛ, καὶ τὸ ΖΛ <lb n="1799,001,,,,10,100,27"></lb> |
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12367 ἄρα μέσον ἐστίν.</s> <s>καὶ παρὰ ῥητὴν τὴν ΖΕ παράκειται <lb n="1799,001,,,,10,100,28"></lb> |
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12368 πλάτος ποιοῦν τὴν ΖΜ· ῥητὴ ἄρα ἐστὶν ἡ ΖΜ καὶ <lb n="1799,001,,,,10,100,29"></lb> |
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12369 ἀσύμμετρος τῇ ΓΔ μήκει.</s> <s>καὶ ἐπεὶ τὸ μὲν συγκείμενον <lb n="1799,001,,,,10,100,30"></lb> |
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12370 ἐκ τῶν ἀπὸ τῶν ΑΗ, ΗΒ ῥητόν ἐστιν, τὸ δὲ δὶς ὑπὸ <lb n="1799,001,,,,10,100,31"></lb> |
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12371 τῶν ΑΗ, ΗΒ μέσον, ἀσύμμετρα [ἄρα] ἐστὶ τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,100,32"></lb> |
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12372 ΑΗ, ΗΒ τῷ δὶς ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>ἴσον δέ [ἐστι] τὸ <lb n="1799,001,,,,10,100,33"></lb> |
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12373 ΓΛ τοῖς ἀπὸ τῶν ΑΗ, ΗΒ, τῷ δὲ δὶς ὑπὸ τῶν ΑΗ, ΗΒ <lb n="1799,001,,,,10,100,34"></lb> |
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12374 ἴσον τὸ ΖΛ· ἀσύμμετρον ἄρα [ἐστὶ] τὸ ΓΛ τῷ ΖΛ.</s> <s>ὡς <lb n="1799,001,,,,10,100,35"></lb> |
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12375 δὲ τὸ ΓΛ πρὸς τὸ ΖΛ, οὕτως ἐστὶν ἡ ΓΜ πρὸς τὴν ΜΖ· <lb n="1799,001,,,,10,100,36"></lb> |
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12376 ἀσύμμετρος ἄρα ἐστὶν ἡ ΓΜ τῇ ΜΖ μήκει.</s> <s>καί εἰσιν <lb n="1799,001,,,,10,100,37"></lb> |
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12377 ἀμφότεραι ῥηταί· αἱ ἄρα ΓΜ, ΜΖ ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,100,38"></lb> |
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12378 μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΓΖ.</s> <lb n="1799,001,,,,10,100,39"></lb> |
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12379 <s>Λέγω [δή], ὅτι καὶ τετάρτη.</s> <lb n="1799,001,,,,10,100,40"></lb> |
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12380 <s>Ἐπεὶ γὰρ αἱ ΑΗ, ΗΒ δυνάμει εἰσὶν ἀσύμμετροι, <lb n="1799,001,,,,10,100,41"></lb> |
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12381 ἀσύμμετρον ἄρα καὶ τὸ ἀπὸ τῆς ΑΗ τῷ ἀπὸ τῆς ΗΒ.</s> <lb n="1799,001,,,,10,100,42"></lb> |
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12382 <s>καί ἐστι τῷ μὲν ἀπὸ τῆς ΑΗ ἴσον τὸ ΓΘ, τῷ δὲ ἀπὸ <lb n="1799,001,,,,10,100,43"></lb> |
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12383 τῆς ΗΒ ἴσον τὸ ΚΛ· ἀσύμμετρον ἄρα ἐστὶ τὸ ΓΘ τῷ <lb n="1799,001,,,,10,100,44"></lb> |
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12384 ΚΛ.</s> <s>ὡς δὲ τὸ ΓΘ πρὸς τὸ ΚΛ, οὕτως ἐστὶν ἡ ΓΚ πρὸς <lb n="1799,001,,,,10,100,45"></lb> |
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12385 τὴν ΚΜ· ἀσύμμετρος ἄρα ἐστὶν ἡ ΓΚ τῇ ΚΜ μήκει.</s> <s>καὶ <pb></pb> <lb n="1799,001,,,,10,100,46"></lb> |
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12386 ἐπεὶ τῶν ἀπὸ τῶν ΑΗ, ΗΒ μέσον ἀνάλογόν ἐστι τὸ ὑπὸ <lb n="1799,001,,,,10,100,47"></lb> |
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12387 τῶν ΑΗ, ΗΒ, καί ἐστιν ἴσον τὸ μὲν ἀπὸ τῆς ΑΗ τῷ <lb n="1799,001,,,,10,100,48"></lb> |
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12388 ΓΘ, τὸ δὲ ἀπὸ τῆς ΗΒ τῷ ΚΛ, τὸ δὲ ὑπὸ τῶν ΑΗ, ΗΒ <lb n="1799,001,,,,10,100,49"></lb> |
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12389 τῷ ΝΛ, τῶν ἄρα ΓΘ, ΚΛ μέσον ἀνάλογόν ἐστι τὸ ΝΛ· <lb n="1799,001,,,,10,100,50"></lb> |
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12390 ἔστιν ἄρα ὡς τὸ ΓΘ πρὸς τὸ ΝΛ, οὕτως τὸ ΝΛ πρὸς <lb n="1799,001,,,,10,100,51"></lb> |
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12391 τὸ ΚΛ.</s> <s>ἀλλ' ὡς μὲν τὸ ΓΘ πρὸς τὸ ΝΛ, οὕτως ἐστὶν ἡ <lb n="1799,001,,,,10,100,52"></lb> |
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12392 ΓΚ πρὸς τὴν ΝΜ, ὡς δὲ τὸ ΝΛ πρὸς τὸ ΚΛ, οὕτως <lb n="1799,001,,,,10,100,53"></lb> |
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12393 ἐστὶν ἡ ΝΜ πρὸς τὴν ΚΜ· ὡς ἄρα ἡ ΓΚ πρὸς τὴν ΜΝ, <lb n="1799,001,,,,10,100,54"></lb> |
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12394 οὕτως ἐστὶν ἡ ΜΝ πρὸς τὴν ΚΜ· τὸ ἄρα ὑπὸ τῶν ΓΚ, <lb n="1799,001,,,,10,100,55"></lb> |
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12395 ΚΜ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΜΝ, τουτέστι τῷ τετάρτῳ <lb n="1799,001,,,,10,100,56"></lb> |
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12396 μέρει τοῦ ἀπὸ τῆς ΖΜ.</s> <s>ἐπεὶ οὖν δύο εὐθεῖαι ἄνισοί εἰσιν <lb n="1799,001,,,,10,100,57"></lb> |
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12397 αἱ ΓΜ, ΜΖ, καὶ τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΜΖ <lb n="1799,001,,,,10,100,58"></lb> |
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12398 ἴσον παρὰ τὴν ΓΜ παραβέβληται ἐλλεῖπον εἴδει τετρα-<lb n="1799,001,,,,10,100,59"></lb> |
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12399 γώνῳ τὸ ὑπὸ τῶν ΓΚ, ΚΜ καὶ εἰς ἀσύμμετρα αὐτὴν <lb n="1799,001,,,,10,100,60"></lb> |
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12400 διαιρεῖ, ἡ ἄρα ΓΜ τῆς ΜΖ μεῖζον δύναται τῷ ἀπὸ <lb n="1799,001,,,,10,100,61"></lb> |
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12401 ἀσυμμέτρου ἑαυτῇ.</s> <s>καί ἐστιν ὅλη ἡ ΓΜ σύμμετρος μήκει <lb n="1799,001,,,,10,100,62"></lb> |
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12402 τῇ ἐκκειμένῃ ῥητῇ τῇ ΓΔ· ἡ ἄρα ΓΖ ἀποτομή ἐστι τε-<lb n="1799,001,,,,10,100,63"></lb> |
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12403 τάρτη.</s> <lb n="1799,001,,,,10,100,64"></lb> |
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12404 <s>Τὸ ἄρα ἀπὸ ἐλάσσονος καὶ τὰ ἑξῆς.</s> <lb n="1799,001,,,,10,100,65"></lb> |
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12405 <s>Τὸ ἀπὸ τῆς μετὰ ῥητοῦ μέσον τὸ ὅλον ποιούσης παρὰ <lb n="1799,001,,,,10,101,1"></lb> |
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12406 ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν πέμπτην.</s> <s><pb></pb> <lb n="1799,001,,,,10,101,2"></lb> |
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12407 Ἔστω ἡ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα ἡ ΑΒ, ῥητὴ <lb n="1799,001,,,,10,101,3"></lb> |
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12408 δὲ ἡ ΓΔ, καὶ τῷ ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν ΓΔ παρα-<lb n="1799,001,,,,10,101,4"></lb> |
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12409 βεβλήσθω τὸ ΓΕ πλάτος ποιοῦν τὴν ΓΖ· λέγω, ὅτι <lb n="1799,001,,,,10,101,5"></lb> |
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12410 ἡ ΓΖ ἀποτομή ἐστι πέμπτη.</s> <lb n="1799,001,,,,10,101,6"></lb> |
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12411 <s>Ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΗ· αἱ ἄρα <lb n="1799,001,,,,10,101,7"></lb> |
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12412 ΑΗ, ΗΒ εὐθεῖαι δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ <lb n="1799,001,,,,10,101,8"></lb> |
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12413 μὲν συγκείμενον ἐκ τῶν ἀπ' αὐ- <lb n="1799,001,,,,10,101,9"></lb> |
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12414 τῶν τετραγώνων μέσον, τὸ δὲ <lb n="1799,001,,,,10,101,10"></lb> |
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12415 δὶς ὑπ' αὐτῶν ῥητόν.</s> <s>καὶ τῷ <lb n="1799,001,,,,10,101,11"></lb> |
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12416 μὲν ἀπὸ τῆς ΑΗ ἴσον παρὰ τὴν <lb n="1799,001,,,,10,101,12"></lb> |
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12417 ΓΔ παραβεβλήσθω τὸ ΓΘ, τῷ <lb n="1799,001,,,,10,101,13"></lb> |
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12418 δὲ ἀπὸ τῆς ΗΒ ἴσον τὸ ΚΛ· <lb n="1799,001,,,,10,101,14"></lb> |
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12419 ὅλον ἄρα τὸ ΓΛ ἴσον ἐστὶ τοῖς <lb n="1799,001,,,,10,101,15"></lb> |
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12420 ἀπὸ τῶν ΑΗ, ΗΒ.</s> <s>τὸ δὲ συγκείμενον ἐκ τῶν ἀπὸ τῶν <lb n="1799,001,,,,10,101,16"></lb> |
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12421 ΑΗ, ΗΒ ἅμα μέσον ἐστίν· μέσον ἄρα ἐστὶ τὸ ΓΛ.</s> <s>καὶ <lb n="1799,001,,,,10,101,17"></lb> |
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12422 παρὰ ῥητὴν τὴν ΓΔ παράκειται πλάτος ποιοῦν τὴν ΓΜ· <lb n="1799,001,,,,10,101,18"></lb> |
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12423 ῥητὴ ἄρα ἐστὶν ἡ ΓΜ καὶ ἀσύμμετρος τῇ ΓΔ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,10,101,19"></lb> |
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12424 ὅλον τὸ ΓΛ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΑΗ, ΗΒ, ὧν τὸ <lb n="1799,001,,,,10,101,20"></lb> |
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12425 ΓΕ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ, λοιπὸν ἄρα τὸ ΖΛ ἴσον <lb n="1799,001,,,,10,101,21"></lb> |
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12426 ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>τετμήσθω οὖν ἡ ΖΜ <lb n="1799,001,,,,10,101,22"></lb> |
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12427 δίχα κατὰ τὸ Ν, καὶ ἤχθω διὰ τοῦ Ν ὁποτέρᾳ τῶν ΓΔ, <lb n="1799,001,,,,10,101,23"></lb> |
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12428 ΜΛ παράλληλος ἡ ΝΞ· ἑκάτερον ἄρα τῶν ΖΞ, ΝΛ ἴσον <lb n="1799,001,,,,10,101,24"></lb> |
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12429 ἐστὶ τῷ ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>καὶ ἐπεὶ τὸ δὶς ὑπὸ τῶν ΑΗ, <lb n="1799,001,,,,10,101,25"></lb> |
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12430 ΗΒ ῥητόν ἐστι καί [ἐστιν] ἴσον τῷ ΖΛ, ῥητὸν ἄρα ἐστὶ <lb n="1799,001,,,,10,101,26"></lb> |
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12431 τὸ ΖΛ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΕΖ παράκειται πλάτος <pb></pb> <lb n="1799,001,,,,10,101,27"></lb> |
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12432 ποιοῦν τὴν ΖΜ· ῥητὴ ἄρα ἐστὶν ἡ ΖΜ καὶ σύμμετρος <lb n="1799,001,,,,10,101,28"></lb> |
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12433 τῇ ΓΔ μήκει.</s> <s>καὶ ἐπεὶ τὸ μὲν ΓΛ μέσον ἐστίν, τὸ δὲ ΖΛ <lb n="1799,001,,,,10,101,29"></lb> |
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12434 ῥητόν, ἀσύμμετρον ἄρα ἐστὶ τὸ ΓΛ τῷ ΖΛ.</s> <s>ὡς δὲ τὸ <lb n="1799,001,,,,10,101,30"></lb> |
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12435 ΓΛ πρὸς τὸ ΖΛ, οὕτως ἡ ΓΜ πρὸς τὴν ΜΖ· ἀσύμμετρος <lb n="1799,001,,,,10,101,31"></lb> |
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12436 ἄρα ἐστὶν ἡ ΓΜ τῇ ΜΖ μήκει.</s> <s>καί εἰσιν ἀμφότεραι <lb n="1799,001,,,,10,101,32"></lb> |
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12437 ῥηταί· αἱ ἄρα ΓΜ, ΜΖ ῥηταί εἰσι δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,101,33"></lb> |
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12438 μετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΓΖ.</s> <lb n="1799,001,,,,10,101,34"></lb> |
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12439 <s>Λέγω δή, ὅτι καὶ πέμπτη.</s> <lb n="1799,001,,,,10,101,35"></lb> |
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12440 <s>Ὁμοίως γὰρ δείξομεν, ὅτι τὸ ὑπὸ τῶν ΓΚΜ ἴσον ἐστὶ τῷ <lb n="1799,001,,,,10,101,36"></lb> |
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12441 ἀπὸ τῆς ΝΜ, τουτέστι τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΖΜ.</s> <lb n="1799,001,,,,10,101,37"></lb> |
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12442 <s>καὶ ἐπεὶ ἀσύμμετρόν ἐστι τὸ ἀπὸ τῆς ΑΗ τῷ ἀπὸ τῆς <lb n="1799,001,,,,10,101,38"></lb> |
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12443 ΗΒ, ἴσον δὲ τὸ μὲν ἀπὸ τῆς ΑΗ τῷ ΓΘ, τὸ δὲ ἀπὸ τῆς <lb n="1799,001,,,,10,101,39"></lb> |
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12444 ΗΒ τῷ ΚΛ, ἀσύμμετρον ἄρα τὸ ΓΘ τῷ ΚΛ.</s> <s>ὡς δὲ τὸ <lb n="1799,001,,,,10,101,40"></lb> |
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12445 ΓΘ πρὸς τὸ ΚΛ, οὕτως ἡ ΓΚ πρὸς τὴν ΚΜ· ἀσύμμετρος <lb n="1799,001,,,,10,101,41"></lb> |
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12446 ἄρα ἡ ΓΚ τῇ ΚΜ μήκει.</s> <s>ἐπεὶ οὖν δύο εὐθεῖαι ἄνισοί <lb n="1799,001,,,,10,101,42"></lb> |
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12447 εἰσιν αἱ ΓΜ, ΜΖ, καὶ τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς <lb n="1799,001,,,,10,101,43"></lb> |
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12448 ΖΜ ἴσον παρὰ τὴν ΓΜ παραβέβληται ἐλλεῖπον εἴδει <lb n="1799,001,,,,10,101,44"></lb> |
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12449 τετραγώνῳ καὶ εἰς ἀσύμμετρα αὐτὴν διαιρεῖ, ἡ ἄρα ΓΜ <lb n="1799,001,,,,10,101,45"></lb> |
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12450 τῆς ΜΖ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <s>καί <lb n="1799,001,,,,10,101,46"></lb> |
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12451 ἐστιν ἡ προσαρμόζουσα ἡ ΖΜ σύμμετρος τῇ ἐκκειμένῃ <lb n="1799,001,,,,10,101,47"></lb> |
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12452 ῥητῇ τῇ ΓΔ· ἡ ἄρα ΓΖ ἀποτομή ἐστι πέμπτη· ὅπερ ἔδει <lb n="1799,001,,,,10,101,48"></lb> |
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12453 δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,101,49"></lb> |
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12454 Τὸ ἀπὸ τῆς μετὰ μέσου μέσον τὸ ὅλον ποιούσης παρὰ <lb n="1799,001,,,,10,102,1"></lb> |
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12455 ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν ἕκτην.</s> <lb n="1799,001,,,,10,102,2"></lb> |
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12456 <s>Ἔστω ἡ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα ἡ ΑΒ, <lb n="1799,001,,,,10,102,3"></lb> |
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12457 ῥητὴ δὲ ἡ ΓΔ, καὶ τῷ ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν ΓΔ <lb n="1799,001,,,,10,102,4"></lb> |
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12458 παραβεβλήσθω τὸ ΓΕ πλάτος ποιοῦν τὴν ΓΖ· λέγω, ὅτι <lb n="1799,001,,,,10,102,5"></lb> |
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12459 ἡ ΓΖ ἀποτομή ἐστιν ἕκτη.</s> <lb n="1799,001,,,,10,102,6"></lb> |
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12460 <s>Ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΗ· αἱ ἄρα ΑΗ, <lb n="1799,001,,,,10,102,7"></lb> |
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12461 ΗΒ δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τό τε συγκεί-<lb n="1799,001,,,,10,102,8"></lb> |
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12462 μενον ἐκ τῶν ἀπ' αὐτῶν τετρα- <lb n="1799,001,,,,10,102,9"></lb> |
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12463 γώνων μέσον καὶ τὸ δὶς ὑπὸ τῶν <lb n="1799,001,,,,10,102,10"></lb> |
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12464 ΑΗ, ΗΒ μέσον καὶ ἀσύμμετρον <lb n="1799,001,,,,10,102,11"></lb> |
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12465 τὰ ἀπὸ τῶν ΑΗ, ΗΒ τῷ δὶς <lb n="1799,001,,,,10,102,12"></lb> |
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12466 ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>παραβεβλή-<lb n="1799,001,,,,10,102,13"></lb> |
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12467 σθω οὖν παρὰ τὴν ΓΔ τῷ μὲν <lb n="1799,001,,,,10,102,14"></lb> |
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12468 ἀπὸ τῆς ΑΗ ἴσον τὸ ΓΘ πλάτος <lb n="1799,001,,,,10,102,15"></lb> |
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12469 ποιοῦν τὴν ΓΚ, τῷ δὲ ἀπὸ τῆς ΒΗ τὸ ΚΛ· ὅλον ἄρα τὸ <lb n="1799,001,,,,10,102,16"></lb> |
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12470 ΓΛ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΑΗ, ΗΒ· μέσον ἄρα [ἐστὶ] <lb n="1799,001,,,,10,102,17"></lb> |
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12471 καὶ τὸ ΓΛ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΓΔ παράκειται πλάτος <lb n="1799,001,,,,10,102,18"></lb> |
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12472 ποιοῦν τὴν ΓΜ· ῥητὴ ἄρα ἐστὶν ἡ ΓΜ καὶ ἀσύμμετρος <lb n="1799,001,,,,10,102,19"></lb> |
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12473 τῇ ΓΔ μήκει.</s> <s>ἐπεὶ οὖν τὸ ΓΛ ἴσον ἐστὶ τοῖς ἀπὸ τῶν <lb n="1799,001,,,,10,102,20"></lb> |
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12474 ΑΗ, ΗΒ, ὧν τὸ ΓΕ ἴσον τῷ ἀπὸ τῆς ΑΒ, λοιπὸν ἄρα <lb n="1799,001,,,,10,102,21"></lb> |
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12475 τὸ ΖΛ ἴσον ἐστὶ δὶς ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>καί ἐστι τὸ δὶς <lb n="1799,001,,,,10,102,22"></lb> |
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12476 ὑπὸ τῶν ΑΗ, ΗΒ μέσον· καὶ τὸ ΖΛ ἄρα μέσον ἐστίν.</s> <lb n="1799,001,,,,10,102,23"></lb> |
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12477 <s>καὶ παρὰ ῥητὴν τὴν ΖΕ παράκειται πλάτος ποιοῦν τὴν <pb></pb> <lb n="1799,001,,,,10,102,24"></lb> |
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12478 ΖΜ· ῥητὴ ἄρα ἐστὶν ἡ ΖΜ καὶ ἀσύμμετρος τῇ ΓΔ μήκει.</s> <lb n="1799,001,,,,10,102,25"></lb> |
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12479 <s>καὶ ἐπεὶ τὰ ἀπὸ τῶν ΑΗ, ΗΒ ἀσύμμετρά ἐστι τῷ δὶς <lb n="1799,001,,,,10,102,26"></lb> |
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12480 ὑπὸ τῶν ΑΗ, ΗΒ, καί ἐστι τοῖς μὲν ἀπὸ τῶν ΑΗ, ΗΒ <lb n="1799,001,,,,10,102,27"></lb> |
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12481 ἴσον τὸ ΓΛ, τῷ δὲ δὶς ὑπὸ τῶν ΑΗ, ΗΒ ἴσον τὸ ΖΛ, <lb n="1799,001,,,,10,102,28"></lb> |
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12482 ἀσύμμετρον ἄρα [ἐστὶ] τὸ ΓΛ τῷ ΖΛ.</s> <s>ὡς δὲ τὸ ΓΛ <lb n="1799,001,,,,10,102,29"></lb> |
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12483 πρὸς τὸ ΖΛ, οὕτως ἐστὶν ἡ ΓΜ πρὸς τὴν ΜΖ· ἀσύμ-<lb n="1799,001,,,,10,102,30"></lb> |
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12484 μετρος ἄρα ἐστὶν ἡ ΓΜ τῇ ΜΖ μήκει.</s> <s>καί εἰσιν ἀμφότεραι <lb n="1799,001,,,,10,102,31"></lb> |
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12485 ῥηταί.</s> <s>αἱ ΓΜ, ΜΖ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,102,32"></lb> |
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12486 μετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΓΖ.</s> <lb n="1799,001,,,,10,102,33"></lb> |
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12487 <s>Λέγω δή, ὅτι καὶ ἕκτη.</s> <lb n="1799,001,,,,10,102,34"></lb> |
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12488 <s>Ἐπεὶ γὰρ τὸ ΖΛ ἴσον ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΗ, ΗΒ, <lb n="1799,001,,,,10,102,35"></lb> |
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12489 τετμήσθω δίχα ἡ ΖΜ κατὰ τὸ Ν, καὶ ἤχθω διὰ τοῦ Ν <lb n="1799,001,,,,10,102,36"></lb> |
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12490 τῇ ΓΔ παράλληλος ἡ ΝΞ· ἑκάτερον ἄρα τῶν ΖΞ, ΝΛ <lb n="1799,001,,,,10,102,37"></lb> |
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12491 ἴσον ἐστὶ τῷ ὑπὸ τῶν ΑΗ, ΗΒ.</s> <s>καὶ ἐπεὶ αἱ ΑΗ, ΗΒ <lb n="1799,001,,,,10,102,38"></lb> |
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12492 δυνάμει εἰσὶν ἀσύμμετροι, ἀσύμμετρον ἄρα ἐστὶ τὸ ἀπὸ <lb n="1799,001,,,,10,102,39"></lb> |
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12493 τῆς ΑΗ τῷ ἀπὸ τῆς ΗΒ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΗ <lb n="1799,001,,,,10,102,40"></lb> |
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12494 ἴσον ἐστὶ τὸ ΓΘ, τῷ δὲ ἀπὸ τῆς ΗΒ ἴσον ἐστὶ τὸ ΚΛ· <lb n="1799,001,,,,10,102,41"></lb> |
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12495 ἀσύμμετρον ἄρα ἐστὶ τὸ ΓΘ τῷ ΚΛ.</s> <s>ὡς δὲ τὸ ΓΘ πρὸς <lb n="1799,001,,,,10,102,42"></lb> |
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12496 τὸ ΚΛ, οὕτως ἐστὶν ἡ ΓΚ πρὸς τὴν ΚΜ· ἀσύμμετρος <lb n="1799,001,,,,10,102,43"></lb> |
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12497 ἄρα ἐστὶν ἡ ΓΚ τῇ ΚΜ.</s> <s>καὶ ἐπεὶ τῶν ἀπὸ τῶν ΑΗ, <lb n="1799,001,,,,10,102,44"></lb> |
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12498 ΗΒ μέσον ἀνάλογόν ἐστι τὸ ὑπὸ τῶν ΑΗ, ΗΒ, καί ἐστι <lb n="1799,001,,,,10,102,45"></lb> |
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12499 τῷ μὲν ἀπὸ τῆς ΑΗ ἴσον τὸ ΓΘ, τῷ δὲ ἀπὸ τῆς ΗΒ <lb n="1799,001,,,,10,102,46"></lb> |
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12500 ἴσον τὸ ΚΛ, τῷ δὲ ὑπὸ τῶν ΑΗ, ΗΒ ἴσον τὸ ΝΛ, καὶ <lb n="1799,001,,,,10,102,47"></lb> |
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12501 τῶν ἄρα ΓΘ, ΚΛ μέσον ἀνάλογόν ἐστι τὸ ΝΛ· ἔστιν ἄρα <pb></pb> <lb n="1799,001,,,,10,102,48"></lb> |
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12502 ὡς τὸ ΓΘ πρὸς τὸ ΝΛ, οὕτως τὸ ΝΛ πρὸς τὸ ΚΛ.</s> <s>καὶ <lb n="1799,001,,,,10,102,49"></lb> |
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12503 διὰ τὰ αὐτὰ ἡ ΓΜ τῆς ΜΖ μεῖζον δύναται τῷ ἀπὸ <lb n="1799,001,,,,10,102,50"></lb> |
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12504 ἀσυμμέτρου ἑαυτῇ.</s> <s>καὶ οὐδετέρα αὐτῶν σύμμετρός ἐστι <lb n="1799,001,,,,10,102,51"></lb> |
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12505 τῇ ἐκκειμένῃ ῥητῇ τῇ ΓΔ· ἡ ΓΖ ἄρα ἀποτομή ἐστιν <lb n="1799,001,,,,10,102,52"></lb> |
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12506 ἕκτη· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,102,53"></lb> |
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12507 <s>Ἡ τῇ ἀποτομῇ μήκει σύμμετρος ἀποτομή ἐστι καὶ τῇ <lb n="1799,001,,,,10,103,1"></lb> |
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12508 τάξει ἡ αὐτή.</s> <lb n="1799,001,,,,10,103,2"></lb> |
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12509 <s>Ἔστω ἀποτομὴ ἡ ΑΒ, καὶ τῇ ΑΒ μήκει σύμμετρος <lb n="1799,001,,,,10,103,3"></lb> |
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12510 ἔστω ἡ ΓΔ· λέγω, ὅτι καὶ ἡ ΓΔ ἀποτομή ἐστι καὶ τῇ <lb n="1799,001,,,,10,103,4"></lb> |
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12511 τάξει ἡ αὐτὴ τῇ ΑΒ.</s> <lb n="1799,001,,,,10,103,5"></lb> |
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12512 <s>Ἐπεὶ γὰρ ἀποτομή ἐστιν ἡ ΑΒ, ἔστω αὐτῇ προσ-<lb n="1799,001,,,,10,103,6"></lb> |
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12513 αρμόζουσα ἡ ΒΕ· αἱ ΑΕ, ΕΒ ἄρα ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,103,7"></lb> |
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12514 μόνον σύμμετροι.</s> <s>καὶ τῷ τῆς ΑΒ πρὸς τὴν ΓΔ λόγῳ <lb n="1799,001,,,,10,103,8"></lb> |
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12515 ὁ αὐτὸς γεγονέτω ὁ τῆς ΒΕ πρὸς τὴν ΔΖ· καὶ ὡς ἓν <lb n="1799,001,,,,10,103,9"></lb> |
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12516 ἄρα πρὸς ἕν, πάντα [ἐστὶ] πρὸς πάντα· ἔστιν ἄρα καὶ <pb></pb> <lb n="1799,001,,,,10,103,10"></lb> |
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12517 ὡς ὅλη ἡ ΑΕ πρὸς ὅλην τὴν ΓΖ, οὕτως ἡ ΑΒ πρὸς <lb n="1799,001,,,,10,103,11"></lb> |
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12518 τὴν ΓΔ.</s> <s>σύμμετρος δὲ ἡ ΑΒ τῇ ΓΔ μήκει.</s> <s>σύμμετρος <lb n="1799,001,,,,10,103,12"></lb> |
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12519 ἄρα καὶ ἡ ΑΕ μὲν τῇ ΓΖ, ἡ δὲ ΒΕ τῇ ΔΖ.</s> <s>καὶ αἱ ΑΕ, <lb n="1799,001,,,,10,103,13"></lb> |
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12520 ΕΒ ῥηταί εἰσι δυνάμει μόνον σύμμετροι· καὶ αἱ ΓΖ, ΖΔ <lb n="1799,001,,,,10,103,14"></lb> |
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12521 ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι.</s> <s>[ἀποτομὴ ἄρα <lb n="1799,001,,,,10,103,15"></lb> |
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12522 ἐστὶν ἡ ΓΔ.</s> <lb n="1799,001,,,,10,103,16"></lb> |
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12523 <s>Λέγω δή, ὅτι καὶ τῇ τάξει ἡ αὐτὴ τῇ ΑΒ.] <lb n="1799,001,,,,10,103,17"></lb> |
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12524 Ἐπεὶ οὖν ἐστιν ὡς ἡ ΑΕ πρὸς τὴν ΓΖ, οὕτως ἡ ΒΕ <lb n="1799,001,,,,10,103,18"></lb> |
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12525 πρὸς τὴν ΔΖ, ἐναλλὰξ ἄρα ἐστὶν ὡς ἡ ΑΕ πρὸς τὴν ΕΒ, <lb n="1799,001,,,,10,103,19"></lb> |
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12526 οὕτως ἡ ΓΖ πρὸς τὴν ΖΔ.</s> <s>ἤτοι δὴ ἡ ΑΕ τῆς ΕΒ μεῖζον <lb n="1799,001,,,,10,103,20"></lb> |
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12527 δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ ἢ τῷ ἀπὸ ἀσυμμέτρου.</s> <lb n="1799,001,,,,10,103,21"></lb> |
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12528 <s>εἰ μὲν οὖν ἡ ΑΕ τῆς ΕΒ μεῖζον δύναται τῷ ἀπὸ συμ-<lb n="1799,001,,,,10,103,22"></lb> |
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12529 μέτρου ἑαυτῇ, καὶ ἡ ΓΖ τῆς ΖΔ μεῖζον δυνήσεται τῷ ἀπὸ <lb n="1799,001,,,,10,103,23"></lb> |
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12530 συμμέτρου ἑαυτῇ.</s> <s>καὶ εἰ μὲν σύμμετρός ἐστιν ἡ ΑΕ τῇ <lb n="1799,001,,,,10,103,24"></lb> |
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12531 ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΓΖ, εἰ δὲ ἡ ΒΕ, καὶ ἡ ΔΖ, <lb n="1799,001,,,,10,103,25"></lb> |
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12532 εἰ δὲ οὐδετέρα τῶν ΑΕ, ΕΒ, καὶ οὐδετέρα τῶν ΓΖ, <lb n="1799,001,,,,10,103,26"></lb> |
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12533 ΖΔ.</s> <s>εἰ δὲ ἡ ΑΕ [τῆς ΕΒ] μεῖζον δύναται τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,103,27"></lb> |
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12534 μέτρου ἑαυτῇ, καὶ ἡ ΓΖ τῆς ΖΔ μεῖζον δυνήσεται τῷ ἀπὸ <lb n="1799,001,,,,10,103,28"></lb> |
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12535 ἀσυμμέτρου ἑαυτῇ.</s> <s>καὶ εἰ μὲν σύμμετρός ἐστιν ἡ ΑΕ τῇ <pb></pb> <lb n="1799,001,,,,10,103,29"></lb> |
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12536 ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΓΖ, εἰ δὲ ἡ ΒΕ, καὶ ἡ ΔΖ, <lb n="1799,001,,,,10,103,30"></lb> |
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12537 εἰ δὲ οὐδετέρα τῶν ΑΕ, ΕΒ, οὐδετέρα τῶν ΓΖ, ΖΔ.</s> <lb n="1799,001,,,,10,103,31"></lb> |
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12538 <s>Ἀποτομὴ ἄρα ἐστὶν ἡ ΓΔ καὶ τῇ τάξει ἡ αὐτὴ τῇ <lb n="1799,001,,,,10,103,32"></lb> |
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12539 ΑΒ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,103,33"></lb> |
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12540 <s>Ἡ τῇ μέσης ἀποτομῇ σύμμετρος μέσης ἀποτομή <lb n="1799,001,,,,10,104,1"></lb> |
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12541 ἐστι καὶ τῇ τάξει ἡ αὐτή.</s> <lb n="1799,001,,,,10,104,2"></lb> |
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12542 <s>Ἔστω μέσης ἀποτομὴ ἡ ΑΒ, καὶ τῇ ΑΒ μήκει σύμ-<lb n="1799,001,,,,10,104,3"></lb> |
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12543 μετρος ἔστω ἡ ΓΔ· λέγω, ὅτι καὶ ἡ ΓΔ μέσης ἀποτομή <lb n="1799,001,,,,10,104,4"></lb> |
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12544 ἐστι καὶ τῇ τάξει ἡ αὐτὴ τῇ ΑΒ.</s> <lb n="1799,001,,,,10,104,5"></lb> |
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12545 <s>Ἐπεὶ γὰρ μέσης ἀποτομή ἐστιν ἡ ΑΒ, ἔστω αὐτῇ <lb n="1799,001,,,,10,104,6"></lb> |
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12546 προσαρμόζουσα ἡ ΕΒ.</s> <s>αἱ ΑΕ, ΕΒ ἄρα μέσαι <lb n="1799,001,,,,10,104,7"></lb> |
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12547 εἰσὶ δυνάμει μόνον σύμμετροι.</s> <s>καὶ γεγονέτω <lb n="1799,001,,,,10,104,8"></lb> |
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12548 ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΒΕ πρὸς <lb n="1799,001,,,,10,104,9"></lb> |
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12549 τὴν ΔΖ· σύμμετρος ἄρα [ἐστὶ] καὶ ἡ ΑΕ τῇ <lb n="1799,001,,,,10,104,10"></lb> |
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12550 ΓΖ, ἡ δὲ ΒΕ τῇ ΔΖ.</s> <s>αἱ δὲ ΑΕ, ΕΒ μέσαι <lb n="1799,001,,,,10,104,11"></lb> |
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12551 εἰσὶ δυνάμει μόνον σύμμετροι· καὶ αἱ ΓΖ, ΖΔ <lb n="1799,001,,,,10,104,12"></lb> |
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12552 ἄρα μέσαι εἰσὶ δυνάμει μόνον σύμμετροι· <lb n="1799,001,,,,10,104,13"></lb> |
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12553 μέσης ἄρα ἀποτομή ἐστιν ἡ ΓΔ.</s> <s><pb></pb> <lb n="1799,001,,,,10,104,14"></lb> |
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12554 Λέγω δή, ὅτι καὶ τῇ τάξει ἐστὶν ἡ αὐτὴ τῇ ΑΒ.</s> <lb n="1799,001,,,,10,104,15"></lb> |
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12555 <s>Ἐπεὶ [γάρ] ἐστιν ὡς ἡ ΑΕ πρὸς τὴν ΕΒ, οὕτως ἡ ΓΖ <lb n="1799,001,,,,10,104,16"></lb> |
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12556 πρὸς τὴν ΖΔ [ἀλλ' ὡς μὲν ἡ ΑΕ πρὸς τὴν ΕΒ, οὕτως <lb n="1799,001,,,,10,104,17"></lb> |
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12557 τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ὑπὸ τῶν ΑΕ, ΕΒ, ὡς δὲ ἡ ΓΖ <lb n="1799,001,,,,10,104,18"></lb> |
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12558 πρὸς τὴν ΖΔ, οὕτως τὸ ἀπὸ τῆς ΓΖ πρὸς τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,104,19"></lb> |
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12559 ΓΖ, ΖΔ], ἔστιν ἄρα καὶ ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ὑπὸ <lb n="1799,001,,,,10,104,20"></lb> |
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12560 τῶν ΑΕ, ΕΒ, οὕτως τὸ ἀπὸ τῆς ΓΖ πρὸς τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,104,21"></lb> |
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12561 ΓΖ, ΖΔ [καὶ ἐναλλὰξ ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,104,22"></lb> |
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12562 τῆς ΓΖ, οὕτως τὸ ὑπὸ τῶν ΑΕ, ΕΒ πρὸς τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,104,23"></lb> |
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12563 ΓΖ, ΖΔ].</s> <s>σύμμετρον δὲ τὸ ἀπὸ τῆς ΑΕ τῷ ἀπὸ τῆς ΓΖ· <lb n="1799,001,,,,10,104,24"></lb> |
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12564 σύμμετρον ἄρα ἐστὶ καὶ τὸ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν <lb n="1799,001,,,,10,104,25"></lb> |
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12565 ΓΖ, ΖΔ.</s> <s>εἴτε οὖν ῥητόν ἐστι τὸ ὑπὸ τῶν ΑΕ, ΕΒ, <lb n="1799,001,,,,10,104,26"></lb> |
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12566 ῥητὸν ἔσται καὶ τὸ ὑπὸ τῶν ΓΖ, ΖΔ, εἴτε μέσον [ἐστὶ] <lb n="1799,001,,,,10,104,27"></lb> |
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12567 τὸ ὑπὸ τῶν ΑΕ, ΕΒ, μέσον [ἐστὶ] καὶ τὸ ὑπὸ τῶν ΓΖ, <lb n="1799,001,,,,10,104,28"></lb> |
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12568 ΖΔ.</s> <lb n="1799,001,,,,10,104,29"></lb> |
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12569 <s>Μέσης ἄρα ἀποτομή ἐστιν ἡ ΓΔ καὶ τῇ τάξει ἡ αὐτὴ <lb n="1799,001,,,,10,104,30"></lb> |
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12570 τῇ ΑΒ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,104,31"></lb> |
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12571 <s>Ἡ τῇ ἐλάσσονι σύμμετρος ἐλάσσων ἐστίν.</s> <lb n="1799,001,,,,10,105,1"></lb> |
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12572 <s>Ἔστω γὰρ ἐλάσσων ἡ ΑΒ καὶ τῇ ΑΒ σύμμετρος ἡ ΓΔ· <lb n="1799,001,,,,10,105,2"></lb> |
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12573 λέγω, ὅτι καὶ ἡ ΓΔ ἐλάσσων ἐστίν.</s> <s><pb></pb> <lb n="1799,001,,,,10,105,3"></lb> |
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12574 Γεγονέτω γὰρ τὰ αὐτά· καὶ ἐπεὶ αἱ ΑΕ, ΕΒ <lb n="1799,001,,,,10,105,4"></lb> |
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12575 δυνάμει εἰσὶν ἀσύμμετροι, καὶ αἱ ΓΖ, ΖΔ ἄρα <lb n="1799,001,,,,10,105,5"></lb> |
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12576 δυνάμει εἰσὶν ἀσύμμετροι.</s> <s>ἐπεὶ οὖν ἐστιν ὡς ἡ <lb n="1799,001,,,,10,105,6"></lb> |
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12577 ΑΕ πρὸς τὴν ΕΒ, οὕτως ἡ ΓΖ πρὸς τὴν ΖΔ, <lb n="1799,001,,,,10,105,7"></lb> |
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12578 ἔστιν ἄρα καὶ ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,105,8"></lb> |
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12579 τῆς ΕΒ, οὕτως τὸ ἀπὸ τῆς ΓΖ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,105,9"></lb> |
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12580 τῆς ΖΔ.</s> <s>συνθέντι ἄρα ἐστὶν ὡς τὰ ἀπὸ τῶν <lb n="1799,001,,,,10,105,10"></lb> |
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12581 ΑΕ, ΕΒ πρὸς τὸ ἀπὸ τῆς ΕΒ, οὕτως τὰ ἀπὸ τῶν ΓΖ, ΖΔ <lb n="1799,001,,,,10,105,11"></lb> |
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12582 πρὸς τὸ ἀπὸ τῆς ΖΔ [καὶ ἐναλλάξ]· σύμμετρον δέ ἐστι <lb n="1799,001,,,,10,105,12"></lb> |
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12583 τὸ ἀπὸ τῆς ΒΕ τῷ ἀπὸ τῆς ΔΖ· σύμμετρον ἄρα καὶ τὸ <lb n="1799,001,,,,10,105,13"></lb> |
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12584 συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων τῷ <lb n="1799,001,,,,10,105,14"></lb> |
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12585 συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων.</s> <s>ῥητὸν <lb n="1799,001,,,,10,105,15"></lb> |
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12586 δέ ἐστι τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετρα-<lb n="1799,001,,,,10,105,16"></lb> |
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12587 γώνων· ῥητὸν ἄρα ἐστὶ καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,105,17"></lb> |
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12588 τῶν ΓΖ, ΖΔ τετραγώνων.</s> <s>πάλιν, ἐπεί ἐστιν ὡς τὸ ἀπὸ <lb n="1799,001,,,,10,105,18"></lb> |
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12589 τῆς ΑΕ πρὸς τὸ ὑπὸ τῶν ΑΕ, ΕΒ, οὕτως τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,105,19"></lb> |
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12590 ΓΖ πρὸς τὸ ὑπὸ τῶν ΓΖ, ΖΔ, σύμμετρον δὲ τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,105,20"></lb> |
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12591 ΑΕ τετράγωνον τῷ ἀπὸ τῆς ΓΖ τετραγώνῳ, σύμμετρον <lb n="1799,001,,,,10,105,21"></lb> |
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12592 ἄρα ἐστὶ καὶ τὸ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, <lb n="1799,001,,,,10,105,22"></lb> |
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12593 ΖΔ.</s> <s>μέσον δὲ τὸ ὑπὸ τῶν ΑΕ, ΕΒ· μέσον ἄρα καὶ τὸ <pb></pb> <lb n="1799,001,,,,10,105,23"></lb> |
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12594 ὑπὸ τῶν ΓΖ, ΖΔ· αἱ ΓΖ, ΖΔ ἄρα δυνάμει εἰσὶν ἀσύμ-<lb n="1799,001,,,,10,105,24"></lb> |
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12595 μετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ' αὐτῶν <lb n="1799,001,,,,10,105,25"></lb> |
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12596 τετραγώνων ῥητόν, τὸ δ' ὑπ' αὐτῶν μέσον.</s> <lb n="1799,001,,,,10,105,26"></lb> |
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12597 <s>Ἐλάσσων ἄρα ἐστὶν ἡ ΓΔ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,105,27"></lb> |
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12598 <s>Ἡ τῇ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιούσῃ σύμμετρος <lb n="1799,001,,,,10,106,1"></lb> |
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12599 μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσά ἐστιν.</s> <lb n="1799,001,,,,10,106,2"></lb> |
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12600 <s>Ἔστω μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα ἡ ΑΒ καὶ <lb n="1799,001,,,,10,106,3"></lb> |
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12601 τῇ ΑΒ σύμμετρος ἡ ΓΔ· λέγω, ὅτι καὶ ἡ ΓΔ μετὰ <lb n="1799,001,,,,10,106,4"></lb> |
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12602 ῥητοῦ μέσον τὸ ὅλον ποιοῦσά ἐστιν.</s> <lb n="1799,001,,,,10,106,5"></lb> |
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12603 <s>Ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΕ· αἱ ΑΕ, ΕΒ <lb n="1799,001,,,,10,106,6"></lb> |
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12604 ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον <lb n="1799,001,,,,10,106,7"></lb> |
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12605 ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων μέσον, τὸ δ' <lb n="1799,001,,,,10,106,8"></lb> |
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12606 ὑπ' αὐτῶν ῥητόν.</s> <s>καὶ τὰ αὐτὰ κατεσκευά-<lb n="1799,001,,,,10,106,9"></lb> |
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12607 σθω.</s> <s>ὁμοίως δὴ δείξομεν τοῖς πρότερον, ὅτι <lb n="1799,001,,,,10,106,10"></lb> |
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12608 αἱ ΓΖ, ΖΔ ἐν τῷ αὐτῷ λόγῳ εἰσὶ ταῖς ΑΕ, <lb n="1799,001,,,,10,106,11"></lb> |
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12609 ΕΒ, καὶ σύμμετρόν ἐστι τὸ συγκείμενον ἐκ <lb n="1799,001,,,,10,106,12"></lb> |
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12610 τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων τῷ συγ-<lb n="1799,001,,,,10,106,13"></lb> |
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12611 κειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώ-<lb n="1799,001,,,,10,106,14"></lb> |
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12612 νων, τὸ δὲ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν <lb n="1799,001,,,,10,106,15"></lb> |
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12613 ΓΖ, ΖΔ· ὥστε καὶ αἱ ΓΖ, ΖΔ δυνάμει εἰσὶν <pb></pb> <lb n="1799,001,,,,10,106,16"></lb> |
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12614 ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ <lb n="1799,001,,,,10,106,17"></lb> |
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12615 τῶν ΓΖ, ΖΔ τετραγώνων μέσον, τὸ δ' ὑπ' αὐτῶν <lb n="1799,001,,,,10,106,18"></lb> |
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12616 ῥητόν.</s> <lb n="1799,001,,,,10,106,19"></lb> |
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12617 <s>Ἡ ΓΔ ἄρα μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσά ἐστιν· <lb n="1799,001,,,,10,106,20"></lb> |
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12618 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,106,21"></lb> |
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12619 <s>Ἡ τῇ μετὰ μέσου μέσον τὸ ὅλον ποιούσῃ σύμμετρος <lb n="1799,001,,,,10,107,1"></lb> |
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12620 καὶ αὐτὴ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν.</s> <lb n="1799,001,,,,10,107,2"></lb> |
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12621 <s>Ἔστω μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα ἡ ΑΒ, καὶ <lb n="1799,001,,,,10,107,3"></lb> |
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12622 τῇ ΑΒ ἔστω σύμμετρος ἡ ΓΔ· λέγω, ὅτι καὶ ἡ ΓΔ μετὰ <lb n="1799,001,,,,10,107,4"></lb> |
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12623 μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν.</s> <lb n="1799,001,,,,10,107,5"></lb> |
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12624 <s>Ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΕ, καὶ τὰ αὐτὰ <lb n="1799,001,,,,10,107,6"></lb> |
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12625 κατεσκευάσθω· αἱ ΑΕ, ΕΒ ἄρα δυνάμει εἰσὶν ἀσύμ-<lb n="1799,001,,,,10,107,7"></lb> |
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12626 μετροι ποιοῦσαι τό τε συγκείμενον ἐκ τῶν <lb n="1799,001,,,,10,107,8"></lb> |
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12627 ἀπ' αὐτῶν τετραγώνων μέσον καὶ τὸ ὑπ' <lb n="1799,001,,,,10,107,9"></lb> |
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12628 αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τὸ συγκεί-<lb n="1799,001,,,,10,107,10"></lb> |
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12629 μενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων τῷ ὑπ' <lb n="1799,001,,,,10,107,11"></lb> |
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12630 αὐτῶν.</s> <s>καί εἰσιν, ὡς ἐδείχθη, αἱ ΑΕ, ΕΒ σύμ-<lb n="1799,001,,,,10,107,12"></lb> |
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12631 μετροι ταῖς ΓΖ, ΖΔ, καὶ τὸ συγκείμενον ἐκ <lb n="1799,001,,,,10,107,13"></lb> |
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12632 τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων τῷ συγ-<lb n="1799,001,,,,10,107,14"></lb> |
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12633 κειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ, τὸ δὲ <lb n="1799,001,,,,10,107,15"></lb> |
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12634 ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, ΖΔ· καὶ <lb n="1799,001,,,,10,107,16"></lb> |
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12635 αἱ ΓΖ, ΖΔ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι <lb n="1799,001,,,,10,107,17"></lb> |
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12636 τό τε συγκείμενον ἐκ τῶν ἀπ' αὐτῶν τετραγώνων μέσον <lb n="1799,001,,,,10,107,18"></lb> |
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12637 καὶ τὸ ὑπ' αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τὸ συγ- <pb></pb><lb n="1799,001,,,,10,107,19"></lb> |
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12638 κείμενον ἐκ τῶν ἀπ' αὐτῶν [τετραγώνων] τῷ ὑπ' <lb n="1799,001,,,,10,107,20"></lb> |
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12639 αὐτῶν.</s> <lb n="1799,001,,,,10,107,21"></lb> |
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12640 <s>Ἡ ΓΔ ἄρα μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν· <lb n="1799,001,,,,10,107,22"></lb> |
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12641 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,107,23"></lb> |
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12642 <s>Ἀπὸ ῥητοῦ μέσου ἀφαιρουμένου ἡ τὸ λοιπὸν χωρίον <lb n="1799,001,,,,10,108,1"></lb> |
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12643 δυναμένη μία δύο ἀλόγων γίνεται ἤτοι ἀποτομὴ ἢ <lb n="1799,001,,,,10,108,2"></lb> |
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12644 ἐλάσσων.</s> <lb n="1799,001,,,,10,108,3"></lb> |
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12645 <s>Ἀπὸ γὰρ ῥητοῦ τοῦ ΒΓ μέσον ἀφῃρήσθω τὸ ΒΔ· λέγω, <lb n="1799,001,,,,10,108,4"></lb> |
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12646 ὅτι ἡ τὸ λοιπὸν δυναμένη τὸ ΕΓ μία δύο ἀλόγων γίνεται <lb n="1799,001,,,,10,108,5"></lb> |
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12647 ἤτοι ἀποτομὴ ἢ ἐλάσσων.</s> <lb n="1799,001,,,,10,108,6"></lb> |
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12648 <s>Ἐκκείσθω γὰρ ῥητὴ ἡ ΖΗ, καὶ τῷ μὲν ΒΓ ἴσον παρὰ <lb n="1799,001,,,,10,108,7"></lb> |
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12649 τὴν ΖΗ παραβεβλήσθω ὀρθογώνιον παραλληλόγραμμον <lb n="1799,001,,,,10,108,8"></lb> |
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12650 τὸ ΗΘ, τῷ δὲ ΔΒ ἴσον ἀφῃρήσθω τὸ ΗΚ· λοιπὸν ἄρα <lb n="1799,001,,,,10,108,9"></lb> |
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12651 τὸ ΕΓ ἴσον ἐστὶ τῷ ΛΘ.</s> <s>ἐπεὶ οὖν ῥητὸν μέν ἐστι τὸ ΒΓ, <lb n="1799,001,,,,10,108,10"></lb> |
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12652 μέσον δὲ τὸ ΒΔ, ἴσον δὲ τὸ μὲν ΒΓ τῷ ΗΘ, τὸ δὲ ΒΔ <pb></pb> <lb n="1799,001,,,,10,108,11"></lb> |
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12653 τῷ ΗΚ, ῥητὸν μὲν ἄρα ἐστὶ τὸ ΗΘ, μέσον δὲ τὸ ΗΚ.</s> <lb n="1799,001,,,,10,108,12"></lb> |
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12654 <s>καὶ παρὰ ῥητὴν τὴν ΖΗ παράκειται· ῥητὴ μὲν ἄρα ἡ <lb n="1799,001,,,,10,108,13"></lb> |
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12655 ΖΘ καὶ σύμμετρος τῇ ΖΗ μήκει, ῥητὴ δὲ ἡ ΖΚ καὶ <lb n="1799,001,,,,10,108,14"></lb> |
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12656 ἀσύμμετρος τῇ ΖΗ μήκει· ἀσύμμετρος ἄρα ἐστὶν ἡ <lb n="1799,001,,,,10,108,15"></lb> |
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12657 ΖΘ τῇ ΖΚ μήκει.</s> <s>αἱ ΖΘ, ΖΚ ἄρα ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,108,16"></lb> |
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12658 μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΚΘ, προσαρμό-<lb n="1799,001,,,,10,108,17"></lb> |
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12659 ζουσα δὲ αὐτῇ ἡ ΚΖ.</s> <s>ἤτοι δὴ ἡ ΘΖ τῆς ΖΚ μεῖζον <lb n="1799,001,,,,10,108,18"></lb> |
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12660 δύναται τῷ ἀπὸ συμμέτρου ἢ οὔ.</s> <lb n="1799,001,,,,10,108,19"></lb> |
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12661 <s>Δυνάσθω πρότερον τῷ ἀπὸ συμμέτρου.</s> <s>καί ἐστιν ὅλη <lb n="1799,001,,,,10,108,20"></lb> |
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12662 ἡ ΘΖ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ· <lb n="1799,001,,,,10,108,21"></lb> |
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12663 ἀποτομὴ ἄρα πρώτη ἐστὶν ἡ ΚΘ.</s> <s>τὸ δ' ὑπὸ ῥητῆς καὶ <lb n="1799,001,,,,10,108,22"></lb> |
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12664 ἀποτομῆς πρώτης περιεχόμενον ἡ δυναμένη ἀποτομή <lb n="1799,001,,,,10,108,23"></lb> |
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12665 ἐστιν.</s> <s>ἡ ἄρα τὸ ΛΘ, τουτέστι τὸ ΕΓ, δυναμένη ἀποτομή <lb n="1799,001,,,,10,108,24"></lb> |
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12666 ἐστιν.</s> <lb n="1799,001,,,,10,108,25"></lb> |
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12667 <s>Εἰ δὲ ἡ ΘΖ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,108,26"></lb> |
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12668 μέτρου ἑαυτῇ, καί ἐστιν ὅλη ἡ ΖΘ σύμμετρος τῇ ἐκ-<lb n="1799,001,,,,10,108,27"></lb> |
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12669 κειμένῃ ῥητῇ μήκει τῇ ΖΗ, ἀποτομὴ τετάρτη ἐστὶν ἡ <lb n="1799,001,,,,10,108,28"></lb> |
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12670 ΚΘ.</s> <s>τὸ δ' ὑπὸ ῥητῆς καὶ ἀποτομῆς τετάρτης περι-<lb n="1799,001,,,,10,108,29"></lb> |
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12671 εχόμενον ἡ δυναμένη ἐλάσσων ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,108,30"></lb> |
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12672 Ἀπὸ μέσου ῥητοῦ ἀφαιρουμένου ἄλλαι δύο ἄλογοι <lb n="1799,001,,,,10,109,1"></lb> |
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12673 γίνονται ἤτοι μέσης ἀποτομὴ πρώτη ἢ μετὰ ῥητοῦ μέσον <lb n="1799,001,,,,10,109,2"></lb> |
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12674 τὸ ὅλον ποιοῦσα.</s> <lb n="1799,001,,,,10,109,3"></lb> |
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12675 <s>Ἀπὸ γὰρ μέσου τοῦ ΒΓ ῥητὸν ἀφῃρήσθω τὸ ΒΔ.</s> <lb n="1799,001,,,,10,109,4"></lb> |
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12676 <s>λέγω, ὅτι ἡ τὸ λοιπὸν τὸ ΕΓ δυναμένη μία δύο ἀλόγων <lb n="1799,001,,,,10,109,5"></lb> |
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12677 γίνεται ἤτοι μέσης ἀποτομὴ πρώτη ἢ μετὰ ῥητοῦ μέσον <lb n="1799,001,,,,10,109,6"></lb> |
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12678 τὸ ὅλον ποιοῦσα.</s> <lb n="1799,001,,,,10,109,7"></lb> |
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12679 <s>Ἐκκείσθω γὰρ ῥητὴ ἡ ΖΗ, καὶ παραβεβλήσθω ὁμοίως <lb n="1799,001,,,,10,109,8"></lb> |
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12680 τὰ χωρία.</s> <s>ἔστι δὴ ἀκολούθως ῥητὴ μὲν ἡ ΖΘ καὶ ἀσύμ-<lb n="1799,001,,,,10,109,9"></lb> |
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12681 μετρος τῇ ΖΗ μήκει, ῥητὴ <lb n="1799,001,,,,10,109,10"></lb> |
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12682 δὲ ἡ ΚΖ καὶ σύμμετρος τῇ <lb n="1799,001,,,,10,109,11"></lb> |
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12683 ΖΗ μήκει· αἱ ΖΘ, ΖΚ ἄρα <lb n="1799,001,,,,10,109,12"></lb> |
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12684 ῥηταί εἰσι δυνάμει μόνον σύμ-<lb n="1799,001,,,,10,109,13"></lb> |
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12685 μετροι· ἀποτομὴ ἄρα ἐστὶν <lb n="1799,001,,,,10,109,14"></lb> |
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12686 ἡ ΚΘ, προσαρμόζουσα δὲ <lb n="1799,001,,,,10,109,15"></lb> |
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12687 ταύτῃ ἡ ΖΚ.</s> <s>ἤτοι δὴ ἡ ΘΖ <lb n="1799,001,,,,10,109,16"></lb> |
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12688 τῆς ΖΚ μεῖζον δύναται τῷ <lb n="1799,001,,,,10,109,17"></lb> |
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12689 ἀπὸ συμμέτρου ἑαυτῇ ἢ τῷ <lb n="1799,001,,,,10,109,18"></lb> |
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12690 ἀπὸ ἀσυμμέτρου.</s> <lb n="1799,001,,,,10,109,19"></lb> |
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12691 <s>Εἰ μὲν οὖν ἡ ΘΖ τῆς <lb n="1799,001,,,,10,109,20"></lb> |
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12692 ΖΚ μεῖζον δύναται τῷ ἀπὸ <lb n="1799,001,,,,10,109,21"></lb> |
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12693 συμμέτρου ἑαυτῇ, καί ἐστιν ἡ προσαρμόζουσα ἡ ΖΚ <lb n="1799,001,,,,10,109,22"></lb> |
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12694 σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ, ἀπο-<lb n="1799,001,,,,10,109,23"></lb> |
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12695 τομὴ δευτέρα ἐστὶν ἡ ΚΘ.</s> <s>ῥητὴ δὲ ἡ ΖΗ· ὥστε ἡ <pb></pb> <lb n="1799,001,,,,10,109,24"></lb> |
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12696 τὸ ΛΘ, τουτέστι τὸ ΕΓ, δυναμένη μέσης ἀποτομὴ πρώτη <lb n="1799,001,,,,10,109,25"></lb> |
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12697 ἐστίν.</s> <lb n="1799,001,,,,10,109,26"></lb> |
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12698 <s>Εἰ δὲ ἡ ΘΖ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,109,27"></lb> |
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12699 μέτρου, καί ἐστιν ἡ προσαρμόζουσα ἡ ΖΚ σύμμετρος <lb n="1799,001,,,,10,109,28"></lb> |
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12700 τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ, ἀποτομὴ πέμπτη ἐστὶν <lb n="1799,001,,,,10,109,29"></lb> |
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12701 ἡ ΚΘ· ὥστε ἡ τὸ ΕΓ δυναμένη μετὰ ῥητοῦ μέσον τὸ <lb n="1799,001,,,,10,109,30"></lb> |
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12702 ὅλον ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,109,31"></lb> |
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12703 <s>Ἀπὸ μέσου μέσου ἀφαιρουμένου ἀσυμμέτρου τῷ ὅλῳ <lb n="1799,001,,,,10,110,1"></lb> |
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12704 αἱ λοιπαὶ δύο ἄλογοι γίνονται ἤτοι μέσης ἀποτομὴ <lb n="1799,001,,,,10,110,2"></lb> |
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12705 δευτέρα ἢ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα.</s> <lb n="1799,001,,,,10,110,3"></lb> |
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12706 <s>Ἀφῃρήσθω γὰρ ὡς ἐπὶ τῶν προκειμένων καταγραφῶν <lb n="1799,001,,,,10,110,4"></lb> |
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12707 ἀπὸ μέσου τοῦ ΒΓ μέσον τὸ ΒΔ ἀσύμμετρον τῷ ὅλῳ· <lb n="1799,001,,,,10,110,5"></lb> |
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12708 λέγω, ὅτι ἡ τὸ ΕΓ δυναμένη μία ἐστὶ δύο ἀλόγων ἤτοι <lb n="1799,001,,,,10,110,6"></lb> |
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12709 μέσης ἀποτομὴ δευτέρα ἢ μετὰ μέσου μέσον τὸ ὅλον <lb n="1799,001,,,,10,110,7"></lb> |
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12710 ποιοῦσα.</s> <lb n="1799,001,,,,10,110,8"></lb> |
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12711 <s>Ἐπεὶ γὰρ μέσον ἐστὶν ἑκάτερον τῶν ΒΓ, ΒΔ, καὶ <lb n="1799,001,,,,10,110,9"></lb> |
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12712 ἀσύμμετρον τὸ Β*γ τῷ ΒΔ, ἔσται ἀκολούθως ῥητὴ <lb n="1799,001,,,,10,110,10"></lb> |
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12713 ἑκατέρα τῶν ΖΘ, ΖΚ καὶ ἀσύμμετρος τῇ ΖΗ μήκει.</s> <lb n="1799,001,,,,10,110,11"></lb> |
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12714 <s>καὶ ἐπεὶ ἀσύμμετρόν ἐστι τὸ ΒΓ τῷ ΒΔ, τουτέστι τὸ <lb n="1799,001,,,,10,110,12"></lb> |
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12715 ΗΘ τῷ ΗΚ, ἀσύμμετρος καὶ ἡ ΘΖ τῇ ΖΚ· αἱ ΖΘ, ΖΚ <pb></pb> <lb n="1799,001,,,,10,110,13"></lb> |
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12716 ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα <lb n="1799,001,,,,10,110,14"></lb> |
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12717 ἐστὶν ἡ ΚΘ [προσαρμόζουσα δὲ ἡ ΖΚ.</s> <s>ἤτοι δὴ ἡ ΖΘ <lb n="1799,001,,,,10,110,15"></lb> |
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12718 τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἢ τῷ ἀπὸ <lb n="1799,001,,,,10,110,16"></lb> |
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12719 ἀσυμμέτρου ἑαυτῇ].</s> <lb n="1799,001,,,,10,110,17"></lb> |
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12720 <s>Εἰ μὲν δὴ ἡ ΖΘ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ συμ-<lb n="1799,001,,,,10,110,18"></lb> |
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12721 μέτρου ἑαυτῇ, καὶ οὐθετέρα τῶν ΖΘ, ΖΚ σύμμετρός ἐστι <lb n="1799,001,,,,10,110,19"></lb> |
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12722 τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ, ἀποτομὴ τρίτη ἐστὶν <lb n="1799,001,,,,10,110,20"></lb> |
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12723 ἡ ΚΘ.</s> <s>ῥητὴ δὲ ἡ ΚΛ, τὸ <lb n="1799,001,,,,10,110,21"></lb> |
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12724 δ' ὑπὸ ῥητῆς καὶ ἀποτομῆς <lb n="1799,001,,,,10,110,22"></lb> |
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12725 τρίτης περιεχόμενον ὀρθο-<lb n="1799,001,,,,10,110,23"></lb> |
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12726 γώνιον ἄλογόν ἐστιν, καὶ <lb n="1799,001,,,,10,110,24"></lb> |
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12727 ἡ δυναμένη αὐτὸ ἄλογός <lb n="1799,001,,,,10,110,25"></lb> |
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12728 ἐστιν, καλεῖται δὲ μέσης <lb n="1799,001,,,,10,110,26"></lb> |
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12729 ἀποτομὴ δευτέρα· ὥστε ἡ <lb n="1799,001,,,,10,110,27"></lb> |
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12730 τὸ ΛΘ, τουτέστι τὸ ΕΓ, <lb n="1799,001,,,,10,110,28"></lb> |
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12731 δυναμένη μέσης ἀποτομή <lb n="1799,001,,,,10,110,29"></lb> |
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12732 ἐστι δευτέρα.</s> <lb n="1799,001,,,,10,110,30"></lb> |
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12733 <s>Εἰ δὲ ἡ ΖΘ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,110,31"></lb> |
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12734 μέτρου ἑαυτῇ [μήκει], καὶ οὐθετέρα τῶν ΘΖ, ΖΚ <lb n="1799,001,,,,10,110,32"></lb> |
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12735 σύμμετρός ἐστι τῇ ΖΗ μήκει, ἀποτομὴ ἕκτη ἐστὶν ἡ <lb n="1799,001,,,,10,110,33"></lb> |
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12736 ΚΘ.</s> <s>τὸ δ' ὑπὸ ῥητῆς καὶ ἀποτομῆς ἕκτης ἡ δυναμένη <lb n="1799,001,,,,10,110,34"></lb> |
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12737 ἐστὶ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα.</s> <s>ἡ τὸ ΛΘ ἄρα, <lb n="1799,001,,,,10,110,35"></lb> |
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12738 τουτέστι τὸ ΕΓ, δυναμένη μετὰ μέσου μέσον τὸ ὅλον <lb n="1799,001,,,,10,110,36"></lb> |
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12739 ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,110,37"></lb> |
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12740 Ἡ ἀποτομὴ οὐκ ἔστιν ἡ αὐτὴ τῇ ἐκ δύο ὀνομάτων.</s> <lb n="1799,001,,,,10,111,1"></lb> |
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12741 <s>Ἔστω ἀποτομὴ ἡ ΑΒ· λέγω, ὅτι ἡ ΑΒ οὐκ ἔστιν ἡ <lb n="1799,001,,,,10,111,2"></lb> |
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12742 αὐτὴ τῇ ἐκ δύο ὀνομάτων.</s> <lb n="1799,001,,,,10,111,3"></lb> |
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12743 <s>Εἰ γὰρ δυνατόν, ἔστω· καὶ ἐκκείσθω ῥητὴ ἡ ΔΓ, καὶ <lb n="1799,001,,,,10,111,4"></lb> |
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12744 τῷ ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω <lb n="1799,001,,,,10,111,5"></lb> |
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12745 ὀρθογώνιον τὸ ΓΕ πλάτος ποιοῦν τὴν ΔΕ.</s> <s>ἐπεὶ οὖν <lb n="1799,001,,,,10,111,6"></lb> |
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12746 ἀποτομή ἐστιν ἡ ΑΒ, ἀπο- <lb n="1799,001,,,,10,111,7"></lb> |
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12747 τομὴ πρώτη ἐστὶν ἡ ΔΕ.</s> <s>ἔστω <lb n="1799,001,,,,10,111,8"></lb> |
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12748 αὐτῇ προσαρμόζουσα ἡ ΕΖ· <lb n="1799,001,,,,10,111,9"></lb> |
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12749 αἱ ΔΖ, ΖΕ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,111,10"></lb> |
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12750 δυνάμει μόνον σύμμετροι, καὶ <lb n="1799,001,,,,10,111,11"></lb> |
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12751 ἡ ΔΖ τῆς ΖΕ μεῖζον δύνα-<lb n="1799,001,,,,10,111,12"></lb> |
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12752 ται τῷ ἀπὸ συμμέτρου ἑαυτῇ, <lb n="1799,001,,,,10,111,13"></lb> |
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12753 καὶ ἡ ΔΖ σύμμετρός ἐστι <lb n="1799,001,,,,10,111,14"></lb> |
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12754 τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ <lb n="1799,001,,,,10,111,15"></lb> |
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12755 ΔΓ.</s> <s>πάλιν, ἐπεὶ ἐκ δύο ὀνο-<lb n="1799,001,,,,10,111,16"></lb> |
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12756 μάτων ἐστὶν ἡ ΑΒ, ἐκ δύο <lb n="1799,001,,,,10,111,17"></lb> |
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12757 ἄρα ὀνομάτων πρώτη ἐστὶν ἡ <lb n="1799,001,,,,10,111,18"></lb> |
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12758 ΔΕ.</s> <s>διῃρήσθω εἰς τὰ ὀνόματα κατὰ τὸ Η, καὶ ἔστω <lb n="1799,001,,,,10,111,19"></lb> |
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12759 μεῖζον ὄνομα τὸ ΔΗ· αἱ ΔΗ, ΗΕ ἄρα ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,111,20"></lb> |
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12760 μόνον σύμμετροι, καὶ ἡ ΔΗ τῆς ΗΕ μεῖζον δύναται τῷ <lb n="1799,001,,,,10,111,21"></lb> |
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12761 ἀπὸ συμμέτρου ἑαυτῇ, καὶ τὸ μεῖζον ἡ ΔΗ σύμμετρός <lb n="1799,001,,,,10,111,22"></lb> |
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12762 ἐστι τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΔΓ.</s> <s>καὶ ἡ ΔΖ ἄρα τῇ <pb></pb> <lb n="1799,001,,,,10,111,23"></lb> |
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12763 ΔΗ σύμμετρός ἐστι μήκει· καὶ λοιπὴ ἄρα ἡ ΗΖ σύμ-<lb n="1799,001,,,,10,111,24"></lb> |
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12764 μετρός ἐστι τῇ ΔΖ μήκει.</s> <s>[ἐπεὶ οὖν σύμμετρός ἐστιν ἡ <lb n="1799,001,,,,10,111,25"></lb> |
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12765 ΔΖ τῇ ΗΖ, ῥητὴ δέ ἐστιν ἡ ΔΖ, ῥητὴ ἄρα ἐστὶ καὶ ἡ <lb n="1799,001,,,,10,111,26"></lb> |
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12766 ΗΖ.</s> <s>ἐπεὶ οὖν σύμμετρός ἐστιν ἡ ΔΖ τῇ ΗΖ μήκει] <lb n="1799,001,,,,10,111,27"></lb> |
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12767 ἀσύμμετρος δὲ ἡ ΔΖ τῇ ΕΖ μήκει· ἀσύμμετρος ἄρα ἐστὶ <lb n="1799,001,,,,10,111,28"></lb> |
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12768 καὶ ἡ ΖΗ τῇ ΕΖ μήκει.</s> <s>αἱ ΗΖ, ΖΕ ἄρα ῥηταί [εἰσι] <lb n="1799,001,,,,10,111,29"></lb> |
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12769 δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΕΗ.</s> <lb n="1799,001,,,,10,111,30"></lb> |
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12770 <s>ἀλλὰ καὶ ῥητή· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,10,111,31"></lb> |
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12771 <s>Ἡ ἄρα ἀποτομὴ οὐκ ἔστιν ἡ αὐτὴ τῇ ἐκ δύο ὀνο-<lb n="1799,001,,,,10,111,32"></lb> |
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12772 μάτων· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,111,33"></lb> |
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12773 <s>[Πόρισμα] <lb n="1799,001,,,,10,111,34"></lb> |
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12774 Ἡ ἀποτομὴ καὶ αἱ μετ' αὐτὴν ἄλογοι οὔτε τῇ μέσῃ <lb n="1799,001,,,,10,111,35"></lb> |
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12775 οὔτε ἀλλήλαις εἰσὶν αἱ αὐταί.</s> <lb n="1799,001,,,,10,111,36"></lb> |
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12776 <s>Τὸ μὲν γὰρ ἀπὸ μέσης παρὰ ῥητὴν παραβαλλόμενον <lb n="1799,001,,,,10,111,37"></lb> |
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12777 πλάτος ποιεῖ ῥητὴν καὶ ἀσύμμετρον τῇ, παρ' ἣν παρά-<lb n="1799,001,,,,10,111,38"></lb> |
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12778 κειται, μήκει, τὸ δὲ ἀπὸ ἀποτομῆς παρὰ ῥητὴν παρα-<lb n="1799,001,,,,10,111,39"></lb> |
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12779 βαλλόμενον πλάτος ποιεῖ ἀποτομὴν πρώτην, τὸ δὲ ἀπὸ <lb n="1799,001,,,,10,111,40"></lb> |
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12780 μέσης ἀποτομῆς πρώτης παρὰ ῥητὴν παραβαλλόμενον <lb n="1799,001,,,,10,111,41"></lb> |
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12781 πλάτος ποιεῖ ἀποτομὴν δευτέραν, τὸ δὲ ἀπὸ μέσης ἀπο-<lb n="1799,001,,,,10,111,42"></lb> |
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12782 τομῆς δευτέρας παρὰ ῥητὴν παραβαλλόμενον πλάτος <pb></pb> <lb n="1799,001,,,,10,111,43"></lb> |
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12783 ποιεῖ ἀποτομὴν τρίτην, τὸ δὲ ἀπὸ ἐλάσσονος παρὰ ῥητὴν <lb n="1799,001,,,,10,111,44"></lb> |
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12784 παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν τετάρτην, τὸ <lb n="1799,001,,,,10,111,45"></lb> |
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12785 δὲ ἀπὸ τῆς μετὰ ῥητοῦ μέσον τὸ ὅλον ποιούσης παρὰ <lb n="1799,001,,,,10,111,46"></lb> |
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12786 ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν πέμπτην, <lb n="1799,001,,,,10,111,47"></lb> |
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12787 τὸ δὲ ἀπὸ τῆς μετὰ μέσου μέσον τὸ ὅλον ποιούσης παρὰ <lb n="1799,001,,,,10,111,48"></lb> |
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12788 ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν ἕκτην.</s> <lb n="1799,001,,,,10,111,49"></lb> |
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12789 <s>ἐπεὶ οὖν τὰ εἰρημένα πλάτη διαφέρει τοῦ τε πρώτου καὶ <lb n="1799,001,,,,10,111,50"></lb> |
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12790 ἀλλήλων, τοῦ μὲν πρώτου, ὅτι ῥητή ἐστιν, ἀλλήλων δέ, <lb n="1799,001,,,,10,111,51"></lb> |
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12791 ἐπεὶ τῇ τάξει οὐκ εἰσὶν αἱ αὐταί, δῆλον, ὡς καὶ αὐταὶ αἱ <lb n="1799,001,,,,10,111,52"></lb> |
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12792 ἄλογοι διαφέρουσιν ἀλλήλων.</s> <s>καὶ ἐπεὶ δέδεικται ἡ <lb n="1799,001,,,,10,111,53"></lb> |
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12793 ἀποτομὴ οὐκ οὖσα ἡ αὐτὴ τῇ ἐκ δύο ὀνομάτων, ποιοῦσι <lb n="1799,001,,,,10,111,54"></lb> |
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12794 δὲ πλάτη παρὰ ῥητὴν παραβαλλόμεναι αἱ μετὰ τὴν <lb n="1799,001,,,,10,111,55"></lb> |
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12795 ἀποτομὴν ἀποτομὰς ἀκολούθως ἑκάστη τῇ τάξει τῇ καθ' <lb n="1799,001,,,,10,111,56"></lb> |
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12796 αὑτήν, αἱ δὲ μετὰ τὴν ἐκ δύο ὀνομάτων τὰς ἐκ δύο <lb n="1799,001,,,,10,111,57"></lb> |
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12797 ὀνομάτων καὶ αὐταὶ τῇ τάξει ἀκολούθως, ἕτεραι ἄρα εἰσὶν <lb n="1799,001,,,,10,111,58"></lb> |
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12798 αἱ μετὰ τὴν ἀποτομὴν καὶ ἕτεραι αἱ μετὰ τὴν ἐκ δύο <lb n="1799,001,,,,10,111,59"></lb> |
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12799 ὀνομάτων, ὡς εἶναι τῇ τάξει πάσας ἀλόγους <ιγ>, <lb n="1799,001,,,,10,111,60"></lb> |
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12800 Μέσην, <lb n="1799,001,,,,10,111,61"></lb> |
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12801 Ἐκ δύο ὀνομάτων, <lb n="1799,001,,,,10,111,62"></lb> |
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12802 Ἐκ δύο μέσων πρώτην, <lb n="1799,001,,,,10,111,63"></lb> |
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12803 Ἐκ δύο μέσων δευτέραν, <lb n="1799,001,,,,10,111,64"></lb> |
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12804 Μείζονα, <lb n="1799,001,,,,10,111,65"></lb> |
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12805 Ῥητὸν καὶ μέσον δυναμένην, <lb n="1799,001,,,,10,111,66"></lb> |
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12806 Δύο μέσα δυναμένην, <lb n="1799,001,,,,10,111,67"></lb> |
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12807 Ἀποτομήν, <lb n="1799,001,,,,10,111,68"></lb> |
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12808 Μέσης ἀποτομὴν πρώτην, <pb></pb> <lb n="1799,001,,,,10,111,69"></lb> |
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12809 Μέσης ἀποτομὴν δευτέραν, <lb n="1799,001,,,,10,111,70"></lb> |
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12810 Ἐλάσσονα, <lb n="1799,001,,,,10,111,71"></lb> |
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12811 Μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσαν, <lb n="1799,001,,,,10,111,72"></lb> |
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12812 Μετὰ μέσου μέσον τὸ ὅλον ποιοῦσαν.</s> <lb n="1799,001,,,,10,111,73"></lb> |
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12813 <s>Τὸ ἀπὸ ῥητῆς παρὰ τὴν ἐκ δύο ὀνομάτων παραβαλλό-<lb n="1799,001,,,,10,112,1"></lb> |
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12814 μενον πλάτος ποιεῖ ἀποτομήν, ἧς τὰ ὀνόματα σύμμετρά <lb n="1799,001,,,,10,112,2"></lb> |
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12815 ἐστι τοῖς τῆς ἐκ δύο ὀνομάτων ὀνόμασι καὶ ἔτι ἐν τῷ <lb n="1799,001,,,,10,112,3"></lb> |
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12816 αὐτῷ λόγῳ, καὶ ἔτι ἡ γινομένη ἀποτομὴ τὴν αὐτὴν ἕξει <lb n="1799,001,,,,10,112,4"></lb> |
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12817 τάξιν τῇ ἐκ δύο ὀνομάτων.</s> <lb n="1799,001,,,,10,112,5"></lb> |
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12818 <s>Ἔστω ῥητὴ μὲν ἡ Α, ἐκ δύο ὀνομάτων δὲ ἡ ΒΓ, ἧς <lb n="1799,001,,,,10,112,6"></lb> |
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12819 μεῖζον ὄνομα ἔστω ἡ ΔΓ, καὶ τῷ ἀπὸ τῆς Α ἴσον ἔστω τὸ <lb n="1799,001,,,,10,112,7"></lb> |
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12820 ὑπὸ τῶν ΒΓ, ΕΖ· λέγω, ὅτι ἡ ΕΖ ἀποτομή ἐστιν, ἧς <lb n="1799,001,,,,10,112,8"></lb> |
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12821 τὰ ὀνόματα σύμμετρά ἐστι τοῖς ΓΔ, ΔΒ, καὶ ἐν τῷ αὐτῷ <lb n="1799,001,,,,10,112,9"></lb> |
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12822 λόγῳ, καὶ ἔτι ἡ ΕΖ τὴν αὐτὴν ἕξει τάξιν τῇ ΒΓ.</s> <lb n="1799,001,,,,10,112,10"></lb> |
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12823 <s>Ἔστω γὰρ πάλιν τῷ ἀπὸ τῆς Α ἴσον τὸ ὑπὸ τῶν ΒΔ, <lb n="1799,001,,,,10,112,11"></lb> |
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12824 Η.</s> <s>ἐπεὶ οὖν τὸ ὑπὸ τῶν ΒΓ, ΕΖ ἴσον ἐστὶ τῷ ὑπὸ τῶν <lb n="1799,001,,,,10,112,12"></lb> |
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12825 ΒΔ, Η, ἔστιν ἄρα ὡς ἡ ΓΒ πρὸς τὴν ΒΔ, οὕτως ἡ Η <pb></pb> <lb n="1799,001,,,,10,112,13"></lb> |
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12826 πρὸς τὴν ΕΖ.</s> <s>μείζων δὲ ἡ ΓΒ τῆς ΒΔ· μείζων ἄρα ἐστὶ <lb n="1799,001,,,,10,112,14"></lb> |
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12827 καὶ ἡ Η τῆς ΕΖ.</s> <s>ἔστω τῇ Η ἴση ἡ ΕΘ· ἔστιν ἄρα ὡς <lb n="1799,001,,,,10,112,15"></lb> |
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12828 ἡ ΓΒ πρὸς τὴν ΒΔ, οὕτως ἡ ΘΕ πρὸς τὴν ΕΖ· διελόντι <lb n="1799,001,,,,10,112,16"></lb> |
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12829 ἄρα ἐστὶν ὡς ἡ ΓΔ πρὸς τὴν ΒΔ, οὕτως ἡ ΘΖ πρὸς τὴν <lb n="1799,001,,,,10,112,17"></lb> |
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12830 ΖΕ.</s> <s>γεγονέτω ὡς ἡ ΘΖ πρὸς τὴν ΖΕ, οὕτως ἡ ΖΚ <lb n="1799,001,,,,10,112,18"></lb> |
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12831 πρὸς τὴν ΚΕ· καὶ ὅλη ἄρα ἡ ΘΚ πρὸς ὅλην τὴν ΚΖ ἐστιν, <lb n="1799,001,,,,10,112,19"></lb> |
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12832 ὡς ἡ ΖΚ πρὸς ΚΕ· ὡς γὰρ ἓν τῶν ἡγουμένων πρὸς ἓν <lb n="1799,001,,,,10,112,20"></lb> |
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12833 τῶν ἑπομένων, οὕτως ἅπαντα τὰ ἡγούμενα πρὸς ἅπαντα <lb n="1799,001,,,,10,112,21"></lb> |
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12834 τὰ ἑπόμενα.</s> <s>ὡς δὲ ἡ ΖΚ πρὸς ΚΕ, οὕτως ἐστὶν ἡ ΓΔ <lb n="1799,001,,,,10,112,22"></lb> |
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12835 πρὸς τὴν ΔΒ· καὶ ὡς ἄρα ἡ ΘΚ πρὸς ΚΖ, οὕτως ἡ <lb n="1799,001,,,,10,112,23"></lb> |
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12836 ΓΔ πρὸς τὴν ΔΒ.</s> <s>σύμμετρον δὲ τὸ ἀπὸ τῆς ΓΔ τῷ ἀπὸ <lb n="1799,001,,,,10,112,24"></lb> |
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12837 τῆς ΔΒ· σύμμετρον ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς ΘΚ τῷ <lb n="1799,001,,,,10,112,25"></lb> |
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12838 ἀπὸ τῆς ΚΖ.</s> <s>καί ἐστιν ὡς τὸ ἀπὸ τῆς ΘΚ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,10,112,26"></lb> |
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12839 τῆς ΚΖ, οὕτως ἡ ΘΚ πρὸς τὴν ΚΕ, ἐπεὶ αἱ τρεῖς αἱ <lb n="1799,001,,,,10,112,27"></lb> |
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12840 ΘΚ, ΚΖ, ΚΕ ἀνάλογόν εἰσιν.</s> <s>σύμμετρος ἄρα ἡ ΘΚ τῇ <lb n="1799,001,,,,10,112,28"></lb> |
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12841 ΚΕ μήκει· ὥστε καὶ ἡ ΘΕ τῇ ΕΚ σύμμετρός ἐστι μήκει.</s> <lb n="1799,001,,,,10,112,29"></lb> |
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12842 <s>καὶ ἐπεὶ τὸ ἀπὸ τῆς Α ἴσον ἐστὶ τῷ ὑπὸ τῶν ΕΘ, ΒΔ, <pb></pb> <lb n="1799,001,,,,10,112,30"></lb> |
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12843 ῥητὸν δέ ἐστι τὸ ἀπὸ τῆς Α, ῥητὸν ἄρα ἐστὶ καὶ τὸ ὑπὸ <lb n="1799,001,,,,10,112,31"></lb> |
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12844 τῶν ΕΘ, ΒΔ.</s> <s>καὶ παρὰ ῥητὴν τὴν ΒΔ παράκειται· ῥητὴ <lb n="1799,001,,,,10,112,32"></lb> |
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12845 ἄρα ἐστὶν ἡ ΕΘ καὶ σύμμετρος τῇ ΒΔ μήκει· ὥστε καὶ <lb n="1799,001,,,,10,112,33"></lb> |
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12846 ἡ σύμμετρος αὐτῇ ἡ ΕΚ ῥητή ἐστι καὶ σύμμετρος τῇ <lb n="1799,001,,,,10,112,34"></lb> |
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12847 ΒΔ μήκει.</s> <s>ἐπεὶ οὖν ἐστιν ὡς ἡ ΓΔ πρὸς ΔΒ, οὕτως ἡ <lb n="1799,001,,,,10,112,35"></lb> |
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12848 ΖΚ πρὸς ΚΕ, αἱ δὲ ΓΔ, ΔΒ δυνάμει μόνον εἰσὶ σύμ-<lb n="1799,001,,,,10,112,36"></lb> |
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12849 μετροι, καὶ αἱ ΖΚ, ΚΕ δυνάμει μόνον εἰσὶ σύμμετροι.</s> <lb n="1799,001,,,,10,112,37"></lb> |
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12850 <s>ῥητὴ δέ ἐστιν ἡ ΚΕ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΖΚ.</s> <s>αἱ ΖΚ, ΚΕ <lb n="1799,001,,,,10,112,38"></lb> |
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12851 ἄρα ῥηταὶ δυνάμει μόνον εἰσὶ σύμμετροι· ἀποτομὴ ἄρα <lb n="1799,001,,,,10,112,39"></lb> |
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12852 ἐστὶν ἡ ΕΖ.</s> <lb n="1799,001,,,,10,112,40"></lb> |
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12853 <s>Ἤτοι δὲ ἡ ΓΔ τῆς ΔΒ μεῖζον δύναται τῷ ἀπὸ συμ-<lb n="1799,001,,,,10,112,41"></lb> |
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12854 μέτρου ἑαυτῇ ἢ τῷ ἀπὸ ἀσυμμέτρου.</s> <lb n="1799,001,,,,10,112,42"></lb> |
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12855 <s>Εἰ μὲν οὖν ἡ ΓΔ τῆς ΔΒ μεῖζον δύναται τῷ ἀπὸ <lb n="1799,001,,,,10,112,43"></lb> |
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12856 συμμέτρου [ἑαυτῇ], καὶ ἡ ΖΚ τῆς ΚΕ μεῖζον δυνή-<lb n="1799,001,,,,10,112,44"></lb> |
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12857 σεται τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <s>καὶ εἰ μὲν σύμμετρός <lb n="1799,001,,,,10,112,45"></lb> |
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12858 ἐστιν ἡ ΓΔ τῇ ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΖΚ· εἰ δὲ <lb n="1799,001,,,,10,112,46"></lb> |
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12859 ἡ ΒΔ, καὶ ἡ ΚΕ· εἰ δὲ οὐδετέρα τῶν ΓΔ, ΔΒ, καὶ <lb n="1799,001,,,,10,112,47"></lb> |
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12860 οὐδετέρα τῶν ΖΚ, ΚΕ.</s> <lb n="1799,001,,,,10,112,48"></lb> |
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12861 <s>Εἰ δὲ ἡ ΓΔ τῆς ΔΒ μεῖζον δύναται τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,112,49"></lb> |
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12862 μέτρου ἑαυτῇ, καὶ ἡ ΖΚ τῆς ΚΕ μεῖζον δυνήσεται τῷ <pb></pb> <lb n="1799,001,,,,10,112,50"></lb> |
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12863 ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <s>καὶ εἰ μὲν ἡ ΓΔ σύμμετρός ἐστι <lb n="1799,001,,,,10,112,51"></lb> |
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12864 τῇ ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΖΚ· εἰ δὲ ἡ ΒΔ, καὶ ἡ <lb n="1799,001,,,,10,112,52"></lb> |
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12865 ΚΕ· εἰ δὲ οὐδετέρα τῶν ΓΔ, ΔΒ, καὶ οὐδετέρα τῶν ΖΚ, <lb n="1799,001,,,,10,112,53"></lb> |
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12866 ΚΕ· ὥστε ἀποτομή ἐστιν ἡ ΖΕ, ἧς τὰ ὀνόματα τὰ ΖΚ, <lb n="1799,001,,,,10,112,54"></lb> |
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12867 ΚΕ σύμμετρά ἐστι τοῖς τῆς ἐκ δύο ὀνομάτων ὀνόμασι <lb n="1799,001,,,,10,112,55"></lb> |
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12868 τοῖς ΓΔ, ΔΒ καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ τὴν αὐτὴν τάξιν <lb n="1799,001,,,,10,112,56"></lb> |
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12869 ἔχει τῇ ΒΓ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,112,57"></lb> |
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12870 <s>Τὸ ἀπὸ ῥητῆς παρὰ ἀποτομὴν παραβαλλόμενον πλάτος <lb n="1799,001,,,,10,113,1"></lb> |
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12871 ποιεῖ τὴν ἐκ δύο ὀνομάτων, ἧς τὰ ὀνόματα σύμμετρά ἐστι <lb n="1799,001,,,,10,113,2"></lb> |
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12872 τοῖς τῆς ἀποτομῆς ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, ἔτι <lb n="1799,001,,,,10,113,3"></lb> |
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12873 δὲ ἡ γινομένη ἐκ δύο ὀνομάτων τὴν αὐτὴν τάξιν ἔχει τῇ <lb n="1799,001,,,,10,113,4"></lb> |
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12874 ἀποτομῇ.</s> <lb n="1799,001,,,,10,113,5"></lb> |
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12875 <s>Ἔστω ῥητὴ μὲν ἡ Α, ἀποτομὴ δὲ ἡ ΒΔ, καὶ τῷ ἀπὸ <lb n="1799,001,,,,10,113,6"></lb> |
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12876 τῆς Α ἴσον ἔστω τὸ ὑπὸ τῶν ΒΔ, ΚΘ, ὥστε τὸ ἀπὸ τῆς <lb n="1799,001,,,,10,113,7"></lb> |
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12877 Α ῥητῆς παρὰ τὴν ΒΔ ἀποτομὴν <lb n="1799,001,,,,10,113,8"></lb> |
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12878 παραβαλλόμενον πλάτος ποιεῖ τὴν ΚΘ· <lb n="1799,001,,,,10,113,9"></lb> |
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12879 λέγω, ὅτι ἐκ δύο ὀνομάτων ἐστὶν ἡ ΚΘ, <lb n="1799,001,,,,10,113,10"></lb> |
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12880 ἧς τὰ ὀνόματα σύμμετρά ἐστι τοῖς τῆς <lb n="1799,001,,,,10,113,11"></lb> |
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12881 ΒΔ ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ <lb n="1799,001,,,,10,113,12"></lb> |
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12882 ἔτι ἡ ΚΘ τὴν αὐτὴν ἔχει τάξιν τῇ ΒΔ.</s> <lb n="1799,001,,,,10,113,13"></lb> |
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12883 <s>Ἔστω γὰρ τῇ ΒΔ προσαρμόζουσα <lb n="1799,001,,,,10,113,14"></lb> |
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12884 ἡ ΔΓ· αἱ ΒΓ, ΓΔ ἄρα ῥηταί εἰσι <lb n="1799,001,,,,10,113,15"></lb> |
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12885 δυνάμει μόνον σύμμετροι.</s> <s>καὶ τῷ ἀπὸ <pb></pb> <lb n="1799,001,,,,10,113,16"></lb> |
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12886 τῆς Α ἴσον ἔστω καὶ τὸ ὑπὸ τῶν ΒΓ, Η.</s> <s>ῥητὸν δὲ τὸ <lb n="1799,001,,,,10,113,17"></lb> |
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12887 ἀπὸ τῆς Α· ῥητὸν ἄρα καὶ τὸ ὑπὸ τῶν ΒΓ, Η.</s> <s>καὶ παρὰ <lb n="1799,001,,,,10,113,18"></lb> |
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12888 ῥητὴν τὴν ΒΓ παραβέβληται· ῥητὴ ἄρα ἐστὶν ἡ Η καὶ <lb n="1799,001,,,,10,113,19"></lb> |
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12889 σύμμετρος τῇ ΒΓ μήκει.</s> <s>ἐπεὶ οὖν τὸ ὑπὸ τῶν ΒΓ, Η <lb n="1799,001,,,,10,113,20"></lb> |
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12890 ἴσον ἐστὶ τῷ ὑπὸ τῶν ΒΔ, ΚΘ, ἀνάλογον ἄρα ἐστὶν ὡς <lb n="1799,001,,,,10,113,21"></lb> |
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12891 ἡ ΓΒ πρὸς ΒΔ, οὕτως ἡ ΚΘ πρὸς Η.</s> <s>μείζων δὲ ἡ ΒΓ <lb n="1799,001,,,,10,113,22"></lb> |
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12892 τῆς ΒΔ· μείζων ἄρα καὶ ἡ ΚΘ τῆς Η.</s> <s>κείσθω τῇ Η ἴση <lb n="1799,001,,,,10,113,23"></lb> |
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12893 ἡ ΚΕ· σύμμετρος ἄρα ἐστὶν ἡ ΚΕ τῇ ΒΓ μήκει.</s> <s>καὶ <lb n="1799,001,,,,10,113,24"></lb> |
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12894 ἐπεί ἐστιν ὡς ἡ ΓΒ πρὸς ΒΔ, οὕτως ἡ ΘΚ πρὸς ΚΕ, <lb n="1799,001,,,,10,113,25"></lb> |
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12895 ἀναστρέψαντι ἄρα ἐστὶν ὡς ἡ ΒΓ πρὸσ τὴν ΓΔ, οὕτως <lb n="1799,001,,,,10,113,26"></lb> |
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12896 ἡ ΚΘ πρὸς ΘΕ.</s> <s>γεγονέτω ὡς ἡ ΚΘ πρὸς ΘΕ, οὕτως ἡ <lb n="1799,001,,,,10,113,27"></lb> |
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12897 ΘΖ πρὸς ΖΕ· καὶ λοιπὴ ἄρα ἡ ΚΖ πρὸς ΖΘ ἐστιν, ὡς <lb n="1799,001,,,,10,113,28"></lb> |
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12898 ἡ ΚΘ πρὸς ΘΕ, τουτέστιν [ὡσ] ἡ ΒΓ πρὸς ΓΔ.</s> <s>αἱ δὲ <lb n="1799,001,,,,10,113,29"></lb> |
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12899 ΒΓ, ΓΔ δυνάμει μόνον [εἰσὶ] σύμμετροι· καὶ αἱ ΚΖ, <lb n="1799,001,,,,10,113,30"></lb> |
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12900 ΖΘ ἄρα δυνάμει μόνον εἰσὶ σύμμετροι.</s> <s>καὶ ἐπεί ἐστιν ὡς <lb n="1799,001,,,,10,113,31"></lb> |
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12901 ἡ ΚΘ πρὸς ΘΕ, ἡ ΚΖ πρὸς ΖΘ, ἀλλ' ὡς ἡ ΚΘ πρὸς <lb n="1799,001,,,,10,113,32"></lb> |
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12902 ΘΕ, ἡ ΘΖ πρὸς ΖΕ, καὶ ὡς ἄρα ἡ ΚΖ πρὸς ΖΘ, ἡ ΘΖ <lb n="1799,001,,,,10,113,33"></lb> |
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12903 πρὸς ΖΕ· ὥστε καὶ ὡς ἡ πρώτη πρὸς τὴν τρίτην, τὸ ἀπὸ <lb n="1799,001,,,,10,113,34"></lb> |
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12904 τῆς πρώτης πρὸς τὸ ἀπὸ τῆς δευτέρας· καὶ ὡς ἄρα ἡ ΚΖ <lb n="1799,001,,,,10,113,35"></lb> |
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12905 πρὸς ΖΕ, οὕτως τὸ ἀπὸ τῆς ΚΖ πρὸς τὸ ἀπὸ τῆς ΖΘ.</s> <lb n="1799,001,,,,10,113,36"></lb> |
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12906 <s>σύμμετρον δέ ἐστι τὸ ἀπὸ τῆς ΚΖ τῷ ἀπὸ τῆς ΖΘ· αἱ <pb></pb> <lb n="1799,001,,,,10,113,37"></lb> |
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12907 γὰρ ΚΖ, ΖΘ δυνάμει εἰσὶ σύμμετροι· σύμμετρος ἄρα <lb n="1799,001,,,,10,113,38"></lb> |
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12908 ἐστὶ καὶ ἡ ΚΖ τῇ ΖΕ μήκει· ὥστε ἡ ΚΖ καὶ τῇ ΚΕ <lb n="1799,001,,,,10,113,39"></lb> |
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12909 σύμμετρός [ἐστι] μήκει.</s> <s>ῥητὴ δέ ἐστιν ἡ ΚΕ καὶ σύμ-<lb n="1799,001,,,,10,113,40"></lb> |
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12910 μετρος τῇ ΒΓ μήκει· ῥητὴ ἄρα καὶ ἡ ΚΖ καὶ σύμμετρος <lb n="1799,001,,,,10,113,41"></lb> |
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12911 τῇ ΒΓ μήκει.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ ΒΓ πρὸς ΓΔ, οὕτως <lb n="1799,001,,,,10,113,42"></lb> |
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12912 ἡ ΚΖ πρὸς ΖΘ, ἐναλλὰξ ὡς ἡ ΒΓ πρὸς ΚΖ, οὕτως ἡ ΔΓ <lb n="1799,001,,,,10,113,43"></lb> |
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12913 πρὸς ΖΘ.</s> <s>σύμμετρος δὲ ἡ ΒΓ τῇ ΚΖ· σύμμετρος ἄρα καὶ <lb n="1799,001,,,,10,113,44"></lb> |
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12914 ἡ ΖΘ τῇ ΓΔ μήκει.</s> <s>αἱ ΒΓ, ΓΔ δὲ ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,113,45"></lb> |
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12915 μόνον σύμμετροι· καὶ αἱ ΚΖ, ΖΘ ἄρα ῥηταί εἰσι δυνάμει <lb n="1799,001,,,,10,113,46"></lb> |
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12916 μόνον σύμμετροι· ἐκ δύο ὀνομάτων ἐστὶν ἄρα ἡ ΚΘ.</s> <lb n="1799,001,,,,10,113,47"></lb> |
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12917 <s>Εἰ μὲν οὖν ἡ ΒΓ τῆς ΓΔ μεῖζον δύναται τῷ ἀπὸ <lb n="1799,001,,,,10,113,48"></lb> |
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12918 συμμέτρου ἑαυτῇ, καὶ ἡ ΚΖ τῆς ΖΘ μεῖζον δυνήσεται <lb n="1799,001,,,,10,113,49"></lb> |
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12919 τῷ ἀπὸ συμμέτρου ἑαυτῇ.</s> <s>καὶ εἰ μὲν σύμμετρός ἐστιν ἡ <lb n="1799,001,,,,10,113,50"></lb> |
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12920 ΒΓ τῇ ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΚΖ, εἰ δὲ ἡ ΓΔ <lb n="1799,001,,,,10,113,51"></lb> |
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12921 σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΖΘ, <lb n="1799,001,,,,10,113,52"></lb> |
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12922 εἰ δὲ οὐδετέρα τῶν ΒΓ, ΓΔ, οὐδετέρα τῶν ΚΖ, ΖΘ.</s> <lb n="1799,001,,,,10,113,53"></lb> |
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12923 <s>Εἰ δὲ ἡ ΒΓ τῆς ΓΔ μεῖζον δύναται τῷ ἀπὸ ἀσυμ-<lb n="1799,001,,,,10,113,54"></lb> |
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12924 μέτρου ἑαυτῇ, καὶ ἡ ΚΖ τῆς ΖΘ μεῖζον δυνήσεται τῷ <lb n="1799,001,,,,10,113,55"></lb> |
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12925 ἀπὸ ἀσυμμέτρου ἑαυτῇ.</s> <s>καὶ εἰ μὲν σύμμετρός ἐστιν ἡ ΒΓ <pb></pb> <lb n="1799,001,,,,10,113,56"></lb> |
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12926 τῇ ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΚΖ, εἰ δὲ ἡ ΓΔ, καὶ ἡ <lb n="1799,001,,,,10,113,57"></lb> |
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12927 ΖΘ, εἰ δὲ οὐδετέρα τῶν ΒΓ, ΓΔ, οὐδετέρα τῶν ΚΖ, ΖΘ.</s> <lb n="1799,001,,,,10,113,58"></lb> |
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12928 <s>Ἐκ δύο ἄρα ὀνομάτων ἐστὶν ἡ ΚΘ, ἧς τὰ ὀνόματα τὰ <lb n="1799,001,,,,10,113,59"></lb> |
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12929 ΚΖ, ΖΘ σύμμετρά [ἐστι] τοῖς τῆς ἀποτομῆς ὀνόμασι <lb n="1799,001,,,,10,113,60"></lb> |
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12930 τοῖς ΒΓ, ΓΔ καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ ἔτι ἡ ΚΘ τῇ ΒΓ <lb n="1799,001,,,,10,113,61"></lb> |
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12931 τὴν αὐτὴν ἕξει τάξιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,10,113,62"></lb> |
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12932 <s>Ἐὰν χωρίον περιέχηται ὑπὸ ἀποτομῆς καὶ τῆς ἐκ δύο <lb n="1799,001,,,,10,114,1"></lb> |
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12933 ὀνομάτων, ἧς τὰ ὀνόματα σύμμετρά τέ ἐστι τοῖς τῆς <lb n="1799,001,,,,10,114,2"></lb> |
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12934 ἀποτομῆς ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, ἡ τὸ χωρίον <lb n="1799,001,,,,10,114,3"></lb> |
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12935 δυναμένη ῥητή ἐστιν.</s> <lb n="1799,001,,,,10,114,4"></lb> |
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12936 <s>Περιεχέσθω γὰρ χωρίον τὸ ὑπὸ τῶν ΑΒ, ΓΔ ὑπὸ ἀπο-<lb n="1799,001,,,,10,114,5"></lb> |
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12937 τομῆς τῆς ΑΒ καὶ τῆς ἐκ δύο ὀνομάτων τῆς ΓΔ, ἧς μεῖζον <lb n="1799,001,,,,10,114,6"></lb> |
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12938 ὄνομα ἔστω τὸ ΓΕ, καὶ ἔστω τὰ ὀνό-<lb n="1799,001,,,,10,114,7"></lb> |
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12939 ματα τῆς ἐκ δύο ὀνομάτων τὰ ΓΕ, ΕΔ <lb n="1799,001,,,,10,114,8"></lb> |
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12940 σύμμετρά τε τοῖς τῆς ἀποτομῆς ὀνό-<lb n="1799,001,,,,10,114,9"></lb> |
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12941 μασι τοῖς ΑΖ, ΖΒ καὶ ἐν τῷ αὐτῷ <lb n="1799,001,,,,10,114,10"></lb> |
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12942 λόγῳ, καὶ ἔστω ἡ τὸ ὑπὸ τῶν ΑΒ, <lb n="1799,001,,,,10,114,11"></lb> |
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12943 ΓΔ δυναμένη ἡ Η· λέγω, ὅτι ῥητή <lb n="1799,001,,,,10,114,12"></lb> |
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12944 ἐστιν ἡ Η.</s> <lb n="1799,001,,,,10,114,13"></lb> |
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12945 <s>Ἐκκείσθω γὰρ ῥητὴ ἡ Θ, καὶ τῷ <lb n="1799,001,,,,10,114,14"></lb> |
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12946 ἀπὸ τῆς Θ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω πλάτος ποιοῦν <lb n="1799,001,,,,10,114,15"></lb> |
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12947 τὴν ΚΛ· ἀποτομὴ ἄρα ἐστὶν ἡ ΚΛ, ἧς τὰ ὀνόματα ἔστω <lb n="1799,001,,,,10,114,16"></lb> |
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12948 τὰ ΚΜ, ΜΛ σύμμετρα τοῖς τῆς ἐκ δύο ὀνομάτων ὀνόμασι <pb></pb> <lb n="1799,001,,,,10,114,17"></lb> |
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12949 τοῖς ΓΕ, ΕΔ καὶ ἐν τῷ αὐτῷ λόγῳ.</s> <s>ἀλλὰ καὶ αἱ ΓΕ, ΕΔ <lb n="1799,001,,,,10,114,18"></lb> |
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12950 σύμμετροί τέ εἰσι ταῖς ΑΖ, ΖΒ καὶ ἐν τῷ αὐτῷ λόγῳ· <lb n="1799,001,,,,10,114,19"></lb> |
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12951 ἔστιν ἄρα ὡς ἡ ΑΖ πρὸς τὴν ΖΒ, οὕτως ἡ ΚΜ πρὸς ΜΛ.</s> <lb n="1799,001,,,,10,114,20"></lb> |
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12952 <s>ἐναλλὰξ ἄρα ἐστὶν ὡς ἡ ΑΖ πρὸς τὴν ΚΜ, οὕτως ἡ ΒΖ <lb n="1799,001,,,,10,114,21"></lb> |
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12953 πρὸς τὴν ΛΜ· καὶ λοιπὴ ἄρα ἡ ΑΒ πρὸς λοιπὴν τὴν ΚΛ <lb n="1799,001,,,,10,114,22"></lb> |
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12954 ἐστιν ὡς ἡ ΑΖ πρὸς ΚΜ.</s> <s>σύμμετρος δὲ ἡ ΑΖ τῇ ΚΜ· <lb n="1799,001,,,,10,114,23"></lb> |
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12955 σύμμετρος ἄρα ἐστὶ καὶ ἡ ΑΒ τῇ ΚΛ.</s> <s>καί ἐστιν ὡς ἡ ΑΒ <lb n="1799,001,,,,10,114,24"></lb> |
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12956 πρὸς ΚΛ, οὕτως τὸ ὑπὸ τῶν ΓΔ, ΑΒ πρὸς τὸ ὑπὸ τῶν <lb n="1799,001,,,,10,114,25"></lb> |
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12957 ΓΔ, ΚΛ· σύμμετρον ἄρα ἐστὶ καὶ τὸ ὑπὸ τῶν ΓΔ, ΑΒ <lb n="1799,001,,,,10,114,26"></lb> |
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12958 τῷ ὑπὸ τῶν ΓΔ, ΚΛ.</s> <s>ἴσον δὲ τὸ ὑπὸ τῶν ΓΔ, ΚΛ τῷ <lb n="1799,001,,,,10,114,27"></lb> |
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12959 ἀπὸ τῆς Θ· σύμμετρον ἄρα ἐστὶ τὸ ὑπὸ τῶν ΓΔ, ΑΒ <lb n="1799,001,,,,10,114,28"></lb> |
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12960 τῷ ἀπὸ τῆς Θ.</s> <s>τῷ δὲ ὑπὸ τῶν ΓΔ, ΑΒ ἴσον ἐστὶ τὸ ἀπὸ <lb n="1799,001,,,,10,114,29"></lb> |
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12961 τῆς Η· σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς Η τῷ ἀπὸ τῆς Θ.</s> <lb n="1799,001,,,,10,114,30"></lb> |
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12962 <s>ῥητὸν δὲ τὸ ἀπὸ τῆς Θ· ῥητὸν ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς Η· <lb n="1799,001,,,,10,114,31"></lb> |
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12963 ῥητὴ ἄρα ἐστὶν ἡ Η.</s> <s>καὶ δύναται τὸ ὑπὸ τῶν ΓΔ, ΑΒ.</s> <lb n="1799,001,,,,10,114,32"></lb> |
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12964 <s>Ἐὰν ἄρα χωρίον περιέχηται ὑπὸ ἀποτομῆς καὶ τῆς ἐκ <lb n="1799,001,,,,10,114,33"></lb> |
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12965 δύο ὀνομάτων, ἧς τὰ ὀνόματα σύμμετρά ἐστι τοῖς τῆς ἀπο-<lb n="1799,001,,,,10,114,34"></lb> |
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12966 τομῆς ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, ἡ τὸ χωρίον δυνα-<lb n="1799,001,,,,10,114,35"></lb> |
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12967 μένη ῥητή ἐστιν.</s> <lb n="1799,001,,,,10,114,36"></lb> |
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12968 <s>Πόρισμα <lb n="1799,001,,,,10,114,37"></lb> |
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12969 Καὶ γέγονεν ἡμῖν καὶ διὰ τούτου φανερόν, ὅτι δυνατόν <lb n="1799,001,,,,10,114,38"></lb> |
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12970 ἐστι ῥητὸν χωρίον ὑπὸ ἀλόγων εὐθειῶν περιέχεσθαι.</s> <lb n="1799,001,,,,10,114,39"></lb> |
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12971 <s>ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,10,114,40"></lb> |
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12972 Ἀπὸ μέσης ἄπειροι ἄλογοι γίνονται, καὶ οὐδεμία <lb n="1799,001,,,,10,115,1"></lb> |
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12973 οὐδεμιᾷ τῶν πρότερον ἡ αὐτή.</s> <lb n="1799,001,,,,10,115,2"></lb> |
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12974 <s>Ἔστω μέση ἡ Α· λέγω, ὅτι ἀπὸ τῆς Α ἄπειροι ἄλογοι <lb n="1799,001,,,,10,115,3"></lb> |
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12975 γίνονται, καὶ οὐδεμία οὐδεμιᾷ τῶν πρότερον ἡ αὐτή.</s> <lb n="1799,001,,,,10,115,4"></lb> |
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12976 <s>Ἐκκείσθω ῥητὴ ἡ Β, καὶ τῷ ὑπὸ τῶν Β, Α ἴσον ἔστω <lb n="1799,001,,,,10,115,5"></lb> |
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12977 τὸ ἀπὸ τῆς Γ· ἄλογος ἄρα ἐστὶν ἡ Γ· τὸ γὰρ ὑπὸ ἀλόγου <lb n="1799,001,,,,10,115,6"></lb> |
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12978 καὶ ῥητῆς ἄλογόν ἐστιν.</s> <s>καὶ οὐδε-<lb n="1799,001,,,,10,115,7"></lb> |
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12979 μιᾷ τῶν πρότερον ἡ αὐτή· τὸ γὰρ <lb n="1799,001,,,,10,115,8"></lb> |
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12980 ἀπ' οὐδεμιᾶς τῶν πρότερον παρὰ ῥη-<lb n="1799,001,,,,10,115,9"></lb> |
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12981 τὴν παραβαλλόμενον πλάτος ποιεῖ <lb n="1799,001,,,,10,115,10"></lb> |
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12982 μέσην.</s> <s>πάλιν δὴ τῷ ὑπὸ τῶν Β, Γ <lb n="1799,001,,,,10,115,11"></lb> |
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12983 ἴσον ἔστω τὸ ἀπὸ τῆς Δ· ἄλογον <lb n="1799,001,,,,10,115,12"></lb> |
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12984 ἄρα ἐστὶ τὸ ἀπὸ τῆς Δ.</s> <s>ἄλογος ἄρα ἐστὶν ἡ Δ· καὶ οὐδεμιᾷ <lb n="1799,001,,,,10,115,13"></lb> |
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12985 τῶν πρότερον ἡ αὐτή· τὸ γὰρ ἀπ' οὐδεμιᾶς τῶν πρότερον <lb n="1799,001,,,,10,115,14"></lb> |
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12986 παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ τὴν Γ.</s> <s>ὁμοίως <lb n="1799,001,,,,10,115,15"></lb> |
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12987 δὴ τῆς τοιαύτης τάξεως ἐπ' ἄπειρον προβαινούσης <lb n="1799,001,,,,10,115,16"></lb> |
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12988 φανερόν, ὅτι ἀπὸ τῆς μέσης ἄπειροι ἄλογοι γίνονται, καὶ <lb n="1799,001,,,,10,115,17"></lb> |
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12989 οὐδεμία οὐδεμιᾷ τῶν πρότερον ἡ αὐτή· ὅπερ ἔδει δεῖξαι].</s> <s><pb></pb> <lb n="1799,001,,,,10,115,18"></lb> |
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12990 </s></p> |
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12991 </chap> |
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12992 <chap><p type="main"><s>Στερεόν ἐστι τὸ μῆκος καὶ πλάτος καὶ βάθος ἔχον.</s> <lb n="1799,001,,,,11;HOR,1,1"></lb> |
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12993 <s> |
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12994 Στερεοῦ δὲ πέρας ἐπιφάνεια.</s> <lb n="1799,001,,,,11;HOR,2,1"></lb> |
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12995 <s>Εὐθεῖα πρὸς ἐπίπεδον ὀρθή ἐστιν, ὅταν πρὸς πάσας <lb n="1799,001,,,,11;HOR,3,1"></lb> |
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12996 τὰς ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ [ὑποκειμένῳ] <lb n="1799,001,,,,11;HOR,3,2"></lb> |
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12997 ἐπιπέδῳ ὀρθὰς ποιῇ γωνίας.</s> <lb n="1799,001,,,,11;HOR,3,3"></lb> |
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12998 <s>Ἐπίπεδον πρὸς ἐπίπεδον ὀρθόν ἐστιν, ὅταν αἱ τῇ κοι-<lb n="1799,001,,,,11;HOR,4,1"></lb> |
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12999 νῇ τομῇ τῶν ἐπιπέδων πρὸς ὀρθὰς ἀγόμεναι εὐθεῖαι ἐν ἑνὶ <lb n="1799,001,,,,11;HOR,4,2"></lb> |
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13000 τῶν ἐπιπέδων τῷ λοιπῷ ἐπιπέδῳ πρὸς ὀρθὰς ὦσιν.</s> <lb n="1799,001,,,,11;HOR,4,3"></lb> |
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13001 <s>Εὐθείας πρὸς ἐπίπεδον κλίσις ἐστίν, ὅταν ἀπὸ τοῦ <lb n="1799,001,,,,11;HOR,5,1"></lb> |
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13002 μετεώρου πέρατος τῆς εὐθείας ἐπὶ τὸ ἐπίπεδον κάθετος <lb n="1799,001,,,,11;HOR,5,2"></lb> |
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13003 ἀχθῇ, καὶ ἀπὸ τοῦ γενομένου σημείου ἐπὶ τὸ ἐν τῷ ἐπιπέδῳ <lb n="1799,001,,,,11;HOR,5,3"></lb> |
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13004 πέρας τῆς εὐθείας εὐθεῖα ἐπιζευχθῇ, ἡ περιεχομένη γωνία <lb n="1799,001,,,,11;HOR,5,4"></lb> |
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13005 ὑπὸ τῆς ἀχθείσης καὶ τῆς ἐφεστώσης.</s> <lb n="1799,001,,,,11;HOR,5,5"></lb> |
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13006 <s>Ἐπιπέδου πρὸς ἐπίπεδον κλίσις ἐστὶν ἡ περιεχομένη <lb n="1799,001,,,,11;HOR,6,1"></lb> |
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13007 ὀξεῖα γωνία ὑπὸ τῶν πρὸς ὀρθὰς τῇ κοινῇ τομῇ ἀγομένων <lb n="1799,001,,,,11;HOR,6,2"></lb> |
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13008 πρὸς τῷ αὐτῷ σημείῳ ἐν ἑκατέρῳ τῶν ἐπιπέδων.</s> <s><pb></pb> <lb n="1799,001,,,,11;HOR,6,3"></lb> |
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13009 Ἐπίπεδον πρὸς ἐπίπεδον ὁμοίως κεκλίσθαι λέγεται <lb n="1799,001,,,,11;HOR,7,1"></lb> |
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13010 καὶ ἕτερον πρὸς ἕτερον, ὅταν αἱ εἰρημέναι τῶν κλίσεων γω-<lb n="1799,001,,,,11;HOR,7,2"></lb> |
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13011 νίαι ἴσαι ἀλλήλαις ὦσιν.</s> <lb n="1799,001,,,,11;HOR,7,3"></lb> |
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13012 <s>Παράλληλα ἐπίπεδά ἐστι τὰ ἀσύμπτωτα.</s> <lb n="1799,001,,,,11;HOR,8,1"></lb> |
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13013 <s>Ὅμοια στερεὰ σχήματά ἐστι τὰ ὑπὸ ὁμοίων ἐπιπέ-<lb n="1799,001,,,,11;HOR,9,1"></lb> |
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13014 δων περιεχόμενα ἴσων τὸ πλῆθος.</s> <lb n="1799,001,,,,11;HOR,9,2"></lb> |
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13015 <s>Ἴσα δὲ καὶ ὅμοια στερεὰ σχήματά ἐστι τὰ ὑπὸ ὁμοίων <lb n="1799,001,,,,11;HOR,10,1"></lb> |
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13016 ἐπιπέδων περιεχόμενα ἴσων τῷ πλήθει καὶ τῷ μεγέθει.</s> <lb n="1799,001,,,,11;HOR,10,2"></lb> |
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13017 <s>Στερεὰ γωνία ἐστὶν ἡ ὑπὸ πλειόνων ἢ δύο γραμ-<lb n="1799,001,,,,11;HOR,11,1"></lb> |
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13018 μῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐν τῇ αὐτῇ ἐπιφανείᾳ <lb n="1799,001,,,,11;HOR,11,2"></lb> |
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13019 οὐσῶν πρὸς πάσαις ταῖς γραμμαῖς κλίσις.</s> <s>Ἄλλως· στερεὰ <lb n="1799,001,,,,11;HOR,11,3"></lb> |
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13020 γωνία ἐστὶν ἡ ὑπὸ πλειόνων ἢ δύο γωνιῶν ἐπιπέδων περι-<lb n="1799,001,,,,11;HOR,11,4"></lb> |
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13021 εχομένη μὴ οὐσῶν ἐν τῷ αὐτῷ ἐπιπέδῳ πρὸς ἑνὶ σημείῳ <lb n="1799,001,,,,11;HOR,11,5"></lb> |
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13022 συνισταμένων.</s> <lb n="1799,001,,,,11;HOR,11,6"></lb> |
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13023 <s>Πυραμίς ἐστι σχῆμα στερεὸν ἐπιπέδοις περιεχόμε-<lb n="1799,001,,,,11;HOR,12,1"></lb> |
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13024 νον ἀπὸ ἑνὸς ἐπιπέδου πρὸς ἑνὶ σημείῳ συνεστώς.</s> <lb n="1799,001,,,,11;HOR,12,2"></lb> |
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13025 <s>Πρίσμα ἐστὶ σχῆμα στερεὸν ἐπιπέδοις περιεχόμε-<lb n="1799,001,,,,11;HOR,13,1"></lb> |
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13026 νον, ὧν δύο τὰ ἀπεναντίον ἴσα τε καὶ ὅμοιά ἐστι καὶ παράλ-<lb n="1799,001,,,,11;HOR,13,2"></lb> |
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13027 ληλα, τὰ δὲ λοιπὰ παραλληλόγραμμα.</s> <lb n="1799,001,,,,11;HOR,13,3"></lb> |
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13028 <s>Σφαῖρά ἐστιν, ὅταν ἡμικυκλίου μενούσης τῆς δια-<lb n="1799,001,,,,11;HOR,14,1"></lb> |
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13029 μέτρου περιενεχθὲν τὸ ἡμικύκλιον εἰς τὸ αὐτὸ πάλιν ἀπο-<lb n="1799,001,,,,11;HOR,14,2"></lb> |
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13030 κατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, τὸ περιληφθὲν σχῆμα.</s> <lb n="1799,001,,,,11;HOR,14,3"></lb> |
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13031 <s>Ἄξων δὲ τῆς σφαίρας ἐστὶν ἡ μένουσα εὐθεῖα, περὶ <lb n="1799,001,,,,11;HOR,15,1"></lb> |
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13032 ἣν τὸ ἡμικύκλιον στρέφεται.</s> <lb n="1799,001,,,,11;HOR,15,2"></lb> |
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13033 <s>Κέντρον δὲ τῆς σφαίρας ἐστὶ τὸ αὐτό, ὃ καὶ τοῦ <lb n="1799,001,,,,11;HOR,16,1"></lb> |
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13034 ἡμικυκλίου.</s> <s><pb></pb> <lb n="1799,001,,,,11;HOR,16,2"></lb> |
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13035 Διάμετρος δὲ τῆς σφαίρας ἐστὶν εὐθεῖά τις διὰ τοῦ <lb n="1799,001,,,,11;HOR,17,1"></lb> |
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13036 κέντρου ἠγμένη καὶ περατουμένη ἐφ' ἑκάτερα τὰ μέρη <lb n="1799,001,,,,11;HOR,17,2"></lb> |
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13037 ὑπὸ τῆς ἐπιφανείας τῆς σφαίρας.</s> <lb n="1799,001,,,,11;HOR,17,3"></lb> |
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13038 <s>Κῶνός ἐστιν, ὅταν ὀρθογωνίου τριγώνου μενούσης <lb n="1799,001,,,,11;HOR,18,1"></lb> |
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13039 μιᾶς πλευρᾶς τῶν περὶ τὴν ὀρθὴν γωνίαν περιενεχθὲν τὸ <lb n="1799,001,,,,11;HOR,18,2"></lb> |
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13040 τρίγωνον εἰς τὸ αὐτὸ πάλιν ἀποκατασταθῇ, ὅθεν ἤρξατο <lb n="1799,001,,,,11;HOR,18,3"></lb> |
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13041 φέρεσθαι, τὸ περιληφθὲν σχῆμα.</s> <s>κἂν μὲν ἡ μένουσα εὐθεῖα <lb n="1799,001,,,,11;HOR,18,4"></lb> |
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13042 ἴση ᾖ τῇ λοιπῇ [τῇ] περὶ τὴν ὀρθὴν περιφερομένῃ, ὀρθο-<lb n="1799,001,,,,11;HOR,18,5"></lb> |
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13043 γώνιος ἔσται ὁ κῶνος, ἐὰν δὲ ἐλάττων, ἀμβλυγώνιος, ἐὰν <lb n="1799,001,,,,11;HOR,18,6"></lb> |
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13044 δὲ μείζων, ὀξυγώνιος.</s> <lb n="1799,001,,,,11;HOR,18,7"></lb> |
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13045 <s>Ἄξων δὲ τοῦ κώνου ἐστὶν ἡ μένουσα εὐθεῖα, περὶ <lb n="1799,001,,,,11;HOR,19,1"></lb> |
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13046 ἣν τὸ τρίγωνον στρέφεται.</s> <lb n="1799,001,,,,11;HOR,19,2"></lb> |
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13047 <s>Βάσις δὲ ὁ κύκλος ὁ ὑπὸ τῆς περιφερομένης εὐθείας <lb n="1799,001,,,,11;HOR,20,1"></lb> |
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13048 γραφόμενος.</s> <lb n="1799,001,,,,11;HOR,20,2"></lb> |
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13049 <s>Κύλινδρός ἐστιν, ὅταν ὀρθογωνίου παραλληλο-<lb n="1799,001,,,,11;HOR,21,1"></lb> |
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13050 γράμμου μενούσης μιᾶς πλευρᾶς τῶν περὶ τὴν ὀρθὴν <lb n="1799,001,,,,11;HOR,21,2"></lb> |
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13051 γωνίαν περιενεχθὲν τὸ παραλληλόγραμμον εἰς τὸ αὐτὸ <lb n="1799,001,,,,11;HOR,21,3"></lb> |
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13052 πάλιν ἀποκατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, τὸ περι-<lb n="1799,001,,,,11;HOR,21,4"></lb> |
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13053 ληφθὲν σχῆμα.</s> <lb n="1799,001,,,,11;HOR,21,5"></lb> |
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13054 <s>Ἄξων δὲ τοῦ κυλίνδρου ἐστὶν ἡ μένουσα εὐθεῖα, <lb n="1799,001,,,,11;HOR,22,1"></lb> |
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13055 περὶ ἣν τὸ παραλληλόγραμμον στρέφεται.</s> <lb n="1799,001,,,,11;HOR,22,2"></lb> |
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13056 <s>Βάσεις δὲ οἱ κύκλοι οἱ ὑπὸ τῶν ἀπεναντίον περι-<lb n="1799,001,,,,11;HOR,23,1"></lb> |
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13057 αγομένων δύο πλευρῶν γραφόμενοι.</s> <lb n="1799,001,,,,11;HOR,23,2"></lb> |
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13058 <s>Ὅμοιοι κῶνοι καὶ κύλινδροί εἰσιν, ὧν οἵ τε ἄξονες <lb n="1799,001,,,,11;HOR,24,1"></lb> |
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13059 καὶ αἱ διάμετροι τῶν βάσεων ἀνάλογόν εἰσιν.</s> <lb n="1799,001,,,,11;HOR,24,2"></lb> |
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13060 <s>Κύβος ἐστὶ σχῆμα στερεὸν ὑπὸ ἓξ τετραγώνων <lb n="1799,001,,,,11;HOR,25,1"></lb> |
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13061 ἴσων περιεχόμενον.</s> <s><pb></pb> <lb n="1799,001,,,,11;HOR,25,2"></lb> |
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13062 Ὀκτάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ ὀκτὼ τριγώ-<lb n="1799,001,,,,11;HOR,26,1"></lb> |
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13063 νων ἴσων καὶ ἰσοπλεύρων περιεχόμενον.</s> <lb n="1799,001,,,,11;HOR,26,2"></lb> |
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13064 <s>Εἰκοσάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ εἴκοσι τρι-<lb n="1799,001,,,,11;HOR,27,1"></lb> |
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13065 γώνων ἴσων καὶ ἰσοπλεύρων περιεχόμενον.</s> <lb n="1799,001,,,,11;HOR,27,2"></lb> |
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13066 <s>Δωδεκάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ δώδεκα <lb n="1799,001,,,,11;HOR,28,1"></lb> |
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13067 πενταγώνων ἴσων καὶ ἰσοπλεύρων καὶ ἰσογωνίων περιεχό-<lb n="1799,001,,,,11;HOR,28,2"></lb> |
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13068 μενον.</s> <lb n="1799,001,,,,11;HOR,28,3"></lb> |
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13069 <s></s></p> |
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13070 </chap> |
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13071 <chap><p type="main"><s>Εὐθείας γραμμῆς μέρος μέν τι οὐκ ἔστιν ἐν τῷ ὑπο-<lb n="1799,001,,,,11,1,1"></lb> |
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13072 |
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13073 κειμένῳ, ἐπιπέδῳ, μέρος δέ τι ἐν μετεωροτέρῳ.</s> <lb n="1799,001,,,,11,1,2"></lb> |
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13074 <s>Εἰ γὰρ δυνατόν, εὐθείας γραμμῆς τῆς <lb n="1799,001,,,,11,1,3"></lb> |
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13075 ΑΒΓ μέρος μέν τι τὸ ΑΒ ἔστω ἐν τῷ ὑπο-<lb n="1799,001,,,,11,1,4"></lb> |
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13076 κειμένῳ ἐπιπέδῳ, μέρος δέ τι τὸ ΒΓ ἐν <lb n="1799,001,,,,11,1,5"></lb> |
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13077 μετεωροτέρῳ.</s> <lb n="1799,001,,,,11,1,6"></lb> |
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13078 <s>Ἔσται δή τις τῇ ΑΒ συνεχὴς εὐθεῖα ἐπ' <lb n="1799,001,,,,11,1,7"></lb> |
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13079 εὐθείας ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ.</s> <s>ἔστω <lb n="1799,001,,,,11,1,8"></lb> |
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13080 ἡ ΒΔ· δύο ἄρα εὐθειῶν τῶν ΑΒΓ, ΑΒΔ <lb n="1799,001,,,,11,1,9"></lb> |
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13081 κοινὸν τμῆμά ἐστιν ἡ ΑΒ· ὅπερ ἐστὶν ἀδύ-<lb n="1799,001,,,,11,1,10"></lb> |
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13082 νατον, ἐπειδήπερ ἐὰν κέντρῳ τῷ Β καὶ διαστήματι τῷ ΑΒ <lb n="1799,001,,,,11,1,11"></lb> |
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13083 κύκλον γράψωμεν, αἱ διάμετροι ἀνίσους ἀπολήψονται τοῦ <lb n="1799,001,,,,11,1,12"></lb> |
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13084 κύκλου περιφερείας.</s> <s><pb></pb> <lb n="1799,001,,,,11,1,13"></lb> |
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13085 Εὐθείας ἄρα γραμμῆς μέρος μέν τι οὐκ ἔστιν ἐν τῷ <lb n="1799,001,,,,11,1,14"></lb> |
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13086 ὑποκειμένῳ ἐπιπέδῳ, τὸ δὲ ἐν μετεωροτέρῳ· ὅπερ ἔδει <lb n="1799,001,,,,11,1,15"></lb> |
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13087 δεῖξαι.</s> <lb n="1799,001,,,,11,1,16"></lb> |
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13088 <s>Ἐὰν δύο εὐθεῖαι τέμνωσιν ἀλλήλας, ἐν ἑνί εἰσιν ἐπιπέδῳ, <lb n="1799,001,,,,11,2,1"></lb> |
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13089 καὶ πᾶν τρίγωνον ἐν ἑνί ἐστιν ἐπιπέδῳ.</s> <lb n="1799,001,,,,11,2,2"></lb> |
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13090 <s>Δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΓΔ τεμνέτωσαν ἀλλήλας κατὰ <lb n="1799,001,,,,11,2,3"></lb> |
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13091 τὸ Ε σημεῖον· λέγω, ὅτι αἱ ΑΒ, ΓΔ ἐν ἑνί εἰσιν ἐπιπέδῳ, <lb n="1799,001,,,,11,2,4"></lb> |
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13092 καὶ πᾶν τρίγωνον ἐν ἑνί ἐστιν ἐπιπέδῳ.</s> <lb n="1799,001,,,,11,2,5"></lb> |
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13093 <s>Εἰλήφθω γὰρ ἐπὶ τῶν ΕΓ, ΕΒ τυχόντα σημεῖα τὰ Ζ, <lb n="1799,001,,,,11,2,6"></lb> |
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13094 Η, καὶ ἐπεζεύχθωσαν αἱ ΓΒ, ΖΗ, καὶ διήχθωσαν αἱ <lb n="1799,001,,,,11,2,7"></lb> |
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13095 ΖΘ, ΗΚ· λέγω πρῶτον, ὅτι τὸ ΕΓΒ <lb n="1799,001,,,,11,2,8"></lb> |
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13096 τρίγωνον ἐν ἑνί ἐστιν ἐπιπέδῳ.</s> <s>εἰ γάρ <lb n="1799,001,,,,11,2,9"></lb> |
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13097 ἐστι τοῦ ΕΓΒ τριγώνου μέρος ἤτοι τὸ <lb n="1799,001,,,,11,2,10"></lb> |
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13098 ΖΘΓ ἢ τὸ ΗΒΚ ἐν τῷ ὑποκειμένῳ [ἐπι-<lb n="1799,001,,,,11,2,11"></lb> |
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13099 πέδῳ], τὸ δὲ λοιπὸν ἐν ἄλλῳ, ἔσται καὶ <lb n="1799,001,,,,11,2,12"></lb> |
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13100 μιᾶς τῶν ΕΓ, ΕΒ εὐθειῶν μέρος μέν τι <lb n="1799,001,,,,11,2,13"></lb> |
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13101 ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ, τὸ δὲ ἐν ἄλλῳ.</s> <lb n="1799,001,,,,11,2,14"></lb> |
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13102 <s>εἰ δὲ τοῦ ΕΓΒ τριγώνου τὸ ΖΓΒΗ μέρος ᾖ ἐν τῷ ὑπο-<lb n="1799,001,,,,11,2,15"></lb> |
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13103 κειμένῳ ἐπιπέδῳ, τὸ δὲ λοιπὸν ἐν ἄλλῳ, ἔσται καὶ ἀμφοτέ-<lb n="1799,001,,,,11,2,16"></lb> |
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13104 ρων τῶν ΕΓ, ΕΒ εὐθειῶν μέρος μέν τι ἐν τῷ ὑποκειμένῳ <lb n="1799,001,,,,11,2,17"></lb> |
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13105 ἐπιπέδῳ, τὸ δὲ ἐν ἄλλῳ· ὅπερ ἄτοπον ἐδείχθη.</s> <s>τὸ ἄρα <lb n="1799,001,,,,11,2,18"></lb> |
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13106 ΕΓΒ τρίγωνον ἐν ἑνί ἐστιν ἐπιπέδῳ.</s> <s>ἐν ᾧ δέ ἐστι τὸ ΕΓΒ <lb n="1799,001,,,,11,2,19"></lb> |
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13107 τρίγωνον, ἐν τούτῳ καὶ ἑκατέρα τῶν ΕΓ, ΕΒ, ἐν ᾧ δὲ <lb n="1799,001,,,,11,2,20"></lb> |
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13108 ἑκατέρα τῶν ΕΓ, ΕΒ, ἐν τούτῳ καὶ αἱ ΑΒ, ΓΔ.</s> <s>αἱ ΑΒ, <pb></pb> <lb n="1799,001,,,,11,2,21"></lb> |
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13109 ΓΔ ἄρα εὐθεῖαι ἐν ἑνί εἰσιν ἐπιπέδῳ, καὶ πᾶν τρίγωνον ἐν <lb n="1799,001,,,,11,2,22"></lb> |
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13110 ἑνί ἐστιν ἐπιπέδῳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,2,23"></lb> |
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13111 <s>Ἐὰν δύο ἐπίπεδα τέμνῃ ἄλληλα, ἡ κοινὴ αὐτῶν τομὴ <lb n="1799,001,,,,11,3,1"></lb> |
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13112 εὐθεῖά ἐστιν.</s> <lb n="1799,001,,,,11,3,2"></lb> |
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13113 <s>Δύο γὰρ ἐπίπεδα τὰ ΑΒ, ΒΓ τεμνέτω ἄλληλα, κοινὴ δὲ <lb n="1799,001,,,,11,3,3"></lb> |
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13114 αὐτῶν τομὴ ἔστω ἡ ΔΒ γραμμή· λέγω, ὅτι ἡ ΔΒ γραμμὴ <lb n="1799,001,,,,11,3,4"></lb> |
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13115 εὐθεῖά ἐστιν.</s> <lb n="1799,001,,,,11,3,5"></lb> |
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13116 <s>Εἰ γὰρ μή, ἐπεζεύχθω ἀπὸ τοῦ Δ ἐπὶ τὸ Β ἐν μὲν τῷ <lb n="1799,001,,,,11,3,6"></lb> |
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13117 ΑΒ ἐπιπέδῳ εὐθεῖα ἡ ΔΕΒ, ἐν δὲ τῷ ΒΓ ἐπιπέδῳ εὐθεῖα <lb n="1799,001,,,,11,3,7"></lb> |
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13118 ἡ ΔΖΒ.</s> <s>ἔσται δὴ δύο εὐθειῶν τῶν ΔΕΒ, <lb n="1799,001,,,,11,3,8"></lb> |
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13119 ΔΖΒ τὰ αὐτὰ πέρατα, καὶ περιέξουσι <lb n="1799,001,,,,11,3,9"></lb> |
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13120 δηλαδὴ χωρίον· ὅπερ ἄτοπον.</s> <s>οὐκ ἄρα αἱ <lb n="1799,001,,,,11,3,10"></lb> |
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13121 ΔΕΒ, ΔΖΒ εὐθεῖαί εἰσιν.</s> <s>ὁμοίως δὴ δείξο-<lb n="1799,001,,,,11,3,11"></lb> |
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13122 μεν, ὅτι οὐδὲ ἄλλη τις ἀπὸ τοῦ Δ ἐπὶ τὸ <lb n="1799,001,,,,11,3,12"></lb> |
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13123 Β ἐπιζευγνυμένη εὐθεῖα ἔσται πλὴν τῆς <lb n="1799,001,,,,11,3,13"></lb> |
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13124 ΔΒ κοινῆς τομῆς τῶν ΑΒ, ΒΓ ἐπιπέδων.</s> <lb n="1799,001,,,,11,3,14"></lb> |
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13125 <s>Ἐὰν ἄρα δύο ἐπίπεδα τέμνῃ ἄλληλα, ἡ κοινὴ αὐτῶν <lb n="1799,001,,,,11,3,15"></lb> |
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13126 τομὴ εὐθεῖά ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,3,16"></lb> |
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13127 <s>Ἐὰν εὐθεῖα δύο εὐθείαις τεμνούσαις ἀλλήλας πρὸς <lb n="1799,001,,,,11,4,1"></lb> |
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13128 ὀρθὰς ἐπὶ τῆς κοινῆς τομῆς ἐπισταθῇ, καὶ τῷ δι' αὐτῶν <lb n="1799,001,,,,11,4,2"></lb> |
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13129 ἐπιπέδῳ πρὸς ὀρθὰς ἔσται.</s> <s><pb></pb> <lb n="1799,001,,,,11,4,3"></lb> |
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13130 Εὐθεῖα γάρ τις ἡ ΕΖ δύο εὐθείαις ταῖς ΑΒ, ΓΔ <lb n="1799,001,,,,11,4,4"></lb> |
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13131 τεμνούσαις ἀλλήλας κατὰ τὸ Ε σημεῖον ἀπὸ τοῦ Ε πρὸς <lb n="1799,001,,,,11,4,5"></lb> |
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13132 ὀρθὰς ἐφεστάτω· λέγω, ὅτι ἡ ΕΖ καὶ τῷ διὰ τῶν ΑΒ, <lb n="1799,001,,,,11,4,6"></lb> |
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13133 ΓΔ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <lb n="1799,001,,,,11,4,7"></lb> |
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13134 <s>Ἀπειλήφθωσαν γὰρ αἱ ΑΕ, ΕΒ, ΓΕ, ΕΔ ἴσαι ἀλλή-<lb n="1799,001,,,,11,4,8"></lb> |
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13135 λαις, καὶ διήχθω τις διὰ τοῦ Ε, ὡς ἔτυχεν, ἡ ΗΕΘ, καὶ <lb n="1799,001,,,,11,4,9"></lb> |
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13136 ἐπεζεύχθωσαν αἱ ΑΔ, ΓΒ, καὶ <lb n="1799,001,,,,11,4,10"></lb> |
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13137 ἔτι ἀπὸ τυχόντος τοῦ Ζ ἐπεζεύ-<lb n="1799,001,,,,11,4,11"></lb> |
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13138 χθωσαν αἱ ΖΑ, ΖΗ, ΖΔ, ΖΓ, <lb n="1799,001,,,,11,4,12"></lb> |
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13139 ΖΘ, ΖΒ.</s> <s>καὶ ἐπεὶ δύο αἱ ΑΕ, <lb n="1799,001,,,,11,4,13"></lb> |
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13140 ΕΔ δυσὶ ταῖς ΓΕ, ΕΒ ἴσαι εἰσὶ καὶ <lb n="1799,001,,,,11,4,14"></lb> |
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13141 γωνίας ἴσας περιέχουσιν, βάσις ἄρα <lb n="1799,001,,,,11,4,15"></lb> |
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13142 ἡ ΑΔ βάσει τῇ ΓΒ ἴση ἐστίν, καὶ <lb n="1799,001,,,,11,4,16"></lb> |
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13143 τὸ ΑΕΔ τρίγωνον τῷ ΓΕΒ τριγώνῳ ἴσον ἔσται· ὥστε καὶ <lb n="1799,001,,,,11,4,17"></lb> |
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13144 γωνία ἡ ὑπὸ ΔΑΕ γωνίᾳ τῇ ὑπὸ ΕΒΓ ἴση [ἐστίν].</s> <s>ἔστι δὲ <lb n="1799,001,,,,11,4,18"></lb> |
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13145 καὶ ἡ ὑπὸ ΑΕΗ γωνία τῇ ὑπὸ ΒΕΘ ἴση.</s> <s>δύο δὴ τρίγωνά <lb n="1799,001,,,,11,4,19"></lb> |
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13146 ἐστι τὰ ΑΗΕ, ΒΕΘ τὰς δύο γωνίας δυσὶ γωνίαις ἴσας <lb n="1799,001,,,,11,4,20"></lb> |
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13147 ἔχοντα ἑκατέραν ἑκατέρᾳ καὶ μίαν πλευρὰν μιᾳ̂ πλευρᾷ <lb n="1799,001,,,,11,4,21"></lb> |
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13148 ἴσην τὴν πρὸς ταῖς ἴσαις γωνίαις τὴν ΑΕ τῇ ΕΒ· καὶ τὰς <lb n="1799,001,,,,11,4,22"></lb> |
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13149 λοιπὰς ἄρα πλευρὰς ταῖς λοιπαῖς πλευραῖς ἴσας ἕξουσιν.</s> <lb n="1799,001,,,,11,4,23"></lb> |
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13150 <s>ἴση ἄρα ἡ μὲν ΗΕ τῇ ΕΘ, ἡ δὲ ΑΗ τῇ ΒΘ.</s> <s>καὶ ἐπεὶ ἴση <lb n="1799,001,,,,11,4,24"></lb> |
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13151 ἐστὶν ἡ ΑΕ τῇ ΕΒ, κοινὴ δὲ καὶ πρὸς ὀρθὰς ἡ ΖΕ, βάσις <lb n="1799,001,,,,11,4,25"></lb> |
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13152 ἄρα ἡ ΖΑ βάσει τῇ ΖΒ ἐστιν ἴση.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΖΓ <lb n="1799,001,,,,11,4,26"></lb> |
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13153 τῇ ΖΔ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΔ τῇ ΓΒ, ἔστι δὲ <lb n="1799,001,,,,11,4,27"></lb> |
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13154 καὶ ἡ ΖΑ τῇ ΖΒ ἴση, δύο δὴ αἱ ΖΑ, ΑΔ δυσὶ ταῖς ΖΒ, ΒΓ <pb></pb> <lb n="1799,001,,,,11,4,28"></lb> |
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13155 ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ βάσις ἡ ΖΔ βάσει τῇ ΖΓ <lb n="1799,001,,,,11,4,29"></lb> |
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13156 ἐδείχθη ἴση· καὶ γωνία ἄρα ἡ ὑπὸ ΖΑΔ γωνίᾳ τῇ ὑπὸ <lb n="1799,001,,,,11,4,30"></lb> |
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13157 ΖΒΓ ἴση ἐστίν.</s> <s>καὶ ἐπεὶ πάλιν ἐδείχθη ἡ ΑΗ τῇ ΒΘ ἴση, <lb n="1799,001,,,,11,4,31"></lb> |
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13158 ἀλλὰ μὴν καὶ ἡ ΖΑ τῇ ΖΒ ἴση, δύο δὴ αἱ ΖΑ, ΑΗ δυσὶ <lb n="1799,001,,,,11,4,32"></lb> |
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13159 ταῖς ΖΒ, ΒΘ ἴσαι εἰσίν.</s> <s>καὶ γωνία ἡ ὑπὸ ΖΑΗ ἐδεί-<lb n="1799,001,,,,11,4,33"></lb> |
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13160 χθη ἴση τῇ ὑπὸ ΖΒΘ· βάσις ἄρα ἡ ΖΗ βάσει τῇ ΖΘ <lb n="1799,001,,,,11,4,34"></lb> |
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13161 ἐστιν ἴση.</s> <s>καὶ ἐπεὶ πάλιν ἴση ἐδείχθη ἡ ΗΕ τῇ ΕΘ, κοινὴ <lb n="1799,001,,,,11,4,35"></lb> |
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13162 δὲ ἡ ΕΖ, δύο δὴ αἱ ΗΕ, ΕΖ δυσὶ ταῖς ΘΕ, ΕΖ ἴσαι <lb n="1799,001,,,,11,4,36"></lb> |
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13163 εἰσίν· καὶ βάσις ἡ ΖΗ βάσει τῇ ΖΘ ἴση· γωνία ἄρα ἡ ὑπὸ <lb n="1799,001,,,,11,4,37"></lb> |
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13164 ΗΕΖ γωνίᾳ τῇ ὑπὸ ΘΕΖ ἴση ἐστίν.</s> <s>ὀρθὴ ἄρα ἑκατέρα <lb n="1799,001,,,,11,4,38"></lb> |
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13165 τῶν ὑπὸ ΗΕΖ, ΘΕΖ γωνιῶν.</s> <s>ἡ ΖΕ ἄρα πρὸς τὴν ΗΘ <lb n="1799,001,,,,11,4,39"></lb> |
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13166 τυχόντως διὰ τοῦ Ε ἀχθεῖσαν ὀρθή ἐστιν.</s> <s>ὁμοίως δὴ δεί-<lb n="1799,001,,,,11,4,40"></lb> |
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13167 ξομεν, ὅτι ἡ ΖΕ καὶ πρὸς πάσας τὰς ἁπτομένας αὐτῆς <lb n="1799,001,,,,11,4,41"></lb> |
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13168 εὐθείας καὶ οὔσας ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ ὀρθὰς ποιή-<lb n="1799,001,,,,11,4,42"></lb> |
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13169 σει γωνίας.</s> <s>εὐθεῖα δὲ πρὸς ἐπίπεδον ὀρθή ἐστιν, ὅταν πρὸς <lb n="1799,001,,,,11,4,43"></lb> |
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13170 πάσας τὰς ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ αὐτῷ <lb n="1799,001,,,,11,4,44"></lb> |
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13171 ἐπιπέδῳ ὀρθὰς ποιῇ γωνίας· ἡ ΖΕ ἄρα τῷ ὑποκειμένῳ <lb n="1799,001,,,,11,4,45"></lb> |
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13172 ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <s>τὸ δὲ ὑποκείμενον ἐπίπεδόν <lb n="1799,001,,,,11,4,46"></lb> |
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13173 ἐστι τὸ διὰ τῶν ΑΒ, ΓΔ εὐθειῶν.</s> <s>ἡ ΖΕ ἄρα πρὸς ὀρθάς <lb n="1799,001,,,,11,4,47"></lb> |
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13174 ἐστι τῷ διὰ τῶν ΑΒ, ΓΔ ἐπιπέδῳ.</s> <lb n="1799,001,,,,11,4,48"></lb> |
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13175 <s>Ἐὰν ἄρα εὐθεῖα δύο εὐθείαις τεμνούσαις ἀλλήλας πρὸς <lb n="1799,001,,,,11,4,49"></lb> |
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13176 ὀρθὰς ἐπὶ τῆς κοινῆς τομῆς ἐπισταθῇ, καὶ τῷ δι' αὐτῶν <lb n="1799,001,,,,11,4,50"></lb> |
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13177 ἐπιπέδῳ πρὸς ὀρθὰς ἔσται· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,4,51"></lb> |
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13178 Ἐὰν εὐθεῖα τρισὶν εὐθείαις ἁπτομέναις ἀλλήλων πρὸς <lb n="1799,001,,,,11,5,1"></lb> |
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13179 ὀρθὰς ἐπὶ τῆς κοινῆς τομῆς ἐπισταθῇ, αἱ τρεῖς εὐθεῖαι ἐν <lb n="1799,001,,,,11,5,2"></lb> |
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13180 ἑνί εἰσιν ἐπιπέδῳ.</s> <lb n="1799,001,,,,11,5,3"></lb> |
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13181 <s>Εὐθεῖα γάρ τις ἡ ΑΒ τρισὶν εὐθείαις ταῖς ΒΓ, ΒΔ, <lb n="1799,001,,,,11,5,4"></lb> |
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13182 ΒΕ πρὸς ὀρθὰς ἐπὶ τῆς κατὰ τὸ Β ἁφῆς ἐφεστάτω· λέγω, <lb n="1799,001,,,,11,5,5"></lb> |
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13183 ὅτι αἱ ΒΓ, ΒΔ, ΒΕ ἐν ἑνί εἰσιν ἐπιπέδῳ.</s> <lb n="1799,001,,,,11,5,6"></lb> |
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13184 <s>Μὴ γάρ, ἀλλ' εἰ δυνατόν, ἔστωσαν αἱ μὲν ΒΔ, ΒΕ ἐν <lb n="1799,001,,,,11,5,7"></lb> |
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13185 τῷ ὑποκειμένῳ ἐπιπέδῳ, ἡ δὲ ΒΓ ἐν μετεωροτέρῳ, καὶ <lb n="1799,001,,,,11,5,8"></lb> |
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13186 ἐκβεβλήσθω τὸ διὰ τῶν ΑΒ, ΒΓ <lb n="1799,001,,,,11,5,9"></lb> |
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13187 ἐπίπεδον· κοινὴν δὴ τομὴν ποιήσει <lb n="1799,001,,,,11,5,10"></lb> |
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13188 ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ εὐθεῖαν.</s> <lb n="1799,001,,,,11,5,11"></lb> |
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13189 <s>ποιείτω τὴν ΒΖ.</s> <s>ἐν ἑνὶ ἄρα εἰσὶν <lb n="1799,001,,,,11,5,12"></lb> |
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13190 ἐπιπέδῳ τῷ διηγμένῳ διὰ τῶν ΑΒ, <lb n="1799,001,,,,11,5,13"></lb> |
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13191 ΒΓ αἱ τρεῖς εὐθεῖαι αἱ ΑΒ, ΒΓ, <lb n="1799,001,,,,11,5,14"></lb> |
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13192 ΒΖ.</s> <s>καὶ ἐπεὶ ἡ ΑΒ ὀρθή ἐστι πρὸς <lb n="1799,001,,,,11,5,15"></lb> |
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13193 ἑκατέραν τῶν ΒΔ, ΒΕ, καὶ τῷ διὰ <lb n="1799,001,,,,11,5,16"></lb> |
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13194 τῶν ΒΔ, ΒΕ ἄρα ἐπιπέδῳ ὀρθή ἐστιν ἡ ΑΒ.</s> <s>τὸ δὲ διὰ τῶν <lb n="1799,001,,,,11,5,17"></lb> |
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13195 ΒΔ, ΒΕ ἐπίπεδον τὸ ὑποκείμενόν ἐστιν· ἡ ΑΒ ἄρα ὀρθή <lb n="1799,001,,,,11,5,18"></lb> |
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13196 ἐστι πρὸς τὸ ὑποκείμενον ἐπίπεδον.</s> <s>ὥστε καὶ πρὸς πάσας <lb n="1799,001,,,,11,5,19"></lb> |
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13197 τὰς ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ ὑποκειμένῳ <lb n="1799,001,,,,11,5,20"></lb> |
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13198 ἐπιπέδῳ ὀρθὰς ποιήσει γωνίας ἡ ΑΒ.</s> <s>ἅπτεται δὲ αὐτῆς ἡ <lb n="1799,001,,,,11,5,21"></lb> |
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13199 ΒΖ οὖσα ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ· ἡ ἄρα ὑπὸ ΑΒΖ <lb n="1799,001,,,,11,5,22"></lb> |
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13200 γωνία ὀρθή ἐστιν.</s> <s>ὑπόκειται δὲ καὶ ἡ ὑπὸ ΑΒΓ ὀρθή· <lb n="1799,001,,,,11,5,23"></lb> |
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13201 ἴση ἄρα ἡ ὑπὸ ΑΒΖ γωνία τῇ ὑπὸ ΑΒΓ.</s> <s>καί εἰσιν ἐν ἑνὶ <lb n="1799,001,,,,11,5,24"></lb> |
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13202 ἐπιπέδῳ· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ἡ ΒΓ εὐθεῖα ἐν <pb></pb> <lb n="1799,001,,,,11,5,25"></lb> |
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13203 μετεωροτέρῳ ἐστὶν ἐπιπέδῳ· αἱ τρεῖς ἄρα εὐθεῖαι αἱ ΒΓ, <lb n="1799,001,,,,11,5,26"></lb> |
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13204 ΒΔ, ΒΕ ἐν ἑνί εἰσιν ἐπιπέδῳ.</s> <lb n="1799,001,,,,11,5,27"></lb> |
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13205 <s>Ἐὰν ἄρα εὐθεῖα τρισὶν εὐθείαις ἁπτομέναις ἀλλή-<lb n="1799,001,,,,11,5,28"></lb> |
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13206 λων ἐπὶ τῆς ἁφῆς πρὸς ὀρθὰς ἐπισταθῇ, αἱ τρεῖς εὐθεῖαι <lb n="1799,001,,,,11,5,29"></lb> |
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13207 ἐν ἑνί εἰσιν ἐπιπέδῳ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,5,30"></lb> |
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13208 <s>Ἐὰν δύο εὐθεῖαι τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθὰς ὦσιν, <lb n="1799,001,,,,11,6,1"></lb> |
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13209 παράλληλοι ἔσονται αἱ εὐθεῖαι.</s> <lb n="1799,001,,,,11,6,2"></lb> |
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13210 <s>Δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΓΔ τῷ ὑποκειμένῳ ἐπιπέδῳ <lb n="1799,001,,,,11,6,3"></lb> |
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13211 πρὸς ὀρθὰς ἔστωσαν· λέγω, ὅτι παράλληλός ἐστιν ἡ ΑΒ <lb n="1799,001,,,,11,6,4"></lb> |
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13212 τῇ ΓΔ.</s> <lb n="1799,001,,,,11,6,5"></lb> |
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13213 <s>Συμβαλλέτωσαν γὰρ τῷ ὑποκειμένῳ ἐπιπέδῳ κατὰ τὰ <lb n="1799,001,,,,11,6,6"></lb> |
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13214 Β, Δ σημεῖα, καὶ ἐπεζεύχθω ἡ ΒΔ εὐθεῖα, καὶ ἤχθω τῇ <lb n="1799,001,,,,11,6,7"></lb> |
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13215 ΒΔ πρὸς ὀρθὰς ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ ἡ ΔΕ, καὶ <lb n="1799,001,,,,11,6,8"></lb> |
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13216 κείσθω τῇ ΑΒ ἴση ἡ ΔΕ, καὶ ἐπεζεύ-<lb n="1799,001,,,,11,6,9"></lb> |
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13217 χθωσαν αἱ ΒΕ, ΑΕ, ΑΔ.</s> <lb n="1799,001,,,,11,6,10"></lb> |
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13218 <s>Καὶ ἐπεὶ ἡ ΑΒ ὀρθή ἐστι πρὸς τὸ <lb n="1799,001,,,,11,6,11"></lb> |
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13219 ὑποκείμενον ἐπίπεδον, καὶ πρὸς πάσας <lb n="1799,001,,,,11,6,12"></lb> |
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13220 [ἄρα] τὰς ἁπτομένας αὐτῆς εὐθείας <lb n="1799,001,,,,11,6,13"></lb> |
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13221 καὶ οὔσας ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ <lb n="1799,001,,,,11,6,14"></lb> |
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13222 ὀρθὰς ποιήσει γωνίας.</s> <s>ἅπτεται δὲ τῆς <lb n="1799,001,,,,11,6,15"></lb> |
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13223 ΑΒ ἑκατέρα τῶν ΒΔ, ΒΕ οὖσα ἐν <lb n="1799,001,,,,11,6,16"></lb> |
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13224 τῷ ὑποκειμένῳ ἐπιπέδῳ· ὀρθὴ ἄρα ἐστὶν ἑκατέρα τῶν ὑπὸ <lb n="1799,001,,,,11,6,17"></lb> |
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13225 ΑΒΔ, ΑΒΕ γωνιῶν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἑκατέρα τῶν <lb n="1799,001,,,,11,6,18"></lb> |
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13226 ὑπὸ ΓΔΒ, ΓΔΕ ὀρθή ἐστιν.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒ τῇ <lb n="1799,001,,,,11,6,19"></lb> |
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13227 ΔΕ, κοινὴ δὲ ἡ ΒΔ, δύο δὴ αἱ ΑΒ, ΒΔ δυσὶ ταῖς ΕΔ, ΔΒ <pb></pb> <lb n="1799,001,,,,11,6,20"></lb> |
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13228 ἴσαι εἰσίν· καὶ γωνίας ὀρθὰς περιέχουσιν· βάσις ἄρα ἡ ΑΔ <lb n="1799,001,,,,11,6,21"></lb> |
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13229 βάσει τῇ ΒΕ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒ τῇ ΔΕ, <lb n="1799,001,,,,11,6,22"></lb> |
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13230 ἀλλὰ καὶ ἡ ΑΔ τῇ ΒΕ, δύο δὴ αἱ ΑΒ, ΒΕ δυσὶ ταῖς ΕΔ, <lb n="1799,001,,,,11,6,23"></lb> |
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13231 ΔΑ ἴσαι εἰσίν· καὶ βάσις αὐτῶν κοινὴ ἡ ΑΕ· γωνία ἄρα <lb n="1799,001,,,,11,6,24"></lb> |
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13232 ἡ ὑπὸ ΑΒΕ γωνίᾳ τῇ ὑπὸ ΕΔΑ ἐστιν ἴση.</s> <s>ὀρθὴ δὲ ἡ ὑπὸ <lb n="1799,001,,,,11,6,25"></lb> |
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13233 ΑΒΕ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΕΔΑ· ἡ ΕΔ ἄρα πρὸς τὴν ΔΑ <lb n="1799,001,,,,11,6,26"></lb> |
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13234 ὀρθή ἐστιν.</s> <s>ἔστι δὲ καὶ πρὸς ἑκατέραν τῶν ΒΔ, ΔΓ ὀρθή.</s> <lb n="1799,001,,,,11,6,27"></lb> |
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13235 <s>ἡ ΕΔ ἄρα τρισὶν εὐθείαις ταῖς ΒΔ, ΔΑ, ΔΓ πρὸς ὀρθὰς <lb n="1799,001,,,,11,6,28"></lb> |
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13236 ἐπὶ τῆς ἁφῆς ἐφέστηκεν· αἱ τρεῖς ἄρα εὐθεῖαι αἱ ΒΔ, ΔΑ, <lb n="1799,001,,,,11,6,29"></lb> |
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13237 ΔΓ ἐν ἑνί εἰσιν ἐπιπέδῳ.</s> <s>ἐν ᾧ δὲ αἱ ΔΒ, ΔΑ, ἐν τούτῳ καὶ <lb n="1799,001,,,,11,6,30"></lb> |
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13238 ἡ ΑΒ· πᾶν γὰρ τρίγωνον ἐν ἑνί ἐστιν ἐπιπέδῳ· αἱ ἄρα ΑΒ, <lb n="1799,001,,,,11,6,31"></lb> |
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13239 ΒΔ, ΔΓ εὐθεῖαι ἐν ἑνί εἰσιν ἐπιπέδῳ.</s> <s>καί ἐστιν ὀρθὴ ἑκα-<lb n="1799,001,,,,11,6,32"></lb> |
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13240 τέρα τῶν ὑπὸ ΑΒΔ, ΒΔΓ γωνιῶν· παράλληλος ἄρα ἐστὶν <lb n="1799,001,,,,11,6,33"></lb> |
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13241 ἡ ΑΒ τῇ ΓΔ.</s> <lb n="1799,001,,,,11,6,34"></lb> |
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13242 <s>Ἐὰν ἄρα δύο εὐθεῖαι τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθὰς ὦσιν, <lb n="1799,001,,,,11,6,35"></lb> |
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13243 παράλληλοι ἔσονται αἱ εὐθεῖαι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,6,36"></lb> |
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13244 <s>Ἐὰν ὦσι δύο εὐθεῖαι παράλληλοι, ληφθῇ δὲ ἐφ' ἑκατέρας <lb n="1799,001,,,,11,7,1"></lb> |
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13245 αὐτῶν τυχόντα σημεῖα, ἡ ἐπὶ τὰ σημεῖα ἐπιζευγνυμένη <lb n="1799,001,,,,11,7,2"></lb> |
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13246 εὐθεῖα ἐν τῷ αὐτῷ ἐπιπέδῳ ἐστὶ ταῖς παραλλήλοις.</s> <lb n="1799,001,,,,11,7,3"></lb> |
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13247 <s>Ἔστωσαν δύο εὐθεῖαι παράλληλοι αἱ ΑΒ, ΓΔ, καὶ εἰλή-<lb n="1799,001,,,,11,7,4"></lb> |
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13248 φθω ἐφ' ἑκατέρας αὐτῶν τυχόντα σημεῖα τὰ Ε, Ζ· λέγω, <lb n="1799,001,,,,11,7,5"></lb> |
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13249 ὅτι ἡ ἐπὶ τὰ Ε, Ζ σημεῖα ἐπιζευγνυμένη εὐθεῖα ἐν τῷ <lb n="1799,001,,,,11,7,6"></lb> |
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13250 αὐτῷ ἐπιπέδῳ ἐστὶ ταῖς παραλλήλοις.</s> <s><pb></pb> <lb n="1799,001,,,,11,7,7"></lb> |
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13251 Μὴ γάρ, ἀλλ' εἰ δυνατόν, ἔστω <lb n="1799,001,,,,11,7,8"></lb> |
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13252 ἐν μετεωροτέρῳ ὡς ἡ ΕΗΖ, <lb n="1799,001,,,,11,7,9"></lb> |
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13253 καὶ διήχθω διὰ τῆς ΕΗΖ ἐπίπε-<lb n="1799,001,,,,11,7,10"></lb> |
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13254 δον· τομὴν δὴ ποιήσει ἐν τῷ ὑπο-<lb n="1799,001,,,,11,7,11"></lb> |
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13255 κειμένῳ ἐπιπέδῳ εὐθεῖαν.</s> <s>ποιεί-<lb n="1799,001,,,,11,7,12"></lb> |
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13256 τω ὡς τὴν ΕΖ· δύο ἄρα εὐθεῖαι <lb n="1799,001,,,,11,7,13"></lb> |
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13257 αἱ ΕΗΖ, ΕΖ χωρίον περιέξουσιν· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ <lb n="1799,001,,,,11,7,14"></lb> |
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13258 ἄρα ἡ ἀπὸ τοῦ Ε ἐπὶ τὸ Ζ ἐπιζευγνυμένη εὐθεῖα ἐν μετεωρο-<lb n="1799,001,,,,11,7,15"></lb> |
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13259 τέρῳ ἐστὶν ἐπιπέδῳ· ἐν τῷ διὰ τῶν ΑΒ, ΓΔ ἄρα παραλλήλων <lb n="1799,001,,,,11,7,16"></lb> |
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13260 ἐστὶν ἐπιπέδῳ ἡ ἀπὸ τοῦ Ε ἐπὶ τὸ Ζ ἐπιζευγνυμένη εὐθεῖα.</s> <lb n="1799,001,,,,11,7,17"></lb> |
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13261 <s>Ἐὰν ἄρα ὦσι δύο εὐθεῖαι παράλληλοι, ληφθῇ δὲ ἐφ' <lb n="1799,001,,,,11,7,18"></lb> |
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13262 ἑκατέρας αὐτῶν τυχόντα σημεῖα, ἡ ἐπὶ τὰ σημεῖα ἐπιζευ-<lb n="1799,001,,,,11,7,19"></lb> |
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13263 γνυμένη εὐθεῖα ἐν τῷ αὐτῷ ἐπιπέδῳ ἐστὶ ταῖς παραλ-<lb n="1799,001,,,,11,7,20"></lb> |
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13264 λήλοις· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,7,21"></lb> |
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13265 <s>Ἐὰν ὦσι δύο εὐθεῖαι παράλληλοι, ἡ δὲ ἑτέρα αὐτῶν <lb n="1799,001,,,,11,8,1"></lb> |
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13266 ἐπιπέδῳ τινὶ πρὸς ὀρθὰς ᾖ, καὶ ἡ λοιπὴ τῷ αὐτῷ <lb n="1799,001,,,,11,8,2"></lb> |
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13267 ἐπιπέδῳ πρὸς ὀρθὰς ἔσται.</s> <lb n="1799,001,,,,11,8,3"></lb> |
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13268 <s>Ἔστωσαν δύο εὐθεῖαι παράλληλοι αἱ <lb n="1799,001,,,,11,8,4"></lb> |
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13269 ΑΒ, ΓΔ, ἡ δὲ ἑτέρα αὐτῶν ἡ ΑΒ τῷ <lb n="1799,001,,,,11,8,5"></lb> |
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13270 ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθὰς ἔστω· <lb n="1799,001,,,,11,8,6"></lb> |
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13271 λέγω, ὅτι καὶ ἡ λοιπὴ ἡ ΓΔ τῷ αὐτῷ <lb n="1799,001,,,,11,8,7"></lb> |
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13272 ἐπιπέδῳ πρὸς ὀρθὰς ἔσται.</s> <lb n="1799,001,,,,11,8,8"></lb> |
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13273 <s>Συμβαλλέτωσαν γὰρ αἱ ΑΒ, ΓΔ τῷ <lb n="1799,001,,,,11,8,9"></lb> |
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13274 ὑποκειμένῳ ἐπιπέδῳ κατὰ τὰ Β, Δ ση-<lb n="1799,001,,,,11,8,10"></lb> |
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13275 μεῖα, καὶ ἐπεζεύχθω ἡ ΒΔ· αἱ ΑΒ, ΓΔ, <lb n="1799,001,,,,11,8,11"></lb> |
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13276 ΒΔ ἄρα ἐν ἑνί εἰσιν ἐπιπέδῳ.</s> <s>ἤχθω τῇ <pb></pb> <lb n="1799,001,,,,11,8,12"></lb> |
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13277 ΒΔ πρὸς ὀρθὰς ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ ἡ ΔΕ, καὶ κείσθω <lb n="1799,001,,,,11,8,13"></lb> |
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13278 τῇ ΑΒ ἴση ἡ ΔΕ, καὶ ἐπεζεύχθωσαν αἱ ΒΕ, ΑΕ, ΑΔ.</s> <s>καὶ <lb n="1799,001,,,,11,8,14"></lb> |
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13279 ἐπεὶ ἡ ΑΒ ὀρθή ἐστι πρὸσ τὸ ὑποκείμενον ἐπίπεδον, καὶ <lb n="1799,001,,,,11,8,15"></lb> |
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13280 πρὸς πάσας ἄρα τὰς ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας <lb n="1799,001,,,,11,8,16"></lb> |
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13281 ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν ἡ ΑΒ· ὀρθὴ <lb n="1799,001,,,,11,8,17"></lb> |
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13282 ἄρα [ἐστὶν] ἑκατέρα τῶν ὑπὸ ΑΒΔ, ΑΒΕ γωνιῶν.</s> <s>καὶ <lb n="1799,001,,,,11,8,18"></lb> |
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13283 ἐπεὶ εἰς παραλλήλους τὰς ΑΒ, ΓΔ εὐθεῖα ἐμπέπτωκεν ἡ <lb n="1799,001,,,,11,8,19"></lb> |
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13284 ΒΔ, αἱ ἄρα ὑπὸ ΑΒΔ, ΓΔΒ γωνίαι δυσὶν ὀρθαῖς ἴσαι <lb n="1799,001,,,,11,8,20"></lb> |
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13285 εἰσίν.</s> <s>ὀρθὴ δὲ ἡ ὑπὸ ΑΒΔ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΓΔΒ· ἡ <lb n="1799,001,,,,11,8,21"></lb> |
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13286 ΓΔ ἄρα πρὸς τὴν ΒΔ ὀρθή ἐστιν.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒ <lb n="1799,001,,,,11,8,22"></lb> |
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13287 τῇ ΔΕ, κοινὴ δὲ ἡ ΒΔ, δύο δὴ αἱ ΑΒ, ΒΔ δυσὶ ταῖς ΕΔ, <lb n="1799,001,,,,11,8,23"></lb> |
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13288 ΔΒ ἴσαι εἰσίν· καὶ γωνία ἡ ὑπὸ ΑΒΔ γωνίᾳ τῇ ὑπὸ ΕΔΒ <lb n="1799,001,,,,11,8,24"></lb> |
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13289 ἴση· ὀρθὴ γὰρ ἑκατέρα· βάσις ἄρα ἡ ΑΔ βάσει τῇ ΒΕ ἴση.</s> <lb n="1799,001,,,,11,8,25"></lb> |
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13290 <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ μὲν ΑΒ τῇ ΔΕ, ἡ δὲ ΒΕ τῇ ΑΔ, δύο <lb n="1799,001,,,,11,8,26"></lb> |
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13291 δὴ αἱ ΑΒ, ΒΕ δυσὶ ταῖς ΕΔ, ΔΑ ἴσαι εἰσὶν ἑκατέρα ἑκα-<lb n="1799,001,,,,11,8,27"></lb> |
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13292 τέρᾳ.</s> <s>καὶ βάσις αὐτῶν κοινὴ ἡ ΑΕ· γωνία ἄρα ἡ ὑπὸ ΑΒΕ <lb n="1799,001,,,,11,8,28"></lb> |
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13293 γωνίᾳ τῇ ὑπὸ ΕΔΑ ἐστιν ἴση.</s> <s>ὀρθὴ δὲ ἡ ὑπὸ ΑΒΕ· ὀρθὴ <lb n="1799,001,,,,11,8,29"></lb> |
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13294 ἄρα καὶ ἡ ὑπὸ ΕΔΑ· ἡ ΕΔ ἄρα πρὸς τὴν ΑΔ ὀρθή ἐστιν.</s> <lb n="1799,001,,,,11,8,30"></lb> |
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13295 <s>ἔστι δὲ καὶ πρὸς τὴν ΔΒ ὀρθή· ἡ ΕΔ ἄρα καὶ τῷ διὰ <lb n="1799,001,,,,11,8,31"></lb> |
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13296 τῶν ΒΔ, ΔΑ ἐπιπέδῳ ὀρθή ἐστιν.</s> <s>καὶ πρὸς πάσας ἄρα <lb n="1799,001,,,,11,8,32"></lb> |
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13297 τὰς ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ διὰ τῶν <lb n="1799,001,,,,11,8,33"></lb> |
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13298 ΒΔΑ ἐπιπέδῳ ὀρθὰς ποιήσει γωνίας ἡ ΕΔ.</s> <s>ἐν δὲ τῷ διὰ <lb n="1799,001,,,,11,8,34"></lb> |
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13299 τῶν ΒΔΑ ἐπιπέδῳ ἐστὶν ἡ ΔΓ, ἐπειδήπερ ἐν τῷ διὰ τῶν <lb n="1799,001,,,,11,8,35"></lb> |
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13300 ΒΔΑ ἐπιπέδῳ εἰσὶν αἱ ΑΒ, ΒΔ, ἐν ᾧ δὲ αἱ ΑΒ, ΒΔ, ἐν <lb n="1799,001,,,,11,8,36"></lb> |
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13301 τούτῳ ἐστὶ καὶ ἡ ΔΓ.</s> <s>ἡ ΕΔ ἄρα τῇ ΔΓ πρὸς ὀρθάς ἐστιν· <pb></pb> <lb n="1799,001,,,,11,8,37"></lb> |
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13302 ὥστε καὶ ἡ ΓΔ τῇ ΔΕ πρὸς ὀρθάς ἐστιν.</s> <s>ἔστι δὲ καὶ ἡ ΓΔ <lb n="1799,001,,,,11,8,38"></lb> |
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13303 τῇ ΒΔ πρὸς ὀρθάς.</s> <s>ἡ ΓΔ ἄρα δύο εὐθείαις τεμνούσαις <lb n="1799,001,,,,11,8,39"></lb> |
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13304 ἀλλήλας ταῖς ΔΕ, ΔΒ ἀπὸ τῆς κατὰ τὸ Δ τομῆς πρὸς <lb n="1799,001,,,,11,8,40"></lb> |
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13305 ὀρθὰς ἐφέστηκεν· ὥστε ἡ ΓΔ καὶ τῷ διὰ τῶν ΔΕ, ΔΒ <lb n="1799,001,,,,11,8,41"></lb> |
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13306 ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <s>τὸ δὲ διὰ τῶν ΔΕ, ΔΒ ἐπί-<lb n="1799,001,,,,11,8,42"></lb> |
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13307 πεδον τὸ ὑποκείμενόν ἐστιν· ἡ ΓΔ ἄρα τῷ ὑποκειμένῳ <lb n="1799,001,,,,11,8,43"></lb> |
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13308 ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <lb n="1799,001,,,,11,8,44"></lb> |
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13309 <s>Ἐὰν ἄρα ὦσι δύο εὐθεῖαι παράλληλοι, ἡ δὲ μία αὐτῶν <lb n="1799,001,,,,11,8,45"></lb> |
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13310 ἐπιπέδῳ τινὶ πρὸς ὀρθὰς ᾖ, καὶ ἡ λοιπὴ τῷ αὐτῷ ἐπιπέδῳ <lb n="1799,001,,,,11,8,46"></lb> |
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13311 πρὸς ὀρθὰς ἔσται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,8,47"></lb> |
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13312 <s>Αἱ τῇ αὐτῇ εὐθείᾳ παράλληλοι καὶ μὴ οὖσαι αὐτῇ ἐν <lb n="1799,001,,,,11,9,1"></lb> |
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13313 τῷ αὐτῷ ἐπιπέδῳ καὶ ἀλλήλαις εἰσὶ παράλληλοι.</s> <lb n="1799,001,,,,11,9,2"></lb> |
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13314 <s>Ἔστω γὰρ ἑκατέρα τῶν ΑΒ, <lb n="1799,001,,,,11,9,3"></lb> |
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13315 ΓΔ τῇ ΕΖ παράλληλος μὴ <lb n="1799,001,,,,11,9,4"></lb> |
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13316 οὖσαι αὐτῇ ἐν τῷ αὐτῷ ἐπι-<lb n="1799,001,,,,11,9,5"></lb> |
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13317 πέδῳ· λέγω, ὅτι παράλληλός <lb n="1799,001,,,,11,9,6"></lb> |
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13318 ἐστιν ἡ ΑΒ τῇ ΓΔ.</s> <lb n="1799,001,,,,11,9,7"></lb> |
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13319 <s>Εἰλήφθω γὰρ ἐπὶ τῆς ΕΖ <lb n="1799,001,,,,11,9,8"></lb> |
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13320 τυχὸν σημεῖον τὸ Η, καὶ ἀπ' <lb n="1799,001,,,,11,9,9"></lb> |
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13321 αὐτοῦ τῇ ΕΖ ἐν μὲν τῷ διὰ τῶν ΕΖ, ΑΒ ἐπιπέδῳ πρὸς <lb n="1799,001,,,,11,9,10"></lb> |
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13322 ὀρθὰς ἤχθω ἡ ΗΘ, ἐν δὲ τῷ διὰ τῶν ΖΕ, ΓΔ τῇ ΕΖ <lb n="1799,001,,,,11,9,11"></lb> |
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13323 πάλιν πρὸς ὀρθὰς ἤχθω ἡ ΗΚ.</s> <s>καὶ ἐπεὶ ἡ ΕΖ πρὸς ἑκα-<lb n="1799,001,,,,11,9,12"></lb> |
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13324 τέραν τῶν ΗΘ, ΗΚ ὀρθή ἐστιν, ἡ ΕΖ ἄρα καὶ τῷ διὰ <pb></pb> <lb n="1799,001,,,,11,9,13"></lb> |
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13325 τῶν ΗΘ, ΗΚ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <s>καί ἐστιν ἡ ΕΖ <lb n="1799,001,,,,11,9,14"></lb> |
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13326 τῇ ΑΒ παράλληλος· καὶ ἡ ΑΒ ἄρα τῷ διὰ τῶν ΘΗΚ ἐπι-<lb n="1799,001,,,,11,9,15"></lb> |
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13327 πέδῳ πρὸς ὀρθάς ἐστιν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΓΔ τῷ διὰ <lb n="1799,001,,,,11,9,16"></lb> |
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13328 τῶν ΘΗΚ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν· ἑκατέρα ἄρα τῶν <lb n="1799,001,,,,11,9,17"></lb> |
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13329 ΑΒ, ΓΔ τῷ διὰ τῶν ΘΗΚ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <lb n="1799,001,,,,11,9,18"></lb> |
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13330 <s>ἐὰν δὲ δύο εὐθεῖαι τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθὰς ὦσιν, <lb n="1799,001,,,,11,9,19"></lb> |
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13331 παράλληλοί εἰσιν αἱ εὐθεῖαι· παράλληλος ἄρα ἐστὶν ἡ <lb n="1799,001,,,,11,9,20"></lb> |
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13332 ΑΒ τῇ ΓΔ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,9,21"></lb> |
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13333 <s>Ἐὰν δύο εὐθεῖαι ἁπτόμεναι ἀλλήλων παρὰ δύο εὐθείας <lb n="1799,001,,,,11,10,1"></lb> |
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13334 ἁπτομένας ἀλλήλων ὦσι μὴ ἐν τῷ αὐτῷ ἐπιπέδῳ, ἴσας <lb n="1799,001,,,,11,10,2"></lb> |
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13335 γωνίας περιέξουσιν.</s> <lb n="1799,001,,,,11,10,3"></lb> |
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13336 <s>Δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΒΓ ἁπτόμεναι ἀλλήλων παρὰ <lb n="1799,001,,,,11,10,4"></lb> |
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13337 δύο εὐθείας τὰς ΔΕ, ΕΖ ἁπτομένας ἀλλήλων ἔστωσαν <lb n="1799,001,,,,11,10,5"></lb> |
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13338 μὴ ἐν τῷ αὐτῷ ἐπιπέδῳ· <lb n="1799,001,,,,11,10,6"></lb> |
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13339 λέγω, ὅτι ἴση ἐστὶν ἡ ὑπὸ <lb n="1799,001,,,,11,10,7"></lb> |
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13340 ΑΒΓ γωνία τῇ ὑπὸ ΔΕΖ.</s> <lb n="1799,001,,,,11,10,8"></lb> |
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13341 <s>Ἀπειλήφθωσαν γὰρ αἱ <lb n="1799,001,,,,11,10,9"></lb> |
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13342 ΒΑ, ΒΓ, ΕΔ, ΕΖ ἴσαι <lb n="1799,001,,,,11,10,10"></lb> |
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13343 ἀλλήλαις, καὶ ἐπεζεύχθωσαν <lb n="1799,001,,,,11,10,11"></lb> |
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13344 αἱ ΑΔ, ΓΖ, ΒΕ, ΑΓ, ΔΖ.</s> <lb n="1799,001,,,,11,10,12"></lb> |
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13345 <s>καὶ ἐπεὶ ἡ ΒΑ τῇ ΕΔ ἴση <lb n="1799,001,,,,11,10,13"></lb> |
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13346 ἐστὶ καὶ παράλληλος, καὶ ἡ <pb></pb> <lb n="1799,001,,,,11,10,14"></lb> |
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13347 ΑΔ ἄρα τῇ ΒΕ ἴση ἐστὶ καὶ παράλληλος.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,11,10,15"></lb> |
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13348 ἡ ΓΖ τῇ ΒΕ ἴση ἐστὶ καὶ παράλληλος· ἑκατέρα ἄρα τῶν <lb n="1799,001,,,,11,10,16"></lb> |
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13349 ΑΔ, ΓΖ τῇ ΒΕ ἴση ἐστὶ καὶ παράλληλος.</s> <s>αἱ δὲ τῇ αὐτῇ <lb n="1799,001,,,,11,10,17"></lb> |
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13350 εὐθείᾳ παράλληλοι καὶ μὴ οὖσαι αὐτῇ ἐν τῷ αὐτῷ ἐπιπέδῳ <lb n="1799,001,,,,11,10,18"></lb> |
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13351 καὶ ἀλλήλαις εἰσὶ παράλληλοι· παράλληλος ἄρα ἐστὶν ἡ <lb n="1799,001,,,,11,10,19"></lb> |
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13352 ΑΔ τῇ ΓΖ καὶ ἴση.</s> <s>καὶ ἐπιζευγνύουσιν αὐτὰς αἱ ΑΓ, ΔΖ· <lb n="1799,001,,,,11,10,20"></lb> |
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13353 καὶ ἡ ΑΓ ἄρα τῇ ΔΖ ἴση ἐστὶ καὶ παράλληλος.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,11,10,21"></lb> |
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13354 δύο αἱ ΑΒ, ΒΓ δυσὶ ταῖς ΔΕ, ΕΖ ἴσαι εἰσίν, καὶ βάσις <lb n="1799,001,,,,11,10,22"></lb> |
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13355 ἡ ΑΓ βάσει τῇ ΔΖ ἴση, γωνία ἄρα ἡ ὑπὸ ΑΒΓ γωνίᾳ <lb n="1799,001,,,,11,10,23"></lb> |
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13356 τῇ ὑπὸ ΔΕΖ ἐστιν ἴση.</s> <lb n="1799,001,,,,11,10,24"></lb> |
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13357 <s>Ἐὰν ἄρα δύο εὐθεῖαι ἁπτόμεναι ἀλλήλων παρὰ δύο <lb n="1799,001,,,,11,10,25"></lb> |
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13358 εὐθείας ἁπτομένας ἀλλήλων ὦσι μὴ ἐν τῷ αὐτῷ ἐπιπέδῳ, <lb n="1799,001,,,,11,10,26"></lb> |
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13359 ἴσας γωνίας περιέξουσιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,10,27"></lb> |
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13360 <s>Ἀπὸ τοῦ δοθέντος σημείου μετεώρου ἐπὶ τὸ δοθὲν ἐπί-<lb n="1799,001,,,,11,11,1"></lb> |
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13361 πεδον κάθετον εὐθεῖαν γραμμὴν ἀγαγεῖν.</s> <lb n="1799,001,,,,11,11,2"></lb> |
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13362 <s>Ἔστω τὸ μὲν δοθὲν σημεῖον μετ-<lb n="1799,001,,,,11,11,3"></lb> |
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13363 έωρον τὸ Α, τὸ δὲ δοθὲν ἐπίπεδον <lb n="1799,001,,,,11,11,4"></lb> |
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13364 τὸ ὑποκείμενον· δεῖ δὴ ἀπὸ τοῦ Α <lb n="1799,001,,,,11,11,5"></lb> |
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13365 σημείου ἐπὶ τὸ ὑποκείμενον ἐπί-<lb n="1799,001,,,,11,11,6"></lb> |
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13366 πεδον κάθετον εὐθεῖαν γραμμὴν <lb n="1799,001,,,,11,11,7"></lb> |
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13367 ἀγαγεῖν.</s> <s><pb></pb> <lb n="1799,001,,,,11,11,8"></lb> |
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13368 Διήχθω γάρ τις ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ εὐθεῖα, <lb n="1799,001,,,,11,11,9"></lb> |
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13369 ὡς ἔτυχεν, ἡ ΒΓ, καὶ ἤχθω ἀπὸ τοῦ Α σημείου ἐπὶ τὴν <lb n="1799,001,,,,11,11,10"></lb> |
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13370 ΒΓ κάθετος ἡ ΑΔ.</s> <s>εἰ μὲν οὖν ἡ ΑΔ κάθετός ἐστι καὶ <lb n="1799,001,,,,11,11,11"></lb> |
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13371 ἐπὶ τὸ ὑποκείμενον ἐπίπεδον, γεγονὸς ἂν εἴη τὸ ἐπιταχ-<lb n="1799,001,,,,11,11,12"></lb> |
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13372 θέν.</s> <s>εἰ δὲ οὔ, ἤχθω ἀπὸ τοῦ Δ σημείου τῇ ΒΓ ἐν τῷ ὑπο-<lb n="1799,001,,,,11,11,13"></lb> |
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13373 κειμένῳ ἐπιπέδῳ πρὸς ὀρθὰς ἡ ΔΕ, καὶ ἤχθω ἀπὸ τοῦ Α <lb n="1799,001,,,,11,11,14"></lb> |
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13374 ἐπὶ τὴν ΔΕ κάθετος ἡ ΑΖ, καὶ διὰ τοῦ Ζ σημείου τῇ ΒΓ <lb n="1799,001,,,,11,11,15"></lb> |
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13375 παράλληλος ἤχθω ἡ ΗΘ.</s> <lb n="1799,001,,,,11,11,16"></lb> |
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13376 <s>Καὶ ἐπεὶ ἡ ΒΓ ἑκατέρᾳ τῶν ΔΑ, ΔΕ πρὸς ὀρθάς ἐστιν, <lb n="1799,001,,,,11,11,17"></lb> |
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13377 ἡ ΒΓ ἄρα καὶ τῷ διὰ τῶν ΕΔΑ ἐπιπέδῳ πρὸς ὀρθάς <lb n="1799,001,,,,11,11,18"></lb> |
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13378 ἐστιν.</s> <s>καί ἐστιν αὐτῇ παράλληλος ἡ ΗΘ· ἐὰν δὲ ὦσι δύο <lb n="1799,001,,,,11,11,19"></lb> |
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13379 εὐθεῖαι παράλληλοι, ἡ δὲ μία αὐτῶν ἐπιπέδῳ τινὶ πρὸς <lb n="1799,001,,,,11,11,20"></lb> |
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13380 ὀρθὰς ᾖ, καὶ ἡ λοιπὴ τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθὰς ἔσται· <lb n="1799,001,,,,11,11,21"></lb> |
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13381 καὶ ἡ ΗΘ ἄρα τῷ διὰ τῶν ΕΔ, ΔΑ ἐπιπέδῳ πρὸς ὀρθάς <lb n="1799,001,,,,11,11,22"></lb> |
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13382 ἐστιν.</s> <s>καὶ πρὸς πάσας ἄρα τὰς ἁπτομένας αὐτῆς εὐθείας <lb n="1799,001,,,,11,11,23"></lb> |
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13383 καὶ οὔσας ἐν τῷ διὰ τῶν ΕΔ, ΔΑ ἐπιπέδῳ ὀρθή ἐστιν <lb n="1799,001,,,,11,11,24"></lb> |
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13384 ἡ ΗΘ.</s> <s>ἅπτεται δὲ αὐτῆς ἡ ΑΖ οὖσα ἐν τῷ διὰ τῶν ΕΔ, <lb n="1799,001,,,,11,11,25"></lb> |
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13385 ΔΑ ἐπιπέδῳ· ἡ ΗΘ ἄρα ὀρθή ἐστι πρὸς τὴν ΖΑ· ὥστε <lb n="1799,001,,,,11,11,26"></lb> |
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13386 καὶ ἡ ΖΑ ὀρθή ἐστι πρὸς τὴν ΘΗ.</s> <s>ἔστι δὲ ἡ ΑΖ καὶ <lb n="1799,001,,,,11,11,27"></lb> |
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13387 πρὸς τὴν ΔΕ ὀρθή· ἡ ΑΖ ἄρα πρὸς ἑκατέραν τῶν ΗΘ, <lb n="1799,001,,,,11,11,28"></lb> |
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13388 ΔΕ ὀρθή ἐστιν.</s> <s>ἐὰν δὲ εὐθεῖα δυσὶν εὐθείαις τεμνούσαις <lb n="1799,001,,,,11,11,29"></lb> |
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13389 ἀλλήλας ἐπὶ τῆς τομῆς πρὸς ὀρθὰς ἐπισταθῇ, καὶ τῷ δι' <lb n="1799,001,,,,11,11,30"></lb> |
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13390 αὐτῶν ἐπιπέδῳ πρὸς ὀρθὰς ἔσται· ἡ ΖΑ ἄρα τῷ διὰ τῶν <pb></pb> <lb n="1799,001,,,,11,11,31"></lb> |
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13391 ΕΔ, ΗΘ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <s>τὸ δὲ διὰ τῶν ΕΔ, <lb n="1799,001,,,,11,11,32"></lb> |
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13392 ΗΘ ἐπίπεδόν ἐστι τὸ ὑποκείμενον· ἡ ΑΖ ἄρα τῷ ὑποκει-<lb n="1799,001,,,,11,11,33"></lb> |
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13393 μένῳ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <lb n="1799,001,,,,11,11,34"></lb> |
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13394 <s>Ἀπὸ τοῦ ἄρα δοθέντος σημείου μετεώρου τοῦ Α ἐπὶ τὸ <lb n="1799,001,,,,11,11,35"></lb> |
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13395 ὑποκείμενον ἐπίπεδον κάθετος εὐθεῖα γραμμὴ ἦκται ἡ <lb n="1799,001,,,,11,11,36"></lb> |
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13396 ΑΖ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,11,11,37"></lb> |
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13397 <s>Τῷ δοθέντι ἐπιπέδῳ ἀπὸ τοῦ πρὸς αὐτῷ δοθέντος ση-<lb n="1799,001,,,,11,12,1"></lb> |
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13398 μείου πρὸς ὀρθὰς εὐθεῖαν γραμμὴν ἀναστῆσαι.</s> <lb n="1799,001,,,,11,12,2"></lb> |
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13399 <s>Ἔστω τὸ μὲν δοθὲν ἐπίπεδον τὸ <lb n="1799,001,,,,11,12,3"></lb> |
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13400 ὑποκείμενον, τὸ δὲ πρὸς αὐτῷ σημεῖον <lb n="1799,001,,,,11,12,4"></lb> |
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13401 τὸ Α· δεῖ δὴ ἀπὸ τοῦ Α σημείου τῷ <lb n="1799,001,,,,11,12,5"></lb> |
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13402 ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθὰς εὐ-<lb n="1799,001,,,,11,12,6"></lb> |
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13403 θεῖαν γραμμὴν ἀναστῆσαι.</s> <lb n="1799,001,,,,11,12,7"></lb> |
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13404 <s>Νενοήσθω τι σημεῖον μετέωρον <lb n="1799,001,,,,11,12,8"></lb> |
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13405 τὸ Β, καὶ ἀπὸ τοῦ Β ἐπὶ τὸ ὑποκεί-<lb n="1799,001,,,,11,12,9"></lb> |
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13406 μενον ἐπίπεδον κάθετος ἤχθω ἡ ΒΓ, καὶ διὰ τοῦ Α <lb n="1799,001,,,,11,12,10"></lb> |
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13407 σημείου τῇ ΒΓ παράλληλος ἤχθω ἡ ΑΔ.</s> <lb n="1799,001,,,,11,12,11"></lb> |
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13408 <s>Ἐπεὶ οὖν δύο εὐθεῖαι παράλληλοί εἰσιν αἱ ΑΔ, ΓΒ, ἡ δὲ <lb n="1799,001,,,,11,12,12"></lb> |
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13409 μία αὐτῶν ἡ ΒΓ τῷ ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν, <lb n="1799,001,,,,11,12,13"></lb> |
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13410 καὶ ἡ λοιπὴ ἄρα ἡ ΑΔ τῷ ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθάς <lb n="1799,001,,,,11,12,14"></lb> |
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13411 ἐστιν.</s> <lb n="1799,001,,,,11,12,15"></lb> |
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13412 <s>Τῷ ἄρα δοθέντι ἐπιπέδῳ ἀπὸ τοῦ πρὸς αὐτῷ σημείου <lb n="1799,001,,,,11,12,16"></lb> |
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13413 τοῦ Α πρὸς ὀρθὰς ἀνέσταται ἡ ΑΔ· ὅπερ ἔδει ποιῆσαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,12,17"></lb> |
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13414 Ἀπὸ τοῦ αὐτοῦ σημείου τῷ αὐτῷ ἐπιπέδῳ δύο εὐθεῖαι <lb n="1799,001,,,,11,13,1"></lb> |
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13415 πρὸς ὀρθὰς οὐκ ἀναστήσονται ἐπὶ τὰ αὐτὰ μέρη.</s> <lb n="1799,001,,,,11,13,2"></lb> |
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13416 <s>Εἰ γὰρ δυνατόν, ἀπὸ τοῦ αὐτοῦ σημείου τοῦ Α τῷ ὑπο-<lb n="1799,001,,,,11,13,3"></lb> |
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13417 κειμένῳ ἐπιπέδῳ δύο εὐθεῖαι αἱ ΑΒ, ΑΓ πρὸς ὀρθὰς <lb n="1799,001,,,,11,13,4"></lb> |
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13418 ἀνεστάτωσαν ἐπὶ τὰ αὐτὰ μέρη, καὶ διήχθω τὸ διὰ τῶν <lb n="1799,001,,,,11,13,5"></lb> |
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13419 ΒΑ, ΑΓ ἐπίπεδον· τομὴν δὴ ποιήσει <lb n="1799,001,,,,11,13,6"></lb> |
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13420 διὰ τοῦ Α ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ <lb n="1799,001,,,,11,13,7"></lb> |
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13421 εὐθεῖαν.</s> <s>ποιείτω τὴν ΔΑΕ· αἱ ἄρα <lb n="1799,001,,,,11,13,8"></lb> |
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13422 ΑΒ, ΑΓ, ΔΑΕ εὐθεῖαι ἐν ἑνί εἰσιν <lb n="1799,001,,,,11,13,9"></lb> |
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13423 ἐπιπέδῳ.</s> <s>καὶ ἐπεὶ ἡ ΓΑ τῷ ὑποκει-<lb n="1799,001,,,,11,13,10"></lb> |
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13424 μένῳ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν, <lb n="1799,001,,,,11,13,11"></lb> |
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13425 καὶ πρὸς πάσας ἄρα τὰς ἁπτομένας <lb n="1799,001,,,,11,13,12"></lb> |
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13426 αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ ὑπο-<lb n="1799,001,,,,11,13,13"></lb> |
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13427 κειμένῳ ἐπιπέδῳ ὀρθὰς ποιήσει γωνίας.</s> <s>ἅπτεται δὲ αὐτῆς <lb n="1799,001,,,,11,13,14"></lb> |
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13428 ἡ ΔΑΕ οὖσα ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ· ἡ ἄρα ὑπὸ ΓΑΕ <lb n="1799,001,,,,11,13,15"></lb> |
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13429 γωνία ὀρθή ἐστιν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ ΒΑΕ ὀρθή <lb n="1799,001,,,,11,13,16"></lb> |
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13430 ἐστιν· ἴση ἄρα ἡ ὑπὸ ΓΑΕ τῇ ὑπὸ ΒΑΕ.</s> <s>καί εἰσιν ἐν ἑνὶ <lb n="1799,001,,,,11,13,17"></lb> |
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13431 ἐπιπέδῳ· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,11,13,18"></lb> |
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13432 <s>Οὐκ ἄρα ἀπὸ τοῦ αὐτοῦ σημείου τῷ αὐτῷ ἐπιπέδῳ <lb n="1799,001,,,,11,13,19"></lb> |
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13433 δύο εὐθεῖαι πρὸς ὀρθὰς ἀνασταθήσονται ἐπὶ τὰ αὐτὰ <lb n="1799,001,,,,11,13,20"></lb> |
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13434 μέρη· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,13,21"></lb> |
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13435 <s>Πρὸς ἃ ἐπίπεδα ἡ αὐτὴ εὐθεῖα ὀρθή ἐστιν, παράλληλα <lb n="1799,001,,,,11,14,1"></lb> |
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13436 ἔσται τὰ ἐπίπεδα.</s> <s><pb></pb> <lb n="1799,001,,,,11,14,2"></lb> |
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13437 Εὐθεῖα γάρ τις ἡ ΑΒ πρὸς ἑκάτερον τῶν ΓΔ, ΕΖ ἐπι-<lb n="1799,001,,,,11,14,3"></lb> |
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13438 πέδων πρὸς ὀρθὰς ἔστω· λέγω, ὅτι παράλληλά ἐστι τὰ <lb n="1799,001,,,,11,14,4"></lb> |
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13439 ἐπίπεδα.</s> <lb n="1799,001,,,,11,14,5"></lb> |
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13440 <s>Εἰ γὰρ μή, ἐκβαλλόμενα συμπεσοῦνται.</s> <s>συμπιπτέ-<lb n="1799,001,,,,11,14,6"></lb> |
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13441 τωσαν· ποιήσουσι δὴ κοινὴν τομὴν εὐθεῖαν.</s> <s>ποιείτωσαν <lb n="1799,001,,,,11,14,7"></lb> |
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13442 τὴν ΗΘ, καὶ εἰλήφθω ἐπὶ τῆς ΗΘ τυχὸν σημεῖον τὸ Κ, <lb n="1799,001,,,,11,14,8"></lb> |
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13443 καὶ ἐπεζεύχθωσαν αἱ ΑΚ, ΒΚ.</s> <s>καὶ ἐπεὶ ἡ ΑΒ ὀρθή ἐστι <lb n="1799,001,,,,11,14,9"></lb> |
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13444 πρὸς τὸ ΕΖ ἐπίπεδον, καὶ πρὸς τὴν ΒΚ ἄρα εὐθεῖαν οὖσαν <lb n="1799,001,,,,11,14,10"></lb> |
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13445 ἐν τῷ ΕΖ ἐκβληθέντι ἐπιπέδῳ ὀρθή ἐστιν ἡ ΑΒ· ἡ ἄρα <lb n="1799,001,,,,11,14,11"></lb> |
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13446 ὑπὸ ΑΒΚ γωνία ὀρθή ἐστιν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ <lb n="1799,001,,,,11,14,12"></lb> |
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13447 ΒΑΚ ὀρθή ἐστιν.</s> <s>τριγώνου δὴ τοῦ ΑΒΚ αἱ δύο γωνίαι <lb n="1799,001,,,,11,14,13"></lb> |
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13448 αἱ ὑπὸ ΑΒΚ, ΒΑΚ δυσὶν ὀρθαῖς εἰσιν ἴσαι· ὅπερ ἐστὶν <lb n="1799,001,,,,11,14,14"></lb> |
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13449 ἀδύνατον.</s> <s>οὐκ ἄρα τὰ ΓΔ, ΕΖ ἐπίπεδα ἐκβαλλόμενα συμ-<lb n="1799,001,,,,11,14,15"></lb> |
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13450 πεσοῦνται· παράλληλα ἄρα ἐστὶ τὰ ΓΔ, ΕΖ ἐπίπεδα.</s> <lb n="1799,001,,,,11,14,16"></lb> |
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13451 <s>Πρὸς ἃ ἐπίπεδα ἄρα ἡ αὐτὴ εὐθεῖα ὀρθή ἐστιν, παράλ-<lb n="1799,001,,,,11,14,17"></lb> |
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13452 ληλά ἐστι τὰ ἐπίπεδα· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,14,18"></lb> |
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13453 Ἐὰν δύο εὐθεῖαι ἁπτόμεναι ἀλλήλων παρὰ δύο εὐθείας <lb n="1799,001,,,,11,15,1"></lb> |
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13454 ἁπτομένας ἀλλήλων ὦσι μὴ ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι, <lb n="1799,001,,,,11,15,2"></lb> |
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13455 παράλληλά ἐστι τὰ δι' αὐτῶν ἐπίπεδα.</s> <lb n="1799,001,,,,11,15,3"></lb> |
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13456 <s>Δύο γὰρ εὐθεῖαι ἁπτόμεναι ἀλλήλων αἱ ΑΒ, ΒΓ παρὰ <lb n="1799,001,,,,11,15,4"></lb> |
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13457 δύο εὐθείας ἁπτομένας ἀλλήλων τὰς ΔΕ, ΕΖ ἔστωσαν μὴ <lb n="1799,001,,,,11,15,5"></lb> |
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13458 ἐν τῳ̂ αὐτῷ ἐπιπέδῳ οὖσαι· λέγω, ὅτι ἐκβαλλόμενα τὰ διὰ <lb n="1799,001,,,,11,15,6"></lb> |
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13459 τῶν ΑΒ, ΒΓ, ΔΕ, ΕΖ ἐπίπεδα οὐ συμπεσεῖται ἀλλήλοις.</s> <lb n="1799,001,,,,11,15,7"></lb> |
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13460 <s>Ἤχθω γὰρ ἀπὸ τοῦ Β ση- <lb n="1799,001,,,,11,15,8"></lb> |
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13461 μείου ἐπὶ τὸ διὰ τῶν ΔΕ, ΕΖ <lb n="1799,001,,,,11,15,9"></lb> |
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13462 ἐπίπεδον κάθετος ἡ ΒΗ καὶ συμ-<lb n="1799,001,,,,11,15,10"></lb> |
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13463 βαλλέτω τῷ ἐπιπέδῳ κατὰ τὸ <lb n="1799,001,,,,11,15,11"></lb> |
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13464 Η σημεῖον, καὶ διὰ τοῦ Η τῇ μὲν <lb n="1799,001,,,,11,15,12"></lb> |
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13465 ΕΔ παράλληλος ἤχθω ἡ ΗΘ, τῇ <lb n="1799,001,,,,11,15,13"></lb> |
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13466 δὲ ΕΖ ἡ ΗΚ.</s> <s>καὶ ἐπεὶ ἡ ΒΗ <lb n="1799,001,,,,11,15,14"></lb> |
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13467 ὀρθή ἐστι πρὸς τὸ διὰ τῶν ΔΕ, <lb n="1799,001,,,,11,15,15"></lb> |
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13468 ΕΖ ἐπίπεδον, καὶ πρὸς πάσας <lb n="1799,001,,,,11,15,16"></lb> |
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13469 ἄρα τὰς ἁπτομένας αὐτῆς εὐθείας <lb n="1799,001,,,,11,15,17"></lb> |
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13470 καὶ οὔσας ἐν τῷ διὰ τῶν ΔΕ, <lb n="1799,001,,,,11,15,18"></lb> |
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13471 ΕΖ ἐπιπέδῳ ὀρθὰς ποιήσει γωνίας.</s> <s>ἅπτεται δὲ αὐτῆς <lb n="1799,001,,,,11,15,19"></lb> |
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13472 ἑκατέρα τῶν ΗΘ, ΗΚ οὖσα ἐν τῷ διὰ τῶν ΔΕ, ΕΖ ἐπιπέ-<lb n="1799,001,,,,11,15,20"></lb> |
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13473 δῳ· ὀρθὴ ἄρα ἐστὶν ἑκατέρα τῶν ὑπὸ ΒΗΘ, ΒΗΚ γω-<lb n="1799,001,,,,11,15,21"></lb> |
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13474 νιῶν.</s> <s>καὶ ἐπεὶ παράλληλός ἐστιν ἡ ΒΑ τῇ ΗΘ, αἱ ἄρα <lb n="1799,001,,,,11,15,22"></lb> |
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13475 ὑπὸ ΗΒΑ, ΒΗΘ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.</s> <s>ὀρθὴ δὲ <lb n="1799,001,,,,11,15,23"></lb> |
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13476 ἡ ὑπὸ ΒΗΘ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΗΒΑ· ἡ ΗΒ ἄρα τῇ <lb n="1799,001,,,,11,15,24"></lb> |
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13477 ΒΑ πρὸς ὀρθάς ἐστιν.</s> <s>διὰ τὰ αὐτὰ δὴ ἡ ΗΒ καὶ τῇ ΒΓ <pb></pb> <lb n="1799,001,,,,11,15,25"></lb> |
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13478 ἐστι πρὸς ὀρθάς.</s> <s>ἐπεὶ οὖν εὐθεῖα ἡ ΗΒ δυσὶν εὐθείαις ταῖς <lb n="1799,001,,,,11,15,26"></lb> |
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13479 ΒΑ, ΒΓ τεμνούσαις ἀλλήλας πρὸς ὀρθὰς ἐφέστηκεν, ἡ <lb n="1799,001,,,,11,15,27"></lb> |
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13480 ΗΒ ἄρα καὶ τῷ διὰ τῶν ΒΑ, ΒΓ ἐπιπέδῳ πρὸς ὀρθάς <lb n="1799,001,,,,11,15,28"></lb> |
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13481 ἐστιν.</s> <s>[διὰ τὰ αὐτὰ δὴ ἡ ΒΗ καὶ τῷ διὰ τῶν ΗΘ, ΗΚ <lb n="1799,001,,,,11,15,29"></lb> |
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13482 ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <s>τὸ δὲ διὰ τῶν ΗΘ, ΗΚ ἐπί-<lb n="1799,001,,,,11,15,30"></lb> |
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13483 πεδόν ἐστι τὸ διὰ τῶν ΔΕ, ΕΖ· ἡ ΒΗ ἄρα τῷ διὰ τῶν <lb n="1799,001,,,,11,15,31"></lb> |
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13484 ΔΕ, ΕΖ ἐπιπέδῳ ἐστὶ πρὸς ὀρθάς.</s> <s>ἐδείχθη δὲ ἡ ΗΒ καὶ <lb n="1799,001,,,,11,15,32"></lb> |
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13485 τῷ διὰ τῶν ΑΒ, ΒΓ ἐπιπέδῳ πρὸς ὀρθάσ].</s> <s>πρὸς ἃ δὲ ἐπί-<lb n="1799,001,,,,11,15,33"></lb> |
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13486 πεδα ἡ αὐτὴ εὐθεῖα ὀρθή ἐστιν, παράλληλά ἐστι τὰ ἐπί-<lb n="1799,001,,,,11,15,34"></lb> |
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13487 πεδα· παράλληλον ἄρα ἐστὶ τὸ διὰ τῶν ΑΒ, ΒΓ ἐπίπεδον <lb n="1799,001,,,,11,15,35"></lb> |
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13488 τῷ διὰ τῶν ΔΕ, ΕΖ.</s> <lb n="1799,001,,,,11,15,36"></lb> |
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13489 <s>Ἐὰν ἄρα δύο εὐθεῖαι ἁπτόμεναι ἀλλήλων παρὰ δύο <lb n="1799,001,,,,11,15,37"></lb> |
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13490 εὐθείας ἁπτομένας ἀλλήλων ὦσι μὴ ἐν τῷ αὐτῷ ἐπιπέδῳ, <lb n="1799,001,,,,11,15,38"></lb> |
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13491 παράλληλά ἐστι τὰ δι' αὐτῶν ἐπίπεδα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,15,39"></lb> |
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13492 <s>Ἐὰν δύο ἐπίπεδα παράλληλα ὑπὸ ἐπιπέδου τινὸς τέμνη-<lb n="1799,001,,,,11,16,1"></lb> |
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13493 ται, αἱ κοιναὶ αὐτῶν τομαὶ παράλληλοί εἰσιν.</s> <lb n="1799,001,,,,11,16,2"></lb> |
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13494 <s>Δύο γὰρ ἐπίπεδα παράλληλα τὰ ΑΒ, ΓΔ ὑπὸ ἐπιπέδου <lb n="1799,001,,,,11,16,3"></lb> |
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13495 τοῦ ΕΖΗΘ τεμνέσθω, κοιναὶ δὲ αὐτῶν τομαὶ ἔστωσαν <lb n="1799,001,,,,11,16,4"></lb> |
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13496 αἱ ΕΖ, ΗΘ· λέγω, ὅτι παράλληλός ἐστιν ἡ ΕΖ τῇ ΗΘ.</s> <s><pb></pb> <lb n="1799,001,,,,11,16,5"></lb> |
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13497 Εἰ γὰρ μή, ἐκβαλλόμεναι αἱ ΕΖ, ΗΘ ἤτοι ἐπὶ τὰ Ζ, Θ <lb n="1799,001,,,,11,16,6"></lb> |
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13498 μέρη ἢ ἐπὶ τὰ Ε, Η συμπεσοῦνται.</s> <s>ἐκβεβλήσθωσαν ὡς <lb n="1799,001,,,,11,16,7"></lb> |
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13499 ἐπὶ τὰ Ζ, Θ μέρη καὶ συμπιπτέτωσαν πρότερον κατὰ τὸ Κ. <lb n="1799,001,,,,11,16,8"></lb> |
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13500 καὶ ἐπεὶ ἡ ΕΖΚ ἐν τῷ ΑΒ ἐστιν ἐπιπέδῳ, καὶ πάντα ἄρα <lb n="1799,001,,,,11,16,9"></lb> |
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13501 τὰ ἐπὶ τῆς ΕΖΚ σημεῖα ἐν τῷ ΑΒ ἐστιν ἐπιπέδῳ.</s> <s>ἓν δὲ <lb n="1799,001,,,,11,16,10"></lb> |
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13502 τῶν ἐπὶ τῆς ΕΖΚ εὐθείας σημείων ἐστὶ τὸ Κ· τὸ Κ ἄρα <lb n="1799,001,,,,11,16,11"></lb> |
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13503 ἐν τῷ ΑΒ ἐστιν ἐπιπέδῳ.</s> <s>διὰ τὰ αὐτὰ δὴ τὸ Κ καὶ ἐν τῷ <lb n="1799,001,,,,11,16,12"></lb> |
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13504 ΓΔ ἐστιν ἐπιπέδῳ· τὰ ΑΒ, ΓΔ ἄρα ἐπίπεδα ἐκβαλλό-<lb n="1799,001,,,,11,16,13"></lb> |
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13505 μενα συμπεσοῦνται.</s> <s>οὐ συμπίπτουσι δὲ διὰ τὸ παράλληλα <lb n="1799,001,,,,11,16,14"></lb> |
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13506 ὑποκεῖσθαι· οὐκ ἄρα αἱ ΕΖ, ΗΘ εὐθεῖαι ἐκβαλλόμεναι <lb n="1799,001,,,,11,16,15"></lb> |
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13507 ἐπὶ τὰ Ζ, Θ μέρη συμπεσοῦνται.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι <lb n="1799,001,,,,11,16,16"></lb> |
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13508 αἱ ΕΖ, ΗΘ εὐθεῖαι οὐδὲ ἐπὶ τὰ Ε, Η μέρη ἐκβαλλόμεναι <lb n="1799,001,,,,11,16,17"></lb> |
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13509 συμπεσοῦνται.</s> <s>αἱ δὲ ἐπὶ μηδέτερα τὰ μέρη συμπίπτουσαι <lb n="1799,001,,,,11,16,18"></lb> |
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13510 παράλληλοί εἰσιν.</s> <s>παράλληλος ἄρα ἐστὶν ἡ ΕΖ τῇ ΗΘ.</s> <s><pb></pb> <lb n="1799,001,,,,11,16,19"></lb> |
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13511 Ἐὰν ἄρα δύο ἐπίπεδα παράλληλα ὑπὸ ἐπιπέδου τινὸς <lb n="1799,001,,,,11,16,20"></lb> |
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13512 τέμνηται, αἱ κοιναὶ αὐτῶν τομαὶ παράλληλοί εἰσιν· ὅπερ <lb n="1799,001,,,,11,16,21"></lb> |
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13513 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,16,22"></lb> |
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13514 <s>Ἐὰν δύο εὐθεῖαι ὑπὸ παραλλήλων ἐπιπέδων τέμνωνται <lb n="1799,001,,,,11,17,1"></lb> |
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13515 εἰς τοὺς αὐτοὺς λόγους τμηθήσονται.</s> <lb n="1799,001,,,,11,17,2"></lb> |
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13516 <s>Δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΓΔ ὑπὸ παραλλήλων ἐπιπέ-<lb n="1799,001,,,,11,17,3"></lb> |
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13517 δων τῶν ΗΘ, ΚΛ, ΜΝ τεμνέσθωσαν κατὰ τὰ Α, Ε, Β, <lb n="1799,001,,,,11,17,4"></lb> |
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13518 Γ, Ζ, Δ σημεῖα· λέγω, ὅτι ἐστὶν ὡς ἡ ΑΕ εὐθεῖα πρὸς τὴν <lb n="1799,001,,,,11,17,5"></lb> |
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13519 ΕΒ, οὕτως ἡ ΓΖ πρὸς τὴν ΖΔ.</s> <lb n="1799,001,,,,11,17,6"></lb> |
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13520 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΑΓ, ΒΔ, ΑΔ, καὶ συμβαλλέτω <lb n="1799,001,,,,11,17,7"></lb> |
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13521 ἡ ΑΔ τῷ ΚΛ ἐπιπέδῳ κατὰ τὸ Ξ σημεῖον, καὶ ἐπεζεύχθω-<lb n="1799,001,,,,11,17,8"></lb> |
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13522 σαν αἱ ΕΞ, ΞΖ.</s> <s>καὶ ἐπεὶ δύο ἐπίπεδα παράλληλα τὰ ΚΛ, <lb n="1799,001,,,,11,17,9"></lb> |
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13523 ΜΝ ὑπὸ ἐπιπέδου τοῦ ΕΒΔΞ τέμνεται, αἱ κοιναὶ αὐτῶν <pb></pb> <lb n="1799,001,,,,11,17,10"></lb> |
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13524 τομαὶ αἱ ΕΞ, ΒΔ παράλληλοί εἰσιν.</s> <s>διὰ τὰ αὐτὰ δὴ ἐπεὶ <lb n="1799,001,,,,11,17,11"></lb> |
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13525 δύο ἐπίπεδα παράλληλα τὰ ΗΘ, ΚΛ ὑπὸ ἐπιπέδου τοῦ <lb n="1799,001,,,,11,17,12"></lb> |
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13526 ΑΞΖΓ τέμνεται, αἱ κοιναὶ αὐτῶν τομαὶ αἱ ΑΓ, ΞΖ <lb n="1799,001,,,,11,17,13"></lb> |
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13527 παράλληλοί εἰσιν.</s> <s>καὶ ἐπεὶ τριγώνου τοῦ ΑΒΔ παρὰ μίαν <lb n="1799,001,,,,11,17,14"></lb> |
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13528 τῶν πλευρῶν τὴν ΒΔ εὐθεῖα ἦκται ἡ ΕΞ, ἀνάλογον ἄρα <lb n="1799,001,,,,11,17,15"></lb> |
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13529 ἐστὶν ὡς ἡ ΑΕ πρὸς ΕΒ, οὕτως ἡ ΑΞ πρὸς ΞΔ.</s> <s>πάλιν <lb n="1799,001,,,,11,17,16"></lb> |
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13530 ἐπεὶ τριγώνου τοῦ ΑΔΓ παρὰ μίαν τῶν πλευρῶν τὴν ΑΓ <lb n="1799,001,,,,11,17,17"></lb> |
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13531 εὐθεῖα ἦκται ἡ ΞΖ, ἀνάλογόν ἐστιν ὡς ἡ ΑΞ πρὸς ΞΔ, <lb n="1799,001,,,,11,17,18"></lb> |
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13532 οὕτως ἡ ΓΖ πρὸς ΖΔ.</s> <s>ἐδείχθη δὲ καὶ ὡς ἡ ΑΞ πρὸς ΞΔ, <lb n="1799,001,,,,11,17,19"></lb> |
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13533 οὕτως ἡ ΑΕ πρὸς ΕΒ· καὶ ὡς ἄρα ἡ ΑΕ πρὸς ΕΒ, οὕτως <lb n="1799,001,,,,11,17,20"></lb> |
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13534 ἡ ΓΖ πρὸς ΖΔ.</s> <lb n="1799,001,,,,11,17,21"></lb> |
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13535 <s>Ἐὰν ἄρα δύο εὐθεῖαι ὑπὸ παραλλήλων ἐπιπέδων τέμ-<lb n="1799,001,,,,11,17,22"></lb> |
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13536 νωνται, εἰς τοὺς αὐτοὺς λόγους τμηθήσονται· ὅπερ ἔδει <lb n="1799,001,,,,11,17,23"></lb> |
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13537 δεῖξαι.</s> <lb n="1799,001,,,,11,17,24"></lb> |
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13538 <s>Ἐὰν εὐθεῖα ἐπιπέδῳ τινὶ πρὸς ὀρθὰς ᾖ, καὶ πάντα τὰ δι' <lb n="1799,001,,,,11,18,1"></lb> |
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13539 αὐτῆς ἐπίπεδα τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθὰς ἔσται.</s> <lb n="1799,001,,,,11,18,2"></lb> |
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13540 <s>Εὐθεῖα γάρ τις ἡ ΑΒ τῷ ὑποκειμένῳ ἐπιπέδῳ πρὸς <lb n="1799,001,,,,11,18,3"></lb> |
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13541 ὀρθὰς ἔστω· λέγω, ὅτι καὶ πάντα τὰ διὰ τῆς ΑΒ ἐπίπεδα <lb n="1799,001,,,,11,18,4"></lb> |
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13542 τῷ ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <s><pb></pb> <lb n="1799,001,,,,11,18,5"></lb> |
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13543 Ἐκβεβλήσθω γὰρ διὰ τῆς ΑΒ ἐπίπεδον τὸ ΔΕ, καὶ <lb n="1799,001,,,,11,18,6"></lb> |
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13544 ἔστω κοινὴ τομὴ τοῦ ΔΕ ἐπιπέδου καὶ τοῦ ὑποκειμένου <lb n="1799,001,,,,11,18,7"></lb> |
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13545 ἡ ΓΕ, καὶ εἰλήφθω ἐπὶ τῆς ΓΕ τυχὸν σημεῖον τὸ Ζ, καὶ <lb n="1799,001,,,,11,18,8"></lb> |
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13546 ἀπὸ τοῦ Ζ τῇ ΓΕ πρὸς ὀρθὰς <lb n="1799,001,,,,11,18,9"></lb> |
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13547 ἤχθω ἐν τῷ ΔΕ ἐπιπέδῳ ἡ ΖΗ.</s> <lb n="1799,001,,,,11,18,10"></lb> |
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13548 <s>καὶ ἐπεὶ ἡ ΑΒ πρὸς τὸ ὑποκείμενον <lb n="1799,001,,,,11,18,11"></lb> |
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13549 ἐπίπεδον ὀρθή ἐστιν, καὶ πρὸς <lb n="1799,001,,,,11,18,12"></lb> |
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13550 πάσας ἄρα τὰς ἁπτομένας αὐτῆς <lb n="1799,001,,,,11,18,13"></lb> |
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13551 εὐθείας καὶ οὔσας ἐν τῷ ὑποκει-<lb n="1799,001,,,,11,18,14"></lb> |
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13552 μένῳ ἐπιπέδῳ ὀρθή ἐστιν ἡ ΑΒ· <lb n="1799,001,,,,11,18,15"></lb> |
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13553 ὥστε καὶ πρὸς τὴν ΓΕ ὀρθή ἐστιν· ἡ ἄρα ὑπὸ ΑΒΖ γωνία <lb n="1799,001,,,,11,18,16"></lb> |
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13554 ὀρθή ἐστιν.</s> <s>ἔστι δὲ καὶ ἡ ὑπὸ ΗΖΒ ὀρθή· παράλληλος <lb n="1799,001,,,,11,18,17"></lb> |
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13555 ἄρα ἐστὶν ἡ ΑΒ τῇ ΖΗ.</s> <s>ἡ δὲ ΑΒ τῷ ὑποκειμένῳ ἐπιπέδῳ <lb n="1799,001,,,,11,18,18"></lb> |
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13556 πρὸς ὀρθάς ἐστιν· καὶ ἡ ΖΗ ἄρα τῷ ὑποκειμένῳ ἐπιπέδῳ <lb n="1799,001,,,,11,18,19"></lb> |
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13557 πρὸς ὀρθάς ἐστιν.</s> <s>καὶ ἐπίπεδον πρὸς ἐπίπεδον ὀρθόν ἐστιν, <lb n="1799,001,,,,11,18,20"></lb> |
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13558 ὅταν αἱ τῇ κοινῇ τομῇ τῶν ἐπιπέδων πρὸς ὀρθὰς ἀγόμεναι <lb n="1799,001,,,,11,18,21"></lb> |
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13559 εὐθεῖαι ἐν ἑνὶ τῶν ἐπιπέδων τῷ λοιπῷ ἐπιπέδῳ πρὸς ὀρθὰς <lb n="1799,001,,,,11,18,22"></lb> |
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13560 ὦσιν.</s> <s>καὶ τῇ κοινῇ τομῇ τῶν ἐπιπέδων τῇ ΓΕ ἐν ἑνὶ τῶν <lb n="1799,001,,,,11,18,23"></lb> |
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13561 ἐπιπέδων τῷ ΔΕ πρὸς ὀρθὰς ἀχθεῖσα ἡ ΖΗ ἐδείχθη τῷ <lb n="1799,001,,,,11,18,24"></lb> |
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13562 ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθάς· τὸ ἄρα ΔΕ ἐπίπεδον <lb n="1799,001,,,,11,18,25"></lb> |
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13563 ὀρθόν ἐστι πρὸς τὸ ὑποκείμενον.</s> <s>ὁμοίως δὴ δειχθήσεται <lb n="1799,001,,,,11,18,26"></lb> |
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13564 καὶ πάντα τὰ διὰ τῆς ΑΒ ἐπίπεδα ὀρθὰ τυγχάνοντα πρὸς <lb n="1799,001,,,,11,18,27"></lb> |
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13565 τὸ ὑποκείμενον ἐπίπεδον.</s> <lb n="1799,001,,,,11,18,28"></lb> |
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13566 <s>Ἐὰν ἄρα εὐθεῖα ἐπιπέδῳ τινὶ πρὸς ὀρθὰς ᾖ, καὶ πάντα <lb n="1799,001,,,,11,18,29"></lb> |
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13567 τὰ δι' αὐτῆς ἐπίπεδα τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθὰς ἔσται· <lb n="1799,001,,,,11,18,30"></lb> |
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13568 ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,18,31"></lb> |
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13569 Ἐὰν δύο ἐπίπεδα τέμνοντα ἄλληλα ἐπιπέδῳ τινὶ πρὸς <lb n="1799,001,,,,11,19,1"></lb> |
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13570 ὀρθὰς ᾖ, καὶ ἡ κοινὴ αὐτῶν τομὴ τῷ αὐτῷ ἐπιπέδῳ πρὸς <lb n="1799,001,,,,11,19,2"></lb> |
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13571 ὀρθὰς ἔσται.</s> <lb n="1799,001,,,,11,19,3"></lb> |
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13572 <s>Δύο γὰρ ἐπίπεδα τὰ ΑΒ, ΒΓ τῷ ὑποκειμένῳ ἐπιπέδῳ <lb n="1799,001,,,,11,19,4"></lb> |
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13573 πρὸς ὀρθὰς ἔστω, κοινὴ δὲ αὐτῶν τομὴ ἔστω ἡ ΒΔ· λέγω, <lb n="1799,001,,,,11,19,5"></lb> |
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13574 ὅτι ἡ ΒΔ τῷ ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <lb n="1799,001,,,,11,19,6"></lb> |
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13575 <s>Μὴ γάρ, καὶ ἤχθωσαν ἀπὸ <lb n="1799,001,,,,11,19,7"></lb> |
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13576 τοῦ Δ σημείου ἐν μὲν τῷ ΑΒ <lb n="1799,001,,,,11,19,8"></lb> |
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13577 ἐπιπέδῳ τῇ ΑΔ εὐθείᾳ πρὸς <lb n="1799,001,,,,11,19,9"></lb> |
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13578 ὀρθὰς ἡ ΔΕ, ἐν δὲ τῷ ΒΓ <lb n="1799,001,,,,11,19,10"></lb> |
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13579 ἐπιπέδῳ τῇ ΓΔ πρὸς ὀρθὰς ἡ <lb n="1799,001,,,,11,19,11"></lb> |
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13580 ΔΖ.</s> <s>καὶ ἐπεὶ τὸ ΑΒ ἐπίπεδον <lb n="1799,001,,,,11,19,12"></lb> |
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13581 ὀρθόν ἐστι πρὸς τὸ ὑποκείμε-<lb n="1799,001,,,,11,19,13"></lb> |
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13582 νον, καὶ τῇ κοινῇ αὐτῶν τομῇ <lb n="1799,001,,,,11,19,14"></lb> |
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13583 τῇ ΑΔ πρὸς ὀρθὰς ἐν τῷ ΑΒ <lb n="1799,001,,,,11,19,15"></lb> |
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13584 ἐπιπέδῳ ἦκται ἡ ΔΕ, ἡ ΔΕ <lb n="1799,001,,,,11,19,16"></lb> |
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13585 ἄρα ὀρθή ἐστι πρὸς τὸ ὑποκεί-<lb n="1799,001,,,,11,19,17"></lb> |
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13586 μενον ἐπίπεδον.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ ΔΖ ὀρθή <lb n="1799,001,,,,11,19,18"></lb> |
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13587 ἐστι πρὸς τὸ ὑποκείμενον ἐπίπεδον.</s> <s>ἀπὸ τοῦ αὐτοῦ ἄρα <lb n="1799,001,,,,11,19,19"></lb> |
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13588 σημείου τοῦ Δ τῷ ὑποκειμένῳ ἐπιπέδῳ δύο εὐθεῖαι πρὸς <lb n="1799,001,,,,11,19,20"></lb> |
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13589 ὀρθὰς ἀνεσταμέναι εἰσὶν ἐπὶ τὰ αὐτὰ μέρη· ὅπερ ἐστὶν <lb n="1799,001,,,,11,19,21"></lb> |
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13590 ἀδύνατον.</s> <s>οὐκ ἄρα τῷ ὑποκειμένῳ ἐπιπέδῳ ἀπὸ τοῦ Δ <pb></pb> <lb n="1799,001,,,,11,19,22"></lb> |
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13591 σημείου ἀνασταθήσεται πρὸς ὀρθὰς πλὴν τῆς ΔΒ κοινῆς <lb n="1799,001,,,,11,19,23"></lb> |
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13592 τομῆς τῶν ΑΒ, ΒΓ ἐπιπέδων.</s> <lb n="1799,001,,,,11,19,24"></lb> |
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13593 <s>Ἐὰν ἄρα δύο ἐπίπεδα τέμνοντα ἄλληλα ἐπιπέδῳ τινὶ <lb n="1799,001,,,,11,19,25"></lb> |
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13594 πρὸς ὀρθὰς ᾖ, καὶ ἡ κοινὴ αὐτῶν τομὴ τῷ αὐτῷ ἐπιπέδῳ <lb n="1799,001,,,,11,19,26"></lb> |
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13595 πρὸς ὀρθὰς ἔσται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,19,27"></lb> |
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13596 <s>Ἐὰν στερεὰ γωνία ὑπὸ τριῶν γωνιῶν ἐπιπέδων περιέχη-<lb n="1799,001,,,,11,20,1"></lb> |
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13597 ται, δύο ὁποιαιοῦν τῆς λοιπῆς μείζονές εἰσι πάντῃ μετα-<lb n="1799,001,,,,11,20,2"></lb> |
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13598 λαμβανόμεναι.</s> <lb n="1799,001,,,,11,20,3"></lb> |
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13599 <s>Στερεὰ γὰρ γωνία ἡ πρὸς τῷ Α ὑπὸ τριῶν γωνιῶν ἐπι-<lb n="1799,001,,,,11,20,4"></lb> |
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13600 πέδων τῶν ὑπὸ ΒΑΓ, ΓΑΔ, ΔΑΒ περιεχέσθω· λέγω, <lb n="1799,001,,,,11,20,5"></lb> |
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13601 ὅτι τῶν ὑπὸ ΒΑΓ, ΓΑΔ, <lb n="1799,001,,,,11,20,6"></lb> |
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13602 ΔΑΒ γωνιῶν δύο ὁποι-<lb n="1799,001,,,,11,20,7"></lb> |
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13603 αιοῦν τῆς λοιπῆς μείζο-<lb n="1799,001,,,,11,20,8"></lb> |
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13604 νές εἰσι πάντῃ μεταλαμ-<lb n="1799,001,,,,11,20,9"></lb> |
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13605 βανόμεναι.</s> <lb n="1799,001,,,,11,20,10"></lb> |
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13606 <s>Εἰ μὲν οὖν αἱ ὑπὸ <lb n="1799,001,,,,11,20,11"></lb> |
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13607 ΒΑΓ, ΓΑΔ, ΔΑΒ γω-<lb n="1799,001,,,,11,20,12"></lb> |
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13608 νίαι ἴσαι ἀλλήλαις εἰσίν, <lb n="1799,001,,,,11,20,13"></lb> |
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13609 φανερόν, ὅτι δύο ὁποιαιοῦν τῆς λοιπῆς μείζονές εἰσιν.</s> <s>εἰ <lb n="1799,001,,,,11,20,14"></lb> |
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13610 δὲ οὔ, ἔστω μείζων ἡ ὑπὸ ΒΑΓ, καὶ συνεστάτω πρὸς <lb n="1799,001,,,,11,20,15"></lb> |
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13611 τῇ ΑΒ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Α τῇ <lb n="1799,001,,,,11,20,16"></lb> |
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13612 ὑπὸ ΔΑΒ γωνίᾳ ἐν τῷ διὰ τῶν ΒΑΓ ἐπιπέδῳ ἴση ἡ ὑπὸ <pb></pb> <lb n="1799,001,,,,11,20,17"></lb> |
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13613 ΒΑΕ, καὶ κείσθω τῇ ΑΔ ἴση ἡ ΑΕ, καὶ διὰ τοῦ Ε σημείου <lb n="1799,001,,,,11,20,18"></lb> |
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13614 διαχθεῖσα ἡ ΒΕΓ τεμνέτω τὰς ΑΒ, ΑΓ εὐθείας κατὰ <lb n="1799,001,,,,11,20,19"></lb> |
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13615 τὰ Β, Γ σημεῖα, καὶ ἐπεζεύχθωσαν αἱ ΔΒ, ΔΓ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,11,20,20"></lb> |
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13616 ἴση ἐστὶν ἡ ΔΑ τῇ ΑΕ, κοινὴ δὲ ἡ ΑΒ, δύο δυσὶν ἴσαι· καὶ <lb n="1799,001,,,,11,20,21"></lb> |
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13617 γωνία ἡ ὑπὸ ΔΑΒ γωνίᾳ τῇ ὑπὸ ΒΑΕ ἴση· βάσις ἄρα ἡ <lb n="1799,001,,,,11,20,22"></lb> |
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13618 ΔΒ βάσει τῇ ΒΕ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ δύο αἱ ΒΔ, ΔΓ τῆς <lb n="1799,001,,,,11,20,23"></lb> |
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13619 ΒΓ μείζονές εἰσιν, ὧν ἡ ΔΒ τῇ ΒΕ ἐδείχθη ἴση, λοιπὴ <lb n="1799,001,,,,11,20,24"></lb> |
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13620 ἄρα ἡ ΔΓ λοιπῆς τῆς ΕΓ μείζων ἐστίν.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν <lb n="1799,001,,,,11,20,25"></lb> |
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13621 ἡ ΔΑ τῇ ΑΕ, κοινὴ δὲ ἡ ΑΓ, καὶ βάσις ἡ ΔΓ βάσεως τῆς <lb n="1799,001,,,,11,20,26"></lb> |
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13622 ΕΓ μείζων ἐστίν, γωνία ἄρα ἡ ὑπὸ ΔΑΓ γωνίας τῆς ὑπὸ <lb n="1799,001,,,,11,20,27"></lb> |
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13623 ΕΑΓ μείζων ἐστίν.</s> <s>ἐδείχθη δὲ καὶ ἡ ὑπὸ ΔΑΒ τῇ ὑπὸ <lb n="1799,001,,,,11,20,28"></lb> |
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13624 ΒΑΕ ἴση· αἱ ἄρα ὑπὸ ΔΑΒ, ΔΑΓ τῆς ὑπὸ ΒΑΓ μεί-<lb n="1799,001,,,,11,20,29"></lb> |
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13625 ζονές εἰσιν.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ λοιπαὶ σύνδυο <lb n="1799,001,,,,11,20,30"></lb> |
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13626 λαμβανόμεναι τῆς λοιπῆς μείζονές εἰσιν.</s> <lb n="1799,001,,,,11,20,31"></lb> |
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13627 <s>Ἐὰν ἄρα στερεὰ γωνία ὑπὸ τριῶν γωνιῶν ἐπιπέδων <lb n="1799,001,,,,11,20,32"></lb> |
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13628 περιέχηται, δύο ὁποιαιοῦν τῆς λοιπῆς μείζονές εἰσι πάντῃ <lb n="1799,001,,,,11,20,33"></lb> |
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13629 μεταλαμβανόμεναι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,20,34"></lb> |
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13630 <s>Ἅπασα στερεὰ γωνία ὑπὸ ἐλασσόνων [ἢ] τεσσάρων <lb n="1799,001,,,,11,21,1"></lb> |
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13631 ὀρθῶν γωνιῶν ἐπιπέδων περιέχεται.</s> <lb n="1799,001,,,,11,21,2"></lb> |
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13632 <s>Ἔστω στερεὰ γωνία ἡ πρὸς τῷ Α περιεχομένη ὑπὸ <lb n="1799,001,,,,11,21,3"></lb> |
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13633 ἐπιπέδων γωνιῶν τῶν ὑπὸ ΒΑΓ, ΓΑΔ, ΔΑΒ· λέγω, <pb></pb> <lb n="1799,001,,,,11,21,4"></lb> |
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13634 ὅτι αἱ ὑπὸ ΒΑΓ, ΓΑΔ, ΔΑΒ τεσσάρων ὀρθῶν ἐλάσ-<lb n="1799,001,,,,11,21,5"></lb> |
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13635 σονές εἰσιν.</s> <lb n="1799,001,,,,11,21,6"></lb> |
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13636 <s>Εἰλήφθω γὰρ ἐφ' ἑκάστης τῶν ΑΒ, ΑΓ, ΑΔ τυχόντα <lb n="1799,001,,,,11,21,7"></lb> |
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13637 σημεῖα τὰ Β, Γ, Δ, καὶ ἐπεζεύχθωσαν αἱ ΒΓ, ΓΔ, ΔΒ.</s> <lb n="1799,001,,,,11,21,8"></lb> |
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13638 <s>καὶ ἐπεὶ στερεὰ γωνία ἡ πρὸς τῷ Β ὑπὸ τριῶν γωνιῶν ἐπι-<lb n="1799,001,,,,11,21,9"></lb> |
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13639 πέδων περιέχεται τῶν ὑπὸ ΓΒΑ, <lb n="1799,001,,,,11,21,10"></lb> |
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13640 ΑΒΔ, ΓΒΔ, δύο ὁποιαιοῦν τῆς <lb n="1799,001,,,,11,21,11"></lb> |
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13641 λοιπῆς μείζονές εἰσιν· αἱ ἄρα <lb n="1799,001,,,,11,21,12"></lb> |
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13642 ὑπὸ ΓΒΑ, ΑΒΔ τῆς ὑπὸ ΓΒΔ <lb n="1799,001,,,,11,21,13"></lb> |
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13643 μείζονές εἰσιν.</s> <s>διὰ τὰ αὐτὰ δὴ <lb n="1799,001,,,,11,21,14"></lb> |
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13644 καὶ αἱ μὲν ὑπὸ ΒΓΑ, ΑΓΔ τῆς <lb n="1799,001,,,,11,21,15"></lb> |
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13645 ὑπὸ ΒΓΔ μείζονές εἰσιν, αἱ δὲ <lb n="1799,001,,,,11,21,16"></lb> |
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13646 ὑπὸ ΓΔΑ, ΑΔΒ τῆς ὑπὸ ΓΔΒ μείζονές εἰσιν· αἱ ἓξ ἄρα <lb n="1799,001,,,,11,21,17"></lb> |
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13647 γωνίαι αἱ ὑπὸ ΓΒΑ, ΑΒΔ, ΒΓΑ, ΑΓΔ, ΓΔΑ, ΑΔΒ <lb n="1799,001,,,,11,21,18"></lb> |
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13648 τριῶν τῶν ὑπὸ ΓΒΔ, ΒΓΔ, ΓΔΒ μείζονές εἰσιν.</s> <s>ἀλλὰ αἱ <lb n="1799,001,,,,11,21,19"></lb> |
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13649 τρεῖς αἱ ὑπὸ ΓΒΔ, ΒΔΓ, ΒΓΔ δυσὶν ὀρθαῖς ἴσαι εἰσίν· <lb n="1799,001,,,,11,21,20"></lb> |
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13650 αἱ ἓξ ἄρα αἱ ὑπὸ ΓΒΑ, ΑΒΔ, ΒΓΑ, ΑΓΔ, ΓΔΑ, ΑΔΒ <lb n="1799,001,,,,11,21,21"></lb> |
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13651 δύο ὀρθῶν μείζονές εἰσιν.</s> <s>καὶ ἐπεὶ ἑκάστου τῶν ΑΒΓ, <lb n="1799,001,,,,11,21,22"></lb> |
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13652 ΑΓΔ, ΑΔΒ τριγώνων αἱ τρεῖς γωνίαι δυσὶν ὀρθαῖς ἴσαι <lb n="1799,001,,,,11,21,23"></lb> |
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13653 εἰσίν, αἱ ἄρα τῶν τριῶν τριγώνων ἐννέα γωνίαι αἱ ὑπὸ <lb n="1799,001,,,,11,21,24"></lb> |
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13654 ΓΒΑ, ΑΓΒ, ΒΑΓ, ΑΓΔ, ΓΔΑ, ΓΑΔ, ΑΔΒ, ΔΒΑ, <lb n="1799,001,,,,11,21,25"></lb> |
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13655 ΒΑΔ ἓξ ὀρθαῖς ἴσαι εἰσίν, ὧν αἱ ὑπὸ ΑΒΓ, ΒΓΑ, ΑΓΔ, <lb n="1799,001,,,,11,21,26"></lb> |
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13656 ΓΔΑ, ΑΔΒ, ΔΒΑ ἓξ γωνίαι δύο ὀρθῶν εἰσι μείζονες· <pb></pb> <lb n="1799,001,,,,11,21,27"></lb> |
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13657 λοιπαὶ ἄρα αἱ ὑπὸ ΒΑΓ, ΓΑΔ, ΔΑΒ τρεῖς [γωνίαι] <lb n="1799,001,,,,11,21,28"></lb> |
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13658 περιέχουσαι τὴν στερεὰν γωνίαν τεσσάρων ὀρθῶν ἐλάσ-<lb n="1799,001,,,,11,21,29"></lb> |
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13659 σονές εἰσιν.</s> <lb n="1799,001,,,,11,21,30"></lb> |
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13660 <s>Ἅπασα ἄρα στερεὰ γωνία ὑπὸ ἐλασσόνων [ἢ] τεσσάρων <lb n="1799,001,,,,11,21,31"></lb> |
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13661 ὀρθῶν γωνιῶν ἐπιπέδων περιέχεται· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,21,32"></lb> |
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13662 <s>Ἐὰν ὦσι τρεῖς γωνίαι ἐπίπεδοι, ὧν αἱ δύο τῆς λοιπῆς <lb n="1799,001,,,,11,22,1"></lb> |
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13663 μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, περιέχωσι δὲ αὐ-<lb n="1799,001,,,,11,22,2"></lb> |
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13664 τὰς ἴσαι εὐθεῖαι, δυνατόν ἐστιν ἐκ τῶν ἐπιζευγνυουσῶν <lb n="1799,001,,,,11,22,3"></lb> |
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13665 τὰς ἴσας εὐθείας τρίγωνον συστήσασθαι. <lb n="1799,001,,,,11,22,4"></lb> |
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13666 Ἔστωσαν τρεῖς γωνίαι ἐπίπεδοι αἱ ὑπὸ ΑΒΓ, ΔΕΖ, <lb n="1799,001,,,,11,22,5"></lb> |
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13667 ΗΘΚ, ὧν αἱ δύο τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμ-<lb n="1799,001,,,,11,22,6"></lb> |
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13668 βανόμεναι, αἱ μὲν ὑπὸ ΑΒΓ, ΔΕΖ τῆς ὑπὸ ΗΘΚ, αἱ δὲ <lb n="1799,001,,,,11,22,7"></lb> |
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13669 ὑπὸ ΔΕΖ, ΗΘΚ τῆς ὑπὸ ΑΒΓ, καὶ ἔτι αἱ ὑπὸ ΗΘΚ, <lb n="1799,001,,,,11,22,8"></lb> |
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13670 ΑΒΓ τῆς ὑπὸ ΔΕΖ, καὶ ἔστωσαν ἴσαι αἱ ΑΒ, ΒΓ, ΔΕ, <lb n="1799,001,,,,11,22,9"></lb> |
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13671 ΕΖ, ΗΘ, ΘΚ εὐθεῖαι, καὶ ἐπεζεύχθωσαν αἱ ΑΓ, ΔΖ, <lb n="1799,001,,,,11,22,10"></lb> |
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13672 ΗΚ· λέγω, ὅτι δυνατόν ἐστιν ἐκ τῶν ἴσων ταῖς ΑΓ, ΔΖ, <pb></pb> <lb n="1799,001,,,,11,22,11"></lb> |
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13673 ΗΚ τρίγωνον συστήσασθαι, τουτέστιν ὅτι τῶν ΑΓ, ΔΖ, <lb n="1799,001,,,,11,22,12"></lb> |
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13674 ΗΚ δύο ὁποιαιοῦν τῆς λοιπῆς μείζονές εἰσιν.</s> <lb n="1799,001,,,,11,22,13"></lb> |
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13675 <s>Εἰ μὲν οὖν αἱ ὑπὸ ΑΒΓ, ΔΕΖ, ΗΘΚ γωνίαι ἴσαι <lb n="1799,001,,,,11,22,14"></lb> |
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13676 ἀλλήλαις εἰσίν, φανερόν, ὅτι καὶ τῶν ΑΓ, ΔΖ, ΗΚ ἴσων <lb n="1799,001,,,,11,22,15"></lb> |
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13677 γινομένων δυνατόν ἐστιν ἐκ τῶν ἴσων ταῖς ΑΓ, ΔΖ, ΗΚ <lb n="1799,001,,,,11,22,16"></lb> |
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13678 τρίγωνον συστήσασθαι.</s> <s>εἰ δὲ οὔ, ἔστωσαν ἄνισοι, καὶ συν-<lb n="1799,001,,,,11,22,17"></lb> |
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13679 εστάτω πρὸς τῇ ΘΚ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ <lb n="1799,001,,,,11,22,18"></lb> |
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13680 Θ τῇ ὑπὸ ΑΒΓ γωνίᾳ ἴση ἡ ὑπὸ ΚΘΛ· καὶ κείσθω μιᾷ <lb n="1799,001,,,,11,22,19"></lb> |
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13681 τῶν ΑΒ, ΒΓ, ΔΕ, ΕΖ, ΗΘ, ΘΚ ἴση ἡ ΘΛ, καὶ ἐπεζεύχ-<lb n="1799,001,,,,11,22,20"></lb> |
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13682 θωσαν αἱ ΚΛ, ΗΛ.</s> <s>καὶ ἐπεὶ δύο αἱ ΑΒ, ΒΓ δυσὶ ταῖς <lb n="1799,001,,,,11,22,21"></lb> |
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13683 ΚΘ, ΘΛ ἴσαι εἰσίν, καὶ γωνία <lb n="1799,001,,,,11,22,22"></lb> |
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13684 ἡ πρὸς τῷ Β γωνίᾳ τῇ ὑπὸ ΚΘΛ <lb n="1799,001,,,,11,22,23"></lb> |
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13685 ἴση, βάσις ἄρα ἡ ΑΓ βάσει τῇ <lb n="1799,001,,,,11,22,24"></lb> |
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13686 ΚΛ ἴση.</s> <s>καὶ ἐπεὶ αἱ ὑπὸ ΑΒΓ, <lb n="1799,001,,,,11,22,25"></lb> |
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13687 ΗΘΚ τῆς ὑπὸ ΔΕΖ μείζονές <lb n="1799,001,,,,11,22,26"></lb> |
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13688 εἰσιν, ἴση δὲ ἡ ὑπὸ ΑΒΓ τῇ ὑπὸ <lb n="1799,001,,,,11,22,27"></lb> |
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13689 ΚΘΛ, ἡ ἄρα ὑπὸ ΗΘΛ τῆς ὑπὸ <lb n="1799,001,,,,11,22,28"></lb> |
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13690 ΔΕΖ μείζων ἐστίν.</s> <s>καὶ ἐπεὶ δύο αἱ ΗΘ, ΘΛ δύο ταῖς ΔΕ, <lb n="1799,001,,,,11,22,29"></lb> |
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13691 ΕΖ ἴσαι εἰσίν, καὶ γωνία ἡ ὑπὸ ΗΘΛ γωνίας τῆς ὑπὸ ΔΕΖ <lb n="1799,001,,,,11,22,30"></lb> |
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13692 μείζων, βάσις ἄρα ἡ ΗΛ βάσεως τῆς ΔΖ μείζων ἐστίν.</s> <lb n="1799,001,,,,11,22,31"></lb> |
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13693 <s>ἀλλὰ αἱ ΗΚ, ΚΛ τῆς ΗΛ μείζονές εἰσιν.</s> <s>πολλῷ ἄρα αἱ <lb n="1799,001,,,,11,22,32"></lb> |
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13694 ΗΚ, ΚΛ τῆς ΔΖ μείζονές εἰσιν.</s> <s>ἴση δὲ ἡ ΚΛ τῇ ΑΓ· αἱ <pb></pb> <lb n="1799,001,,,,11,22,33"></lb> |
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13695 ΑΓ, ΗΚ ἄρα τῆς λοιπῆς τῆς ΔΖ μείζονές εἰσιν.</s> <s>ὁμοίως <lb n="1799,001,,,,11,22,34"></lb> |
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13696 δὴ δείξομεν, ὅτι καὶ αἱ μὲν ΑΓ, ΔΖ τῆς ΗΚ μείζονές <lb n="1799,001,,,,11,22,35"></lb> |
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13697 εἰσιν, καὶ ἔτι αἱ ΔΖ, ΗΚ τῆς ΑΓ μείζονές εἰσιν.</s> <s>δυνατὸν <lb n="1799,001,,,,11,22,36"></lb> |
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13698 ἄρα ἐστὶν ἐκ τῶν ἴσων ταῖς ΑΓ, ΔΖ, ΗΚ τρίγωνον συστή-<lb n="1799,001,,,,11,22,37"></lb> |
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13699 σασθαι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,22,38"></lb> |
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13700 <s>Ἐκ τριῶν γωνιῶν ἐπιπέδων, ὧν αἱ δύο τῆς λοιπῆς μεί-<lb n="1799,001,,,,11,23,1"></lb> |
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13701 ζονές εἰσι πάντῃ μεταλαμβανόμεναι, στερεὰν γωνίαν συ-<lb n="1799,001,,,,11,23,2"></lb> |
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13702 στήσασθαι· δεῖ δὴ τὰς τρεῖς τεσσάρων ὀρθῶν ἐλάσσονας <lb n="1799,001,,,,11,23,3"></lb> |
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13703 εἶναι.</s> <lb n="1799,001,,,,11,23,4"></lb> |
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13704 <s>Ἔστωσαν αἱ δοθεῖσαι τρεῖς γωνίαι ἐπίπεδοι αἱ ὑπὸ <lb n="1799,001,,,,11,23,5"></lb> |
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13705 ΑΒΓ, ΔΕΖ, ΗΘΚ, ὧν αἱ δύο τῆς λοιπῆς μείζονεσ ἔστω-<lb n="1799,001,,,,11,23,6"></lb> |
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13706 σαν πάντῃ μεταλαμβανόμεναι, ἔτι δὲ αἱ τρεῖς τεσσάρων <lb n="1799,001,,,,11,23,7"></lb> |
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13707 ὀρθῶν ἐλάσσονες· δεῖ δὴ ἐκ τῶν ἴσων ταῖς ὑπὸ ΑΒΓ, ΔΕΖ, <lb n="1799,001,,,,11,23,8"></lb> |
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13708 ΗΘΚ στερεὰν γωνίαν συστήσασθαι. <pb></pb> <lb n="1799,001,,,,11,23,9"></lb> |
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13709 Ἀπειλήφθωσαν ἴσαι αἱ ΑΒ, ΒΓ, ΔΕ, ΕΖ, ΗΘ, ΘΚ, <lb n="1799,001,,,,11,23,10"></lb> |
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13710 καὶ ἐπεζεύχθωσαν αἱ ΑΓ, ΔΖ, ΗΚ· δυνατὸν ἄρα ἐστὶν <lb n="1799,001,,,,11,23,11"></lb> |
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13711 ἐκ τῶν ἴσων ταῖς ΑΓ, ΔΖ, ΗΚ τρίγωνον συστήσασθαι.</s> <lb n="1799,001,,,,11,23,12"></lb> |
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13712 <s>συνεστάτω τὸ ΛΜΝ, ὥστε ἴσην εἶναι τὴν μὲν ΑΓ τῇ <lb n="1799,001,,,,11,23,13"></lb> |
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13713 ΛΜ, τὴν δὲ ΔΖ τῇ ΜΝ, καὶ ἔτι τὴν ΗΚ τῇ ΝΛ, καὶ <lb n="1799,001,,,,11,23,14"></lb> |
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13714 περιγεγράφθω περὶ τὸ ΛΜΝ τρίγωνον κύκλος ὁ ΛΜΝ <lb n="1799,001,,,,11,23,15"></lb> |
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13715 καὶ εἰλήφθω αὐτοῦ τὸ κέντρον καὶ ἔστω τὸ Ξ, καὶ ἐπε-<lb n="1799,001,,,,11,23,16"></lb> |
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13716 ζεύχθωσαν αἱ ΛΞ, ΜΞ, ΝΞ· λέγω, ὅτι ἡ ΑΒ μείζων <lb n="1799,001,,,,11,23,17"></lb> |
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13717 ἐστὶ τῆς ΛΞ.</s> <s>εἰ γὰρ μή, ἤτοι ἴση ἐστὶν ἡ ΑΒ τῇ ΛΞ ἢ <lb n="1799,001,,,,11,23,18"></lb> |
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13718 ἐλάττων.</s> <s>ἔστω πρότερον ἴση.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒ τῇ <lb n="1799,001,,,,11,23,19"></lb> |
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13719 ΛΞ, ἀλλὰ ἡ μὲν ΑΒ τῇ ΒΓ ἐστιν ἴση, ἡ δὲ ΞΛ τῇ ΞΜ, <lb n="1799,001,,,,11,23,20"></lb> |
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13720 δύο δὴ αἱ ΑΒ, ΒΓ δύο ταῖς ΛΞ, ΞΜ ἴσαι εἰσὶν ἑκατέρα <lb n="1799,001,,,,11,23,21"></lb> |
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13721 ἑκατέρᾳ· καὶ βάσις ἡ ΑΓ βάσει τῇ ΛΜ ὑπόκειται ἴση· <lb n="1799,001,,,,11,23,22"></lb> |
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13722 γωνία ἄρα ἡ ὑπὸ ΑΒΓ γωνίᾳ τῇ ὑπὸ ΛΞΜ ἐστιν ἴση.</s> <lb n="1799,001,,,,11,23,23"></lb> |
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13723 <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ μὲν ὑπὸ <lb n="1799,001,,,,11,23,24"></lb> |
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13724 ΔΕΖ τῇ ὑπὸ ΜΞΝ ἐστιν ἴση, <lb n="1799,001,,,,11,23,25"></lb> |
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13725 καὶ ἔτι ἡ ὑπὸ ΗΘΚ τῇ ὑπὸ <lb n="1799,001,,,,11,23,26"></lb> |
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13726 ΝΞΛ· αἱ ἄρα τρεῖς αἱ ὑπὸ <lb n="1799,001,,,,11,23,27"></lb> |
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13727 ΑΒΓ, ΔΕΖ, ΗΘΚ γωνίαι <lb n="1799,001,,,,11,23,28"></lb> |
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13728 τρισὶ ταῖς ὑπὸ ΛΞΜ, ΜΞΝ, <lb n="1799,001,,,,11,23,29"></lb> |
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13729 ΝΞΛ εἰσιν ἴσαι.</s> <s>ἀλλὰ αἱ τρεῖς <lb n="1799,001,,,,11,23,30"></lb> |
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13730 αἱ ὑπὸ ΛΞΜ, ΜΞΝ, ΝΞΛ <pb></pb> <lb n="1799,001,,,,11,23,31"></lb> |
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13731 τέτταρσιν ὀρθαῖς εἰσιν ἴσαι· καὶ αἱ τρεῖς ἄρα αἱ ὑπὸ <lb n="1799,001,,,,11,23,32"></lb> |
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13732 ΑΒΓ, ΔΕΖ, ΗΘΚ τέτταρσιν ὀρθαῖς ἴσαι εἰσίν.</s> <s>ὑπό-<lb n="1799,001,,,,11,23,33"></lb> |
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13733 κεινται δὲ καὶ τεσσάρων ὀρθῶν ἐλάσσονες· ὅπερ ἄτο-<lb n="1799,001,,,,11,23,34"></lb> |
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13734 πον.</s> <s>οὐκ ἄρα ἡ ΑΒ τῇ ΛΞ ἴση ἐστίν.</s> <s>λέγω δή, ὅτι οὐδὲ <lb n="1799,001,,,,11,23,35"></lb> |
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13735 ἐλάττων ἐστὶν ἡ ΑΒ τῆς ΛΞ.</s> <s>εἰ γὰρ δυνατόν, ἔστω· καὶ <lb n="1799,001,,,,11,23,36"></lb> |
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13736 κείσθω τῇ μὲν ΑΒ ἴση ἡ ΞΟ, τῇ δὲ ΒΓ ἴση ἡ ΞΠ, καὶ <lb n="1799,001,,,,11,23,37"></lb> |
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13737 ἐπεζεύχθω ἡ ΟΠ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒ τῇ ΒΓ, ἴση <lb n="1799,001,,,,11,23,38"></lb> |
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13738 ἐστὶ καὶ ἡ ΞΟ τῇ ΞΠ· ὥστε καὶ λοιπὴ ἡ ΛΟ τῇ ΠΜ <lb n="1799,001,,,,11,23,39"></lb> |
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13739 ἐστιν ἴση.</s> <s>παράλληλος ἄρα ἐστὶν ἡ ΛΜ τῇ ΟΠ, καὶ <lb n="1799,001,,,,11,23,40"></lb> |
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13740 ἰσογώνιον τὸ ΛΜΞ τῷ ΟΠΞ· ἔστιν ἄρα ὡς ἡ ΞΛ πρὸς <lb n="1799,001,,,,11,23,41"></lb> |
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13741 ΛΜ, οὕτως ἡ ΞΟ πρὸς ΟΠ· ἐναλλὰξ ὡς ἡ ΛΞ πρὸς ΞΟ, <lb n="1799,001,,,,11,23,42"></lb> |
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13742 οὕτως ἡ ΛΜ πρὸς ΟΠ.</s> <s>μείζων δὲ ἡ ΛΞ τῆς ΞΟ· μείζων <lb n="1799,001,,,,11,23,43"></lb> |
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13743 ἄρα καὶ ἡ ΛΜ τῆς ΟΠ.</s> <s>ἀλλὰ ἡ ΛΜ κεῖται τῇ ΑΓ ἴση· <lb n="1799,001,,,,11,23,44"></lb> |
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13744 καὶ ἡ ΑΓ ἄρα τῆς ΟΠ μείζων ἐστίν.</s> <s>ἐπεὶ οὖν δύο αἱ ΑΒ, <lb n="1799,001,,,,11,23,45"></lb> |
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13745 ΒΓ δυσὶ ταῖς ΟΞ, ΞΠ ἴσαι εἰσίν, καὶ βάσις ἡ ΑΓ βάσεως <lb n="1799,001,,,,11,23,46"></lb> |
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13746 τῆς ΟΠ μείζων ἐστίν, γωνία ἄρα ἡ ὑπὸ ΑΒΓ γωνίας τῆς <lb n="1799,001,,,,11,23,47"></lb> |
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13747 ὑπὸ ΟΞΠ μείζων ἐστίν.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ <lb n="1799,001,,,,11,23,48"></lb> |
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13748 μὲν ὑπὸ ΔΕΖ τῆς ὑπὸ ΜΞΝ μείζων ἐστίν, ἡ δὲ ὑπὸ ΗΘΚ <lb n="1799,001,,,,11,23,49"></lb> |
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13749 τῆς ὑπὸ ΝΞΛ.</s> <s>αἱ ἄρα τρεῖς γωνίαι αἱ ὑπὸ ΑΒΓ, ΔΕΖ, <lb n="1799,001,,,,11,23,50"></lb> |
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13750 ΗΘΚ τριῶν τῶν ὑπὸ ΛΞΜ, ΜΞΝ, ΝΞΛ μείζονές εἰσιν.</s> <lb n="1799,001,,,,11,23,51"></lb> |
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13751 <s>ἀλλὰ αἱ ὑπὸ ΑΒΓ, ΔΕΖ, ΗΘΚ τεσσάρων ὀρθῶν ἐλάσ-<lb n="1799,001,,,,11,23,52"></lb> |
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13752 σονες ὑπόκεινται· πολλῷ ἄρα αἱ ὑπὸ ΛΞΜ, ΜΞΝ, <pb></pb> <lb n="1799,001,,,,11,23,53"></lb> |
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13753 ΝΞΛ τεσσάρων ὀρθῶν ἐλάσσονές εἰσιν.</s> <s>ἀλλὰ καὶ ἴσαι· <lb n="1799,001,,,,11,23,54"></lb> |
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13754 ὅπερ ἐστὶν ἄτοπον.</s> <s>οὐκ ἄρα ἡ ΑΒ ἐλάσσων ἐστὶ τῆς ΛΞ.</s> <lb n="1799,001,,,,11,23,55"></lb> |
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13755 <s>ἐδείχθη δέ, ὅτι οὐδὲ ἴση· μείζων ἄρα ἡ ΑΒ τῆς ΛΞ.</s> <s>ἀν-<lb n="1799,001,,,,11,23,56"></lb> |
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13756 εστάτω δὴ ἀπὸ τοῦ Ξ σημείου τῷ τοῦ ΛΜΝ κύκλου ἐπι-<lb n="1799,001,,,,11,23,57"></lb> |
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13757 πέδῳ πρὸς ὀρθὰς ἡ ΞΡ, καὶ ᾧ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΑΒ <lb n="1799,001,,,,11,23,58"></lb> |
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13758 τετράγωνον τοῦ ἀπὸ τῆς ΛΞ, ἐκείνῳ ἴσον ἔστω τὸ ἀπὸ <lb n="1799,001,,,,11,23,59"></lb> |
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13759 τῆς ΞΡ, καὶ ἐπεζεύχθωσαν αἱ ΡΛ, ΡΜ, ΡΝ.</s> <s>καὶ ἐπεὶ ἡ <lb n="1799,001,,,,11,23,60"></lb> |
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13760 ΡΞ ὀρθή ἐστι πρὸς τὸ τοῦ ΛΜΝ κύκλου ἐπίπεδον, καὶ <lb n="1799,001,,,,11,23,61"></lb> |
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13761 πρὸς ἑκάστην ἄρα τῶν ΛΞ, ΜΞ, ΝΞ ὀρθή ἐστιν ἡ ΡΞ.</s> <lb n="1799,001,,,,11,23,62"></lb> |
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13762 <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΛΞ τῇ ΞΜ, κοινὴ δὲ καὶ πρὸς ὀρθὰς <lb n="1799,001,,,,11,23,63"></lb> |
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13763 ἡ ΞΡ, βάσις ἄρα ἡ ΡΛ βάσει τῇ ΡΜ ἐστιν ἴση.</s> <s>διὰ τὰ <lb n="1799,001,,,,11,23,64"></lb> |
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13764 αὐτὰ δὴ καὶ ἡ ΡΝ ἑκατέρᾳ τῶν ΡΛ, ΡΜ ἐστιν ἴση· αἱ <lb n="1799,001,,,,11,23,65"></lb> |
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13765 τρεῖς ἄρα αἱ ΡΛ, ΡΜ, ΡΝ ἴσαι ἀλλήλαις εἰσίν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,11,23,66"></lb> |
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13766 ᾧ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΛΞ, ἐκείνῳ <lb n="1799,001,,,,11,23,67"></lb> |
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13767 ἴσον ὑπόκειται τὸ ἀπὸ τῆς ΞΡ, τὸ ἄρα ἀπὸ τῆς ΑΒ ἴσον <lb n="1799,001,,,,11,23,68"></lb> |
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13768 ἐστὶ τοῖς ἀπὸ τῶν ΛΞ, ΞΡ.</s> <s>τοῖς δὲ ἀπὸ τῶν ΛΞ, ΞΡ <lb n="1799,001,,,,11,23,69"></lb> |
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13769 ἴσον ἐστὶ τὸ ἀπὸ τῆς ΛΡ· ὀρθὴ γὰρ ἡ ὑπὸ ΛΞΡ· τὸ ἄρα <lb n="1799,001,,,,11,23,70"></lb> |
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13770 ἀπὸ τῆς ΑΒ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΡΛ· ἴση ἄρα ἡ ΑΒ τῇ <lb n="1799,001,,,,11,23,71"></lb> |
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13771 ΡΛ.</s> <s>ἀλλὰ τῇ μὲν ΑΒ ἴση ἐστὶν ἑκάστη τῶν ΒΓ, ΔΕ, ΕΖ, <lb n="1799,001,,,,11,23,72"></lb> |
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13772 ΗΘ, ΘΚ, τῇ δὲ ΡΛ ἴση ἑκατέρα τῶν ΡΜ, ΡΝ· ἑκάστη <lb n="1799,001,,,,11,23,73"></lb> |
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13773 ἄρα τῶν ΑΒ, ΒΓ, ΔΕ, ΕΖ, ΗΘ, ΘΚ ἑκάστῃ τῶν ΡΛ, <lb n="1799,001,,,,11,23,74"></lb> |
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13774 ΡΜ, ΡΝ ἴση ἐστίν.</s> <s>καὶ ἐπεὶ δύο αἱ ΛΡ, ΡΜ δυσὶ ταῖς <lb n="1799,001,,,,11,23,75"></lb> |
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13775 ΑΒ, ΒΓ ἴσαι εἰσίν, καὶ βάσις ἡ ΛΜ βάσει τῇ ΑΓ ὑπό-<lb n="1799,001,,,,11,23,76"></lb> |
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13776 κειται ἴση, γωνία ἄρα ἡ ὑπὸ ΛΡΜ γωνίᾳ τῇ ὑπὸ ΑΒΓ <pb></pb> <lb n="1799,001,,,,11,23,77"></lb> |
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13777 ἐστιν ἴση.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ μὲν ὑπὸ ΜΡΝ τῇ ὑπὸ <lb n="1799,001,,,,11,23,78"></lb> |
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13778 ΔΕΖ ἐστιν ἴση, ἡ δὲ ὑπὸ ΛΡΝ τῇ ὑπὸ ΗΘΚ.</s> <lb n="1799,001,,,,11,23,79"></lb> |
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13779 <s>Ἐκ τριῶν ἄρα γωνιῶν ἐπιπέδων τῶν ὑπὸ ΛΡΜ, ΜΡΝ, <lb n="1799,001,,,,11,23,80"></lb> |
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13780 ΛΡΝ, αἵ εἰσιν ἴσαι τρισὶ ταῖς δοθείσαις ταῖς ὑπὸ ΑΒΓ, <lb n="1799,001,,,,11,23,81"></lb> |
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13781 ΔΕΖ, ΗΘΚ, στερεὰ γωνία συνέσταται ἡ πρὸς τῷ Ρ περι-<lb n="1799,001,,,,11,23,82"></lb> |
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13782 εχομένη ὑπὸ τῶν ΛΡΜ, ΜΡΝ, ΛΡΝ γωνιῶν· ὅπερ ἔδει <lb n="1799,001,,,,11,23,83"></lb> |
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13783 ποιῆσαι.</s> <lb n="1799,001,,,,11,23,84"></lb> |
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13784 <s>Λῆμμα <lb n="1799,001,,,,11,23,85"></lb> |
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13785 Ὃν δὲ τρόπον, ᾧ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ <lb n="1799,001,,,,11,23,86"></lb> |
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13786 τῆς ΛΞ, ἐκείνῳ ἴσον λαβεῖν ἔστι τὸ ἀπὸ τῆς ΞΡ, δείξο-<lb n="1799,001,,,,11,23,87"></lb> |
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13787 μεν οὕτως.</s> <s>ἐκκείσθωσαν αἱ ΑΒ, ΛΞ εὐθεῖαι, καὶ ἔστω <lb n="1799,001,,,,11,23,88"></lb> |
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13788 μείζων ἡ ΑΒ, καὶ γεγράφθω ἐπ' αὐτῆς ἡμικύκλιον τὸ <lb n="1799,001,,,,11,23,89"></lb> |
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13789 ΑΒΓ, καὶ εἰς τὸ ΑΒΓ ἡμικύκλιον ἐνηρμόσθω τῇ ΛΞ <lb n="1799,001,,,,11,23,90"></lb> |
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13790 εὐθείᾳ μὴ μείζονι οὔσῃ τῆς ΑΒ διαμέτρου ἴση ἡ ΑΓ, καὶ <lb n="1799,001,,,,11,23,91"></lb> |
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13791 ἐπεζεύχθω ἡ ΓΒ.</s> <s>ἐπεὶ οὖν ἐν ἡμικυκλίῳ τῷ ΑΓΒ γωνία <lb n="1799,001,,,,11,23,92"></lb> |
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13792 ἐστὶν ἡ ὑπὸ ΑΓΒ, ὀρθὴ ἄρα ἐστὶν ἡ ὑπὸ ΑΓΒ.</s> <s>τὸ ἄρα <pb></pb> <lb n="1799,001,,,,11,23,93"></lb> |
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13793 ἀπὸ τῆς ΑΒ ἴσον ἐστὶ τοῖς ἀπὸ τῶν <lb n="1799,001,,,,11,23,94"></lb> |
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13794 ΑΓ, ΓΒ.</s> <s>ὥστε τὸ ἀπὸ τῆς ΑΒ τοῦ <lb n="1799,001,,,,11,23,95"></lb> |
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13795 ἀπὸ τῆς ΑΓ μεῖζόν ἐστι τῷ ἀπὸ τῆς <lb n="1799,001,,,,11,23,96"></lb> |
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13796 ΓΒ.</s> <s>ἴση δὲ ἡ ΑΓ τῇ ΛΞ.</s> <s>τὸ ἄρα <lb n="1799,001,,,,11,23,97"></lb> |
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13797 ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΛΞ μεῖζόν <lb n="1799,001,,,,11,23,98"></lb> |
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13798 ἐστι τῷ ἀπὸ τῆς ΓΒ.</s> <s>ἐὰν οὖν τῇ ΒΓ ἴσην τὴν ΞΡ ἀπο-<lb n="1799,001,,,,11,23,99"></lb> |
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13799 λάβωμεν, ἔσται τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΛΞ μεῖζον <lb n="1799,001,,,,11,23,100"></lb> |
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13800 τῷ ἀπὸ τῆς ΞΡ· ὅπερ προέκειτο ποιῆσαι.</s> <lb n="1799,001,,,,11,23,101"></lb> |
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13801 <s>Ἐὰν στερεὸν ὑπὸ παραλλήλων ἐπιπέδων περιέχηται, τὰ <lb n="1799,001,,,,11,24,1"></lb> |
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13802 ἀπεναντίον αὐτοῦ ἐπίπεδα ἴσα τε καὶ παραλληλόγραμμά <lb n="1799,001,,,,11,24,2"></lb> |
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13803 ἐστιν.</s> <lb n="1799,001,,,,11,24,3"></lb> |
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13804 <s>Στερεὸν γὰρ τὸ ΓΔΘΗ ὑπὸ παραλλήλων ἐπιπέδων <lb n="1799,001,,,,11,24,4"></lb> |
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13805 περιεχέσθω τῶν ΑΓ, ΗΖ, ΑΘ, ΔΖ, ΒΖ, ΑΕ· λέγω, <lb n="1799,001,,,,11,24,5"></lb> |
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13806 ὅτι τὰ ἀπεναντίον αὐτοῦ ἐπίπεδα ἴσα τε καὶ παραλληλό-<lb n="1799,001,,,,11,24,6"></lb> |
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13807 γραμμά ἐστιν.</s> <lb n="1799,001,,,,11,24,7"></lb> |
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13808 <s>Ἐπεὶ γὰρ δύο ἐπίπεδα παράλληλα τὰ ΒΗ, ΓΕ ὑπὸ <lb n="1799,001,,,,11,24,8"></lb> |
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13809 ἐπιπέδου τοῦ ΑΓ τέμνεται, αἱ κοιναὶ αὐτῶν τομαὶ παράλ- <pb></pb><lb n="1799,001,,,,11,24,9"></lb> |
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13810 ληλοί εἰσιν.</s> <s>παράλληλος ἄρα ἐστὶν ἡ ΑΒ τῇ ΔΓ.</s> <s>πάλιν, <lb n="1799,001,,,,11,24,10"></lb> |
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13811 ἐπεὶ δύο ἐπίπεδα παράλληλα τὰ ΒΖ, ΑΕ ὑπὸ ἐπιπέδου <lb n="1799,001,,,,11,24,11"></lb> |
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13812 τοῦ ΑΓ τέμνεται, αἱ κοιναὶ αὐτῶν τομαὶ παράλληλοί <lb n="1799,001,,,,11,24,12"></lb> |
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13813 εἰσιν.</s> <s>παράλληλος ἄρα <lb n="1799,001,,,,11,24,13"></lb> |
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13814 ἐστὶν ἡ ΒΓ τῇ ΑΔ.</s> <s>ἐδεί-<lb n="1799,001,,,,11,24,14"></lb> |
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13815 χθη δὲ καὶ ἡ ΑΒ τῇ ΔΓ <lb n="1799,001,,,,11,24,15"></lb> |
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13816 παράλληλος· παραλληλό-<lb n="1799,001,,,,11,24,16"></lb> |
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13817 γραμμον ἄρα ἐστὶ τὸ ΑΓ.</s> <lb n="1799,001,,,,11,24,17"></lb> |
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13818 <s>ὁμοίως δὴ δείξομεν, ὅτι <lb n="1799,001,,,,11,24,18"></lb> |
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13819 καὶ ἕκαστον τῶν ΔΖ, <lb n="1799,001,,,,11,24,19"></lb> |
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13820 ΖΗ, ΗΒ, ΒΖ, ΑΕ πα-<lb n="1799,001,,,,11,24,20"></lb> |
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13821 ραλληλόγραμμόν ἐστιν.</s> <lb n="1799,001,,,,11,24,21"></lb> |
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13822 <s>Ἐπεζεύχθωσαν αἱ ΑΘ, ΔΖ.</s> <s>καὶ ἐπεὶ παράλληλός ἐστιν <lb n="1799,001,,,,11,24,22"></lb> |
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13823 ἡ μὲν ΑΒ τῇ ΔΓ, ἡ δὲ ΒΘ τῇ ΓΖ, δύο δὴ αἱ ΑΒ, ΒΘ <lb n="1799,001,,,,11,24,23"></lb> |
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13824 ἁπτόμεναι ἀλλήλων παρὰ δύο εὐθείας τὰς ΔΓ, ΓΖ ἁπτο-<lb n="1799,001,,,,11,24,24"></lb> |
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13825 μένας ἀλλήλων εἰσὶν οὐκ ἐν τῷ αὐτῷ ἐπιπέδῳ· ἴσας ἄρα <lb n="1799,001,,,,11,24,25"></lb> |
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13826 γωνίας περιέξουσιν· ἴση ἄρα ἡ ὑπὸ ΑΒΘ γωνία τῇ ὑπὸ <lb n="1799,001,,,,11,24,26"></lb> |
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13827 ΔΓΖ.</s> <s>καὶ ἐπεὶ δύο αἱ ΑΒ, ΒΘ δυσὶ ταῖς ΔΓ, ΓΖ ἴσαι <lb n="1799,001,,,,11,24,27"></lb> |
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13828 εἰσίν, καὶ γωνία ἡ ὑπὸ ΑΒΘ γωνίᾳ τῇ ὑπὸ ΔΓΖ ἐστιν <lb n="1799,001,,,,11,24,28"></lb> |
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13829 ἴση, βάσις ἄρα ἡ ΑΘ βάσει τῇ ΔΖ ἐστιν ἴση, καὶ τὸ ΑΒΘ <lb n="1799,001,,,,11,24,29"></lb> |
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13830 τρίγωνον τῷ ΔΓΖ τριγώνῳ ἴσον ἐστίν.</s> <s>καί ἐστι τοῦ μὲν <lb n="1799,001,,,,11,24,30"></lb> |
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13831 ΑΒΘ διπλάσιον τὸ ΒΗ παραλληλόγραμμον, τοῦ δὲ ΔΓΖ <lb n="1799,001,,,,11,24,31"></lb> |
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13832 διπλάσιον τὸ ΓΕ παραλληλόγραμμον· ἴσον ἄρα τὸ ΒΗ <lb n="1799,001,,,,11,24,32"></lb> |
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13833 παραλληλόγραμμον τῷ ΓΕ παραλληλογράμμῳ.</s> <s>ὁμοίως <lb n="1799,001,,,,11,24,33"></lb> |
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13834 δὴ δείξομεν, ὅτι καὶ τὸ μὲν ΑΓ τῷ ΗΖ ἐστιν ἴσον, τὸ δὲ <lb n="1799,001,,,,11,24,34"></lb> |
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13835 ΑΕ τῷ ΒΖ.</s> <s><pb></pb> <lb n="1799,001,,,,11,24,35"></lb> |
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13836 Ἐὰν ἄρα στερεὸν ὑπὸ παραλλήλων ἐπιπέδων περιέχη-<lb n="1799,001,,,,11,24,36"></lb> |
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13837 ται, τὰ ἀπεναντίον αὐτοῦ ἐπίπεδα ἴσα τε καὶ παραλληλό-<lb n="1799,001,,,,11,24,37"></lb> |
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13838 γραμμά ἐστιν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,24,38"></lb> |
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13839 <s>Ἐὰν στερεὸν παραλληλεπίπεδον ἐπιπέδῳ τμηθῇ παραλ-<lb n="1799,001,,,,11,25,1"></lb> |
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13840 λήλῳ ὄντι τοῖς ἀπεναντίον ἐπιπέδοις, ἔσται ὡς ἡ βάσις πρὸς <lb n="1799,001,,,,11,25,2"></lb> |
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13841 τὴν βάσιν, οὕτως τὸ στερεὸν πρὸς τὸ στερεόν.</s> <lb n="1799,001,,,,11,25,3"></lb> |
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13842 <s>Στερεὸν γὰρ παραλληλεπίπεδον τὸ ΑΒΓΔ ἐπιπέδῳ τῷ <lb n="1799,001,,,,11,25,4"></lb> |
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13843 ΖΗ τετμήσθω παραλλήλῳ ὄντι τοῖς ἀπεναντίον ἐπιπέ-<lb n="1799,001,,,,11,25,5"></lb> |
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13844 δοις τοῖς ΡΑ, ΔΘ· λέγω, ὅτι ἐστὶν ὡς ἡ ΑΕΖΦ βάσις <lb n="1799,001,,,,11,25,6"></lb> |
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13845 πρὸς τὴν ΕΘΓΖ βάσιν, οὕτως τὸ ΑΒΖΥ στερεὸν πρὸς <lb n="1799,001,,,,11,25,7"></lb> |
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13846 τὸ ΕΗΓΔ στερεόν. <lb n="1799,001,,,,11,25,8"></lb> |
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13847 Ἐκβεβλήσθω γὰρ ἡ ΑΘ ἐφ' ἑκάτερα τὰ μέρη, καὶ κείσ-<lb n="1799,001,,,,11,25,9"></lb> |
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13848 θωσαν τῇ μὲν ΑΕ ἴσαι ὁσαιδηποτοῦν αἱ ΑΚ, ΚΛ, τῇ δὲ <lb n="1799,001,,,,11,25,10"></lb> |
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13849 ΕΘ ἴσαι ὁσαιδηποτοῦν αἱ ΘΜ, ΜΝ, καὶ συμπεπληρώσθω <pb></pb> <lb n="1799,001,,,,11,25,11"></lb> |
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13850 τὰ ΛΟ, ΚΦ, ΘΧ, ΜΣ παραλληλόγραμμα καὶ τὰ ΛΠ, <lb n="1799,001,,,,11,25,12"></lb> |
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13851 ΚΡ, ΔΜ, ΜΤ στερεά.</s> <s>καὶ ἐπεὶ ἴσαι εἰσὶν αἱ ΛΚ, ΚΑ, <lb n="1799,001,,,,11,25,13"></lb> |
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13852 ΑΕ εὐθεῖαι ἀλλήλαις, ἴσα ἐστὶ καὶ τὰ μὲν ΛΟ, ΚΦ, ΑΖ <lb n="1799,001,,,,11,25,14"></lb> |
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13853 παραλληλόγραμμα ἀλλήλοις, τὰ δὲ ΚΞ, ΚΒ, ΑΗ ἀλλή-<lb n="1799,001,,,,11,25,15"></lb> |
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13854 λοις καὶ ἔτι τὰ ΛΨ, ΚΠ, ΑΡ ἀλλήλοις· ἀπεναντίον γάρ.</s> <lb n="1799,001,,,,11,25,16"></lb> |
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13855 <s>διὰ τὰ αὐτὰ δὴ καὶ τὰ μὲν ΕΓ, ΘΧ, ΜΣ παραλληλό-<lb n="1799,001,,,,11,25,17"></lb> |
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13856 γραμμα ἴσα εἰσὶν ἀλλήλοις, τὰ δὲ ΘΗ, ΘΙ, ΙΝ ἴσα εἰσὶν <lb n="1799,001,,,,11,25,18"></lb> |
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13857 ἀλλήλοις, καὶ ἔτι τὰ ΔΘ, ΜΩ, ΝΤ· τρία ἄρα ἐπίπεδα <lb n="1799,001,,,,11,25,19"></lb> |
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13858 τῶν ΛΠ, ΚΡ, ΑΥ στερεῶν τρισὶν ἐπιπέδοις ἐστὶν ἴσα.</s> <lb n="1799,001,,,,11,25,20"></lb> |
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13859 <s>ἀλλὰ τὰ τρία τρισὶ τοῖς ἀπεναντίον ἐστὶν ἴσα· τὰ ἄρα τρία <lb n="1799,001,,,,11,25,21"></lb> |
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13860 στερεὰ τὰ ΛΠ, ΚΡ, ΑΥ ἴσα ἀλλήλοις ἐστίν.</s> <s>διὰ τὰ αὐτὰ <lb n="1799,001,,,,11,25,22"></lb> |
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13861 δὴ καὶ τὰ τρία στερεὰ τὰ ΕΔ, ΔΜ, ΜΤ ἴσα ἀλλήλοις <lb n="1799,001,,,,11,25,23"></lb> |
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13862 ἐστίν· ὁσαπλασίων ἄρα ἐστὶν ἡ ΛΖ βάσις τῆς ΑΖ βάσεως, <lb n="1799,001,,,,11,25,24"></lb> |
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13863 τοσαυταπλάσιόν ἐστι καὶ τὸ ΛΥ στερεὸν τοῦ ΑΥ στερεοῦ.</s> <lb n="1799,001,,,,11,25,25"></lb> |
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13864 <s>διὰ τὰ αὐτὰ δὴ ὁσαπλασίων ἐστὶν ἡ ΝΖ βάσις τῆς ΖΘ <lb n="1799,001,,,,11,25,26"></lb> |
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13865 βάσεως, τοσαυταπλάσιόν ἐστι καὶ τὸ ΝΥ στερεὸν τοῦ ΘΥ <lb n="1799,001,,,,11,25,27"></lb> |
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13866 στερεοῦ.</s> <s>καὶ εἰ ἴση ἐστὶν ἡ ΛΖ βάσις τῇ ΝΖ βάσει, ἴσον <lb n="1799,001,,,,11,25,28"></lb> |
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13867 ἐστὶ καὶ τὸ ΛΥ στερεὸν τῷ ΝΥ στερεῷ, καὶ εἰ ὑπερέχει ἡ <lb n="1799,001,,,,11,25,29"></lb> |
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13868 ΛΖ βάσις τῆς ΝΖ βάσεως, ὑπερέχει καὶ τὸ ΛΥ στερεὸν <pb></pb> <lb n="1799,001,,,,11,25,30"></lb> |
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13869 τοῦ ΝΥ στερεοῦ, καὶ εἰ ἐλλείπει, ἐλλείπει.</s> <s>τεσσάρων δὴ <lb n="1799,001,,,,11,25,31"></lb> |
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13870 ὄντων μεγεθῶν, δύο μὲν βάσεων τῶν ΑΖ, ΖΘ, δύο δὲ <lb n="1799,001,,,,11,25,32"></lb> |
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13871 στερεῶν τῶν ΑΥ, ΥΘ, εἴληπται ἰσάκις πολλαπλάσια τῆς <lb n="1799,001,,,,11,25,33"></lb> |
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13872 μὲν ΑΖ βάσεως καὶ τοῦ ΑΥ στερεοῦ ἥ τε ΛΖ βάσις καὶ τὸ <lb n="1799,001,,,,11,25,34"></lb> |
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13873 ΛΥ στερεόν, τῆς δὲ ΘΖ βάσεως καὶ τοῦ ΘΥ στερεοῦ ἥ <lb n="1799,001,,,,11,25,35"></lb> |
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13874 τε ΝΖ βάσις καὶ τὸ ΝΥ στερεόν, καὶ δέδεικται, ὅτι εἰ ὑπερ-<lb n="1799,001,,,,11,25,36"></lb> |
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13875 έχει ἡ ΛΖ βάσις τῆς ΖΝ βάσεως, ὑπερέχει καὶ τὸ ΛΥ <lb n="1799,001,,,,11,25,37"></lb> |
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13876 στερεὸν τοῦ ΝΥ [στερεοῦ], καὶ εἰ ἴση, ἴσον, καὶ εἰ ἐλλεί-<lb n="1799,001,,,,11,25,38"></lb> |
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13877 πει, ἐλλείπει.</s> <s>ἔστιν ἄρα ὡς ἡ ΑΖ βάσις πρὸς τὴν ΖΘ <lb n="1799,001,,,,11,25,39"></lb> |
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13878 βάσιν, οὕτως τὸ ΑΥ στερεὸν πρὸς τὸ ΥΘ στερεόν· ὅπερ <lb n="1799,001,,,,11,25,40"></lb> |
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13879 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,25,41"></lb> |
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13880 <s>Πρὸς τῇ δοθείσῃ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῇ <lb n="1799,001,,,,11,26,1"></lb> |
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13881 δοθείσῃ στερεᾷ γωνίᾳ ἴσην στερεὰν γωνίαν συστήσασθαι.</s> <lb n="1799,001,,,,11,26,2"></lb> |
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13882 <s>Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ πρὸς αὐτῇ <lb n="1799,001,,,,11,26,3"></lb> |
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13883 δοθὲν σημεῖον τὸ Α, ἡ δὲ δοθεῖσα στερεὰ γωνία ἡ πρὸς τῷ <lb n="1799,001,,,,11,26,4"></lb> |
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13884 Δ περιεχομένη ὑπὸ τῶν ὑπὸ ΕΔΓ, ΕΔΖ, ΖΔΓ γωνιῶν <lb n="1799,001,,,,11,26,5"></lb> |
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13885 ἐπιπέδων· δεῖ δὴ πρὸς τῇ ΑΒ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ <lb n="1799,001,,,,11,26,6"></lb> |
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13886 σημείῳ τῷ Α τῇ πρὸς τῷ Δ στερεᾷ γωνίᾳ ἴσην στερεὰν <lb n="1799,001,,,,11,26,7"></lb> |
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13887 γωνίαν συστήσασθαι.</s> <lb n="1799,001,,,,11,26,8"></lb> |
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13888 <s>Εἰλήφθω γὰρ ἐπὶ τῆς ΔΖ τυχὸν σημεῖον τὸ Ζ, καὶ <lb n="1799,001,,,,11,26,9"></lb> |
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13889 ἤχθω ἀπὸ τοῦ Ζ ἐπὶ τὸ διὰ τῶν ΕΔ, ΔΓ ἐπίπεδον κάθετος <lb n="1799,001,,,,11,26,10"></lb> |
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13890 ἡ ΖΗ, καὶ συμβαλλέτω τῷ ἐπιπέδῳ κατὰ τὸ Η, καὶ ἐπε-<lb n="1799,001,,,,11,26,11"></lb> |
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13891 ζεύχθω ἡ ΔΗ, καὶ συνεστάτω πρὸς τῇ ΑΒ εὐθείᾳ καὶ τῷ <pb></pb> <lb n="1799,001,,,,11,26,12"></lb> |
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13892 πρὸς αὐτῇ σημείῳ τῷ Α τῇ μὲν ὑπὸ ΕΔΓ γωνίᾳ ἴση ἡ ὑπὸ <lb n="1799,001,,,,11,26,13"></lb> |
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13893 ΒΑΛ, τῇ δὲ ὑπὸ ΕΔΗ ἴση ἡ ὑπὸ ΒΑΚ, καὶ κείσθω τῇ <lb n="1799,001,,,,11,26,14"></lb> |
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13894 ΔΗ ἴση ἡ ΑΚ, καὶ ἀνεστάτω ἀπὸ τοῦ Κ σημείου τῷ διὰ <lb n="1799,001,,,,11,26,15"></lb> |
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13895 τῶν ΒΑΛ ἐπιπέδῳ πρὸς ὀρθὰς ἡ ΚΘ, καὶ κείσθω ἴση <lb n="1799,001,,,,11,26,16"></lb> |
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13896 τῇ ΗΖ ἡ ΚΘ, καὶ ἐπεζεύχθω ἡ ΘΑ· λέγω, ὅτι ἡ πρὸς <lb n="1799,001,,,,11,26,17"></lb> |
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13897 τῷ Α στερεὰ γωνία περιεχομένη ὑπὸ τῶν ΒΑΛ, ΒΑΘ, <lb n="1799,001,,,,11,26,18"></lb> |
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13898 ΘΑΛ γωνιῶν ἴση ἐστὶ τῇ πρὸς τῷ Δ στερεᾷ γωνίᾳ τῇ <lb n="1799,001,,,,11,26,19"></lb> |
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13899 περιεχομένῃ ὑπὸ τῶν ΕΔΓ, ΕΔΖ, ΖΔΓ γωνιῶν.</s> <lb n="1799,001,,,,11,26,20"></lb> |
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13900 <s>Ἀπειλήφθωσαν γὰρ ἴσαι αἱ ΑΒ, ΔΕ, καὶ ἐπεζεύχ-<lb n="1799,001,,,,11,26,21"></lb> |
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13901 θωσαν αἱ ΘΒ, ΚΒ, ΖΕ, ΗΕ.</s> <s>καὶ ἐπεὶ ἡ ΖΗ ὀρθή ἐστι <lb n="1799,001,,,,11,26,22"></lb> |
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13902 πρὸς τὸ ὑποκείμενον ἐπίπεδον, καὶ πρὸς πάσας ἄρα τὰς <lb n="1799,001,,,,11,26,23"></lb> |
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13903 ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ ὑποκειμένῳ <lb n="1799,001,,,,11,26,24"></lb> |
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13904 ἐπιπέδῳ ὀρθὰς ποιήσει γωνίας· ὀρθὴ ἄρα ἐστὶν ἑκατέρα <lb n="1799,001,,,,11,26,25"></lb> |
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13905 τῶν ὑπὸ ΖΗΔ, ΖΗΕ γωνιῶν.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἑκα-<lb n="1799,001,,,,11,26,26"></lb> |
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13906 τέρα τῶν ὑπὸ ΘΚΑ, ΘΚΒ γωνιῶν ὀρθή ἐστιν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,11,26,27"></lb> |
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13907 δύο αἱ ΚΑ, ΑΒ δύο ταῖς ΗΔ, ΔΕ ἴσαι εἰσὶν ἑκατέρα ἑκα-<lb n="1799,001,,,,11,26,28"></lb> |
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13908 τέρᾳ, καὶ γωνίας ἴσας περιέχουσιν, βάσις ἄρα ἡ ΚΒ βάσει <pb></pb> <lb n="1799,001,,,,11,26,29"></lb> |
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13909 τῇ ΗΕ ἴση ἐστίν.</s> <s>ἔστι δὲ καὶ ἡ ΚΘ τῇ ΗΖ ἴση· καὶ γω-<lb n="1799,001,,,,11,26,30"></lb> |
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13910 νίας ὀρθὰς περιέχουσιν· ἴση ἄρα καὶ ἡ ΘΒ τῇ ΖΕ.</s> <s>πάλιν <lb n="1799,001,,,,11,26,31"></lb> |
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13911 ἐπεὶ δύο αἱ ΑΚ, ΚΘ δυσὶ ταῖς ΔΗ, ΗΖ ἴσαι εἰσίν, καὶ <lb n="1799,001,,,,11,26,32"></lb> |
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13912 γωνίας ὀρθὰς περιέχουσιν, βάσις ἄρα ἡ ΑΘ βάσει τῇ <lb n="1799,001,,,,11,26,33"></lb> |
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13913 ΖΔ ἴση ἐστίν.</s> <s>ἔστι δὲ καὶ ἡ ΑΒ τῇ ΔΕ ἴση· δύο <lb n="1799,001,,,,11,26,34"></lb> |
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13914 δὴ αἱ ΘΑ, ΑΒ δύο ταῖς ΔΖ, ΔΕ ἴσαι εἰσίν.</s> <s>καὶ βάσις ἡ <lb n="1799,001,,,,11,26,35"></lb> |
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13915 ΘΒ βάσει τῇ ΖΕ ἴση· γωνία ἄρα ἡ ὑπὸ ΒΑΘ γωνίᾳ τῇ <lb n="1799,001,,,,11,26,36"></lb> |
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13916 ὑπὸ ΕΔΖ ἐστιν ἴση.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ ΘΑΛ τῇ ὑπὸ <lb n="1799,001,,,,11,26,37"></lb> |
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13917 ΖΔΓ ἐστιν ἴση [ἐπειδήπερ ἐὰν ἀπολάβωμεν ἴσας τὰς ΑΛ, <lb n="1799,001,,,,11,26,38"></lb> |
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13918 ΔΓ καὶ ἐπιζεύξωμεν τὰς ΚΛ, ΘΛ, ΗΓ, ΖΓ, ἐπεὶ ὅλη ἡ <lb n="1799,001,,,,11,26,39"></lb> |
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13919 ὑπὸ ΒΑΛ ὅλῃ τῇ ὑπὸ ΕΔΓ ἐστιν ἴση, ὧν ἡ ὑπὸ ΒΑΚ τῇ <lb n="1799,001,,,,11,26,40"></lb> |
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13920 ὑπὸ ΕΔΗ ὑπόκειται ἴση, λοιπὴ ἄρα ἡ ὑπὸ ΚΑΛ λοιπῇ τῇ <lb n="1799,001,,,,11,26,41"></lb> |
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13921 ὑπὸ ΗΔΓ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ δύο αἱ ΚΑ, ΑΛ δυσὶ ταῖς <lb n="1799,001,,,,11,26,42"></lb> |
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13922 ΗΔ, ΔΓ ἴσαι εἰσίν, καὶ γωνίας ἴσας περιέχουσιν, βάσις <lb n="1799,001,,,,11,26,43"></lb> |
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13923 ἄρα ἡ ΚΛ βάσει τῇ ΗΓ ἐστιν ἴση.</s> <s>ἔστι δὲ καὶ ἡ ΚΘ τῇ <lb n="1799,001,,,,11,26,44"></lb> |
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13924 ΗΖ ἴση· δύο δὴ αἱ ΛΚ, ΚΘ δυσὶ ταῖς ΓΗ, ΗΖ εἰσιν <lb n="1799,001,,,,11,26,45"></lb> |
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13925 ἴσαι· καὶ γωνίας ὀρθὰς περιέχουσιν· βάσις ἄρα ἡ ΘΛ βάσει <lb n="1799,001,,,,11,26,46"></lb> |
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13926 τῇ ΖΓ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ δύο αἱ ΘΑ, ΑΛ δυσὶ ταῖς ΖΔ, <lb n="1799,001,,,,11,26,47"></lb> |
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13927 ΔΓ εἰσιν ἴσαι, καὶ βάσις ἡ ΘΛ βάσει τῇ ΖΓ ἐστιν ἴση, <lb n="1799,001,,,,11,26,48"></lb> |
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13928 γωνία ἄρα ἡ ὑπὸ ΘΑΛ γωνίᾳ τῇ ὑπὸ ΖΔΓ ἐστιν ἴση].</s> <lb n="1799,001,,,,11,26,49"></lb> |
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13929 <s>ἔστι δὲ καὶ ἡ ὑπὸ ΒΑΛ τῇ ὑπὸ ΕΔΓ ἴση.</s> <s><pb></pb> <lb n="1799,001,,,,11,26,50"></lb> |
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13930 Πρὸς ἄρα τῇ δοθείσῃ εὐθείᾳ τῇ ΑΒ καὶ τῷ πρὸς αὐτῇ <lb n="1799,001,,,,11,26,51"></lb> |
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13931 σημείῳ τῷ Α τῇ δοθείσῃ στερεᾷ γωνίᾳ τῇ πρὸς τῷ Δ ἴση <lb n="1799,001,,,,11,26,52"></lb> |
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13932 συνέσταται· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,11,26,53"></lb> |
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13933 <s>Ἀπὸ τῆς δοθείσης εὐθείας τῷ δοθέντι στερεῷ παραλ-<lb n="1799,001,,,,11,27,1"></lb> |
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13934 ληλεπιπέδῳ ὅμοιόν τε καὶ ὁμοίως κείμενον στερεὸν παραλ-<lb n="1799,001,,,,11,27,2"></lb> |
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13935 ληλεπίπεδον ἀναγράψαι.</s> <lb n="1799,001,,,,11,27,3"></lb> |
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13936 <s>Ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ δοθὲν στερεὸν <lb n="1799,001,,,,11,27,4"></lb> |
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13937 παραλληλεπίπεδον τὸ ΓΔ· δεῖ δὴ ἀπὸ τῆς δοθείσης εὐθείας <lb n="1799,001,,,,11,27,5"></lb> |
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13938 τῆς ΑΒ τῷ δοθέντι στερεῷ παραλληλεπιπέδῳ τῷ ΓΔ <lb n="1799,001,,,,11,27,6"></lb> |
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13939 ὅμοιόν τε καὶ ὁμοίως κείμενον στερεὸν παραλληλεπίπεδον <lb n="1799,001,,,,11,27,7"></lb> |
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13940 ἀναγράψαι.</s> <lb n="1799,001,,,,11,27,8"></lb> |
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13941 <s>Συνεστάτω γὰρ πρὸς τῇ ΑΒ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ <lb n="1799,001,,,,11,27,9"></lb> |
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13942 σημείῳ τῷ Α τῇ πρὸς τῷ Γ στερεᾷ γωνίᾳ ἴση ἡ περιεχο-<lb n="1799,001,,,,11,27,10"></lb> |
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13943 μένη ὑπὸ τῶν ΒΑΘ, ΘΑΚ, ΚΑΒ, ὥστε ἴσην εἶναι τὴν <lb n="1799,001,,,,11,27,11"></lb> |
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13944 μὲν ὑπὸ ΒΑΘ γωνίαν τῇ ὑπὸ ΕΓΖ, τὴν δὲ ὑπὸ ΒΑΚ τῇ <lb n="1799,001,,,,11,27,12"></lb> |
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13945 ὑπὸ ΕΓΗ, τὴν δὲ ὑπὸ ΚΑΘ τῇ ὑπὸ ΗΓΖ· καὶ γεγονέτω <lb n="1799,001,,,,11,27,13"></lb> |
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13946 ὡς μὲν ἡ ΕΓ πρὸς τὴν ΓΗ, οὕτως ἡ ΒΑ πρὸς τὴν ΑΚ, <lb n="1799,001,,,,11,27,14"></lb> |
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13947 ὡς δὲ ἡ ΗΓ πρὸς τὴν ΓΖ, οὕτως ἡ ΚΑ πρὸς τὴν ΑΘ.</s> <s>καὶ <lb n="1799,001,,,,11,27,15"></lb> |
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13948 δι' ἴσου ἄρα ἐστὶν ὡς ἡ ΕΓ πρὸς τὴν ΓΖ, οὕτως ἡ ΒΑ <pb></pb> <lb n="1799,001,,,,11,27,16"></lb> |
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13949 πρὸς τὴν ΑΘ.</s> <s>καὶ συμπεπληρώσθω τὸ ΘΒ παραλληλό-<lb n="1799,001,,,,11,27,17"></lb> |
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13950 γραμμον καὶ τὸ ΑΛ στερεόν.</s> <lb n="1799,001,,,,11,27,18"></lb> |
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13951 <s>Καὶ ἐπεί ἐστιν ὡς ἡ ΕΓ πρὸς τὴν ΓΗ, οὕτως ἡ ΒΑ πρὸς <lb n="1799,001,,,,11,27,19"></lb> |
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13952 τὴν ΑΚ, καὶ περὶ ἴσας γωνίας τὰς ὑπὸ ΕΓΗ, ΒΑΚ αἱ <lb n="1799,001,,,,11,27,20"></lb> |
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13953 πλευραὶ ἀνάλογόν εἰσιν, ὅμοιον ἄρα ἐστὶ τὸ ΗΕ παραλ-<lb n="1799,001,,,,11,27,21"></lb> |
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13954 ληλόγραμμον τῷ ΚΒ παραλληλογράμμῳ.</s> <s>διὰ τὰ αὐτὰ <lb n="1799,001,,,,11,27,22"></lb> |
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13955 δὴ καὶ τὸ μὲν ΚΘ παραλληλόγραμμον τῷ ΗΖ παραλλη- <lb n="1799,001,,,,11,27,23"></lb> |
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13956 λογράμμῳ ὅμοιόν ἐστι καὶ ἔτι τὸ ΖΕ τῷ ΘΒ· τρία ἄρα <lb n="1799,001,,,,11,27,24"></lb> |
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13957 παραλληλόγραμμα τοῦ ΓΔ στερεοῦ τρισὶ παραλληλο-<lb n="1799,001,,,,11,27,25"></lb> |
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13958 γράμμοις τοῦ ΑΛ στερεοῦ ὅμοιά ἐστιν.</s> <s>ἀλλὰ τὰ μὲν τρία <lb n="1799,001,,,,11,27,26"></lb> |
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13959 τρισὶ τοῖς ἀπεναντίον ἴσα τέ ἐστι καὶ ὅμοια, τὰ δὲ τρία <lb n="1799,001,,,,11,27,27"></lb> |
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13960 τρισὶ τοῖς ἀπεναντίον ἴσα τέ ἐστι καὶ ὅμοια· ὅλον ἄρα τὸ <lb n="1799,001,,,,11,27,28"></lb> |
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13961 ΓΔ στερεὸν ὅλῳ τῷ ΑΛ στερεῷ ὅμοιόν ἐστιν.</s> <s><pb></pb> <lb n="1799,001,,,,11,27,29"></lb> |
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13962 Ἀπὸ τῆς δοθείσης ἄρα εὐθείας τῆς ΑΒ τῷ δοθέντι <lb n="1799,001,,,,11,27,30"></lb> |
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13963 στερεῷ παραλληλεπιπέδῳ τῷ ΓΔ ὅμοιόν τε καὶ ὁμοίως <lb n="1799,001,,,,11,27,31"></lb> |
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13964 κείμενον ἀναγέγραπται τὸ ΑΛ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,11,27,32"></lb> |
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13965 <s>Ἐὰν στερεὸν παραλληλεπίπεδον ἐπιπέδῳ τμηθῇ κατὰ <lb n="1799,001,,,,11,28,1"></lb> |
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13966 τὰς διαγωνίους τῶν ἀπεναντίον ἐπιπέδων, δίχα τμηθήσε-<lb n="1799,001,,,,11,28,2"></lb> |
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13967 ται τὸ στερεὸν ὑπὸ τοῦ ἐπιπέδου.</s> <lb n="1799,001,,,,11,28,3"></lb> |
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13968 <s>Στερεὸν γὰρ παραλληλεπίπεδον τὸ ΑΒ ἐπιπέδῳ τῷ <lb n="1799,001,,,,11,28,4"></lb> |
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13969 ΓΔΕΖ τετμήσθω κατὰ τὰς διαγωνίους τῶν ἀπεναντίον <lb n="1799,001,,,,11,28,5"></lb> |
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13970 ἐπιπέδων τὰς ΓΖ, ΔΕ· λέγω, <lb n="1799,001,,,,11,28,6"></lb> |
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13971 ὅτι δίχα τμηθήσεται τὸ ΑΒ στε-<lb n="1799,001,,,,11,28,7"></lb> |
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13972 ρεὸν ὑπὸ τοῦ ΓΔΕΖ ἐπιπέδου.</s> <lb n="1799,001,,,,11,28,8"></lb> |
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13973 <s>Ἐπεὶ γὰρ ἴσον ἐστὶ τὸ μὲν ΓΗΖ <lb n="1799,001,,,,11,28,9"></lb> |
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13974 τρίγωνον τῷ ΓΖΒ τριγώνῳ, τὸ <lb n="1799,001,,,,11,28,10"></lb> |
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13975 δὲ ΑΔΕ τῷ ΔΕΘ, ἔστι δὲ καὶ τὸ <lb n="1799,001,,,,11,28,11"></lb> |
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13976 μὲν ΓΑ παραλληλόγραμμον τῷ <lb n="1799,001,,,,11,28,12"></lb> |
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13977 ΕΒ ἴσον· ἀπεναντίον γάρ· τὸ δὲ <lb n="1799,001,,,,11,28,13"></lb> |
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13978 ΗΕ τῷ ΓΘ, καὶ τὸ πρίσμα ἄρα τὸ περιεχόμενον ὑπὸ δύο <lb n="1799,001,,,,11,28,14"></lb> |
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13979 μὲν τριγώνων τῶν ΓΗΖ, ΑΔΕ, τριῶν δὲ παραλληλο-<lb n="1799,001,,,,11,28,15"></lb> |
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13980 γράμμων τῶν ΗΕ, ΑΓ, ΓΕ ἴσον ἐστὶ τῷ πρίσματι τῷ <lb n="1799,001,,,,11,28,16"></lb> |
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13981 περιεχομένῳ ὑπὸ δύο μὲν τριγώνων τῶν ΓΖΒ, ΔΕΘ, <lb n="1799,001,,,,11,28,17"></lb> |
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13982 τριῶν δὲ παραλληλογράμμων τῶν ΓΘ, ΒΕ, ΓΕ· ὑπὸ γὰρ <lb n="1799,001,,,,11,28,18"></lb> |
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13983 ἴσων ἐπιπέδων περιέχονται τῷ τε πλήθει καὶ τῷ μεγέθει.</s> <lb n="1799,001,,,,11,28,19"></lb> |
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13984 <s>ὥστε ὅλον τὸ ΑΒ στερεὸν δίχα τέτμηται ὑπὸ τοῦ ΓΔΕΖ <lb n="1799,001,,,,11,28,20"></lb> |
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13985 ἐπιπέδου· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,28,21"></lb> |
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13986 Τὰ ἐπὶ τῆς αὐτῆς βάσεως ὄντα στερεὰ παραλληλεπίπεδα <lb n="1799,001,,,,11,29,1"></lb> |
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13987 καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι ἐπὶ τῶν αὐτῶν <lb n="1799,001,,,,11,29,2"></lb> |
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13988 εἰσιν εὐθειῶν, ἴσα ἀλλήλοις ἐστίν.</s> <lb n="1799,001,,,,11,29,3"></lb> |
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13989 <s>Ἔστω ἐπὶ τῆς αὐτῆς βάσεως τῆς ΑΒ στερεὰ παραλληλ-<lb n="1799,001,,,,11,29,4"></lb> |
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13990 επίπεδα τὰ ΓΜ, ΓΝ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι <lb n="1799,001,,,,11,29,5"></lb> |
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13991 αἱ ΑΗ, ΑΖ, ΛΜ, ΛΝ, <lb n="1799,001,,,,11,29,6"></lb> |
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13992 ΓΔ, ΓΕ, ΒΘ, ΒΚ ἐπὶ τῶν <lb n="1799,001,,,,11,29,7"></lb> |
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13993 αὐτῶν εὐθειῶν ἔστωσαν <lb n="1799,001,,,,11,29,8"></lb> |
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13994 τῶν ΖΝ, ΔΚ· λέγω, ὅτι <lb n="1799,001,,,,11,29,9"></lb> |
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13995 ἴσον ἐστὶ τὸ ΓΜ στερεὸν <lb n="1799,001,,,,11,29,10"></lb> |
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13996 τῷ ΓΝ στερεῷ.</s> <lb n="1799,001,,,,11,29,11"></lb> |
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13997 <s>Ἐπεὶ γὰρ παραλληλό-<lb n="1799,001,,,,11,29,12"></lb> |
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13998 γραμμόν ἐστιν ἑκάτερον <lb n="1799,001,,,,11,29,13"></lb> |
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13999 τῶν ΓΘ, ΓΚ, ἴση ἐστὶν ἡ ΓΒ ἑκατέρᾳ τῶν ΔΘ, ΕΚ· ὥστε <lb n="1799,001,,,,11,29,14"></lb> |
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14000 καὶ ἡ ΔΘ τῇ ΕΚ ἐστιν ἴση.</s> <s>κοινὴ ἀφῃρήσθω ἡ ΕΘ· λοιπὴ <lb n="1799,001,,,,11,29,15"></lb> |
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14001 ἄρα ἡ ΔΕ λοιπῇ τῇ ΘΚ ἐστιν ἴση.</s> <s>ὥστε καὶ τὸ μὲν ΔΓΕ τρί-<lb n="1799,001,,,,11,29,16"></lb> |
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14002 γωνον τῷ ΘΒΚ τριγώνῳ ἴσον ἐστίν, τὸ δὲ ΔΗ παραλληλό-<lb n="1799,001,,,,11,29,17"></lb> |
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14003 γραμμον τῷ ΘΝ παραλληλογράμμῳ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,11,29,18"></lb> |
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14004 τὸ ΑΖΗ τρίγωνον τῷ ΜΛΝ τριγώνῳ ἴσον ἐστίν.</s> <s>ἔστι δὲ <lb n="1799,001,,,,11,29,19"></lb> |
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14005 καὶ τὸ μὲν ΓΖ παραλληλόγραμμον τῷ ΒΜ παραλληλο-<lb n="1799,001,,,,11,29,20"></lb> |
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14006 γράμμῳ ἴσον, τὸ δὲ ΓΗ τῷ ΒΝ· ἀπεναντίον γάρ· καὶ τὸ <lb n="1799,001,,,,11,29,21"></lb> |
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14007 πρίσμα ἄρα τὸ περιεχόμενον ὑπὸ δύο μὲν τριγώνων τῶν <lb n="1799,001,,,,11,29,22"></lb> |
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14008 ΑΖΗ, ΔΓΕ, τριῶν δὲ παραλληλογράμμων τῶν ΑΔ, <lb n="1799,001,,,,11,29,23"></lb> |
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14009 ΔΗ, ΓΗ ἴσον ἐστὶ τῷ πρίσματι τῷ περιεχομένῳ ὑπὸ δύο <pb></pb> <lb n="1799,001,,,,11,29,24"></lb> |
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14010 μὲν τριγώνων τῶν ΜΛΝ, ΘΒ*κ, τριῶν δὲ παραλληλο-<lb n="1799,001,,,,11,29,25"></lb> |
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14011 γράμμων τῶν ΒΜ, ΘΝ, ΒΝ.</s> <s>κοινὸν προσκείσθω τὸ στερε-<lb n="1799,001,,,,11,29,26"></lb> |
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14012 όν, οὗ βάσις μὲν τὸ ΑΒ παραλληλόγραμμον, ἀπεναντίον <lb n="1799,001,,,,11,29,27"></lb> |
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14013 δὲ τὸ ΗΕΘΜ· ὅλον ἄρα τὸ ΓΜ στερεὸν παραλληλεπίπε-<lb n="1799,001,,,,11,29,28"></lb> |
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14014 δον ὅλῳ τῷ ΓΝ στερεῷ παραλληλεπιπέδῳ ἴσον ἐστίν.</s> <lb n="1799,001,,,,11,29,29"></lb> |
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14015 <s>Τὰ ἄρα ἐπὶ τῆς αὐτῆς βάσεως ὄντα στερεὰ παραλ-<lb n="1799,001,,,,11,29,30"></lb> |
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14016 ληλεπίπεδα καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι ἐπὶ <lb n="1799,001,,,,11,29,31"></lb> |
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14017 τῶν αὐτῶν εἰσιν εὐθειῶν, ἴσα ἀλλήλοις ἐστίν· ὅπερ ἔδει <lb n="1799,001,,,,11,29,32"></lb> |
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14018 δεῖξαι.</s> <lb n="1799,001,,,,11,29,33"></lb> |
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14019 <s>Τὰ ἐπὶ τῆς αὐτῆς βάσεως ὄντα στερεὰ παραλληλεπί-<lb n="1799,001,,,,11,30,1"></lb> |
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14020 πεδα καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι οὐκ εἰσὶν ἐπὶ <lb n="1799,001,,,,11,30,2"></lb> |
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14021 τῶν αὐτῶν εὐθειῶν, ἴσα ἀλλήλοις ἐστίν.</s> <lb n="1799,001,,,,11,30,3"></lb> |
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14022 <s>Ἔστω ἐπὶ τῆς αὐτῆς βά- <lb n="1799,001,,,,11,30,4"></lb> |
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14023 σεως τῆς ΑΒ στερεὰ παραλ-<lb n="1799,001,,,,11,30,5"></lb> |
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14024 ληλεπίπεδα τὰ ΓΜ, ΓΝ <lb n="1799,001,,,,11,30,6"></lb> |
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14025 ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ <lb n="1799,001,,,,11,30,7"></lb> |
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14026 ἐφεστῶσαι αἱ ΑΖ, ΑΗ, <lb n="1799,001,,,,11,30,8"></lb> |
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14027 ΛΜ, ΛΝ, ΓΔ, ΓΕ, ΒΘ, <lb n="1799,001,,,,11,30,9"></lb> |
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14028 ΒΚ μὴ ἔστωσαν ἐπὶ τῶν <lb n="1799,001,,,,11,30,10"></lb> |
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14029 αὐτῶν εὐθειῶν· λέγω, ὅτι <lb n="1799,001,,,,11,30,11"></lb> |
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14030 ἴσον ἐστὶ τὸ ΓΜ στερεὸν <lb n="1799,001,,,,11,30,12"></lb> |
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14031 τῷ ΓΝ στερεῷ.</s> <lb n="1799,001,,,,11,30,13"></lb> |
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14032 <s>Ἐκβεβλήσθωσαν γὰρ αἱ ΝΚ, ΔΘ καὶ συμπιπτέτωσαν <lb n="1799,001,,,,11,30,14"></lb> |
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14033 ἀλλήλαις κατὰ τὸ Ρ, καὶ ἔτι ἐκβεβλήσθωσαν αἱ ΖΜ, ΗΕ <lb n="1799,001,,,,11,30,15"></lb> |
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14034 ἐπὶ τὰ Ο, Π, καὶ ἐπεζεύχθωσαν αἱ ΑΞ, ΛΟ, ΓΠ, ΒΡ.</s> <lb n="1799,001,,,,11,30,16"></lb> |
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14035 <s>ἴσον δή ἐστι τὸ ΓΜ στερεόν, οὗ βάσις μὲν τὸ ΑΓΒΛ <pb></pb> <lb n="1799,001,,,,11,30,17"></lb> |
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14036 παραλληλόγραμμον, ἀπεναντίον δὲ τὸ ΖΔΘΜ, τῷ ΓΟ <lb n="1799,001,,,,11,30,18"></lb> |
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14037 στερεῷ, οὗ βάσις μὲν τὸ ΑΓΒΛ παραλληλόγραμμον, <lb n="1799,001,,,,11,30,19"></lb> |
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14038 ἀπεναντίον δὲ τὸ ΞΠΡΟ· ἐπί τε γὰρ τῆς αὐτῆς βάσεώς <lb n="1799,001,,,,11,30,20"></lb> |
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14039 εἰσι τῆς ΑΓΒΛ καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι <lb n="1799,001,,,,11,30,21"></lb> |
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14040 αἱ ΑΖ, ΑΞ, ΛΜ, ΛΟ, ΓΔ, ΓΠ, ΒΘ, ΒΡ ἐπὶ τῶν αὐτῶν <lb n="1799,001,,,,11,30,22"></lb> |
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14041 εἰσιν εὐθειῶν τῶν ΖΟ, ΔΡ.</s> <s>ἀλλὰ τὸ ΓΟ στερεόν, οὗ βάσις <lb n="1799,001,,,,11,30,23"></lb> |
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14042 μέν ἐστι τὸ ΑΓΒΛ παραλληλόγραμμον, ἀπεναντίον δὲ τὸ <lb n="1799,001,,,,11,30,24"></lb> |
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14043 ΞΠΡΟ, ἴσον ἐστὶ τῷ ΓΝ στερεῷ, οὗ βάσις μὲν τὸ ΑΓΒΛ <lb n="1799,001,,,,11,30,25"></lb> |
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14044 παραλληλόγραμμον, ἀπεναντίον δὲ τὸ ΗΕΚΝ· ἐπί τε <lb n="1799,001,,,,11,30,26"></lb> |
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14045 γὰρ πάλιν τῆς αὐτῆς βάσεώς εἰσι τῆς ΑΓΒΛ καὶ ὑπὸ τὸ <lb n="1799,001,,,,11,30,27"></lb> |
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14046 αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι αἱ ΑΗ, ΑΞ, ΓΕ, ΓΠ, ΛΝ, <lb n="1799,001,,,,11,30,28"></lb> |
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14047 ΛΟ, ΒΚ, ΒΡ ἐπὶ τῶν αὐτῶν εἰσιν εὐθειῶν τῶν ΗΠ, ΝΡ.</s> <lb n="1799,001,,,,11,30,29"></lb> |
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14048 <s>ὥστε καὶ τὸ ΓΜ στερεὸν ἴσον ἐστὶ τῷ ΓΝ στερεῷ.</s> <lb n="1799,001,,,,11,30,30"></lb> |
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14049 <s>Τὰ ἄρα ἐπὶ τῆς αὐτῆς βάσεως στερεὰ παραλληλεπίπεδα <lb n="1799,001,,,,11,30,31"></lb> |
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14050 καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι οὐκ εἰσὶν ἐπὶ τῶν <lb n="1799,001,,,,11,30,32"></lb> |
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14051 αὐτῶν εὐθειῶν, ἴσα ἀλλήλοις ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,30,33"></lb> |
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14052 Τὰ ἐπὶ ἴσων βάσεων ὄντα στερεὰ παραλληλεπίπεδα καὶ <lb n="1799,001,,,,11,31,1"></lb> |
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14053 ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλήλοις ἐστίν.</s> <lb n="1799,001,,,,11,31,2"></lb> |
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14054 <s>Ἔστω ἐπὶ ἴσων βάσεων τῶν ΑΒ, ΓΔ στερεὰ παραλληλ-<lb n="1799,001,,,,11,31,3"></lb> |
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14055 επίπεδα τὰ ΑΕ, ΓΖ ὑπὸ τὸ αὐτὸ ὕψος· λέγω, ὅτι ἴσον <lb n="1799,001,,,,11,31,4"></lb> |
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14056 ἐστὶ τὸ ΑΕ στερεὸν τῷ ΓΖ στερεῷ.</s> <lb n="1799,001,,,,11,31,5"></lb> |
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14057 <s>Ἔστωσαν δὴ πρότερον αἱ ἐφεστηκυῖαι αἱ ΘΚ, ΒΕ, ΑΗ, <lb n="1799,001,,,,11,31,6"></lb> |
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14058 ΛΜ, ΟΠ, ΔΖ, ΓΞ, ΡΣ πρὸς ὀρθὰς ταῖς ΑΒ, ΓΔ βάσε-<lb n="1799,001,,,,11,31,7"></lb> |
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14059 σιν, καὶ ἐκβεβλήσθω ἐπ' εὐθείας τῇ ΓΡ εὐθεῖα ἡ ΡΤ, καὶ <lb n="1799,001,,,,11,31,8"></lb> |
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14060 συνεστάτω πρὸς τῇ ΡΤ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ <lb n="1799,001,,,,11,31,9"></lb> |
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14061 τῷ Ρ τῇ ὑπὸ ΑΛΒ γωνίᾳ ἴση ἡ ὑπὸ ΤΡΥ, καὶ κείσθω <lb n="1799,001,,,,11,31,10"></lb> |
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14062 τῇ μὲν ΑΛ ἴση ἡ ΡΤ, τῇ δὲ ΛΒ ἴση ἡ ΡΥ, καὶ συμπεπλη-<lb n="1799,001,,,,11,31,11"></lb> |
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14063 ρώσθω ἥ τε ΡΧ βάσις καὶ τὸ ΨΥ στερεόν.</s> <s>καὶ ἐπεὶ δύο αἱ <lb n="1799,001,,,,11,31,12"></lb> |
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14064 ΤΡ, ΡΥ δυσὶ ταῖς ΑΛ, ΛΒ ἴσαι εἰσίν, καὶ γωνίας ἴσας <lb n="1799,001,,,,11,31,13"></lb> |
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14065 περιέχουσιν, ἴσον ἄρα καὶ ὅμοιον τὸ ΡΧ παραλληλόγραμ-<lb n="1799,001,,,,11,31,14"></lb> |
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14066 μον τῷ ΘΛ παραλληλογράμμῳ.</s> <s>καὶ ἐπεὶ πάλιν ἴση μὲν <lb n="1799,001,,,,11,31,15"></lb> |
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14067 ἡ ΑΛ τῇ ΡΤ, ἡ δὲ ΛΜ τῇ ΡΣ, καὶ γωνίας ὀρθὰς περι-<lb n="1799,001,,,,11,31,16"></lb> |
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14068 έχουσιν, ἴσον ἄρα καὶ ὅμοιόν ἐστι τὸ ΡΨ παραλληλό-<lb n="1799,001,,,,11,31,17"></lb> |
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14069 γραμμον τῷ ΑΜ παραλληλογράμμῳ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,11,31,18"></lb> |
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14070 τὸ ΛΕ τῷ ΣΥ ἴσον τέ ἐστι καὶ ὅμοιον· τρία ἄρα παραλ-<lb n="1799,001,,,,11,31,19"></lb> |
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14071 ληλόγραμμα τοῦ ΑΕ στερεοῦ τρισὶ παραλληλογράμμοις <pb></pb> <lb n="1799,001,,,,11,31,20"></lb> |
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14072 τοῦ ΨΥ στερεοῦ ἴσα τέ ἐστι καὶ ὅμοια.</s> <s>ἀλλὰ τὰ μὲν τρία <lb n="1799,001,,,,11,31,21"></lb> |
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14073 τρισὶ τοῖς ἀπεναντίον ἴσα τέ ἐστι καὶ ὅμοια, τὰ δὲ τρία <lb n="1799,001,,,,11,31,22"></lb> |
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14074 τρισὶ τοῖς ἀπεναντίον· ὅλον ἄρα τὸ ΑΕ στερεὸν παραλ-<lb n="1799,001,,,,11,31,23"></lb> |
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14075 ληλεπίπεδον ὅλῳ τῷ ΨΥ στερεῷ παραλληλεπιπέδῳ ἴσον <lb n="1799,001,,,,11,31,24"></lb> |
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14076 ἐστίν.</s> <s>διήχθωσαν αἱ ΔΡ, ΧΥ καὶ συμπιπτέτωσαν ἀλλή-<lb n="1799,001,,,,11,31,25"></lb> |
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14077 λαις κατὰ τὸ Ω, καὶ διὰ τοῦ Τ τῇ ΔΩ παράλληλος ἤχθω ἡ <lb n="1799,001,,,,11,31,26"></lb> |
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14078 σθω τὰ ΩΨ, ΡΙ στερεά.</s> <s>ἴσον δή ἐστι τὸ ΨΩ στερεόν, οὗ <pb></pb> <lb n="1799,001,,,,11,31,28"></lb> |
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14079 βάσις μέν ἐστι τὸ ΡΨ παραλληλόγραμμον, ἀπεναντίον δὲ <lb n="1799,001,,,,11,31,29"></lb> |
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14080 τὸ Ω#4, τῷ ΨΥ στερεῷ, οὗ βάσις μὲν τὸ ΡΨ παραλληλό-<lb n="1799,001,,,,11,31,30"></lb> |
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14081 γραμμον, ἀπεναντίον δὲ τὸ ΥΦ· ἐπί τε γὰρ τῆς αὐτῆς βά-<lb n="1799,001,,,,11,31,31"></lb> |
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14082 σεώς εἰσι τῆς ΡΨ καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι <lb n="1799,001,,,,11,31,32"></lb> |
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14083 αἱ ΡΩ, ΡΥ, Τ#5, ΤΧ, Σ#2, Σο%24, Ψ#4, ΨΦ ἐπὶ τῶν αὐτῶν <lb n="1799,001,,,,11,31,33"></lb> |
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14084 εἰσιν εὐθειῶν τῶν ΩΧ, #2Φ.</s> <s>ἀλλὰ τὸ ΨΥ στερεὸν τῷ ΑΕ <lb n="1799,001,,,,11,31,34"></lb> |
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14085 ἐστιν ἴσον· καὶ τὸ ΨΩ ἄρα στερεὸν τῷ ΑΕ στερεῷ ἐστιν <lb n="1799,001,,,,11,31,35"></lb> |
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14086 ἴσον.</s> <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΡΥΧΤ παραλληλόγραμμον τῷ <lb n="1799,001,,,,11,31,36"></lb> |
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14087 ΩΤ παραλληλογράμμῳ· ἐπί τε γὰρ τῆς αὐτῆς βάσεώς <lb n="1799,001,,,,11,31,37"></lb> |
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14088 εἰσι τῆς ΡΤ καὶ ἐν ταῖς αὐταῖς παραλλήλοις ταῖς ΡΤ, <lb n="1799,001,,,,11,31,38"></lb> |
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14089 ΩΧ· ἀλλὰ τὸ ΡΥΧΤ τῷ ΓΔ ἐστιν ἴσον, ἐπεὶ καὶ τῷ ΑΒ, <lb n="1799,001,,,,11,31,39"></lb> |
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14090 καὶ τὸ ΩΤ ἄρα παραλληλόγραμμον τῷ ΓΔ ἐστιν ἴσον.</s> <lb n="1799,001,,,,11,31,40"></lb> |
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14091 <s>ἄλλο δὲ τὸ ΔΤ· ἔστιν ἄρα ὡς ἡ ΓΔ βάσις πρὸς τὴν ΔΤ, <lb n="1799,001,,,,11,31,41"></lb> |
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14092 οὕτως ἡ ΩΤ πρὸς τὴν ΔΤ.</s> <s>καὶ ἐπεὶ στερεὸν παραλληλ-<lb n="1799,001,,,,11,31,42"></lb> |
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14093 επίπεδον τὸ ΓΙ ἐπιπέδῳ τῷ ΡΖ τέτμηται παραλλήλῳ ὄντι <lb n="1799,001,,,,11,31,43"></lb> |
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14094 τοῖς ἀπεναντίον ἐπιπέδοις, ἔστιν ὡς ἡ ΓΔ βάσις πρὸς τὴν <lb n="1799,001,,,,11,31,44"></lb> |
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14095 ΔΤ βάσιν, οὕτως τὸ ΓΖ στερεὸν πρὸς τὸ ΡΙ στερεόν.</s> <s>διὰ <lb n="1799,001,,,,11,31,45"></lb> |
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14096 τὰ αὐτὰ δή, ἐπεὶ στερεὸν παραλληλεπίπεδον τὸ ΩΙ ἐπιπέδῳ <lb n="1799,001,,,,11,31,46"></lb> |
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14097 τῷ ΡΨ τέτμηται παραλλήλῳ ὄντι τοῖς ἀπεναντίον ἐπι-<lb n="1799,001,,,,11,31,47"></lb> |
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14098 πέδοις, ἔστιν ὡς ἡ ΩΤ βάσις πρὸς τὴν ΤΔ βάσιν, οὕτως <lb n="1799,001,,,,11,31,48"></lb> |
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14099 τὸ ΩΨ στερεὸν πρὸς τὸ ΡΙ.</s> <s>ἀλλ' ὡς ἡ ΓΔ βάσις πρὸς τὴν <lb n="1799,001,,,,11,31,49"></lb> |
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14100 ΔΤ, οὕτως ἡ ΩΤ πρὸς τὴν ΔΤ· καὶ ὡς ἄρα τὸ ΓΖ στε- <pb></pb><lb n="1799,001,,,,11,31,50"></lb> |
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14101 ρεὸν πρὸς τὸ ΡΙ στερεόν, οὕτως τὸ ΩΨ στερεὸν πρὸς τὸ <lb n="1799,001,,,,11,31,51"></lb> |
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14102 ΡΙ.</s> <s>ἑκάτερον ἄρα τῶν ΓΖ, ΩΨ στερεῶν πρὸς τὸ ΡΙ τὸν <lb n="1799,001,,,,11,31,52"></lb> |
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14103 αὐτὸν ἔχει λόγον· ἴσον ἄρα ἐστὶ τὸ ΓΖ στερεὸν τῷ ΩΨ <lb n="1799,001,,,,11,31,53"></lb> |
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14104 στερεῷ.</s> <s>ἀλλὰ τὸ ΩΨ τῷ ΑΕ ἐδείχθη ἴσον· καὶ τὸ ΑΕ <lb n="1799,001,,,,11,31,54"></lb> |
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14105 ἄρα τῷ ΓΖ ἐστιν ἴσον. <lb n="1799,001,,,,11,31,55"></lb> |
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14106 Μὴ ἔστωσαν δὴ αἱ ἐφεστηκυῖαι αἱ ΑΗ, ΘΚ, ΒΕ, ΛΜ, <lb n="1799,001,,,,11,31,56"></lb> |
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14107 ΓΝ, ΟΠ, ΔΖ, ΡΣ πρὸς ὀρθὰς ταῖς ΑΒ, ΓΔ βάσεσιν· <lb n="1799,001,,,,11,31,57"></lb> |
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14108 λέγω πάλιν, ὅτι ἴσον τὸ ΑΕ στερεὸν τῷ ΓΖ στερεῷ.</s> <s>ἤχθω-<lb n="1799,001,,,,11,31,58"></lb> |
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14109 σαν γὰρ ἀπὸ τῶν Κ, Ε, Η, Μ, Π, Ζ, Ν, Σ σημείων ἐπὶ τὸ <lb n="1799,001,,,,11,31,59"></lb> |
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14110 ὑποκείμενον ἐπίπεδον κάθετοι αἱ ΚΞ, ΕΤ, ΗΥ, ΜΦ, <lb n="1799,001,,,,11,31,60"></lb> |
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14111 ΠΧ, ΖΨ, ΝΩ, ΣΙ, καὶ συμβαλλέτωσαν τῷ ἐπιπέδῳ κατὰ <lb n="1799,001,,,,11,31,61"></lb> |
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14112 τὰ Ξ, Τ, Υ, Φ, Χ, Ψ, Ω, Ι σημεῖα, καὶ ἐπεζεύχθωσαν αἱ <lb n="1799,001,,,,11,31,62"></lb> |
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14113 ΞΤ, ΞΥ, ΥΦ, ΤΦ, ΧΨ, ΧΩ, ΩΙ, ΙΨ.</s> <s>ἴσον δή ἐστι τὸ <lb n="1799,001,,,,11,31,63"></lb> |
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14114 ΚΦ στερεὸν τῷ ΠΙ στερεῷ· ἐπί τε γὰρ ἴσων βάσεών εἰσι <pb></pb> <lb n="1799,001,,,,11,31,64"></lb> |
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14115 τῶν ΚΜ, ΠΣ καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι πρὸς <lb n="1799,001,,,,11,31,65"></lb> |
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14116 ὀρθάς εἰσι ταῖς βάσεσιν.</s> <s>ἀλλὰ τὸ μὲν ΚΦ στερεὸν τῷ ΑΕ <lb n="1799,001,,,,11,31,66"></lb> |
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14117 στερεῷ ἐστιν ἴσον, τὸ δὲ ΠΙ τῷ ΓΖ· ἐπί τε γὰρ τῆς αὐτῆς <lb n="1799,001,,,,11,31,67"></lb> |
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14118 βάσεώς εἰσι καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι οὔκ <lb n="1799,001,,,,11,31,68"></lb> |
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14119 εἰσιν ἐπὶ τῶν αὐτῶν εὐθειῶν.</s> <s>καὶ τὸ ΑΕ ἄρα στερεὸν τῷ <lb n="1799,001,,,,11,31,69"></lb> |
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14120 ΓΖ στερεῷ ἐστιν ἴσον.</s> <lb n="1799,001,,,,11,31,70"></lb> |
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14121 <s>Τὰ ἄρα ἐπὶ ἴσων βάσεων ὄντα στερεὰ παραλληλεπίπεδα <lb n="1799,001,,,,11,31,71"></lb> |
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14122 καὶ ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλήλοις ἐστίν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,31,72"></lb> |
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14123 <s>Τὰ ὑπὸ τὸ αὐτὸ ὕψος ὄντα στερεὰ παραλληλεπίπεδα <lb n="1799,001,,,,11,32,1"></lb> |
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14124 πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις. <lb n="1799,001,,,,11,32,2"></lb> |
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14125 Ἔστω ὑπὸ τὸ αὐτὸ ὕψος στερεὰ παραλληλεπίπεδα τὰ <lb n="1799,001,,,,11,32,3"></lb> |
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14126 ΑΒ, ΓΔ· λέγω, ὅτι τὰ ΑΒ, ΓΔ στερεὰ παραλληλεπίπεδα <lb n="1799,001,,,,11,32,4"></lb> |
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14127 πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις, τουτέστιν ὅτι ἐστὶν ὡς ἡ <lb n="1799,001,,,,11,32,5"></lb> |
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14128 ΑΕ βάσις πρὸς τὴν ΓΖ βάσιν, οὕτως τὸ ΑΒ στερεὸν πρὸς <lb n="1799,001,,,,11,32,6"></lb> |
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14129 τὸ ΓΔ στερεόν.</s> <lb n="1799,001,,,,11,32,7"></lb> |
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14130 <s>Παραβεβλήσθω γὰρ παρὰ τὴν ΖΗ τῷ ΑΕ ἴσον τὸ ΖΘ, <lb n="1799,001,,,,11,32,8"></lb> |
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14131 καὶ ἀπὸ βάσεως μὲν τῆς ΖΘ, ὕψους δὲ τοῦ αὐτοῦ τῷ ΓΔ <pb></pb> <lb n="1799,001,,,,11,32,9"></lb> |
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14132 στερεὸν παραλληλεπίπεδον συμπεπληρώσθω τὸ ΗΚ.</s> <s>ἴσον <lb n="1799,001,,,,11,32,10"></lb> |
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14133 δή ἐστι τὸ ΑΒ στερεὸν τῷ ΗΚ στερεῷ· ἐπί τε γὰρ ἴσων <lb n="1799,001,,,,11,32,11"></lb> |
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14134 βάσεών εἰσι τῶν ΑΕ, ΖΘ καὶ ὑπὸ τὸ αὐτὸ ὕψος.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,11,32,12"></lb> |
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14135 στερεὸν παραλληλεπίπεδον τὸ ΓΚ ἐπιπέδῳ τῷ ΔΗ τέτμη-<lb n="1799,001,,,,11,32,13"></lb> |
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14136 ται παραλλήλῳ ὄντι τοῖς ἀπεναντίον ἐπιπέδοις, ἔστιν ἄρα <lb n="1799,001,,,,11,32,14"></lb> |
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14137 ὡς ἡ ΓΖ βάσις πρὸς τὴν ΖΘ βάσιν, οὕτως τὸ ΓΔ στερεὸν <lb n="1799,001,,,,11,32,15"></lb> |
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14138 πρὸς τὸ ΔΘ στερεόν.</s> <s>ἴση δὲ ἡ μὲν ΖΘ βάσις τῇ ΑΕ βάσει, <lb n="1799,001,,,,11,32,16"></lb> |
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14139 τὸ δὲ ΗΚ στερεὸν τῷ ΑΒ στερεῷ· ἔστιν ἄρα καὶ ὡς ἡ ΑΕ <lb n="1799,001,,,,11,32,17"></lb> |
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14140 βάσις πρὸς τὴν ΓΖ βάσιν, οὕτως τὸ ΑΒ στερεὸν πρὸς τὸ <lb n="1799,001,,,,11,32,18"></lb> |
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14141 ΓΔ στερεόν.</s> <lb n="1799,001,,,,11,32,19"></lb> |
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14142 <s>Τὰ ἄρα ὑπὸ τὸ αὐτὸ ὕψος ὄντα στερεὰ παραλληλεπίπεδα <lb n="1799,001,,,,11,32,20"></lb> |
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14143 πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,32,21"></lb> |
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14144 <s>Τὰ ὅμοια στερεὰ παραλληλεπίπεδα πρὸς ἄλληλα ἐν τρι-<lb n="1799,001,,,,11,33,1"></lb> |
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14145 πλασίονι λόγῳ εἰσὶ τῶν ὁμολόγων πλευρῶν.</s> <lb n="1799,001,,,,11,33,2"></lb> |
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14146 <s>Ἔστω ὅμοια στερεὰ παραλληλεπίπεδα <lb n="1799,001,,,,11,33,3"></lb> |
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14147 τὰ ΑΒ, ΓΔ, ὁμόλογος δὲ ἔστω ἡ ΑΕ τῇ <lb n="1799,001,,,,11,33,4"></lb> |
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14148 ΓΖ· λέγω, ὅτι τὸ ΑΒ στερεὸν πρὸς τὸ ΓΔ <lb n="1799,001,,,,11,33,5"></lb> |
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14149 στερεὸν τριπλασίονα λόγον ἔχει, ἤπερ ἡ <lb n="1799,001,,,,11,33,6"></lb> |
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14150 ΑΕ πρὸς τὴν ΓΖ.</s> <lb n="1799,001,,,,11,33,7"></lb> |
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14151 <s>Ἐκβεβλήσθωσαν γὰρ ἐπ' εὐθείας ταῖς ΑΕ, <lb n="1799,001,,,,11,33,8"></lb> |
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14152 ΗΕ, ΘΕ αἱ ΕΚ, ΕΛ, ΕΜ, καὶ κείσθω τῇ μὲν ΓΖ ἴση ἡ ΕΚ, <lb n="1799,001,,,,11,33,9"></lb> |
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14153 τῇ δὲ ΖΝ ἴση ἡ ΕΛ, καὶ ἔτι τῇ ΖΡ ἴση ἡ ΕΜ, καὶ συμπε-<lb n="1799,001,,,,11,33,10"></lb> |
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14154 πληρώσθω τὸ ΚΛ παραλληλόγραμμον καὶ τὸ ΚΟ στερεόν.</s> <s><pb></pb> <lb n="1799,001,,,,11,33,11"></lb> |
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14155 Καὶ ἐπεὶ δύο αἱ ΚΕ, ΕΛ δυσὶ ταῖς ΓΖ, ΖΝ ἴσαι εἰσίν, <lb n="1799,001,,,,11,33,12"></lb> |
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14156 ἀλλὰ καὶ γωνία ἡ ὑπὸ ΚΕΛ γωνίᾳ τῇ ὑπὸ ΓΖΝ ἐστιν ἴση, <lb n="1799,001,,,,11,33,13"></lb> |
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14157 ἐπειδήπερ καὶ ἡ ὑπὸ ΑΕΗ τῇ ὑπὸ ΓΖΝ ἐστιν ἴση διὰ τὴν <lb n="1799,001,,,,11,33,14"></lb> |
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14158 ὁμοιότητα τῶν ΑΒ, ΓΔ στερεῶν, ἴσον ἄρα ἐστὶ [καὶ ὅμοι-<lb n="1799,001,,,,11,33,15"></lb> |
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14159 ον] τὸ ΚΛ παραλληλόγραμμον τῷ ΓΝ παραλληλογράμμῳ.</s> <lb n="1799,001,,,,11,33,16"></lb> |
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14160 <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ μὲν ΚΜ παραλληλόγραμμον ἴσον <lb n="1799,001,,,,11,33,17"></lb> |
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14161 ἐστὶ καὶ ὅμοιον τῷ ΓΡ [παραλληλογράμμῳ] καὶ ἔτι τὸ <lb n="1799,001,,,,11,33,18"></lb> |
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14162 ΕΟ τῷ ΔΖ· τρία ἄρα παραλληλόγραμμα τοῦ ΚΟ στερεοῦ <lb n="1799,001,,,,11,33,19"></lb> |
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14163 τρισὶ παραλληλογράμμοις τοῦ ΓΔ στερεοῦ ἴσα ἐστὶ καὶ <lb n="1799,001,,,,11,33,20"></lb> |
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14164 ὅμοια.</s> <s>ἀλλὰ τὰ μὲν τρία τρισὶ τοῖς ἀπεναντίον ἴσα ἐστὶ καὶ <lb n="1799,001,,,,11,33,21"></lb> |
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14165 ὅμοια, τὰ δὲ τρία τρισὶ τοῖς ἀπεναντίον ἴσα ἐστὶ καὶ ὅμοια· <lb n="1799,001,,,,11,33,22"></lb> |
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14166 ὅλον ἄρα τὸ ΚΟ στερεὸν ὅλῳ τῷ ΓΔ στερεῷ ἴσον ἐστὶ καὶ <pb></pb> <lb n="1799,001,,,,11,33,23"></lb> |
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14167 ὅμοιον.</s> <s>συμπεπληρώσθω τὸ ΗΚ παραλληλόγραμμον, καὶ <lb n="1799,001,,,,11,33,24"></lb> |
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14168 ἀπὸ βάσεων μὲν τῶν ΗΚ, ΚΛ παραλληλογράμμων, ὕψους <lb n="1799,001,,,,11,33,25"></lb> |
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14169 δὲ τοῦ αὐτοῦ τῷ ΑΒ στερεὰ συμπεπληρώσθω τὰ ΕΞ, ΛΠ.</s> <lb n="1799,001,,,,11,33,26"></lb> |
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14170 <s>καὶ ἐπεὶ διὰ τὴν ὁμοιότητα τῶν ΑΒ, ΓΔ στερεῶν ἐστιν <lb n="1799,001,,,,11,33,27"></lb> |
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14171 ὡς ἡ ΑΕ πρὸς τὴν ΓΖ, οὕτως ἡ ΕΗ πρὸς τὴν ΖΝ, καὶ ἡ <lb n="1799,001,,,,11,33,28"></lb> |
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14172 ΕΘ πρὸς τὴν ΖΡ, ἴση δὲ ἡ μέν ΓΖ τῇ ΕΚ, ἡ δὲ ΖΝ τῇ <lb n="1799,001,,,,11,33,29"></lb> |
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14173 ΕΛ, ἡ δὲ ΖΡ τῇ ΕΜ, ἔστιν ἄρα ὡς ἡ ΑΕ πρὸς τὴν ΕΚ, <lb n="1799,001,,,,11,33,30"></lb> |
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14174 οὕτως ἡ ΗΕ πρὸς τὴν ΕΛ καὶ ἡ ΘΕ πρὸς τὴν ΕΜ.</s> <s>ἀλλ' <lb n="1799,001,,,,11,33,31"></lb> |
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14175 ὡς μὲν ἡ ΑΕ πρὸς τὴν ΕΚ, οὕτως τὸ ΑΗ [παραλληλό-<lb n="1799,001,,,,11,33,32"></lb> |
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14176 γραμμον] πρὸς τὸ ΗΚ παραλληλόγραμμον, ὡς δὲ ἡ ΗΕ <lb n="1799,001,,,,11,33,33"></lb> |
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14177 πρὸς τὴν ΕΛ, οὕτως τὸ ΗΚ πρὸς τὸ ΚΛ, ὡς δὲ ἡ ΘΕ <lb n="1799,001,,,,11,33,34"></lb> |
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14178 πρὸς ΕΜ, οὕτως τὸ ΠΕ πρὸς τὸ ΚΜ· καὶ ὡς ἄρα τὸ <lb n="1799,001,,,,11,33,35"></lb> |
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14179 ΑΗ παραλληλόγραμμον πρὸς τὸ ΗΚ, οὕτως τὸ ΗΚ πρὸς <lb n="1799,001,,,,11,33,36"></lb> |
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14180 τὸ ΚΛ καὶ τὸ ΠΕ πρὸς τὸ ΚΜ.</s> <s>ἀλλ' ὡς μὲν τὸ ΑΗ πρὸς <lb n="1799,001,,,,11,33,37"></lb> |
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14181 τὸ ΗΚ, οὕτως τὸ ΑΒ στερεὸν πρὸς τὸ ΕΞ στερεόν, ὡς δὲ <lb n="1799,001,,,,11,33,38"></lb> |
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14182 τὸ ΗΚ πρὸς τὸ Κ*λ, οὕτως τὸ ΞΕ στερεὸν πρὸς τὸ ΠΛ <lb n="1799,001,,,,11,33,39"></lb> |
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14183 στερεόν, ὡς δὲ τὸ ΠΕ πρὸς τὸ ΚΜ, οὕτως τὸ ΠΛ στερεὸν <lb n="1799,001,,,,11,33,40"></lb> |
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14184 πρὸς τὸ ΚΟ στερεόν· καὶ ὡς ἄρα τὸ ΑΒ στερεὸν πρὸς τὸ <lb n="1799,001,,,,11,33,41"></lb> |
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14185 ΕΞ, οὕτως τὸ ΕΞ πρὸς τὸ ΠΛ καὶ τὸ ΠΛ πρὸς τὸ ΚΟ.</s> <lb n="1799,001,,,,11,33,42"></lb> |
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14186 <s>ἐὰν δὲ τέσσαρα μεγέθη κατὰ τὸ συνεχὲς ἀνάλογον ᾖ, τὸ <lb n="1799,001,,,,11,33,43"></lb> |
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14187 πρῶτον πρὸς τὸ τέταρτον τριπλασίονα λόγον ἔχει ἤπερ <lb n="1799,001,,,,11,33,44"></lb> |
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14188 πρὸς τὸ δεύτερον· τὸ ΑΒ ἄρα στερεὸν πρὸς τὸ ΚΟ τριπλα-<lb n="1799,001,,,,11,33,45"></lb> |
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14189 σίονα λόγον ἔχει ἤπερ τὸ ΑΒ πρὸς τὸ ΕΞ.</s> <s>ἀλλ' ὡς τὸ ΑΒ <lb n="1799,001,,,,11,33,46"></lb> |
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14190 πρὸς τὸ ΕΞ, οὕτως τὸ ΑΗ παραλληλόγραμμον πρὸς τὸ <lb n="1799,001,,,,11,33,47"></lb> |
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14191 ΗΚ καὶ ἡ ΑΕ εὐθεῖα πρὸς τὴν ΕΚ· ὥστε καὶ τὸ ΑΒ στε- <pb></pb><lb n="1799,001,,,,11,33,48"></lb> |
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14192 ρεὸν πρὸς τὸ ΚΟ τριπλασίονα λόγον ἔχει ἤπερ ἡ ΑΕ πρὸς <lb n="1799,001,,,,11,33,49"></lb> |
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14193 τὴν ΕΚ.</s> <s>ἴσον δὲ τὸ [μὲν] ΚΟ στερεὸν τῷ ΓΔ στερεῷ, ἡ <lb n="1799,001,,,,11,33,50"></lb> |
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14194 δὲ ΕΚ εὐθεῖα τῇ ΓΖ· καὶ τὸ ΑΒ ἄρα στερεὸν πρὸς τὸ ΓΔ <lb n="1799,001,,,,11,33,51"></lb> |
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14195 στερεὸν τριπλασίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος αὐτοῦ <lb n="1799,001,,,,11,33,52"></lb> |
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14196 πλευρὰ ἡ ΑΕ πρὸς τὴν ὁμόλογον πλευρὰν τὴν ΓΖ.</s> <lb n="1799,001,,,,11,33,53"></lb> |
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14197 <s>Τὰ ἄρα ὅμοια στερεὰ παραλληλεπίπεδα ἐν τριπλασίονι <lb n="1799,001,,,,11,33,54"></lb> |
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14198 λόγῳ ἐστὶ τῶν ὁμολόγων πλευρῶν· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,33,55"></lb> |
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14199 <s>Πόρισμα <lb n="1799,001,,,,11,33,56"></lb> |
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14200 Ἐκ δὴ τούτου φανερόν, ὅτι ἐὰν τέσσαρες εὐθεῖαι ἀνάλο-<lb n="1799,001,,,,11,33,57"></lb> |
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14201 γον ὦσιν, ἔσται ὡς ἡ πρώτη πρὸς τὴν τετάρτην, οὕτω τὸ <lb n="1799,001,,,,11,33,58"></lb> |
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14202 ἀπὸ τῆς πρώτης στερεὸν παραλληλεπίπεδον πρὸς τὸ ἀπὸ <lb n="1799,001,,,,11,33,59"></lb> |
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14203 τῆς δευτέρας τὸ ὅμοιον καὶ ὁμοίως ἀναγραφόμενον, ἐπείπερ <lb n="1799,001,,,,11,33,60"></lb> |
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14204 καὶ ἡ πρώτη πρὸς τὴν τετάρτην τριπλασίονα λόγον ἔχει <lb n="1799,001,,,,11,33,61"></lb> |
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14205 ἤπερ πρὸς τὴν δευτέραν.</s> <lb n="1799,001,,,,11,33,62"></lb> |
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14206 <s>Τῶν ἴσων στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν <lb n="1799,001,,,,11,34,1"></lb> |
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14207 αἱ βάσεις τοῖς ὕψεσιν· καὶ ὧν στερεῶν παραλληλεπιπέδων <lb n="1799,001,,,,11,34,2"></lb> |
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14208 ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, ἴσα ἐστὶν ἐκεῖνα.</s> <lb n="1799,001,,,,11,34,3"></lb> |
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14209 <s>Ἔστω ἴσα στερεὰ παραλληλεπίπεδα τὰ ΑΒ, ΓΔ· λέγω, <lb n="1799,001,,,,11,34,4"></lb> |
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14210 ὅτι τῶν ΑΒ, ΓΔ στερεῶν παραλληλεπιπέδων ἀντιπεπόν-<lb n="1799,001,,,,11,34,5"></lb> |
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14211 θασιν αἱ βάσεις τοῖς ὕψεσιν, καί ἐστιν ὡς ἡ ΕΘ βάσις πρὸς <lb n="1799,001,,,,11,34,6"></lb> |
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14212 τὴν ΝΠ βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ ὕψος πρὸς τὸ <lb n="1799,001,,,,11,34,7"></lb> |
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14213 τοῦ ΑΒ στερεοῦ ὕψος.</s> <s><pb></pb> <lb n="1799,001,,,,11,34,8"></lb> |
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14214 Ἔστωσαν γὰρ πρότερον αἱ ἐφεστηκυῖαι αἱ ΑΗ, ΕΖ, <lb n="1799,001,,,,11,34,9"></lb> |
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14215 ΛΒ, ΘΚ, ΓΜ, ΝΞ, ΟΔ, ΠΡ πρὸς ὀρθὰς ταῖς βάσεσιν <lb n="1799,001,,,,11,34,10"></lb> |
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14216 αὐτῶν· λέγω, ὅτι ἐστὶν ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ <lb n="1799,001,,,,11,34,11"></lb> |
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14217 βάσιν, οὕτως ἡ ΓΜ πρὸς τὴν ΑΗ. <lb n="1799,001,,,,11,34,12"></lb> |
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14218 Εἰ μὲν οὖν ἴση ἐστιν ἡ ΕΘ βάσις τῇ ΝΠ βάσει, ἔστι δὲ <lb n="1799,001,,,,11,34,13"></lb> |
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14219 καὶ τὸ ΑΒ στερεὸν τῷ ΓΔ στερεῷ ἴσον, ἔσται καὶ ἡ ΓΜ <lb n="1799,001,,,,11,34,14"></lb> |
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14220 τῇ ΑΗ ἴση.</s> <s>τὰ γὰρ ὑπὸ τὸ αὐτὸ ὕψος στερεὰ παραλληλ-<lb n="1799,001,,,,11,34,15"></lb> |
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14221 επίπεδα πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις [εἰ γὰρ τῶν ΕΘ, <lb n="1799,001,,,,11,34,16"></lb> |
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14222 ΝΠ βάσεων ἴσων οὐσῶν μὴ εἴη τὰ ΑΗ, ΓΜ ὕψη ἴσα, <pb></pb> <lb n="1799,001,,,,11,34,17"></lb> |
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14223 οὐδ' ἄρα τὸ ΑΒ στερεὸν ἴσον ἔσται τῷ ΓΔ.</s> <s>ὑπόκειται δὲ <lb n="1799,001,,,,11,34,18"></lb> |
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14224 ἴσον· οὐκ ἄρα ἄνισόν ἐστι τὸ ΓΜ ὕψος τῷ ΑΗ ὕψει· ἴσον <lb n="1799,001,,,,11,34,19"></lb> |
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14225 ἄρα].</s> <s>καὶ ἔσται ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ, οὕτως ἡ <lb n="1799,001,,,,11,34,20"></lb> |
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14226 ΓΜ πρὸς τὴν ΑΗ, καὶ φανερόν, ὅτι τῶν ΑΒ, ΓΔ στερεῶν <lb n="1799,001,,,,11,34,21"></lb> |
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14227 παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν.</s> <lb n="1799,001,,,,11,34,22"></lb> |
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14228 <s>Μὴ ἔστω δὴ ἴση ἡ ΕΘ βάσις τῇ ΝΠ βάσει, ἀλλ' ἔστω <lb n="1799,001,,,,11,34,23"></lb> |
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14229 μείζων ἡ ΕΘ.</s> <s>ἔστι δὲ καὶ τὸ ΑΒ στερεὸν τῷ ΓΔ στερεῷ <lb n="1799,001,,,,11,34,24"></lb> |
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14230 ἴσον· μείζων ἄρα ἐστὶ καὶ ἡ ΓΜ τῆς ΑΗ [εἰ γὰρ μή, οὐδ' <lb n="1799,001,,,,11,34,25"></lb> |
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14231 ἄρα πάλιν τὰ ΑΒ, ΓΔ στερεὰ ἴσα ἔσται· ὑπόκειται δὲ <lb n="1799,001,,,,11,34,26"></lb> |
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14232 ἴσα].</s> <s>κείσθω οὖν τῇ ΑΗ ἴση ἡ ΓΤ, καὶ συμπεπληρώσθω <lb n="1799,001,,,,11,34,27"></lb> |
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14233 ἀπὸ βάσεως μὲν τῆς ΝΠ, ὕψους δὲ τοῦ ΓΤ, στερεὸν παραλ-<lb n="1799,001,,,,11,34,28"></lb> |
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14234 ληλεπίπεδον τὸ ΦΓ.</s> <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΑΒ στερεὸν τῷ <lb n="1799,001,,,,11,34,29"></lb> |
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14235 ΓΔ στερεῷ, ἔξωθεν δὲ τὸ ΓΦ, τὰ δὲ ἴσα πρὸς τὸ αὐτὸ τὸν <lb n="1799,001,,,,11,34,30"></lb> |
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14236 αὐτὸν ἔχει λόγον, ἔστιν ἄρα ὡς τὸ ΑΒ στερεὸν πρὸς τὸ ΓΦ <lb n="1799,001,,,,11,34,31"></lb> |
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14237 στερεόν, οὕτως τὸ ΓΔ στερεὸν πρὸς τὸ ΓΦ στερεόν.</s> <s>ἀλλ' <lb n="1799,001,,,,11,34,32"></lb> |
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14238 ὡς μὲν τὸ ΑΒ στερεὸν πρὸς τὸ ΓΦ στερεόν, οὕτως ἡ ΕΘ <lb n="1799,001,,,,11,34,33"></lb> |
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14239 βάσις πρὸς τὴν ΝΠ βάσιν· ἰσοϋψῆ γὰρ τὰ ΑΒ, ΓΦ στε-<lb n="1799,001,,,,11,34,34"></lb> |
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14240 ρεά· ὡς δὲ τὸ ΓΔ στερεὸν πρὸς τὸ ΓΦ στερεόν, οὕτως ἡ <lb n="1799,001,,,,11,34,35"></lb> |
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14241 ΜΠ βάσις πρὸς τὴν ΤΠ βάσιν καὶ ἡ ΓΜ πρὸς τὴν ΓΤ· <lb n="1799,001,,,,11,34,36"></lb> |
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14242 καὶ ὡς ἄρα ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως ἡ ΜΓ <lb n="1799,001,,,,11,34,37"></lb> |
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14243 πρὸς τὴν ΓΤ.</s> <s>ἴση δὲ ἡ ΓΤ τῇ ΑΗ· καὶ ὡς ἄρα ἡ ΕΘ βάσις <lb n="1799,001,,,,11,34,38"></lb> |
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14244 πρὸς τὴν ΝΠ βάσιν, οὕτως ἡ ΜΓ πρὸς τὴν ΑΗ.</s> <s>τῶν ΑΒ, <pb></pb> <lb n="1799,001,,,,11,34,39"></lb> |
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14245 ΓΔ ἄρα στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ <lb n="1799,001,,,,11,34,40"></lb> |
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14246 βάσεις τοῖς ὕψεσιν.</s> <lb n="1799,001,,,,11,34,41"></lb> |
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14247 <s>Πάλιν δὴ τῶν ΑΒ, ΓΔ στερεῶν παραλληλεπιπέδων <lb n="1799,001,,,,11,34,42"></lb> |
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14248 ἀντιπεπονθέτωσαν αἱ βάσεις τοῖς ὕψεσιν, καὶ ἔστω ὡς <lb n="1799,001,,,,11,34,43"></lb> |
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14249 ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ <lb n="1799,001,,,,11,34,44"></lb> |
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14250 ὕψος πρὸς τὸ τοῦ ΑΒ στερεοῦ ὕψος· λέγω, ὅτι ἴσον ἐστὶ τὸ <lb n="1799,001,,,,11,34,45"></lb> |
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14251 ΑΒ στερεὸν τῷ ΓΔ στερεῷ.</s> <lb n="1799,001,,,,11,34,46"></lb> |
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14252 <s>Ἔστωσαν [γὰρ] πάλιν αἱ ἐφεστηκυῖαι πρὸς ὀρθὰς ταῖς <lb n="1799,001,,,,11,34,47"></lb> |
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14253 βάσεσιν, καὶ εἰ μὲν ἴση ἐστὶν ἡ ΕΘ βάσις τῇ ΝΠ βάσει, <lb n="1799,001,,,,11,34,48"></lb> |
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14254 καί ἐστιν ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως τὸ <lb n="1799,001,,,,11,34,49"></lb> |
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14255 τοῦ ΓΔ στερεοῦ ὕψος πρὸς τὸ τοῦ ΑΒ στερεοῦ ὕψος, ἴσον <lb n="1799,001,,,,11,34,50"></lb> |
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14256 ἄρα ἐστὶ καὶ τὸ τοῦ ΓΔ στερεοῦ ὕψος τῷ τοῦ ΑΒ στερεοῦ <lb n="1799,001,,,,11,34,51"></lb> |
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14257 ὕψει.</s> <s>τὰ δὲ ἐπὶ ἴσων βάσεων στερεὰ παραλληλεπίπεδα καὶ <lb n="1799,001,,,,11,34,52"></lb> |
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14258 ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλήλοις ἐστίν· ἴσον ἄρα ἐστὶ τὸ ΑΒ <lb n="1799,001,,,,11,34,53"></lb> |
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14259 στερεὸν τῷ ΓΔ στερεῷ.</s> <lb n="1799,001,,,,11,34,54"></lb> |
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14260 <s>Μὴ ἔστω δὴ ἡ ΕΘ βάσις τῇ ΝΠ [βάσει] ἴση, ἀλλ' ἔστω <lb n="1799,001,,,,11,34,55"></lb> |
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14261 μείζων ἡ ΕΘ· μεῖζον ἄρα ἐστὶ καὶ τὸ τοῦ ΓΔ στερεοῦ <lb n="1799,001,,,,11,34,56"></lb> |
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14262 ὕψος τοῦ τοῦ ΑΒ στερεοῦ ὕψους, τουτέστιν ἡ ΓΜ τῆς <lb n="1799,001,,,,11,34,57"></lb> |
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14263 ΑΗ.</s> <s>κείσθω τῇ ΑΗ ἴση πάλιν ἡ ΓΤ, καὶ συμπεπλη-<lb n="1799,001,,,,11,34,58"></lb> |
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14264 ρώσθω ὁμοίως τὸ ΓΦ στερεόν.</s> <s>ἐπεί ἐστιν ὡς ἡ ΕΘ βάσις <lb n="1799,001,,,,11,34,59"></lb> |
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14265 πρὸς τὴν ΝΠ βάσιν, οὕτως ἡ ΜΓ πρὸς τὴν ΑΗ, ἴση δὲ ἡ <lb n="1799,001,,,,11,34,60"></lb> |
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14266 ΑΗ τῇ ΓΤ, ἔστιν ἄρα ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, <lb n="1799,001,,,,11,34,61"></lb> |
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14267 οὕτως ἡ ΓΜ πρὸς τὴν ΓΤ.</s> <s>ἀλλ' ὡς μὲν ἡ ΕΘ [βάσισ] πρὸς <pb></pb> <lb n="1799,001,,,,11,34,62"></lb> |
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14268 τὴν ΝΠ βάσιν, οὕτως τὸ ΑΒ στερεὸν πρὸς τὸ ΓΦ στερεόν· <lb n="1799,001,,,,11,34,63"></lb> |
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14269 ἰσοϋψῆ γάρ ἐστι τὰ ΑΒ, ΓΦ στερεά· ὡς δὲ ἡ ΓΜ πρὸς <lb n="1799,001,,,,11,34,64"></lb> |
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14270 τὴν ΓΤ, οὕτως ἥ τε ΜΠ βάσις πρὸς τὴν ΠΤ βάσιν καὶ <lb n="1799,001,,,,11,34,65"></lb> |
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14271 τὸ ΓΔ στερεὸν πρὸς τὸ ΓΦ στερεόν.</s> <s>καὶ ὡς ἄρα τὸ ΑΒ <lb n="1799,001,,,,11,34,66"></lb> |
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14272 στερεὸν πρὸς τὸ ΓΦ στερεόν, οὕτως τὸ ΓΔ στερεὸν πρὸς <lb n="1799,001,,,,11,34,67"></lb> |
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14273 τὸ ΓΦ στερεόν· ἑκάτερον ἄρα τῶν ΑΒ, ΓΔ πρὸς τὸ ΓΦ <lb n="1799,001,,,,11,34,68"></lb> |
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14274 τὸν αὐτὸν ἔχει λόγον.</s> <s>ἴσον ἄρα ἐστὶ τὸ ΑΒ στερεὸν τῷ <lb n="1799,001,,,,11,34,69"></lb> |
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14275 ΓΔ στερεῷ [ὅπερ ἔδει δεῖξαι].</s> <lb n="1799,001,,,,11,34,70"></lb> |
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14276 <s>Μὴ ἔστωσαν δὴ αἱ ἐφεστηκυῖαι αἱ ΖΕ, ΒΛ, ΗΑ, ΘΚ, <lb n="1799,001,,,,11,34,71"></lb> |
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14277 ΞΝ, ΔΟ, ΜΓ, ΡΠ πρὸς ὀρθὰς ταῖς βάσεσιν αὐτῶν, καὶ <lb n="1799,001,,,,11,34,72"></lb> |
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14278 ἤχθωσαν ἀπὸ τῶν Ζ, Η, Β, Κ, Ξ, Μ, Δ, Ρ σημείων ἐπὶ <lb n="1799,001,,,,11,34,73"></lb> |
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14279 τὰ διὰ τῶν ΕΘ, ΝΠ ἐπίπεδα κάθετοι καὶ συμβαλλέτωσαν <lb n="1799,001,,,,11,34,74"></lb> |
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14280 τοῖς ἐπιπέδοις κατὰ τὰ Σ, Τ, Υ, Φ, Χ, Ψ, Ω, #2, καὶ συμ-<lb n="1799,001,,,,11,34,75"></lb> |
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14281 πεπληρώσθω τὰ ΖΦ, ΞΩ στερεά· λέγω, ὅτι καὶ οὕτως <lb n="1799,001,,,,11,34,76"></lb> |
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14282 ἴσων ὄντων τῶν ΑΒ, ΓΔ στερεῶν ἀντιπεπόνθασιν αἱ <lb n="1799,001,,,,11,34,77"></lb> |
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14283 βάσεις τοῖς ὕψεσιν, καί ἐστιν ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ <lb n="1799,001,,,,11,34,78"></lb> |
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14284 βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ ὕψος πρὸς τὸ τοῦ ΑΒ <lb n="1799,001,,,,11,34,79"></lb> |
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14285 στερεοῦ ὕψος.</s> <lb n="1799,001,,,,11,34,80"></lb> |
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14286 <s>Ἐπεὶ ἴσον ἐστὶ τὸ ΑΒ στερεὸν τῷ ΓΔ στερεῷ, ἀλλὰ τὸ <lb n="1799,001,,,,11,34,81"></lb> |
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14287 μὲν ΑΒ τῷ ΒΤ ἐστιν ἴσον· ἐπί τε γὰρ τῆς αὐτῆς βάσεώς <lb n="1799,001,,,,11,34,82"></lb> |
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14288 εἰσι τῆς ΖΚ καὶ ὑπὸ τὸ αὐτὸ ὕψος [ὧν αἱ ἐφεστῶσαι οὐκ <lb n="1799,001,,,,11,34,83"></lb> |
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14289 εἰσὶν ἐπὶ τῶν αὐτῶν εὐθειῶν]· τὸ δὲ ΓΔ στερεὸν τῷ ΔΨ <pb></pb> <lb n="1799,001,,,,11,34,84"></lb> |
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14290 ἐστιν ἴσον· ἐπί τε γὰρ πάλιν τῆς αὐτῆς βάσεώς εἰσι τῆς <lb n="1799,001,,,,11,34,85"></lb> |
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14291 ΡΞ καὶ ὑπὸ τὸ αὐτὸ ὕψος [ὧν αἱ ἐφεστῶσαι οὐκ εἰσὶν ἐπὶ <lb n="1799,001,,,,11,34,86"></lb> |
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14292 τῶν αὐτῶν εὐθειῶν]· καὶ τὸ ΒΤ ἄρα στερεὸν τῷ ΔΨ <lb n="1799,001,,,,11,34,87"></lb> |
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14293 στερεῷ ἴσον ἐστίν [τῶν δὲ ἴσων στερεῶν παραλληλεπι-<lb n="1799,001,,,,11,34,88"></lb> |
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14294 πέδων, ὧν τὰ ὕψη πρὸς ὀρθάς ἐστι ταῖς βάσεσιν αὐτῶν, <lb n="1799,001,,,,11,34,89"></lb> |
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14295 ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν].</s> <s>ἔστιν ἄρα ὡς ἡ <lb n="1799,001,,,,11,34,90"></lb> |
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14296 ΖΚ βάσις πρὸς τὴν ΞΡ βάσιν, οὕτως τὸ τοῦ ΔΨ στερεοῦ <lb n="1799,001,,,,11,34,91"></lb> |
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14297 ὕψος πρὸς τὸ τοῦ ΒΤ στερεοῦ ὕψος.</s> <s>ἴση δὲ ἡ μὲν ΖΚ βάσις <lb n="1799,001,,,,11,34,92"></lb> |
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14298 τῇ ΕΘ βάσει, ἡ δὲ ΞΡ βάσις τῇ ΝΠ βάσει· ἔστιν ἄρα ὡς <lb n="1799,001,,,,11,34,93"></lb> |
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14299 ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως τὸ τοῦ ΔΨ στερεοῦ <lb n="1799,001,,,,11,34,94"></lb> |
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14300 ὕψος πρὸς τὸ τοῦ ΒΤ στερεοῦ ὕψος.</s> <s>τὰ δ' αὐτὰ ὕψη <pb></pb> <lb n="1799,001,,,,11,34,95"></lb> |
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14301 ἐστὶ τῶν ΔΨ, ΒΤ στερεῶν καὶ τῶν ΔΓ, ΒΑ· ἔστιν ἄρα <lb n="1799,001,,,,11,34,96"></lb> |
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14302 ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως τὸ τοῦ ΔΓ <lb n="1799,001,,,,11,34,97"></lb> |
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14303 στερεοῦ ὕψος πρὸς τὸ τοῦ ΑΒ στερεοῦ ὕψος.</s> <s>τῶν ΑΒ, ΓΔ <lb n="1799,001,,,,11,34,98"></lb> |
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14304 ἄρα στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις <lb n="1799,001,,,,11,34,99"></lb> |
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14305 τοῖς ὕψεσιν.</s> <lb n="1799,001,,,,11,34,100"></lb> |
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14306 <s>Πάλιν δὴ τῶν ΑΒ, ΓΔ στερεῶν παραλληλεπιπέδων <lb n="1799,001,,,,11,34,101"></lb> |
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14307 ἀντιπεπονθέτωσαν αἱ βάσεις τοῖς ὕψεσιν, καὶ ἔστω ὡς ἡ <lb n="1799,001,,,,11,34,102"></lb> |
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14308 ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ <lb n="1799,001,,,,11,34,103"></lb> |
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14309 ὕψος πρὸς τὸ τοῦ ΑΒ στερεοῦ ὕψος· λέγω, ὅτι ἴσον ἐστὶ τὸ <lb n="1799,001,,,,11,34,104"></lb> |
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14310 ΑΒ στερεὸν τῷ ΓΔ στερεῷ.</s> <lb n="1799,001,,,,11,34,105"></lb> |
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14311 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεί ἐστιν ὡς ἡ <lb n="1799,001,,,,11,34,106"></lb> |
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14312 ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ <lb n="1799,001,,,,11,34,107"></lb> |
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14313 ὕψος πρὸς τὸ τοῦ ΑΒ στερεοῦ ὕψος, ἴση δὲ ἡ μὲν ΕΘ βά-<lb n="1799,001,,,,11,34,108"></lb> |
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14314 σις τῇ ΖΚ βάσει, ἡ δὲ ΝΠ τῇ ΞΡ, ἔστιν ἄρα ὡς ἡ ΖΚ <lb n="1799,001,,,,11,34,109"></lb> |
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14315 βάσις πρὸς τὴν ΞΡ βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ ὕψος <lb n="1799,001,,,,11,34,110"></lb> |
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14316 πρὸς τὸ τοῦ ΑΒ στερεοῦ ὕψος.</s> <s>τὰ δ' αὐτὰ ὕψη ἐστὶ τῶν <lb n="1799,001,,,,11,34,111"></lb> |
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14317 ΑΒ, ΓΔ στερεῶν καὶ τῶν ΒΤ, ΔΨ· ἔστιν ἄρα ὡς ἡ ΖΚ <lb n="1799,001,,,,11,34,112"></lb> |
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14318 βάσις πρὸς τὴν ΞΡ βάσιν, οὕτως τὸ τοῦ ΔΨ στερεοῦ <lb n="1799,001,,,,11,34,113"></lb> |
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14319 ὕψος πρὸς τὸ τοῦ ΒΤ στερεοῦ ὕψος.</s> <s>τῶν ΒΤ, ΔΨ ἄρα στε-<lb n="1799,001,,,,11,34,114"></lb> |
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14320 ρεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς <lb n="1799,001,,,,11,34,115"></lb> |
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14321 ὕψεσιν [ὧν δὲ στερεῶν παραλληλεπιπέδων τὰ ὕψη πρὸς <lb n="1799,001,,,,11,34,116"></lb> |
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14322 ὀρθάς ἐστι ταῖς βάσεσιν αὐτῶν, ἀντιπεπόνθασι δὲ αἱ βάσεις <lb n="1799,001,,,,11,34,117"></lb> |
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14323 τοῖς ὕψεσιν, ἴσα ἐστὶν ἐκεῖνα]· ἴσον ἄρα ἐστὶ τὸ ΒΤ στερεὸν <lb n="1799,001,,,,11,34,118"></lb> |
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14324 τῷ ΔΨ στερεῷ.</s> <s>ἀλλὰ τὸ μὲν ΒΤ τῷ ΒΑ ἴσον ἐστίν· ἐπί <lb n="1799,001,,,,11,34,119"></lb> |
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14325 τε γὰρ τῆς αὐτῆς βάσεως [εἰσι] τῆς ΖΚ καὶ ὑπὸ τὸ αὐτὸ <lb n="1799,001,,,,11,34,120"></lb> |
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14326 ὕψος [ὧν αἱ ἐφεστῶσαι οὐκ εἰσὶν ἐπὶ τῶν αὐτῶν εὐθειῶν].</s> <lb n="1799,001,,,,11,34,121"></lb> |
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14327 <s>τὸ δὲ ΔΨ στερεὸν τῷ ΔΓ στερεῷ ἴσον ἐστίν [ἐπί τε γὰρ <pb></pb> <lb n="1799,001,,,,11,34,122"></lb> |
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14328 πάλιν τῆς αὐτῆς βάσεώς εἰσι τῆς ΞΡ καὶ ὑπὸ τὸ αὐτὸ ὕψος <lb n="1799,001,,,,11,34,123"></lb> |
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14329 καὶ οὐκ ἐν ταῖς αὐταῖς εὐθείαισ].</s> <s>καὶ τὸ ΑΒ ἄρα στερεὸν <lb n="1799,001,,,,11,34,124"></lb> |
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14330 τῷ ΓΔ στερεῷ ἐστιν ἴσον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,34,125"></lb> |
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14331 <s>Ἐὰν ὦσι δύο γωνίαι ἐπίπεδοι ἴσαι, ἐπὶ δὲ τῶν κορυφῶν <lb n="1799,001,,,,11,35,1"></lb> |
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14332 αὐτῶν μετέωροι εὐθεῖαι ἐπισταθῶσιν ἴσας γωνίας περι-<lb n="1799,001,,,,11,35,2"></lb> |
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14333 έχουσαι μετὰ τῶν ἐξ ἀρχῆς εὐθειῶν ἑκατέραν ἑκατέρᾳ, ἐπὶ <lb n="1799,001,,,,11,35,3"></lb> |
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14334 δὲ τῶν μετεώρων ληφθῇ τυχόντα σημεῖα, καὶ ἀπ' αὐτῶν <lb n="1799,001,,,,11,35,4"></lb> |
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14335 ἐπὶ τὰ ἐπίπεδα, ἐν οἷς εἰσιν αἱ ἐξ ἀρχῆς γωνίαι, κάθετοι <lb n="1799,001,,,,11,35,5"></lb> |
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14336 ἀχθῶσιν, ἀπὸ δὲ τῶν γενομένων σημείων ἐν τοῖς ἐπιπέ-<lb n="1799,001,,,,11,35,6"></lb> |
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14337 δοις ἐπὶ τὰς ἐξ ἀρχῆς γωνίας ἐπιζευχθῶσιν εὐθεῖαι, ἴσας <lb n="1799,001,,,,11,35,7"></lb> |
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14338 γωνίας περιέξουσι μετὰ τῶν μετεώρων.</s> <lb n="1799,001,,,,11,35,8"></lb> |
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14339 <s>Ἔστωσαν δύο γωνίαι εὐθύ-<lb n="1799,001,,,,11,35,9"></lb> |
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14340 γραμμοι ἴσαι αἱ ὑπὸ ΒΑΓ, <lb n="1799,001,,,,11,35,10"></lb> |
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14341 ΕΔΖ, ἀπὸ δὲ τῶν Α, Δ σημείων <lb n="1799,001,,,,11,35,11"></lb> |
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14342 μετέωροι εὐθεῖαι ἐφεστάτωσαν <lb n="1799,001,,,,11,35,12"></lb> |
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14343 αἱ ΑΗ, ΔΜ ἴσας γωνίας περι-<lb n="1799,001,,,,11,35,13"></lb> |
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14344 έχουσαι μετὰ τῶν ἐξ ἀρχῆς <lb n="1799,001,,,,11,35,14"></lb> |
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14345 εὐθειῶν ἑκατέραν ἑκατέρᾳ, τὴν <lb n="1799,001,,,,11,35,15"></lb> |
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14346 μὲν ὑπὸ ΜΔΕ τῇ ὑπὸ ΗΑΒ, <lb n="1799,001,,,,11,35,16"></lb> |
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14347 τὴν δὲ ὑπὸ ΜΔΖ τῇ ὑπὸ ΗΑΓ, <lb n="1799,001,,,,11,35,17"></lb> |
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14348 καὶ εἰλήφθω ἐπὶ τῶν ΑΗ, ΔΜ τυχόντα σημεῖα τὰ Η, Μ, <lb n="1799,001,,,,11,35,18"></lb> |
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14349 καὶ ἤχθωσαν ἀπὸ τῶν Η, Μ σημείων ἐπὶ τὰ διὰ τῶν <lb n="1799,001,,,,11,35,19"></lb> |
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14350 ΒΑΓ, ΕΔΖ ἐπίπεδα κάθετοι αἱ ΗΛ, ΜΝ, καὶ συμβαλ- <pb></pb><lb n="1799,001,,,,11,35,20"></lb> |
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14351 λέτωσαν τοῖς ἐπιπέδοις κατὰ τὰ Ν, Λ, καὶ ἐπεζεύχθωσαν <lb n="1799,001,,,,11,35,21"></lb> |
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14352 αἱ ΛΑ, ΝΔ· λέγω, ὅτι ἴση ἐστὶν ἡ ὑπὸ ΗΑΛ γωνία τῇ <lb n="1799,001,,,,11,35,22"></lb> |
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14353 ὑπὸ ΜΔΝ γωνίᾳ.</s> <lb n="1799,001,,,,11,35,23"></lb> |
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14354 <s>Κείσθω τῇ ΔΜ ἴση ἡ ΑΘ, καὶ ἤχθω διὰ τοῦ Θ σημείου <lb n="1799,001,,,,11,35,24"></lb> |
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14355 τῇ ΗΛ παράλληλος ἡ ΘΚ.</s> <s>ἡ δὲ ΗΛ κάθετός ἐστιν ἐπὶ τὸ <lb n="1799,001,,,,11,35,25"></lb> |
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14356 διὰ τῶν ΒΑΓ ἐπίπεδον· καὶ ἡ ΘΚ ἄρα κάθετός ἐστιν <lb n="1799,001,,,,11,35,26"></lb> |
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14357 ἐπὶ τὸ διὰ τῶν ΒΑΓ ἐπίπεδον.</s> <s>ἤχθωσαν ἀπὸ τῶν Κ, Ν <lb n="1799,001,,,,11,35,27"></lb> |
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14358 σημείων ἐπὶ τὰς ΑΒ, ΑΓ, ΔΖ, ΔΕ εὐθείας κάθετοι αἱ <lb n="1799,001,,,,11,35,28"></lb> |
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14359 ΚΓ, ΝΖ, ΚΒ, ΝΕ, καὶ ἐπεζεύχθωσαν αἱ ΘΓ, ΓΒ, ΜΖ, <lb n="1799,001,,,,11,35,29"></lb> |
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14360 ΖΕ.</s> <s>ἐπεὶ τὸ ἀπὸ τῆς ΘΑ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΘΚ, ΚΑ, <lb n="1799,001,,,,11,35,30"></lb> |
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14361 τῷ δὲ ἀπὸ τῆς ΚΑ ἴσα ἐστὶ τὰ ἀπὸ τῶν ΚΓ, ΓΑ, καὶ τὸ <lb n="1799,001,,,,11,35,31"></lb> |
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14362 ἀπὸ τῆς ΘΑ ἄρα ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΘΚ, ΚΓ, ΓΑ.</s> <lb n="1799,001,,,,11,35,32"></lb> |
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14363 <s>τοῖς δὲ ἀπὸ τῶν ΘΚ, ΚΓ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΘΓ· τὸ <lb n="1799,001,,,,11,35,33"></lb> |
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14364 ἄρα ἀπὸ τῆς ΘΑ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΘΓ, ΓΑ.</s> <s>ὀρθὴ <lb n="1799,001,,,,11,35,34"></lb> |
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14365 ἄρα ἐστὶν ἡ ὑπὸ ΘΓΑ γωνία.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ <lb n="1799,001,,,,11,35,35"></lb> |
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14366 ΔΖΜ γωνία ὀρθή ἐστιν.</s> <s>ἴση ἄρα ἐστὶν ἡ ὑπὸ ΑΓΘ γωνία <lb n="1799,001,,,,11,35,36"></lb> |
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14367 τῇ ὑπὸ ΔΖΜ.</s> <s>ἔστι δὲ καὶ ἡ ὑπὸ ΘΑΓ τῇ ὑπὸ ΜΔΖ ἴση.</s> <lb n="1799,001,,,,11,35,37"></lb> |
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14368 <s>δύο δὴ τρίγωνά ἐστι τὰ ΜΔΖ, ΘΑΓ δύο γωνίας δυσὶ γω-<lb n="1799,001,,,,11,35,38"></lb> |
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14369 νίαις ἴσας ἔχοντα ἑκατέραν ἑκατέρᾳ καὶ μίαν πλευρὰν μιᾷ <lb n="1799,001,,,,11,35,39"></lb> |
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14370 πλευρᾷ ἴσην τὴν ὑποτείνουσαν ὑπὸ μίαν τῶν ἴσων γωνιῶν <lb n="1799,001,,,,11,35,40"></lb> |
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14371 τὴν ΘΑ τῇ ΜΔ· καὶ τὰς λοιπὰς ἄρα πλευρὰς ταῖς λοιπαῖς <lb n="1799,001,,,,11,35,41"></lb> |
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14372 πλευραῖς ἴσας ἕξει ἑκατέραν ἑκατέρᾳ.</s> <s>ἴση ἄρα ἐστὶν ἡ ΑΓ <lb n="1799,001,,,,11,35,42"></lb> |
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14373 τῇ ΔΖ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ ΑΒ τῇ ΔΕ ἐστιν ἴση <pb></pb> <lb n="1799,001,,,,11,35,43"></lb> |
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14374 [οὕτως· ἐπεζεύχθωσαν αἱ ΘΒ, ΜΕ.</s> <s>καὶ ἐπεὶ τὸ ἀπὸ <lb n="1799,001,,,,11,35,44"></lb> |
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14375 τῆς ΑΘ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΑΚ, ΚΘ, τῷ δὲ ἀπὸ τῆς <lb n="1799,001,,,,11,35,45"></lb> |
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14376 ΑΚ ἴσα ἐστὶ τὰ ἀπὸ τῶν ΑΒ, ΒΚ, τὰ ἄρα ἀπὸ τῶν ΑΒ, <lb n="1799,001,,,,11,35,46"></lb> |
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14377 ΒΚ, ΚΘ ἴσα ἐστὶ τῷ ἀπὸ ΑΘ.</s> <s>ἀλλὰ τοῖς ἀπὸ τῶν ΒΚ, <lb n="1799,001,,,,11,35,47"></lb> |
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14378 ΚΘ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΒΘ· ὀρθὴ γὰρ ἡ ὑπὸ ΘΚΒ <lb n="1799,001,,,,11,35,48"></lb> |
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14379 γωνία διὰ τὸ καὶ τὴν ΘΚ κάθετον εἶναι ἐπὶ τὸ ὑποκείμε-<lb n="1799,001,,,,11,35,49"></lb> |
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14380 νον ἐπίπεδον· τὸ ἄρα ἀπὸ τῆς ΑΘ ἴσον ἐστὶ τοῖς ἀπὸ τῶν <lb n="1799,001,,,,11,35,50"></lb> |
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14381 ΑΒ, ΒΘ· ὀρθὴ ἄρα ἐστὶν ἡ ὑπὸ ΑΒΘ γωνία.</s> <s>διὰ τὰ αὐτὰ <lb n="1799,001,,,,11,35,51"></lb> |
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14382 δὴ καὶ ἡ ὑπὸ ΔΕΜ γωνία ὀρθή ἐστιν.</s> <s>ἔστι δὲ καὶ ἡ ὑπὸ <lb n="1799,001,,,,11,35,52"></lb> |
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14383 ΒΑΘ γωνία τῇ ὑπὸ ΕΔΜ ἴση· ὑπόκεινται γάρ· καὶ ἔστιν <lb n="1799,001,,,,11,35,53"></lb> |
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14384 ἡ ΑΘ τῇ ΔΜ ἴση· ἴση ἄρα ἐστὶ καὶ ἡ ΑΒ τῇ ΔΕ].</s> <s>ἐπεὶ οὖν <lb n="1799,001,,,,11,35,54"></lb> |
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14385 ἴση ἐστὶν ἡ μὲν ΑΓ τῇ ΔΖ, ἡ δὲ ΑΒ τῇ ΔΕ, δύο δὴ αἱ <lb n="1799,001,,,,11,35,55"></lb> |
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14386 ΓΑ, ΑΒ δυσὶ ταῖς ΖΔ, ΔΕ ἴσαι εἰσίν.</s> <s>ἀλλὰ καὶ γωνία <lb n="1799,001,,,,11,35,56"></lb> |
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14387 ἡ ὑπὸ ΓΑΒ γωνίᾳ τῇ ὑπὸ ΖΔΕ ἐστιν ἴση· βάσις ἄρα ἡ <lb n="1799,001,,,,11,35,57"></lb> |
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14388 ΒΓ βάσει τῇ ΕΖ ἴση ἐστὶ καὶ τὸ τρίγωνον τῷ τριγώνῳ <lb n="1799,001,,,,11,35,58"></lb> |
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14389 καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις· ἴση ἄρα ἡ ὑπὸ <lb n="1799,001,,,,11,35,59"></lb> |
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14390 ΑΓΒ γωνία τῇ ὑπὸ ΔΖΕ.</s> <s>ἔστι δὲ καὶ ὀρθὴ ἡ ὑπὸ ΑΓΚ <lb n="1799,001,,,,11,35,60"></lb> |
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14391 ὀρθῇ τῇ ὑπὸ ΔΖΝ ἴση· καὶ λοιπὴ ἄρα ἡ ὑπὸ ΒΓΚ λοιπῇ <lb n="1799,001,,,,11,35,61"></lb> |
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14392 τῇ ὑπὸ ΕΖΝ ἐστιν ἴση.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ ΓΒΚ <lb n="1799,001,,,,11,35,62"></lb> |
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14393 τῇ ὑπὸ ΖΕΝ ἐστιν ἴση.</s> <s>δύο δὴ τρίγωνά ἐστι τὰ ΒΓΚ, <lb n="1799,001,,,,11,35,63"></lb> |
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14394 ΕΖΝ [τὰσ] δύο γωνίας δυσὶ γωνίαις ἴσας ἔχοντα ἑκατέ-<lb n="1799,001,,,,11,35,64"></lb> |
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14395 ραν ἑκατέρᾳ καὶ μίαν πλευρὰν μιᾷ πλευρᾷ ἴσην τὴν πρὸς <lb n="1799,001,,,,11,35,65"></lb> |
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14396 ταῖς ἴσαις γωνίαις τὴν ΒΓ τῇ ΕΖ· καὶ τὰς λοιπὰς ἄρα <pb></pb> <lb n="1799,001,,,,11,35,66"></lb> |
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14397 πλευρὰς ταῖς λοιπαῖς πλευραῖς ἴσας ἕξουσιν.</s> <s>ἴση ἄρα ἐστὶν <lb n="1799,001,,,,11,35,67"></lb> |
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14398 ἡ ΓΚ τῇ ΖΝ.</s> <s>ἔστι δὲ καὶ ἡ ΑΓ τῇ ΔΖ ἴση· δύο δὴ αἱ ΑΓ, <lb n="1799,001,,,,11,35,68"></lb> |
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14399 ΓΚ δυσὶ ταῖς ΔΖ, ΖΝ ἴσαι εἰσίν· καὶ ὀρθὰς γωνίας περι-<lb n="1799,001,,,,11,35,69"></lb> |
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14400 έχουσιν.</s> <s>βάσις ἄρα ἡ ΑΚ βάσει τῇ ΔΝ ἴση ἐστίν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,11,35,70"></lb> |
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14401 ἴση ἐστὶν ἡ ΑΘ τῇ ΔΜ, ἴσον ἐστὶ καὶ τὸ ἀπὸ τῆς ΑΘ τῷ <lb n="1799,001,,,,11,35,71"></lb> |
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14402 ἀπὸ τῆς ΔΜ.</s> <s>ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΘ ἴσα ἐστὶ τὰ ἀπὸ <lb n="1799,001,,,,11,35,72"></lb> |
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14403 τῶν ΑΚ, ΚΘ· ὀρθὴ γὰρ ἡ ὑπὸ ΑΚΘ· τῷ δὲ ἀπὸ τῆς ΔΜ <lb n="1799,001,,,,11,35,73"></lb> |
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14404 ἴσα τὰ ἀπὸ τῶν ΔΝ, ΝΜ· ὀρθὴ γὰρ ἡ ὑπὸ ΔΝΜ· τὰ ἄρα <lb n="1799,001,,,,11,35,74"></lb> |
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14405 ἀπὸ τῶν ΑΚ, ΚΘ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΔΝ, ΝΜ, ὧν τὸ <lb n="1799,001,,,,11,35,75"></lb> |
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14406 ἀπὸ τῆς ΑΚ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΔΝ· λοιπὸν ἄρα τὸ ἀπὸ <lb n="1799,001,,,,11,35,76"></lb> |
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14407 τῆς ΚΘ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΝΜ· ἴση ἄρα ἡ ΘΚ τῇ ΜΝ.</s> <lb n="1799,001,,,,11,35,77"></lb> |
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14408 <s>καὶ ἐπεὶ δύο αἱ ΘΑ, ΑΚ δυσὶ ταῖς ΜΔ, ΔΝ ἴσαι εἰσὶν ἑκα-<lb n="1799,001,,,,11,35,78"></lb> |
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14409 τέρα ἑκατέρᾳ, καὶ βάσις ἡ ΘΚ βάσει τῇ ΜΝ ἐδείχθη ἴση, <lb n="1799,001,,,,11,35,79"></lb> |
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14410 γωνία ἄρα ἡ ὑπὸ ΘΑΚ γωνίᾳ τῇ ὑπὸ ΜΔΝ ἐστιν ἴση.</s> <lb n="1799,001,,,,11,35,80"></lb> |
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14411 <s>Ἐὰν ἄρα ὠ̂σι δύο γωνίαι ἐπίπεδοι ἴσαι καὶ τὰ ἑξῆς τῆς <lb n="1799,001,,,,11,35,81"></lb> |
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14412 προτάσεως [ὅπερ ἔδει δεῖξαι].</s> <lb n="1799,001,,,,11,35,82"></lb> |
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14413 <s>Πόρισμα <lb n="1799,001,,,,11,35,83"></lb> |
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14414 Ἐκ δὴ τούτου φανερόν, ὅτι, ἐὰν ὦσι δύο γωνίαι ἐπίπεδοι <lb n="1799,001,,,,11,35,84"></lb> |
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14415 ἴσαι, ἐπισταθῶσι δὲ ἐπ' αὐτῶν μετέωροι εὐθεῖαι ἴσαι ἴσας <lb n="1799,001,,,,11,35,85"></lb> |
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14416 γωνίας περιέχουσαι μετὰ τῶν ἐξ ἀρχῆς εὐθειῶν ἑκατέραν <lb n="1799,001,,,,11,35,86"></lb> |
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14417 ἑκατέρᾳ, αἱ ἀπ' αὐτῶν κάθετοι ἀγόμεναι ἐπὶ τὰ ἐπίπεδα, ἐν <lb n="1799,001,,,,11,35,87"></lb> |
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14418 οἷς εἰσιν αἱ ἐξ ἀρχῆς γωνίαι, ἴσαι ἀλλήλαις εἰσίν.</s> <s>ὅπερ ἔδει <lb n="1799,001,,,,11,35,88"></lb> |
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14419 δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,35,89"></lb> |
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14420 Ἐὰν τρεῖς εὐθεῖαι ἀνάλογον ὦσιν, τὸ ἐκ τῶν τριῶν στε-<lb n="1799,001,,,,11,36,1"></lb> |
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14421 ρεὸν παραλληλεπίπεδον ἴσον ἐστὶ τῷ ἀπὸ τῆς μέσης στε-<lb n="1799,001,,,,11,36,2"></lb> |
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14422 ρεῷ παραλληλεπιπέδῳ ἰσοπλεύρῳ μέν, ἰσογωνίῳ δὲ τῷ <lb n="1799,001,,,,11,36,3"></lb> |
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14423 προειρημένῳ.</s> <lb n="1799,001,,,,11,36,4"></lb> |
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14424 <s>Ἔστωσαν τρεῖς εὐθεῖαι ἀνάλογον αἱ Α, Β, Γ, ὡς ἡ Α <lb n="1799,001,,,,11,36,5"></lb> |
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14425 πρὸς τὴν Β, οὕτως ἡ Β πρὸς τὴν Γ· λέγω, ὅτι τὸ ἐκ τῶν <lb n="1799,001,,,,11,36,6"></lb> |
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14426 Α, Β, Γ στερεὸν ἴσον ἐστὶ τῷ ἀπὸ τῆς Β στερεῷ ἰσοπλεύρῳ <lb n="1799,001,,,,11,36,7"></lb> |
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14427 μέν, ἰσογωνίῳ δὲ τῷ προειρημένῳ. <lb n="1799,001,,,,11,36,8"></lb> |
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14428 Ἐκκείσθω στερεὰ γωνία ἡ πρὸς τῷ Ε περιεχομένη ὑπὸ <lb n="1799,001,,,,11,36,9"></lb> |
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14429 τῶν ὑπὸ ΔΕΗ, ΗΕΖ, ΖΕΔ, καὶ κείσθω τῇ μὲν Β ἴση <lb n="1799,001,,,,11,36,10"></lb> |
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14430 ἑκάστη τῶν ΔΕ, ΗΕ, ΕΖ, καὶ συμπεπληρώσθω τὸ ΕΚ <lb n="1799,001,,,,11,36,11"></lb> |
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14431 στερεὸν παραλληλεπίπεδον, τῇ δὲ Α ἴση ἡ ΛΜ, καὶ συν-<lb n="1799,001,,,,11,36,12"></lb> |
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14432 εστάτω πρὸς τῇ ΛΜ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ <pb></pb> <lb n="1799,001,,,,11,36,13"></lb> |
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14433 Λ τῇ πρὸς τῷ Ε στερεᾷ γωνίᾳ ἴση στερεὰ γωνία ἡ περι-<lb n="1799,001,,,,11,36,14"></lb> |
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14434 εχομένη ὑπὸ τῶν ΝΛΞ, ΞΛΜ, ΜΛΝ, καὶ κείσθω τῇ <lb n="1799,001,,,,11,36,15"></lb> |
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14435 μὲν Β ἴση ἡ ΛΞ, τῇ δὲ Γ ἴση ἡ ΛΝ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ Α <lb n="1799,001,,,,11,36,16"></lb> |
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14436 πρὸς τὴν Β, οὕτως ἡ Β πρὸς τὴν Γ, ἴση δὲ ἡ μὲν Α τῇ <lb n="1799,001,,,,11,36,17"></lb> |
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14437 ΛΜ, ἡ δὲ Β ἑκατέρᾳ τῶν ΛΞ, ΕΔ, ἡ δὲ Γ τῇ ΛΝ, ἔστιν <lb n="1799,001,,,,11,36,18"></lb> |
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14438 ἄρα ὡς ἡ ΛΜ πρὸς τὴν ΕΖ, οὕτως ἡ ΔΕ πρὸς τὴν ΛΝ.</s> <lb n="1799,001,,,,11,36,19"></lb> |
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14439 <s>καὶ περὶ ἴσας γωνίας τὰς ὑπὸ ΝΛΜ, ΔΕΖ αἱ πλευραὶ <lb n="1799,001,,,,11,36,20"></lb> |
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14440 ἀντιπεπόνθασιν· ἴσον ἄρα ἐστὶ τὸ ΜΝ παραλληλόγραμ-<lb n="1799,001,,,,11,36,21"></lb> |
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14441 μον τῷ ΔΖ παραλληλογράμμῳ.</s> <s>καὶ ἐπεὶ δύο γωνίαι ἐπί-<lb n="1799,001,,,,11,36,22"></lb> |
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14442 πεδοι εὐθύγραμμοι ἴσαι εἰσὶν αἱ ὑπὸ ΔΕΖ, ΝΛΜ, καὶ <lb n="1799,001,,,,11,36,23"></lb> |
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14443 ἐπ' αὐτῶν μετέωροι εὐθεῖαι ἐφεστᾶσιν αἱ ΛΞ, ΕΗ ἴσαι <lb n="1799,001,,,,11,36,24"></lb> |
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14444 τε ἀλλήλαις καὶ ἴσας γωνίας περιέχουσαι μετὰ τῶν ἐξ <lb n="1799,001,,,,11,36,25"></lb> |
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14445 ἀρχῆς εὐθειῶν ἑκατέραν ἑκατέρᾳ, αἱ ἄρα ἀπὸ τῶν Η, Ξ <lb n="1799,001,,,,11,36,26"></lb> |
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14446 σημείων κάθετοι ἀγόμεναι ἐπὶ τὰ διὰ τῶν ΝΛΜ, ΔΕΖ <lb n="1799,001,,,,11,36,27"></lb> |
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14447 ἐπίπεδα ἴσαι ἀλλήλαις εἰσίν· ὥστε τὰ ΛΘ, ΕΚ στερεὰ <lb n="1799,001,,,,11,36,28"></lb> |
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14448 ὑπὸ τὸ αὐτὸ ὕψος ἐστίν.</s> <s>τὰ δὲ ἐπὶ ἴσων βάσεων στερεὰ <lb n="1799,001,,,,11,36,29"></lb> |
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14449 παραλληλεπίπεδα καὶ ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλήλοις <lb n="1799,001,,,,11,36,30"></lb> |
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14450 ἐστίν· ἴσον ἄρα ἐστὶ τὸ ΘΛ στερεὸν τῷ ΕΚ στερεῷ.</s> <s>καί <lb n="1799,001,,,,11,36,31"></lb> |
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14451 ἐστι τὸ μὲν ΛΘ τὸ ἐκ τῶν Α, Β, Γ στερεόν, τὸ δὲ ΕΚ τὸ <lb n="1799,001,,,,11,36,32"></lb> |
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14452 ἀπὸ τῆς Β στερεόν· τὸ ἄρα ἐκ τῶν Α, Β, Γ στερεὸν παραλ-<lb n="1799,001,,,,11,36,33"></lb> |
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14453 ληλεπίπεδον ἴσον ἐστὶ τῷ ἀπὸ τῆς Β στερεῷ ἰσοπλεύρῳ <lb n="1799,001,,,,11,36,34"></lb> |
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14454 μέν, ἰσογωνίῳ δὲ τῷ προειρημένῳ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,36,35"></lb> |
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14455 Ἐὰν τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, καὶ τὰ ἀπ' αὐτῶν <lb n="1799,001,,,,11,37,1"></lb> |
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14456 στερεὰ παραλληλεπίπεδα ὅμοιά τε καὶ ὁμοίως ἀναγραφό-<lb n="1799,001,,,,11,37,2"></lb> |
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14457 μενα ἀνάλογον ἔσται· καὶ ἐὰν τὰ ἀπ' αὐτῶν στερεὰ παραλ-<lb n="1799,001,,,,11,37,3"></lb> |
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14458 ληλεπίπεδα ὅμοιά τε καὶ ὁμοίως ἀναγραφόμενα ἀνάλογον <lb n="1799,001,,,,11,37,4"></lb> |
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14459 ᾖ, καὶ αὐταὶ αἱ εὐθεῖαι ἀνάλογον ἔσονται. <lb n="1799,001,,,,11,37,5"></lb> |
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14460 Ἔστωσαν τέσσαρες εὐθεῖαι ἀνάλογον αἱ ΑΒ, ΓΔ, ΕΖ, <lb n="1799,001,,,,11,37,6"></lb> |
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14461 ΗΘ, ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΗΘ, <lb n="1799,001,,,,11,37,7"></lb> |
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14462 καὶ ἀναγεγράφθωσαν ἀπὸ τῶν ΑΒ, ΓΔ, ΕΖ, ΗΘ ὅμοιά <lb n="1799,001,,,,11,37,8"></lb> |
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14463 τε καὶ ὁμοίως κείμενα στερεὰ παραλληλεπίπεδα τὰ ΚΑ, <lb n="1799,001,,,,11,37,9"></lb> |
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14464 ΛΓ, ΜΕ, ΝΗ· λέγω, ὅτι ἐστὶν ὡς τὸ ΚΑ πρὸς τὸ ΛΓ, <lb n="1799,001,,,,11,37,10"></lb> |
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14465 οὕτως τὸ ΜΕ πρὸς τὸ ΝΗ.</s> <lb n="1799,001,,,,11,37,11"></lb> |
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14466 <s>Ἐπεὶ γὰρ ὅμοιόν ἐστι τὸ ΚΑ στερεὸν παραλληλεπίπε-<lb n="1799,001,,,,11,37,12"></lb> |
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14467 δον τῷ ΛΓ, τὸ ΚΑ ἄρα πρὸς τὸ ΛΓ τριπλασίονα λόγον <lb n="1799,001,,,,11,37,13"></lb> |
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14468 ἔχει ἤπερ ἡ ΑΒ πρὸς τὴν ΓΔ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ ΜΕ <lb n="1799,001,,,,11,37,14"></lb> |
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14469 πρὸς τὸ ΝΗ τριπλασίονα λόγον ἔχει ἤπερ ἡ ΕΖ πρὸς τὴν <lb n="1799,001,,,,11,37,15"></lb> |
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14470 ΗΘ.</s> <s>καί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς <lb n="1799,001,,,,11,37,16"></lb> |
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14471 τὴν ΗΘ.</s> <s>καὶ ὡς ἄρα τὸ ΑΚ πρὸς τὸ ΛΓ, οὕτως τὸ ΜΕ <lb n="1799,001,,,,11,37,17"></lb> |
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14472 πρὸς τὸ ΝΗ.</s> <s><pb></pb> <lb n="1799,001,,,,11,37,18"></lb> |
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14473 Ἀλλὰ δὴ ἔστω ὡς τὸ ΑΚ στερεὸν πρὸς τὸ ΛΓ στερεόν, <lb n="1799,001,,,,11,37,19"></lb> |
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14474 οὕτως τὸ ΜΕ στερεὸν πρὸς τὸ ΝΗ· λέγω, ὅτι ἐστὶν ὡς ἡ <lb n="1799,001,,,,11,37,20"></lb> |
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14475 ΑΒ εὐθεῖα πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΗΘ.</s> <lb n="1799,001,,,,11,37,21"></lb> |
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14476 <s>Ἐπεὶ γὰρ πάλιν τὸ ΚΑ πρὸς τὸ ΛΓ τριπλασίονα λόγον <lb n="1799,001,,,,11,37,22"></lb> |
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14477 ἔχει ἤπερ ἡ ΑΒ πρὸς τὴν ΓΔ, ἔχει δὲ καὶ τὸ ΜΕ πρὸς τὸ <lb n="1799,001,,,,11,37,23"></lb> |
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14478 ΝΗ τριπλασίονα λόγον ἤπερ ἡ ΕΖ πρὸς τὴν ΗΘ, καί <lb n="1799,001,,,,11,37,24"></lb> |
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14479 ἐστιν ὡς τὸ ΚΑ πρὸς τὸ ΛΓ, οὕτως τὸ ΜΕ πρὸς τὸ ΝΗ, <lb n="1799,001,,,,11,37,25"></lb> |
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14480 καὶ ὡς ἄρα ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν <lb n="1799,001,,,,11,37,26"></lb> |
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14481 ΗΘ.</s> <lb n="1799,001,,,,11,37,27"></lb> |
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14482 <s>Ἐὰν ἄρα τέσσαρες εὐθεῖαι ἀνάλογον ὦσι καὶ τὰ ἑξῆς <lb n="1799,001,,,,11,37,28"></lb> |
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14483 τῆς προτάσεως· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,11,37,29"></lb> |
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14484 <s>Ἐὰν κύβου τῶν ἀπεναντίον ἐπιπέδων αἱ πλευραὶ δίχα <lb n="1799,001,,,,11,38,1"></lb> |
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14485 τμηθῶσιν, διὰ δὲ τῶν τομῶν ἐπίπεδα ἐκβληθῇ, ἡ κοινὴ <lb n="1799,001,,,,11,38,2"></lb> |
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14486 τομὴ τῶν ἐπιπέδων καὶ ἡ τοῦ κύβου διάμετρος δίχα τέμ-<lb n="1799,001,,,,11,38,3"></lb> |
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14487 νουσιν ἀλλήλας.</s> <lb n="1799,001,,,,11,38,4"></lb> |
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14488 <s>Κύβου γὰρ τοῦ ΑΖ τῶν ἀπεναντίον ἐπιπέδων τῶν ΓΖ, <lb n="1799,001,,,,11,38,5"></lb> |
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14489 ΑΘ αἱ πλευραὶ δίχα τετμήσθωσαν κατὰ τὰ Κ, Λ, Μ, Ν, <lb n="1799,001,,,,11,38,6"></lb> |
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14490 Ξ, Π, Ο, Ρ σημεῖα, διὰ δὲ τῶν τομῶν ἐπίπεδα ἐκβεβλήσθω <pb></pb> <lb n="1799,001,,,,11,38,7"></lb> |
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14491 τὰ ΚΝ, ΞΡ, κοινὴ δὲ τομὴ τῶν ἐπιπέδων ἔστω ἡ ΥΣ, τοῦ <lb n="1799,001,,,,11,38,8"></lb> |
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14492 δὲ ΑΖ κύβου διαγώνιος ἡ ΔΗ.</s> <s>λέγω, ὅτι ἴση ἐστὶν ἡ μὲν <lb n="1799,001,,,,11,38,9"></lb> |
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14493 ΥΤ τῇ ΤΣ, ἡ δὲ ΔΤ τῇ ΤΗ.</s> <lb n="1799,001,,,,11,38,10"></lb> |
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14494 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΔΥ, ΥΕ, ΒΣ, ΣΗ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,11,38,11"></lb> |
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14495 παράλληλός ἐστιν ἡ ΔΞ τῇ ΟΕ, αἱ ἐναλλὰξ γωνίαι αἱ ὑπὸ <lb n="1799,001,,,,11,38,12"></lb> |
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14496 ΔΞΥ, ΥΟΕ ἴσαι ἀλλήλαις εἰσίν.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ μὲν <lb n="1799,001,,,,11,38,13"></lb> |
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14497 ΔΞ τῇ ΟΕ, ἡ δὲ ΞΥ τῇ ΥΟ, καὶ γωνίας ἴσας περιέχουσιν, <lb n="1799,001,,,,11,38,14"></lb> |
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14498 βάσις ἄρα ἡ ΔΥ τῇ ΥΕ ἐστιν ἴση, καὶ τὸ ΔΞΥ τρίγωνον <lb n="1799,001,,,,11,38,15"></lb> |
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14499 τῷ ΟΥΕ τριγώνῳ ἐστὶν ἴσον καὶ αἱ λοιπαὶ γωνίαι ταῖς <lb n="1799,001,,,,11,38,16"></lb> |
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14500 λοιπαῖς γωνίαις ἴσαι· ἴση ἄρα ἡ ὑπὸ ΞΥΔ γωνία τῇ ὑπὸ <lb n="1799,001,,,,11,38,17"></lb> |
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14501 ΟΥΕ γωνίᾳ.</s> <s>διὰ δὴ τοῦτο εὐθεῖά ἐστιν ἡ ΔΥΕ.</s> <s>διὰ τὰ <lb n="1799,001,,,,11,38,18"></lb> |
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14502 αὐτὰ δὴ καὶ ἡ ΒΣΗ εὐθεῖά ἐστιν, καὶ ἴση ἡ ΒΣ τῇ ΣΗ.</s> <lb n="1799,001,,,,11,38,19"></lb> |
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14503 <s>καὶ ἐπεὶ ἡ ΓΑ τῇ ΔΒ ἴση ἐστὶ καὶ παράλληλος, ἀλλὰ ἡ <lb n="1799,001,,,,11,38,20"></lb> |
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14504 ΓΑ καὶ τῇ ΕΗ ἴση τέ ἐστι καὶ παράλληλος, καὶ ἡ ΔΒ <lb n="1799,001,,,,11,38,21"></lb> |
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14505 ἄρα τῇ ΕΗ ἴση τέ ἐστι καὶ παράλληλος.</s> <s>καὶ ἐπιζευ-<lb n="1799,001,,,,11,38,22"></lb> |
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14506 γνύουσιν αὐτὰς εὐθεῖαι αἱ ΔΕ, ΒΗ· παράλληλος ἄρα ἐστὶν <lb n="1799,001,,,,11,38,23"></lb> |
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14507 ἡ ΔΕ τῇ ΒΗ.</s> <s>ἴση ἄρα ἡ μὲν ὑπὸ ΕΔΤ γωνία τῇ ὑπὸ ΒΗΤ· <pb></pb> <lb n="1799,001,,,,11,38,24"></lb> |
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14508 ἐναλλὰξ γάρ· ἡ δὲ ὑπὸ ΔΤΥ τῇ ὑπὸ ΗΤΣ.</s> <s>δύο δὴ τρίγωνά <lb n="1799,001,,,,11,38,25"></lb> |
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14509 ἐστι τὰ ΔΤΥ, ΗΤΣ τὰς δύο γωνίας ταῖς δυσὶ γωνίαις ἴσας <lb n="1799,001,,,,11,38,26"></lb> |
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14510 ἔχοντα καὶ μίαν πλευρὰν μιᾷ πλευρᾷ ἴσην τὴν ὑποτείνου-<lb n="1799,001,,,,11,38,27"></lb> |
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14511 σαν ὑπὸ μίαν τῶν ἴσων γωνιῶν τὴν ΔΥ τῇ ΗΣ· ἡμίσειαι <lb n="1799,001,,,,11,38,28"></lb> |
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14512 γάρ εἰσι τῶν ΔΕ, ΒΗ· καὶ τὰς λοιπὰς πλευρὰς ταῖς λοι-<lb n="1799,001,,,,11,38,29"></lb> |
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14513 παῖς πλευραῖς ἴσας ἕξει.</s> <s>ἴση ἄρα ἡ μὲν ΔΤ τῇ ΤΗ, ἡ δὲ <lb n="1799,001,,,,11,38,30"></lb> |
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14514 ΥΤ τῇ ΤΣ.</s> <lb n="1799,001,,,,11,38,31"></lb> |
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14515 <s>Ἐὰν ἄρα κύβου τῶν ἀπεναντίον ἐπιπέδων αἱ πλευραὶ <lb n="1799,001,,,,11,38,32"></lb> |
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14516 δίχα τμηθῶσιν, διὰ δὲ τῶν τομῶν ἐπίπεδα ἐκβληθῇ, ἡ <lb n="1799,001,,,,11,38,33"></lb> |
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14517 κοινὴ τομὴ τῶν ἐπιπέδων καὶ ἡ τοῦ κύβου διάμετρος δίχα <lb n="1799,001,,,,11,38,34"></lb> |
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14518 τέμνουσιν ἀλλήλας· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,38,35"></lb> |
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14519 Ἐὰν ᾖ δύο πρίσματα ἰσοϋψῆ, καὶ τὸ μὲν ἔχῃ βάσιν παραλ-<lb n="1799,001,,,,11,39,1"></lb> |
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14520 ληλόγραμμον, τὸ δὲ τρίγωνον, διπλάσιον δὲ ᾖ τὸ παραλ-<lb n="1799,001,,,,11,39,2"></lb> |
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14521 ληλόγραμμον τοῦ τριγώνου, ἴσα ἔσται τὰ πρίσματα.</s> <lb n="1799,001,,,,11,39,3"></lb> |
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14522 <s>Ἔστω δύο πρίσματα ἰσοϋψῆ τὰ ΑΒΓΔΕΖ, ΗΘΚΛ <lb n="1799,001,,,,11,39,4"></lb> |
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14523 ΜΝ, καὶ τὸ μὲν ἐχέτω βάσιν τὸ ΑΖ παραλληλόγραμμον, <lb n="1799,001,,,,11,39,5"></lb> |
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14524 τὸ δὲ τὸ ΗΘΚ τρίγωνον, διπλάσιον δὲ ἔστω τὸ ΑΖ παραλ-<lb n="1799,001,,,,11,39,6"></lb> |
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14525 ληλόγραμμον τοῦ ΗΘΚ τριγώνου· λέγω, ὅτι ἴσον ἐστὶ <lb n="1799,001,,,,11,39,7"></lb> |
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14526 τὸ ΑΒΓΔΕΖ πρίσμα τῷ ΗΘΚΛΜΝ πρίσματι.</s> <lb n="1799,001,,,,11,39,8"></lb> |
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14527 <s>Συμπεπληρώσθω γὰρ <lb n="1799,001,,,,11,39,9"></lb> |
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14528 τὰ ΑΞ, ΗΟ στερεά.</s> <s>ἐπεὶ <lb n="1799,001,,,,11,39,10"></lb> |
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14529 διπλάσιόν ἐστι τὸ ΑΖ <lb n="1799,001,,,,11,39,11"></lb> |
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14530 παραλληλόγραμμον τοῦ <lb n="1799,001,,,,11,39,12"></lb> |
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14531 ΗΘΚ τριγώνου, ἔστι δὲ <lb n="1799,001,,,,11,39,13"></lb> |
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14532 καὶ τὸ ΘΚ παραλληλό-<lb n="1799,001,,,,11,39,14"></lb> |
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14533 γραμμον διπλάσιον τοῦ <lb n="1799,001,,,,11,39,15"></lb> |
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14534 ΗΘΚ τριγώνου, ἴσον ἄρα ἐστὶ τὸ ΑΖ παραλληλόγραμμον <lb n="1799,001,,,,11,39,16"></lb> |
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14535 τῷ ΘΚ παραλληλογράμμῳ.</s> <s>τὰ δὲ ἐπὶ ἴσων βάσεων ὄντα <lb n="1799,001,,,,11,39,17"></lb> |
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14536 στερεὰ παραλληλεπίπεδα καὶ ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλή-<lb n="1799,001,,,,11,39,18"></lb> |
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14537 λοις ἐστίν· ἴσον ἄρα ἐστὶ τὸ ΑΞ στερεὸν τῷ ΗΟ στερεῷ.</s> <lb n="1799,001,,,,11,39,19"></lb> |
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14538 <s>καί ἐστι τοῦ μὲν ΑΞ στερεοῦ ἥμισυ τὸ ΑΒΓΔΕΖ πρίσμα, <lb n="1799,001,,,,11,39,20"></lb> |
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14539 τοῦ δὲ ΗΟ στερεοῦ ἥμισυ τὸ ΗΘΚΛΜΝ πρίσμα· <lb n="1799,001,,,,11,39,21"></lb> |
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14540 ἴσον ἄρα ἐστὶ τὸ ΑΒΓΔΕΖ πρίσμα τῷ ΗΘΚΛΜΝ <lb n="1799,001,,,,11,39,22"></lb> |
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14541 πρίσματι.</s> <s><pb></pb> <lb n="1799,001,,,,11,39,23"></lb> |
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14542 Ἐὰν ἄρα ᾖ δύο πρίσματα ἰσοϋψῆ, καὶ τὸ μὲν ἔχῃ βάσιν <lb n="1799,001,,,,11,39,24"></lb> |
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14543 παραλληλόγραμμον, τὸ δὲ τρίγωνον, διπλάσιον δὲ ᾖ τὸ <lb n="1799,001,,,,11,39,25"></lb> |
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14544 παραλληλόγραμμον τοῦ τριγώνου, ἴσα ἐστὶ τὰ πρίσματα· <lb n="1799,001,,,,11,39,26"></lb> |
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14545 ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,11,39,27"></lb> |
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14546 </s></p> |
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14547 </chap> |
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14548 <chap><p type="main"><s>Τὰ ἐν τοῖς κύκλοις ὅμοια πολύγωνα πρὸς ἄλληλά ἐστιν <lb n="1799,001,,,,12,1,1"></lb> |
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14549 |
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14550 ὡς τὰ ἀπὸ τῶν διαμέτρων τετράγωνα.</s> <lb n="1799,001,,,,12,1,2"></lb> |
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14551 <s>Ἔστωσαν κύκλοι οἱ ΑΒΓ, ΖΗΘ, καὶ ἐν αὐτοῖς ὅμοια <lb n="1799,001,,,,12,1,3"></lb> |
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14552 πολύγωνα ἔστω τὰ ΑΒΓΔΕ, ΖΗΘΚΛ, διάμετροι δὲ <lb n="1799,001,,,,12,1,4"></lb> |
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14553 τῶν κύκλων ἔστωσαν αἱ ΒΜ, ΗΝ· λέγω, ὅτι ἐστὶν ὡς τὸ <lb n="1799,001,,,,12,1,5"></lb> |
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14554 ἀπὸ τῆς ΒΜ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΗΝ τετράγω-<lb n="1799,001,,,,12,1,6"></lb> |
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14555 νον, οὕτως τὸ ΑΒΓΔΕ πολύγωνον πρὸς τὸ ΖΗΘΚΛ <lb n="1799,001,,,,12,1,7"></lb> |
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14556 πολύγωνον.</s> <lb n="1799,001,,,,12,1,8"></lb> |
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14557 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΒΕ, ΑΜ, ΗΛ, ΖΝ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,12,1,9"></lb> |
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14558 ὅμοιον τὸ ΑΒΓΔΕ πολύγωνον τῷ ΖΗΘΚΛ πολυγώνῳ, <pb></pb> <lb n="1799,001,,,,12,1,10"></lb> |
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14559 ἴση ἐστὶ καὶ ἡ ὑπὸ ΒΑΕ γωνία τῇ ὑπὸ ΗΖΛ, καί ἐστιν <lb n="1799,001,,,,12,1,11"></lb> |
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14560 ὡς ἡ ΒΑ πρὸς τὴν ΑΕ, οὕτως ἡ ΗΖ πρὸς τὴν ΖΛ.</s> <s>δύο <lb n="1799,001,,,,12,1,12"></lb> |
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14561 δὴ τρίγωνά ἐστι τὰ ΒΑΕ, ΗΖΛ μίαν γωνίαν μιᾷ γωνίᾳ <lb n="1799,001,,,,12,1,13"></lb> |
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14562 ἴσην ἔχοντα τὴν ὑπὸ ΒΑΕ τῇ ὑπὸ ΗΖΛ, περὶ δὲ τὰς ἴσας <lb n="1799,001,,,,12,1,14"></lb> |
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14563 γωνίας τὰς πλευρὰς ἀνάλογον· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΕ <lb n="1799,001,,,,12,1,15"></lb> |
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14564 τρίγωνον τῷ ΖΗΛ τριγώνῳ.</s> <s>ἴση ἄρα ἐστὶν ἡ ὑπὸ ΑΕΒ <lb n="1799,001,,,,12,1,16"></lb> |
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14565 γωνία τῇ ὑπὸ ΖΛΗ.</s> <s>ἀλλ' ἡ μὲν ὑπὸ ΑΕΒ τῇ ὑπὸ ΑΜΒ <lb n="1799,001,,,,12,1,17"></lb> |
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14566 ἐστιν ἴση· ἐπὶ γὰρ τῆς αὐτῆς περιφερείας βεβήκασιν· ἡ δὲ <lb n="1799,001,,,,12,1,18"></lb> |
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14567 ὑπὸ ΖΛΗ τῇ ὑπὸ ΖΝΗ· καὶ ἡ ὑπὸ ΑΜΒ ἄρα τῇ ὑπὸ <lb n="1799,001,,,,12,1,19"></lb> |
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14568 ΖΝΗ ἐστιν ἴση.</s> <s>ἔστι δὲ καὶ ὀρθὴ ἡ ὑπὸ ΒΑΜ ὀρθῇ τῇ <lb n="1799,001,,,,12,1,20"></lb> |
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14569 ὑπὸ ΗΖΝ ἴση· καὶ ἡ λοιπὴ ἄρα τῇ λοιπῇ ἐστιν ἴση.</s> <s>ἰσο-<lb n="1799,001,,,,12,1,21"></lb> |
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14570 γώνιον ἄρα ἐστὶ τὸ ΑΒΜ τρίγωνον τῷ ΖΗΝ τριγώνῳ.</s> <lb n="1799,001,,,,12,1,22"></lb> |
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14571 <s>ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΒΜ πρὸς τὴν ΗΝ, οὕτως ἡ ΒΑ <lb n="1799,001,,,,12,1,23"></lb> |
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14572 πρὸς τὴν ΗΖ.</s> <s>ἀλλὰ τοῦ μὲν τῆς ΒΜ πρὸς τὴν ΗΝ λόγου <lb n="1799,001,,,,12,1,24"></lb> |
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14573 διπλασίων ἐστὶν ὁ τοῦ ἀπὸ τῆς ΒΜ τετραγώνου πρὸς τὸ <lb n="1799,001,,,,12,1,25"></lb> |
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14574 ἀπὸ τῆς ΗΝ τετράγωνον, τοῦ δὲ τῆς ΒΑ πρὸς τὴν ΗΖ <lb n="1799,001,,,,12,1,26"></lb> |
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14575 διπλασίων ἐστὶν ὁ τοῦ ΑΒΓΔΕ πολυγώνου πρὸς τὸ ΖΗ <lb n="1799,001,,,,12,1,27"></lb> |
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14576 ΘΚΛ πολύγωνον· καὶ ὡς ἄρα τὸ ἀπὸ τῆς ΒΜ τετράγωνον <lb n="1799,001,,,,12,1,28"></lb> |
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14577 πρὸς τὸ ἀπὸ τῆς ΗΝ τετράγωνον, οὕτως τὸ ΑΒΓΔΕ <lb n="1799,001,,,,12,1,29"></lb> |
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14578 πολύγωνον πρὸς τὸ ΖΗΘΚΛ πολύγωνον.</s> <lb n="1799,001,,,,12,1,30"></lb> |
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14579 <s>Τὰ ἄρα ἐν τοῖς κύκλοις ὅμοια πολύγωνα πρὸς ἄλληλά <lb n="1799,001,,,,12,1,31"></lb> |
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14580 ἐστιν ὡς τὰ ἀπὸ τῶν διαμέτρων τετράγωνα· ὅπερ ἔδει <lb n="1799,001,,,,12,1,32"></lb> |
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14581 δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,12,1,33"></lb> |
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14582 Οἱ κύκλοι πρὸς ἀλλήλους εἰσὶν ὡς τὰ ἀπὸ τῶν δια-<lb n="1799,001,,,,12,2,1"></lb> |
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14583 μέτρων τετράγωνα.</s> <lb n="1799,001,,,,12,2,2"></lb> |
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14584 <s>Ἔστωσαν κύκλοι οἱ ΑΒΓΔ, ΕΖΗΘ, διάμετροι δὲ αὐ-<lb n="1799,001,,,,12,2,3"></lb> |
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14585 τῶν [ἔστωσαν] αἱ ΒΔ, ΖΘ· λέγω, ὅτι ἐστὶν ὡς ὁ ΑΒΓΔ <lb n="1799,001,,,,12,2,4"></lb> |
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14586 κύκλος πρὸς τὸν ΕΖΗΘ κύκλον, οὕτως τὸ ἀπὸ τῆς ΒΔ <lb n="1799,001,,,,12,2,5"></lb> |
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14587 τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΘ τετράγωνον.</s> <lb n="1799,001,,,,12,2,6"></lb> |
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14588 <s>Εἰ γὰρ μή ἐστιν ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ, <lb n="1799,001,,,,12,2,7"></lb> |
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14589 οὕτως τὸ ἀπὸ τῆς ΒΔ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΘ, <lb n="1799,001,,,,12,2,8"></lb> |
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14590 ἔσται ὡς τὸ ἀπὸ τῆς ΒΔ πρὸς τὸ ἀπὸ τῆς ΖΘ, οὕτως ὁ <lb n="1799,001,,,,12,2,9"></lb> |
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14591 ΑΒΓΔ κύκλος ἤτοι πρὸς ἔλασσόν τι τοῦ ΕΖΗΘ κύκλου <lb n="1799,001,,,,12,2,10"></lb> |
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14592 χωρίον ἢ πρὸς μεῖζον.</s> <s>ἔστω πρότερον πρὸς ἔλασσον τὸ Σ.</s> <lb n="1799,001,,,,12,2,11"></lb> |
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14593 <s>καὶ ἐγγεγράφθω εἰς τὸν ΕΖΗΘ κύκλον τετράγωνον τὸ <pb></pb> <lb n="1799,001,,,,12,2,12"></lb> |
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14594 ΕΖΗΘ· τὸ δὴ ἐγγεγραμμένον τετράγωνον μεῖζόν ἐστιν <lb n="1799,001,,,,12,2,13"></lb> |
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14595 ἢ τὸ ἥμισυ τοῦ ΕΖΗΘ κύκλου, ἐπειδήπερ ἐὰν διὰ τῶν <lb n="1799,001,,,,12,2,14"></lb> |
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14596 Ε, Ζ, Η, Θ σημείων ἐφαπτομένας [εὐθείασ] τοῦ κύκλου <lb n="1799,001,,,,12,2,15"></lb> |
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14597 ἀγάγωμεν, τοῦ περιγραφομένου περὶ τὸν κύκλον τετρα-<lb n="1799,001,,,,12,2,16"></lb> |
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14598 γώνου ἥμισύ ἐστι τὸ ΕΖΗΘ τετράγωνον, τοῦ δὲ περι-<lb n="1799,001,,,,12,2,17"></lb> |
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14599 γραφέντος τετραγώνου ἐλάττων ἐστὶν ὁ κύκλος· ὥστε τὸ <lb n="1799,001,,,,12,2,18"></lb> |
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14600 ΕΖΗΘ ἐγγεγραμμένον τετράγωνον μεῖζόν ἐστι τοῦ ἡμί-<lb n="1799,001,,,,12,2,19"></lb> |
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14601 σεως τοῦ ΕΖΗΘ κύκλου.</s> <s>τετμήσθωσαν δίχα αἱ ΕΖ, <lb n="1799,001,,,,12,2,20"></lb> |
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14602 ΖΗ, ΗΘ, ΘΕ περιφέρειαι κατὰ τὰ Κ, Λ, Μ, Ν σημεῖα, <lb n="1799,001,,,,12,2,21"></lb> |
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14603 καὶ ἐπεζεύχθωσαν αἱ ΕΚ, ΚΖ, ΖΛ, ΛΗ, ΗΜ, ΜΘ, <lb n="1799,001,,,,12,2,22"></lb> |
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14604 ΘΝ, ΝΕ· καὶ ἕκαστον ἄρα τῶν ΕΚΖ, ΖΛΗ, ΗΜΘ, <lb n="1799,001,,,,12,2,23"></lb> |
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14605 ΘΝΕ τριγώνων μεῖζόν ἐστιν ἢ τὸ ἥμισυ τοῦ καθ' ἑαυτὸ <lb n="1799,001,,,,12,2,24"></lb> |
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14606 τμήματος τοῦ κύκλου, ἐπειδήπερ ἐὰν διὰ τῶν Κ, Λ, Μ, Ν <lb n="1799,001,,,,12,2,25"></lb> |
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14607 σημείων ἐφαπτομένας τοῦ κύκλου ἀγάγωμεν καὶ ἀναπλη-<lb n="1799,001,,,,12,2,26"></lb> |
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14608 ρώσωμεν τὰ ἐπὶ τῶν ΕΖ, ΖΗ, ΗΘ, ΘΕ εὐθειῶν παραλ-<lb n="1799,001,,,,12,2,27"></lb> |
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14609 ληλόγραμμα, ἕκαστον τῶν ΕΚΖ, ΖΛΗ, ΗΜΘ, ΘΝΕ <lb n="1799,001,,,,12,2,28"></lb> |
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14610 τριγώνων ἥμισυ ἔσται τοῦ καθ' ἑαυτὸ παραλληλογράμμου, <lb n="1799,001,,,,12,2,29"></lb> |
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14611 ἀλλὰ τὸ καθ' ἑαυτὸ τμῆμα ἔλαττόν ἐστι τοῦ παραλληλο-<lb n="1799,001,,,,12,2,30"></lb> |
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14612 γράμμου· ὥστε ἕκαστον τῶν ΕΚΖ, ΖΛΗ, ΗΜΘ, ΘΝΕ <lb n="1799,001,,,,12,2,31"></lb> |
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14613 τριγώνων μεῖζόν ἐστι τοῦ ἡμίσεως τοῦ καθ' ἑαυτὸ τμή-<lb n="1799,001,,,,12,2,32"></lb> |
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14614 ματος τοῦ κύκλου.</s> <s>τέμνοντες δὴ τὰς ὑπολειπομένας περι-<lb n="1799,001,,,,12,2,33"></lb> |
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14615 φερείας δίχα καὶ ἐπιζευγνύντες εὐθείας καὶ τοῦτο ἀεὶ ποι-<lb n="1799,001,,,,12,2,34"></lb> |
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14616 οῦντες καταλείψομέν τινα ἀποτμήματα τοῦ κύκλου, ἃ ἔσται <lb n="1799,001,,,,12,2,35"></lb> |
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14617 ἐλάσσονα τῆς ὑπεροχῆς, ᾗ ὑπερέχει ὁ ΕΖΗΘ κύκλος τοῦ <pb></pb> <lb n="1799,001,,,,12,2,36"></lb> |
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14618 Σ χωρίου.</s> <s>ἐδείχθη γὰρ ἐν τῷ πρώτῳ θεωρήματι τοῦ <lb n="1799,001,,,,12,2,37"></lb> |
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14619 δεκάτου βιβλίου, ὅτι δύο μεγεθῶν ἀνίσων ἐκκειμένων, <lb n="1799,001,,,,12,2,38"></lb> |
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14620 ἐὰν ἀπὸ τοῦ μείζονος ἀφαιρεθῇ μεῖζον ἢ τὸ ἥμισυ καὶ τοῦ <lb n="1799,001,,,,12,2,39"></lb> |
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14621 καταλειπομένου μεῖζον ἢ τὸ ἥμισυ, καὶ τοῦτο ἀεὶ γίγνηται, <lb n="1799,001,,,,12,2,40"></lb> |
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14622 λειφθήσεταί τι μέγεθος, ὃ ἔσται ἔλασσον τοῦ ἐκκειμένου <lb n="1799,001,,,,12,2,41"></lb> |
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14623 ἐλάσσονος μεγέθους.</s> <s>λελείφθω οὖν, καὶ ἔστω τὰ ἐπὶ τῶν <lb n="1799,001,,,,12,2,42"></lb> |
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14624 ΕΚ, ΚΖ, ΖΛ, ΛΗ, ΗΜ, ΜΘ, ΘΝ, ΝΕ τμήματα τοῦ <lb n="1799,001,,,,12,2,43"></lb> |
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14625 ΕΖΗΘ κύκλου ἐλάττονα τῆς ὑπεροχῆς, ᾗ ὑπερέχει ὁ <lb n="1799,001,,,,12,2,44"></lb> |
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14626 ΕΖΗΘ κύκλος τοῦ Σ χωρίου.</s> <s>λοιπὸν ἄρα τὸ ΕΚΖΛΗ <lb n="1799,001,,,,12,2,45"></lb> |
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14627 ΜΘΝ πολύγωνον μεῖζόν ἐστι τοῦ Σ χωρίου.</s> <s>ἐγγεγράφθω <lb n="1799,001,,,,12,2,46"></lb> |
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14628 καὶ εἰς τὸν ΑΒΓΔ κύκλον τῷ ΕΚΖΛΗΜΘΝ πολυγώνῳ <lb n="1799,001,,,,12,2,47"></lb> |
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14629 ὅμοιον πολύγωνον τὸ ΑΞΒΟΓΠΔΡ· ἔστιν ἄρα ὡς τὸ <lb n="1799,001,,,,12,2,48"></lb> |
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14630 ἀπὸ τῆς ΒΔ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΘ τετράγωνον, <lb n="1799,001,,,,12,2,49"></lb> |
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14631 οὕτως τὸ ΑΞΒΟΓΠΔΡ πολύγωνον πρὸς τὸ ΕΚΖΛ <lb n="1799,001,,,,12,2,50"></lb> |
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14632 ΗΜΘΝ πολύγωνον.</s> <s>ἀλλὰ καὶ ὡς τὸ ἀπὸ τῆς ΒΔ τετρά-<lb n="1799,001,,,,12,2,51"></lb> |
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14633 γωνον πρὸς τὸ ἀπὸ τῆς ΖΘ, οὕτως ὁ ΑΒΓΔ κύκλος πρὸς <lb n="1799,001,,,,12,2,52"></lb> |
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14634 τὸ Σ χωρίον· καὶ ὡς ἄρα ὁ ΑΒΓΔ κύκλος πρὸς τὸ Σ <lb n="1799,001,,,,12,2,53"></lb> |
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14635 χωρίον, οὕτως τὸ ΑΞΒΟΓΠΔΡ πολύγωνον πρὸς τὸ <lb n="1799,001,,,,12,2,54"></lb> |
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14636 ΕΚΖΛΗΜΘΝ πολύγωνον· ἐναλλὰξ ἄρα ὡς ὁ ΑΒΓΔ <lb n="1799,001,,,,12,2,55"></lb> |
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14637 κύκλος πρὸς τὸ ἐν αὐτῷ πολύγωνον, οὕτως τὸ Σ χωρίον <lb n="1799,001,,,,12,2,56"></lb> |
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14638 πρὸς τὸ ΕΚΖΛΗΜΘΝ πολύγωνον.</s> <s>μείζων δὲ ὁ ΑΒΓΔ <lb n="1799,001,,,,12,2,57"></lb> |
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14639 κύκλος τοῦ ἐν αὐτῷ πολυγώνου· μεῖζον ἄρα καὶ τὸ Σ <lb n="1799,001,,,,12,2,58"></lb> |
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14640 χωρίον τοῦ ΕΚΖΛΗΜΘΝ πολυγώνου.</s> <s>ἀλλὰ καὶ ἔλατ-<lb n="1799,001,,,,12,2,59"></lb> |
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14641 τον· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ἐστὶν ὡς τὸ ἀπὸ τῆς <lb n="1799,001,,,,12,2,60"></lb> |
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14642 ΒΔ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΘ, οὕτως ὁ ΑΒΓΔ <pb></pb> <lb n="1799,001,,,,12,2,61"></lb> |
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14643 κύκλος πρὸς ἔλασσόν τι τοῦ ΕΖΗΘ κύκλου χωρίον.</s> <s>ὁμοί-<lb n="1799,001,,,,12,2,62"></lb> |
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14644 ως δὴ δείξομεν, ὅτι οὐδὲ ὡς τὸ ἀπὸ ΖΘ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,12,2,63"></lb> |
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14645 ΒΔ, οὕτως ὁ ΕΖΗΘ κύκλος πρὸς ἔλασσόν τι τοῦ ΑΒΓΔ <lb n="1799,001,,,,12,2,64"></lb> |
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14646 κύκλου χωρίον.</s> <lb n="1799,001,,,,12,2,65"></lb> |
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14647 <s>Λέγω δή, ὅτι οὐδὲ ὡς τὸ ἀπὸ τῆς ΒΔ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,12,2,66"></lb> |
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14648 ΖΘ, οὕτως ὁ ΑΒΓΔ κύκλος πρὸς μεῖζόν τι τοῦ ΕΖΗΘ <lb n="1799,001,,,,12,2,67"></lb> |
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14649 κύκλου χωρίον.</s> <lb n="1799,001,,,,12,2,68"></lb> |
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14650 <s>Εἰ γὰρ δυνατόν, ἔστω πρὸς μεῖζον τὸ Σ.</s> <s>ἀνάπαλιν ἄρα <lb n="1799,001,,,,12,2,69"></lb> |
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14651 [ἐστὶν] ὡς τὸ ἀπὸ τῆς ΖΘ τετράγωνον πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,12,2,70"></lb> |
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14652 ΔΒ, οὕτως τὸ Σ χωρίον πρὸς τὸν ΑΒΓΔ κύκλον.</s> <s>ἀλλ' ὡς <lb n="1799,001,,,,12,2,71"></lb> |
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14653 τὸ Σ χωρίον πρὸς τὸν ΑΒΓΔ κύκλον, οὕτως ὁ ΕΖΗΘ <lb n="1799,001,,,,12,2,72"></lb> |
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14654 κύκλος πρὸς ἔλαττόν τι τοῦ ΑΒΓΔ κύκλου χωρίον· καὶ <lb n="1799,001,,,,12,2,73"></lb> |
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14655 ὡς ἄρα τὸ ἀπὸ τῆς ΖΘ πρὸς τὸ ἀπὸ τῆς ΒΔ, οὕτως ὁ <lb n="1799,001,,,,12,2,74"></lb> |
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14656 ΕΖΗΘ κύκλος πρὸς ἔλασσόν τι τοῦ ΑΒΓΔ κύκλου χω-<lb n="1799,001,,,,12,2,75"></lb> |
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14657 ρίον· ὅπερ ἀδύνατον ἐδείχθη.</s> <s>οὐκ ἄρα ἐστὶν ὡς τὸ ἀπὸ τῆς <lb n="1799,001,,,,12,2,76"></lb> |
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14658 ΒΔ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΘ, οὕτως ὁ ΑΒΓΔ <lb n="1799,001,,,,12,2,77"></lb> |
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14659 κύκλος πρὸς μεῖζόν τι τοῦ ΕΖΗΘ κύκλου χωρίον.</s> <s>ἐδείχθη <lb n="1799,001,,,,12,2,78"></lb> |
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14660 δέ, ὅτι οὐδὲ πρὸς ἔλασσον· ἔστιν ἄρα ὡς τὸ ἀπὸ τῆς ΒΔ <lb n="1799,001,,,,12,2,79"></lb> |
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14661 τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΘ, οὕτως ὁ ΑΒΓΔ κύκλος <lb n="1799,001,,,,12,2,80"></lb> |
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14662 πρὸς τὸν ΕΖΗΘ κύκλον.</s> <lb n="1799,001,,,,12,2,81"></lb> |
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14663 <s>Οἱ ἄρα κύκλοι πρὸς ἀλλήλους εἰσὶν ὡς τὰ ἀπὸ τῶν δια-<lb n="1799,001,,,,12,2,82"></lb> |
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14664 μέτρων τετράγωνα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,2,83"></lb> |
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14665 <s>Λῆμμα <lb n="1799,001,,,,12,2,84"></lb> |
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14666 Λέγω δή, ὅτι τοῦ Σ χωρίου μείζονος ὄντος τοῦ ΕΖΗΘ <lb n="1799,001,,,,12,2,85"></lb> |
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14667 κύκλου ἐστὶν ὡς τὸ Σ χωρίον πρὸς τὸν ΑΒΓΔ κύκλον, <pb></pb> <lb n="1799,001,,,,12,2,86"></lb> |
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14668 οὕτως ὁ ΕΖΗΘ κύκλος πρὸς ἔλαττόν τι τοῦ ΑΒΓΔ <lb n="1799,001,,,,12,2,87"></lb> |
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14669 κύκλου χωρίον.</s> <lb n="1799,001,,,,12,2,88"></lb> |
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14670 <s>Γεγονέτω γὰρ ὡς τὸ Σ χωρίον πρὸς τὸν ΑΒΓΔ κύκλον, <lb n="1799,001,,,,12,2,89"></lb> |
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14671 οὕτως ὁ ΕΖΗΘ κύκλος πρὸς τὸ Τ χωρίον.</s> <s>λέγω, ὅτι <lb n="1799,001,,,,12,2,90"></lb> |
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14672 ἔλαττόν ἐστι τὸ Τ χωρίον τοῦ ΑΒΓΔ κύκλου.</s> <s>ἐπεὶ γάρ <lb n="1799,001,,,,12,2,91"></lb> |
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14673 ἐστιν ὡς τὸ Σ χωρίον πρὸς τὸν ΑΒΓΔ κύκλον, οὕτως ὁ <lb n="1799,001,,,,12,2,92"></lb> |
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14674 ΕΖΗΘ κύκλος πρὸς τὸ Τ χωρίον, ἐναλλάξ ἐστιν ὡς τὸ <lb n="1799,001,,,,12,2,93"></lb> |
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14675 Σ χωρίον πρὸς τὸν ΕΖΗΘ κύκλον, οὕτως ὁ ΑΒΓΔ κύ-<lb n="1799,001,,,,12,2,94"></lb> |
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14676 κλος πρὸς τὸ Τ χωρίον.</s> <s>μεῖζον δὲ τὸ Σ χωρίον του̂ ΕΖΗΘ <lb n="1799,001,,,,12,2,95"></lb> |
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14677 κύκλου· μείζων ἄρα καὶ ὁ ΑΒΓΔ κύκλος τοῦ Τ χωρίου.</s> <lb n="1799,001,,,,12,2,96"></lb> |
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14678 <s>ὥστε ἐστὶν ὡς τὸ Σ χωρίον πρὸς τὸν ΑΒΓΔ κύκλον, <lb n="1799,001,,,,12,2,97"></lb> |
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14679 οὕτως ὁ ΕΖΗΘ κύκλος πρὸς ἔλαττόν τι τοῦ ΑΒΓΔ <lb n="1799,001,,,,12,2,98"></lb> |
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14680 κύκλου χωρίον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,2,99"></lb> |
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14681 <s>Πᾶσα πυραμὶς τρίγωνον ἔχουσα βάσιν διαιρεῖται εἰς <lb n="1799,001,,,,12,3,1"></lb> |
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14682 δύο πυραμίδας ἴσας τε καὶ ὁμοίας ἀλλήλαις καὶ [ὁμοίασ] <lb n="1799,001,,,,12,3,2"></lb> |
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14683 τῇ ὅλῃ τριγώνους ἐχούσας βάσεις καὶ εἰς δύο πρίσματα <lb n="1799,001,,,,12,3,3"></lb> |
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14684 ἴσα· καὶ τὰ δύο πρίσματα μείζονά ἐστιν ἢ τὸ ἥμισυ τῆς <lb n="1799,001,,,,12,3,4"></lb> |
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14685 ὅλης πυραμίδος.</s> <lb n="1799,001,,,,12,3,5"></lb> |
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14686 <s>Ἔστω πυραμίς, ἧς βάσις μέν ἐστι τὸ ΑΒΓ τρίγωνον, <lb n="1799,001,,,,12,3,6"></lb> |
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14687 κορυφὴ δὲ τὸ Δ σημεῖον· λέγω, ὅτι ἡ ΑΒΓΔ πυραμὶς <lb n="1799,001,,,,12,3,7"></lb> |
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14688 διαιρεῖται εἰς δύο πυραμίδας ἴσας ἀλλήλαις τριγώνους <pb></pb> <lb n="1799,001,,,,12,3,8"></lb> |
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14689 βάσεις ἐχούσας καὶ ὁμοίας τῇ ὅλῃ καὶ εἰς δύο πρίσματα <lb n="1799,001,,,,12,3,9"></lb> |
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14690 ἴσα· καὶ τὰ δύο πρίσματα μείζονά ἐστιν ἢ τὸ ἥμισυ τῆς <lb n="1799,001,,,,12,3,10"></lb> |
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14691 ὅλης πυραμίδος.</s> <lb n="1799,001,,,,12,3,11"></lb> |
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14692 <s>Τετμήσθωσαν γὰρ αἱ ΑΒ, ΒΓ, ΓΑ, ΑΔ, ΔΒ, ΔΓ <lb n="1799,001,,,,12,3,12"></lb> |
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14693 δίχα κατὰ τὰ Ε, Ζ, Η, Θ, Κ, Λ σημεῖα, καὶ ἐπεζεύχθω-<lb n="1799,001,,,,12,3,13"></lb> |
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14694 σαν αἱ ΘΕ, ΕΗ, ΗΘ, ΘΚ, ΚΛ, ΛΘ, ΚΖ, ΖΗ.</s> <s>ἐπεὶ ἴση <lb n="1799,001,,,,12,3,14"></lb> |
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14695 ἐστὶν ἡ μὲν ΑΕ τῇ ΕΒ, ἡ δὲ ΑΘ τῇ ΔΘ, παράλληλος ἄρα <lb n="1799,001,,,,12,3,15"></lb> |
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14696 ἐστὶν ἡ ΕΘ τῇ ΔΒ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΘΚ τῇ ΑΒ <lb n="1799,001,,,,12,3,16"></lb> |
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14697 παράλληλός ἐστιν.</s> <s>παραλληλόγραμμον ἄρα ἐστὶ τὸ ΘΕ <lb n="1799,001,,,,12,3,17"></lb> |
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14698 ΒΚ· ἴση ἄρα ἐστὶν ἡ ΘΚ τῇ ΕΒ.</s> <s>ἀλλὰ ἡ ΕΒ τῇ ΕΑ ἐστιν <lb n="1799,001,,,,12,3,18"></lb> |
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14699 ἴση· καὶ ἡ ΑΕ ἄρα τῇ ΘΚ ἐστιν ἴση.</s> <s>ἔστι δὲ καὶ ἡ ΑΘ <lb n="1799,001,,,,12,3,19"></lb> |
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14700 τῇ ΘΔ ἴση· δύο δὴ αἱ ΕΑ, ΑΘ δυσὶ ταῖς ΚΘ, ΘΔ ἴσαι <lb n="1799,001,,,,12,3,20"></lb> |
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14701 εἰσὶν ἑκατέρα ἑκατέρᾳ· καὶ γωνία ἡ ὑπὸ ΕΑΘ γωνίᾳ τῇ <lb n="1799,001,,,,12,3,21"></lb> |
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14702 ὑπὸ ΚΘΔ ἴση· βάσις ἄρα ἡ ΕΘ βάσει τῇ ΚΔ ἐστιν ἴση.</s> <lb n="1799,001,,,,12,3,22"></lb> |
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14703 <s>ἴσον ἄρα καὶ ὅμοιόν ἐστι τὸ ΑΕΘ τρίγωνον τῷ ΘΚΔ <lb n="1799,001,,,,12,3,23"></lb> |
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14704 τριγώνῳ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ ΑΘΗ τρίγωνον τῷ ΘΛΔ <lb n="1799,001,,,,12,3,24"></lb> |
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14705 τριγώνῳ ἴσον τέ ἐστι καὶ ὅμοιον.</s> <s>καὶ ἐπεὶ δύο εὐθεῖαι ἁπτό-<lb n="1799,001,,,,12,3,25"></lb> |
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14706 μεναι ἀλλήλων αἱ ΕΘ, ΘΗ παρὰ δύο εὐθείας ἁπτομένας <lb n="1799,001,,,,12,3,26"></lb> |
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14707 ἀλλήλων τὰς ΚΔ, ΔΛ εἰσιν οὐκ ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι, <lb n="1799,001,,,,12,3,27"></lb> |
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14708 ἴσας γωνίας περιέξουσιν.</s> <s>ἴση ἄρα ἐστὶν ἡ ὑπὸ ΕΘΗ γω-<lb n="1799,001,,,,12,3,28"></lb> |
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14709 νία τῇ ὑπὸ ΚΔΛ γωνίᾳ.</s> <s>καὶ ἐπεὶ δύο εὐθεῖαι αἱ ΕΘ, ΘΗ <lb n="1799,001,,,,12,3,29"></lb> |
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14710 δυσὶ ταῖς ΚΔ, ΔΛ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ, καὶ γωνία <lb n="1799,001,,,,12,3,30"></lb> |
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14711 ἡ ὑπὸ ΕΘΗ γωνίᾳ τῇ ὑπὸ ΚΔΛ ἐστιν ἴση, βάσις ἄρα ἡ <pb></pb> <lb n="1799,001,,,,12,3,31"></lb> |
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14712 ΕΗ βάσει τῇ ΚΛ [ἐστιν] ἴση· ἴσον ἄρα καὶ ὅμοιόν ἐστι τὸ <lb n="1799,001,,,,12,3,32"></lb> |
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14713 ΕΘΗ τρίγωνον τῷ ΚΔΛ τριγώνῳ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,12,3,33"></lb> |
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14714 τὸ ΑΕΗ τρίγωνον τῷ ΘΚΛ τριγώνῳ ἴσον τε καὶ ὅμοιόν <lb n="1799,001,,,,12,3,34"></lb> |
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14715 ἐστιν.</s> <s>ἡ ἄρα πυραμίς, ἧς βάσις μέν ἐστι τὸ ΑΕΗ τρίγω-<lb n="1799,001,,,,12,3,35"></lb> |
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14716 νον, κορυφὴ δὲ τὸ Θ σημεῖον, ἴση καὶ ὁμοία ἐστὶ πυρα-<lb n="1799,001,,,,12,3,36"></lb> |
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14717 μίδι, ἧς βάσις μέν ἐστι τὸ ΘΚΛ τρίγωνον, κορυφὴ δὲ τὸ <lb n="1799,001,,,,12,3,37"></lb> |
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14718 Δ σημεῖον.</s> <s>καὶ ἐπεὶ τριγώνου τοῦ ΑΔΒ παρὰ μίαν τῶν <lb n="1799,001,,,,12,3,38"></lb> |
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14719 πλευρῶν τὴν ΑΒ ἦκται ἡ ΘΚ, ἰσογώνιόν ἐστι τὸ ΑΔΒ <lb n="1799,001,,,,12,3,39"></lb> |
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14720 τρίγωνον τῷ ΔΘΚ τριγώνῳ, καὶ τὰς πλευρὰς ἀνάλογον <lb n="1799,001,,,,12,3,40"></lb> |
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14721 ἔχουσιν· ὅμοιον ἄρα ἐστὶ τὸ ΑΔΒ τρίγωνον τῷ ΔΘΚ <lb n="1799,001,,,,12,3,41"></lb> |
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14722 τριγώνῳ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ μὲν ΔΒΓ τρίγωνον τῷ <lb n="1799,001,,,,12,3,42"></lb> |
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14723 ΔΚΛ τριγώνῳ ὅμοιόν ἐστιν, τὸ δὲ ΑΔΓ τῷ ΔΛΘ.</s> <s>καὶ <lb n="1799,001,,,,12,3,43"></lb> |
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14724 ἐπεὶ δύο εὐθεῖαι ἁπτόμεναι ἀλλήλων αἱ ΒΑ, ΑΓ παρὰ δύο <lb n="1799,001,,,,12,3,44"></lb> |
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14725 εὐθείας ἁπτομένας ἀλλήλων τὰς ΚΘ, ΘΛ εἰσιν οὐκ ἐν τῷ <lb n="1799,001,,,,12,3,45"></lb> |
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14726 αὐτῷ ἐπιπέδῳ, ἴσας γωνίας περιέξουσιν.</s> <s>ἴση ἄρα ἐστὶν ἡ <lb n="1799,001,,,,12,3,46"></lb> |
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14727 ὑπὸ ΒΑΓ γωνία τῇ ὑπὸ ΚΘΛ.</s> <s>καί ἐστιν ὡς ἡ ΒΑ πρὸς <lb n="1799,001,,,,12,3,47"></lb> |
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14728 τὴν ΑΓ, οὕτως ἡ ΚΘ πρὸς τὴν ΘΛ· ὅμοιον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,12,3,48"></lb> |
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14729 ΑΒΓ τρίγωνον τῷ ΘΚΛ τριγώνῳ.</s> <s>καὶ πυραμὶς ἄρα, ἧς <lb n="1799,001,,,,12,3,49"></lb> |
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14730 βάσις μέν ἐστι τὸ ΑΒΓ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, <lb n="1799,001,,,,12,3,50"></lb> |
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14731 ὁμοία ἐστὶ πυραμίδι, ἧς βάσις μέν ἐστι τὸ ΘΚΛ τρί-<lb n="1799,001,,,,12,3,51"></lb> |
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14732 γωνον, κορυφὴ δὲ τὸ Δ σημεῖον.</s> <s>ἀλλὰ πυραμίς, ἧς βάσις <lb n="1799,001,,,,12,3,52"></lb> |
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14733 μὲν [ἐστι] τὸ ΘΚΛ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, <pb></pb> <lb n="1799,001,,,,12,3,53"></lb> |
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14734 ὁμοία ἐδείχθη πυραμίδι, ἧς βάσις μέν ἐστι τὸ ΑΕΗ τρίγω-<lb n="1799,001,,,,12,3,54"></lb> |
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14735 νον, κορυφὴ δὲ τὸ Θ σημεῖον [ὥστε καὶ πυραμίς, ἧς βάσις <lb n="1799,001,,,,12,3,55"></lb> |
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14736 μὲν τὸ ΑΒΓ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, ὁμοία <lb n="1799,001,,,,12,3,56"></lb> |
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14737 ἐστὶ πυραμίδι, ἧς βάσις μὲν τὸ ΑΕΗ τρίγωνον, κορυφὴ <lb n="1799,001,,,,12,3,57"></lb> |
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14738 δὲ τὸ Θ σημεῖον].</s> <s>ἑκατέρα ἄρα τῶν ΑΕΗΘ, ΘΚΛΔ <lb n="1799,001,,,,12,3,58"></lb> |
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14739 πυραμίδων ὁμοία ἐστὶ τῇ ὅλῃ <lb n="1799,001,,,,12,3,59"></lb> |
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14740 τῇ ΑΒΓΔ πυραμίδι.</s> <s>̄Καὶ <lb n="1799,001,,,,12,3,60"></lb> |
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14741 ἐπεὶ ἴση ἐστὶν ἡ ΒΖ τῇ ΖΓ, <lb n="1799,001,,,,12,3,61"></lb> |
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14742 διπλάσιόν ἐστι τὸ ΕΒΖΗ <lb n="1799,001,,,,12,3,62"></lb> |
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14743 παραλληλόγραμμον τοῦ ΗΖΓ <lb n="1799,001,,,,12,3,63"></lb> |
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14744 τριγώνου.</s> <s>καὶ ἐπεί, ἐὰν ᾖ <lb n="1799,001,,,,12,3,64"></lb> |
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14745 δύο πρίσματα ἰσοϋψῆ, καὶ τὸ <lb n="1799,001,,,,12,3,65"></lb> |
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14746 μὲν ἔχῃ βάσιν παραλληλό-<lb n="1799,001,,,,12,3,66"></lb> |
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14747 γραμμον, τὸ δὲ τρίγωνον, δι-<lb n="1799,001,,,,12,3,67"></lb> |
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14748 πλάσιον δὲ ᾖ τὸ παραλληλό-<lb n="1799,001,,,,12,3,68"></lb> |
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14749 γραμμον τοῦ τριγώνου, ἴσα <lb n="1799,001,,,,12,3,69"></lb> |
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14750 ἐστὶ τὰ πρίσματα, ἴσον ἄρα <lb n="1799,001,,,,12,3,70"></lb> |
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14751 ἐστὶ τὸ πρίσμα τὸ περιεχόμενον ὑπὸ δύο μὲν τριγώνων <lb n="1799,001,,,,12,3,71"></lb> |
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14752 τῶν ΒΚΖ, ΕΘΗ, τριῶν δὲ παραλληλογράμμων τῶν <lb n="1799,001,,,,12,3,72"></lb> |
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14753 ΕΒΖΗ, ΕΒΚΘ, ΘΚΖΗ τῷ πρίσματι τῷ περιεχομένῳ <lb n="1799,001,,,,12,3,73"></lb> |
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14754 ὑπὸ δύο μὲν τριγώνων τῶν ΗΖΓ, ΘΚΛ, τριῶν δὲ παραλ-<lb n="1799,001,,,,12,3,74"></lb> |
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14755 ληλογράμμων τῶν ΚΖΓΛ, ΛΓΗΘ, ΘΚΖΗ.</s> <s>καὶ φανε-<lb n="1799,001,,,,12,3,75"></lb> |
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14756 ρόν, ὅτι ἑκάτερον τῶν πρισμάτων, οὗ τε βάσις τὸ ΕΒΖΗ <lb n="1799,001,,,,12,3,76"></lb> |
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14757 παραλληλόγραμμον, ἀπεναντίον δὲ ἡ ΘΚ εὐθεῖα, καὶ οὗ <lb n="1799,001,,,,12,3,77"></lb> |
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14758 βάσις τὸ ΗΖΓ τρίγωνον, ἀπεναντίον δὲ τὸ ΘΚΛ τρίγωνον, <lb n="1799,001,,,,12,3,78"></lb> |
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14759 μεῖζόν ἐστιν ἑκατέρας τῶν πυραμίδων, ὧν βάσεις μὲν <lb n="1799,001,,,,12,3,79"></lb> |
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14760 τὰ ΑΕΗ, ΘΚΛ τρίγωνα, κορυφαὶ δὲ τὰ Θ, Δ σημεῖα, <pb></pb> <lb n="1799,001,,,,12,3,80"></lb> |
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14761 ἐπειδήπερ [καὶ] ἐὰν ἐπιζεύξωμεν τὰς ΕΖ, ΕΚ εὐθείας, <lb n="1799,001,,,,12,3,81"></lb> |
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14762 τὸ μὲν πρίσμα, οὗ βάσις τὸ ΕΒΖΗ παραλληλόγραμμον, <lb n="1799,001,,,,12,3,82"></lb> |
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14763 ἀπεναντίον δὲ ἡ ΘΚ εὐθεῖα, μεῖζόν ἐστι τῆς πυραμίδος, <lb n="1799,001,,,,12,3,83"></lb> |
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14764 ἧς βάσις τὸ ΕΒΖ τρίγωνον, κορυφὴ δὲ τὸ Κ σημεῖον.</s> <s>ἀλλ' <lb n="1799,001,,,,12,3,84"></lb> |
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14765 ἡ πυραμίς, ἧς βάσις τὸ ΕΒΖ τρίγωνον, κορυφὴ δὲ τὸ Κ <lb n="1799,001,,,,12,3,85"></lb> |
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14766 σημεῖον, ἴση ἐστὶ πυραμίδι, ἧς βάσις τὸ ΑΕΗ τρίγωνον, <lb n="1799,001,,,,12,3,86"></lb> |
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14767 κορυφὴ δὲ τὸ Θ σημεῖον· ὑπὸ γὰρ ἴσων καὶ ὁμοίων ἐπιπέ-<lb n="1799,001,,,,12,3,87"></lb> |
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14768 δων περιέχονται.</s> <s>ὥστε καὶ τὸ πρίσμα, οὗ βάσις μὲν τὸ <lb n="1799,001,,,,12,3,88"></lb> |
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14769 ΕΒΖΗ παραλληλόγραμμον, ἀπεναντίον δὲ ἡ ΘΚ εὐ-<lb n="1799,001,,,,12,3,89"></lb> |
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14770 θεῖα, μεῖζόν ἐστι πυραμίδος, ἧς βάσις μὲν τὸ ΑΕΗ τρίγω-<lb n="1799,001,,,,12,3,90"></lb> |
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14771 νον, κορυφὴ δὲ τὸ Θ σημεῖον.</s> <s>ἴσον δὲ τὸ μὲν πρίσμα, οὗ <lb n="1799,001,,,,12,3,91"></lb> |
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14772 βάσις τὸ ΕΒΖΗ παραλληλόγραμμον, ἀπεναντίον δὲ ἡ <lb n="1799,001,,,,12,3,92"></lb> |
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14773 ΘΚ εὐθεῖα, τῷ πρίσματι, οὗ βάσις μὲν τὸ ΗΖΓ τρίγω-<lb n="1799,001,,,,12,3,93"></lb> |
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14774 νον, ἀπεναντίον δὲ τὸ ΘΚΛ τρίγωνον· ἡ δὲ πυραμίς, ἧς <lb n="1799,001,,,,12,3,94"></lb> |
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14775 βάσις τὸ ΑΕΗ τρίγωνον, κορυφὴ δὲ τὸ Θ σημεῖον, ἴση <lb n="1799,001,,,,12,3,95"></lb> |
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14776 ἐστὶ πυραμίδι, ἧς βάσις τὸ ΘΚΛ τρίγωνον, κορυφὴ δὲ τὸ <lb n="1799,001,,,,12,3,96"></lb> |
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14777 Δ σημεῖον.</s> <s>τὰ ἄρα εἰρημένα δύο πρίσματα μείζονά ἐστι <lb n="1799,001,,,,12,3,97"></lb> |
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14778 τῶν εἰρημένων δύο πυραμίδων, ὧν βάσεις μὲν τὰ ΑΕΗ, <lb n="1799,001,,,,12,3,98"></lb> |
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14779 ΘΚΛ τρίγωνα, κορυφαὶ δὲ τὰ Θ, Δ σημεῖα.</s> <lb n="1799,001,,,,12,3,99"></lb> |
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14780 <s>Ἡ ἄρα ὅλη πυραμίς, ἧς βάσις τὸ ΑΒΓ τρίγωνον, κορυ-<lb n="1799,001,,,,12,3,100"></lb> |
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14781 φὴ δὲ τὸ Δ σημεῖον, διῄρηται εἴς τε δύο πυραμίδας ἴσας <lb n="1799,001,,,,12,3,101"></lb> |
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14782 ἀλλήλαις [καὶ ὁμοίας τῇ ὅλῃ] καὶ εἰς δύο πρίσματα ἴσα, <lb n="1799,001,,,,12,3,102"></lb> |
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14783 καὶ τὰ δύο πρίσματα μείζονά ἐστιν ἢ τὸ ἥμισυ τῆς ὅλης <lb n="1799,001,,,,12,3,103"></lb> |
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14784 πυραμίδος· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,3,104"></lb> |
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14785 <s>Ἐὰν ὦσι δύο πυραμίδες ὑπὸ τὸ αὐτὸ ὕψος τριγώνους <lb n="1799,001,,,,12,4,1"></lb> |
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14786 ἔχουσαι βάσεις, διαιρεθῇ δὲ ἑκατέρα αὐτῶν εἴς τε δύο <lb n="1799,001,,,,12,4,2"></lb> |
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14787 πυραμίδας ἴσας ἀλλήλαις καὶ ὁμοίας τῇ ὅλῃ καὶ εἰς δύο <pb></pb> <lb n="1799,001,,,,12,4,3"></lb> |
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14788 πρίσματα ἴσα, ἔσται ὡς ἡ τῆς μιᾶς πυραμίδος βάσις πρὸς <lb n="1799,001,,,,12,4,4"></lb> |
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14789 τὴν τῆς ἑτέρας πυραμίδος βάσιν, οὕτως τὰ ἐν τῇ μιᾷ πυρα-<lb n="1799,001,,,,12,4,5"></lb> |
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14790 μίδι πρίσματα πάντα πρὸς τὰ ἐν τῇ ἑτέρᾳ πυραμίδι πρίσ-<lb n="1799,001,,,,12,4,6"></lb> |
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14791 ματα πάντα ἰσοπληθῆ. <lb n="1799,001,,,,12,4,7"></lb> |
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14792 Ἔστωσαν δύο πυραμίδες ὑπὸ τὸ αὐτὸ ὕψος τριγώνους <lb n="1799,001,,,,12,4,8"></lb> |
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14793 ἔχουσαι βάσεις τὰς ΑΒΓ, ΔΕΖ, κορυφὰς δὲ τὰ Η, Θ <lb n="1799,001,,,,12,4,9"></lb> |
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14794 σημεῖα, καὶ διῃρήσθω ἑκατέρα αὐτῶν εἴς τε δύο πυραμί-<lb n="1799,001,,,,12,4,10"></lb> |
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14795 δας ἴσας ἀλλήλαις καὶ ὁμοίας τῇ ὅλῃ καὶ εἰς δύο πρίσματα <lb n="1799,001,,,,12,4,11"></lb> |
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14796 ἴσα· λέγω, ὅτι ἐστὶν ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βά-<lb n="1799,001,,,,12,4,12"></lb> |
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14797 σιν, οὕτως τὰ ἐν τῇ ΑΒΓΗ πυραμίδι πρίσματα πάντα πρὸς <lb n="1799,001,,,,12,4,13"></lb> |
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14798 τὰ ἐν τῇ ΔΕΖΘ πυραμίδι πρίσματα ἰσοπληθῆ.</s> <lb n="1799,001,,,,12,4,14"></lb> |
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14799 <s>Ἐπεὶ γὰρ ἴση ἐστὶν ἡ μὲν ΒΞ τῇ ΞΓ, ἡ δὲ ΑΛ τῇ ΛΓ, <pb></pb> <lb n="1799,001,,,,12,4,15"></lb> |
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14800 παράλληλος ἄρα ἐστὶν ἡ ΛΞ τῇ ΑΒ καὶ ὅμοιον τὸ ΑΒΓ <lb n="1799,001,,,,12,4,16"></lb> |
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14801 τρίγωνον τῷ ΛΞΓ τριγώνῳ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ ΔΕΖ <lb n="1799,001,,,,12,4,17"></lb> |
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14802 τρίγωνον τῷ ΡΦΖ τριγώνῳ ὅμοιόν ἐστιν.</s> <s>καὶ ἐπεὶ διπλα-<lb n="1799,001,,,,12,4,18"></lb> |
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14803 σίων ἐστὶν ἡ μὲν ΒΓ τῆς ΓΞ, ἡ δὲ ΕΖ τῆς ΖΦ, ἔστιν ἄρα <lb n="1799,001,,,,12,4,19"></lb> |
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14804 ὡς ἡ ΒΓ πρὸς τὴν ΓΞ, οὕτως ἡ ΕΖ πρὸς τὴν ΖΦ.</s> <s>καὶ <lb n="1799,001,,,,12,4,20"></lb> |
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14805 ἀναγέγραπται ἀπὸ μὲν τῶν ΒΓ, ΓΞ ὅμοιά τε καὶ ὁμοίως <lb n="1799,001,,,,12,4,21"></lb> |
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14806 κείμενα εὐθύγραμμα τὰ ΑΒΓ, ΛΞΓ, ἀπὸ δὲ τῶν ΕΖ, <lb n="1799,001,,,,12,4,22"></lb> |
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14807 ΖΦ ὅμοιά τε καὶ ὁμοίως κείμενα [εὐθύγραμμα] τὰ ΔΕΖ, <lb n="1799,001,,,,12,4,23"></lb> |
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14808 ΡΦΖ.</s> <s>ἔστιν ἄρα ὡς τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΛΞΓ <lb n="1799,001,,,,12,4,24"></lb> |
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14809 τρίγωνον, οὕτως τὸ ΔΕΖ τρίγωνον πρὸς τὸ ΡΦΖ τρίγω-<lb n="1799,001,,,,12,4,25"></lb> |
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14810 νον· ἐναλλὰξ ἄρα ἐστὶν ὡς τὸ ΑΒΓ τρίγωνον πρὸς τὸ <lb n="1799,001,,,,12,4,26"></lb> |
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14811 ΔΕΖ [τρίγωνον], οὕτως τὸ ΛΞΓ [τρίγωνον] πρὸς τὸ <lb n="1799,001,,,,12,4,27"></lb> |
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14812 ΡΦΖ τρίγωνον.</s> <s>ἀλλ' ὡς τὸ ΛΞΓ τρίγωνον πρὸς τὸ <lb n="1799,001,,,,12,4,28"></lb> |
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14813 ΡΦΖ τρίγωνον, οὕτως τὸ πρίσμα, οὗ βάσις μὲν [ἐστι] τὸ <lb n="1799,001,,,,12,4,29"></lb> |
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14814 ΛΞΓ τρίγωνον, ἀπεναντίον δὲ τὸ ΟΜΝ, πρὸς τὸ πρίσμα, <lb n="1799,001,,,,12,4,30"></lb> |
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14815 οὗ βάσις μὲν τὸ ΡΦΖ τρίγωνον, ἀπεναντίον δὲ τὸ ΣΤΥ· <lb n="1799,001,,,,12,4,31"></lb> |
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14816 καὶ ὡς ἄρα τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΔΕΖ τρίγωνον, <lb n="1799,001,,,,12,4,32"></lb> |
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14817 οὕτως τὸ πρίσμα, οὗ βάσις μὲν τὸ ΛΞΓ τρίγωνον, ἀπεν-<lb n="1799,001,,,,12,4,33"></lb> |
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14818 αντίον δὲ τὸ ΟΜΝ, πρὸς τὸ πρίσμα, οὗ βάσις μὲν τὸ <lb n="1799,001,,,,12,4,34"></lb> |
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14819 ΡΦΖ τρίγωνον, ἀπεναντίον δὲ τὸ ΣΤΥ.</s> <s>ὡς δὲ τὰ εἰρη-<lb n="1799,001,,,,12,4,35"></lb> |
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14820 μένα πρίσματα πρὸς ἄλληλα, οὕτως τὸ πρίσμα, οὗ βάσις <lb n="1799,001,,,,12,4,36"></lb> |
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14821 μὲν τὸ ΚΒΞΛ παραλληλόγραμμον, ἀπεναντίον δὲ ἡ ΟΜ <lb n="1799,001,,,,12,4,37"></lb> |
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14822 εὐθεῖα, πρὸς τὸ πρίσμα, οὗ βάσις μὲν τὸ ΠΕΦΡ παραλ-<lb n="1799,001,,,,12,4,38"></lb> |
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14823 ληλόγραμμον, ἀπεναντίον δὲ ἡ ΣΤ εὐθεῖα.</s> <s>καὶ τὰ δύο ἄρα <pb></pb> <lb n="1799,001,,,,12,4,39"></lb> |
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14824 πρίσματα, οὗ τε βάσις μὲν τὸ ΚΒΞΛ παραλληλόγραμμον, <lb n="1799,001,,,,12,4,40"></lb> |
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14825 ἀπεναντίον δὲ ἡ ΟΜ, καὶ οὗ βάσις μὲν τὸ ΛΞΓ, ἀπεναν-<lb n="1799,001,,,,12,4,41"></lb> |
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14826 τίον δὲ τὸ ΟΜΝ, πρὸς τὰ πρίσματα, οὗ τε βάσις μὲν τὸ <lb n="1799,001,,,,12,4,42"></lb> |
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14827 ΠΕΦΡ, ἀπεναντίον δὲ ἡ ΣΤ εὐθεῖα, καὶ οὗ βάσις μὲν τὸ <lb n="1799,001,,,,12,4,43"></lb> |
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14828 ΡΦΖ τρίγωνον, ἀπεναντίον δὲ τὸ ΣΤΥ.</s> <s>καὶ ὡς ἄρα ἡ <lb n="1799,001,,,,12,4,44"></lb> |
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14829 ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὰ εἰρημένα δύο <lb n="1799,001,,,,12,4,45"></lb> |
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14830 πρίσματα πρὸς τὰ εἰρημένα δύο πρίσματα.</s> <lb n="1799,001,,,,12,4,46"></lb> |
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14831 <s>Καὶ ὁμοίως, ἐὰν διαιρεθῶσιν αἱ ΟΜΝΗ, ΣΤΥΘ πυρα-<lb n="1799,001,,,,12,4,47"></lb> |
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14832 μίδες εἴς τε δύο πρίσματα καὶ δύο πυραμίδας, ἔσται ὡς ἡ <lb n="1799,001,,,,12,4,48"></lb> |
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14833 ΟΜΝ βάσις πρὸς τὴν ΣΤΥ βάσιν, οὕτως τὰ ἐν τῇ ΟΜ <lb n="1799,001,,,,12,4,49"></lb> |
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14834 ΝΗ πυραμίδι δύο πρίσματα πρὸς τὰ ἐν τῇ ΣΤΥΘ πυρα-<lb n="1799,001,,,,12,4,50"></lb> |
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14835 μίδι δύο πρίσματα.</s> <s>ἀλλ' ὡς ἡ ΟΜΝ βάσις πρὸς τὴν ΣΤΥ <lb n="1799,001,,,,12,4,51"></lb> |
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14836 βάσιν, οὕτως ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν· ἴσον <lb n="1799,001,,,,12,4,52"></lb> |
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14837 γὰρ ἑκάτερον τῶν ΟΜΝ, ΣΤΥ τριγώνων ἑκατέρῳ τῶν <lb n="1799,001,,,,12,4,53"></lb> |
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14838 ΛΞΓ, ΡΦΖ.</s> <s>καὶ ὡς ἄρα ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ <lb n="1799,001,,,,12,4,54"></lb> |
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14839 βάσιν, οὕτως τὰ τέσσαρα πρίσματα πρὸς τὰ τέσσαρα πρίσ-<lb n="1799,001,,,,12,4,55"></lb> |
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14840 ματα.</s> <s>ὁμοίως δὲ κἂν τὰς ὑπολειπομένας πυραμίδας διέλω-<lb n="1799,001,,,,12,4,56"></lb> |
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14841 μεν εἴς τε δύο πυραμίδας καὶ εἰς δύο πρίσματα, ἔσται ὡς <lb n="1799,001,,,,12,4,57"></lb> |
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14842 ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὰ ἐν τῇ ΑΒ <lb n="1799,001,,,,12,4,58"></lb> |
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14843 ΓΗ πυραμίδι πρίσματα πάντα πρὸς τὰ ἐν τῇ ΔΕΖΘ <lb n="1799,001,,,,12,4,59"></lb> |
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14844 πυραμίδι πρίσματα πάντα ἰσοπληθῆ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,4,60"></lb> |
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14845 <s>Λῆμμα <lb n="1799,001,,,,12,4,61"></lb> |
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14846 Ὅτι δέ ἐστιν ὡς τὸ ΛΞΓ τρίγωνον πρὸς τὸ ΡΦΖ τρί-<lb n="1799,001,,,,12,4,62"></lb> |
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14847 γωνον, οὕτως τὸ πρίσμα, οὗ βάσις τὸ ΛΞΓ τρίγωνον, <lb n="1799,001,,,,12,4,63"></lb> |
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14848 ἀπεναντίον δὲ τὸ ΟΜΝ, πρὸς τὸ πρίσμα, οὗ βάσις μὲν τὸ <lb n="1799,001,,,,12,4,64"></lb> |
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14849 ΡΦΖ [τρίγωνον], ἀπεναντίον δὲ τὸ ΣΤΥ, οὕτω δεικτέον.</s> <lb n="1799,001,,,,12,4,65"></lb> |
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14850 <s>Ἐπὶ γὰρ τῆς αὐτῆς καταγραφῆς νενοήσθωσαν ἀπὸ τῶν <lb n="1799,001,,,,12,4,66"></lb> |
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14851 Η, Θ κάθετοι ἐπὶ τὰ ΑΒΓ, ΔΕΖ ἐπίπεδα, ἴσαι δηλαδὴ <pb></pb> <lb n="1799,001,,,,12,4,67"></lb> |
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14852 τυγχάνουσαι διὰ τὸ ἰσοϋψεῖς ὑποκεῖσθαι τὰς πυραμίδας.</s> <lb n="1799,001,,,,12,4,68"></lb> |
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14853 <s>καὶ ἐπεὶ δύο εὐθεῖαι ἥ τε ΗΓ καὶ ἡ ἀπὸ τοῦ Η κάθετος <lb n="1799,001,,,,12,4,69"></lb> |
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14854 ὑπὸ παραλλήλων ἐπιπέδων τῶν ΑΒΓ, ΟΜΝ τέμνονται, <lb n="1799,001,,,,12,4,70"></lb> |
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14855 εἰς τοὺς αὐτοὺς λόγους τμηθήσονται.</s> <s>καὶ τέτμηται ἡ ΗΓ <lb n="1799,001,,,,12,4,71"></lb> |
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14856 δίχα ὑπὸ τοῦ ΟΜΝ ἐπιπέδου κατὰ τὸ Ν· καὶ ἡ ἀπὸ τοῦ Η <lb n="1799,001,,,,12,4,72"></lb> |
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14857 ἄρα κάθετος ἐπὶ τὸ ΑΒΓ ἐπίπεδον δίχα τμηθήσεται ὑπὸ <lb n="1799,001,,,,12,4,73"></lb> |
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14858 τοῦ ΟΜΝ ἐπιπέδου.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ἀπὸ τοῦ Θ κάθε-<lb n="1799,001,,,,12,4,74"></lb> |
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14859 τος ἐπὶ τὸ ΔΕΖ ἐπίπεδον δίχα τμηθήσεται ὑπὸ τοῦ ΣΤΥ <lb n="1799,001,,,,12,4,75"></lb> |
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14860 ἐπιπέδου.</s> <s>καί εἰσιν ἴσαι αἱ ἀπὸ τῶν Η, Θ κάθετοι ἐπὶ τὰ <lb n="1799,001,,,,12,4,76"></lb> |
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14861 ΑΒΓ, ΔΕΖ ἐπίπεδα· ἴσαι ἄρα καὶ αἱ ἀπὸ τῶν ΟΜΝ, <lb n="1799,001,,,,12,4,77"></lb> |
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14862 ΣΤΥ τριγώνων ἐπὶ τὰ ΑΒΓ, ΔΕΖ κάθετοι.</s> <s>ἰσοϋψῆ ἄρα <lb n="1799,001,,,,12,4,78"></lb> |
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14863 [ἐστὶ] τὰ πρίσματα, ὧν βάσεις μέν εἰσι τὰ ΛΞΓ, ΡΦΖ <lb n="1799,001,,,,12,4,79"></lb> |
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14864 τρίγωνα, ἀπεναντίον δὲ τὰ ΟΜΝ, ΣΤΥ.</s> <s>ὥστε καὶ τὰ <lb n="1799,001,,,,12,4,80"></lb> |
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14865 στερεὰ παραλληλεπίπεδα τὰ ἀπὸ τῶν εἰρημένων πρισμά-<lb n="1799,001,,,,12,4,81"></lb> |
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14866 των ἀναγραφόμενα ἰσοϋψῆ καὶ πρὸς ἄλληλα [εἰσὶν] ὡς αἱ <lb n="1799,001,,,,12,4,82"></lb> |
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14867 βάσεις· καὶ τὰ ἡμίση ἄρα ἐστὶν ὡς ἡ ΛΞΓ βάσις πρὸς τὴν <lb n="1799,001,,,,12,4,83"></lb> |
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14868 ΡΦΖ βάσιν, οὕτως τὰ εἰρημένα πρίσματα πρὸς ἄλληλα· <lb n="1799,001,,,,12,4,84"></lb> |
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14869 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,4,85"></lb> |
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14870 <s>Αἱ ὑπὸ τὸ αὐτὸ ὕψος οὖσαι πυραμίδες καὶ τριγώνους <lb n="1799,001,,,,12,5,1"></lb> |
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14871 ἔχουσαι βάσεις πρὸς ἀλλήλας εἰσὶν ὡς αἱ βάσεις.</s> <lb n="1799,001,,,,12,5,2"></lb> |
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14872 <s>Ἔστωσαν ὑπὸ τὸ αὐτὸ ὕψος πυραμίδες, ὧν βάσεις μὲν <lb n="1799,001,,,,12,5,3"></lb> |
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14873 τὰ ΑΒΓ, ΔΕΖ τρίγωνα, κορυφαὶ δὲ τὰ Η, Θ σημεῖα· <lb n="1799,001,,,,12,5,4"></lb> |
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14874 λέγω, ὅτι ἐστὶν ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, <lb n="1799,001,,,,12,5,5"></lb> |
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14875 οὕτως ἡ ΑΒΓΗ πυραμὶς πρὸς τὴν ΔΕΖΘ πυραμίδα.</s> <s><pb></pb> <lb n="1799,001,,,,12,5,6"></lb> |
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14876 Εἰ γὰρ μή ἐστιν ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, <lb n="1799,001,,,,12,5,7"></lb> |
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14877 οὕτως ἡ ΑΒΓΗ πυραμὶς πρὸς τὴν ΔΕΖΘ πυραμίδα, <lb n="1799,001,,,,12,5,8"></lb> |
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14878 ἔσται ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως ἡ <lb n="1799,001,,,,12,5,9"></lb> |
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14879 ΑΒΓΗ πυραμὶς ἤτοι πρὸς ἔλασσόν τι τῆς ΔΕΖΘ πυρα-<lb n="1799,001,,,,12,5,10"></lb> |
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14880 μίδος στερεὸν ἢ πρὸς μεῖζον.</s> <s>ἔστω πρότερον πρὸς ἔλασ-<lb n="1799,001,,,,12,5,11"></lb> |
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14881 σον τὸ Χ, καὶ διῃρήσθω ἡ ΔΕΖΘ πυραμὶς εἴς τε δύο πυρα-<lb n="1799,001,,,,12,5,12"></lb> |
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14882 μίδας ἴσας ἀλλήλαις καὶ ὁμοίας τῇ ὅλῃ καὶ εἰς δύο πρίσματα <lb n="1799,001,,,,12,5,13"></lb> |
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14883 ἴσα· τὰ δὴ δύο πρίσματα μείζονά ἐστιν ἢ τὸ ἥμισυ τῆς <lb n="1799,001,,,,12,5,14"></lb> |
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14884 ὅλης πυραμίδος.</s> <s>καὶ πάλιν αἱ ἐκ τῆς διαιρέσεως γινόμεναι <lb n="1799,001,,,,12,5,15"></lb> |
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14885 πυραμίδες ὁμοίως διῃρήσθωσαν, καὶ τοῦτο ἀεὶ γινέσθω, <lb n="1799,001,,,,12,5,16"></lb> |
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14886 ἕως οὗ λειφθῶσί τινες πυραμίδες ἀπὸ τῆς ΔΕΖΘ πυρα-<lb n="1799,001,,,,12,5,17"></lb> |
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14887 μίδος, αἵ εἰσιν ἐλάττονες τῆς ὑπεροχῆς, ᾗ ὑπερέχει ἡ ΔΕ <lb n="1799,001,,,,12,5,18"></lb> |
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14888 ΖΘ πυραμὶς τοῦ Χ στερεοῦ.</s> <s>λελείφθωσαν καὶ ἔστωσαν <lb n="1799,001,,,,12,5,19"></lb> |
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14889 λόγου ἕνεκεν αἱ ΔΠΡΣ, ΣΤΥΘ· λοιπὰ ἄρα τὰ ἐν τῇ ΔΕ <lb n="1799,001,,,,12,5,20"></lb> |
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14890 ΖΘ πυραμίδι πρίσματα μείζονά ἐστι τοῦ Χ στερεοῦ.</s> <s>δι- <pb></pb><lb n="1799,001,,,,12,5,21"></lb> |
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14891 ῃρήσθω καὶ ἡ ΑΒΓΗ πυραμὶς ὁμοίως καὶ ἰσοπληθῶς τῇ <lb n="1799,001,,,,12,5,22"></lb> |
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14892 ΔΕΖΘ πυραμίδι· ἔστιν ἄρα ὡς ἡ ΑΒΓ βάσις πρὸς τὴν <lb n="1799,001,,,,12,5,23"></lb> |
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14893 ΔΕΖ βάσιν, οὕτως τὰ ἐν τῇ ΑΒΓΗ πυραμίδι πρίσματα <lb n="1799,001,,,,12,5,24"></lb> |
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14894 πρὸς τὰ ἐν τῇ ΔΕΖΘ πυραμίδι πρίσματα.</s> <s>ἀλλὰ καὶ ὡς ἡ <lb n="1799,001,,,,12,5,25"></lb> |
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14895 ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως ἡ ΑΒΓΗ πυρα-<lb n="1799,001,,,,12,5,26"></lb> |
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14896 μὶς πρὸς τὸ Χ στερεόν· καὶ ὡς ἄρα ἡ ΑΒΓΗ πυραμὶς <lb n="1799,001,,,,12,5,27"></lb> |
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14897 πρὸς τὸ Χ στερεόν, οὕτως τὰ ἐν τῇ ΑΒΓΗ πυραμίδι πρίσ-<lb n="1799,001,,,,12,5,28"></lb> |
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14898 ματα πρὸς τὰ ἐν τῇ ΔΕΖΘ πυραμίδι πρίσματα· ἐναλλὰξ <lb n="1799,001,,,,12,5,29"></lb> |
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14899 ἄρα ὡς ἡ ΑΒΓΗ πυραμὶς πρὸς τὰ ἐν αὐτῇ πρίσματα, <lb n="1799,001,,,,12,5,30"></lb> |
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14900 οὕτως τὸ Χ στερεὸν πρὸς τὰ ἐν τῇ ΔΕΖΘ πυραμίδι πρίσ-<lb n="1799,001,,,,12,5,31"></lb> |
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14901 ματα.</s> <s>μείζων δὲ ἡ ΑΒΓΗ πυραμὶς τῶν ἐν αὐτῇ πρισμά-<lb n="1799,001,,,,12,5,32"></lb> |
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14902 των· μεῖζον ἄρα καὶ τὸ Χ στερεὸν τῶν ἐν τῇ ΔΕΖΘ πυρα-<lb n="1799,001,,,,12,5,33"></lb> |
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14903 μίδι πρισμάτων.</s> <s>ἀλλὰ καὶ ἔλαττον· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,12,5,34"></lb> |
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14904 <s>οὐκ ἄρα ἐστὶν ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕ-<lb n="1799,001,,,,12,5,35"></lb> |
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14905 τως ἡ ΑΒΓΗ πυραμὶς πρὸς ἔλασσόν τι τῆς ΔΕΖΘ πυρα-<lb n="1799,001,,,,12,5,36"></lb> |
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14906 μίδος στερεόν.</s> <s>ὁμοίως δὴ δειχθήσεται, <lb n="1799,001,,,,12,5,37"></lb> |
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14907 ὅτι οὐδὲ ὡς ἡ ΔΕΖ βάσις πρὸς τὴν <lb n="1799,001,,,,12,5,38"></lb> |
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14908 ΑΒΓ βάσιν, οὕτως ἡ ΔΕΖΘ πυραμὶς <lb n="1799,001,,,,12,5,39"></lb> |
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14909 πρὸς ἔλαττόν τι τῆς ΑΒΓΗ πυραμί-<lb n="1799,001,,,,12,5,40"></lb> |
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14910 δος στερεόν.</s> <lb n="1799,001,,,,12,5,41"></lb> |
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14911 <s>Λέγω δή, ὅτι οὐκ ἔστιν οὐδὲ ὡς ἡ ΑΒΓ βάσις πρὸς τὴν <lb n="1799,001,,,,12,5,42"></lb> |
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14912 ΔΕΖ βάσιν, οὕτως ἡ ΑΒΓΗ πυραμὶς πρὸς μεῖζόν τι τῆς <lb n="1799,001,,,,12,5,43"></lb> |
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14913 ΔΕΖΘ πυραμίδος στερεόν.</s> <lb n="1799,001,,,,12,5,44"></lb> |
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14914 <s>Εἰ γὰρ δυνατόν, ἔστω πρὸς μεῖζον τὸ Χ· ἀνάπαλιν ἄρα <lb n="1799,001,,,,12,5,45"></lb> |
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14915 ἐστὶν ὡς ἡ ΔΕΖ βάσις πρὸς τὴν ΑΒΓ βάσιν, οὕτως τὸ <lb n="1799,001,,,,12,5,46"></lb> |
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14916 Χ στερεὸν πρὸς τὴν ΑΒΓΗ πυραμίδα.</s> <s>ὡς δὲ τὸ Χ στερεὸν <lb n="1799,001,,,,12,5,47"></lb> |
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14917 πρὸς τὴν ΑΒΓΗ πυραμίδα, οὕτως ἡ ΔΕΖΘ πυραμὶς πρὸς <lb n="1799,001,,,,12,5,48"></lb> |
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14918 ἔλασσόν τι τῆς ΑΒΓΗ πυραμίδος, ὡς ἔμπροσθεν ἐδείχθη· <pb></pb> <lb n="1799,001,,,,12,5,49"></lb> |
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14919 καὶ ὡς ἄρα ἡ ΔΕΖ βάσις πρὸς τὴν ΑΒΓ βάσιν, οὕτως ἡ <lb n="1799,001,,,,12,5,50"></lb> |
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14920 ΔΕΖΘ πυραμὶς πρὸς ἔλασσόν τι τῆς ΑΒΓΗ πυραμίδος· <lb n="1799,001,,,,12,5,51"></lb> |
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14921 ὅπερ ἄτοπον ἐδείχθη.</s> <s>οὐκ ἄρα ἐστὶν ὡς ἡ ΑΒΓ βάσις <lb n="1799,001,,,,12,5,52"></lb> |
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14922 πρὸς τὴν ΔΕΖ βάσιν, οὕτως ἡ ΑΒΓΗ πυραμὶς πρὸς μεῖ-<lb n="1799,001,,,,12,5,53"></lb> |
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14923 ζόν τι τῆς ΔΕΖΘ πυραμίδος στερεόν.</s> <s>ἐδείχθη δέ, ὅτι <lb n="1799,001,,,,12,5,54"></lb> |
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14924 οὐδὲ πρὸς ἔλασσον.</s> <s>ἔστιν ἄρα ὡς ἡ ΑΒΓ βάσις πρὸς τὴν <lb n="1799,001,,,,12,5,55"></lb> |
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14925 ΔΕΖ βάσιν, οὕτως ἡ ΑΒΓΗ πυραμὶς πρὸς τὴν ΔΕΖΘ <lb n="1799,001,,,,12,5,56"></lb> |
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14926 πυραμίδα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,5,57"></lb> |
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14927 <s>Αἱ ὑπὸ τὸ αὐτὸ ὕψος οὖσαι πυραμίδες καὶ πολυγώνους <lb n="1799,001,,,,12,6,1"></lb> |
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14928 ἔχουσαι βάσεις πρὸς ἀλλήλας εἰσὶν ὡς αἱ βάσεις.</s> <lb n="1799,001,,,,12,6,2"></lb> |
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14929 <s>Ἔστωσαν ὑπὸ τὸ αὐτὸ ὕψος πυραμίδες, ὧν [αἱ] βάσεις <lb n="1799,001,,,,12,6,3"></lb> |
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14930 μὲν τὰ ΑΒΓΔΕ, ΖΗΘΚΛ πολύγωνα, κορυφαὶ δὲ τὰ Μ, <lb n="1799,001,,,,12,6,4"></lb> |
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14931 Ν σημεῖα· λέγω, ὅτι ἐστὶν ὡς ἡ ΑΒΓΔΕ βάσις πρὸς τὴν <lb n="1799,001,,,,12,6,5"></lb> |
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14932 ΖΗΘΚΛ βάσιν, οὕτως ἡ ΑΒΓΔΕΜ πυραμὶς πρὸς τὴν <lb n="1799,001,,,,12,6,6"></lb> |
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14933 ΖΗΘΚΛΝ πυραμίδα.</s> <lb n="1799,001,,,,12,6,7"></lb> |
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14934 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΑΓ, ΑΔ, ΖΘ, ΖΚ.</s> <s>ἐπεὶ οὖν <lb n="1799,001,,,,12,6,8"></lb> |
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14935 δύο πυραμίδες εἰσὶν αἱ ΑΒΓΜ, ΑΓΔΜ τριγώνους ἔχου-<lb n="1799,001,,,,12,6,9"></lb> |
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14936 σαι βάσεις καὶ ὕψος ἴσον, πρὸς ἀλλήλας εἰσὶν ὡς αἱ βάσεις· <lb n="1799,001,,,,12,6,10"></lb> |
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14937 ἔστιν ἄρα ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΑΓΔ βάσιν, οὕτως <lb n="1799,001,,,,12,6,11"></lb> |
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14938 ἡ ΑΒΓΜ πυραμὶς πρὸς τὴν ΑΓΔΜ πυραμίδα.</s> <s>καὶ συν-<lb n="1799,001,,,,12,6,12"></lb> |
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14939 θέντι ὡς ἡ ΑΒΓΔ βάσις πρὸς τὴν ΑΓΔ βάσιν, οὕτως ἡ <pb></pb> <lb n="1799,001,,,,12,6,13"></lb> |
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14940 ΑΒΓΔΜ πυραμὶς πρὸς τὴν ΑΓΔΜ πυραμίδα.</s> <s>ἀλλὰ καὶ <lb n="1799,001,,,,12,6,14"></lb> |
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14941 ὡς ἡ ΑΓΔ βάσις πρὸς τὴν ΑΔΕ βάσιν, οὕτως ἡ ΑΓΔΜ <lb n="1799,001,,,,12,6,15"></lb> |
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14942 πυραμὶς πρὸς τὴν ΑΔΕΜ πυραμίδα.</s> <s>δι' ἴσου ἄρα ὡς ἡ <lb n="1799,001,,,,12,6,16"></lb> |
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14943 ΑΒΓΔ βάσις πρὸς τὴν ΑΔΕ βάσιν, οὕτως ἡ ΑΒΓΔΜ <lb n="1799,001,,,,12,6,17"></lb> |
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14944 πυραμὶς πρὸς τὴν ΑΔΕΜ πυραμίδα.</s> <s>καὶ συνθέντι πάλιν, <lb n="1799,001,,,,12,6,18"></lb> |
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14945 ὡς ἡ ΑΒΓΔΕ βάσις πρὸς τὴν ΑΔΕ βάσιν, οὕτως ἡ <lb n="1799,001,,,,12,6,19"></lb> |
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14946 ΑΒΓΔΕΜ πυραμὶς πρὸς τὴν ΑΔΕΜ πυραμίδα.</s> <s>ὁμοίως <lb n="1799,001,,,,12,6,20"></lb> |
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14947 δὴ δειχθήσεται, ὅτι καὶ ὡς ἡ ΖΗΘΚΛ βάσις πρὸς τὴν <lb n="1799,001,,,,12,6,21"></lb> |
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14948 ΖΗΘ βάσιν, οὕτως καὶ ἡ ΖΗΘΚΛΝ πυραμὶς πρὸς τὴν <lb n="1799,001,,,,12,6,22"></lb> |
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14949 ΖΗΘΝ πυραμίδα.</s> <s>καὶ ἐπεὶ δύο πυραμίδες εἰσὶν αἱ ΑΔ <lb n="1799,001,,,,12,6,23"></lb> |
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14950 ΕΜ, ΖΗΘΝ τριγώνους ἔχουσαι βάσεις καὶ ὕψος ἴσον, <lb n="1799,001,,,,12,6,24"></lb> |
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14951 ἔστιν ἄρα ὡς ἡ ΑΔΕ βάσις πρὸς τὴν ΖΗΘ βάσιν, οὕτως <lb n="1799,001,,,,12,6,25"></lb> |
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14952 ἡ ΑΔΕΜ πυραμὶς πρὸς τὴν ΖΗΘΝ πυραμίδα.</s> <s>ἀλλ' ὡς <pb></pb> <lb n="1799,001,,,,12,6,26"></lb> |
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14953 ἡ ΑΔΕ βάσις πρὸς τὴν ΑΒΓΔΕ βάσιν, οὕτως ἦν ἡ ΑΔ <lb n="1799,001,,,,12,6,27"></lb> |
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14954 ΕΜ πυραμὶς πρὸς τὴν ΑΒΓΔΕΜ πυραμίδα.</s> <s>καὶ δι' ἴσου <lb n="1799,001,,,,12,6,28"></lb> |
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14955 ἄρα ὡς ἡ ΑΒΓΔΕ βάσις πρὸς τὴν ΖΗΘ βάσιν, οὕτως ἡ <lb n="1799,001,,,,12,6,29"></lb> |
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14956 ΑΒΓΔΕΜ πυραμὶς πρὸς τὴν ΖΗΘΝ πυραμίδα.</s> <s>ἀλλὰ <lb n="1799,001,,,,12,6,30"></lb> |
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14957 μὴν καὶ ὡς ἡ ΖΗΘ βάσις πρὸς τὴν ΖΗΘΚΛ βάσιν, οὕτως <lb n="1799,001,,,,12,6,31"></lb> |
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14958 ἦν καὶ ἡ ΖΗΘΝ πυραμὶς πρὸς τὴν ΖΗΘΚΛΝ πυραμίδα.</s> <lb n="1799,001,,,,12,6,32"></lb> |
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14959 <s>καὶ δι' ἴσου ἄρα ὡς ἡ ΑΒΓΔΕ βάσις πρὸς τὴν ΖΗΘΚΛ <lb n="1799,001,,,,12,6,33"></lb> |
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14960 βάσιν, οὕτως ἡ ΑΒΓΔΕΜ πυραμὶς πρὸς τὴν ΖΗΘΚΛΝ <lb n="1799,001,,,,12,6,34"></lb> |
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14961 πυραμίδα· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,6,35"></lb> |
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14962 <s>Πᾶν πρίσμα τρίγωνον ἔχον βάσιν διαιρεῖται εἰς τρεῖς <lb n="1799,001,,,,12,7,1"></lb> |
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14963 πυραμίδας ἴσας ἀλλήλαις τριγώνους βάσεις ἐχούσας.</s> <lb n="1799,001,,,,12,7,2"></lb> |
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14964 <s>Ἔστω πρίσμα, οὗ βάσις μὲν τὸ ΑΒΓ τρίγωνον, ἀπεναν-<lb n="1799,001,,,,12,7,3"></lb> |
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14965 τίον δὲ τὸ ΔΕΖ· λέγω, ὅτι τὸ ΑΒΓΔΕΖ πρίσμα διαιρεῖ-<lb n="1799,001,,,,12,7,4"></lb> |
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14966 ται εἰς τρεῖς πυραμίδας ἴσας ἀλλήλαις τριγώνους ἐχούσας <lb n="1799,001,,,,12,7,5"></lb> |
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14967 βάσεις.</s> <lb n="1799,001,,,,12,7,6"></lb> |
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14968 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΒΔ, ΕΓ, ΓΔ.</s> <s>ἐπεὶ παραλληλό-<lb n="1799,001,,,,12,7,7"></lb> |
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14969 γραμμόν ἐστι τὸ ΑΒΕΔ, διάμετρος δὲ αὐτοῦ ἐστιν ἡ ΒΔ, <lb n="1799,001,,,,12,7,8"></lb> |
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14970 ἴσον ἄρα ἐστὶ τὸ ΑΒΔ τρίγωνον τῷ ΕΒΔ τριγώνῳ· καὶ <lb n="1799,001,,,,12,7,9"></lb> |
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14971 ἡ πυραμὶς ἄρα, ἧς βάσις μὲν τὸ ΑΒΔ τρίγωνον, κορυφὴ <lb n="1799,001,,,,12,7,10"></lb> |
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14972 δὲ τὸ Γ σημεῖον, ἴση ἐστὶ πυραμίδι, ἧς βάσις μέν ἐστι τὸ <lb n="1799,001,,,,12,7,11"></lb> |
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14973 ΔΕΒ τρίγωνον, κορυφὴ δὲ τὸ Γ σημεῖον.</s> <s>ἀλλὰ ἡ πυραμίς, <lb n="1799,001,,,,12,7,12"></lb> |
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14974 ἧς βάσις μέν ἐστι τὸ ΔΕΒ τρίγωνον, κορυφὴ δὲ τὸ Γ <pb></pb> <lb n="1799,001,,,,12,7,13"></lb> |
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14975 σημεῖον, ἡ αὐτή ἐστι πυραμίδι, ἧς βάσις μέν ἐστι τὸ ΕΒΓ <lb n="1799,001,,,,12,7,14"></lb> |
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14976 τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον· ὑπὸ γὰρ τῶν αὐτῶν ἐπι-<lb n="1799,001,,,,12,7,15"></lb> |
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14977 πέδων περιέχεται.</s> <s>καὶ πυραμὶς ἄρα, ἧς βάσις μέν ἐστι τὸ <lb n="1799,001,,,,12,7,16"></lb> |
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14978 ΑΒΔ τρίγωνον, κορυφὴ δὲ τὸ Γ σημεῖον, ἴση ἐστὶ πυρα-<lb n="1799,001,,,,12,7,17"></lb> |
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14979 μίδι, ἧς βάσις μέν ἐστι τὸ ΕΒΓ τρίγωνον, κορυφὴ δὲ τὸ <lb n="1799,001,,,,12,7,18"></lb> |
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14980 Δ σημεῖον.</s> <s>πάλιν, ἐπεὶ παραλλη-<lb n="1799,001,,,,12,7,19"></lb> |
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14981 λόγραμμόν ἐστι τὸ ΖΓΒΕ, διά-<lb n="1799,001,,,,12,7,20"></lb> |
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14982 μετρος δέ ἐστιν αὐτοῦ ἡ ΓΕ, <lb n="1799,001,,,,12,7,21"></lb> |
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14983 ἴσον ἐστὶ τὸ ΓΕΖ τρίγωνον τῷ <lb n="1799,001,,,,12,7,22"></lb> |
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14984 ΓΒΕ τριγώνῳ.</s> <s>καὶ πυραμὶς <lb n="1799,001,,,,12,7,23"></lb> |
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14985 ἄρα, ἧς βάσις μέν ἐστι τὸ ΒΓΕ <lb n="1799,001,,,,12,7,24"></lb> |
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14986 τρίγωνον, κορυφὴ δὲ τὸ Δ ση-<lb n="1799,001,,,,12,7,25"></lb> |
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14987 μεῖον, ἴση ἐστὶ πυραμίδι, ἧς βά-<lb n="1799,001,,,,12,7,26"></lb> |
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14988 σις μέν ἐστι τὸ ΕΓΖ τρίγωνον, <lb n="1799,001,,,,12,7,27"></lb> |
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14989 κορυφὴ δὲ τὸ Δ σημεῖον.</s> <s>ἡ δὲ πυ-<lb n="1799,001,,,,12,7,28"></lb> |
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14990 ραμίς, ἧς βάσις μέν ἐστι τὸ ΒΓΕ <lb n="1799,001,,,,12,7,29"></lb> |
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14991 τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, ἴση ἐδείχθη πυρα-<lb n="1799,001,,,,12,7,30"></lb> |
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14992 μίδι, ἧς βάσις μέν ἐστι τὸ ΑΒΔ τρίγωνον, κορυφὴ δὲ τὸ <lb n="1799,001,,,,12,7,31"></lb> |
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14993 Γ σημεῖον· καὶ πυραμὶς ἄρα, ἧς βάσις μέν ἐστι τὸ ΓΕΖ <lb n="1799,001,,,,12,7,32"></lb> |
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14994 τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, ἴση ἐστὶ πυραμίδι, ἧς <lb n="1799,001,,,,12,7,33"></lb> |
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14995 βάσις μὲν [ἐστι] τὸ ΑΒΔ τρίγωνον, κορυφὴ δὲ τὸ Γ σημεῖ-<lb n="1799,001,,,,12,7,34"></lb> |
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14996 ον· διῄρηται ἄρα τὸ ΑΒΓΔΕΖ πρίσμα εἰς τρεῖς πυρα-<lb n="1799,001,,,,12,7,35"></lb> |
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14997 μίδας ἴσας ἀλλήλαις τριγώνους ἐχούσας βάσεις.</s> <lb n="1799,001,,,,12,7,36"></lb> |
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14998 <s>Καὶ ἐπεὶ πυραμίς, ἧς βάσις μέν ἐστι τὸ ΑΒΔ τρίγωνον, <lb n="1799,001,,,,12,7,37"></lb> |
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14999 κορυφὴ δὲ τὸ Γ σημεῖον, ἡ αὐτή ἐστι πυραμίδι, ἧς βάσις <lb n="1799,001,,,,12,7,38"></lb> |
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15000 τὸ ΓΑΒ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον· ὑπὸ γὰρ τῶν <lb n="1799,001,,,,12,7,39"></lb> |
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15001 αὐτῶν ἐπιπέδων περιέχονται· ἡ δὲ πυραμίς, ἧς βάσις τὸ <lb n="1799,001,,,,12,7,40"></lb> |
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15002 ΑΒΔ τρίγωνον, κορυφὴ δὲ τὸ Γ σημεῖον, τρίτον ἐδείχθη <pb></pb> <lb n="1799,001,,,,12,7,41"></lb> |
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15003 τοῦ πρίσματος, οὗ βάσις τὸ ΑΒΓ τρίγωνον, ἀπεναντίον <lb n="1799,001,,,,12,7,42"></lb> |
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15004 δὲ τὸ ΔΕΖ, καὶ ἡ πυραμὶς ἄρα, ἧς βάσις τὸ ΑΒΓ τρίγω-<lb n="1799,001,,,,12,7,43"></lb> |
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15005 νον, κορυφὴ δὲ τὸ Δ σημεῖον, τρίτον ἐστὶ τοῦ πρίσματος <lb n="1799,001,,,,12,7,44"></lb> |
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15006 τοῦ ἔχοντος βάσιν τὴν αὐτὴν τὸ ΑΒΓ τρίγωνον, ἀπεναν-<lb n="1799,001,,,,12,7,45"></lb> |
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15007 τίον δὲ τὸ ΔΕΖ.</s> <lb n="1799,001,,,,12,7,46"></lb> |
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15008 <s>Πόρισμα <lb n="1799,001,,,,12,7,47"></lb> |
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15009 Ἐκ δὴ τούτου φανερόν, ὅτι πᾶσα πυραμὶς τρίτον μέρος <lb n="1799,001,,,,12,7,48"></lb> |
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15010 ἐστὶ τοῦ πρίσματος τοῦ τὴν αὐτὴν βάσιν ἔχοντος αὐτῇ καὶ <lb n="1799,001,,,,12,7,49"></lb> |
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15011 ὕψος ἴσον [ἐπειδήπερ κἂν ἕτερόν τι σχῆμα εὐθύγραμμον <lb n="1799,001,,,,12,7,50"></lb> |
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15012 ἔχῃ ἡ βάσις τοῦ πρίσματος, τοιοῦτο καὶ τὸ ἀπεναντίον, <lb n="1799,001,,,,12,7,51"></lb> |
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15013 καὶ διαιρεῖται εἰς πρίσματα τρίγωνα ἔχοντα τὰς βάσεις <lb n="1799,001,,,,12,7,52"></lb> |
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15014 καὶ τὰ ἀπεναντίον, καὶ ὡς ἡ ὅλη βάσις πρὸς ἕκαστον]· <lb n="1799,001,,,,12,7,53"></lb> |
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15015 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,7,54"></lb> |
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15016 <s>Αἱ ὅμοιαι πυραμίδες καὶ τριγώνους ἔχουσαι βάσεις ἐν <lb n="1799,001,,,,12,8,1"></lb> |
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15017 τριπλασίονι λόγῳ εἰσὶ τῶν ὁμολόγων πλευρῶν. <pb></pb> <lb n="1799,001,,,,12,8,2"></lb> |
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15018 Ἔστωσαν ὅμοιαι καὶ ὁμοίως κείμεναι πυραμίδες, ὧν <lb n="1799,001,,,,12,8,3"></lb> |
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15019 βάσεις μέν εἰσι τὰ ΑΒΓ, ΔΕΖ τρίγωνα, κορυφαὶ δὲ τὰ Η, <lb n="1799,001,,,,12,8,4"></lb> |
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15020 Θ σημεῖα· λέγω, ὅτι ἡ ΑΒΓΗ πυραμὶς πρὸς τὴν ΔΕΖΘ <lb n="1799,001,,,,12,8,5"></lb> |
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15021 πυραμίδα τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΓ πρὸς τὴν <lb n="1799,001,,,,12,8,6"></lb> |
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15022 ΕΖ.</s> <lb n="1799,001,,,,12,8,7"></lb> |
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15023 <s>Συμπεπληρώσθω γὰρ τὰ ΒΗΜΛ, ΕΘΠΟ στερεὰ <lb n="1799,001,,,,12,8,8"></lb> |
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15024 παραλληλεπίπεδα.</s> <s>καὶ ἐπεὶ ὁμοία ἐστὶν ἡ ΑΒΓΗ πυρα-<lb n="1799,001,,,,12,8,9"></lb> |
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15025 μὶς τῇ ΔΕΖΘ πυραμίδι, ἴση ἄρα ἐστὶν ἡ μὲν ὑπὸ ΑΒΓ <lb n="1799,001,,,,12,8,10"></lb> |
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15026 γωνία τῇ ὑπὸ ΔΕΖ γωνίᾳ, ἡ δὲ ὑπὸ ΗΒΓ τῇ ὑπὸ ΘΕΖ, <lb n="1799,001,,,,12,8,11"></lb> |
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15027 ἡ δὲ ὑπὸ ΑΒΗ τῇ ὑπὸ ΔΕΘ, καί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν <lb n="1799,001,,,,12,8,12"></lb> |
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15028 ΔΕ, οὕτως ἡ ΒΓ πρὸς τὴν ΕΖ, καὶ ἡ ΒΗ πρὸς τὴν ΕΘ.</s> <lb n="1799,001,,,,12,8,13"></lb> |
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15029 <s>καὶ ἐπεί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΔΕ, οὕτως ἡ ΒΓ πρὸς <lb n="1799,001,,,,12,8,14"></lb> |
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15030 τὴν ΕΖ, καὶ περὶ ἴσας γωνίας αἱ πλευραὶ ἀνάλογόν εἰσιν, <lb n="1799,001,,,,12,8,15"></lb> |
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15031 ὅμοιον ἄρα ἐστὶ τὸ ΒΜ παραλληλόγραμμον τῷ ΕΠ παραλ-<lb n="1799,001,,,,12,8,16"></lb> |
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15032 ληλογράμμῳ.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ τὸ μὲν ΒΝ τῷ ΕΡ ὅμοιόν <lb n="1799,001,,,,12,8,17"></lb> |
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15033 ἐστι, τὸ δὲ ΒΚ τῷ ΕΞ· τὰ τρία ἄρα τὰ ΜΒ, ΒΚ, ΒΝ <lb n="1799,001,,,,12,8,18"></lb> |
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15034 τρισὶ τοῖς ΕΠ, ΕΞ, ΕΡ ὅμοιά ἐστιν.</s> <s>ἀλλὰ τὰ μὲν τρία τὰ <lb n="1799,001,,,,12,8,19"></lb> |
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15035 ΜΒ, ΒΚ, ΒΝ τρισὶ τοῖς ἀπεναντίον ἴσα τε καὶ ὅμοιά <lb n="1799,001,,,,12,8,20"></lb> |
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15036 ἐστιν, τὰ δὲ τρία τὰ ΕΠ, ΕΞ, ΕΡ τρισὶ τοῖς ἀπεναντίον <lb n="1799,001,,,,12,8,21"></lb> |
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15037 ἴσα τε καὶ ὅμοιά ἐστιν.</s> <s>τὰ ΒΗΜΛ, ΕΘΠΟ ἄρα στερεὰ <lb n="1799,001,,,,12,8,22"></lb> |
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15038 ὑπὸ ὁμοίων ἐπιπέδων ἴσων τὸ πλῆθος περιέχεται.</s> <s>ὅμοιον <lb n="1799,001,,,,12,8,23"></lb> |
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15039 ἄρα ἐστὶ τὸ ΒΗΜΛ στερεὸν τῷ ΕΘΠΟ στερεῷ.</s> <s>τὰ δὲ <lb n="1799,001,,,,12,8,24"></lb> |
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15040 ὅμοια στερεὰ παραλληλεπίπεδα ἐν τριπλασίονι λόγῳ ἐστὶ <lb n="1799,001,,,,12,8,25"></lb> |
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15041 τῶν ὁμολόγων πλευρῶν.</s> <s>τὸ ΒΗΜΛ ἄρα στερεὸν πρὸς τὸ <lb n="1799,001,,,,12,8,26"></lb> |
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15042 ΕΘΠΟ στερεὸν τριπλασίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος <lb n="1799,001,,,,12,8,27"></lb> |
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15043 πλευρὰ ἡ ΒΓ πρὸς τὴν ὁμόλογον πλευρὰν τὴν ΕΖ.</s> <s>ὡς δὲ <pb></pb> <lb n="1799,001,,,,12,8,28"></lb> |
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15044 τὸ ΒΗΜΛ στερεὸν πρὸς τὸ ΕΘΠΟ στερεόν, οὕτως ἡ <lb n="1799,001,,,,12,8,29"></lb> |
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15045 ΑΒΓΗ πυραμὶς πρὸς τὴν ΔΕΖΘ πυραμίδα, ἐπειδήπερ <lb n="1799,001,,,,12,8,30"></lb> |
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15046 ἡ πυραμὶς ἕκτον μέρος ἐστὶ τοῦ στερεοῦ διὰ τὸ καὶ τὸ πρίσ-<lb n="1799,001,,,,12,8,31"></lb> |
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15047 μα ἥμισυ ὂν τοῦ στερεοῦ παραλληλεπιπέδου τριπλάσιον <lb n="1799,001,,,,12,8,32"></lb> |
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15048 εἶναι τῆς πυραμίδος.</s> <s>καὶ ἡ ΑΒΓΗ ἄρα πυραμὶς πρὸς τὴν <lb n="1799,001,,,,12,8,33"></lb> |
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15049 ΔΕΖΘ πυραμίδα τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΓ <lb n="1799,001,,,,12,8,34"></lb> |
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15050 πρὸς τὴν ΕΖ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,8,35"></lb> |
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15051 <s>Πόρισμα <lb n="1799,001,,,,12,8,36"></lb> |
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15052 Ἐκ δὴ τούτου φανερόν, ὅτι καὶ αἱ πολυγώνους ἔχουσαι <lb n="1799,001,,,,12,8,37"></lb> |
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15053 βάσεις ὅμοιαι πυραμίδες πρὸς ἀλλήλας ἐν τριπλασίονι <lb n="1799,001,,,,12,8,38"></lb> |
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15054 λόγῳ εἰσὶ τῶν ὁμολόγων πλευρῶν.</s> <s>διαιρεθεισῶν γὰρ <lb n="1799,001,,,,12,8,39"></lb> |
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15055 αὐτῶν εἰς τὰς ἐν αὐταῖς πυραμίδας τριγώνους βάσεις ἐχού-<lb n="1799,001,,,,12,8,40"></lb> |
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15056 σας τῷ καὶ τὰ ὅμοια πολύγωνα τῶν βάσεων εἰς ὅμοια τρί-<lb n="1799,001,,,,12,8,41"></lb> |
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15057 γωνα διαιρεῖσθαι καὶ ἴσα τῷ πλήθει καὶ ὁμόλογα τοῖς ὅλοις <lb n="1799,001,,,,12,8,42"></lb> |
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15058 ἔσται ὡς [ἡ] ἐν τῇ ἑτέρᾳ μία πυραμὶς τρίγωνον ἔχουσα <lb n="1799,001,,,,12,8,43"></lb> |
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15059 βάσιν πρὸς τὴν ἐν τῇ ἑτέρᾳ μίαν πυραμίδα τρίγωνον ἔχου-<lb n="1799,001,,,,12,8,44"></lb> |
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15060 σαν βάσιν, οὕτως καὶ ἅπασαι αἱ ἐν τῇ ἑτέρᾳ πυραμίδι <lb n="1799,001,,,,12,8,45"></lb> |
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15061 πυραμίδες τριγώνους ἔχουσαι βάσεις πρὸς τὰς ἐν τῇ ἑτέρᾳ <lb n="1799,001,,,,12,8,46"></lb> |
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15062 πυραμίδι πυραμίδας τριγώνους βάσεις ἐχούσας, τουτέστιν <lb n="1799,001,,,,12,8,47"></lb> |
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15063 αὐτὴ ἡ πολύγωνον βάσιν ἔχουσα πυραμὶς πρὸς τὴν πολύ-<lb n="1799,001,,,,12,8,48"></lb> |
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15064 γωνον βάσιν ἔχουσαν πυραμίδα.</s> <s>ἡ δὲ τρίγωνον βάσιν ἔχου-<lb n="1799,001,,,,12,8,49"></lb> |
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15065 σα πυραμὶς πρὸς τὴν τρίγωνον βάσιν ἔχουσαν ἐν τριπλα- <pb></pb><lb n="1799,001,,,,12,8,50"></lb> |
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15066 σίονι λόγῳ ἐστὶ τῶν ὁμολόγων πλευρῶν· καὶ ἡ πολύγωνον <lb n="1799,001,,,,12,8,51"></lb> |
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15067 ἄρα βάσιν ἔχουσα πρὸς τὴν ὁμοίαν βάσιν ἔχουσαν τριπλα-<lb n="1799,001,,,,12,8,52"></lb> |
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15068 σίονα λόγον ἔχει ἤπερ ἡ πλευρὰ πρὸς τὴν πλευράν.</s> <lb n="1799,001,,,,12,8,53"></lb> |
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15069 <s>Τῶν ἴσων πυραμίδων καὶ τριγώνους βάσεις ἐχουσῶν <lb n="1799,001,,,,12,9,1"></lb> |
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15070 ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν· καὶ ὧν πυραμίδων <lb n="1799,001,,,,12,9,2"></lb> |
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15071 τριγώνους βάσεις ἐχουσῶν ἀντιπεπόνθασιν αἱ βάσεις τοῖς <lb n="1799,001,,,,12,9,3"></lb> |
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15072 ὕψεσιν, ἴσαι εἰσὶν ἐκεῖναι.</s> <lb n="1799,001,,,,12,9,4"></lb> |
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15073 <s>Ἔστωσαν γὰρ ἴσαι πυραμίδες τριγώνους βάσεις ἔχουσαι <lb n="1799,001,,,,12,9,5"></lb> |
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15074 τὰς ΑΒΓ, ΔΕΖ, κορυφὰς δὲ τὰ Η, Θ σημεῖα· λέγω, ὅτι <lb n="1799,001,,,,12,9,6"></lb> |
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15075 τῶν ΑΒΓΗ, ΔΕΖΘ πυραμίδων ἀντιπεπόνθασιν αἱ βά-<lb n="1799,001,,,,12,9,7"></lb> |
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15076 σεις τοῖς ὕψεσιν, καί ἐστιν ὡς ἡ ΑΒΓ βάσις πρὸς τὴν <lb n="1799,001,,,,12,9,8"></lb> |
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15077 ΔΕΖ βάσιν, οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς <lb n="1799,001,,,,12,9,9"></lb> |
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15078 τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος.</s> <lb n="1799,001,,,,12,9,10"></lb> |
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15079 <s>Συμπεπληρώσθω γὰρ τὰ ΒΗΜΛ, ΕΘΠΟ στερεὰ <lb n="1799,001,,,,12,9,11"></lb> |
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15080 παραλληλεπίπεδα.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒΓΗ πυραμὶς <lb n="1799,001,,,,12,9,12"></lb> |
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15081 τῇ ΔΕΖΘ πυραμίδι, καί <lb n="1799,001,,,,12,9,13"></lb> |
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15082 ἐστι τῆς μὲν ΑΒΓΗ πυρα-<lb n="1799,001,,,,12,9,14"></lb> |
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15083 μίδος ἑξαπλάσιον τὸ ΒΗ <lb n="1799,001,,,,12,9,15"></lb> |
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15084 ΜΛ στερεόν, τῆς δὲ ΔΕΖΘ <lb n="1799,001,,,,12,9,16"></lb> |
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15085 πυραμίδος ἑξαπλάσιον τὸ <lb n="1799,001,,,,12,9,17"></lb> |
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15086 ΕΘΠΟ στερεόν, ἴσον ἄρα <lb n="1799,001,,,,12,9,18"></lb> |
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15087 ἐστὶ τὸ ΒΗΜΛ στερεὸν τῷ <lb n="1799,001,,,,12,9,19"></lb> |
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15088 ΕΘΠΟ στερεῷ.</s> <s>τῶν δὲ <lb n="1799,001,,,,12,9,20"></lb> |
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15089 ἴσων στερεῶν παραλληλεπι-<lb n="1799,001,,,,12,9,21"></lb> |
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15090 πέδων ἀντιπεπόνθασιν αἱ <pb></pb> <lb n="1799,001,,,,12,9,22"></lb> |
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15091 βάσεις τοῖς ὕψεσιν· ἔστιν ἄρα ὡς ἡ ΒΜ βάσις πρὸς τὴν ΕΠ <lb n="1799,001,,,,12,9,23"></lb> |
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15092 βάσιν, οὕτως τὸ τοῦ ΕΘΠΟ στερεοῦ ὕψος πρὸς τὸ τοῦ <lb n="1799,001,,,,12,9,24"></lb> |
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15093 ΒΗΜΛ στερεοῦ ὕψος.</s> <s>ἀλλ' ὡς ἡ ΒΜ βάσις πρὸς τὴν ΕΠ, <lb n="1799,001,,,,12,9,25"></lb> |
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15094 οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΔΕΖ τρίγωνον.</s> <s>καὶ ὡς <lb n="1799,001,,,,12,9,26"></lb> |
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15095 ἄρα τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΔΕΖ τρίγωνον, οὕτως τὸ <lb n="1799,001,,,,12,9,27"></lb> |
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15096 τοῦ ΕΘΠΟ στερεοῦ ὕψος πρὸς τὸ τοῦ ΒΗΜΛ στερεοῦ <lb n="1799,001,,,,12,9,28"></lb> |
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15097 ὕψος.</s> <s>ἀλλὰ τὸ μὲν τοῦ ΕΘΠΟ στερεοῦ ὕψος τὸ αὐτό ἐστι <lb n="1799,001,,,,12,9,29"></lb> |
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15098 τῷ τῆς ΔΕΖΘ πυραμίδος ὕψει, τὸ δὲ τοῦ ΒΗΜΛ στε-<lb n="1799,001,,,,12,9,30"></lb> |
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15099 ρεοῦ ὕψος τὸ αὐτό ἐστι τῷ τῆς ΑΒΓΗ πυραμίδος ὕψει· <lb n="1799,001,,,,12,9,31"></lb> |
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15100 ἔστιν ἄρα ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως <lb n="1799,001,,,,12,9,32"></lb> |
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15101 τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ <lb n="1799,001,,,,12,9,33"></lb> |
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15102 πυραμίδος ὕψος.</s> <s>τῶν ΑΒΓΗ, ΔΕΖΘ ἄρα πυραμίδων <lb n="1799,001,,,,12,9,34"></lb> |
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15103 ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν.</s> <lb n="1799,001,,,,12,9,35"></lb> |
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15104 <s>Ἀλλὰ δὴ τῶν ΑΒΓΗ, ΔΕΖΘ πυραμίδων ἀντιπεπον-<lb n="1799,001,,,,12,9,36"></lb> |
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15105 θέτωσαν αἱ βάσεις τοῖς ὕψεσιν, καὶ ἔστω ὡς ἡ ΑΒΓ <lb n="1799,001,,,,12,9,37"></lb> |
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15106 βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ τῆς ΔΕΖΘ πυρα-<lb n="1799,001,,,,12,9,38"></lb> |
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15107 μίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος· λέγω, <lb n="1799,001,,,,12,9,39"></lb> |
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15108 ὅτι ἴση ἐστὶν ἡ ΑΒΓΗ πυραμὶς τῇ ΔΕΖΘ πυραμίδι.</s> <lb n="1799,001,,,,12,9,40"></lb> |
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15109 <s>Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεί ἐστιν ὡς ἡ <lb n="1799,001,,,,12,9,41"></lb> |
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15110 ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ τῆς ΔΕΖΘ <lb n="1799,001,,,,12,9,42"></lb> |
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15111 πυραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος, ἀλλ' <lb n="1799,001,,,,12,9,43"></lb> |
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15112 ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ ΒΜ <lb n="1799,001,,,,12,9,44"></lb> |
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15113 παραλληλόγραμμον πρὸς τὸ ΕΠ παραλληλόγραμμον, καὶ <lb n="1799,001,,,,12,9,45"></lb> |
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15114 ὡς ἄρα τὸ ΒΜ παραλληλόγραμμον πρὸς τὸ ΕΠ παραλ-<lb n="1799,001,,,,12,9,46"></lb> |
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15115 ληλόγραμμον, οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς <lb n="1799,001,,,,12,9,47"></lb> |
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15116 τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος.</s> <s>ἀλλὰ τὸ [μὲν] τῆς ΔΕΖΘ <lb n="1799,001,,,,12,9,48"></lb> |
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15117 πυραμίδος ὕψος τὸ αὐτό ἐστι τῷ τοῦ ΕΘΠΟ παραλληλ-<lb n="1799,001,,,,12,9,49"></lb> |
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15118 επιπέδου ὕψει, τὸ δὲ τῆς ΑΒΓΗ πυραμίδος ὕψος τὸ αὐτό <pb></pb> <lb n="1799,001,,,,12,9,50"></lb> |
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15119 ἐστι τῷ τοῦ ΒΗΜΛ παραλληλεπιπέδου ὕψει· ἔστιν ἄρα ὡς <lb n="1799,001,,,,12,9,51"></lb> |
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15120 ἡ ΒΜ βάσις πρὸς τὴν ΕΠ βάσιν, οὕτως τὸ τοῦ ΕΘΠΟ <lb n="1799,001,,,,12,9,52"></lb> |
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15121 παραλληλεπιπέδου ὕψος πρὸς τὸ τοῦ ΒΗΜΛ παραλληλ-<lb n="1799,001,,,,12,9,53"></lb> |
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15122 επιπέδου ὕψος.</s> <s>ὧν δὲ στερεῶν παραλληλεπιπέδων ἀντι-<lb n="1799,001,,,,12,9,54"></lb> |
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15123 πεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, ἴσα ἐστὶν ἐκεῖνα· ἴσον <lb n="1799,001,,,,12,9,55"></lb> |
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15124 ἄρα ἐστὶ τὸ ΒΗΜΛ στερεὸν παραλληλεπίπεδον τῷ ΕΘ <lb n="1799,001,,,,12,9,56"></lb> |
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15125 ΠΟ στερεῷ παραλληλεπιπέδῳ.</s> <s>καί ἐστι τοῦ μὲν ΒΗΜΛ <lb n="1799,001,,,,12,9,57"></lb> |
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15126 ἕκτον μέρος ἡ ΑΒΓΗ πυραμίς, τοῦ δὲ ΕΘΠΟ παραλ-<lb n="1799,001,,,,12,9,58"></lb> |
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15127 ληλεπιπέδου ἕκτον μέρος ἡ ΔΕΖΘ πυραμίς· ἴση ἄρα ἡ <lb n="1799,001,,,,12,9,59"></lb> |
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15128 ΑΒΓΗ πυραμὶς τῇ ΔΕΖΘ πυραμίδι.</s> <lb n="1799,001,,,,12,9,60"></lb> |
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15129 <s>Τῶν ἄρα ἴσων πυραμίδων καὶ τριγώνους βάσεις ἐχου-<lb n="1799,001,,,,12,9,61"></lb> |
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15130 σῶν ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν· καὶ ὧν πυρα-<lb n="1799,001,,,,12,9,62"></lb> |
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15131 μίδων τριγώνους βάσεις ἐχουσῶν ἀντιπεπόνθασιν αἱ βά-<lb n="1799,001,,,,12,9,63"></lb> |
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15132 σεις τοῖς ὕψεσιν, ἴσαι εἰσὶν ἐκεῖναι· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,9,64"></lb> |
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15133 <s>Πᾶς κῶνος κυλίνδρου τρίτον μέρος ἐστὶ τοῦ τὴν αὐτὴν <lb n="1799,001,,,,12,10,1"></lb> |
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15134 βάσιν ἔχοντος αὐτῷ καὶ ὕψος ἴσον.</s> <lb n="1799,001,,,,12,10,2"></lb> |
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15135 <s>Ἐχέτω γὰρ κῶνος κυλίνδρῳ βάσιν τε τὴν αὐτὴν τὸν <lb n="1799,001,,,,12,10,3"></lb> |
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15136 ΑΒΓΔ κύκλον καὶ ὕψος ἴσον· λέγω, ὅτι ὁ κῶνος τοῦ κυλίν-<lb n="1799,001,,,,12,10,4"></lb> |
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15137 δρου τρίτον ἐστὶ μέρος, τουτέστιν ὅτι ὁ κύλινδρος τοῦ κώ-<lb n="1799,001,,,,12,10,5"></lb> |
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15138 νου τριπλασίων ἐστίν.</s> <lb n="1799,001,,,,12,10,6"></lb> |
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15139 <s>Εἰ γὰρ μή ἐστιν ὁ κύλινδρος τοῦ κώνου τριπλασίων, <lb n="1799,001,,,,12,10,7"></lb> |
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15140 ἔσται ὁ κύλινδρος τοῦ κώνου ἤτοι μείζων ἢ τριπλασίων <pb></pb> <lb n="1799,001,,,,12,10,8"></lb> |
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15141 ἢ ἐλάσσων ἢ τριπλασίων.</s> <s>ἔστω πρότερον μείζων ἢ τριπλα-<lb n="1799,001,,,,12,10,9"></lb> |
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15142 σίων, καὶ ἐγγεγράφθω εἰς τὸν ΑΒΓΔ κύκλον τετράγωνον <lb n="1799,001,,,,12,10,10"></lb> |
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15143 τὸ ΑΒΓΔ· τὸ δὴ ΑΒΓΔ τετράγωνον μεῖζόν ἐστιν ἢ τὸ <lb n="1799,001,,,,12,10,11"></lb> |
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15144 ἥμισυ τοῦ ΑΒΓΔ κύκλου.</s> <s>καὶ ἀνεστάτω ἀπὸ τοῦ ΑΒΓΔ <lb n="1799,001,,,,12,10,12"></lb> |
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15145 τετραγώνου πρίσμα ἰσο- <lb n="1799,001,,,,12,10,13"></lb> |
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15146 υψὲς τῷ κυλίνδρῳ.</s> <s>τὸ δὴ <lb n="1799,001,,,,12,10,14"></lb> |
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15147 ἀνιστάμενον πρίσμα μεῖ-<lb n="1799,001,,,,12,10,15"></lb> |
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15148 ζόν ἐστιν ἢ τὸ ἥμισυ τοῦ <lb n="1799,001,,,,12,10,16"></lb> |
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15149 κυλίνδρου, ἐπειδήπερ κἂν <lb n="1799,001,,,,12,10,17"></lb> |
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15150 περὶ τὸν ΑΒΓΔ κύκλον <lb n="1799,001,,,,12,10,18"></lb> |
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15151 τετράγωνον περιγράψω-<lb n="1799,001,,,,12,10,19"></lb> |
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15152 μεν, τὸ ἐγγεγραμμένον <lb n="1799,001,,,,12,10,20"></lb> |
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15153 εἰς τὸν ΑΒΓΔ κύκλον τε-<lb n="1799,001,,,,12,10,21"></lb> |
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15154 τράγωνον ἥμισύ ἐστι τοῦ <lb n="1799,001,,,,12,10,22"></lb> |
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15155 περιγεγραμμένου· καί ἐστι <lb n="1799,001,,,,12,10,23"></lb> |
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15156 τὰ ἀπ' αὐτῶν ἀνιστά-<lb n="1799,001,,,,12,10,24"></lb> |
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15157 μενα στερεὰ παραλληλεπίπεδα πρίσματα ἰσοϋψῆ· τὰ <lb n="1799,001,,,,12,10,25"></lb> |
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15158 δὲ ὑπὸ τὸ αὐτὸ ὕψος ὄντα στερεὰ παραλληλεπίπεδα <lb n="1799,001,,,,12,10,26"></lb> |
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15159 πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις· καὶ τὸ ἐπὶ τοῦ <lb n="1799,001,,,,12,10,27"></lb> |
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15160 ΑΒΓΔ ἄρα τετραγώνου ἀνασταθὲν πρίσμα ἥμισύ ἐστι <lb n="1799,001,,,,12,10,28"></lb> |
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15161 τοῦ ἀνασταθέντος πρίσματος ἀπὸ τοῦ περὶ τὸν ΑΒΓΔ <lb n="1799,001,,,,12,10,29"></lb> |
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15162 κύκλον περιγραφέντος τετραγώνου· καί ἐστιν ὁ κύλιν-<lb n="1799,001,,,,12,10,30"></lb> |
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15163 δρος ἐλάττων τοῦ πρίσματος τοῦ ἀνασταθέντος ἀπὸ τοῦ <lb n="1799,001,,,,12,10,31"></lb> |
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15164 περὶ τὸν ΑΒΓΔ κύκλον περιγραφέντος τετραγώνου· τὸ <lb n="1799,001,,,,12,10,32"></lb> |
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15165 ἄρα πρίσμα τὸ ἀνασταθὲν ἀπὸ τοῦ ΑΒΓΔ τετραγώνου <pb></pb> <lb n="1799,001,,,,12,10,33"></lb> |
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15166 ἰσοϋψὲς τῷ κυλίνδρῳ μεῖζόν ἐστι τοῦ ἡμίσεως τοῦ κυλίν-<lb n="1799,001,,,,12,10,34"></lb> |
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15167 δρου.</s> <s>τετμήσθωσαν αἱ ΑΒ, ΒΓ, ΓΔ, ΔΑ περιφέρειαι <lb n="1799,001,,,,12,10,35"></lb> |
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15168 δίχα κατὰ τὰ Ε, Ζ, Η, Θ σημεῖα, καὶ ἐπεζεύχθωσαν αἱ <lb n="1799,001,,,,12,10,36"></lb> |
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15169 ΑΕ, ΕΒ, ΒΖ, ΖΓ, ΓΗ, ΗΔ, ΔΘ, ΘΑ· καὶ ἕκαστον <lb n="1799,001,,,,12,10,37"></lb> |
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15170 ἄρα τῶν ΑΕΒ, ΒΖΓ, ΓΗΔ, ΔΘΑ τριγώνων μεῖζόν <lb n="1799,001,,,,12,10,38"></lb> |
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15171 ἐστιν ἢ τὸ ἥμισυ τοῦ καθ' ἑαυτὸ τμήματος τοῦ ΑΒΓΔ <lb n="1799,001,,,,12,10,39"></lb> |
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15172 κύκλου, ὡς ἔμπροσθεν ἐδείκνυμεν.</s> <s>ἀνεστάτω ἐφ' ἑκάστου <lb n="1799,001,,,,12,10,40"></lb> |
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15173 τῶν ΑΕΒ, ΒΖΓ, ΓΗΔ, ΔΘΑ τριγώνων πρίσματα ἰσο-<lb n="1799,001,,,,12,10,41"></lb> |
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15174 υψῆ τῷ κυλίνδρῳ· καὶ ἕκαστον ἄρα τῶν ἀνασταθέντων <lb n="1799,001,,,,12,10,42"></lb> |
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15175 πρισμάτων μεῖζόν ἐστιν ἢ τὸ ἥμισυ μέρος τοῦ καθ' ἑαυτὸ <lb n="1799,001,,,,12,10,43"></lb> |
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15176 τμήματος τοῦ κυλίνδρου, ἐπειδήπερ ἐὰν διὰ τῶν Ε, Ζ, Η, <lb n="1799,001,,,,12,10,44"></lb> |
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15177 Θ σημείων παραλλήλους ταῖς ΑΒ, ΒΓ, ΓΔ, ΔΑ ἀγάγω-<lb n="1799,001,,,,12,10,45"></lb> |
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15178 μεν, καὶ συμπληρώσωμεν τὰ ἐπὶ τῶν ΑΒ, ΒΓ, ΓΔ, ΔΑ <lb n="1799,001,,,,12,10,46"></lb> |
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15179 παραλληλόγραμμα, καὶ ἀπ' αὐτῶν ἀναστήσωμεν στερεὰ <lb n="1799,001,,,,12,10,47"></lb> |
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15180 παραλληλεπίπεδα ἰσοϋψῆ τῷ κυλίνδρῳ, ἑκάστου τῶν <lb n="1799,001,,,,12,10,48"></lb> |
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15181 ἀνασταθέντων ἡμίση ἐστὶ τὰ πρίσματα τὰ ἐπὶ τῶν ΑΕΒ, <lb n="1799,001,,,,12,10,49"></lb> |
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15182 ΒΖΓ, ΓΗΔ, ΔΘΑ τριγώνων· καί ἐστι τὰ τοῦ κυλίνδρου <lb n="1799,001,,,,12,10,50"></lb> |
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15183 τμήματα ἐλάττονα τῶν ἀνασταθέντων στερεῶν παραλληλ-<lb n="1799,001,,,,12,10,51"></lb> |
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15184 επιπέδων· ὥστε καὶ τὰ ἐπὶ τῶν ΑΕΒ, ΒΖΓ, ΓΗΔ, <lb n="1799,001,,,,12,10,52"></lb> |
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15185 ΔΘΑ τριγώνων πρίσματα μείζονά ἐστιν ἢ τὸ ἥμισυ τῶν <lb n="1799,001,,,,12,10,53"></lb> |
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15186 καθ' ἑαυτὰ τοῦ κυλίνδρου τμημάτων.</s> <s>τέμνοντες δὴ τὰς <lb n="1799,001,,,,12,10,54"></lb> |
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15187 ὑπολειπομένας περιφερείας δίχα καὶ ἐπιζευγνύντες εὐ-<lb n="1799,001,,,,12,10,55"></lb> |
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15188 θείας καὶ ἀνιστάντες ἐφ' ἑκάστου τῶν τριγώνων πρίσματα <lb n="1799,001,,,,12,10,56"></lb> |
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15189 ἰσοϋψῆ τῷ κυλίνδρῳ καὶ τοῦτο ἀεὶ ποιοῦντες καταλείψο-<lb n="1799,001,,,,12,10,57"></lb> |
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15190 μέν τινα ἀποτμήματα τοῦ κυλίνδρου, ἃ ἔσται ἐλάττονα τῆς <lb n="1799,001,,,,12,10,58"></lb> |
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15191 ὑπεροχῆς, ᾗ ὑπερέχει ὁ κύλινδρος τοῦ τριπλασίου τοῦ κώ-<lb n="1799,001,,,,12,10,59"></lb> |
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15192 νου.</s> <s>λελείφθω, καὶ ἔστω τὰ ΑΕ, ΕΒ, ΒΖ, ΖΓ, ΓΗ, ΗΔ, <pb></pb> <lb n="1799,001,,,,12,10,60"></lb> |
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15193 ΔΘ, ΘΑ· λοιπὸν ἄρα τὸ πρίσμα, οὗ βάσις μὲν τὸ ΑΕΒΖ <lb n="1799,001,,,,12,10,61"></lb> |
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15194 ΓΗΔΘ πολύγωνον, ὕψος δὲ τὸ αὐτὸ τῷ κυλίνδρῳ, μεῖ-<lb n="1799,001,,,,12,10,62"></lb> |
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15195 ζόν ἐστιν ἢ τριπλάσιον τοῦ κώνου.</s> <s>ἀλλὰ τὸ πρίσμα, οὗ <lb n="1799,001,,,,12,10,63"></lb> |
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15196 βάσις μέν ἐστι τὸ ΑΕΒΖΓΗΔΘ πολύγωνον, ὕψος δὲ τὸ <lb n="1799,001,,,,12,10,64"></lb> |
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15197 αὐτὸ τῷ κυλίνδρῳ, τριπλάσιόν ἐστι τῆς πυραμίδος, ἧς <lb n="1799,001,,,,12,10,65"></lb> |
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15198 βάσις μέν ἐστι τὸ ΑΕΒΖΓΗΔΘ πολύγωνον, κορυφὴ δὲ <lb n="1799,001,,,,12,10,66"></lb> |
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15199 ἡ αὐτὴ τῷ κώνῳ· καὶ ἡ πυραμὶς ἄρα, ἧς βάσις μὲν [ἐστι] <lb n="1799,001,,,,12,10,67"></lb> |
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15200 τὸ ΑΕΒΖΓΗΔΘ πολύγωνον, κορυφὴ δὲ ἡ αὐτὴ τῷ <lb n="1799,001,,,,12,10,68"></lb> |
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15201 κώνῳ, μείζων ἐστὶ τοῦ κώνου τοῦ βάσιν ἔχοντος τὸν ΑΒ <lb n="1799,001,,,,12,10,69"></lb> |
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15202 ΓΔ κύκλον.</s> <s>ἀλλὰ καὶ ἐλάττων· ἐμπεριέχεται γὰρ ὑπ' <lb n="1799,001,,,,12,10,70"></lb> |
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15203 αὐτοῦ· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα ἐστὶν ὁ κύλινδρος <lb n="1799,001,,,,12,10,71"></lb> |
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15204 τοῦ κώνου μείζων ἢ τριπλάσιος.</s> <lb n="1799,001,,,,12,10,72"></lb> |
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15205 <s>Λέγω δή, ὅτι οὐδὲ ἐλάττων ἐστὶν ἢ τριπλάσιος ὁ κύλιν-<lb n="1799,001,,,,12,10,73"></lb> |
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15206 δρος τοῦ κώνου.</s> <lb n="1799,001,,,,12,10,74"></lb> |
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15207 <s>Εἰ γὰρ δυνατόν, ἔστω ἐλάττων ἢ τριπλάσιος ὁ κύλινδρος <lb n="1799,001,,,,12,10,75"></lb> |
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15208 τοῦ κώνου· ἀνάπαλιν ἄρα ὁ κῶνος τοῦ κυλίνδρου μείζων <lb n="1799,001,,,,12,10,76"></lb> |
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15209 ἐστὶν ἢ τρίτον μέρος.</s> <s>ἐγγεγράφθω δὴ εἰς τὸν ΑΒΓΔ <lb n="1799,001,,,,12,10,77"></lb> |
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15210 κύκλον τετράγωνον τὸ ΑΒΓΔ· τὸ ΑΒΓΔ ἄρα τετράγω-<lb n="1799,001,,,,12,10,78"></lb> |
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15211 νον μεῖζόν ἐστιν ἢ τὸ ἥμισυ τοῦ ΑΒΓΔ κύκλου.</s> <s>καὶ ἀν-<lb n="1799,001,,,,12,10,79"></lb> |
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15212 εστάτω ἀπὸ τοῦ ΑΒΓΔ τετραγώνου πυραμὶς τὴν αὐτὴν <lb n="1799,001,,,,12,10,80"></lb> |
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15213 κορυφὴν ἔχουσα τῷ κώνῳ· ἡ ἄρα ἀνασταθεῖσα πυραμὶς <lb n="1799,001,,,,12,10,81"></lb> |
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15214 μείζων ἐστὶν ἢ τὸ ἥμισυ μέρος τοῦ κώνου, ἐπειδήπερ, ὡς <lb n="1799,001,,,,12,10,82"></lb> |
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15215 ἔμπροσθεν ἐδείκνυμεν, ὅτι ἐὰν περὶ τὸν κύκλον τετράγω-<lb n="1799,001,,,,12,10,83"></lb> |
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15216 νον περιγράψωμεν, ἔσται τὸ ΑΒΓΔ τετράγωνον ἥμισυ <lb n="1799,001,,,,12,10,84"></lb> |
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15217 τοῦ περὶ τὸν κύκλον περιγεγραμμένου τετραγώνου· καὶ <pb></pb> <lb n="1799,001,,,,12,10,85"></lb> |
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15218 ἐὰν ἀπὸ τῶν τετραγώνων στερεὰ παραλληλεπίπεδα ἀνα-<lb n="1799,001,,,,12,10,86"></lb> |
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15219 στήσωμεν ἰσοϋψῆ τῷ κώνῳ, ἃ καὶ καλεῖται πρίσματα, <lb n="1799,001,,,,12,10,87"></lb> |
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15220 ἔσται τὸ ἀνασταθὲν ἀπὸ τοῦ ΑΒΓΔ τετραγώνου ἥμισυ <lb n="1799,001,,,,12,10,88"></lb> |
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15221 τοῦ ἀνασταθέντος ἀπὸ τοῦ περὶ τὸν κύκλον περιγραφέντος <lb n="1799,001,,,,12,10,89"></lb> |
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15222 τετραγώνου· πρὸς ἄλληλα γάρ εἰσιν ὡς αἱ βάσεις.</s> <s>ὥστε <lb n="1799,001,,,,12,10,90"></lb> |
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15223 καὶ τὰ τρίτα· καὶ πυραμὶς ἄρα, ἧς βάσις τὸ ΑΒΓΔ τετρά-<lb n="1799,001,,,,12,10,91"></lb> |
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15224 γωνον, ἥμισύ ἐστι τῆς πυραμίδος τῆς ἀνασταθείσης ἀπὸ <lb n="1799,001,,,,12,10,92"></lb> |
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15225 τοῦ περὶ τὸν κύκλον περιγραφέντος τετραγώνου.</s> <s>καί ἐστι <lb n="1799,001,,,,12,10,93"></lb> |
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15226 μείζων ἡ πυραμὶς ἡ ἀνασταθεῖσα ἀπὸ τοῦ περὶ τὸν κύκλον <lb n="1799,001,,,,12,10,94"></lb> |
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15227 τετραγώνου τοῦ κώνου· ἐμπεριέχει γὰρ αὐτόν.</s> <s>ἡ ἄρα <lb n="1799,001,,,,12,10,95"></lb> |
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15228 πυραμὶς, ἧς βάσις τὸ ΑΒΓΔ τετράγωνον, κορυφὴ δὲ ἡ <lb n="1799,001,,,,12,10,96"></lb> |
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15229 αὐτὴ τῷ κώνῳ, μείζων ἐστὶν ἢ τὸ ἥμισυ τοῦ κώνου.</s> <lb n="1799,001,,,,12,10,97"></lb> |
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15230 <s>τετμήσθωσαν αἱ ΑΒ, ΒΓ, ΓΔ, ΔΑ περιφέρειαι δίχα <lb n="1799,001,,,,12,10,98"></lb> |
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15231 κατὰ τὰ Ε, Ζ, Η, Θ σημεῖα, καὶ ἐπεζεύχθωσαν αἱ ΑΕ, <lb n="1799,001,,,,12,10,99"></lb> |
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15232 ΕΒ, ΒΖ, ΖΓ, ΓΗ, ΗΔ, ΔΘ, ΘΑ· καὶ ἕκαστον ἄρα τῶν <lb n="1799,001,,,,12,10,100"></lb> |
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15233 ΑΕΒ, ΒΖΓ, ΓΗΔ, ΔΘΑ τριγώνων μεῖζόν ἐστιν ἢ τὸ <lb n="1799,001,,,,12,10,101"></lb> |
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15234 ἥμισυ μέρος τοῦ καθ' ἑαυτὸ τμήματος τοῦ ΑΒΓΔ κύκλου.</s> <lb n="1799,001,,,,12,10,102"></lb> |
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15235 <s>καὶ ἀνεστάτωσαν ἐφ' ἑκάστου τῶν ΑΕΒ, ΒΖΓ, ΓΗΔ, <lb n="1799,001,,,,12,10,103"></lb> |
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15236 ΔΘΑ τριγώνων πυραμίδες τὴν αὐτὴν κορυφὴν ἔχουσαι <lb n="1799,001,,,,12,10,104"></lb> |
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15237 τῷ κώνῳ· καὶ ἑκάστη ἄρα τῶν ἀνασταθεισῶν πυραμίδων <lb n="1799,001,,,,12,10,105"></lb> |
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15238 κατὰ τὸν αὐτὸν τρόπον μείζων ἐστὶν ἢ τὸ ἥμισυ μέρος τοῦ <lb n="1799,001,,,,12,10,106"></lb> |
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15239 καθ' ἑαυτὴν τμήματος τοῦ κώνου.</s> <s>τέμνοντες δὴ τὰς ὑπο-<lb n="1799,001,,,,12,10,107"></lb> |
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15240 λειπομένας περιφερείας δίχα καὶ ἐπιζευγνύντες εὐθείας <lb n="1799,001,,,,12,10,108"></lb> |
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15241 καὶ ἀνιστάντες ἐφ' ἑκάστου τῶν τριγώνων πυραμίδα τὴν <lb n="1799,001,,,,12,10,109"></lb> |
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15242 αὐτὴν κορυφὴν ἔχουσαν τῷ κώνῳ καὶ τοῦτο ἀεὶ ποιοῦντες <lb n="1799,001,,,,12,10,110"></lb> |
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15243 καταλείψομέν τινα ἀποτμήματα τοῦ κώνου, ἃ ἔσται ἐλάτ-<lb n="1799,001,,,,12,10,111"></lb> |
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15244 τονα τῆς ὑπεροχῆς, ᾗ ὑπερέχει ὁ κῶνος τοῦ τρίτου μέρους <pb></pb> <lb n="1799,001,,,,12,10,112"></lb> |
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15245 τοῦ κυλίνδρου.</s> <s>λελείφθω, καὶ ἔστω τὰ ἐπὶ τῶν ΑΕ, ΕΒ, <lb n="1799,001,,,,12,10,113"></lb> |
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15246 ΒΖ, ΖΓ, ΓΗ, ΗΔ, ΔΘ, ΘΑ· λοιπὴ ἄρα ἡ πυραμίς, ἧς <lb n="1799,001,,,,12,10,114"></lb> |
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15247 βάσις μέν ἐστι τὸ ΑΕΒΖΓΗΔΘ πολύγωνον, κορυφὴ δὲ <lb n="1799,001,,,,12,10,115"></lb> |
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15248 ἡ αὐτὴ τῷ κώνῳ, μείζων ἐστὶν ἢ τρίτον μέρος τοῦ κυλίν-<lb n="1799,001,,,,12,10,116"></lb> |
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15249 δρου.</s> <s>ἀλλ' ἡ πυραμίς, ἧς βάσις μέν ἐστι τὸ ΑΕΒΖΓ <lb n="1799,001,,,,12,10,117"></lb> |
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15250 ΗΔΘ πολύγωνον, κορυφὴ δὲ ἡ αὐτὴ τῷ κώνῳ, τρίτον <lb n="1799,001,,,,12,10,118"></lb> |
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15251 ἐστὶ μέρος τοῦ πρίσματος, οὗ βάσις μέν ἐστι τὸ ΑΕΒΖΓ <lb n="1799,001,,,,12,10,119"></lb> |
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15252 ΗΔΘ πολύγωνον, ὕψος δὲ τὸ αὐτὸ τῷ κυλίνδρῳ· τὸ ἄρα <lb n="1799,001,,,,12,10,120"></lb> |
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15253 πρίσμα, οὗ βάσις μέν ἐστι τὸ ΑΕΒΖΓΗΔΘ πολύγωνον, <lb n="1799,001,,,,12,10,121"></lb> |
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15254 ὕψος δὲ τὸ αὐτὸ τῷ κυλίνδρῳ, μεῖζόν ἐστι τοῦ κυλίνδρου, <lb n="1799,001,,,,12,10,122"></lb> |
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15255 οὗ βάσις ἐστὶν ὁ ΑΒΓΔ κύκλος.</s> <s>ἀλλὰ καὶ ἔλαττον· ἐμ-<lb n="1799,001,,,,12,10,123"></lb> |
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15256 περιέχεται γὰρ ὑπ' αὐτοῦ· ὅπερ ἐστὶν ἀδύνατον.</s> <s>οὐκ ἄρα <lb n="1799,001,,,,12,10,124"></lb> |
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15257 ὁ κύλινδρος τοῦ κώνου ἐλάττων ἐστὶν ἢ τριπλάσιος.</s> <s>ἐδείχ-<lb n="1799,001,,,,12,10,125"></lb> |
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15258 θη δέ, ὅτι οὐδὲ μείζων ἢ τριπλάσιος· τριπλάσιος ἄρα ὁ <lb n="1799,001,,,,12,10,126"></lb> |
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15259 κύλινδρος τοῦ κώνου· ὥστε ὁ κῶνος τρίτον ἐστὶ μέρος τοῦ <lb n="1799,001,,,,12,10,127"></lb> |
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15260 κυλίνδρου.</s> <lb n="1799,001,,,,12,10,128"></lb> |
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15261 <s>Πᾶς ἄρα κῶνος κυλίνδρου τρίτον μέρος ἐστὶ τοῦ τὴν <lb n="1799,001,,,,12,10,129"></lb> |
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15262 αὐτὴν βάσιν ἔχοντος αὐτῷ καὶ ὕψος ἴσον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,10,130"></lb> |
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15263 <s>Οἱ ὑπὸ τὸ αὐτὸ ὕψος ὄντες κῶνοι καὶ κύλινδροι πρὸς <lb n="1799,001,,,,12,11,1"></lb> |
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15264 ἀλλήλους εἰσὶν ὡς αἱ βάσεις.</s> <lb n="1799,001,,,,12,11,2"></lb> |
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15265 <s>Ἔστωσαν ὑπὸ τὸ αὐτὸ ὕψος κῶνοι καὶ κύλινδροι, ὧν <lb n="1799,001,,,,12,11,3"></lb> |
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15266 βάσεις μὲν [εἰσιν] οἱ ΑΒΓΔ, ΕΖΗΘ κύκλοι, ἄξονες δὲ <lb n="1799,001,,,,12,11,4"></lb> |
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15267 οἱ ΚΛ, ΜΝ, διάμετροι δὲ τῶν βάσεων αἱ ΑΓ, ΕΗ· λέγω, <lb n="1799,001,,,,12,11,5"></lb> |
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15268 ὅτι ἐστὶν ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ κύκλον, <lb n="1799,001,,,,12,11,6"></lb> |
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15269 οὕτως ὁ ΑΛ κῶνος πρὸς τὸν ΕΝ κῶνον.</s> <s><pb></pb> <lb n="1799,001,,,,12,11,7"></lb> |
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15270 Εἰ γὰρ μή, ἔσται ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ <lb n="1799,001,,,,12,11,8"></lb> |
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15271 κύκλον, οὕτως ὁ ΑΛ κῶνος ἤτοι πρὸς ἔλασσόν τι τοῦ ΕΝ <lb n="1799,001,,,,12,11,9"></lb> |
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15272 κώνου στερεὸν ἢ πρὸς μεῖζον.</s> <s>ἔστω πρότερον πρὸς ἔλασ-<lb n="1799,001,,,,12,11,10"></lb> |
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15273 σον τὸ Ξ, καὶ ᾧ ἔλασσόν ἐστι τὸ Ξ στερεὸν τοῦ ΕΝ κώνου, <lb n="1799,001,,,,12,11,11"></lb> |
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15274 ἐκείνῳ ἴσον ἔστω τὸ Ψ στερεόν· ὁ ΕΝ κῶνος ἄρα ἴσος <lb n="1799,001,,,,12,11,12"></lb> |
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15275 ἐστὶ τοῖς Ξ, Ψ στερεοῖς.</s> <s>ἐγγεγράφθω εἰς τὸν ΕΖΗΘ <lb n="1799,001,,,,12,11,13"></lb> |
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15276 κύκλον τετράγωνον τὸ ΕΖΗΘ· τὸ ἄρα τετράγωνον μεῖ-<lb n="1799,001,,,,12,11,14"></lb> |
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15277 ζόν ἐστιν ἢ τὸ ἥμισυ τοῦ κύκλου.</s> <s>ἀνεστάτω ἀπὸ τοῦ ΕΖ <lb n="1799,001,,,,12,11,15"></lb> |
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15278 ΗΘ τετραγώνου πυραμὶς ἰσοϋψὴς τῷ κώνῳ· ἡ ἄρα ἀνα-<lb n="1799,001,,,,12,11,16"></lb> |
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15279 σταθεῖσα πυραμὶς μείζων ἐστὶν ἢ τὸ ἥμισυ τοῦ κώνου, <lb n="1799,001,,,,12,11,17"></lb> |
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15280 ἐπειδήπερ ἐὰν περιγράψωμεν περὶ τὸν κύκλον τετράγω-<lb n="1799,001,,,,12,11,18"></lb> |
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15281 νον, καὶ ἀπ' αὐτοῦ ἀναστήσωμεν πυραμίδα ἰσοϋψῆ τῷ <lb n="1799,001,,,,12,11,19"></lb> |
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15282 κώνῳ, ἡ ἐγγραφεῖσα πυραμὶς ἥμισύ ἐστι τῆς περιγρα-<lb n="1799,001,,,,12,11,20"></lb> |
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15283 φείσης· πρὸς ἀλλήλας γάρ εἰσιν ὡς αἱ βάσεις· ἐλάττων δὲ <lb n="1799,001,,,,12,11,21"></lb> |
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15284 ὁ κῶνος τῆς περιγραφείσης πυραμίδος.</s> <s>τετμήσθωσαν αἱ <lb n="1799,001,,,,12,11,22"></lb> |
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15285 ΕΖ, ΖΗ, ΗΘ, ΘΕ περιφέρειαι δίχα κατὰ τὰ Ο, Π, Ρ, <lb n="1799,001,,,,12,11,23"></lb> |
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15286 Σ σημεῖα, καὶ ἐπεζεύχθωσαν αἱ ΘΟ, ΟΕ, ΕΠ, ΠΖ, <lb n="1799,001,,,,12,11,24"></lb> |
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15287 ΖΡ, ΡΗ, ΗΣ, ΣΘ.</s> <s>ἕκαστον ἄρα τῶν ΘΟΕ, ΕΠΖ, <lb n="1799,001,,,,12,11,25"></lb> |
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15288 ΖΡΗ, ΗΣΘ τριγώνων μεῖζόν ἐστιν ἢ τὸ ἥμισυ τοῦ καθ' <lb n="1799,001,,,,12,11,26"></lb> |
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15289 ἑαυτὸ τμήματος τοῦ κύκλου.</s> <s>ἀνεστάτω ἐφ' ἑκάστου τῶν <lb n="1799,001,,,,12,11,27"></lb> |
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15290 ΘΟΕ, ΕΠΖ, ΖΡΗ, ΗΣΘ τριγώνων πυραμὶς ἰσοϋψὴς <lb n="1799,001,,,,12,11,28"></lb> |
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15291 τῷ κώνῳ· καὶ ἑκάστη ἄρα τῶν ἀνασταθεισῶν πυραμίδων <lb n="1799,001,,,,12,11,29"></lb> |
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15292 μείζων ἐστὶν ἢ τὸ ἥμισυ τοῦ καθ' ἑαυτὴν τμήματος τοῦ <pb></pb> <lb n="1799,001,,,,12,11,30"></lb> |
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15293 κώνου.</s> <s>τέμνοντες δὴ τὰς ὑπολειπομένας περιφερείας δίχα <lb n="1799,001,,,,12,11,31"></lb> |
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15294 καὶ ἐπιζευγνύντες εὐθείας καὶ ἀνιστάντες ἐπὶ ἑκάστου τῶν <lb n="1799,001,,,,12,11,32"></lb> |
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15295 τριγώνων πυραμίδας ἰσοϋψεῖς τῷ κώνῳ καὶ ἀεὶ τοῦτο <lb n="1799,001,,,,12,11,33"></lb> |
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15296 ποιοῦντες καταλείψομέν τινα ἀποτμήματα τοῦ κώνου, ἃ <lb n="1799,001,,,,12,11,34"></lb> |
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15297 ἔσται ἐλάσσονα τοῦ Ψ στερεοῦ.</s> <s>λελείφθω, καὶ ἔστω τὰ ἐπὶ <lb n="1799,001,,,,12,11,35"></lb> |
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15298 τῶν ΘΟΕ, ΕΠΖ, ΖΡΗ, ΗΣΘ· λοιπὴ ἄρα ἡ πυραμίς, <lb n="1799,001,,,,12,11,36"></lb> |
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15299 ἧς βάσις τὸ ΘΟΕΠΖΡΗΣ πολύγωνον, ὕψος δὲ τὸ αὐτὸ <lb n="1799,001,,,,12,11,37"></lb> |
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15300 τῷ κώνῳ, μείζων ἐστὶ τοῦ Ξ στερεοῦ.</s> <s>ἐγγεγράφθω καὶ <lb n="1799,001,,,,12,11,38"></lb> |
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15301 εἰς τὸν ΑΒΓΔ κύκλον τῷ ΘΟΕΠΖΡΗΣ πολυγώνῳ <lb n="1799,001,,,,12,11,39"></lb> |
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15302 ὅμοιόν τε καὶ ὁμοίως κείμενον πολύγωνον τὸ ΔΤΑΥΒ <lb n="1799,001,,,,12,11,40"></lb> |
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15303 ΦΓΧ, καὶ ἀνεστάτω ἐπ' αὐτοῦ πυραμὶς ἰσοϋψὴς τῷ ΑΛ <lb n="1799,001,,,,12,11,41"></lb> |
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15304 κώνῳ.</s> <s>ἐπεὶ οὖν ἐστιν ὡς τὸ ἀπὸ τῆς ΑΓ πρὸς τὸ ἀπὸ τῆς <lb n="1799,001,,,,12,11,42"></lb> |
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15305 ΕΗ, οὕτως τὸ ΔΤΑΥΒΦΓΧ πολύγωνον πρὸς τὸ ΘΟΕ <lb n="1799,001,,,,12,11,43"></lb> |
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15306 ΠΖΡΗΣ πολύγωνον, ὡς δὲ τὸ ἀπὸ τῆς ΑΓ πρὸς τὸ <lb n="1799,001,,,,12,11,44"></lb> |
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15307 ἀπὸ τῆς ΕΗ, οὕτως ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ <lb n="1799,001,,,,12,11,45"></lb> |
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15308 κύκλον, καὶ ὡς ἄρα ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ <pb></pb> <lb n="1799,001,,,,12,11,46"></lb> |
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15309 κύκλον, οὕτως τὸ ΔΤΑΥΒΦΓΧ πολύγωνον πρὸς τὸ <lb n="1799,001,,,,12,11,47"></lb> |
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15310 ΘΟΕΠΖΡΗΣ πολύγωνον.</s> <s>ὡς δὲ ὁ ΑΒΓΔ κύκλος πρὸς <lb n="1799,001,,,,12,11,48"></lb> |
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15311 τὸν ΕΖΗΘ κύκλον, οὕτως ὁ ΑΛ κῶνος πρὸς τὸ Ξ στερεόν, <lb n="1799,001,,,,12,11,49"></lb> |
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15312 ὡς δὲ τὸ ΔΤΑΥΒ*φΓΧ πολύγωνον πρὸς τὸ ΘΟΕΠΖ <lb n="1799,001,,,,12,11,50"></lb> |
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15313 ΡΗΣ πολύγωνον, οὕτως ἡ πυραμίς, ἧς βάσις μὲν τὸ <lb n="1799,001,,,,12,11,51"></lb> |
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15314 ΔΤΑΥΒΦΓΧ πολύγωνον, κορυφὴ δὲ τὸ Λ σημεῖον, πρὸς <lb n="1799,001,,,,12,11,52"></lb> |
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15315 τὴν πυραμίδα, ἧς βάσις μὲν τὸ ΘΟΕΠΖΡΗΣ πολύγω-<lb n="1799,001,,,,12,11,53"></lb> |
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15316 νον, κορυφὴ δὲ τὸ Ν σημεῖον.</s> <s>καὶ ὡς ἄρα ὁ ΑΛ κῶνος <lb n="1799,001,,,,12,11,54"></lb> |
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15317 πρὸς τὸ Ξ στερεόν, οὕτως ἡ πυραμίς, ἧς βάσις μὲν τὸ <lb n="1799,001,,,,12,11,55"></lb> |
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15318 ΔΤΑΥΒΦΓΧ πολύγωνον, κορυφὴ δὲ τὸ Λ σημεῖον, πρὸς <lb n="1799,001,,,,12,11,56"></lb> |
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15319 τὴν πυραμίδα, ἧς βάσις μὲν τὸ ΘΟΕΠΖΡΗΣ πολύ-<lb n="1799,001,,,,12,11,57"></lb> |
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15320 γωνον, κορυφὴ δὲ τὸ Ν σημεῖον· ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ <lb n="1799,001,,,,12,11,58"></lb> |
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15321 ΑΛ κῶνος πρὸς τὴν ἐν αὐτῷ πυραμίδα, οὕτως τὸ Ξ στερεὸν <lb n="1799,001,,,,12,11,59"></lb> |
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15322 πρὸς τὴν ἐν τῷ ΕΝ κώνῳ πυραμίδα.</s> <s>μείζων δὲ ὁ ΑΛ <lb n="1799,001,,,,12,11,60"></lb> |
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15323 κῶνος τῆς ἐν αὐτῷ πυραμίδος· μεῖζον ἄρα καὶ τὸ Ξ στε-<lb n="1799,001,,,,12,11,61"></lb> |
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15324 ρεὸν τῆς ἐν τῷ ΕΝ κώνῳ πυραμίδος.</s> <s>ἀλλὰ καὶ ἔλασσον· <lb n="1799,001,,,,12,11,62"></lb> |
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15325 ὅπερ ἄτοπον.</s> <s>οὐκ ἄρα ἐστὶν ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν <lb n="1799,001,,,,12,11,63"></lb> |
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15326 ΕΖΗΘ κύκλον, οὕτως ὁ ΑΛ κῶνος πρὸς ἔλασσόν τι τοῦ <lb n="1799,001,,,,12,11,64"></lb> |
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15327 ΕΝ κώνου στερεόν.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι οὐδέ ἐστιν <lb n="1799,001,,,,12,11,65"></lb> |
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15328 ὡς ὁ ΕΖΗΘ κύκλος πρὸς τὸν ΑΒΓΔ κύκλον, οὕτως ὁ <lb n="1799,001,,,,12,11,66"></lb> |
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15329 ΕΝ κῶνος πρὸς ἔλασσόν τι τοῦ ΑΛ κώνου στερεόν.</s> <lb n="1799,001,,,,12,11,67"></lb> |
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15330 <s>Λέγω δή, ὅτι οὐδέ ἐστιν ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν <lb n="1799,001,,,,12,11,68"></lb> |
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15331 ΕΖΗΘ κύκλον, οὕτως ὁ ΑΛ κῶνος πρὸς μεῖζόν τι τοῦ <lb n="1799,001,,,,12,11,69"></lb> |
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15332 ΕΝ κώνου στερεόν.</s> <lb n="1799,001,,,,12,11,70"></lb> |
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15333 <s>Εἰ γὰρ δυνατόν, ἔστω πρὸς μεῖζον τὸ Ξ· ἀνάπαλιν ἄρα <lb n="1799,001,,,,12,11,71"></lb> |
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15334 ἐστὶν ὡς ὁ ΕΖΗΘ κύκλος πρὸς τὸν ΑΒΓΔ κύκλον, οὕ-<lb n="1799,001,,,,12,11,72"></lb> |
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15335 τως τὸ Ξ στερεὸν πρὸς τὸν ΑΛ κῶνον.</s> <s>ἀλλ' ὡς τὸ Ξ στε-<lb n="1799,001,,,,12,11,73"></lb> |
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15336 ρεὸν πρὸς τὸν ΑΛ κῶνον, οὕτως ὁ ΕΝ κῶνος πρὸς ἔλασ- <pb></pb><lb n="1799,001,,,,12,11,74"></lb> |
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15337 σόν τι τοῦ ΑΛ κώνου στερεόν· καὶ ὡς ἄρα ὁ ΕΖΗΘ <lb n="1799,001,,,,12,11,75"></lb> |
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15338 κύκλος πρὸς τὸν ΑΒΓΔ κύκλον, οὕτως ὁ ΕΝ κῶνος πρὸς <lb n="1799,001,,,,12,11,76"></lb> |
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15339 ἔλασσόν τι τοῦ ΑΛ κώνου στερεόν· ὅπερ ἀδύνατον ἐδείχθη.</s> <lb n="1799,001,,,,12,11,77"></lb> |
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15340 <s>οὐκ ἄρα ἐστὶν ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ <lb n="1799,001,,,,12,11,78"></lb> |
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15341 κύκλον, οὕτως ὁ ΑΛ κῶνος πρὸς μεῖζόν τι τοῦ ΕΝ <lb n="1799,001,,,,12,11,79"></lb> |
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15342 κώνου στερεόν.</s> <s>ἐδείχθη δέ, ὅτι οὐδὲ πρὸς ἔλασσον· ἔστιν <lb n="1799,001,,,,12,11,80"></lb> |
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15343 ἄρα ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ κύκλον, οὕτως <lb n="1799,001,,,,12,11,81"></lb> |
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15344 ὁ ΑΛ κῶνος πρὸς τὸν ΕΝ κῶνον.</s> <lb n="1799,001,,,,12,11,82"></lb> |
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15345 <s>Ἀλλ' ὡς ὁ κῶνος πρὸς τὸν κῶνον, ὁ κύλινδρος πρὸς τὸν <lb n="1799,001,,,,12,11,83"></lb> |
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15346 κύλινδρον· τριπλασίων γὰρ ἑκάτερος ἑκατέρου.</s> <s>καὶ ὡς <lb n="1799,001,,,,12,11,84"></lb> |
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15347 ἄρα ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ κύκλον, οὕτως οἱ <lb n="1799,001,,,,12,11,85"></lb> |
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15348 ἐπ' αὐτῶν ἰσοϋψεῖς [τοῖς κώνοισ] κύλινδροι.</s> <lb n="1799,001,,,,12,11,86"></lb> |
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15349 <s>Οἱ ἄρα ὑπὸ τὸ αὐτὸ ὕψος ὄντες κῶνοι καὶ κύλινδροι <lb n="1799,001,,,,12,11,87"></lb> |
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15350 πρὸς ἀλλήλους εἰσὶν ὡς αἱ βάσεις· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,11,88"></lb> |
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15351 <s>Οἱ ὅμοιοι κῶνοι καὶ κύλινδροι πρὸς ἀλλήλους ἐν τρι-<lb n="1799,001,,,,12,12,1"></lb> |
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15352 πλασίονι λόγῳ εἰσὶ τῶν ἐν ταῖς βάσεσι διαμέτρων.</s> <lb n="1799,001,,,,12,12,2"></lb> |
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15353 <s>Ἔστωσαν ὅμοιοι κῶνοι καὶ κύλινδροι, ὧν βάσεις μὲν <lb n="1799,001,,,,12,12,3"></lb> |
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15354 οἱ ΑΒΓΔ, ΕΖΗΘ κύκλοι, διάμετροι δὲ τῶν βάσεων αἱ <lb n="1799,001,,,,12,12,4"></lb> |
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15355 ΒΔ, ΖΘ, ἄξονες δὲ τῶν κώνων καὶ κυλίνδρων οἱ ΚΛ, <lb n="1799,001,,,,12,12,5"></lb> |
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15356 ΜΝ· λέγω, ὅτι ὁ κῶνος, οὗ βάσις μὲν [ἐστιν] ὁ ΑΒΓΔ <lb n="1799,001,,,,12,12,6"></lb> |
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15357 κύκλος, κορυφὴ δὲ τὸ Λ σημεῖον, πρὸς τὸν κῶνον, οὗ <lb n="1799,001,,,,12,12,7"></lb> |
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15358 βάσις μὲν [ἐστιν] ὁ ΕΖΗΘ κύκλος, κορυφὴ δὲ τὸ Ν σημεῖ-<lb n="1799,001,,,,12,12,8"></lb> |
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15359 ον, τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΔ πρὸς τὴν ΖΘ.</s> <s><pb></pb> <lb n="1799,001,,,,12,12,9"></lb> |
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15360 Εἰ γὰρ μὴ ἔχει ὁ ΑΒΓΔΛ κῶνος πρὸς τὸν ΕΖΗΘΝ <lb n="1799,001,,,,12,12,10"></lb> |
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15361 κῶνον τριπλασίονα λόγον ἤπερ ἡ ΒΔ πρὸς τὴν ΖΘ, ἕξει <lb n="1799,001,,,,12,12,11"></lb> |
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15362 ὁ ΑΒΓΔΛ κῶνος ἢ πρὸς ἔλασσόν τι τοῦ ΕΖΗΘΝ <lb n="1799,001,,,,12,12,12"></lb> |
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15363 κώνου στερεὸν τριπλασίονα λόγον ἢ πρὸς μεῖζον.</s> <s>ἐχέτω <lb n="1799,001,,,,12,12,13"></lb> |
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15364 πρότερον πρὸς ἔλασσον τὸ Ξ, καὶ ἐγγεγράφθω εἰς τὸν <lb n="1799,001,,,,12,12,14"></lb> |
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15365 ΕΖΗΘ κύκλον τετράγωνον τὸ ΕΖΗΘ· τὸ ἄρα ΕΖΗΘ <lb n="1799,001,,,,12,12,15"></lb> |
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15366 τετράγωνον μεῖζόν ἐστιν ἢ τὸ ἥμισυ τοῦ ΕΖΗΘ κύκλου.</s> <lb n="1799,001,,,,12,12,16"></lb> |
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15367 <s>καὶ ἀνεστάτω ἐπὶ τοῦ ΕΖΗΘ τετραγώνου πυραμὶς τὴν <lb n="1799,001,,,,12,12,17"></lb> |
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15368 αὐτὴν κορυφὴν ἔχουσα τῷ κώνῳ· ἡ ἄρα ἀνασταθεῖσα <lb n="1799,001,,,,12,12,18"></lb> |
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15369 πυραμὶς μείζων ἐστὶν ἢ τὸ ἥμισυ μέρος τοῦ κώνου.</s> <s>τετμή-<lb n="1799,001,,,,12,12,19"></lb> |
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15370 σθωσαν δὴ αἱ ΕΖ, ΖΗ, ΗΘ, ΘΕ περιφέρειαι δίχα κατὰ <lb n="1799,001,,,,12,12,20"></lb> |
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15371 τὰ Ο, Π, Ρ, Σ σημεῖα, καὶ ἐπεζεύχθωσαν αἱ ΕΟ, ΟΖ, <lb n="1799,001,,,,12,12,21"></lb> |
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15372 ΖΠ, ΠΗ, ΗΡ, ΡΘ, ΘΣ, ΣΕ.</s> <s>καὶ ἕκαστον ἄρα τῶν <lb n="1799,001,,,,12,12,22"></lb> |
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15373 ΕΟΖ, ΖΠΗ, ΗΡΘ, ΘΣΕ τριγώνων μεῖζόν ἐστιν ἢ τὸ <pb></pb> <lb n="1799,001,,,,12,12,23"></lb> |
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15374 ἥμισυ μέρος τοῦ καθ' ἑαυτὸ τμήματος τοῦ ΕΖΗΘ <lb n="1799,001,,,,12,12,24"></lb> |
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15375 κύκλου.</s> <s>καὶ ἀνεστάτω ἐφ' ἑκάστου τῶν ΕΟΖ, ΖΠΗ, <lb n="1799,001,,,,12,12,25"></lb> |
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15376 ΗΡΘ, ΘΣΕ τριγώνων πυραμὶς τὴν αὐτὴν κορυφὴν <lb n="1799,001,,,,12,12,26"></lb> |
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15377 ἔχουσα τῷ κώνῳ· καὶ ἑκάστη ἄρα τῶν ἀνασταθεισῶν <lb n="1799,001,,,,12,12,27"></lb> |
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15378 πυραμίδων μείζων ἐστὶν ἢ τὸ ἥμισυ μέρος τοῦ καθ' ἑαυ-<lb n="1799,001,,,,12,12,28"></lb> |
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15379 τὴν τμήματος τοῦ κώνου.</s> <s>τέμνοντες δὴ τὰς ὑπολειπομέ-<lb n="1799,001,,,,12,12,29"></lb> |
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15380 νας περιφερείας δίχα καὶ ἐπιζευγνύντες εὐθείας καὶ ἀνιστάν-<lb n="1799,001,,,,12,12,30"></lb> |
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15381 τες ἐφ' ἑκάστου τῶν τριγώνων πυραμίδας τὴν αὐτὴν <lb n="1799,001,,,,12,12,31"></lb> |
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15382 κορυφὴν ἐχούσας τῷ κώνῳ καὶ τοῦτο ἀεὶ ποιοῦντες κατα-<lb n="1799,001,,,,12,12,32"></lb> |
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15383 λείψομέν τινα ἀποτμήματα τοῦ κώνου, ἃ ἔσται ἐλάσσονα <lb n="1799,001,,,,12,12,33"></lb> |
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15384 τῆς ὑπεροχῆς, ᾗ ὑπερέχει ὁ ΕΖΗΘΝ κῶνος τοῦ Ξ στε-<lb n="1799,001,,,,12,12,34"></lb> |
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15385 ρεοῦ.</s> <s>λελείφθω, καὶ ἔστω τὰ ἐπὶ τῶν ΕΟ, ΟΖ, ΖΠ, ΠΗ, <lb n="1799,001,,,,12,12,35"></lb> |
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15386 ΗΡ, ΡΘ, ΘΣ, ΣΕ· λοιπὴ ἄρα ἡ πυραμίς, ἧς βάσις μέν <lb n="1799,001,,,,12,12,36"></lb> |
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15387 ἐστι τὸ ΕΟΖΠΗΡΘΣ πολύγωνον, κορυφὴ δὲ τὸ Ν <lb n="1799,001,,,,12,12,37"></lb> |
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15388 σημεῖον, μείζων ἐστὶ τοῦ Ξ στερεοῦ.</s> <s>ἐγγεγράφθω καὶ εἰς <lb n="1799,001,,,,12,12,38"></lb> |
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15389 τὸν ΑΒΓΔ κύκλον τῷ ΕΟΖΠΗΡΘΣ πολυγώνῳ ὅμοιόν <lb n="1799,001,,,,12,12,39"></lb> |
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15390 τε καὶ ὁμοίως κείμενον πολύγωνον τὸ ΑΤΒΥΓΦΔΧ, <lb n="1799,001,,,,12,12,40"></lb> |
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15391 καὶ ἀνεστάτω ἐπὶ τοῦ ΑΤΒΥΓΦΔΧ πολυγώνου πυρα-<lb n="1799,001,,,,12,12,41"></lb> |
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15392 μὶς τὴν αὐτὴν κορυφὴν ἔχουσα τῷ κώνῳ, καὶ τῶν μὲν <lb n="1799,001,,,,12,12,42"></lb> |
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15393 περιεχόντων τὴν πυραμίδα, ἧς βάσις μέν ἐστι τὸ ΑΤΒΥ <lb n="1799,001,,,,12,12,43"></lb> |
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15394 ΓΦΔΧ πολύγωνον, κορυφὴ δὲ τὸ Λ σημεῖον, ἓν τρίγω-<lb n="1799,001,,,,12,12,44"></lb> |
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15395 νον ἔστω τὸ ΛΒΤ, τῶν δὲ περιεχόντων τὴν πυραμίδα, ἧς <lb n="1799,001,,,,12,12,45"></lb> |
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15396 βάσις μέν ἐστι τὸ ΕΟΖΠΗΡΘΣ πολύγωνον, κορυφὴ δὲ <lb n="1799,001,,,,12,12,46"></lb> |
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15397 τὸ Ν σημεῖον, ἓν τρίγωνον ἔστω τὸ ΝΖΟ, καὶ ἐπεζεύχθω-<lb n="1799,001,,,,12,12,47"></lb> |
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15398 σαν αἱ ΚΤ, ΜΟ.</s> <s>καὶ ἐπεὶ ὅμοιός ἐστιν ὁ ΑΒΓΔΛ κῶνος <lb n="1799,001,,,,12,12,48"></lb> |
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15399 τῷ ΕΖΗΘΝ κώνῳ, ἔστιν ἄρα ὡς ἡ ΒΔ πρὸς τὴν ΖΘ, <lb n="1799,001,,,,12,12,49"></lb> |
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15400 οὕτως ὁ ΚΛ ἄξων πρὸς τὸν ΜΝ ἄξονα.</s> <s>ὡς δὲ ἡ ΒΔ πρὸς <pb></pb> <lb n="1799,001,,,,12,12,50"></lb> |
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15401 τὴν ΖΘ, οὕτως ἡ ΒΚ πρὸς τὴν ΖΜ· καὶ ὡς ἄρα ἡ ΒΚ <lb n="1799,001,,,,12,12,51"></lb> |
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15402 πρὸς τὴν ΖΜ, οὕτως ἡ ΚΛ πρὸς τὴν ΜΝ.</s> <s>καὶ ἐναλλὰξ <lb n="1799,001,,,,12,12,52"></lb> |
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15403 ὡς ἡ ΒΚ πρὸς τὴν ΚΛ, οὕτως ἡ ΖΜ πρὸς τὴν ΜΝ.</s> <s>καὶ <lb n="1799,001,,,,12,12,53"></lb> |
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15404 περὶ ἴσας γωνίας τὰς ὑπὸ ΒΚΛ, ΖΜΝ αἱ πλευραὶ ἀνάλο-<lb n="1799,001,,,,12,12,54"></lb> |
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15405 γόν εἰσιν· ὅμοιον ἄρα ἐστὶ τὸ ΒΚΛ τρίγωνον τῷ ΖΜΝ <lb n="1799,001,,,,12,12,55"></lb> |
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15406 τριγώνῳ.</s> <s>πάλιν, ἐπεί ἐστιν ὡς ἡ ΒΚ πρὸς τὴν ΚΤ, οὕτως <lb n="1799,001,,,,12,12,56"></lb> |
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15407 ἡ ΖΜ πρὸς τὴν ΜΟ, καὶ περὶ ἴσας γωνίας τὰς ὑπὸ ΒΚΤ, <lb n="1799,001,,,,12,12,57"></lb> |
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15408 ΖΜΟ, ἐπειδήπερ, ὃ μέρος ἐστὶν ἡ ὑπὸ ΒΚΤ γωνία τῶν <lb n="1799,001,,,,12,12,58"></lb> |
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15409 πρὸς τῷ Κ κέντρῳ τεσσάρων ὀρθῶν, τὸ αὐτὸ μέρος ἐστὶ <lb n="1799,001,,,,12,12,59"></lb> |
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15410 καὶ ἡ ὑπὸ ΖΜΟ γωνία τῶν πρὸς τῷ Μ κέντρῳ τεσσάρων <lb n="1799,001,,,,12,12,60"></lb> |
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15411 ὀρθῶν· ἐπεὶ οὖν περὶ ἴσας γωνίας αἱ πλευραὶ ἀνάλογόν <lb n="1799,001,,,,12,12,61"></lb> |
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15412 εἰσιν, ὅμοιον ἄρα ἐστὶ τὸ ΒΚΤ τρίγωνον τῷ ΖΜΟ τρι-<lb n="1799,001,,,,12,12,62"></lb> |
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15413 γώνῳ.</s> <s>πάλιν, ἐπεὶ ἐδείχθη ὡς ἡ ΒΚ πρὸς τὴν ΚΛ, οὕτως <lb n="1799,001,,,,12,12,63"></lb> |
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15414 ἡ ΖΜ πρὸς τὴν ΜΝ, ἴση δὲ ἡ μὲν ΒΚ τῇ ΚΤ, ἡ δὲ ΖΜ <lb n="1799,001,,,,12,12,64"></lb> |
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15415 τῇ ΟΜ, ἔστιν ἄρα ὡς ἡ ΤΚ πρὸς τὴν ΚΛ, οὕτως ἡ ΟΜ <lb n="1799,001,,,,12,12,65"></lb> |
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15416 πρὸς τὴν ΜΝ.</s> <s>καὶ περὶ ἴσας γωνίας τὰς ὑπὸ ΤΚΛ, <lb n="1799,001,,,,12,12,66"></lb> |
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15417 ΟΜΝ· ὀρθαὶ γάρ· αἱ πλευραὶ ἀνάλογόν εἰσιν· ὅμοιον <lb n="1799,001,,,,12,12,67"></lb> |
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15418 ἄρα ἐστὶ τὸ ΛΚΤ τρίγωνον τῷ ΝΜΟ τριγώνῳ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,12,12,68"></lb> |
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15419 διὰ τὴν ὁμοιότητα τῶν ΛΚΒ, ΝΜΖ τριγώνων ἐστὶν ὡς <lb n="1799,001,,,,12,12,69"></lb> |
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15420 ἡ ΛΒ πρὸς τὴν ΒΚ, οὕτως ἡ ΝΖ πρὸς τὴν ΖΜ, διὰ δὲ <lb n="1799,001,,,,12,12,70"></lb> |
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15421 τὴν ὁμοιότητα τῶν ΒΚΤ, ΖΜΟ τριγώνων ἐστὶν ὡς ἡ <lb n="1799,001,,,,12,12,71"></lb> |
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15422 ΚΒ πρὸς τὴν ΒΤ, οὕτως ἡ ΜΖ πρὸς τὴν ΖΟ, δι' ἴσου <pb></pb> <lb n="1799,001,,,,12,12,72"></lb> |
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15423 ἄρα ὡς ἡ ΛΒ πρὸς τὴν ΒΤ, οὕτως ἡ ΝΖ πρὸς τὴν ΖΟ.</s> <lb n="1799,001,,,,12,12,73"></lb> |
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15424 <s>πάλιν, ἐπεὶ διὰ τὴν ὁμοιότητα τῶν ΛΤΚ, ΝΟΜ τριγώ-<lb n="1799,001,,,,12,12,74"></lb> |
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15425 νων ἐστὶν ὡς ἡ ΛΤ πρὸς τὴν ΤΚ, οὕτως ἡ ΝΟ πρὸς τὴν <lb n="1799,001,,,,12,12,75"></lb> |
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15426 ΟΜ, διὰ δὲ τὴν ὁμοιότητα τῶν ΤΚΒ, ΟΜΖ τριγώνων <lb n="1799,001,,,,12,12,76"></lb> |
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15427 ἐστὶν ὡς ἡ ΚΤ πρὸς τὴν ΤΒ, οὕτως ἡ ΜΟ πρὸς τὴν ΟΖ, <lb n="1799,001,,,,12,12,77"></lb> |
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15428 δι' ἴσου ἄρα ὡς ἡ ΛΤ πρὸς τὴν ΤΒ, οὕτως ἡ ΝΟ πρὸς <lb n="1799,001,,,,12,12,78"></lb> |
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15429 τὴν ΟΖ.</s> <s>ἐδείχθη δὲ καὶ ὡς ἡ ΤΒ πρὸς τὴν ΒΛ, οὕτως ἡ <lb n="1799,001,,,,12,12,79"></lb> |
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15430 ΟΖ πρὸς τὴν ΖΝ.</s> <s>δι' ἴσου ἄρα ὡς ἡ ΤΛ πρὸς τὴν ΛΒ, <lb n="1799,001,,,,12,12,80"></lb> |
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15431 οὕτως ἡ ΟΝ πρὸς τὴν ΝΖ.</s> <s>τῶν ΛΤΒ, ΝΟΖ ἄρα τριγώ-<lb n="1799,001,,,,12,12,81"></lb> |
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15432 νων ἀνάλογόν εἰσιν αἱ πλευραί· ἰσογώνια ἄρα ἐστὶ τὰ <lb n="1799,001,,,,12,12,82"></lb> |
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15433 ΛΤΒ, ΝΟΖ τρίγωνα· ὥστε καὶ ὅμοια.</s> <s>καὶ πυραμὶς ἄρα, <lb n="1799,001,,,,12,12,83"></lb> |
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15434 ἧς βάσις μὲν τὸ ΒΚΤ τρίγωνον, κορυφὴ δὲ τὸ Λ σημεῖον, <lb n="1799,001,,,,12,12,84"></lb> |
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15435 ὁμοία ἐστὶ πυραμίδι, ἧς βάσις μὲν τὸ ΖΜΟ τρίγωνον, <lb n="1799,001,,,,12,12,85"></lb> |
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15436 κορυφὴ δὲ τὸ Ν σημεῖον· ὑπὸ γὰρ ὁμοίων ἐπιπέδων περι-<lb n="1799,001,,,,12,12,86"></lb> |
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15437 έχονται ἴσων τὸ πλῆθος.</s> <s>αἱ δὲ ὅμοιαι πυραμίδες καὶ τριγώ-<lb n="1799,001,,,,12,12,87"></lb> |
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15438 νους ἔχουσαι βάσεις ἐν τριπλασίονι λόγῳ εἰσὶ τῶν ὁμο-<lb n="1799,001,,,,12,12,88"></lb> |
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15439 λόγων πλευρῶν.</s> <s>ἡ ἄρα ΒΚΤΛ πυραμὶς πρὸς τὴν ΖΜΟΝ <lb n="1799,001,,,,12,12,89"></lb> |
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15440 πυραμίδα τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΚ πρὸς τὴν <lb n="1799,001,,,,12,12,90"></lb> |
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15441 ΖΜ.</s> <s>ὁμοίως δὴ ἐπιζευγνύντες ἀπὸ τῶν Α, Χ, Δ, Φ, Γ, <lb n="1799,001,,,,12,12,91"></lb> |
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15442 Υ ἐπὶ τὸ Κ εὐθείας καὶ ἀπὸ τῶν Ε, Σ, Θ, Ρ, Η, Π ἐπὶ τὸ <lb n="1799,001,,,,12,12,92"></lb> |
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15443 Μ καὶ ἀνιστάντες ἐφ' ἑκάστου τῶν τριγώνων πυραμίδας <lb n="1799,001,,,,12,12,93"></lb> |
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15444 τὴν αὐτὴν κορυφὴν ἐχούσας τοῖς κώνοις δείξομεν, ὅτι καὶ <lb n="1799,001,,,,12,12,94"></lb> |
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15445 ἑκάστη τῶν ὁμοταγῶν πυραμίδων πρὸς ἑκάστην ὁμοταγῆ <lb n="1799,001,,,,12,12,95"></lb> |
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15446 πυραμίδα τριπλασίονα λόγον ἕξει ἤπερ ἡ ΒΚ ὁμόλογος <lb n="1799,001,,,,12,12,96"></lb> |
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15447 πλευρὰ πρὸς τὴν ΖΜ ὁμόλογον πλευράν, τουτέστιν ἤπερ <lb n="1799,001,,,,12,12,97"></lb> |
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15448 ἡ ΒΔ πρὸς τὴν ΖΘ.</s> <s>καὶ ὡς ἓν τῶν ἡγουμένων πρὸς ἓν τῶν <pb></pb> <lb n="1799,001,,,,12,12,98"></lb> |
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15449 ἑπομένων, οὕτως ἅπαντα τὰ ἡγούμενα πρὸς ἅπαντα τὰ <lb n="1799,001,,,,12,12,99"></lb> |
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15450 ἑπόμενα· ἔστιν ἄρα καὶ ὡς ἡ ΒΚΤΛ πυραμὶς πρὸς τὴν <lb n="1799,001,,,,12,12,100"></lb> |
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15451 ΖΜΟΝ πυραμίδα, οὕτως ἡ ὅλη πυραμίς, ἧς βάσις τὸ <lb n="1799,001,,,,12,12,101"></lb> |
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15452 ΑΤΒΥΓΦΔΧ πολύγωνον, κορυφὴ δὲ τὸ Λ σημεῖον, <lb n="1799,001,,,,12,12,102"></lb> |
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15453 πρὸς τὴν ὅλην πυραμίδα, ἧς βάσις μὲν τὸ ΕΟΖΠΗΡΘΣ <lb n="1799,001,,,,12,12,103"></lb> |
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15454 πολύγωνον, κορυφὴ δὲ τὸ Ν σημεῖον· ὥστε καὶ πυραμίς, <lb n="1799,001,,,,12,12,104"></lb> |
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15455 ἧς βάσις μὲν τὸ ΑΤΒΥΓΦΔΧ, κορυφὴ δὲ τὸ Λ, πρὸς <lb n="1799,001,,,,12,12,105"></lb> |
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15456 τὴν πυραμίδα, ἧς βάσις [μὲν] τὸ ΕΟΖΠΗΡΘΣ πολύ-<lb n="1799,001,,,,12,12,106"></lb> |
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15457 γωνον, κορυφὴ δὲ τὸ Ν σημεῖον, τριπλασίονα λόγον ἔχει <lb n="1799,001,,,,12,12,107"></lb> |
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15458 ἤπερ ἡ ΒΔ πρὸς τὴν ΖΘ.</s> <s>ὑπόκειται δὲ καὶ ὁ κῶνος, οὗ <lb n="1799,001,,,,12,12,108"></lb> |
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15459 βάσις [μὲν] ὁ ΑΒΓΔ κύκλος, κορυφὴ δὲ τὸ Λ σημεῖον, <lb n="1799,001,,,,12,12,109"></lb> |
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15460 πρὸς τὸ Ξ στερεὸν τριπλασίονα λόγον ἔχων ἤπερ ἡ ΒΔ <lb n="1799,001,,,,12,12,110"></lb> |
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15461 πρὸς τὴν ΖΘ· ἔστιν ἄρα ὡς ὁ κῶνος, οὗ βάσις μέν ἐστιν <lb n="1799,001,,,,12,12,111"></lb> |
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15462 ὁ ΑΒΓΔ κύκλος, κορυφὴ δὲ τὸ Λ, πρὸς τὸ Ξ στερεόν, <lb n="1799,001,,,,12,12,112"></lb> |
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15463 οὕτως ἡ πυραμίς, ἧς βάσις μὲν τὸ ΑΤΒΥΓΦΔΧ [πολύ-<lb n="1799,001,,,,12,12,113"></lb> |
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15464 γωνον], κορυφὴ δὲ τὸ Λ, πρὸς τὴν πυραμίδα, ἧς βάσις <lb n="1799,001,,,,12,12,114"></lb> |
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15465 μέν ἐστι τὸ ΕΟΖΠΗΡΘΣ πολύγωνον, κορυφὴ δὲ τὸ Ν· <lb n="1799,001,,,,12,12,115"></lb> |
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15466 ἐναλλὰξ ἄρα, ὡς ὁ κῶνος, οὗ βάσις μὲν ὁ ΑΒΓΔ κύκλος, <lb n="1799,001,,,,12,12,116"></lb> |
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15467 κορυφὴ δὲ τὸ Λ, πρὸς τὴν ἐν αὐτῷ πυραμίδα, ἧς βάσις <lb n="1799,001,,,,12,12,117"></lb> |
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15468 μὲν τὸ ΑΤΒΥΓΦΔΧ πολύγωνον, κορυφὴ δὲ τὸ Λ, <lb n="1799,001,,,,12,12,118"></lb> |
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15469 οὕτως τὸ Ξ [στερεὸν] πρὸς τὴν πυραμίδα, ἧς βάσις μέν <lb n="1799,001,,,,12,12,119"></lb> |
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15470 ἐστι τὸ ΕΟΖΠΗΡΘΣ πολύγωνον, κορυφὴ δὲ τὸ Ν.</s> <s>μεί-<lb n="1799,001,,,,12,12,120"></lb> |
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15471 ζων δὲ ὁ εἰρημένος κῶνος τῆς ἐν αὐτῷ πυραμίδος· ἐμ-<lb n="1799,001,,,,12,12,121"></lb> |
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15472 περιέχει γὰρ αὐτήν.</s> <s>μεῖζον ἄρα καὶ τὸ Ξ στερεὸν τῆς πυρα-<lb n="1799,001,,,,12,12,122"></lb> |
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15473 μίδος, ἧς βάσις μέν ἐστι τὸ ΕΟΖΠΗΡΘΣ πολύγωνον, <lb n="1799,001,,,,12,12,123"></lb> |
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15474 κορυφὴ δὲ τὸ Ν.</s> <s>ἀλλὰ καὶ ἔλαττον· ὅπερ ἐστὶν ἀδύνατον.</s> <lb n="1799,001,,,,12,12,124"></lb> |
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15475 <s>οὐκ ἄρα ὁ κῶνος, οὗ βάσις ὁ ΑΒΓΔ κύκλος, κορυφὴ δὲ <pb></pb> <lb n="1799,001,,,,12,12,125"></lb> |
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15476 τὸ Λ [σημεῖον], πρὸς ἔλαττόν τι τοῦ κώνου στερεόν, οὗ <lb n="1799,001,,,,12,12,126"></lb> |
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15477 βάσις μὲν ὁ ΕΖΗΘ κύκλος, κορυφὴ δὲ τὸ Ν σημεῖον, τρι-<lb n="1799,001,,,,12,12,127"></lb> |
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15478 πλασίονα λόγον ἔχει ἤπερ ἡ ΒΔ πρὸς τὴν ΖΘ.</s> <s>ὁμοίως <lb n="1799,001,,,,12,12,128"></lb> |
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15479 δὴ δείξομεν, ὅτι οὐδὲ ὁ ΕΖΗΘΝ κῶνος πρὸς ἔλαττόν τι <lb n="1799,001,,,,12,12,129"></lb> |
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15480 τοῦ ΑΒΓΔΛ κώνου στερεὸν τριπλασίονα λόγον ἔχει <lb n="1799,001,,,,12,12,130"></lb> |
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15481 ἤπερ ἡ ΖΘ πρὸς τὴν ΒΔ.</s> <lb n="1799,001,,,,12,12,131"></lb> |
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15482 <s>Λέγω δή, ὅτι οὐδὲ ὁ ΑΒΓΔΛ κῶνος πρὸς μεῖζόν τι <lb n="1799,001,,,,12,12,132"></lb> |
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15483 τοῦ ΕΖΗΘΝ κώνου στερεὸν τριπλασίονα λόγον ἔχει <lb n="1799,001,,,,12,12,133"></lb> |
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15484 ἤπερ ἡ ΒΔ πρὸς τὴν ΖΘ.</s> <lb n="1799,001,,,,12,12,134"></lb> |
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15485 <s>Εἰ γὰρ δυνατόν, ἐχέτω πρὸς μεῖζον τὸ Ξ.</s> <s>ἀνάπαλιν ἄρα <lb n="1799,001,,,,12,12,135"></lb> |
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15486 τὸ Ξ στερεὸν πρὸς τὸν ΑΒΓΔΛ κῶνον τριπλασίονα <lb n="1799,001,,,,12,12,136"></lb> |
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15487 λόγον ἔχει ἤπερ ἡ ΖΘ πρὸς τὴν ΒΔ.</s> <s>ὡς δὲ τὸ Ξ στερεὸν <lb n="1799,001,,,,12,12,137"></lb> |
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15488 πρὸς τὸν ΑΒΓΔΛ κῶνον, οὕτως ὁ ΕΖΗΘΝ κῶνος πρὸς <lb n="1799,001,,,,12,12,138"></lb> |
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15489 ἔλαττόν τι τοῦ ΑΒΓΔΛ κώνου στερεόν.</s> <s>καὶ ὁ ΕΖΗΘΝ <lb n="1799,001,,,,12,12,139"></lb> |
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15490 ἄρα κῶνος πρὸς ἔλαττόν τι τοῦ ΑΒΓΔΛ κώνου στερεὸν <lb n="1799,001,,,,12,12,140"></lb> |
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15491 τριπλασίονα λόγον ἔχει ἤπερ ἡ ΖΘ πρὸς τὴν ΒΔ· ὅπερ <lb n="1799,001,,,,12,12,141"></lb> |
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15492 ἀδύνατον ἐδείχθη.</s> <s>οὐκ ἄρα ὁ ΑΒΓΔΛ κῶνος πρὸς μεῖζόν <lb n="1799,001,,,,12,12,142"></lb> |
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15493 τι τοῦ ΕΖΗΘΝ κώνου στερεὸν τριπλασίονα λόγον ἔχει <lb n="1799,001,,,,12,12,143"></lb> |
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15494 ἤπερ ἡ ΒΔ πρὸς τὴν ΖΘ.</s> <s>ἐδείχθη δέ, ὅτι οὐδὲ πρὸς ἔλατ-<lb n="1799,001,,,,12,12,144"></lb> |
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15495 τον.</s> <s>ὁ ΑΒΓΔΛ ἄρα κῶνος πρὸς τὸν ΕΖΗΘΝ κῶνον <lb n="1799,001,,,,12,12,145"></lb> |
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15496 τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΔ πρὸς τὴν ΖΘ.</s> <lb n="1799,001,,,,12,12,146"></lb> |
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15497 <s>Ὡς δὲ ὁ κῶνος πρὸς τὸν κῶνον, ὁ κύλινδρος πρὸς τὸν <lb n="1799,001,,,,12,12,147"></lb> |
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15498 κύλινδρον· τριπλάσιος γὰρ ὁ κύλινδρος τοῦ κώνου ὁ ἐπὶ <lb n="1799,001,,,,12,12,148"></lb> |
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15499 τῆς αὐτῆς βάσεως τῷ κώνῳ καὶ ἰσοϋψὴς αὐτῷ.</s> <s>καὶ ὁ κύλιν- <pb></pb><lb n="1799,001,,,,12,12,149"></lb> |
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15500 δρος ἄρα πρὸς τὸν κύλινδρον τριπλασίονα λόγον ἔχει ἤπερ <lb n="1799,001,,,,12,12,150"></lb> |
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15501 ἡ ΒΔ πρὸς τὴν ΖΘ.</s> <lb n="1799,001,,,,12,12,151"></lb> |
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15502 <s>Οἱ ἄρα ὅμοιοι κῶνοι καὶ κύλινδροι πρὸς ἀλλήλους ἐν <lb n="1799,001,,,,12,12,152"></lb> |
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15503 τριπλασίονι λόγῳ εἰσὶ τῶν ἐν ταῖς βάσεσι διαμέτρων· <lb n="1799,001,,,,12,12,153"></lb> |
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15504 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,12,154"></lb> |
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15505 <s>Ἐὰν κύλινδρος ἐπιπέδῳ τμηθῇ παραλλήλῳ ὄντι τοῖς <lb n="1799,001,,,,12,13,1"></lb> |
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15506 ἀπεναντίον ἐπιπέδοις, ἔσται ὡς ὁ κύλινδρος πρὸς τὸν κύλιν-<lb n="1799,001,,,,12,13,2"></lb> |
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15507 δρον, οὕτως ὁ ἄξων πρὸς τὸν ἄξονα.</s> <lb n="1799,001,,,,12,13,3"></lb> |
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15508 <s>Κύλινδρος γὰρ ὁ ΑΔ ἐπιπέδῳ τῷ ΗΘ τετμήσθω παραλ-<lb n="1799,001,,,,12,13,4"></lb> |
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15509 λήλῳ ὄντι τοῖς ἀπεναντίον ἐπιπέδοις τοῖς ΑΒ, ΓΔ, καὶ <lb n="1799,001,,,,12,13,5"></lb> |
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15510 συμβαλλέτω τῷ ἄξονι τὸ ΗΘ ἐπίπεδον κατὰ τὸ Κ σημεῖον· <lb n="1799,001,,,,12,13,6"></lb> |
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15511 λέγω, ὅτι ἐστὶν ὡς ὁ ΒΗ κύλινδρος πρὸς τὸν ΗΔ <lb n="1799,001,,,,12,13,7"></lb> |
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15512 κύλινδρον, οὕτως ὁ ΕΚ ἄξων πρὸς τὸν ΚΖ ἄξονα. <lb n="1799,001,,,,12,13,8"></lb> |
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15513 Ἐκβεβλήσθω γὰρ ὁ ΕΖ ἄξων ἐφ' ἑκάτερα τὰ μέρη ἐπὶ <lb n="1799,001,,,,12,13,9"></lb> |
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15514 τὰ Λ, Μ σημεῖα, καὶ ἐκκείσθωσαν τῷ ΕΚ ἄξονι ἴσοι <lb n="1799,001,,,,12,13,10"></lb> |
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15515 ὁσοιδηποτοῦν οἱ ΕΝ, ΝΛ, τῷ δὲ ΖΚ ἴσοι ὁσοιδηποτοῦν <lb n="1799,001,,,,12,13,11"></lb> |
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15516 οἱ ΖΞ, ΞΜ, καὶ νοείσθω ὁ ἐπὶ τοῦ ΛΜ ἄξονος κύλινδρος <lb n="1799,001,,,,12,13,12"></lb> |
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15517 ὁ ΟΧ, οὗ βάσεις οἱ ΟΠ, ΦΧ κύκλοι.</s> <s>καὶ ἐκβεβλήσθω <lb n="1799,001,,,,12,13,13"></lb> |
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15518 διὰ τῶν Ν, Ξ σημείων ἐπίπεδα παράλληλα τοῖς ΑΒ, ΓΔ <lb n="1799,001,,,,12,13,14"></lb> |
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15519 καὶ ταῖς βάσεσι τοῦ ΟΧ κυλίνδρου καὶ ποιείτωσαν τοὺς <pb></pb> <lb n="1799,001,,,,12,13,15"></lb> |
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15520 ΡΣ, ΤΥ κύκλους περὶ τὰ Ν, Ξ κέντρα.</s> <s>καὶ ἐπεὶ οἱ ΛΝ, <lb n="1799,001,,,,12,13,16"></lb> |
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15521 ΝΕ, ΕΚ ἄξονες ἴσοι εἰσὶν ἀλλήλοις, οἱ ἄρα ΠΡ, ΡΒ, ΒΗ <lb n="1799,001,,,,12,13,17"></lb> |
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15522 κύλινδροι πρὸς ἀλλήλους εἰσὶν ὡς αἱ βάσεις.</s> <s>ἴσαι δέ εἰσιν <lb n="1799,001,,,,12,13,18"></lb> |
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15523 αἱ βάσεις· ἴσοι ἄρα καὶ οἱ ΠΡ, ΡΒ, ΒΗ κύλινδροι ἀλλή-<lb n="1799,001,,,,12,13,19"></lb> |
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15524 λοις.</s> <s>ἐπεὶ οὖν οἱ ΛΝ, ΝΕ, ΕΚ ἄξονες ἴσοι εἰσὶν ἀλλή-<lb n="1799,001,,,,12,13,20"></lb> |
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15525 λοις, εἰσὶ δὲ καὶ οἱ ΠΡ, ΡΒ, ΒΗ κύλινδροι ἴσοι ἀλλήλοις, <lb n="1799,001,,,,12,13,21"></lb> |
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15526 καί ἐστιν ἴσον τὸ πλῆθος τῷ πλήθει, ὁσαπλασίων ἄρα ὁ <lb n="1799,001,,,,12,13,22"></lb> |
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15527 ΚΛ ἄξων τοῦ ΕΚ ἄξονος, τοσαυταπλασίων ἔσται καὶ ὁ <lb n="1799,001,,,,12,13,23"></lb> |
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15528 ΠΗ κύλινδρος τοῦ ΗΒ κυλίνδρου.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ <lb n="1799,001,,,,12,13,24"></lb> |
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15529 ὁσαπλασίων ἐστὶν ὁ ΜΚ ἄξων τοῦ ΚΖ ἄξονος, τοσαυτα-<lb n="1799,001,,,,12,13,25"></lb> |
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15530 πλασίων ἐστὶ καὶ ὁ ΧΗ κύλινδρος τοῦ ΗΔ κυλίνδρου.</s> <lb n="1799,001,,,,12,13,26"></lb> |
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15531 <s>καὶ εἰ μὲν ἴσος ἐστὶν ὁ ΚΛ ἄξων τῷ ΚΜ ἄξονι, ἴσος ἔσται <lb n="1799,001,,,,12,13,27"></lb> |
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15532 καὶ ὁ ΠΗ κύλινδρος τῷ ΗΧ κυλίνδρῳ, εἰ δὲ μείζων ὁ <lb n="1799,001,,,,12,13,28"></lb> |
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15533 ἄξων τοῦ ἄξονος, μείζων καὶ ὁ κύλινδρος τοῦ κυλίνδρου, <lb n="1799,001,,,,12,13,29"></lb> |
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15534 καὶ εἰ ἐλάσσων, ἐλάσσων.</s> <s>τεσσάρων δὴ μεγεθῶν ὄντων, <lb n="1799,001,,,,12,13,30"></lb> |
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15535 ἀξόνων μὲν τῶν ΕΚ, ΚΖ, κυλίνδρων δὲ τῶν ΒΗ, ΗΔ, <lb n="1799,001,,,,12,13,31"></lb> |
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15536 εἴληπται ἰσάκις πολλαπλάσια, τοῦ μὲν ΕΚ ἄξονος καὶ τοῦ <lb n="1799,001,,,,12,13,32"></lb> |
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15537 ΒΗ κυλίνδρου ὅ τε ΛΚ ἄξων καὶ ὁ ΠΗ κύλινδρος, τοῦ <lb n="1799,001,,,,12,13,33"></lb> |
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15538 δὲ ΚΖ ἄξονος καὶ τοῦ ΗΔ κυλίνδρου ὅ τε ΚΜ ἄξων καὶ <lb n="1799,001,,,,12,13,34"></lb> |
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15539 ὁ ΗΧ κύλινδρος, καὶ δέδεικται, ὅτι εἰ ὑπερέχει ὁ ΚΛ <lb n="1799,001,,,,12,13,35"></lb> |
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15540 ἄξων τοῦ ΚΜ ἄξονος, ὑπερέχει καὶ ὁ ΠΗ κύλινδρος τοῦ <lb n="1799,001,,,,12,13,36"></lb> |
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15541 ΗΧ κυλίνδρου, καὶ εἰ ἴσος, ἴσος, καὶ εἰ ἐλάσσων, ἐλάσ-<lb n="1799,001,,,,12,13,37"></lb> |
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15542 σων.</s> <s>ἔστιν ἄρα ὡς ὁ ΕΚ ἄξων πρὸς τὸν ΚΖ ἄξονα, οὕτως <lb n="1799,001,,,,12,13,38"></lb> |
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15543 ὁ ΒΗ κύλινδρος πρὸς τὸν ΗΔ κύλινδρον· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,12,13,39"></lb> |
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15544 Οἱ ἐπὶ ἴσων βάσεων ὄντες κῶνοι καὶ κύλινδροι πρὸς <lb n="1799,001,,,,12,14,1"></lb> |
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15545 ἀλλήλους εἰσὶν ὡς τὰ ὕψη.</s> <lb n="1799,001,,,,12,14,2"></lb> |
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15546 <s>Ἔστωσαν γὰρ ἐπὶ ἴσων βάσεων τῶν ΑΒ, ΓΔ κύκλων <lb n="1799,001,,,,12,14,3"></lb> |
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15547 κύλινδροι οἱ ΕΒ, ΖΔ· λέγω, ὅτι ἐστὶν ὡς ὁ ΕΒ κύλιν-<lb n="1799,001,,,,12,14,4"></lb> |
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15548 δρος πρὸς τὸν ΖΔ κύλινδρον, οὕτως ὁ ΗΘ ἄξων πρὸς τὸν <lb n="1799,001,,,,12,14,5"></lb> |
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15549 ΚΛ ἄξονα.</s> <lb n="1799,001,,,,12,14,6"></lb> |
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15550 <s>Ἐκβεβλήσθω γὰρ ὁ ΚΛ ἄξων ἐπὶ τὸ Ν σημεῖον, καὶ <lb n="1799,001,,,,12,14,7"></lb> |
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15551 κείσθω τῷ ΗΘ ἄξονι ἴσος ὁ ΛΝ, καὶ περὶ ἄξονα τὸν ΛΝ <lb n="1799,001,,,,12,14,8"></lb> |
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15552 κύλινδρος νενοήσθω ὁ ΓΜ.</s> <s>ἐπεὶ οὖν οἱ ΕΒ, ΓΜ κύλινδροι <lb n="1799,001,,,,12,14,9"></lb> |
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15553 ὑπὸ τὸ αὐτὸ ὕψος εἰσίν, πρὸς ἀλλήλους εἰσὶν ὡς αἱ βάσεις.</s> <lb n="1799,001,,,,12,14,10"></lb> |
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15554 <s>ἴσαι δέ εἰσιν αἱ βάσεις ἀλλήλαις· ἴσοι ἄρα εἰσὶ καὶ οἱ ΕΒ, <lb n="1799,001,,,,12,14,11"></lb> |
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15555 ΓΜ κύλινδροι.</s> <s>καὶ ἐπεὶ κύλινδρος <lb n="1799,001,,,,12,14,12"></lb> |
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15556 ὁ ΖΜ ἐπιπέδῳ τέτμηται τῷ ΓΔ <lb n="1799,001,,,,12,14,13"></lb> |
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15557 παραλλήλῳ ὄντι τοῖς ἀπεναντίον <lb n="1799,001,,,,12,14,14"></lb> |
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15558 ἐπιπέδοις, ἔστιν ἄρα ὡς ὁ ΓΜ <lb n="1799,001,,,,12,14,15"></lb> |
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15559 κύλινδρος πρὸς τὸν ΖΔ κύλινδρον, <lb n="1799,001,,,,12,14,16"></lb> |
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15560 οὕτως ὁ ΛΝ ἄξων πρὸς τὸν ΚΛ <lb n="1799,001,,,,12,14,17"></lb> |
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15561 ἄξονα.</s> <s>ἴσος δέ ἐστιν ὁ μὲν ΓΜ <lb n="1799,001,,,,12,14,18"></lb> |
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15562 κύλινδρος τῷ ΕΒ κυλίνδρῳ, ὁ δὲ ΛΝ ἄξων τῷ ΗΘ <lb n="1799,001,,,,12,14,19"></lb> |
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15563 ἄξονι· ἔστιν ἄρα ὡς ὁ ΕΒ κύλινδρος πρὸς τὸν ΖΔ <lb n="1799,001,,,,12,14,20"></lb> |
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15564 κύλινδρον, οὕτως ὁ ΗΘ ἄξων πρὸς τὸν ΚΛ ἄξονα.</s> <s>ὡς δὲ <lb n="1799,001,,,,12,14,21"></lb> |
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15565 ὁ ΕΒ κύλινδρος πρὸς τὸν ΖΔ κύλινδρον, οὕτως ὁ ΑΒΗ <lb n="1799,001,,,,12,14,22"></lb> |
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15566 κῶνος πρὸς τὸν ΓΔΚ κῶνον.</s> <s>καὶ ὡς ἄρα ὁ ΗΘ ἄξων <lb n="1799,001,,,,12,14,23"></lb> |
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15567 πρὸς τὸν ΚΛ ἄξονα, οὕτως ὁ ΑΒΗ κῶνος πρὸς τὸν ΓΔΚ <lb n="1799,001,,,,12,14,24"></lb> |
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15568 κῶνον καὶ ὁ ΕΒ κύλινδρος πρὸς τὸν ΖΔ κύλινδρον· ὅπερ <lb n="1799,001,,,,12,14,25"></lb> |
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15569 ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,12,14,26"></lb> |
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15570 Τῶν ἴσων κώνων καὶ κυλίνδρων ἀντιπεπόνθασιν αἱ <lb n="1799,001,,,,12,15,1"></lb> |
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15571 βάσεις τοῖς ὕψεσιν· καὶ ὧν κώνων καὶ κυλίνδρων ἀντι-<lb n="1799,001,,,,12,15,2"></lb> |
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15572 πεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, ἴσοι εἰσὶν ἐκεῖνοι.</s> <lb n="1799,001,,,,12,15,3"></lb> |
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15573 <s>Ἔστωσαν ἴσοι κῶνοι καὶ κύλινδροι, ὧν βάσεις μὲν οἱ <lb n="1799,001,,,,12,15,4"></lb> |
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15574 ΑΒΓΔ, ΕΖΗΘ κύκλοι, διάμετροι δὲ αὐτῶν αἱ ΑΓ, ΕΗ, <lb n="1799,001,,,,12,15,5"></lb> |
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15575 ἄξονες δὲ οἱ ΚΛ, ΜΝ, οἵτινες καὶ ὕψη εἰσὶ τῶν κώνων ἢ <lb n="1799,001,,,,12,15,6"></lb> |
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15576 κυλίνδρων, καὶ συμπεπληρώσθωσαν οἱ ΑΞ, ΕΟ κύλιν-<lb n="1799,001,,,,12,15,7"></lb> |
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15577 δροι.</s> <s>λέγω, ὅτι τῶν ΑΞ, ΕΟ κυλίνδρων ἀντιπεπόνθασιν <lb n="1799,001,,,,12,15,8"></lb> |
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15578 αἱ βάσεις τοῖς ὕψεσιν, καί ἐστιν ὡς ἡ ΑΒΓΔ βάσις πρὸς <lb n="1799,001,,,,12,15,9"></lb> |
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15579 τὴν ΕΖΗΘ βάσιν, οὕτως τὸ ΜΝ ὕψος πρὸς τὸ ΚΛ ὕψος.</s> <lb n="1799,001,,,,12,15,10"></lb> |
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15580 <s>Τὸ γὰρ ΛΚ ὕψος τῷ ΜΝ ὕψει ἤτοι ἴσον ἐστὶν ἢ οὔ.</s> <lb n="1799,001,,,,12,15,11"></lb> |
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15581 <s>ἔστω πρότερον ἴσον.</s> <s>ἔστι δὲ καὶ ὁ ΑΞ κύλινδρος τῷ ΕΟ <lb n="1799,001,,,,12,15,12"></lb> |
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15582 κυλίνδρῳ ἴσος.</s> <s>οἱ δὲ ὑπὸ τὸ αὐτὸ ὕψος ὄντες κῶνοι καὶ <lb n="1799,001,,,,12,15,13"></lb> |
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15583 κύλινδροι πρὸς ἀλλήλους εἰσὶν ὡς αἱ βάσεις· ἴση ἄρα καὶ ἡ <lb n="1799,001,,,,12,15,14"></lb> |
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15584 ΑΒΓΔ βάσις τῇ ΕΖΗΘ βάσει.</s> <s>ὥστε καὶ ἀντιπέπονθεν, <lb n="1799,001,,,,12,15,15"></lb> |
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15585 ὡς ἡ ΑΒΓΔ βάσις πρὸς τὴν ΕΖΗΘ βάσιν, οὕτως τὸ ΜΝ <lb n="1799,001,,,,12,15,16"></lb> |
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15586 ὕψος πρὸς τὸ ΚΛ ὕψος.</s> <s>ἀλλὰ δὴ μὴ ἔστω τὸ ΛΚ ὕψος <lb n="1799,001,,,,12,15,17"></lb> |
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15587 τῷ ΜΝ ἴσον, ἀλλ' ἔστω μεῖζον τὸ ΜΝ, καὶ ἀφῃρήσθω <lb n="1799,001,,,,12,15,18"></lb> |
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15588 ἀπὸ τοῦ ΜΝ ὕψους τῷ ΚΛ ἴσον τὸ ΠΝ, καὶ διὰ τοῦ Π <lb n="1799,001,,,,12,15,19"></lb> |
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15589 σημείου τετμήσθω ὁ ΕΟ κύλινδρος ἐπιπέδῳ τῷ ΤΥΣ <lb n="1799,001,,,,12,15,20"></lb> |
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15590 παραλλήλῳ τοῖς τῶν ΕΖΗΘ, ΡΟ κύκλων ἐπιπέδοις, καὶ <pb></pb> <lb n="1799,001,,,,12,15,21"></lb> |
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15591 ἀπὸ βάσεως μὲν τοῦ ΕΖΗΘ κύκλου, ὕψους δὲ τοῦ ΝΠ <lb n="1799,001,,,,12,15,22"></lb> |
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15592 κύλινδρος νενοήσθω ὁ ΕΣ.</s> <s>καὶ ἐπεὶ ἴσος ἐστὶν ὁ ΑΞ <lb n="1799,001,,,,12,15,23"></lb> |
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15593 κύλινδρος τῷ ΕΟ κυλίνδρῳ, ἔστιν ἄρα ὡς ὁ ΑΞ κύλινδρος <lb n="1799,001,,,,12,15,24"></lb> |
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15594 πρὸς τὸν ΕΣ κύλινδρον, οὕτως ὁ ΕΟ κύλινδρος πρὸς τὸν <lb n="1799,001,,,,12,15,25"></lb> |
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15595 ΕΣ κύλινδρον.</s> <s>ἀλλ' ὡς μὲν ὁ ΑΞ κύλινδρος πρὸς τὸν ΕΣ <lb n="1799,001,,,,12,15,26"></lb> |
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15596 κύλινδρον, οὕτως ἡ ΑΒΓΔ βάσις πρὸς τὴν ΕΖΗΘ· ὑπὸ <lb n="1799,001,,,,12,15,27"></lb> |
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15597 γὰρ τὸ αὐτὸ ὕψος εἰσὶν οἱ ΑΞ, ΕΣ κύλινδροι· ὡς δὲ ὁ ΕΟ <lb n="1799,001,,,,12,15,28"></lb> |
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15598 κύλινδρος πρὸς τὸν ΕΣ, οὕτως τὸ ΜΝ ὕψος πρὸς τὸ ΠΝ <lb n="1799,001,,,,12,15,29"></lb> |
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15599 ὕψος· ὁ γὰρ ΕΟ κύλινδρος ἐπιπέδῳ τέτμηται παραλλήλῳ <lb n="1799,001,,,,12,15,30"></lb> |
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15600 ὄντι τοῖς ἀπεναντίον ἐπιπέδοις.</s> <s>ἔστιν ἄρα καὶ ὡς ἡ ΑΒΓΔ <lb n="1799,001,,,,12,15,31"></lb> |
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15601 βάσις πρὸς τὴν ΕΖΗΘ βάσιν, οὕτως τὸ ΜΝ ὕψος πρὸς <lb n="1799,001,,,,12,15,32"></lb> |
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15602 τὸ ΠΝ ὕψος.</s> <s>ἴσον δὲ τὸ ΠΝ ὕψος τῷ ΚΛ ὕψει· ἔστιν ἄρα <lb n="1799,001,,,,12,15,33"></lb> |
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15603 ὡς ἡ ΑΒΓΔ βάσις πρὸς τὴν ΕΖΗΘ βάσιν, οὕτως τὸ <lb n="1799,001,,,,12,15,34"></lb> |
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15604 ΜΝ ὕψος πρὸς τὸ ΚΛ ὕψος.</s> <s>τῶν ἄρα ΑΞ, ΕΟ κυλίνδρων <lb n="1799,001,,,,12,15,35"></lb> |
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15605 ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν.</s> <lb n="1799,001,,,,12,15,36"></lb> |
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15606 <s>Ἀλλὰ δὴ τῶν ΑΞ, ΕΟ κυλίνδρων ἀντιπεπονθέτωσαν <lb n="1799,001,,,,12,15,37"></lb> |
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15607 αἱ βάσεις τοῖς ὕψεσιν, καὶ ἔστω ὡς ἡ ΑΒΓΔ βάσις πρὸς <lb n="1799,001,,,,12,15,38"></lb> |
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15608 τὴν ΕΖΗΘ βάσιν, οὕτως τὸ ΜΝ ὕψος πρὸς τὸ ΚΛ ὕψος· <lb n="1799,001,,,,12,15,39"></lb> |
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15609 λέγω, ὅτι ἴσος ἐστὶν ὁ ΑΞ κύλινδρος τῷ ΕΟ κυλίνδρῳ.</s> <s><pb></pb> <lb n="1799,001,,,,12,15,40"></lb> |
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15610 Τῶν γὰρ αὐτῶν κατασκευασθέντων ἐπεί ἐστιν ὡς ἡ <lb n="1799,001,,,,12,15,41"></lb> |
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15611 ΑΒΓΔ βάσις πρὸς τὴν ΕΖΗΘ βάσιν, οὕτως τὸ ΜΝ <lb n="1799,001,,,,12,15,42"></lb> |
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15612 ὕψος πρὸς τὸ ΚΛ ὕψος, ἴσον δὲ τὸ ΚΛ ὕψος τῷ ΠΝ ὕψει, <lb n="1799,001,,,,12,15,43"></lb> |
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15613 ἔστιν ἄρα ὡς ἡ ΑΒΓΔ βάσις πρὸς τὴν ΕΖΗΘ βάσιν, <lb n="1799,001,,,,12,15,44"></lb> |
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15614 οὕτως τὸ ΜΝ ὕψος πρὸς τὸ ΠΝ ὕψος.</s> <s>ἀλλ' ὡς μὲν ἡ <lb n="1799,001,,,,12,15,45"></lb> |
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15615 ΑΒΓΔ βάσις πρὸς τὴν ΕΖΗΘ βάσιν, οὕτως ὁ ΑΞ κύλιν-<lb n="1799,001,,,,12,15,46"></lb> |
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15616 δρος πρὸς τὸν ΕΣ κύλινδρον· ὑπὸ γὰρ τὸ αὐτὸ ὕψος εἰσίν· <lb n="1799,001,,,,12,15,47"></lb> |
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15617 ὡς δὲ τὸ ΜΝ ὕψος πρὸς τὸ ΠΝ [ὕψοσ], οὕτως ὁ ΕΟ κύλιν-<lb n="1799,001,,,,12,15,48"></lb> |
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15618 δρος πρὸς τὸν ΕΣ κύλινδρον· ἔστιν ἄρα ὡς ὁ ΑΞ κύλιν-<lb n="1799,001,,,,12,15,49"></lb> |
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15619 δρος πρὸς τὸν ΕΣ κύλινδρον, οὕτως ὁ ΕΟ κύλινδρος πρὸς <lb n="1799,001,,,,12,15,50"></lb> |
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15620 τὸν ΕΣ.</s> <s>ἴσος ἄρα ὁ ΑΞ κύλινδρος τῷ ΕΟ κυλίνδρῳ.</s> <lb n="1799,001,,,,12,15,51"></lb> |
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15621 <s>ὡσαύτως δὲ καὶ ἐπὶ τῶν κώνων· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,15,52"></lb> |
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15622 <s>Δύο κύκλων περὶ τὸ αὐτὸ κέντρον ὄντων εἰς τὸν <lb n="1799,001,,,,12,16,1"></lb> |
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15623 μείζονα κύκλον πολύγωνον ἰσόπλευρόν τε καὶ ἀρτιό-<lb n="1799,001,,,,12,16,2"></lb> |
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15624 πλευρον ἐγγράψαι μὴ ψαῦον τοῦ ἐλάσσονος κύκλου.</s> <lb n="1799,001,,,,12,16,3"></lb> |
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15625 <s>Ἔστωσαν οἱ δοθέντες δύο κύκ- <lb n="1799,001,,,,12,16,4"></lb> |
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15626 λοι οἱ ΑΒΓΔ, ΕΖΗΘ περὶ τὸ αὐτὸ <lb n="1799,001,,,,12,16,5"></lb> |
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15627 κέντρον τὸ Κ· δεῖ δὴ εἰς τὸν μεί-<lb n="1799,001,,,,12,16,6"></lb> |
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15628 ζονα κύκλον τὸν ΑΒΓΔ πολύγω-<lb n="1799,001,,,,12,16,7"></lb> |
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15629 νον ἰσόπλευρόν τε καὶ ἀρτιόπλευρον <lb n="1799,001,,,,12,16,8"></lb> |
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15630 ἐγγράψαι μὴ ψαῦον τοῦ ΕΖΗΘ <lb n="1799,001,,,,12,16,9"></lb> |
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15631 κύκλου.</s> <s><pb></pb> <lb n="1799,001,,,,12,16,10"></lb> |
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15632 Ἤχθω γὰρ διὰ τοῦ Κ κέντρου εὐθεῖα ἡ ΒΚΔ, καὶ ἀπὸ <lb n="1799,001,,,,12,16,11"></lb> |
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15633 τοῦ Η σημείου τῇ ΒΔ εὐθείᾳ πρὸς ὀρθὰς ἤχθω ἡ ΗΑ καὶ <lb n="1799,001,,,,12,16,12"></lb> |
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15634 διήχθω ἐπὶ τὸ Γ· ἡ ΑΓ ἄρα ἐφάπτεται τοῦ ΕΖΗΘ κύκ-<lb n="1799,001,,,,12,16,13"></lb> |
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15635 λου.</s> <s>τέμνοντες δὴ τὴν ΒΑΔ περιφέρειαν δίχα καὶ τὴν ἡμί-<lb n="1799,001,,,,12,16,14"></lb> |
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15636 σειαν αὐτῆς δίχα καὶ τοῦτο ἀεὶ ποιοῦντες καταλείψομεν <lb n="1799,001,,,,12,16,15"></lb> |
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15637 περιφέρειαν ἐλάσσονα τῆς ΑΔ.</s> <s>λελείφθω, καὶ ἔστω ἡ <lb n="1799,001,,,,12,16,16"></lb> |
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15638 ΛΔ, καὶ ἀπὸ τοῦ Λ ἐπὶ τὴν ΒΔ κάθετος ἤχθω ἡ ΛΜ καὶ <lb n="1799,001,,,,12,16,17"></lb> |
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15639 διήχθω ἐπὶ τὸ Ν, καὶ ἐπεζεύχθωσαν αἱ ΛΔ, ΔΝ· ἴση ἄρα <lb n="1799,001,,,,12,16,18"></lb> |
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15640 ἐστὶν ἡ ΛΔ τῇ ΔΝ.</s> <s>καὶ ἐπεὶ παράλληλός ἐστιν ἡ ΛΝ τῇ <lb n="1799,001,,,,12,16,19"></lb> |
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15641 ΑΓ, ἡ δὲ ΑΓ ἐφάπτεται τοῦ ΕΖΗΘ κύκλου, ἡ ΛΝ ἄρα <lb n="1799,001,,,,12,16,20"></lb> |
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15642 οὐκ ἐφάπτεται τοῦ ΕΖΗΘ κύκλου· πολλῷ ἄρα αἱ ΛΔ, <lb n="1799,001,,,,12,16,21"></lb> |
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15643 ΔΝ οὐκ ἐφάπτονται τοῦ ΕΖΗΘ κύκλου.</s> <s>ἐὰν δὴ τῇ ΛΔ <lb n="1799,001,,,,12,16,22"></lb> |
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15644 εὐθείᾳ ἴσας κατὰ τὸ συνεχὲς ἐναρμόσωμεν εἰς τὸν ΑΒΓΔ <lb n="1799,001,,,,12,16,23"></lb> |
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15645 κύκλον, ἐγγραφήσεται εἰς τὸν ΑΒΓΔ κύκλον πολύγωνον <lb n="1799,001,,,,12,16,24"></lb> |
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15646 ἰσόπλευρόν τε καὶ ἀρτιόπλευρον μὴ ψαῦον τοῦ ἐλάσσονος <lb n="1799,001,,,,12,16,25"></lb> |
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15647 κύκλου τοῦ ΕΖΗΘ· ὅπερ ἔδει ποιῆσαι.</s> <lb n="1799,001,,,,12,16,26"></lb> |
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15648 <s>Δύο σφαιρῶν περὶ τὸ αὐτὸ κέντρον οὐσῶν εἰς τὴν μείζονα <lb n="1799,001,,,,12,17,1"></lb> |
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15649 σφαῖραν στερεὸν πολύεδρον ἐγγράψαι μὴ ψαῦον τῆς ἐλάσ-<lb n="1799,001,,,,12,17,2"></lb> |
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15650 σονος σφαίρας κατὰ τὴν ἐπιφάνειαν.</s> <s><pb></pb> <lb n="1799,001,,,,12,17,3"></lb> |
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15651 Νενοήσθωσαν δύο σφαῖραι περὶ τὸ αὐτὸ κέντρον <lb n="1799,001,,,,12,17,4"></lb> |
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15652 τὸ Α· δεῖ δὴ εἰς τὴν μείζονα σφαῖραν στερεὸν πολύεδρον <lb n="1799,001,,,,12,17,5"></lb> |
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15653 ἐγγράψαι μὴ ψαῦον τῆς ἐλάσσονος σφαίρας κατὰ τὴν <lb n="1799,001,,,,12,17,6"></lb> |
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15654 ἐπιφάνειαν.</s> <lb n="1799,001,,,,12,17,7"></lb> |
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15655 <s>Τετμήσθωσαν αἱ σφαῖραι ἐπιπέδῳ τινὶ διὰ τοῦ κέντρου· <lb n="1799,001,,,,12,17,8"></lb> |
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15656 ἔσονται δὴ αἱ τομαὶ κύκλοι, ἐπειδήπερ μενούσης τῆς δια-<lb n="1799,001,,,,12,17,9"></lb> |
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15657 μέτρου καὶ περιφερομένου τοῦ ἡμικυκλίου ἐγίγνετο ἡ σφαῖ-<lb n="1799,001,,,,12,17,10"></lb> |
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15658 ρα· ὥστε καὶ καθ' οἵας ἂν θέσεως ἐπινοήσωμεν τὸ ἡμικύ-<lb n="1799,001,,,,12,17,11"></lb> |
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15659 κλιον, τὸ δι' αὐτοῦ ἐκβαλλόμενον ἐπίπεδον ποιήσει ἐπὶ τῆς <lb n="1799,001,,,,12,17,12"></lb> |
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15660 ἐπιφανείας τῆς σφαίρας κύκλον.</s> <s>καὶ φανερόν, ὅτι καὶ μέγι-<lb n="1799,001,,,,12,17,13"></lb> |
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15661 στον, ἐπειδήπερ ἡ διάμετρος τῆς σφαίρας, ἥτις ἐστὶ καὶ τοῦ <lb n="1799,001,,,,12,17,14"></lb> |
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15662 ἡμικυκλίου διάμετρος δηλαδὴ καὶ τοῦ κύκλου, μείζων <lb n="1799,001,,,,12,17,15"></lb> |
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15663 ἐστὶ πασῶν τῶν εἰς τὸν κύκλον ἢ τὴν σφαῖραν διαγομέ-<lb n="1799,001,,,,12,17,16"></lb> |
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15664 νων [εὐθειῶν].</s> <s>ἔστω οὖν ἐν μὲν τῇ μείζονι σφαίρᾳ κύκλος <lb n="1799,001,,,,12,17,17"></lb> |
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15665 ὁ ΒΓΔΕ, ἐν δὲ τῇ ἐλάσσονι σφαίρᾳ κύκλος ὁ ΖΗΘ, καὶ <lb n="1799,001,,,,12,17,18"></lb> |
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15666 ἤχθωσαν αὐτῶν δύο διάμετροι πρὸς ὀρθὰς ἀλλήλαις αἱ <lb n="1799,001,,,,12,17,19"></lb> |
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15667 ΒΔ, ΓΕ, καὶ δύο κύκλων περὶ τὸ αὐτὸ κέντρον ὄντων τῶν <lb n="1799,001,,,,12,17,20"></lb> |
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15668 ΒΓΔΕ, ΖΗΘ εἰς τὸν μείζονα κύκλον τὸν ΒΓΔΕ πολύ-<lb n="1799,001,,,,12,17,21"></lb> |
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15669 γωνον ἰσόπλευρον καὶ ἀρτιόπλευρον ἐγγεγράφθω μὴ ψαῦον <lb n="1799,001,,,,12,17,22"></lb> |
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15670 τοῦ ἐλάσσονος κύκλου τοῦ ΖΗΘ, οὗ πλευραὶ ἔστωσαν ἐν <lb n="1799,001,,,,12,17,23"></lb> |
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15671 τῷ ΒΕ τεταρτημορίῳ αἱ ΒΚ, ΚΛ, ΛΜ, ΜΕ, καὶ ἐπι-<lb n="1799,001,,,,12,17,24"></lb> |
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15672 ζευχθεῖσα ἡ ΚΑ διήχθω ἐπὶ τὸ Ν, καὶ ἀνεστάτω ἀπὸ τοῦ <lb n="1799,001,,,,12,17,25"></lb> |
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15673 Α σημείου τῷ τοῦ ΒΓΔΕ κύκλου ἐπιπέδῳ πρὸς ὀρθὰς ἡ <lb n="1799,001,,,,12,17,26"></lb> |
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15674 ΑΞ καὶ συμβαλλέτω τῇ ἐπιφανείᾳ τῆς σφαίρας κατὰ τὸ <lb n="1799,001,,,,12,17,27"></lb> |
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15675 Ξ, καὶ διὰ τῆς ΑΞ καὶ ἑκατέρας τῶν ΒΔ, ΚΝ ἐπίπεδα <lb n="1799,001,,,,12,17,28"></lb> |
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15676 ἐκβεβλήσθω· ποιήσουσι δὴ διὰ τὰ εἰρημένα ἐπὶ τῆς ἐπι-<lb n="1799,001,,,,12,17,29"></lb> |
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15677 φανείας τῆς σφαίρας μεγίστους κύκλους.</s> <s>ποιείτωσαν, <pb></pb> <lb n="1799,001,,,,12,17,30"></lb> |
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15678 ὧν ἡμικύκλια ἔστω ἐπὶ τῶν ΒΔ, ΚΝ διαμέτρων τὰ ΒΞΔ, <lb n="1799,001,,,,12,17,31"></lb> |
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15679 ΚΞΝ.</s> <s>καὶ ἐπεὶ ἡ ΞΑ ὀρθή ἐστι πρὸς τὸ τοῦ ΒΓΔΕ <lb n="1799,001,,,,12,17,32"></lb> |
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15680 κύκλου ἐπίπεδον, καὶ πάντα ἄρα τὰ διὰ τῆς ΞΑ ἐπίπεδά <lb n="1799,001,,,,12,17,33"></lb> |
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15681 ἐστιν ὀρθὰ πρὸς τὸ τοῦ ΒΓΔΕ κύκλου ἐπίπεδον· ὥστε <lb n="1799,001,,,,12,17,34"></lb> |
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15682 καὶ τὰ ΒΞΔ, ΚΞΝ ἡμικύκλια ὀρθά ἐστι πρὸς τὸ τοῦ <lb n="1799,001,,,,12,17,35"></lb> |
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15683 ΒΓΔΕ κύκλου ἐπίπεδον.</s> <s>καὶ ἐπεὶ ἴσα ἐστὶ τὰ ΒΕΔ, <lb n="1799,001,,,,12,17,36"></lb> |
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15684 ΒΞΔ, ΚΞΝ ἡμικύκλια· ἐπὶ γὰρ ἴσων εἰσὶ διαμέτρων <pb></pb> <lb n="1799,001,,,,12,17,37"></lb> |
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15685 τῶν ΒΔ, ΚΝ· ἴσα ἐστὶ καὶ τὰ ΒΕ, ΒΞ, ΚΞ τεταρτη-<lb n="1799,001,,,,12,17,38"></lb> |
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15686 μόρια ἀλλήλοις.</s> <s>ὅσαι ἄρα εἰσὶν ἐν τῷ ΒΕ τεταρτημορίῳ <lb n="1799,001,,,,12,17,39"></lb> |
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15687 πλευραὶ τοῦ πολυγώνου, τοσαῦταί εἰσι καὶ ἐν τοῖς ΒΞ, <lb n="1799,001,,,,12,17,40"></lb> |
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15688 ΚΞ τεταρτημορίοις ἴσαι ταῖς ΒΚ, ΚΛ, ΛΜ, ΜΕ εὐθεί-<lb n="1799,001,,,,12,17,41"></lb> |
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15689 αις.</s> <s>ἐγγεγράφθωσαν καὶ ἔστωσαν αἱ ΒΟ, ΟΠ, ΠΡ, ΡΞ, <lb n="1799,001,,,,12,17,42"></lb> |
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15690 ΚΣ, ΣΤ, ΤΥ, ΥΞ, καὶ ἐπεζεύχθωσαν αἱ ΣΟ, ΤΠ, ΥΡ, <lb n="1799,001,,,,12,17,43"></lb> |
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15691 καὶ ἀπὸ τῶν Ο, Σ ἐπὶ τὸ τοῦ ΒΓΔΕ κύκλου ἐπίπεδον <lb n="1799,001,,,,12,17,44"></lb> |
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15692 κάθετοι ἤχθωσαν· πεσοῦνται δὴ ἐπὶ τὰς κοινὰς τομὰς τῶν <lb n="1799,001,,,,12,17,45"></lb> |
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15693 ἐπιπέδων τὰς ΒΔ, ΚΝ, ἐπειδήπερ καὶ τὰ τῶν ΒΞΔ, ΚΞΝ <lb n="1799,001,,,,12,17,46"></lb> |
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15694 ἐπίπεδα ὀρθά ἐστι πρὸς τὸ τοῦ ΒΓΔΕ κύκλου ἐπίπεδον.</s> <lb n="1799,001,,,,12,17,47"></lb> |
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15695 <s>πιπτέτωσαν, καὶ ἔστωσαν αἱ ΟΦ, ΣΧ, καὶ ἐπεζεύχθω ἡ <lb n="1799,001,,,,12,17,48"></lb> |
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15696 ΧΦ.</s> <s>καὶ ἐπεὶ ἐν ἴσοις ἡμικυκλίοις τοῖς ΒΞΔ, ΚΞΝ ἴσαι <lb n="1799,001,,,,12,17,49"></lb> |
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15697 ἀπειλημμέναι εἰσὶν αἱ ΒΟ, ΚΣ, καὶ κάθετοι ἠγμέναι <lb n="1799,001,,,,12,17,50"></lb> |
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15698 εἰσὶν αἱ ΟΦ, ΣΧ, ἴση [ἄρα] ἐστὶν ἡ μὲν ΟΦ τῇ ΣΧ, ἡ δὲ <lb n="1799,001,,,,12,17,51"></lb> |
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15699 ΒΦ τῇ ΚΧ.</s> <s>ἔστι δὲ καὶ ὅλη ἡ ΒΑ ὅλῃ τῇ ΚΑ ἴση· καὶ <lb n="1799,001,,,,12,17,52"></lb> |
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15700 λοιπὴ ἄρα ἡ ΦΑ λοιπῇ τῇ ΧΑ ἐστιν ἴση· ἔστιν ἄρα ὡς ἡ <lb n="1799,001,,,,12,17,53"></lb> |
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15701 ΒΦ πρὸς τὴν ΦΑ, οὕτως ἡ ΚΧ πρὸς τὴν ΧΑ· παράλλη-<lb n="1799,001,,,,12,17,54"></lb> |
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15702 λος ἄρα ἐστὶν ἡ ΧΦ τῇ ΚΒ.</s> <s>καὶ ἐπεὶ ἑκατέρα τῶν ΟΦ, <lb n="1799,001,,,,12,17,55"></lb> |
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15703 ΣΧ ὀρθή ἐστι πρὸς τὸ τοῦ ΒΓΔΕ κύκλου ἐπίπεδον, <lb n="1799,001,,,,12,17,56"></lb> |
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15704 παράλληλος ἄρα ἐστὶν ἡ ΟΦ τῇ ΣΧ.</s> <s>ἐδείχθη δὲ αὐτῇ καὶ <lb n="1799,001,,,,12,17,57"></lb> |
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15705 ἴση· καὶ αἱ ΧΦ, ΣΟ ἄρα ἴσαι εἰσὶ καὶ παράλληλοι.</s> <s>καὶ <lb n="1799,001,,,,12,17,58"></lb> |
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15706 ἐπεὶ παράλληλός ἐστιν ἡ ΧΦ τῇ ΣΟ, ἀλλὰ ἡ ΧΦ τῇ ΚΒ <lb n="1799,001,,,,12,17,59"></lb> |
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15707 ἐστι παράλληλος, καὶ ἡ ΣΟ ἄρα τῇ ΚΒ ἐστι παράλληλος.</s> <lb n="1799,001,,,,12,17,60"></lb> |
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15708 <s>καὶ ἐπιζευγνύουσιν αὐτὰς αἱ ΒΟ, ΚΣ· τὸ ΚΒΟΣ ἄρα <pb></pb> <lb n="1799,001,,,,12,17,61"></lb> |
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15709 τετράπλευρον ἐν ἑνί ἐστιν ἐπιπέδῳ, ἐπειδήπερ, ἐὰν ὦσι <lb n="1799,001,,,,12,17,62"></lb> |
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15710 δύο εὐθεῖαι παράλληλοι, καὶ ἐφ' ἑκατέρας αὐτῶν ληφθῇ <lb n="1799,001,,,,12,17,63"></lb> |
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15711 τυχόντα σημεῖα, ἡ ἐπὶ τὰ σημεῖα ἐπιζευγνυμένη εὐθεῖα ἐν <lb n="1799,001,,,,12,17,64"></lb> |
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15712 τῷ αὐτῷ ἐπιπέδῳ ἐστὶ ταῖς παραλλήλοις.</s> <s>διὰ τὰ αὐτὰ δὴ <lb n="1799,001,,,,12,17,65"></lb> |
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15713 καὶ ἑκάτερον τῶν ΣΟΠΤ, ΤΠΡΥ τετραπλεύρων ἐν ἑνί <lb n="1799,001,,,,12,17,66"></lb> |
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15714 ἐστιν ἐπιπέδῳ.</s> <s>ἔστι δὲ καὶ τὸ ΥΡΞ τρίγωνον ἐν ἑνὶ ἐπι-<lb n="1799,001,,,,12,17,67"></lb> |
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15715 πέδῳ.</s> <s>ἐὰν δὴ νοήσωμεν ἀπὸ τῶν Ο, Σ, Π, Τ, Ρ, Υ σημεί-<lb n="1799,001,,,,12,17,68"></lb> |
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15716 ων ἐπὶ τὸ Α ἐπιζευγνυμένας εὐθείας, συσταθήσεταί τι <lb n="1799,001,,,,12,17,69"></lb> |
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15717 σχῆμα στερεὸν πολύεδρον μεταξὺ τῶν ΒΞ, ΚΞ περι-<lb n="1799,001,,,,12,17,70"></lb> |
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15718 φερειῶν ἐκ πυραμίδων συγκείμενον, ὧν βάσεις μὲν τὰ <lb n="1799,001,,,,12,17,71"></lb> |
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15719 ΚΒΟΣ, ΣΟΠΤ, ΤΠΡΥ τετράπλευρα καὶ τὸ ΥΡΞ <lb n="1799,001,,,,12,17,72"></lb> |
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15720 τρίγωνον, κορυφὴ δὲ τὸ Α σημεῖον.</s> <s>ἐὰν δὲ καὶ ἐπὶ ἑκάστης <lb n="1799,001,,,,12,17,73"></lb> |
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15721 τῶν ΚΛ, ΛΜ, ΜΕ πλευρῶν καθάπερ ἐπὶ τῆς ΒΚ τὰ αὐτὰ <lb n="1799,001,,,,12,17,74"></lb> |
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15722 κατασκευάσωμεν καὶ ἔτι ἐπὶ τῶν λοιπῶν τριῶν τεταρτη-<lb n="1799,001,,,,12,17,75"></lb> |
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15723 μορίων, συσταθήσεταί τι σχῆμα πολύεδρον ἐγγεγραμ-<lb n="1799,001,,,,12,17,76"></lb> |
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15724 μένον εἰς τὴν σφαῖραν πυραμίσι περιεχόμενον, ὧν βάσεις <lb n="1799,001,,,,12,17,77"></lb> |
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15725 [μὲν] τὰ εἰρημένα τετράπλευρα καὶ τὸ ΥΡΞ τρίγωνον καὶ <lb n="1799,001,,,,12,17,78"></lb> |
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15726 τὰ ὁμοταγῆ αὐτοῖς, κορυφὴ δὲ τὸ Α σημεῖον.</s> <lb n="1799,001,,,,12,17,79"></lb> |
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15727 <s>Λέγω, ὅτι τὸ εἰρημένον πολύεδρον οὐκ ἐφάψεται τῆς <lb n="1799,001,,,,12,17,80"></lb> |
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15728 ἐλάσσονος σφαίρας κατὰ τὴν ἐπιφάνειαν, ἐφ' ἧς ἐστιν ὁ <lb n="1799,001,,,,12,17,81"></lb> |
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15729 ΖΗΘ κύκλος.</s> <lb n="1799,001,,,,12,17,82"></lb> |
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15730 <s>Ἤχθω ἀπὸ τοῦ Α σημείου ἐπὶ τὸ τοῦ ΚΒΟΣ τετρα-<lb n="1799,001,,,,12,17,83"></lb> |
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15731 πλεύρου ἐπίπεδον κάθετος ἡ ΑΨ καὶ συμβαλλέτω τῷ <lb n="1799,001,,,,12,17,84"></lb> |
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15732 ἐπιπέδῳ κατὰ τὸ Ψ σημεῖον, καὶ ἐπεζεύχθωσαν αἱ ΨΒ, <lb n="1799,001,,,,12,17,85"></lb> |
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15733 ΨΚ.</s> <s>καὶ ἐπεὶ ἡ ΑΨ ὀρθή ἐστι πρὸς τὸ τοῦ ΚΒΟΣ <pb></pb> <lb n="1799,001,,,,12,17,86"></lb> |
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15734 τετραπλεύρου ἐπίπεδον, καὶ πρὸς πάσας ἄρα τὰς ἁπτο-<lb n="1799,001,,,,12,17,87"></lb> |
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15735 μένας αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ τοῦ τετραπλεύρου <lb n="1799,001,,,,12,17,88"></lb> |
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15736 ἐπιπέδῳ ὀρθή ἐστιν.</s> <s>ἡ ΑΨ ἄρα ὀρθή ἐστι πρὸς ἑκατέραν <lb n="1799,001,,,,12,17,89"></lb> |
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15737 τῶν ΒΨ, ΨΚ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒ τῇ ΑΚ, ἴσον ἐστὶ <lb n="1799,001,,,,12,17,90"></lb> |
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15738 καὶ τὸ ἀπὸ τῆς ΑΒ τῷ ἀπὸ τῆς ΑΚ.</s> <s>καί ἐστι τῷ μὲν ἀπὸ <lb n="1799,001,,,,12,17,91"></lb> |
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15739 τῆς ΑΒ ἴσα τὰ ἀπὸ τῶν ΑΨ, ΨΒ· ὀρθὴ γὰρ ἡ πρὸς <lb n="1799,001,,,,12,17,92"></lb> |
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15740 τῷ Ψ· τῷ δὲ ἀπὸ τῆς ΑΚ ἴσα τὰ ἀπὸ τῶν ΑΨ, ΨΚ.</s> <s>τὰ <lb n="1799,001,,,,12,17,93"></lb> |
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15741 ἄρα ἀπὸ τῶν ΑΨ, ΨΒ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΑΨ, ΨΚ.</s> <lb n="1799,001,,,,12,17,94"></lb> |
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15742 <s>κοινὸν ἀφῃρήσθω τὸ ἀπὸ τῆς ΑΨ· λοιπὸν ἄρα τὸ ἀπὸ τῆς <lb n="1799,001,,,,12,17,95"></lb> |
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15743 ΒΨ λοιπῷ τῷ ἀπὸ τῆς ΨΚ ἴσον ἐστίν· ἴση ἄρα ἡ ΒΨ <lb n="1799,001,,,,12,17,96"></lb> |
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15744 τῇ ΨΚ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ ἀπὸ τοῦ Ψ ἐπὶ <lb n="1799,001,,,,12,17,97"></lb> |
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15745 τὰ Ο, Σ ἐπιζευγνύμεναι εὐθεῖαι ἴσαι εἰσὶν ἑκατέρᾳ τῶν <lb n="1799,001,,,,12,17,98"></lb> |
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15746 ΒΨ, ΨΚ.</s> <s>ὁ ἄρα κέντρῳ τῷ Ψ καὶ διαστήματι ἑνὶ τῶν <lb n="1799,001,,,,12,17,99"></lb> |
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15747 ΨΒ, ΨΚ γραφόμενος κύκλος ἥξει καὶ διὰ τῶν Ο, Σ, καὶ <lb n="1799,001,,,,12,17,100"></lb> |
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15748 ἔσται ἐν κύκλῳ τὸ ΚΒΟΣ τετράπλευρον.</s> <lb n="1799,001,,,,12,17,101"></lb> |
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15749 <s>Καὶ ἐπεὶ μείζων ἐστὶν ἡ ΚΒ τῆς ΧΦ, ἴση δὲ ἡ ΧΦ τῇ <lb n="1799,001,,,,12,17,102"></lb> |
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15750 ΣΟ, μείζων ἄρα ἡ ΚΒ τῆς ΣΟ.</s> <s>ἴση δὲ ἡ ΚΒ ἑκατέρᾳ <lb n="1799,001,,,,12,17,103"></lb> |
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15751 τῶν ΚΣ, ΒΟ· καὶ ἑκατέρα ἄρα τῶν ΚΣ, ΒΟ τῆς ΣΟ <lb n="1799,001,,,,12,17,104"></lb> |
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15752 μείζων ἐστίν.</s> <s>καὶ ἐπεὶ ἐν κύκλῳ τετράπλευρόν ἐστι <lb n="1799,001,,,,12,17,105"></lb> |
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15753 τὸ ΚΒΟΣ, καὶ ἴσαι αἱ ΚΒ, ΒΟ, ΚΣ, καὶ ἐλάττων ἡ <lb n="1799,001,,,,12,17,106"></lb> |
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15754 ΟΣ, καὶ ἐκ τοῦ κέντρου τοῦ κύκλου ἐστὶν ἡ ΒΨ, τὸ <lb n="1799,001,,,,12,17,107"></lb> |
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15755 ἄρα ἀπὸ τῆς ΚΒ τοῦ ἀπὸ τῆς ΒΨ μεῖζόν ἐστιν ἢ <lb n="1799,001,,,,12,17,108"></lb> |
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15756 διπλάσιον.</s> <s>ἤχθω ἀπὸ τοῦ Κ ἐπὶ τὴν ΒΦ κάθετος ἡ ΚΩ.</s> <s><pb></pb> <lb n="1799,001,,,,12,17,109"></lb> |
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15757 καὶ ἐπεὶ ἡ ΒΔ τῆς ΔΩ ἐλάττων ἐστὶν ἢ διπλῆ, καί ἐστιν <lb n="1799,001,,,,12,17,110"></lb> |
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15758 ὡς ἡ ΒΔ πρὸς τὴν ΔΩ, οὕτως τὸ ὑπὸ τῶν ΔΒ, ΒΩ πρὸς <lb n="1799,001,,,,12,17,111"></lb> |
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15759 τὸ ὑπὸ [τῶν] ΔΩ, ΩΒ, ἀναγραφομένου ἀπὸ τῆς ΒΩ <lb n="1799,001,,,,12,17,112"></lb> |
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15760 τετραγώνου καὶ συμπληρουμένου τοῦ ἐπὶ τῆς ΩΔ παραλ-<lb n="1799,001,,,,12,17,113"></lb> |
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15761 ληλογράμμου καὶ τὸ ὑπὸ ΔΒ, ΒΩ ἄρα τοῦ ὑπὸ ΔΩ, ΩΒ <lb n="1799,001,,,,12,17,114"></lb> |
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15762 ἔλαττόν ἐστιν ἢ διπλάσιον.</s> <s>καί ἐστι τῆς ΚΔ ἐπιζευγνυμέ-<lb n="1799,001,,,,12,17,115"></lb> |
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15763 νης τὸ μὲν ὑπὸ ΔΒ, ΒΩ ἴσον τῷ ἀπὸ τῆς ΒΚ, τὸ δὲ ὑπὸ <lb n="1799,001,,,,12,17,116"></lb> |
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15764 τῶν ΔΩ, ΩΒ ἴσον τῷ ἀπὸ τῆς ΚΩ· τὸ ἄρα ἀπὸ τῆς ΚΒ <lb n="1799,001,,,,12,17,117"></lb> |
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15765 τοῦ ἀπὸ τῆς ΚΩ ἔλασσόν ἐστιν ἢ διπλάσιον.</s> <s>ἀλλὰ τὸ ἀπὸ <lb n="1799,001,,,,12,17,118"></lb> |
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15766 τῆς ΚΒ τοῦ ἀπὸ τῆς ΒΨ μεῖζόν ἐστιν ἢ διπλάσιον· μεῖ-<lb n="1799,001,,,,12,17,119"></lb> |
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15767 ζον ἄρα τὸ ἀπὸ τῆς ΚΩ τοῦ ἀπὸ τῆς ΒΨ.</s> <s>καὶ ἐπεὶ ἴση <lb n="1799,001,,,,12,17,120"></lb> |
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15768 ἐστὶν ἡ ΒΑ τῇ ΚΑ, ἴσον ἐστὶ τὸ ἀπὸ τῆς ΒΑ τῷ ἀπὸ τῆς <lb n="1799,001,,,,12,17,121"></lb> |
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15769 ΑΚ.</s> <s>καί ἐστι τῷ μὲν ἀπὸ τῆς ΒΑ ἴσα τὰ ἀπὸ τῶν ΒΨ, <lb n="1799,001,,,,12,17,122"></lb> |
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15770 ΨΑ, τῷ δὲ ἀπὸ τῆς ΚΑ ἴσα τὰ ἀπὸ τῶν ΚΩ, ΩΑ· τὰ <lb n="1799,001,,,,12,17,123"></lb> |
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15771 ἄρα ἀπὸ τῶν ΒΨ, ΨΑ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΚΩ, ΩΑ, <lb n="1799,001,,,,12,17,124"></lb> |
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15772 ὧν τὸ ἀπὸ τῆς ΚΩ μεῖζον τοῦ ἀπὸ τῆς ΒΨ· λοιπὸν ἄρα <lb n="1799,001,,,,12,17,125"></lb> |
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15773 τὸ ἀπὸ τῆς ΩΑ ἔλασσόν ἐστι τοῦ ἀπὸ τῆς ΨΑ.</s> <s>μείζων <lb n="1799,001,,,,12,17,126"></lb> |
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15774 ἄρα ἡ ΑΨ τῆς ΑΩ· πολλῷ ἄρα ἡ ΑΨ μείζων ἐστὶ τῆς <lb n="1799,001,,,,12,17,127"></lb> |
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15775 ΑΗ.</s> <s>καί ἐστιν ἡ μὲν ΑΨ ἐπὶ μίαν τοῦ πολυέδρου βάσιν, <pb></pb> <lb n="1799,001,,,,12,17,128"></lb> |
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15776 ἡ δὲ ΑΗ ἐπὶ τὴν τῆς ἐλάσσονος σφαίρας ἐπιφάνειαν· ὥστε <lb n="1799,001,,,,12,17,129"></lb> |
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15777 τὸ πολύεδρον οὐ ψαύσει τῆς ἐλάσσονος σφαίρας κατὰ τὴν <lb n="1799,001,,,,12,17,130"></lb> |
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15778 ἐπιφάνειαν.</s> <lb n="1799,001,,,,12,17,131"></lb> |
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15779 <s>Δύο ἄρα σφαιρῶν περὶ τὸ αὐτὸ κέντρον οὐσῶν εἰς τὴν <lb n="1799,001,,,,12,17,132"></lb> |
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15780 μείζονα σφαῖραν στερεὸν πολύεδρον ἐγγέγραπται μὴ ψαῦον <lb n="1799,001,,,,12,17,133"></lb> |
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15781 τῆς ἐλάσσονος σφαίρας κατὰ τὴν ἐπιφάνειαν· ὅπερ ἔδει <lb n="1799,001,,,,12,17,134"></lb> |
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15782 ποιῆσαι.</s> <lb n="1799,001,,,,12,17,135"></lb> |
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15783 <s>Πόρισμα <lb n="1799,001,,,,12,17,136"></lb> |
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15784 Ἐὰν δὲ καὶ εἰς ἑτέραν σφαῖραν τῷ ἐν τῇ ΒΓΔΕ σφαίρᾳ <lb n="1799,001,,,,12,17,137"></lb> |
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15785 στερεῷ πολυέδρῳ ὅμοιον στερεὸν πολύεδρον ἐγγραφῇ, τὸ <lb n="1799,001,,,,12,17,138"></lb> |
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15786 ἐν τῇ ΒΓΔΕ σφαίρᾳ στερεὸν πολύεδρον πρὸς τὸ ἐν τῇ <lb n="1799,001,,,,12,17,139"></lb> |
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15787 ἑτέρᾳ σφαίρᾳ στερεὸν πολύεδρον τριπλασίονα λόγον ἔχει, <lb n="1799,001,,,,12,17,140"></lb> |
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15788 ἤπερ ἡ τῆς ΒΓΔΕ σφαίρας διάμετρος πρὸς τὴν τῆς ἑτέρας <lb n="1799,001,,,,12,17,141"></lb> |
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15789 σφαίρας διάμετρον.</s> <s>διαιρεθέντων γὰρ τῶν στερεῶν εἰς <lb n="1799,001,,,,12,17,142"></lb> |
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15790 τὰς ὁμοιοπληθεῖς καὶ ὁμοιοταγεῖς πυραμίδας ἔσονται αἱ <lb n="1799,001,,,,12,17,143"></lb> |
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15791 πυραμίδες ὅμοιαι.</s> <s>αἱ δὲ ὅμοιαι πυραμίδες πρὸς ἀλλήλας <lb n="1799,001,,,,12,17,144"></lb> |
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15792 ἐν τριπλασίονι λόγῳ εἰσὶ τῶν ὁμολόγων πλευρῶν· ἡ ἄρα <lb n="1799,001,,,,12,17,145"></lb> |
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15793 πυραμίς, ἧς βάσις μέν ἐστι τὸ ΚΒΟΣ τετράπλευρον, <lb n="1799,001,,,,12,17,146"></lb> |
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15794 κορυφὴ δὲ τὸ Α σημεῖον, πρὸς τὴν ἐν τῇ ἑτέρᾳ σφαίρᾳ <lb n="1799,001,,,,12,17,147"></lb> |
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15795 ὁμοιοταγῆ πυραμίδα τριπλασίονα λόγον ἔχει, ἤπερ ἡ ὁμό-<lb n="1799,001,,,,12,17,148"></lb> |
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15796 λογος πλευρὰ πρὸς τὴν ὁμόλογον πλευράν, τουτέστιν <lb n="1799,001,,,,12,17,149"></lb> |
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15797 ἤπερ ἡ ΑΒ ἐκ τοῦ κέντρου τῆς σφαίρας τῆς περὶ κέντρον <lb n="1799,001,,,,12,17,150"></lb> |
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15798 τὸ Α πρὸς τὴν ἐκ τοῦ κέντρου τῆς ἑτέρας σφαίρας.</s> <s>ὁμοίως <lb n="1799,001,,,,12,17,151"></lb> |
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15799 καὶ ἑκάστη πυραμὶς τῶν ἐν τῇ περὶ κέντρον τὸ Α σφαίρᾳ <lb n="1799,001,,,,12,17,152"></lb> |
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15800 πρὸς ἑκάστην ὁμοταγῆ πυραμίδα τῶν ἐν τῇ ἑτέρᾳ σφαίρᾳ <lb n="1799,001,,,,12,17,153"></lb> |
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15801 τριπλασίονα λόγον ἕξει, ἤπερ ἡ ΑΒ πρὸς τὴν ἐκ τοῦ κέν- <pb></pb><lb n="1799,001,,,,12,17,154"></lb> |
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15802 τρου τῆς ἑτέρας σφαίρας.</s> <s>καὶ ὡς ἓν τῶν ἡγουμένων πρὸς <lb n="1799,001,,,,12,17,155"></lb> |
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15803 ἓν τῶν ἑπομένων, οὕτως ἅπαντα τὰ ἡγούμενα πρὸς <lb n="1799,001,,,,12,17,156"></lb> |
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15804 ἅπαντα τὰ ἑπόμενα· ὥστε ὅλον τὸ ἐν τῇ περὶ κέντρον τὸ Α <lb n="1799,001,,,,12,17,157"></lb> |
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15805 σφαίρᾳ στερεὸν πολύεδρον πρὸς ὅλον τὸ ἐν τῇ ἑτέρᾳ <lb n="1799,001,,,,12,17,158"></lb> |
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15806 [σφαίρᾳ] στερεὸν πολύεδρον τριπλασίονα λόγον ἕξει, ἤπερ <lb n="1799,001,,,,12,17,159"></lb> |
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15807 ἡ ΑΒ πρὸς τὴν ἐκ τοῦ κέντρου τῆς ἑτέρας σφαίρας, τουτ-<lb n="1799,001,,,,12,17,160"></lb> |
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15808 έστιν ἤπερ ἡ ΒΔ διάμετρος πρὸς τὴν τῆς ἑτέρας σφαίρας <lb n="1799,001,,,,12,17,161"></lb> |
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15809 διάμετρον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,12,17,162"></lb> |
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15810 <s>Αἱ σφαῖραι πρὸς ἀλλήλας ἐν τριπλασίονι λόγῳ εἰσὶ τῶν <lb n="1799,001,,,,12,18,1"></lb> |
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15811 ἰδίων διαμέτρων.</s> <lb n="1799,001,,,,12,18,2"></lb> |
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15812 <s>Νενοήσθωσαν σφαῖραι αἱ ΑΒΓ, ΔΕΖ, διάμετροι δὲ <lb n="1799,001,,,,12,18,3"></lb> |
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15813 αὐτῶν αἱ ΒΓ, ΕΖ· λέγω, ὅτι ἡ ΑΒΓ σφαῖρα πρὸς τὴν <lb n="1799,001,,,,12,18,4"></lb> |
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15814 ΔΕΖ σφαῖραν τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΓ πρὸς <lb n="1799,001,,,,12,18,5"></lb> |
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15815 τὴν ΕΖ. <pb></pb> <lb n="1799,001,,,,12,18,6"></lb> |
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15816 Εἰ γὰρ μὴ ἡ ΑΒΓ σφαῖρα πρὸς τὴν ΔΕΖ σφαῖραν τρι-<lb n="1799,001,,,,12,18,7"></lb> |
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15817 πλασίονα λόγον ἔχει ἤπερ ἡ ΒΓ πρὸς τὴν ΕΖ, ἕξει ἄρα ἡ <lb n="1799,001,,,,12,18,8"></lb> |
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15818 ΑΒΓ σφαῖρα πρὸς ἐλάσσονά τινα τῆς ΔΕΖ σφαίρας τρι-<lb n="1799,001,,,,12,18,9"></lb> |
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15819 πλασίονα λόγον ἢ πρὸς μείζονα ἤπερ ἡ ΒΓ πρὸς τὴν ΕΖ.</s> <lb n="1799,001,,,,12,18,10"></lb> |
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15820 <s>ἐχέτω πρότερον πρὸς ἐλάσσονα τὴν ΗΘΚ, καὶ νενοήσθω <lb n="1799,001,,,,12,18,11"></lb> |
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15821 ἡ ΔΕΖ τῇ ΗΘΚ περὶ τὸ αὐτὸ κέντρον, καὶ ἐγγεγράφθω <lb n="1799,001,,,,12,18,12"></lb> |
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15822 εἰς τὴν μείζονα σφαῖραν τὴν ΔΕΖ στερεὸν πολύεδρον μὴ <lb n="1799,001,,,,12,18,13"></lb> |
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15823 ψαῦον τῆς ἐλάσσονος σφαίρας τῆς ΗΘΚ κατὰ τὴν ἐπι-<lb n="1799,001,,,,12,18,14"></lb> |
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15824 φάνειαν, ἐγγεγράφθω δὲ καὶ εἰς τὴν ΑΒΓ σφαῖραν τῷ ἐν <lb n="1799,001,,,,12,18,15"></lb> |
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15825 τῇ ΔΕΖ σφαίρᾳ στερεῷ πολυέδρῳ ὅμοιον στερεὸν πολύ-<lb n="1799,001,,,,12,18,16"></lb> |
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15826 εδρον· τὸ ἄρα ἐν τῇ ΑΒΓ στερεὸν πολύεδρον πρὸς τὸ ἐν <lb n="1799,001,,,,12,18,17"></lb> |
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15827 τῇ ΔΕΖ στερεὸν πολύεδρον τριπλασίονα λόγον ἔχει ἤπερ <lb n="1799,001,,,,12,18,18"></lb> |
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15828 ἡ ΒΓ πρὸς τὴν ΕΖ.</s> <s>ἔχει δὲ καὶ ἡ ΑΒΓ σφαῖρα πρὸς τὴν <lb n="1799,001,,,,12,18,19"></lb> |
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15829 ΗΘΚ σφαῖραν τριπλασίονα λόγον ἤπερ ἡ ΒΓ πρὸς τὴν <lb n="1799,001,,,,12,18,20"></lb> |
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15830 ΕΖ· ἔστιν ἄρα ὡς ἡ ΑΒΓ σφαῖρα πρὸς τὴν ΗΘΚ σφαῖ-<lb n="1799,001,,,,12,18,21"></lb> |
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15831 ραν, οὕτως τὸ ἐν τῇ ΑΒΓ σφαίρᾳ στερεὸν πολύεδρον πρὸς <lb n="1799,001,,,,12,18,22"></lb> |
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15832 τὸ ἐν τῇ ΔΕΖ σφαίρᾳ στερεὸν πολύεδρον· ἐναλλὰξ [ἄρα] <lb n="1799,001,,,,12,18,23"></lb> |
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15833 ὡς ἡ ΑΒΓ σφαῖρα πρὸς τὸ ἐν αὐτῇ πολύεδρον, οὕτως ἡ <lb n="1799,001,,,,12,18,24"></lb> |
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15834 ΗΘΚ σφαῖρα πρὸς τὸ ἐν τῇ ΔΕΖ σφαίρᾳ στερεὸν πολύ-<lb n="1799,001,,,,12,18,25"></lb> |
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15835 εδρον.</s> <s>μείζων δὲ ἡ ΑΒΓ σφαῖρα τοῦ ἐν αὐτῇ πολυέδρου· <lb n="1799,001,,,,12,18,26"></lb> |
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15836 μείζων ἄρα καὶ ἡ ΗΘΚ σφαῖρα τοῦ ἐν τῇ ΔΕΖ σφαίρᾳ <lb n="1799,001,,,,12,18,27"></lb> |
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15837 πολυέδρου.</s> <s>ἀλλὰ καὶ ἐλάττων· ἐμπεριέχεται γὰρ ὑπ' <lb n="1799,001,,,,12,18,28"></lb> |
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15838 αὐτοῦ.</s> <s>οὐκ ἄρα ἡ ΑΒΓ σφαῖρα πρὸς ἐλάσσονα τῆς ΔΕΖ <lb n="1799,001,,,,12,18,29"></lb> |
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15839 σφαίρας τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΓ διάμετρος πρὸς <lb n="1799,001,,,,12,18,30"></lb> |
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15840 τὴν ΕΖ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι οὐδὲ ἡ ΔΕΖ σφαῖρα πρὸς <lb n="1799,001,,,,12,18,31"></lb> |
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15841 ἐλάσσονα τῆς ΑΒΓ σφαίρας τριπλασίονα λόγον ἔχει ἤπερ <lb n="1799,001,,,,12,18,32"></lb> |
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15842 ἡ ΕΖ πρὸς τὴν ΒΓ.</s> <s><pb></pb> <lb n="1799,001,,,,12,18,33"></lb> |
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15843 Λέγω δή, ὅτι οὐδὲ ἡ ΑΒΓ σφαῖρα πρὸς μείζονά τινα τῆς <lb n="1799,001,,,,12,18,34"></lb> |
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15844 ΔΕΖ σφαίρας τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΓ πρὸς <lb n="1799,001,,,,12,18,35"></lb> |
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15845 τὴν ΕΖ.</s> <lb n="1799,001,,,,12,18,36"></lb> |
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15846 <s>Εἰ γὰρ δυνατόν, ἐχέτω πρὸς μείζονα τὴν ΛΜΝ· ἀνά-<lb n="1799,001,,,,12,18,37"></lb> |
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15847 παλιν ἄρα ἡ ΛΜΝ σφαῖρα πρὸς τὴν ΑΒΓ σφαῖραν τρι-<lb n="1799,001,,,,12,18,38"></lb> |
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15848 πλασίονα λόγον ἔχει ἤπερ ἡ ΕΖ διάμετρος πρὸς τὴν ΒΓ <lb n="1799,001,,,,12,18,39"></lb> |
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15849 διάμετρον.</s> <s>ὡς δὲ ἡ ΛΜΝ σφαῖρα πρὸς τὴν ΑΒΓ σφαῖραν, <lb n="1799,001,,,,12,18,40"></lb> |
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15850 οὕτως ἡ ΔΕΖ σφαῖρα πρὸς ἐλάσσονά τινα τῆς ΑΒΓ σφαί-<lb n="1799,001,,,,12,18,41"></lb> |
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15851 ρας, ἐπειδήπερ μείζων ἐστὶν ἡ ΛΜΝ τῆς ΔΕΖ, ὡς <lb n="1799,001,,,,12,18,42"></lb> |
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15852 ἔμπροσθεν ἐδείχθη.</s> <s>καὶ ἡ ΔΕΖ ἄρα σφαῖρα πρὸς ἐλάσ-<lb n="1799,001,,,,12,18,43"></lb> |
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15853 σονά τινα τῆς ΑΒΓ σφαίρας τριπλασίονα λόγον ἔχει ἤπερ <lb n="1799,001,,,,12,18,44"></lb> |
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15854 ἡ ΕΖ πρὸς τὴν ΒΓ· ὅπερ ἀδύνατον ἐδείχθη.</s> <s>οὐκ ἄρα ἡ <lb n="1799,001,,,,12,18,45"></lb> |
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15855 ΑΒΓ σφαῖρα πρὸς μείζονά τινα τῆς ΔΕΖ σφαίρας τρι-<lb n="1799,001,,,,12,18,46"></lb> |
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15856 πλασίονα λόγον ἔχει ἤπερ ἡ ΒΓ πρὸς τὴν ΕΖ.</s> <s>ἐδείχθη δέ, <lb n="1799,001,,,,12,18,47"></lb> |
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15857 ὅτι οὐδὲ πρὸς ἐλάσσονα.</s> <s>ἡ ἄρα ΑΒΓ σφαῖρα πρὸς τὴν <lb n="1799,001,,,,12,18,48"></lb> |
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15858 ΔΕΖ σφαῖραν τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΓ πρὸς <lb n="1799,001,,,,12,18,49"></lb> |
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15859 τὴν ΕΖ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,12,18,50"></lb> |
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15860 </s></p> |
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15861 </chap> |
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15862 <chap><p type="main"><s>Ἐὰν εὐθεῖα γραμμὴ ἄκρον καὶ μέσον λόγον τμηθῇ, τὸ <lb n="1799,001,,,,13,1,1"></lb> |
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15863 |
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15864 μεῖζον τμῆμα προσλαβὸν τὴν ἡμίσειαν τῆς ὅλης πεντα-<lb n="1799,001,,,,13,1,2"></lb> |
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15865 πλάσιον δύναται τοῦ ἀπὸ τῆς ἡμισείας τετραγώνου.</s> <lb n="1799,001,,,,13,1,3"></lb> |
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15866 <s>Εὐθεῖα γὰρ γραμμὴ ἡ ΑΒ ἄκρον καὶ μέσον λόγον τετμή-<lb n="1799,001,,,,13,1,4"></lb> |
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15867 σθω κατὰ τὸ Γ σημεῖον, καὶ ἔστω μεῖζον τμῆμα τὸ ΑΓ, <lb n="1799,001,,,,13,1,5"></lb> |
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15868 καὶ ἐκβεβλήσθω ἐπ' εὐθείας τῇ ΓΑ εὐθεῖα ἡ ΑΔ, καὶ <lb n="1799,001,,,,13,1,6"></lb> |
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15869 κείσθω τῆς ΑΒ ἡμίσεια ἡ ΑΔ· λέγω, ὅτι πενταπλάσιόν <lb n="1799,001,,,,13,1,7"></lb> |
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15870 ἐστι τὸ ἀπὸ τῆς ΓΔ τοῦ ἀπὸ τῆς ΔΑ. <lb n="1799,001,,,,13,1,8"></lb> |
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15871 Ἀναγεγράφθωσαν γὰρ ἀπὸ τῶν <lb n="1799,001,,,,13,1,9"></lb> |
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15872 ΑΒ, ΔΓ τετράγωνα τὰ ΑΕ, ΔΖ, <lb n="1799,001,,,,13,1,10"></lb> |
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15873 καὶ καταγεγράφθω ἐν τῷ ΔΖ τὸ <lb n="1799,001,,,,13,1,11"></lb> |
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15874 σχῆμα, καὶ διήχθω ἡ ΖΓ ἐπὶ τὸ Η.</s> <lb n="1799,001,,,,13,1,12"></lb> |
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15875 <s>καὶ ἐπεὶ ἡ ΑΒ ἄκρον καὶ μέσον λό-<lb n="1799,001,,,,13,1,13"></lb> |
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15876 γον τέτμηται κατὰ τὸ Γ, τὸ ἄρα <lb n="1799,001,,,,13,1,14"></lb> |
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15877 ὑπὸ τῶν ΑΒΓ ἴσον ἐστὶ τῷ ἀπὸ <lb n="1799,001,,,,13,1,15"></lb> |
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15878 τῆς ΑΓ.</s> <s>καί ἐστι τὸ μὲν ὑπὸ τῶν <lb n="1799,001,,,,13,1,16"></lb> |
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15879 ΑΒΓ τὸ ΓΕ, τὸ δὲ ἀπὸ τῆς ΑΓ <lb n="1799,001,,,,13,1,17"></lb> |
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15880 τὸ ΖΘ· ἴσον ἄρα τὸ ΓΕ τῷ ΖΘ.</s> <s><pb></pb> <lb n="1799,001,,,,13,1,18"></lb> |
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15881 καὶ ἐπεὶ διπλῆ ἐστιν ἡ ΒΑ τῆς ΑΔ, ἴση δὲ ἡ μὲν ΒΑ τῇ <lb n="1799,001,,,,13,1,19"></lb> |
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15882 ΚΑ, ἡ δὲ ΑΔ τῇ ΑΘ, διπλῆ ἄρα καὶ ἡ ΚΑ τῆς ΑΘ.</s> <s>ὡς <lb n="1799,001,,,,13,1,20"></lb> |
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15883 δὲ ἡ ΚΑ πρὸς τὴν ΑΘ, οὕτως τὸ ΓΚ πρὸς τὸ ΓΘ· διπλά-<lb n="1799,001,,,,13,1,21"></lb> |
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15884 σιον ἄρα τὸ ΓΚ τοῦ ΓΘ.</s> <s>εἰσὶ δὲ καὶ τὰ ΛΘ, ΘΓ διπλά-<lb n="1799,001,,,,13,1,22"></lb> |
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15885 σια τοῦ ΓΘ.</s> <s>ἴσον ἄρα τὸ ΚΓ τοῖς ΛΘ, ΘΓ.</s> <s>ἐδείχθη δὲ καὶ <lb n="1799,001,,,,13,1,23"></lb> |
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15886 τὸ ΓΕ τῷ ΘΖ ἴσον· ὅλον ἄρα τὸ ΑΕ τετράγωνον ἴσον <lb n="1799,001,,,,13,1,24"></lb> |
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15887 ἐστὶ τῷ ΜΝΞ γνώμονι.</s> <s>καὶ ἐπεὶ διπλῆ ἐστιν ἡ ΒΑ τῆς <lb n="1799,001,,,,13,1,25"></lb> |
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15888 ΑΔ, τετραπλάσιόν ἐστι τὸ ἀπὸ τῆς ΒΑ τοῦ ἀπὸ τῆς ΑΔ, <lb n="1799,001,,,,13,1,26"></lb> |
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15889 τουτέστι τὸ ΑΕ τοῦ ΔΘ.</s> <s>ἴσον δὲ τὸ ΑΕ τῷ ΜΝΞ γνώ-<lb n="1799,001,,,,13,1,27"></lb> |
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15890 μονι· καὶ ὁ ΜΝΞ ἄρα γνώμων τετραπλάσιός ἐστι τοῦ <lb n="1799,001,,,,13,1,28"></lb> |
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15891 ΑΟ· ὅλον ἄρα τὸ ΔΖ πενταπλάσιόν ἐστι τοῦ ΑΟ.</s> <s>καί <lb n="1799,001,,,,13,1,29"></lb> |
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15892 ἐστι τὸ μὲν ΔΖ τὸ ἀπὸ τῆς ΔΓ, τὸ δὲ ΑΟ τὸ ἀπὸ τῆς ΔΑ· <lb n="1799,001,,,,13,1,30"></lb> |
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15893 τὸ ἄρα ἀπὸ τῆς ΓΔ πενταπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΔΑ.</s> <lb n="1799,001,,,,13,1,31"></lb> |
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15894 <s>Ἐὰν ἄρα εὐθεῖα ἄκρον καὶ μέσον λόγον τμηθῇ, τὸ μεῖ-<lb n="1799,001,,,,13,1,32"></lb> |
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15895 ζον τμῆμα προσλαβὸν τὴν ἡμίσειαν τῆς ὅλης πενταπλά-<lb n="1799,001,,,,13,1,33"></lb> |
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15896 σιον δύναται τοῦ ἀπὸ τῆς ἡμισείας τετραγώνου· ὅπερ ἔδει <lb n="1799,001,,,,13,1,34"></lb> |
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15897 δεῖξαι.</s> <lb n="1799,001,,,,13,1,35"></lb> |
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15898 <s>Ἐὰν εὐθεῖα γραμμὴ τμήματος ἑαυτῆς πενταπλάσιον <lb n="1799,001,,,,13,2,1"></lb> |
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15899 δύνηται, τῆς διπλασίας τοῦ εἰρημένου τμήματος ἄκρον <lb n="1799,001,,,,13,2,2"></lb> |
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15900 καὶ μέσον λόγον τεμνομένης τὸ μεῖζον τμῆμα τὸ λοιπὸν <lb n="1799,001,,,,13,2,3"></lb> |
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15901 μέρος ἐστὶ τῆς ἐξ ἀρχῆς εὐθείας.</s> <lb n="1799,001,,,,13,2,4"></lb> |
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15902 <s>Εὐθεῖα γὰρ γραμμὴ ἡ ΑΒ τμήματος ἑαυτῆς τοῦ ΑΓ <lb n="1799,001,,,,13,2,5"></lb> |
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15903 πενταπλάσιον δυνάσθω, τῆς δὲ ΑΓ διπλῆ ἔστω ἡ ΓΔ· <pb></pb> <lb n="1799,001,,,,13,2,6"></lb> |
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15904 λέγω, ὅτι τῆς ΓΔ ἄκρον καὶ μέσον λόγον τεμνομένης τὸ <lb n="1799,001,,,,13,2,7"></lb> |
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15905 μεῖζον τμῆμά ἐστιν ἡ ΓΒ.</s> <lb n="1799,001,,,,13,2,8"></lb> |
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15906 <s>Ἀναγεγράφθω γὰρ ἀφ' ἑκατέρας τῶν ΑΒ, ΓΔ τετρά-<lb n="1799,001,,,,13,2,9"></lb> |
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15907 γωνα τὰ ΑΖ, ΓΗ, καὶ καταγεγράφθω ἐν τῷ ΑΖ τὸ <lb n="1799,001,,,,13,2,10"></lb> |
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15908 σχῆμα, καὶ διήχθω ἡ ΒΕ.</s> <s>καὶ ἐπεὶ πενταπλάσιόν ἐστι <lb n="1799,001,,,,13,2,11"></lb> |
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15909 τὸ ἀπὸ τῆς ΒΑ τοῦ ἀπὸ τῆς ΑΓ, πενταπλάσιόν ἐστι τὸ <lb n="1799,001,,,,13,2,12"></lb> |
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15910 ΑΖ τοῦ ΑΘ.</s> <s>τετραπλάσιος ἄρα ὁ ΜΝΞ γνώμων τοῦ ΑΘ.</s> <lb n="1799,001,,,,13,2,13"></lb> |
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15911 <s>καὶ ἐπεὶ διπλη̂ ἐστιν ἡ ΔΓ τῆς ΓΑ, τετραπλάσιον ἄρα <lb n="1799,001,,,,13,2,14"></lb> |
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15912 ἐστὶ τὸ ἀπὸ ΔΓ τοῦ ἀπὸ ΓΑ, τουτέστι τὸ ΓΗ τοῦ ΑΘ.</s> <lb n="1799,001,,,,13,2,15"></lb> |
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15913 <s>ἐδείχθη δὲ καὶ ὁ ΜΝΞ γνώμων τετραπλάσιος τοῦ ΑΘ· <lb n="1799,001,,,,13,2,16"></lb> |
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15914 ἴσος ἄρα ὁ ΜΝΞ γνώμων τῷ ΓΗ.</s> <s>καὶ ἐπεὶ διπλῆ ἐστιν <lb n="1799,001,,,,13,2,17"></lb> |
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15915 ἡ ΔΓ τῆς ΓΑ, ἴση δὲ ἡ μὲν ΔΓ τῇ ΓΚ, ἡ δὲ ΑΓ τῇ ΓΘ <lb n="1799,001,,,,13,2,18"></lb> |
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15916 [διπλῆ ἄρα καὶ ἡ ΚΓ τῆς ΓΘ], διπλάσιον ἄρα καὶ τὸ ΚΒ <lb n="1799,001,,,,13,2,19"></lb> |
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15917 τοῦ ΒΘ.</s> <s>εἰσὶ δὲ καὶ τὰ ΛΘ, ΘΒ τοῦ ΘΒ διπλάσια· ἴσον <lb n="1799,001,,,,13,2,20"></lb> |
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15918 ἄρα τὸ ΚΒ τοῖς ΛΘ, ΘΒ. <lb n="1799,001,,,,13,2,21"></lb> |
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15919 ἐδείχθη δὲ καὶ ὅλος ὁ ΜΝΞ <lb n="1799,001,,,,13,2,22"></lb> |
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15920 γνώμων ὅλῳ τῷ ΓΗ ἴσος· καὶ <lb n="1799,001,,,,13,2,23"></lb> |
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15921 λοιπὸν ἄρα τὸ ΘΖ τῷ ΒΗ <lb n="1799,001,,,,13,2,24"></lb> |
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15922 ἐστιν ἴσον.</s> <s>καί ἐστι τὸ μὲν <lb n="1799,001,,,,13,2,25"></lb> |
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15923 ΒΗ τὸ ὑπὸ τῶν ΓΔΒ· ἴση <lb n="1799,001,,,,13,2,26"></lb> |
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15924 γὰρ ἡ ΓΔ τῇ ΔΗ· τὸ δὲ ΘΖ <lb n="1799,001,,,,13,2,27"></lb> |
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15925 τὸ ἀπὸ τῆς ΓΒ· τὸ ἄρα ὑπὸ <lb n="1799,001,,,,13,2,28"></lb> |
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15926 τῶν ΓΔΒ ἴσον ἐστὶ τῷ ἀπὸ <lb n="1799,001,,,,13,2,29"></lb> |
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15927 τῆς ΓΒ.</s> <s>ἔστιν ἄρα ὡς ἡ ΔΓ <lb n="1799,001,,,,13,2,30"></lb> |
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15928 πρὸς τὴν ΓΒ, οὕτως ἡ ΓΒ <lb n="1799,001,,,,13,2,31"></lb> |
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15929 πρὸς τὴν ΒΔ.</s> <s>μείζων δὲ ἡ ΔΓ <pb></pb> <lb n="1799,001,,,,13,2,32"></lb> |
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15930 τῆς ΓΒ· μείζων ἄρα καὶ ἡ ΓΒ τῆς ΒΔ.</s> <s>τῆς ΓΔ ἄρα εὐθείας <lb n="1799,001,,,,13,2,33"></lb> |
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15931 ἄκρον καὶ μέσον λόγον τεμνομένης τὸ μεῖζον τμῆμά <lb n="1799,001,,,,13,2,34"></lb> |
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15932 ἐστιν ἡ ΓΒ.</s> <lb n="1799,001,,,,13,2,35"></lb> |
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15933 <s>Ἐὰν ἄρα εὐθεῖα γραμμὴ τμήματος ἑαυτῆς πενταπλάσιον <lb n="1799,001,,,,13,2,36"></lb> |
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15934 δύνηται, τῆς διπλασίας τοῦ εἰρημένου τμήματος ἄκρον <lb n="1799,001,,,,13,2,37"></lb> |
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15935 καὶ μέσον λόγον τεμνομένης τὸ μεῖζον τμῆμα τὸ λοιπὸν <lb n="1799,001,,,,13,2,38"></lb> |
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15936 μέρος ἐστὶ τῆς ἐξ ἀρχῆς εὐθείας· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,2,39"></lb> |
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15937 <s>Λῆμμα <lb n="1799,001,,,,13,2,40"></lb> |
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15938 Ὅτι δὲ ἡ διπλῆ τῆς ΑΓ μείζων ἐστὶ τῆς ΒΓ, οὕτως <lb n="1799,001,,,,13,2,41"></lb> |
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15939 δεικτέον.</s> <lb n="1799,001,,,,13,2,42"></lb> |
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15940 <s>Εἰ γὰρ μή, ἔστω, εἰ δυνατόν, ἡ ΒΓ διπλῆ τῆς ΓΑ.</s> <lb n="1799,001,,,,13,2,43"></lb> |
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15941 <s>τετραπλάσιον ἄρα τὸ ἀπὸ τῆς ΒΓ τοῦ ἀπὸ τῆς ΓΑ· <lb n="1799,001,,,,13,2,44"></lb> |
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15942 πενταπλάσια ἄρα τὰ ἀπὸ τῶν ΒΓ, ΓΑ τοῦ ἀπὸ τῆς <lb n="1799,001,,,,13,2,45"></lb> |
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15943 ΓΑ.</s> <s>ὑπόκειται δὲ καὶ τὸ ἀπὸ τῆς ΒΑ πενταπλάσιον <lb n="1799,001,,,,13,2,46"></lb> |
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15944 τοῦ ἀπὸ τῆς ΓΑ· τὸ ἄρα ἀπὸ τῆς ΒΑ ἴσον ἐστὶ τοῖς ἀπὸ <lb n="1799,001,,,,13,2,47"></lb> |
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15945 τῶν ΒΓ, ΓΑ· ὅπερ ἀδύνατον.</s> <s>οὐκ ἄρα ἡ ΓΒ διπλασία <lb n="1799,001,,,,13,2,48"></lb> |
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15946 ἐστὶ τῆς ΑΓ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι οὐδὲ ἡ ἐλάττων τῆς <lb n="1799,001,,,,13,2,49"></lb> |
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15947 ΓΒ διπλασίων ἐστὶ τῆς ΓΑ· πολλῷ γὰρ [μεῖζον] τὸ ἄτοπον.</s> <lb n="1799,001,,,,13,2,50"></lb> |
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15948 <s>Ἡ ἄρα τῆς ΑΓ διπλῆ μείζων ἐστὶ τῆς ΓΒ· ὅπερ ἔδει <lb n="1799,001,,,,13,2,51"></lb> |
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15949 δεῖξαι.</s> <lb n="1799,001,,,,13,2,52"></lb> |
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15950 <s>Ἐὰν εὐθεῖα γραμμὴ ἄκρον καὶ μέσον λόγον τμηθῇ, τὸ <lb n="1799,001,,,,13,3,1"></lb> |
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15951 ἔλασσον τμῆμα προσλαβὸν τὴν ἡμίσειαν τοῦ μείζονος τμή-<lb n="1799,001,,,,13,3,2"></lb> |
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15952 ματος πενταπλάσιον δύναται τοῦ ἀπὸ τῆς ἡμισείας τοῦ <lb n="1799,001,,,,13,3,3"></lb> |
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15953 μείζονος τμήματος τετραγώνου.</s> <s><pb></pb> <lb n="1799,001,,,,13,3,4"></lb> |
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15954 Εὐθεῖα γάρ τις ἡ ΑΒ ἄκρον καὶ μέσον λόγον τετμήσθω <lb n="1799,001,,,,13,3,5"></lb> |
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15955 κατὰ τὸ Γ σημεῖον, καὶ ἔστω μεῖζον τμῆμα τὸ ΑΓ, καὶ <lb n="1799,001,,,,13,3,6"></lb> |
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15956 τετμήσθω ἡ ΑΓ δίχα κατὰ τὸ Δ· λέγω, ὅτι πενταπλά-<lb n="1799,001,,,,13,3,7"></lb> |
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15957 σιόν ἐστι τὸ ἀπὸ τῆς ΒΔ τοῦ ἀπὸ τῆς ΔΓ.</s> <lb n="1799,001,,,,13,3,8"></lb> |
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15958 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΕ, <lb n="1799,001,,,,13,3,9"></lb> |
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15959 καὶ καταγεγράφθω διπλοῦν τὸ σχῆμα.</s> <s>ἐπεὶ διπλῆ ἐστιν <lb n="1799,001,,,,13,3,10"></lb> |
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15960 ἡ ΑΓ τῆς ΔΓ, τετραπλάσιον ἄρα τὸ ἀπὸ τῆς ΑΓ τοῦ <lb n="1799,001,,,,13,3,11"></lb> |
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15961 ἀπὸ τῆς ΔΓ, τουτέστι τὸ ΡΣ τοῦ ΖΗ.</s> <s>καὶ ἐπεὶ τὸ ὑπὸ τῶν <lb n="1799,001,,,,13,3,12"></lb> |
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15962 ΑΒΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΓ, καί ἐστι τὸ ὑπὸ τῶν <lb n="1799,001,,,,13,3,13"></lb> |
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15963 ΑΒΓ τὸ ΓΕ, τὸ ἄρα ΓΕ ἴσον ἐστὶ τῷ ΡΣ.</s> <s>τετραπλάσιον <lb n="1799,001,,,,13,3,14"></lb> |
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15964 δὲ τὸ ΡΣ τοῦ ΖΗ· τετραπλάσιον ἄρα καὶ τὸ ΓΕ τοῦ <lb n="1799,001,,,,13,3,15"></lb> |
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15965 ΖΗ.</s> <s>πάλιν ἐπεὶ ἴση ἐστὶν ἡ ΑΔ τῇ ΔΓ, ἴση ἐστὶ καὶ ἡ <lb n="1799,001,,,,13,3,16"></lb> |
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15966 ΘΚ τῇ ΚΖ.</s> <s>ὥστε καὶ τὸ ΗΖ τετράγωνον ἴσον ἐστὶ τῷ ΘΛ <lb n="1799,001,,,,13,3,17"></lb> |
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15967 τετραγώνῳ.</s> <s>ἴση ἄρα ἡ ΗΚ τῇ ΚΛ, τουτέστιν ἡ ΜΝ τῇ <lb n="1799,001,,,,13,3,18"></lb> |
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15968 ΝΕ· ὥστε καὶ τὸ ΜΖ τῷ ΖΕ ἐστιν ἴσον.</s> <s>ἀλλὰ τὸ ΜΖ τῷ <lb n="1799,001,,,,13,3,19"></lb> |
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15969 ΓΗ ἐστιν ἴσον· καὶ τὸ ΓΗ <lb n="1799,001,,,,13,3,20"></lb> |
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15970 ἄρα τῷ ΖΕ ἐστιν ἴσον.</s> <s>κοινὸν <lb n="1799,001,,,,13,3,21"></lb> |
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15971 προσκείσθω τὸ ΓΝ· ὁ ἄρα <lb n="1799,001,,,,13,3,22"></lb> |
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15972 ΞΟΠ γνώμων ἴσος ἐστὶ τῷ <lb n="1799,001,,,,13,3,23"></lb> |
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15973 ΓΕ.</s> <s>ἀλλὰ τὸ ΓΕ τετραπλά-<lb n="1799,001,,,,13,3,24"></lb> |
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15974 σιον ἐδείχθη τοῦ ΗΖ· καὶ ὁ <lb n="1799,001,,,,13,3,25"></lb> |
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15975 ΞΟΠ ἄρα γνώμων τετρα-<lb n="1799,001,,,,13,3,26"></lb> |
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15976 πλάσιός ἐστι τοῦ ΖΗ τετρα-<lb n="1799,001,,,,13,3,27"></lb> |
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15977 γώνου.</s> <s>ὁ ΞΟΠ ἄρα γνώμων <lb n="1799,001,,,,13,3,28"></lb> |
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15978 καὶ τὸ ΖΗ τετράγωνον πεντα- <pb></pb><lb n="1799,001,,,,13,3,29"></lb> |
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15979 πλάσιός ἐστι τοῦ ΖΗ.</s> <s>ἀλλὰ ὁ ΞΟΠ γνώμων καὶ τὸ <lb n="1799,001,,,,13,3,30"></lb> |
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15980 ΖΗ τετράγωνόν ἐστι τὸ ΔΝ.</s> <s>καί ἐστι τὸ μὲν ΔΝ τὸ ἀπὸ <lb n="1799,001,,,,13,3,31"></lb> |
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15981 τῆς ΔΒ, τὸ δὲ ΗΖ τὸ ἀπὸ τῆς ΔΓ.</s> <s>τὸ ἄρα ἀπὸ τῆς ΔΒ <lb n="1799,001,,,,13,3,32"></lb> |
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15982 πενταπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΔΓ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,3,33"></lb> |
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15983 <s>Ἐὰν εὐθεῖα γραμμὴ ἄκρον καὶ μέσον λόγον τμηθῇ, τὸ <lb n="1799,001,,,,13,4,1"></lb> |
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15984 ἀπὸ τῆς ὅλης καὶ τοῦ ἐλάσσονος τμήματος, τὰ συναμφό-<lb n="1799,001,,,,13,4,2"></lb> |
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15985 τερα τετράγωνα, τριπλάσιά ἐστι τοῦ ἀπὸ τοῦ μείζονος <lb n="1799,001,,,,13,4,3"></lb> |
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15986 τμήματος τετραγώνου.</s> <lb n="1799,001,,,,13,4,4"></lb> |
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15987 <s>Ἔστω εὐθεῖα ἡ ΑΒ, καὶ τετμήσθω ἄκρον καὶ μέσον <lb n="1799,001,,,,13,4,5"></lb> |
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15988 λόγον κατὰ τὸ Γ, καὶ ἔστω μεῖζον τμῆμα τὸ ΑΓ· λέγω, <lb n="1799,001,,,,13,4,6"></lb> |
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15989 ὅτι τὰ ἀπὸ τῶν ΑΒ, ΒΓ τριπλάσιά ἐστι τοῦ ἀπὸ τῆς ΓΑ.</s> <lb n="1799,001,,,,13,4,7"></lb> |
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15990 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔΕΒ, <lb n="1799,001,,,,13,4,8"></lb> |
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15991 καὶ καταγεγράφθω τὸ σχῆμα.</s> <s>ἐπεὶ οὖν ἡ ΑΒ ἄκρον καὶ <lb n="1799,001,,,,13,4,9"></lb> |
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15992 μέσον λόγον τέτμηται κατὰ τὸ Γ, καὶ τὸ μεῖζον τμῆμά <lb n="1799,001,,,,13,4,10"></lb> |
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15993 ἐστιν ἡ ΑΓ, τὸ ἄρα ὑπὸ τῶν <lb n="1799,001,,,,13,4,11"></lb> |
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15994 ΑΒΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΓ.</s> <lb n="1799,001,,,,13,4,12"></lb> |
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15995 <s>καί ἐστι τὸ μὲν ὑπὸ τῶν ΑΒΓ τὸ <lb n="1799,001,,,,13,4,13"></lb> |
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15996 ΑΚ, τὸ δὲ ἀπὸ τῆς ΑΓ τὸ ΘΗ· <lb n="1799,001,,,,13,4,14"></lb> |
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15997 ἴσον ἄρα ἐστὶ τὸ ΑΚ τῷ ΘΗ.</s> <lb n="1799,001,,,,13,4,15"></lb> |
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15998 <s>καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΑΖ τῷ ΖΕ, <lb n="1799,001,,,,13,4,16"></lb> |
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15999 κοινὸν προσκείσθω τὸ ΓΚ· ὅλον <lb n="1799,001,,,,13,4,17"></lb> |
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16000 ἄρα τὸ ΑΚ ὅλῳ τῷ ΓΕ ἐστιν <lb n="1799,001,,,,13,4,18"></lb> |
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16001 ἴσον· τὰ ἄρα ΑΚ, ΓΕ τοῦ ΑΚ <pb></pb> <lb n="1799,001,,,,13,4,19"></lb> |
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16002 ἐστι διπλάσια.</s> <s>ἀλλὰ τὰ ΑΚ, ΓΕ ὁ ΛΜΝ γνώμων ἐστὶ <lb n="1799,001,,,,13,4,20"></lb> |
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16003 καὶ τὸ ΓΚ τετράγωνον· ὁ ἄρα ΛΜΝ γνώμων καὶ τὸ ΓΚ <lb n="1799,001,,,,13,4,21"></lb> |
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16004 τετράγωνον διπλάσιά ἐστι τοῦ ΑΚ.</s> <s>ἀλλὰ μὴν καὶ τὸ ΑΚ <lb n="1799,001,,,,13,4,22"></lb> |
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16005 τῷ ΘΗ ἐδείχθη ἴσον· ὁ ἄρα ΛΜΝ γνώμων καὶ [τὸ ΓΚ <lb n="1799,001,,,,13,4,23"></lb> |
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16006 τετράγωνον διπλάσιά ἐστι τοῦ ΘΗ· ὥστε ὁ ΛΜΝ γνώ-<lb n="1799,001,,,,13,4,24"></lb> |
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16007 μων καὶ] τὰ ΓΚ, ΘΗ τετράγωνα τριπλάσιά ἐστι τοῦ ΘΗ <lb n="1799,001,,,,13,4,25"></lb> |
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16008 τετραγώνου.</s> <s>καί ἐστιν ὁ [μὲν] ΛΜΝ γνώμων καὶ τὰ ΓΚ, <lb n="1799,001,,,,13,4,26"></lb> |
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16009 ΘΗ τετράγωνα ὅλον τὸ ΑΕ καὶ τὸ ΓΚ, ἅπερ ἐστὶ τὰ ἀπὸ <lb n="1799,001,,,,13,4,27"></lb> |
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16010 τῶν ΑΒ, ΒΓ τετράγωνα, τὸ δὲ ΗΘ τὸ ἀπὸ τῆς ΑΓ τετρά-<lb n="1799,001,,,,13,4,28"></lb> |
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16011 γωνον.</s> <s>τὰ ἄρα ἀπὸ τῶν ΑΒ, ΒΓ τετράγωνα τριπλάσιά <lb n="1799,001,,,,13,4,29"></lb> |
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16012 ἐστι τοῦ ἀπὸ τῆς ΑΓ τετραγώνου· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,4,30"></lb> |
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16013 <s>Ἐὰν εὐθεῖα γραμμὴ ἄκρον καὶ μέσον λόγον τμηθῇ, καὶ <lb n="1799,001,,,,13,5,1"></lb> |
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16014 προστεθῇ αὐτῇ ἴση τῷ μείζονι τμήματι, ἡ ὅλη εὐθεῖα <lb n="1799,001,,,,13,5,2"></lb> |
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16015 ἄκρον καὶ μέσον λόγον τέτμηται, καὶ τὸ μεῖζον τμῆμά <lb n="1799,001,,,,13,5,3"></lb> |
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16016 ἐστιν ἡ ἐξ ἀρχῆς εὐθεῖα.</s> <lb n="1799,001,,,,13,5,4"></lb> |
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16017 <s>Εὐθεῖα γὰρ γραμμὴ ἡ ΑΒ ἄκρον καὶ μέσον λόγον τε-<lb n="1799,001,,,,13,5,5"></lb> |
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16018 τμήσθω κατὰ τὸ Γ σημεῖον, καὶ ἔστω μεῖζον τμῆμα ἡ ΑΓ, <lb n="1799,001,,,,13,5,6"></lb> |
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16019 καὶ τῇ ΑΓ ἴση [κείσθω] ἡ ΑΔ.</s> <s>λέγω, ὅτι ἡ ΔΒ εὐθεῖα <lb n="1799,001,,,,13,5,7"></lb> |
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16020 ἄκρον καὶ μέσον λόγον τέτμηται κατὰ τὸ Α, καὶ τὸ μεῖζον <lb n="1799,001,,,,13,5,8"></lb> |
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16021 τμῆμά ἐστιν ἡ ἐξ ἀρχῆς εὐθεῖα ἡ ΑΒ.</s> <lb n="1799,001,,,,13,5,9"></lb> |
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16022 <s>Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΕ, <lb n="1799,001,,,,13,5,10"></lb> |
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16023 καὶ καταγεγράφθω τὸ σχῆμα.</s> <s>ἐπεὶ ἡ ΑΒ ἄκρον καὶ μέσον <pb></pb> <lb n="1799,001,,,,13,5,11"></lb> |
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16024 λόγον τέτμηται κατὰ τὸ Γ, τὸ ἄρα ὑπὸ ΑΒΓ ἴσον ἐστὶ <lb n="1799,001,,,,13,5,12"></lb> |
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16025 τῷ ἀπὸ ΑΓ.</s> <s>καί ἐστι τὸ μὲν ὑπὸ ΑΒΓ τὸ ΓΕ, τὸ δὲ ἀπὸ <lb n="1799,001,,,,13,5,13"></lb> |
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16026 τῆς ΑΓ τὸ ΓΘ· ἴσον ἄρα τὸ ΓΕ τῷ ΘΓ.</s> <s>ἀλλὰ τῷ μὲν <lb n="1799,001,,,,13,5,14"></lb> |
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16027 ΓΕ ἴσον ἐστὶ τὸ ΘΕ, τῷ δὲ ΘΓ ἴσον τὸ ΔΘ· καὶ τὸ ΔΘ <lb n="1799,001,,,,13,5,15"></lb> |
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16028 ἄρα ἴσον ἐστὶ τῷ ΘΕ <lb n="1799,001,,,,13,5,16"></lb> |
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16029 [κοινὸν προσκείσθω τὸ <lb n="1799,001,,,,13,5,17"></lb> |
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16030 ΘΒ].</s> <s>ὅλον ἄρα τὸ ΔΚ ὅλῳ <lb n="1799,001,,,,13,5,18"></lb> |
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16031 τῷ ΑΕ ἐστιν ἴσον.</s> <s>καί <lb n="1799,001,,,,13,5,19"></lb> |
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16032 ἐστι τὸ μὲν ΔΚ τὸ ὑπὸ <lb n="1799,001,,,,13,5,20"></lb> |
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16033 τῶν ΒΔ, ΔΑ· ἴση γὰρ ἡ <lb n="1799,001,,,,13,5,21"></lb> |
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16034 ΑΔ τῇ ΔΛ· τὸ δὲ ΑΕ <lb n="1799,001,,,,13,5,22"></lb> |
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16035 τὸ ἀπὸ τῆς ΑΒ· τὸ ἄρα <lb n="1799,001,,,,13,5,23"></lb> |
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16036 ὑπὸ τῶν ΒΔΑ ἴσον ἐστὶ <lb n="1799,001,,,,13,5,24"></lb> |
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16037 τῷ ἀπὸ τῆς ΑΒ.</s> <s>ἔστιν ἄρα ὡς ἡ ΔΒ πρὸς τὴν ΒΑ, <lb n="1799,001,,,,13,5,25"></lb> |
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16038 οὕτως ἡ ΒΑ πρὸς τὴν ΑΔ.</s> <s>μείζων δὲ ἡ ΔΒ τῆς ΒΑ· <lb n="1799,001,,,,13,5,26"></lb> |
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16039 μείζων ἄρα καὶ ἡ ΒΑ τῆς ΑΔ.</s> <lb n="1799,001,,,,13,5,27"></lb> |
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16040 <s>Ἡ ἄρα ΔΒ ἄκρον καὶ μέσον λόγον τέμηται κατὰ τὸ Α, <lb n="1799,001,,,,13,5,28"></lb> |
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16041 καὶ τὸ μεῖζον τμῆμά ἐστιν ἡ ΑΒ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,5,29"></lb> |
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16042 <s>Ἐὰν εὐθεῖα ῥητὴ ἄκρον καὶ μέσον λόγον τμηθῇ, <lb n="1799,001,,,,13,6,1"></lb> |
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16043 ἑκάτερον τῶν τμημάτων ἄλογός ἐστιν ἡ καλουμένη ἀπο-<lb n="1799,001,,,,13,6,2"></lb> |
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16044 τομή.</s> <s><pb></pb> <lb n="1799,001,,,,13,6,3"></lb> |
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16045 Ἔστω εὐθεῖα ῥητὴ ἡ ΑΒ καὶ τετμήσθω ἄκρον καὶ μέσον <lb n="1799,001,,,,13,6,4"></lb> |
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16046 λόγον κατὰ τὸ Γ, καὶ ἔστω μεῖζον τμῆμα ἡ ΑΓ· λέγω, ὅτι <lb n="1799,001,,,,13,6,5"></lb> |
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16047 ἑκατέρα τῶν ΑΓ, ΓΒ ἄλογός ἐστιν ἡ καλουμένη ἀποτομή.</s> <lb n="1799,001,,,,13,6,6"></lb> |
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16048 <s>Ἐκβεβλήσθω γὰρ ἡ ΒΑ, καὶ κείσθω τῆς ΒΑ ἡμίσεια ἡ <lb n="1799,001,,,,13,6,7"></lb> |
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16049 ΑΔ.</s> <s>ἐπεὶ οὖν εὐθεῖα ἡ ΑΒ τέτμηται ἄκρον καὶ μέσον <lb n="1799,001,,,,13,6,8"></lb> |
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16050 λόγον κατὰ τὸ Γ, καὶ τῷ μείζονι τμήματι τῷ ΑΓ πρόσ- <lb n="1799,001,,,,13,6,9"></lb> |
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16051 κειται ἡ ΑΔ ἡμίσεια οὖσα τῆς ΑΒ, τὸ ἄρα ἀπὸ ΓΔ τοῦ ἀπὸ <lb n="1799,001,,,,13,6,10"></lb> |
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16052 ΔΑ πενταπλάσιόν ἐστιν.</s> <s>τὸ ἄρα ἀπὸ ΓΔ πρὸς τὸ ἀπὸ ΔΑ <lb n="1799,001,,,,13,6,11"></lb> |
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16053 λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν· σύμμετρον ἄρα τὸ <lb n="1799,001,,,,13,6,12"></lb> |
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16054 ἀπὸ ΓΔ τῷ ἀπὸ ΔΑ.</s> <s>ῥητὸν δὲ τὸ ἀπὸ ΔΑ· ῥητὴ γὰρ <lb n="1799,001,,,,13,6,13"></lb> |
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16055 [ἐστιν] ἡ ΔΑ ἡμίσεια οὖσα τῆς ΑΒ ῥητῆς οὔσης· ῥητὸν <lb n="1799,001,,,,13,6,14"></lb> |
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16056 ἄρα καὶ τὸ ἀπὸ ΓΔ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΓΔ.</s> <s>καὶ ἐπεὶ τὸ <lb n="1799,001,,,,13,6,15"></lb> |
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16057 ἀπὸ ΓΔ πρὸς τὸ ἀπὸ ΔΑ λόγον οὐκ ἔχει, ὃν τετράγωνος <lb n="1799,001,,,,13,6,16"></lb> |
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16058 ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ἀσύμμετρος ἄρα μήκει <lb n="1799,001,,,,13,6,17"></lb> |
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16059 ἡ ΓΔ τῇ ΔΑ· αἱ ΓΔ, ΔΑ ἄρα ῥηταί εἰσι δυνάμει μόνον <lb n="1799,001,,,,13,6,18"></lb> |
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16060 σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΑΓ.</s> <s>πάλιν, ἐπεὶ ἡ ΑΒ <lb n="1799,001,,,,13,6,19"></lb> |
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16061 ἄκρον καὶ μέσον λόγον τέτμηται, καὶ τὸ μεῖζον τμῆμά <lb n="1799,001,,,,13,6,20"></lb> |
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16062 ἐστιν ἡ ΑΓ, τὸ ἄρα ὑπὸ ΑΒ, ΒΓ τῷ ἀπὸ ΑΓ ἴσον ἐστίν.</s> <lb n="1799,001,,,,13,6,21"></lb> |
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16063 <s>τὸ ἄρα ἀπὸ τῆς ΑΓ ἀποτομῆς παρὰ τὴν ΑΒ ῥητὴν παρα-<lb n="1799,001,,,,13,6,22"></lb> |
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16064 βληθὲν πλάτος ποιεῖ τὴν ΒΓ.</s> <s>τὸ δὲ ἀπὸ ἀποτομῆς παρὰ <lb n="1799,001,,,,13,6,23"></lb> |
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16065 ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν πρώτην· <lb n="1799,001,,,,13,6,24"></lb> |
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16066 ἀποτομὴ ἄρα πρώτη ἐστὶν ἡ ΓΒ.</s> <s>ἐδείχθη δὲ καὶ ἡ ΓΑ <lb n="1799,001,,,,13,6,25"></lb> |
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16067 ἀποτομή.</s> <s><pb></pb> <lb n="1799,001,,,,13,6,26"></lb> |
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16068 Ἐὰν ἄρα εὐθεῖα ῥητὴ ἄκρον καὶ μέσον λόγον τμηθῇ, <lb n="1799,001,,,,13,6,27"></lb> |
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16069 ἑκάτερον τῶν τμημάτων ἄλογός ἐστιν ἡ καλουμένη ἀπο-<lb n="1799,001,,,,13,6,28"></lb> |
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16070 τομή· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,6,29"></lb> |
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16071 <s>Ἐὰν πενταγώνου ἰσοπλεύρου αἱ τρεῖς γωνίαι ἤτοι αἱ <lb n="1799,001,,,,13,7,1"></lb> |
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16072 κατὰ τὸ ἑξῆς ἢ αἱ μὴ κατὰ τὸ ἑξῆς ἴσαι ὦσιν, ἰσογώνιον <lb n="1799,001,,,,13,7,2"></lb> |
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16073 ἔσται τὸ πεντάγωνον.</s> <lb n="1799,001,,,,13,7,3"></lb> |
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16074 <s>Πενταγώνου γὰρ ἰσοπλεύρου τοῦ ΑΒΓΔΕ αἱ τρεῖς <lb n="1799,001,,,,13,7,4"></lb> |
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16075 γωνίαι πρότερον αἱ κατὰ τὸ ἑξῆς αἱ πρὸς τοῖς Α, Β, Γ ἴσαι <lb n="1799,001,,,,13,7,5"></lb> |
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16076 ἀλλήλαις ἔστωσαν· λέγω, ὅτι ἰσογώνιόν ἐστι τὸ ΑΒΓΔΕ <lb n="1799,001,,,,13,7,6"></lb> |
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16077 πεντάγωνον.</s> <lb n="1799,001,,,,13,7,7"></lb> |
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16078 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΑΓ, ΒΕ, ΖΔ.</s> <s>καὶ ἐπεὶ δύο αἱ <lb n="1799,001,,,,13,7,8"></lb> |
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16079 ΓΒ, ΒΑ δυσὶ ταῖς ΒΑ, ΑΕ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ, <lb n="1799,001,,,,13,7,9"></lb> |
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16080 καὶ γωνία ἡ ὑπὸ ΓΒΑ γωνίᾳ τῇ ὑπὸ ΒΑΕ ἐστιν ἴση, <lb n="1799,001,,,,13,7,10"></lb> |
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16081 βάσις ἄρα ἡ ΑΓ βάσει τῇ ΒΕ ἐστιν ἴση, καὶ τὸ ΑΒΓ τρί-<lb n="1799,001,,,,13,7,11"></lb> |
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16082 γωνον τῷ ΑΒΕ τριγώνῳ ἴσον, καὶ αἱ λοιπαὶ γωνίαι ταῖς <lb n="1799,001,,,,13,7,12"></lb> |
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16083 λοιπαῖς γωνίαις ἴσαι ἔσονται, ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑποτεί-<lb n="1799,001,,,,13,7,13"></lb> |
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16084 νουσιν, ἡ μὲν ὑπὸ ΒΓΑ τῇ ὑπὸ ΒΕΑ, ἡ δὲ ὑπὸ ΑΒΕ τῇ <lb n="1799,001,,,,13,7,14"></lb> |
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16085 ὑπὸ ΓΑΒ· ὥστε καὶ πλευρὰ ἡ ΑΖ πλευρᾷ τῇ ΒΖ ἐστιν <lb n="1799,001,,,,13,7,15"></lb> |
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16086 ἴση.</s> <s>ἐδείχθη δὲ καὶ ὅλη ἡ ΑΓ ὅλῃ τῇ ΒΕ ἴση· καὶ <lb n="1799,001,,,,13,7,16"></lb> |
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16087 λοιπὴ ἄρα ἡ ΖΓ λοιπῇ τῇ ΖΕ ἐστιν ἴση.</s> <s>ἔστι δὲ καὶ <lb n="1799,001,,,,13,7,17"></lb> |
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16088 ἡ ΓΔ τῇ ΔΕ ἴση.</s> <s>δύο δὴ αἱ ΖΓ, ΓΔ δυσὶ ταῖς ΖΕ, ΕΔ <lb n="1799,001,,,,13,7,18"></lb> |
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16089 ἴσαι εἰσίν· καὶ βάσις αὐτῶν κοινὴ ἡ ΖΔ· γωνία ἄρα ἡ ὑπὸ <lb n="1799,001,,,,13,7,19"></lb> |
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16090 ΖΓΔ γωνίᾳ τῇ ὑπὸ ΖΕΔ ἐστιν ἴση.</s> <s>ἐδείχθη δὲ καὶ ἡ ὑπὸ <pb></pb> <lb n="1799,001,,,,13,7,20"></lb> |
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16091 ΒΓΑ τῇ ὑπὸ ΑΕΒ ἴση· καὶ ὅλη ἄρα ἡ ὑπὸ ΒΓΔ ὅλῃ <lb n="1799,001,,,,13,7,21"></lb> |
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16092 τῇ ὑπὸ ΑΕΔ ἴση.</s> <s>ἀλλ' ἡ ὑπὸ ΒΓΔ ἴση ὑπόκειται ταῖς <lb n="1799,001,,,,13,7,22"></lb> |
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16093 πρὸς τοῖς Α, Β γωνίαις· καὶ ἡ ὑπὸ ΑΕΔ ἄρα ταῖς <lb n="1799,001,,,,13,7,23"></lb> |
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16094 πρὸς τοῖς Α, Β γωνίαις ἴση ἐστίν. <lb n="1799,001,,,,13,7,24"></lb> |
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16095 ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ ὑπὸ <lb n="1799,001,,,,13,7,25"></lb> |
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16096 ΓΔΕ γωνία ἴση ἐστὶ ταῖς πρὸς τοῖς <lb n="1799,001,,,,13,7,26"></lb> |
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16097 Α, Β, Γ γωνίαις· ἰσογώνιον ἄρα <lb n="1799,001,,,,13,7,27"></lb> |
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16098 ἐστὶ τὸ ΑΒΓΔΕ πεντάγωνον.</s> <lb n="1799,001,,,,13,7,28"></lb> |
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16099 <s>Ἀλλὰ δὴ μὴ ἔστωσαν ἴσαι αἱ <lb n="1799,001,,,,13,7,29"></lb> |
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16100 κατὰ τὸ ἑξῆς γωνίαι, ἀλλ' ἔστω-<lb n="1799,001,,,,13,7,30"></lb> |
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16101 σαν ἴσαι αἱ πρὸς τοῖς Α, Γ, Δ <lb n="1799,001,,,,13,7,31"></lb> |
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16102 σημείοις· λέγω, ὅτι καὶ οὕτως <lb n="1799,001,,,,13,7,32"></lb> |
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16103 ἰσογώνιόν ἐστι τὸ ΑΒΓΔΕ πεν-<lb n="1799,001,,,,13,7,33"></lb> |
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16104 τάγωνον.</s> <lb n="1799,001,,,,13,7,34"></lb> |
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16105 <s>Ἐπεζεύχθω γὰρ ἡ ΒΔ.</s> <s>καὶ ἐπεὶ δύο αἱ ΒΑ, ΑΕ δυσὶ <lb n="1799,001,,,,13,7,35"></lb> |
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16106 ταῖς ΒΓ, ΓΔ ἴσαι εἰσὶ καὶ γωνίας ἴσας περιέχουσιν, βάσις <lb n="1799,001,,,,13,7,36"></lb> |
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16107 ἄρα ἡ ΒΕ βάσει τῇ ΒΔ ἴση ἐστίν, καὶ τὸ ΑΒΕ τρίγωνον <lb n="1799,001,,,,13,7,37"></lb> |
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16108 τῷ ΒΓΔ τριγώνῳ ἴσον ἐστίν, καὶ αἱ λοιπαὶ γωνίαι ταῖς <lb n="1799,001,,,,13,7,38"></lb> |
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16109 λοιπαῖς γωνίαις ἴσαι ἔσονται, ὑφ' ἃς αἱ ἴσαι πλευραὶ ὑπο-<lb n="1799,001,,,,13,7,39"></lb> |
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16110 τείνουσιν· ἴση ἄρα ἐστὶν ἡ ὑπὸ ΑΕΒ γωνία τῇ ὑπὸ ΓΔΒ.</s> <lb n="1799,001,,,,13,7,40"></lb> |
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16111 <s>ἔστι δὲ καὶ ἡ ὑπὸ ΒΕΔ γωνία τῇ ὑπὸ ΒΔΕ ἴση, ἐπεὶ καὶ <lb n="1799,001,,,,13,7,41"></lb> |
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16112 πλευρὰ ἡ ΒΕ πλευρᾷ τῇ ΒΔ ἐστιν ἴση.</s> <s>καὶ ὅλη ἄρα ἡ ὑπὸ <lb n="1799,001,,,,13,7,42"></lb> |
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16113 ΑΕΔ γωνία ὅλῃ τῇ ὑπὸ ΓΔΕ ἐστιν ἴση.</s> <s>ἀλλὰ ἡ ὑπὸ ΓΔΕ <lb n="1799,001,,,,13,7,43"></lb> |
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16114 ταῖς πρὸς τοῖς Α, Γ γωνίαις ὑπόκειται ἴση· καὶ ἡ ὑπὸ <lb n="1799,001,,,,13,7,44"></lb> |
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16115 ΑΕΔ ἄρα γωνία ταῖς πρὸς τοῖς Α, Γ ἴση ἐστίν.</s> <s>διὰ τὰ <lb n="1799,001,,,,13,7,45"></lb> |
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16116 αὐτὰ δὴ καὶ ἡ ὑπὸ ΑΒΓ ἴση ἐστὶ ταῖς πρὸς τοῖς Α, Γ, Δ <lb n="1799,001,,,,13,7,46"></lb> |
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16117 γωνίαις.</s> <s>ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓΔΕ πεντάγωνον· <lb n="1799,001,,,,13,7,47"></lb> |
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16118 ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,13,7,48"></lb> |
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16119 Ἐὰν πενταγώνου ἰσοπλεύρου καὶ ἰσογωνίου τὰς κατὰ <lb n="1799,001,,,,13,8,1"></lb> |
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16120 τὸ ἑξῆς δύο γωνίας ὑποτείνωσιν εὐθεῖαι, ἄκρον καὶ μέσον <lb n="1799,001,,,,13,8,2"></lb> |
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16121 λόγον τέμνουσιν ἀλλήλας, καὶ τὰ μείζονα αὐτῶν τμήματα <lb n="1799,001,,,,13,8,3"></lb> |
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16122 ἴσα ἐστὶ τῇ τοῦ πενταγώνου πλευρᾷ.</s> <lb n="1799,001,,,,13,8,4"></lb> |
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16123 <s>Πενταγώνου γὰρ ἰσοπλεύρου καὶ ἰσογωνίου τοῦ ΑΒΓ <lb n="1799,001,,,,13,8,5"></lb> |
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16124 ΔΕ δύο γωνίας τὰς κατὰ τὸ ἑξῆς τὰς πρὸς τοῖς Α, Β <lb n="1799,001,,,,13,8,6"></lb> |
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16125 ὑποτεινέτωσαν εὐθεῖαι αἱ ΑΓ, <lb n="1799,001,,,,13,8,7"></lb> |
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16126 ΒΕ τέμνουσαι ἀλλήλας κατὰ <lb n="1799,001,,,,13,8,8"></lb> |
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16127 τὸ Θ σημεῖον· λέγω, ὅτι ἑκα-<lb n="1799,001,,,,13,8,9"></lb> |
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16128 τέρα αὐτῶν ἄκρον καὶ μέσον <lb n="1799,001,,,,13,8,10"></lb> |
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16129 λόγον τέτμηται κατὰ τὸ Θ <lb n="1799,001,,,,13,8,11"></lb> |
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16130 σημεῖον, καὶ τὰ μείζονα αὐτῶν <lb n="1799,001,,,,13,8,12"></lb> |
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16131 τμήματα ἴσα ἐστὶ τῇ τοῦ πεν-<lb n="1799,001,,,,13,8,13"></lb> |
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16132 ταγώνου πλευρᾷ.</s> <lb n="1799,001,,,,13,8,14"></lb> |
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16133 <s>Περιγεγράφθω γὰρ περὶ τὸ <lb n="1799,001,,,,13,8,15"></lb> |
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16134 ΑΒΓΔΕ πεντάγωνον κύκλος <lb n="1799,001,,,,13,8,16"></lb> |
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16135 ὁ ΑΒΓΔΕ.</s> <s>καὶ ἐπεὶ δύο εὐθεῖαι αἱ ΕΑ, ΑΒ δυσὶ ταῖς <lb n="1799,001,,,,13,8,17"></lb> |
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16136 ΑΒ, ΒΓ ἴσαι εἰσὶ καὶ γωνίας ἴσας περιέχουσιν, βάσις <lb n="1799,001,,,,13,8,18"></lb> |
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16137 ἄρα ἡ ΒΕ βάσει τῇ ΑΓ ἴση ἐστίν, καὶ τὸ ΑΒΕ τρίγωνον <lb n="1799,001,,,,13,8,19"></lb> |
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16138 τῷ ΑΒΓ τριγώνῳ ἴσον ἐστίν, καὶ αἱ λοιπαὶ γωνίαι ταῖς <lb n="1799,001,,,,13,8,20"></lb> |
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16139 λοιπαῖς γωνίαις ἴσαι ἔσονται ἑκατέρα ἑκατέρᾳ, ὑφ' ἃς αἱ <lb n="1799,001,,,,13,8,21"></lb> |
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16140 ἴσαι πλευραὶ ὑποτείνουσιν.</s> <s>ἴση ἄρα ἐστὶν ἡ ὑπὸ ΒΑΓ <lb n="1799,001,,,,13,8,22"></lb> |
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16141 γωνία τῇ ὑπὸ ΑΒΕ· διπλῆ ἄρα ἡ ὑπὸ ΑΘΕ τῆς ὑπὸ <lb n="1799,001,,,,13,8,23"></lb> |
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16142 ΒΑΘ.</s> <s>ἔστι δὲ καὶ ἡ ὑπὸ ΕΑΓ τῆς ὑπὸ ΒΑΓ διπλῆ, ἐπει- <pb></pb><lb n="1799,001,,,,13,8,24"></lb> |
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16143 δήπερ καὶ περιφέρεια ἡ ΕΔΓ περιφερείας τῆς ΓΒ ἐστι <lb n="1799,001,,,,13,8,25"></lb> |
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16144 διπλῆ· ἴση ἄρα ἡ ὑπὸ ΘΑΕ γωνία τῇ ὑπὸ ΑΘΕ· ὥστε <lb n="1799,001,,,,13,8,26"></lb> |
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16145 καὶ ἡ ΘΕ εὐθεῖα τῇ ΕΑ, τουτέστι τῇ ΑΒ ἐστιν ἴση.</s> <s>καὶ <lb n="1799,001,,,,13,8,27"></lb> |
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16146 ἐπεὶ ἴση ἐστὶν ἡ ΒΑ εὐθεῖα τῇ ΑΕ, ἴση ἐστὶ καὶ γωνία ἡ <lb n="1799,001,,,,13,8,28"></lb> |
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16147 ὑπὸ ΑΒΕ τῇ ὑπὸ ΑΕΒ.</s> <s>ἀλλὰ ἡ ὑπὸ ΑΒΕ τῇ ὑπὸ ΒΑΘ <lb n="1799,001,,,,13,8,29"></lb> |
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16148 ἐδείχθη ἴση· καὶ ἡ ὑπὸ ΒΕΑ ἄρα τῇ ὑπὸ ΒΑΘ ἐστιν ἴση.</s> <lb n="1799,001,,,,13,8,30"></lb> |
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16149 <s>καὶ κοινὴ τῶν δύο τριγώνων τοῦ τε ΑΒΕ καὶ τοῦ ΑΒΘ <lb n="1799,001,,,,13,8,31"></lb> |
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16150 ἐστιν ἡ ὑπὸ ΑΒΕ· λοιπὴ ἄρα ἡ ὑπὸ ΒΑΕ γωνία λοιπῇ <lb n="1799,001,,,,13,8,32"></lb> |
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16151 τῇ ὑπὸ ΑΘΒ ἐστιν ἴση· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΕ τρί-<lb n="1799,001,,,,13,8,33"></lb> |
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16152 γωνον τῷ ΑΒΘ τριγώνῳ· ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΕΒ <lb n="1799,001,,,,13,8,34"></lb> |
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16153 πρὸς τὴν ΒΑ, οὕτως ἡ ΑΒ πρὸς τὴν ΒΘ.</s> <s>ἴση δὲ ἡ ΒΑ <lb n="1799,001,,,,13,8,35"></lb> |
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16154 τῇ ΕΘ· ὡς ἄρα ἡ ΒΕ πρὸς τὴν ΕΘ, οὕτως ἡ ΕΘ πρὸς <lb n="1799,001,,,,13,8,36"></lb> |
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16155 τὴν ΘΒ.</s> <s>μείζων δὲ ἡ ΒΕ τῆς ΕΘ· μείζων ἄρα καὶ ἡ ΕΘ <lb n="1799,001,,,,13,8,37"></lb> |
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16156 τῆς ΘΒ.</s> <s>ἡ ΒΕ ἄρα ἄκρον καὶ μέσον λόγον τέτμηται κατὰ <lb n="1799,001,,,,13,8,38"></lb> |
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16157 τὸ Θ, καὶ τὸ μεῖζον τμῆμα τὸ ΘΕ ἴσον ἐστὶ τῇ τοῦ πεντα-<lb n="1799,001,,,,13,8,39"></lb> |
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16158 γώνου πλευρᾷ.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἡ ΑΓ ἄκρον <lb n="1799,001,,,,13,8,40"></lb> |
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16159 καὶ μέσον λόγον τέτμηται κατὰ τὸ Θ, καὶ τὸ μεῖζον αὐτῆς <lb n="1799,001,,,,13,8,41"></lb> |
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16160 τμῆμα ἡ ΓΘ ἴσον ἐστὶ τῇ τοῦ πενταγώνου πλευρᾷ· ὅπερ <lb n="1799,001,,,,13,8,42"></lb> |
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16161 ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,8,43"></lb> |
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16162 <s>Ἐὰν ἡ τοῦ ἑξαγώνου πλευρὰ καὶ ἡ τοῦ δεκαγώνου τῶν <lb n="1799,001,,,,13,9,1"></lb> |
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16163 εἰς τὸν αὐτὸν κύκλον ἐγγραφομένων συντεθῶσιν, ἡ ὅλη <lb n="1799,001,,,,13,9,2"></lb> |
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16164 εὐθεῖα ἄκρον καὶ μέσον λόγον τέτμηται, καὶ τὸ μεῖζον <lb n="1799,001,,,,13,9,3"></lb> |
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16165 αὐτῆς τμῆμά ἐστιν ἡ τοῦ ἑξαγώνου πλευρά.</s> <s><pb></pb> <lb n="1799,001,,,,13,9,4"></lb> |
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16166 Ἔστω κύκλος ὁ ΑΒΓ, καὶ τῶν εἰς τὸν ΑΒΓ κύκλον <lb n="1799,001,,,,13,9,5"></lb> |
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16167 ἐγγραφομένων σχημάτων, δεκαγώνου μὲν ἔστω πλευρὰ ἡ <lb n="1799,001,,,,13,9,6"></lb> |
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16168 ΒΓ, ἑξαγώνου δὲ ἡ ΓΔ, καὶ ἔστωσαν ἐπ' εὐθείας· λέγω, <lb n="1799,001,,,,13,9,7"></lb> |
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16169 ὅτι ἡ ὅλη εὐθεῖα ἡ ΒΔ ἄκρον καὶ μέσον λόγον τέτμηται, <lb n="1799,001,,,,13,9,8"></lb> |
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16170 καὶ τὸ μεῖζον αὐτῆς τμῆμά ἐστιν ἡ ΓΔ. <lb n="1799,001,,,,13,9,9"></lb> |
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16171 Εἰλήφθω γὰρ τὸ κέντρον τοῦ κύκλου τὸ Ε σημεῖον, καὶ <lb n="1799,001,,,,13,9,10"></lb> |
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16172 ἐπεζεύχθωσαν αἱ ΕΒ, ΕΓ, ΕΔ, καὶ διήχθω ἡ ΒΕ ἐπὶ τὸ <lb n="1799,001,,,,13,9,11"></lb> |
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16173 Α.</s> <s>ἐπεὶ δεκαγώνου ἰσοπλεύρου πλευρά ἐστιν ἡ ΒΓ, πεντα-<lb n="1799,001,,,,13,9,12"></lb> |
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16174 πλασίων ἄρα ἡ ΑΓΒ περιφέρεια τῆς ΒΓ περιφερείας· <lb n="1799,001,,,,13,9,13"></lb> |
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16175 τετραπλασίων ἄρα ἡ ΑΓ περιφέρεια τῆς ΓΒ.</s> <s>ὡς δὲ ἡ ΑΓ <lb n="1799,001,,,,13,9,14"></lb> |
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16176 περιφέρεια πρὸς τὴν ΓΒ, οὕτως ἡ ὑπὸ ΑΕΓ γωνία πρὸς <lb n="1799,001,,,,13,9,15"></lb> |
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16177 τὴν ὑπὸ ΓΕΒ· τετραπλασίων ἄρα ἡ ὑπὸ ΑΕΓ τῆς ὑπὸ <lb n="1799,001,,,,13,9,16"></lb> |
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16178 ΓΕΒ.</s> <s>καὶ ἐπεὶ ἴση ἡ ὑπὸ ΕΒΓ γωνία τῇ ὑπὸ ΕΓΒ, ἡ <lb n="1799,001,,,,13,9,17"></lb> |
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16179 ἄρα ὑπὸ ΑΕΓ γωνία διπλασία ἐστὶ τῆς ὑπὸ ΕΓΒ.</s> <s>καὶ <pb></pb> <lb n="1799,001,,,,13,9,18"></lb> |
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16180 ἐπεὶ ἴση ἐστὶν ἡ ΕΓ εὐθεῖα τῇ ΓΔ· ἑκατέρα γὰρ αὐτῶν <lb n="1799,001,,,,13,9,19"></lb> |
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16181 ἴση ἐστὶ τῇ τοῦ ἑξαγώνου πλευρᾷ τοῦ εἰς τὸν ΑΒΓ κύκλον <lb n="1799,001,,,,13,9,20"></lb> |
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16182 [ἐγγραφομένου]· ἴση ἐστὶ καὶ ἡ ὑπὸ ΓΕΔ γωνία τῇ ὑπὸ <lb n="1799,001,,,,13,9,21"></lb> |
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16183 ΓΔΕ γωνίᾳ· διπλασία ἄρα ἡ ὑπὸ ΕΓΒ γωνία τῆς ὑπὸ <lb n="1799,001,,,,13,9,22"></lb> |
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16184 ΕΔΓ.</s> <s>ἀλλὰ τῆς ὑπὸ ΕΓΒ διπλασία ἐδείχθη ἡ ὑπὸ ΑΕΓ· <lb n="1799,001,,,,13,9,23"></lb> |
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16185 τετραπλασία ἄρα ἡ ὑπὸ ΑΕΓ τῆς ὑπὸ ΕΔΓ.</s> <s>ἐδείχθη δὲ <lb n="1799,001,,,,13,9,24"></lb> |
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16186 καὶ τῆς ὑπὸ ΒΕΓ τετραπλασία ἡ ὑπὸ ΑΕΓ· ἴση ἄρα ἡ <lb n="1799,001,,,,13,9,25"></lb> |
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16187 ὑπὸ ΕΔΓ τῇ ὑπὸ ΒΕΓ.</s> <s>κοινὴ δὲ τῶν δύο τριγώνων, τοῦ <lb n="1799,001,,,,13,9,26"></lb> |
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16188 τε ΒΕΓ καὶ τοῦ ΒΕΔ, ἡ ὑπὸ ΕΒΔ γωνία· καὶ λοιπὴ ἄρα <lb n="1799,001,,,,13,9,27"></lb> |
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16189 ἡ ὑπὸ ΒΕΔ τῇ ὑπὸ ΕΓΒ ἐστιν ἴση· ἰσογώνιον ἄρα ἐστὶ <lb n="1799,001,,,,13,9,28"></lb> |
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16190 τὸ ΕΒΔ τρίγωνον τῷ ΕΒΓ τριγώνῳ.</s> <s>ἀνάλογον ἄρα ἐστὶν <lb n="1799,001,,,,13,9,29"></lb> |
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16191 ὡς ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως ἡ ΕΒ πρὸς τὴν ΒΓ.</s> <s>ἴση <lb n="1799,001,,,,13,9,30"></lb> |
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16192 δὲ ἡ ΕΒ τῇ ΓΔ.</s> <s>ἔστιν ἄρα ὡς ἡ ΒΔ πρὸς τὴν ΔΓ, οὕτως <lb n="1799,001,,,,13,9,31"></lb> |
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16193 ἡ ΔΓ πρὸς τὴν ΓΒ.</s> <s>μείζων δὲ ἡ ΒΔ τῆς ΔΓ· μείζων ἄρα <lb n="1799,001,,,,13,9,32"></lb> |
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16194 καὶ ἡ ΔΓ τῆς ΓΒ.</s> <s>ἡ ΒΔ ἄρα εὐθεῖα ἄκρον καὶ μέσον <lb n="1799,001,,,,13,9,33"></lb> |
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16195 λόγον τέτμηται [κατὰ τὸ Γ], καὶ τὸ μεῖζον τμῆμα αὐτῆς <lb n="1799,001,,,,13,9,34"></lb> |
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16196 ἐστιν ἡ ΔΓ· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,9,35"></lb> |
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16197 <s>Ἐὰν εἰς κύκλον πεντάγωνον ἰσόπλευρον ἐγγραφῇ, ἡ τοῦ <lb n="1799,001,,,,13,10,1"></lb> |
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16198 πενταγώνου πλευρὰ δύναται τήν τε τοῦ ἑξαγώνου καὶ τὴν <lb n="1799,001,,,,13,10,2"></lb> |
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16199 τοῦ δεκαγώνου τῶν εἰς τὸν αὐτὸν κύκλον ἐγγραφομένων.</s> <s><pb></pb> <lb n="1799,001,,,,13,10,3"></lb> |
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16200 Ἔστω κύκλος ὁ ΑΒΓΔΕ, καὶ εἰς τὸν ΑΒΓΔΕ κύ-<lb n="1799,001,,,,13,10,4"></lb> |
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16201 κλον πεντάγωνον ἰσόπλευρον ἐγγεγράφθω τὸ ΑΒΓΔΕ.</s> <lb n="1799,001,,,,13,10,5"></lb> |
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16202 <s>λέγω, ὅτι ἡ τοῦ ΑΒΓΔΕ πενταγώνου πλευρὰ δύναται <lb n="1799,001,,,,13,10,6"></lb> |
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16203 τήν τε τοῦ ἑξαγώνου καὶ τὴν τοῦ δεκαγώνου πλευρὰν τῶν <lb n="1799,001,,,,13,10,7"></lb> |
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16204 εἰς τὸν ΑΒΓΔΕ κύκλον ἐγγραφομένων.</s> <lb n="1799,001,,,,13,10,8"></lb> |
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16205 <s>Εἰλήφθω γὰρ τὸ κέντρον τοῦ κύκλου τὸ Ζ σημεῖον, καὶ <lb n="1799,001,,,,13,10,9"></lb> |
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16206 ἐπιζευχθεῖσα ἡ ΑΖ διήχθω ἐπὶ τὸ Η σημεῖον, καὶ ἐπ-<lb n="1799,001,,,,13,10,10"></lb> |
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16207 εζεύχθω ἡ ΖΒ, καὶ ἀπὸ τοῦ Ζ ἐπὶ τὴν ΑΒ κάθετος ἤχθω <lb n="1799,001,,,,13,10,11"></lb> |
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16208 ἡ ΖΘ, καὶ διήχθω ἐπὶ τὸ Κ, καὶ ἐπεζεύχθωσαν αἱ ΑΚ, <lb n="1799,001,,,,13,10,12"></lb> |
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16209 ΚΒ, καὶ πάλιν ἀπὸ τοῦ Ζ ἐπὶ τὴν ΑΚ κάθετος ἤχθω ἡ <lb n="1799,001,,,,13,10,13"></lb> |
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16210 ΖΛ, καὶ διήχθω ἐπὶ τὸ Μ, καὶ ἐπεζεύχθω ἡ ΚΝ.</s> <s>ἐπεὶ <lb n="1799,001,,,,13,10,14"></lb> |
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16211 ἴση ἐστὶν ἡ ΑΒΓΗ περιφέρεια τῇ ΑΕΔΗ περιφερείᾳ, <lb n="1799,001,,,,13,10,15"></lb> |
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16212 ὧν ἡ ΑΒΓ τῇ ΑΕΔ ἐστιν ἴση, λοιπὴ ἄρα ἡ ΓΗ περι-<lb n="1799,001,,,,13,10,16"></lb> |
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16213 φέρεια λοιπῇ τῇ ΗΔ ἐστιν ἴση.</s> <s>πενταγώνου δὲ ἡ ΓΔ· <lb n="1799,001,,,,13,10,17"></lb> |
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16214 δεκαγώνου ἄρα ἡ ΓΗ.</s> <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΖΑ τῇ ΖΒ, καὶ <lb n="1799,001,,,,13,10,18"></lb> |
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16215 κάθετος ἡ ΖΘ, ἴση ἄρα καὶ ἡ ὑπὸ ΑΖΚ γωνία τῇ ὑπὸ <lb n="1799,001,,,,13,10,19"></lb> |
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16216 ΚΖΒ.</s> <s>ὥστε καὶ περιφέρεια ἡ ΑΚ τῇ ΚΒ ἐστιν ἴση· διπλῆ <pb></pb> <lb n="1799,001,,,,13,10,20"></lb> |
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16217 ἄρα ἡ ΑΒ περιφέρεια τῆς ΒΚ περιφερείας· δεκαγώνου <lb n="1799,001,,,,13,10,21"></lb> |
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16218 ἄρα πλευρά ἐστιν ἡ ΑΚ εὐθεῖα.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΑΚ <lb n="1799,001,,,,13,10,22"></lb> |
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16219 τῆς ΚΜ ἐστι διπλῆ.</s> <s>καὶ ἐπεὶ διπλῆ ἐστιν ἡ ΑΒ περιφέρεια <lb n="1799,001,,,,13,10,23"></lb> |
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16220 τῆς ΒΚ περιφερείας, ἴση δὲ ἡ ΓΔ περιφέρεια τῇ ΑΒ περι-<lb n="1799,001,,,,13,10,24"></lb> |
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16221 φερείᾳ, διπλῆ ἄρα καὶ ἡ ΓΔ περιφέρεια τῆς ΒΚ περιφε-<lb n="1799,001,,,,13,10,25"></lb> |
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16222 ρείας.</s> <s>ἔστι δὲ ἡ ΓΔ περιφέρεια καὶ τῆς ΓΗ διπλῆ· ἴση ἄρα <lb n="1799,001,,,,13,10,26"></lb> |
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16223 ἡ ΓΗ περιφέρεια τῇ ΒΚ περιφερείᾳ.</s> <s>ἀλλὰ ἡ ΒΚ τῆς ΚΜ <lb n="1799,001,,,,13,10,27"></lb> |
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16224 ἐστι διπλῆ, ἐπεὶ καὶ ἡ ΚΑ· καὶ ἡ ΓΗ ἄρα τῆς ΚΜ ἐστι <lb n="1799,001,,,,13,10,28"></lb> |
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16225 διπλῆ.</s> <s>ἀλλὰ μὴν καὶ ἡ ΓΒ περιφέρεια τῆς ΒΚ περιφερείας <lb n="1799,001,,,,13,10,29"></lb> |
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16226 ἐστὶ διπλῆ· ἴση γὰρ ἡ ΓΒ περιφέρεια τῇ ΒΑ.</s> <s>καὶ ὅλη <lb n="1799,001,,,,13,10,30"></lb> |
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16227 ἄρα ἡ ΗΒ περιφέρεια τῆς ΒΜ ἐστι διπλῆ· ὥστε καὶ γωνία <lb n="1799,001,,,,13,10,31"></lb> |
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16228 ἡ ὑπὸ ΗΖΒ γωνίας τῆς ὑπὸ ΒΖΜ [ἐστι] διπλῆ.</s> <s>ἔστι δὲ <lb n="1799,001,,,,13,10,32"></lb> |
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16229 ἡ ὑπὸ ΗΖΒ καὶ τῆς ὑπὸ ΖΑΒ διπλῆ· ἴση γὰρ ἡ ὑπὸ ΖΑΒ <lb n="1799,001,,,,13,10,33"></lb> |
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16230 τῇ ὑπὸ ΑΒΖ.</s> <s>καὶ ἡ ὑπὸ ΒΖΝ ἄρα τῇ ὑπὸ ΖΑΒ ἐστιν <lb n="1799,001,,,,13,10,34"></lb> |
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16231 ἴση.</s> <s>κοινὴ δὲ τῶν δύο τριγώνων, τοῦ τε ΑΒΖ καὶ τοῦ <lb n="1799,001,,,,13,10,35"></lb> |
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16232 ΒΖΝ, ἡ ὑπὸ ΑΒΖ γωνία· λοιπὴ ἄρα ἡ ὑπὸ ΑΖΒ λοιπῇ <lb n="1799,001,,,,13,10,36"></lb> |
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16233 τῇ ὑπὸ ΒΝΖ ἐστιν ἴση· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΖ τρί-<lb n="1799,001,,,,13,10,37"></lb> |
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16234 γωνον τῷ ΒΖΝ τριγώνῳ.</s> <s>ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΑΒ <lb n="1799,001,,,,13,10,38"></lb> |
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16235 εὐθεῖα πρὸς τὴν ΒΖ, οὕτως ἡ ΖΒ πρὸς τὴν ΒΝ· τὸ ἄρα ὑπὸ <lb n="1799,001,,,,13,10,39"></lb> |
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16236 τῶν ΑΒΝ ἴσον ἐστὶ τῷ ἀπὸ ΒΖ.</s> <s>πάλιν ἐπεὶ ἴση ἐστὶν ἡ <lb n="1799,001,,,,13,10,40"></lb> |
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16237 ΑΛ τῇ ΛΚ, κοινὴ δὲ καὶ πρὸς ὀρθὰς ἡ ΛΝ, βάσις ἄρα ἡ <lb n="1799,001,,,,13,10,41"></lb> |
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16238 ΚΝ βάσει τῇ ΑΝ ἐστιν ἴση· καὶ γωνία ἄρα ἡ ὑπὸ ΛΚΝ <lb n="1799,001,,,,13,10,42"></lb> |
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16239 γωνίᾳ τῇ ὑπὸ ΛΑΝ ἐστιν ἴση.</s> <s>ἀλλὰ ἡ ὑπὸ ΛΑΝ τῇ ὑπὸ <pb></pb> <lb n="1799,001,,,,13,10,43"></lb> |
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16240 ΚΒΝ ἐστιν ἴση· καὶ ἡ ὑπὸ ΛΚΝ ἄρα τῇ ὑπὸ ΚΒΝ ἐστιν <lb n="1799,001,,,,13,10,44"></lb> |
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16241 ἴση.</s> <s>καὶ κοινὴ τῶν δύο τριγώνων τοῦ τε ΑΚΒ καὶ τοῦ <lb n="1799,001,,,,13,10,45"></lb> |
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16242 ΑΚΝ ἡ πρὸς τῷ Α.</s> <s>λοιπὴ ἄρα ἡ ὑπὸ ΑΚΒ λοιπῇ τῇ ὑπὸ <lb n="1799,001,,,,13,10,46"></lb> |
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16243 ΚΝΑ ἐστιν ἴση· ἰσογώνιον ἄρα ἐστὶ τὸ ΚΒΑ τρίγωνον <lb n="1799,001,,,,13,10,47"></lb> |
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16244 τῷ ΚΝΑ τριγώνῳ.</s> <s>ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΒΑ εὐθεῖα <lb n="1799,001,,,,13,10,48"></lb> |
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16245 πρὸς τὴν ΑΚ, οὕτως ἡ ΚΑ πρὸς τὴν *αΝ· τὸ ἄρα ὑπὸ τῶν <lb n="1799,001,,,,13,10,49"></lb> |
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16246 ΒΑΝ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΚ.</s> <s>ἐδείχθη δὲ καὶ τὸ ὑπὸ τῶν <lb n="1799,001,,,,13,10,50"></lb> |
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16247 ΑΒΝ ἴσον τῷ ἀπὸ τῆς ΒΖ· τὸ ἄρα ὑπὸ τῶν ΑΒΝ μετὰ <lb n="1799,001,,,,13,10,51"></lb> |
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16248 τοῦ ὑπὸ ΒΑΝ, ὅπερ ἐστὶ τὸ ἀπὸ τῆς ΒΑ, ἴσον ἐστὶ τῷ ἀπὸ <lb n="1799,001,,,,13,10,52"></lb> |
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16249 τῆς ΒΖ μετὰ τοῦ ἀπὸ τῆς ΑΚ.</s> <s>καί ἐστιν ἡ μὲν ΒΑ πεντα-<lb n="1799,001,,,,13,10,53"></lb> |
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16250 γώνου πλευρά, ἡ δὲ ΒΖ ἑξαγώνου, ἡ δὲ ΑΚ δεκαγώνου.</s> <lb n="1799,001,,,,13,10,54"></lb> |
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16251 <s>Ἡ ἄρα τοῦ πενταγώνου πλευρὰ δύναται τήν τε τοῦ ἑξα-<lb n="1799,001,,,,13,10,55"></lb> |
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16252 γώνου καὶ τὴν τοῦ δεκαγώνου τῶν εἰς τὸν αὐτὸν κύκλον <lb n="1799,001,,,,13,10,56"></lb> |
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16253 ἐγγραφομένων· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,10,57"></lb> |
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16254 <s>Ἐὰν εἰς κύκλον ῥητὴν ἔχοντα τὴν διάμετρον πεντάγω-<lb n="1799,001,,,,13,11,1"></lb> |
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16255 νον ἰσόπλευρον ἐγγραφῇ, ἡ τοῦ πενταγώνου πλευρὰ ἄλο-<lb n="1799,001,,,,13,11,2"></lb> |
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16256 γός ἐστιν ἡ καλουμένη ἐλάσσων.</s> <lb n="1799,001,,,,13,11,3"></lb> |
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16257 <s>Εἰς γὰρ κύκλον τὸν ΑΒΓΔΕ ῥητὴν ἔχοντα τὴν διά-<lb n="1799,001,,,,13,11,4"></lb> |
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16258 μετρον πεντάγωνον ἰσόπλευρον ἐγγεγράφθω τὸ ΑΒΓΔΕ· <lb n="1799,001,,,,13,11,5"></lb> |
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16259 λέγω, ὅτι ἡ τοῦ [ΑΒΓΔΕ] πενταγώνου πλευρὰ ἄλογός <lb n="1799,001,,,,13,11,6"></lb> |
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16260 ἐστιν ἡ καλουμένη ἐλάσσων.</s> <s><pb></pb> <lb n="1799,001,,,,13,11,7"></lb> |
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16261 Εἰλήφθω γὰρ τὸ κέντρον τοῦ κύκλου τὸ Ζ σημεῖον, καὶ <lb n="1799,001,,,,13,11,8"></lb> |
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16262 ἐπεζεύχθωσαν αἱ ΑΖ, ΖΒ καὶ διήχθωσαν ἐπὶ τὰ Η, Θ <lb n="1799,001,,,,13,11,9"></lb> |
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16263 σημεῖα, καὶ ἐπεζεύχθω ἡ ΑΓ, καὶ κείσθω τῆς ΑΖ τέταρ-<lb n="1799,001,,,,13,11,10"></lb> |
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16264 τον μέρος ἡ ΖΚ.</s> <s>ῥητὴ δὲ ἡ ΑΖ· ῥητὴ ἄρα καὶ ἡ ΖΚ.</s> <s>ἔστι <lb n="1799,001,,,,13,11,11"></lb> |
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16265 δὲ καὶ ἡ ΒΖ ῥητή· ὅλη ἄρα ἡ ΒΚ ῥητή ἐστιν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,13,11,12"></lb> |
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16266 ἴση ἐστὶν ἡ ΑΓΗ περιφέρεια τῇ ΑΔΗ περιφερείᾳ, ὧν ἡ <lb n="1799,001,,,,13,11,13"></lb> |
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16267 ΑΒΓ τῇ ΑΕΔ ἐστιν ἴση, λοιπὴ ἄρα ἡ ΓΗ λοιπῇ τῇ ΗΔ <lb n="1799,001,,,,13,11,14"></lb> |
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16268 ἐστιν ἴση.</s> <s>καὶ ἐὰν ἐπιζεύξωμεν τὴν ΑΔ, συνάγονται ὀρθαὶ <lb n="1799,001,,,,13,11,15"></lb> |
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16269 αἱ πρὸς τῷ Λ γωνίαι, καὶ διπλῆ ἡ ΓΔ τῆς ΓΛ.</s> <s>διὰ τὰ αὐτὰ <lb n="1799,001,,,,13,11,16"></lb> |
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16270 δὴ καὶ αἱ πρὸς τῷ Μ ὀρθαί εἰσιν, καὶ διπλῆ ἡ ΑΓ τῆς ΓΜ.</s> <lb n="1799,001,,,,13,11,17"></lb> |
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16271 <s>ἐπεὶ οὖν ἴση ἐστὶν ἡ ὑπὸ ΑΛΓ γωνία τῇ ὑπὸ ΑΜΖ, κοινὴ <lb n="1799,001,,,,13,11,18"></lb> |
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16272 δὲ τῶν δύο τριγώνων τοῦ τε ΑΓΛ καὶ τοῦ ΑΜΖ ἡ ὑπὸ <lb n="1799,001,,,,13,11,19"></lb> |
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16273 ΛΑΓ, λοιπὴ ἄρα ἡ ὑπὸ ΑΓΛ λοιπῇ τῇ ὑπὸ ΜΖΑ ἐστιν <pb></pb> <lb n="1799,001,,,,13,11,20"></lb> |
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16274 ἴση· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΓΛ τρίγωνον τῷ ΑΜΖ τρι-<lb n="1799,001,,,,13,11,21"></lb> |
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16275 γώνῳ· ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΛΓ πρὸς ΓΑ, οὕτως ἡ <lb n="1799,001,,,,13,11,22"></lb> |
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16276 ΜΖ πρὸς ΖΑ· καὶ τῶν ἡγουμένων τὰ διπλάσια· ὡς ἄρα <lb n="1799,001,,,,13,11,23"></lb> |
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16277 ἡ τῆς ΛΓ διπλῆ πρὸς τὴν ΓΑ, οὕτως ἡ τῆς ΜΖ διπλῆ <lb n="1799,001,,,,13,11,24"></lb> |
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16278 πρὸς τὴν ΖΑ.</s> <s>ὡς δὲ ἡ τῆς ΜΖ διπλῆ πρὸς τὴν ΖΑ, οὕτως <lb n="1799,001,,,,13,11,25"></lb> |
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16279 ἡ ΜΖ πρὸς τὴν ἡμίσειαν τῆς ΖΑ· καὶ ὡς ἄρα ἡ τῆς ΛΓ <lb n="1799,001,,,,13,11,26"></lb> |
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16280 διπλῆ πρὸς τὴν ΓΑ, οὕτως ἡ ΜΖ πρὸς τὴν ἡμίσειαν τῆς <lb n="1799,001,,,,13,11,27"></lb> |
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16281 ΖΑ.</s> <s>καὶ τῶν ἑπομένων τὰ ἡμίσεα· ὡς ἄρα ἡ τῆς ΛΓ <lb n="1799,001,,,,13,11,28"></lb> |
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16282 διπλῆ πρὸς τὴν ἡμίσειαν τῆς ΓΑ, οὕτως ἡ ΜΖ πρὸς τὸ <lb n="1799,001,,,,13,11,29"></lb> |
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16283 τέταρτον τῆς ΖΑ.</s> <s>καί ἐστι τῆς μὲν ΛΓ διπλῆ ἡ ΔΓ, τῆς <lb n="1799,001,,,,13,11,30"></lb> |
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16284 δὲ ΓΑ ἡμίσεια ἡ ΓΜ, τῆς δὲ ΖΑ τέταρτον μέρος ἡ ΖΚ· <lb n="1799,001,,,,13,11,31"></lb> |
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16285 ἔστιν ἄρα ὡς ἡ ΔΓ πρὸς τὴν ΓΜ, οὕτως ἡ ΜΖ πρὸς τὴν <lb n="1799,001,,,,13,11,32"></lb> |
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16286 ΖΚ.</s> <s>συνθέντι καὶ ὡς συναμφότερος ἡ ΔΓΜ πρὸς τὴν <lb n="1799,001,,,,13,11,33"></lb> |
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16287 ΓΜ, οὕτως ἡ ΜΚ πρὸς ΚΖ· καὶ ὡς ἄρα τὸ ἀπὸ συναμ-<lb n="1799,001,,,,13,11,34"></lb> |
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16288 φοτέρου τῆς ΔΓΜ πρὸς τὸ ἀπὸ ΓΜ, οὕτως τὸ ἀπὸ ΜΚ <lb n="1799,001,,,,13,11,35"></lb> |
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16289 πρὸς τὸ ἀπὸ ΚΖ.</s> <s>καὶ ἐπεὶ τῆς ὑπὸ δύο πλευρὰς τοῦ πεντα-<lb n="1799,001,,,,13,11,36"></lb> |
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16290 γώνου ὑποτεινούσης, οἷον τῆς ΑΓ, ἄκρον καὶ μέσον λόγον <lb n="1799,001,,,,13,11,37"></lb> |
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16291 τεμνομένης τὸ μεῖζον τμῆμα ἴσον ἐστὶ τῇ τοῦ πενταγώνου <lb n="1799,001,,,,13,11,38"></lb> |
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16292 πλευρᾷ, τουτέστι τῇ ΔΓ, τὸ δὲ μεῖζον τμῆμα προσλαβὸν <lb n="1799,001,,,,13,11,39"></lb> |
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16293 τὴν ἡμίσειαν τῆς ὅλης πενταπλάσιον δύναται τοῦ ἀπὸ <lb n="1799,001,,,,13,11,40"></lb> |
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16294 τῆς ἡμισείας τῆς ὅλης, καί ἐστιν ὅλης τῆς ΑΓ ἡμίσεια ἡ <lb n="1799,001,,,,13,11,41"></lb> |
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16295 ΓΜ, τὸ ἄρα ἀπὸ τῆς ΔΓΜ ὡς μιᾶς πενταπλάσιόν ἐστι <lb n="1799,001,,,,13,11,42"></lb> |
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16296 τοῦ ἀπὸ τῆς ΓΜ.</s> <s>ὡς δὲ τὸ ἀπὸ τῆς ΔΓΜ ὡς μιᾶς πρὸς τὸ <lb n="1799,001,,,,13,11,43"></lb> |
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16297 ἀπὸ τῆς ΓΜ, οὕτως ἐδείχθη τὸ ἀπὸ τῆς ΜΚ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,13,11,44"></lb> |
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16298 τῆς ΚΖ· πενταπλάσιον ἄρα τὸ ἀπὸ τῆς ΜΚ τοῦ ἀπὸ τῆς <pb></pb> <lb n="1799,001,,,,13,11,45"></lb> |
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16299 ΚΖ.</s> <s>ῥητὸν δὲ τὸ ἀπὸ τῆς ΚΖ· ῥητὴ γὰρ ἡ διάμετρος· <lb n="1799,001,,,,13,11,46"></lb> |
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16300 ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς ΜΚ· ῥητὴ ἄρα ἐστὶν ἡ ΜΚ <lb n="1799,001,,,,13,11,47"></lb> |
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16301 [δυνάμει μόνον].</s> <s>καὶ ἐπεὶ τετραπλασία ἐστὶν ἡ ΒΖ τῆς <lb n="1799,001,,,,13,11,48"></lb> |
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16302 ΖΚ, πενταπλασία ἄρα ἐστὶν ἡ ΒΚ τῆς ΚΖ· εἰκοσιπεντα-<lb n="1799,001,,,,13,11,49"></lb> |
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16303 πλάσιον ἄρα τὸ ἀπὸ τῆς ΒΚ τοῦ ἀπὸ τῆς ΚΖ.</s> <s>πενταπλά-<lb n="1799,001,,,,13,11,50"></lb> |
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16304 σιον δὲ τὸ ἀπὸ τῆς ΜΚ τοῦ ἀπὸ τῆς ΚΖ· πενταπλάσιον <lb n="1799,001,,,,13,11,51"></lb> |
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16305 ἄρα τὸ ἀπὸ τῆς ΒΚ τοῦ ἀπὸ τῆς ΚΜ· τὸ ἄρα ἀπὸ τῆς ΒΚ <lb n="1799,001,,,,13,11,52"></lb> |
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16306 πρὸς τὸ ἀπὸ ΚΜ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς <lb n="1799,001,,,,13,11,53"></lb> |
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16307 πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΚ <lb n="1799,001,,,,13,11,54"></lb> |
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16308 τῇ ΚΜ μήκει.</s> <s>καί ἐστι ῥητὴ ἑκατέρα αὐτῶν.</s> <s>αἱ ΒΚ, ΚΜ <lb n="1799,001,,,,13,11,55"></lb> |
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16309 ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι.</s> <s>ἐὰν δὲ ἀπὸ ῥητῆς <lb n="1799,001,,,,13,11,56"></lb> |
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16310 ῥητὴ ἀφαιρεθῇ δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ, ἡ <lb n="1799,001,,,,13,11,57"></lb> |
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16311 λοιπὴ ἄλογός ἐστιν ἀποτομή· ἀποτομὴ ἄρα ἐστὶν ἡ ΜΒ, <lb n="1799,001,,,,13,11,58"></lb> |
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16312 προσαρμόζουσα δὲ αὐτῇ ἡ ΜΚ.</s> <s>λέγω δή, ὅτι καὶ τετάρτη.</s> <lb n="1799,001,,,,13,11,59"></lb> |
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16313 <s>ᾧ δὴ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΚ τοῦ ἀπὸ τῆς ΚΜ, ἐκείνῳ <lb n="1799,001,,,,13,11,60"></lb> |
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16314 ἴσον ἔστω τὸ ἀπὸ τῆς Ν· ἡ ΒΚ ἄρα τῆς ΚΜ μεῖζον δύνα-<lb n="1799,001,,,,13,11,61"></lb> |
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16315 ται τῇ Ν.</s> <s>καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΚΖ τῇ ΖΒ, καὶ <lb n="1799,001,,,,13,11,62"></lb> |
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16316 συνθέντι σύμμετρός ἐστιν ἡ ΚΒ τῇ ΖΒ.</s> <s>ἀλλὰ ἡ ΒΖ τῇ ΒΘ <lb n="1799,001,,,,13,11,63"></lb> |
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16317 σύμμετρός ἐστιν· καὶ ἡ ΒΚ ἄρα τῇ ΒΘ σύμμετρός ἐστιν.</s> <lb n="1799,001,,,,13,11,64"></lb> |
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16318 <s>καὶ ἐπεὶ πενταπλάσιόν ἐστι τὸ ἀπὸ τῆς ΒΚ τοῦ ἀπὸ τῆς <lb n="1799,001,,,,13,11,65"></lb> |
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16319 ΚΜ, τὸ ἄρα ἀπὸ τῆς ΒΚ πρὸς τὸ ἀπὸ τῆς ΚΜ λόγον <lb n="1799,001,,,,13,11,66"></lb> |
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16320 ἔχει, ὃν <ε></s> <s>πρὸς ἕν.</s> <s>ἀναστρέψαντι ἄρα τὸ ἀπὸ τῆς ΒΚ πρὸς <pb></pb> <lb n="1799,001,,,,13,11,67"></lb> |
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16321 τὸ ἀπὸ τῆς Ν λόγον ἔχει, ὃν <ε></s> <s>πρὸς <δ>, οὐχ ὃν τετράγωνος <lb n="1799,001,,,,13,11,68"></lb> |
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16322 πρὸς τετράγωνον· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΚ τῇ Ν· ἡ <lb n="1799,001,,,,13,11,69"></lb> |
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16323 ΒΚ ἄρα τῆς ΚΜ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου <lb n="1799,001,,,,13,11,70"></lb> |
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16324 ἑαυτῇ.</s> <s>ἐπεὶ οὖν ὅλη ἡ ΒΚ τῆς προσαρμοζούσης τῆς ΚΜ <lb n="1799,001,,,,13,11,71"></lb> |
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16325 μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, καὶ ὅλη ἡ ΒΚ <lb n="1799,001,,,,13,11,72"></lb> |
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16326 σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῃ̂ τῇ ΒΘ, ἀποτομὴ ἄρα <lb n="1799,001,,,,13,11,73"></lb> |
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16327 τετάρτη ἐστὶν ἡ ΜΒ.</s> <s>τὸ δὲ ὑπὸ ῥητῆς καὶ ἀποτομῆς <lb n="1799,001,,,,13,11,74"></lb> |
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16328 τετάρτης περιεχόμενον ὀρθογώνιον ἄλογόν ἐστιν, καὶ ἡ <lb n="1799,001,,,,13,11,75"></lb> |
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16329 δυναμένη αὐτὸ ἄλογός ἐστιν, καλεῖται δὲ ἐλάττων.</s> <s>δύ-<lb n="1799,001,,,,13,11,76"></lb> |
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16330 ναται δὲ τὸ ὑπὸ τῶν ΘΒΜ ἡ ΑΒ διὰ τὸ ἐπιζευγνυ-<lb n="1799,001,,,,13,11,77"></lb> |
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16331 μένης τῆς ΑΘ ἰσογώνιον γίνεσθαι τὸ ΑΒΘ τρίγωνον τῷ <lb n="1799,001,,,,13,11,78"></lb> |
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16332 ΑΒΜ τριγώνῳ καὶ εἶναι ὡς τὴν ΘΒ πρὸς τὴν ΒΑ, οὕτως <lb n="1799,001,,,,13,11,79"></lb> |
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16333 τὴν ΑΒ πρὸς τὴν ΒΜ.</s> <lb n="1799,001,,,,13,11,80"></lb> |
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16334 <s>Ἡ ἄρα ΑΒ τοῦ πενταγώνου πλευρὰ ἄλογός ἐστιν ἡ <lb n="1799,001,,,,13,11,81"></lb> |
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16335 καλουμένη ἐλάττων· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,11,82"></lb> |
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16336 <s>Ἐὰν εἰς κύκλον τρίγωνον ἰσόπλευρον ἐγγραφῇ, ἡ τοῦ <lb n="1799,001,,,,13,12,1"></lb> |
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16337 τριγώνου πλευρὰ δυνάμει τριπλασίων ἐστὶ τῆς ἐκ τοῦ <lb n="1799,001,,,,13,12,2"></lb> |
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16338 κέντρου τοῦ κύκλου.</s> <lb n="1799,001,,,,13,12,3"></lb> |
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16339 <s>Ἔστω κύκλος ὁ ΑΒΓ, καὶ εἰς αὐτὸν τρίγωνον ἰσόπλευ-<lb n="1799,001,,,,13,12,4"></lb> |
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16340 ρον ἐγγεγράφθω τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ τριγώνου <pb></pb> <lb n="1799,001,,,,13,12,5"></lb> |
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16341 μία πλευρὰ δυνάμει τριπλα- <lb n="1799,001,,,,13,12,6"></lb> |
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16342 σίων ἐστὶ τῆς ἐκ τοῦ κέντρου <lb n="1799,001,,,,13,12,7"></lb> |
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16343 τοῦ ΑΒΓ κύκλου.</s> <lb n="1799,001,,,,13,12,8"></lb> |
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16344 <s>Εἰλήφθω γὰρ τὸ κέντρον <lb n="1799,001,,,,13,12,9"></lb> |
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16345 τοῦ ΑΒΓ κύκλου τὸ Δ, καὶ <lb n="1799,001,,,,13,12,10"></lb> |
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16346 ἐπιζευχθεῖσα ἡ ΑΔ διήχθω <lb n="1799,001,,,,13,12,11"></lb> |
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16347 ἐπὶ τὸ Ε, καὶ ἐπεζεύχθω ἡ <lb n="1799,001,,,,13,12,12"></lb> |
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16348 ΒΕ.</s> <s>καὶ ἐπεὶ ἰσόπλευρόν ἐστι <lb n="1799,001,,,,13,12,13"></lb> |
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16349 τὸ ΑΒΓ τρίγωνον, ἡ ΒΕΓ <lb n="1799,001,,,,13,12,14"></lb> |
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16350 ἄρα περιφέρεια τρίτον μέρος <lb n="1799,001,,,,13,12,15"></lb> |
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16351 ἐστὶ τῆς τοῦ ΑΒΓ κύκλου <lb n="1799,001,,,,13,12,16"></lb> |
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16352 περιφερείας.</s> <s>ἡ ἄρα ΒΕ περιφέρεια ἕκτον ἐστὶ μέρος τῆς <lb n="1799,001,,,,13,12,17"></lb> |
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16353 τοῦ κύκλου περιφερείας· ἑξαγώνου ἄρα ἐστὶν ἡ ΒΕ εὐθεῖα· <lb n="1799,001,,,,13,12,18"></lb> |
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16354 ἴση ἄρα ἐστὶ τῇ ἐκ τοῦ κέντρου τῇ ΔΕ.</s> <s>καὶ ἐπεὶ διπλῆ <lb n="1799,001,,,,13,12,19"></lb> |
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16355 ἐστιν ἡ ΑΕ τῆς ΔΕ, τετραπλάσιόν ἐστι τὸ ἀπὸ τῆς ΑΕ <lb n="1799,001,,,,13,12,20"></lb> |
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16356 τοῦ ἀπὸ τῆς ΕΔ, τουτέστι τοῦ ἀπὸ τῆς ΒΕ.</s> <s>ἴσον δὲ τὸ ἀπὸ <lb n="1799,001,,,,13,12,21"></lb> |
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16357 τῆς ΑΕ τοῖς ἀπὸ τῶν ΑΒ, ΒΕ· τὰ ἄρα ἀπὸ τῶν ΑΒ, ΒΕ <lb n="1799,001,,,,13,12,22"></lb> |
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16358 τετραπλάσιά ἐστι τοῦ ἀπὸ τῆς ΒΕ.</s> <s>διελόντι ἄρα τὸ ἀπὸ <lb n="1799,001,,,,13,12,23"></lb> |
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16359 τῆς ΑΒ τριπλάσιόν ἐστι τοῦ ἀπὸ ΒΕ.</s> <s>ἴση δὲ ἡ ΒΕ τῇ ΔΕ· <lb n="1799,001,,,,13,12,24"></lb> |
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16360 τὸ ἄρα ἀπὸ τῆς ΑΒ τριπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΔΕ.</s> <lb n="1799,001,,,,13,12,25"></lb> |
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16361 <s>Ἡ ἄρα τοῦ τριγώνου πλευρὰ δυνάμει τριπλασία ἐστὶ <lb n="1799,001,,,,13,12,26"></lb> |
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16362 τῆς ἐκ τοῦ κέντρου [τοῦ κύκλου]· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,12,27"></lb> |
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16363 <s>Πυραμίδα συστήσασθαι καὶ σφαίρᾳ περιλαβεῖν τῇ <lb n="1799,001,,,,13,13,1"></lb> |
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16364 δοθείσῃ καὶ δεῖξαι, ὅτι ἡ τῆς σφαίρας διάμετρος δυνάμει <lb n="1799,001,,,,13,13,2"></lb> |
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16365 ἡμιολία ἐστὶ τῆς πλευρᾶς τῆς πυραμίδος.</s> <s><pb></pb> <lb n="1799,001,,,,13,13,3"></lb> |
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16366 Ἐκκείσθω ἡ τῆς δοθείσης σφαίρας διάμετρος ἡ ΑΒ, καὶ <lb n="1799,001,,,,13,13,4"></lb> |
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16367 τετμήσθω κατὰ τὸ Γ σημεῖον, ὥστε διπλασίαν εἶναι τὴν <lb n="1799,001,,,,13,13,5"></lb> |
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16368 ΑΓ τῆς ΓΒ· καὶ γεγράφθω ἐπὶ τῆς ΑΒ ἡμικύκλιον τὸ <lb n="1799,001,,,,13,13,6"></lb> |
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16369 ΑΔΒ, καὶ ἤχθω ἀπὸ τοῦ Γ σημείου τῇ ΑΒ πρὸς ὀρθὰς ἡ <lb n="1799,001,,,,13,13,7"></lb> |
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16370 ΓΔ, καὶ ἐπεζεύχθω ἡ ΔΑ· καὶ ἐκκείσθω κύκλος ὁ ΕΖΗ <lb n="1799,001,,,,13,13,8"></lb> |
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16371 ἴσην ἔχων τὴν ἐκ τοῦ κέντρου τῇ ΔΓ, καὶ ἐγγεγράφθω εἰς <lb n="1799,001,,,,13,13,9"></lb> |
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16372 τὸν ΕΖΗ κύκλον τρίγωνον ἰσόπλευρον τὸ ΕΖΗ· καὶ <lb n="1799,001,,,,13,13,10"></lb> |
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16373 εἰλήφθω τὸ κέντρον τοῦ κύκλου τὸ Θ σημεῖον, καὶ ἐπεζεύ-<lb n="1799,001,,,,13,13,11"></lb> |
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16374 χθωσαν αἱ ΕΘ, ΘΖ, ΘΗ· καὶ ἀνεστάτω ἀπὸ τοῦ Θ ση-<lb n="1799,001,,,,13,13,12"></lb> |
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16375 μείου τῷ τοῦ ΕΖΗ κύκλου ἐπιπέδῳ πρὸς ὀρθὰς ἡ ΘΚ, <lb n="1799,001,,,,13,13,13"></lb> |
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16376 καὶ ἀφῃρήσθω ἀπὸ τῆς ΘΚ τῇ ΑΓ εὐθείᾳ ἴση ἡ ΘΚ, καὶ <lb n="1799,001,,,,13,13,14"></lb> |
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16377 ἐπεζεύχθωσαν αἱ ΚΕ, ΚΖ, ΚΗ.</s> <s>καὶ ἐπεὶ ἡ ΚΘ ὀρθή <lb n="1799,001,,,,13,13,15"></lb> |
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16378 ἐστι πρὸς τὸ τοῦ ΕΖΗ κύκλου <lb n="1799,001,,,,13,13,16"></lb> |
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16379 ἐπίπεδον, καὶ πρὸς πάσας ἄρα <lb n="1799,001,,,,13,13,17"></lb> |
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16380 τὰς ἁπτομένας αὐτῆς εὐθείας <lb n="1799,001,,,,13,13,18"></lb> |
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16381 καὶ οὔσας ἐν τῷ τοῦ ΕΖΗ <lb n="1799,001,,,,13,13,19"></lb> |
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16382 κύκλου ἐπιπέδῳ ὀρθὰς ποιήσει <lb n="1799,001,,,,13,13,20"></lb> |
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16383 γωνίας.</s> <s>ἅπτεται δὲ αὐτῆς ἑκά-<lb n="1799,001,,,,13,13,21"></lb> |
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16384 στη τῶν ΘΕ, ΘΖ, ΘΗ· ἡ ΘΚ <lb n="1799,001,,,,13,13,22"></lb> |
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16385 ἄρα πρὸς ἑκάστην τῶν ΘΕ, <lb n="1799,001,,,,13,13,23"></lb> |
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16386 ΘΖ, ΘΗ ὀρθή ἐστιν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,13,13,24"></lb> |
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16387 ἴση ἐστὶν ἡ μὲν ΑΓ τῇ ΘΚ, ἡ <lb n="1799,001,,,,13,13,25"></lb> |
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16388 δὲ ΓΔ τῇ ΘΕ, καὶ ὀρθὰς γω-<lb n="1799,001,,,,13,13,26"></lb> |
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16389 νίας περιέχουσιν, βάσις ἄρα ἡ <lb n="1799,001,,,,13,13,27"></lb> |
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16390 ΔΑ βάσει τῇ ΚΕ ἐστιν ἴση.</s> <s>διὰ <lb n="1799,001,,,,13,13,28"></lb> |
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16391 τὰ αὐτὰ δὴ καὶ ἑκατέρα τῶν <lb n="1799,001,,,,13,13,29"></lb> |
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16392 ΚΖ, ΚΗ τῇ ΔΑ ἐστιν ἴση· αἱ <pb></pb> <lb n="1799,001,,,,13,13,30"></lb> |
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16393 τρεῖς ἄρα αἱ ΚΕ, ΚΖ, ΚΗ ἴσαι ἀλλήλαις εἰσίν.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,13,13,31"></lb> |
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16394 διπλῆ ἐστιν ἡ ΑΓ τῆς ΓΒ, τριπλῆ ἄρα ἡ ΑΒ τῆς ΒΓ.</s> <s>ὡς <lb n="1799,001,,,,13,13,32"></lb> |
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16395 δὲ ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ τῆς ΑΔ πρὸς τὸ ἀπὸ <lb n="1799,001,,,,13,13,33"></lb> |
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16396 τῆς ΔΓ, ὡς ἑξῆς δειχθήσεται.</s> <s>τριπλάσιον ἄρα τὸ ἀπὸ τῆς <lb n="1799,001,,,,13,13,34"></lb> |
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16397 ΑΔ τοῦ ἀπὸ τῆς ΔΓ.</s> <s>ἔστι δὲ καὶ τὸ ἀπὸ τῆς ΖΕ τοῦ ἀπὸ <lb n="1799,001,,,,13,13,35"></lb> |
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16398 τῆς ΕΘ τριπλάσιον, καί ἐστιν ἴση ἡ ΔΓ τῇ ΕΘ· ἴση ἄρα <lb n="1799,001,,,,13,13,36"></lb> |
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16399 καὶ ἡ ΔΑ τῇ ΕΖ.</s> <s>ἀλλὰ ἡ ΔΑ ἑκάστῃ τῶν ΚΕ, ΚΖ, ΚΗ <lb n="1799,001,,,,13,13,37"></lb> |
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16400 ἐδείχθη ἴση· καὶ ἑκάστη ἄρα τῶν ΕΖ, ΖΗ, ΗΕ ἑκάστῃ <lb n="1799,001,,,,13,13,38"></lb> |
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16401 τῶν ΚΕ, ΚΖ, ΚΗ ἐστιν ἴση· ἰσόπλευρα ἄρα ἐστὶ τὰ τέσ-<lb n="1799,001,,,,13,13,39"></lb> |
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16402 σαρα τρίγωνα τὰ ΕΖΗ, ΚΕΖ, ΚΖΗ, ΚΕΗ.</s> <s>πυραμὶς <lb n="1799,001,,,,13,13,40"></lb> |
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16403 ἄρα συνέσταται ἐκ τεσσάρων τριγώνων ἰσοπλεύρων, ἧς <lb n="1799,001,,,,13,13,41"></lb> |
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16404 βάσις μέν ἐστι τὸ ΕΖΗ τρίγωνον, κορυφὴ δὲ τὸ Κ σημεῖον.</s> <lb n="1799,001,,,,13,13,42"></lb> |
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16405 <s>Δεῖ δὴ αὐτὴν καὶ σφαίρᾳ περιλαβεῖν τῇ δοθείσῃ καὶ δεῖ-<lb n="1799,001,,,,13,13,43"></lb> |
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16406 ξαι, ὅτι ἡ τῆς σφαίρας διάμετρος ἡμιολία ἐστὶ δυνάμει τῆς <lb n="1799,001,,,,13,13,44"></lb> |
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16407 πλευρᾶς τῆς πυραμίδος.</s> <lb n="1799,001,,,,13,13,45"></lb> |
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16408 <s>Ἐκβεβλήσθω γὰρ ἐπ' εὐθείας τῇ ΚΘ εὐθεῖα ἡ ΘΛ, <lb n="1799,001,,,,13,13,46"></lb> |
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16409 καὶ κείσθω τῇ ΓΒ ἴση ἡ ΘΛ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ ΑΓ <lb n="1799,001,,,,13,13,47"></lb> |
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16410 πρὸς τὴν ΓΔ, οὕτως ἡ ΓΔ πρὸς τὴν ΓΒ, ἴση δὲ ἡ μὲν ΑΓ <lb n="1799,001,,,,13,13,48"></lb> |
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16411 τῇ ΚΘ, ἡ δὲ ΓΔ τῇ ΘΕ, ἡ δὲ ΓΒ τῇ ΘΛ, ἔστιν ἄρα ὡς ἡ <lb n="1799,001,,,,13,13,49"></lb> |
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16412 ΚΘ πρὸς τὴν ΘΕ, οὕτως ἡ ΕΘ πρὸς τὴν ΘΛ· τὸ ἄρα ὑπὸ <lb n="1799,001,,,,13,13,50"></lb> |
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16413 τῶν ΚΘ, ΘΛ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΕΘ.</s> <s>καί ἐστιν ὀρθὴ <pb></pb> <lb n="1799,001,,,,13,13,51"></lb> |
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16414 ἑκατέρα τῶν ὑπὸ ΚΘΕ, ΕΘΛ γωνιῶν· τὸ ἄρα ἐπὶ τῆς <lb n="1799,001,,,,13,13,52"></lb> |
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16415 ΚΛ γραφόμενον ἡμικύκλιον ἥξει καὶ διὰ τοῦ Ε [ἐπειδήπερ <lb n="1799,001,,,,13,13,53"></lb> |
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16416 ἐὰν ἐπιζεύξωμεν τὴν ΕΛ, ὀρθὴ γίνεται ἡ ὑπὸ ΛΕΚ γωνία <lb n="1799,001,,,,13,13,54"></lb> |
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16417 διὰ τὸ ἰσογώνιον γίνεσθαι τὸ ΕΛΚ τρίγωνον ἑκατέρῳ τῶν <lb n="1799,001,,,,13,13,55"></lb> |
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16418 ΕΛΘ, ΕΘΚ τριγώνων].</s> <s>ἐὰν δὴ μενούσης τῆς ΚΛ περι-<lb n="1799,001,,,,13,13,56"></lb> |
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16419 ενεχθὲν τὸ ἡμικύκλιον εἰς τὸ αὐτὸ πάλιν ἀποκατασταθῇ, <lb n="1799,001,,,,13,13,57"></lb> |
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16420 ὅθεν ἤρξατο φέρεσθαι, ἥξει καὶ διὰ τῶν Ζ, Η σημείων <lb n="1799,001,,,,13,13,58"></lb> |
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16421 ἐπιζευγνυμένων τῶν ΖΛ, ΛΗ καὶ ὀρθῶν ὁμοίως γινομένων <lb n="1799,001,,,,13,13,59"></lb> |
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16422 τῶν πρὸς τοῖς Ζ, Η γωνιῶν· καὶ ἔσται ἡ πυραμὶς σφαίρᾳ <lb n="1799,001,,,,13,13,60"></lb> |
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16423 περιειλημμένη τῇ δοθείσῃ.</s> <s>ἡ γὰρ ΚΛ τῆς σφαίρας διά-<lb n="1799,001,,,,13,13,61"></lb> |
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16424 μετρος ἴση ἐστὶ τῇ τῆς δοθείσης σφαίρας διαμέτρῳ τῇ ΑΒ, <lb n="1799,001,,,,13,13,62"></lb> |
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16425 ἐπειδήπερ τῇ μὲν ΑΓ ἴση κεῖται ἡ ΚΘ, τῇ δὲ ΓΒ ἡ ΘΛ.</s> <lb n="1799,001,,,,13,13,63"></lb> |
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16426 <s>Λέγω δή, ὅτι ἡ τῆς σφαίρας διάμετρος ἡμιολία ἐστὶ <lb n="1799,001,,,,13,13,64"></lb> |
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16427 δυνάμει τῆς πλευρᾶς τῆς πυραμίδος.</s> <lb n="1799,001,,,,13,13,65"></lb> |
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16428 <s>Ἐπεὶ γὰρ διπλῆ ἐστιν ἡ ΑΓ τῆς ΓΒ, τριπλῆ ἄρα ἐστὶν <lb n="1799,001,,,,13,13,66"></lb> |
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16429 ἡ ΑΒ τῆς ΒΓ· ἀναστρέψαντι ἡμιολία ἄρα ἐστὶν ἡ ΒΑ τῆς <lb n="1799,001,,,,13,13,67"></lb> |
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16430 ΑΓ.</s> <s>ὡς δὲ ἡ ΒΑ πρὸς τὴν ΑΓ, οὕτως τὸ ἀπὸ τῆς ΒΑ <lb n="1799,001,,,,13,13,68"></lb> |
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16431 πρὸς τὸ ἀπὸ τῆς ΑΔ [ἐπειδήπερ ἐπιζευγνυμένης τῆς ΔΒ <lb n="1799,001,,,,13,13,69"></lb> |
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16432 ἐστιν ὡς ἡ ΒΑ πρὸς τὴν ΑΔ, οὕτως ἡ ΔΑ πρὸς τὴν ΑΓ <lb n="1799,001,,,,13,13,70"></lb> |
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16433 διὰ τὴν ὁμοιότητα τῶν ΔΑΒ, ΔΑΓ τριγώνων, καὶ εἶναι <lb n="1799,001,,,,13,13,71"></lb> |
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16434 ὡς τὴν πρώτην πρὸς τὴν τρίτην, οὕτως τὸ ἀπὸ τῆς πρώτης <lb n="1799,001,,,,13,13,72"></lb> |
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16435 πρὸς τὸ ἀπὸ τῆς δευτέρασ].</s> <s>ἡμιόλιον ἄρα καὶ τὸ ἀπὸ τῆς <lb n="1799,001,,,,13,13,73"></lb> |
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16436 ΒΑ τοῦ ἀπὸ τῆς ΑΔ.</s> <s>καί ἐστιν ἡ μὲν ΒΑ ἡ τῆς δοθεί-<lb n="1799,001,,,,13,13,74"></lb> |
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16437 σης σφαίρας διάμετρος, ἡ δὲ ΑΔ ἴση τῇ πλευρᾷ τῆς πυρα-<lb n="1799,001,,,,13,13,75"></lb> |
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16438 μίδος.</s> <s><pb></pb> <lb n="1799,001,,,,13,13,76"></lb> |
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16439 Ἡ ἄρα τῆς σφαίρας διάμετρος ἡμιολία ἐστὶ τῆς πλευ-<lb n="1799,001,,,,13,13,77"></lb> |
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16440 ρᾶς τῆς πυραμίδος· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,13,78"></lb> |
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16441 <s>Λῆμμα <lb n="1799,001,,,,13,13,79"></lb> |
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16442 Δεικτέον, ὅτι ἐστὶν ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ <lb n="1799,001,,,,13,13,80"></lb> |
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16443 ἀπὸ τῆς ΑΔ πρὸς τὸ ἀπὸ τῆς ΔΓ.</s> <lb n="1799,001,,,,13,13,81"></lb> |
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16444 <s>Ἐκκείσθω γὰρ ἡ τοῦ ἡμικυκλίου καταγραφή, καὶ ἐπ-<lb n="1799,001,,,,13,13,82"></lb> |
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16445 εζεύχθω ἡ ΔΒ, καὶ ἀναγεγράφθω ἀπὸ τῆς ΑΓ τετρά-<lb n="1799,001,,,,13,13,83"></lb> |
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16446 γωνον τὸ ΕΓ, καὶ συμπεπλη- <lb n="1799,001,,,,13,13,84"></lb> |
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16447 ρώσθω τὸ ΖΒ παραλληλόγραμ-<lb n="1799,001,,,,13,13,85"></lb> |
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16448 μον.</s> <s>ἐπεὶ οὖν διὰ τὸ ἰσογώνιον <lb n="1799,001,,,,13,13,86"></lb> |
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16449 εἶναι τὸ ΔΑΒ τρίγωνον τῷ <lb n="1799,001,,,,13,13,87"></lb> |
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16450 ΔΑΓ τριγώνῳ ἐστὶν ὡς ἡ ΒΑ <lb n="1799,001,,,,13,13,88"></lb> |
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16451 πρὸς τὴν ΑΔ, οὕτως ἡ ΔΑ <lb n="1799,001,,,,13,13,89"></lb> |
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16452 πρὸς τὴν ΑΓ, τὸ ἄρα ὑπὸ τῶν <lb n="1799,001,,,,13,13,90"></lb> |
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16453 ΒΑ, ΑΓ ἴσον ἐστὶ τῷ ἀπὸ <lb n="1799,001,,,,13,13,91"></lb> |
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16454 τῆς ΑΔ.</s> <s>καὶ ἐπεί ἐστιν ὡς ἡ <lb n="1799,001,,,,13,13,92"></lb> |
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16455 ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ <lb n="1799,001,,,,13,13,93"></lb> |
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16456 ΕΒ πρὸς τὸ ΒΖ, καί ἐστι τὸ <lb n="1799,001,,,,13,13,94"></lb> |
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16457 μὲν ΕΒ τὸ ὑπὸ τῶν ΒΑ, ΑΓ· <lb n="1799,001,,,,13,13,95"></lb> |
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16458 ἴση γὰρ ἡ ΕΑ τῇ ΑΓ· τὸ δὲ ΒΖ τὸ ὑπὸ τῶν ΑΓ, ΓΒ, <lb n="1799,001,,,,13,13,96"></lb> |
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16459 ὡς ἄρα ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ ὑπὸ τῶν ΒΑ, ΑΓ <lb n="1799,001,,,,13,13,97"></lb> |
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16460 πρὸς τὸ ὑπὸ τῶν ΑΓ, ΓΒ.</s> <s>καί ἐστι τὸ μὲν ὑπὸ τῶν ΒΑ, <lb n="1799,001,,,,13,13,98"></lb> |
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16461 ΑΓ ἴσον τῷ ἀπὸ τῆς ΑΔ, τὸ δὲ ὑπὸ τῶν ΑΓΒ ἴσον τῷ <lb n="1799,001,,,,13,13,99"></lb> |
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16462 ἀπὸ τῆς ΔΓ· ἡ γὰρ ΔΓ κάθετος τῶν τῆς βάσεως τμημά-<lb n="1799,001,,,,13,13,100"></lb> |
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16463 των τῶν ΑΓ, ΓΒ μέση ἀνάλογόν ἐστι διὰ τὸ ὀρθὴν εἶναι <lb n="1799,001,,,,13,13,101"></lb> |
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16464 τὴν ὑπὸ ΑΔΒ.</s> <s>ὡς ἄρα ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,13,13,102"></lb> |
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16465 τῆς ΑΔ πρὸς τὸ ἀπὸ τῆς ΔΓ· ὅπερ ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,13,13,103"></lb> |
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16466 Ὀκτάεδρον συστήσασθαι καὶ σφαίρᾳ περιλαβεῖν, ᾗ καὶ <lb n="1799,001,,,,13,14,1"></lb> |
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16467 τὰ πρότερα, καὶ δεῖξαι, ὅτι ἡ τῆς σφαίρας διάμετρος δυνά-<lb n="1799,001,,,,13,14,2"></lb> |
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16468 μει διπλασία ἐστὶ τῆς πλευρᾶς τοῦ ὀκταέδρου.</s> <lb n="1799,001,,,,13,14,3"></lb> |
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16469 <s>Ἐκκείσθω ἡ τῆς δοθείσης σφαίρας διάμετρος ἡ ΑΒ, καὶ <lb n="1799,001,,,,13,14,4"></lb> |
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16470 τετμήσθω δίχα κατὰ τὸ Γ, καὶ γεγράφθω ἐπὶ τῆς ΑΒ <lb n="1799,001,,,,13,14,5"></lb> |
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16471 ἡμικύκλιον τὸ ΑΔΒ, καὶ ἤχθω ἀπὸ τοῦ Γ τῇ ΑΒ πρὸς <lb n="1799,001,,,,13,14,6"></lb> |
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16472 ὀρθὰς ἡ ΓΔ, καὶ ἐπεζεύχθω ἡ ΔΒ, <lb n="1799,001,,,,13,14,7"></lb> |
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16473 καὶ ἐκκείσθω τετράγωνον τὸ ΕΖΗΘ <lb n="1799,001,,,,13,14,8"></lb> |
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16474 ἴσην ἔχον ἑκάστην τῶν πλευρῶν τῇ <lb n="1799,001,,,,13,14,9"></lb> |
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16475 ΔΒ, καὶ ἐπεζεύχθωσαν αἱ ΘΖ, ΕΗ, <lb n="1799,001,,,,13,14,10"></lb> |
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16476 καὶ ἀνεστάτω ἀπὸ τοῦ Κ σημείου τῷ <lb n="1799,001,,,,13,14,11"></lb> |
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16477 τοῦ ΕΖΗΘ τετραγώνου ἐπιπέδῳ πρὸς <lb n="1799,001,,,,13,14,12"></lb> |
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16478 ὀρθὰς εὐθεῖα ἡ ΚΛ καὶ διήχθω ἐπὶ <lb n="1799,001,,,,13,14,13"></lb> |
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16479 τὰ ἕτερα μέρη τοῦ ἐπιπέδου ὡς ἡ ΚΜ, <lb n="1799,001,,,,13,14,14"></lb> |
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16480 καὶ ἀφῃρήσθω ἀφ' ἑκατέρας τῶν ΚΛ, <lb n="1799,001,,,,13,14,15"></lb> |
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16481 ΚΜ μιᾷ τῶν ΕΚ, ΖΚ, ΗΚ, ΘΚ <lb n="1799,001,,,,13,14,16"></lb> |
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16482 ἴση ἑκατέρα τῶν ΚΛ, ΚΜ, καὶ ἐπ-<lb n="1799,001,,,,13,14,17"></lb> |
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16483 εζεύχθωσαν αἱ ΛΕ, ΛΖ, ΛΗ, ΛΘ, ΜΕ, ΜΖ, ΜΗ, ΜΘ.</s> <lb n="1799,001,,,,13,14,18"></lb> |
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16484 <s>καὶ ἐπεὶ ἴση ἐστὶν ἡ ΚΕ τῇ ΚΘ, καί ἐστιν ὀρθὴ ἡ ὑπὸ <lb n="1799,001,,,,13,14,19"></lb> |
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16485 ΕΚΘ γωνία, τὸ ἄρα ἀπὸ τῆς ΘΕ διπλάσιόν ἐστι τοῦ <lb n="1799,001,,,,13,14,20"></lb> |
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16486 ἀπὸ τῆς ΕΚ.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΛΚ τῇ ΚΕ, καί ἐστιν <lb n="1799,001,,,,13,14,21"></lb> |
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16487 ὀρθὴ ἡ ὑπὸ ΛΚΕ γωνία, τὸ ἄρα ἀπὸ τῆς ΕΛ διπλάσιόν <lb n="1799,001,,,,13,14,22"></lb> |
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16488 ἐστι τοῦ ἀπὸ ΕΚ.</s> <s>ἐδείχθη δὲ καὶ τὸ ἀπὸ τῆς ΘΕ διπλά- <pb></pb><lb n="1799,001,,,,13,14,23"></lb> |
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16489 σιον τοῦ ἀπὸ τῆς ΕΚ· τὸ ἄρα ἀπὸ τῆς ΛΕ ἴσον ἐστὶ τῷ <lb n="1799,001,,,,13,14,24"></lb> |
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16490 ἀπὸ τῆς ΕΘ· ἴση ἄρα ἐστὶν ἡ ΛΕ τῇ ΕΘ.</s> <s>διὰ τὰ αὐτὰ δὴ <lb n="1799,001,,,,13,14,25"></lb> |
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16491 καὶ ἡ ΛΘ τῇ ΘΕ ἐστιν ἴση· ἰσόπλευρον ἄρα ἐστὶ τὸ ΛΕΘ <lb n="1799,001,,,,13,14,26"></lb> |
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16492 τρίγωνον.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἕκαστον τῶν λοι-<lb n="1799,001,,,,13,14,27"></lb> |
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16493 πῶν τριγώνων, ὧν βάσεις μέν εἰσιν αἱ τοῦ ΕΖΗΘ τετρα-<lb n="1799,001,,,,13,14,28"></lb> |
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16494 γώνου πλευραί, κορυφαὶ δὲ τὰ Λ, Μ σημεῖα, ἰσόπλευρόν <lb n="1799,001,,,,13,14,29"></lb> |
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16495 ἐστιν· ὀκτάεδρον ἄρα συνέσταται ὑπὸ ὀκτὼ τριγώνων <lb n="1799,001,,,,13,14,30"></lb> |
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16496 ἰσοπλεύρων περιεχόμενον.</s> <lb n="1799,001,,,,13,14,31"></lb> |
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16497 <s>Δεῖ δὴ αὐτὸ καὶ σφαίρᾳ περιλαβεῖν τῇ δοθείσῃ καὶ δεῖ-<lb n="1799,001,,,,13,14,32"></lb> |
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16498 ξαι, ὅτι ἡ τῆς σφαίρας διάμετρος δυνάμει διπλασίων ἐστὶ <lb n="1799,001,,,,13,14,33"></lb> |
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16499 τῆς τοῦ ὀκταέδρου πλευρᾶς.</s> <lb n="1799,001,,,,13,14,34"></lb> |
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16500 <s>Ἐπεὶ γὰρ αἱ τρεῖς αἱ ΛΚ, ΚΜ, ΚΕ ἴσαι ἀλλήλαις εἰσίν, <lb n="1799,001,,,,13,14,35"></lb> |
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16501 τὸ ἄρα ἐπὶ τῆς ΛΜ γραφόμενον ἡμικύκλιον ἥξει καὶ διὰ <lb n="1799,001,,,,13,14,36"></lb> |
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16502 τοῦ Ε.</s> <s>καὶ διὰ τὰ αὐτά, ἐὰν μενούσης τῆς ΛΜ περιενε-<lb n="1799,001,,,,13,14,37"></lb> |
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16503 χθὲν τὸ ἡμικύκλιον εἰς τὸ αὐτὸ ἀποκατασταθῇ, ὅθεν ἤρξα-<lb n="1799,001,,,,13,14,38"></lb> |
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16504 το φέρεσθαι, ἥξει καὶ διὰ τῶν Ζ, Η, Θ σημείων, καὶ ἔσται <lb n="1799,001,,,,13,14,39"></lb> |
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16505 σφαίρᾳ περιειλημμένον τὸ ὀκτάεδρον.</s> <s>λέγω δή, ὅτι καὶ τῇ <lb n="1799,001,,,,13,14,40"></lb> |
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16506 δοθείσῃ.</s> <s>ἐπεὶ γὰρ ἴση ἐστὶν ἡ ΛΚ τῇ ΚΜ, κοινὴ δὲ ἡ ΚΕ, <lb n="1799,001,,,,13,14,41"></lb> |
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16507 καὶ γωνίας ὀρθὰς περιέχουσιν, βάσις ἄρα ἡ ΛΕ βάσει τῇ <lb n="1799,001,,,,13,14,42"></lb> |
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16508 ΕΜ ἐστιν ἴση.</s> <s>καὶ ἐπεὶ ὀρθή ἐστιν ἡ ὑπὸ ΛΕΜ γωνία· <lb n="1799,001,,,,13,14,43"></lb> |
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16509 ἐν ἡμικυκλίῳ γάρ· τὸ ἄρα ἀπὸ τῆς ΛΜ διπλάσιόν ἐστι <lb n="1799,001,,,,13,14,44"></lb> |
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16510 τοῦ ἀπὸ τῆς ΛΕ.</s> <s>πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΑΓ τῇ ΓΒ, <lb n="1799,001,,,,13,14,45"></lb> |
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16511 διπλασία ἐστὶν ἡ ΑΒ τῆς ΒΓ.</s> <s>ὡς δὲ ἡ ΑΒ πρὸς τὴν ΒΓ, <lb n="1799,001,,,,13,14,46"></lb> |
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16512 οὕτως τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΔ· διπλάσιον ἄρα <lb n="1799,001,,,,13,14,47"></lb> |
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16513 ἐστὶ τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΒΔ.</s> <s>ἐδείχθη δὲ καὶ τὸ <lb n="1799,001,,,,13,14,48"></lb> |
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16514 ἀπὸ τῆς ΛΜ διπλάσιον τοῦ ἀπὸ τῆς ΛΕ.</s> <s>καί ἐστιν ἴσον <lb n="1799,001,,,,13,14,49"></lb> |
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16515 τὸ ἀπὸ τῆς ΔΒ τῷ ἀπὸ τῆς ΛΕ· ἴση γὰρ κεῖται ἡ ΕΘ τῇ <pb></pb> <lb n="1799,001,,,,13,14,50"></lb> |
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16516 ΔΒ.</s> <s>ἴσον ἄρα καὶ τὸ ἀπὸ τῆς ΑΒ τῷ ἀπὸ τῆς ΛΜ· ἴση <lb n="1799,001,,,,13,14,51"></lb> |
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16517 ἄρα ἡ ΑΒ τῇ ΛΜ.</s> <s>καί ἐστιν ἡ ΑΒ ἡ τῆς δοθείσης σφαίρας <lb n="1799,001,,,,13,14,52"></lb> |
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16518 διάμετρος· ἡ ΛΜ ἄρα ἴση ἐστὶ τῇ τῆς δοθείσης σφαίρας <lb n="1799,001,,,,13,14,53"></lb> |
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16519 διαμέτρῳ.</s> <lb n="1799,001,,,,13,14,54"></lb> |
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16520 <s>Περιείληπται ἄρα τὸ ὀκτάεδρον τῇ δοθείσῃ σφαίρᾳ.</s> <lb n="1799,001,,,,13,14,55"></lb> |
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16521 <s>καὶ συναποδέδεικται, ὅτι ἡ τῆς σφαίρας διάμετρος δυνάμει <lb n="1799,001,,,,13,14,56"></lb> |
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16522 διπλασίων ἐστὶ τῆς τοῦ ὀκταέδρου πλευρᾶς· ὅπερ ἔδει <lb n="1799,001,,,,13,14,57"></lb> |
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16523 δεῖξαι.</s> <lb n="1799,001,,,,13,14,58"></lb> |
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16524 <s>Κύβον συστήσασθαι καὶ σφαίρᾳ περιλαβεῖν, ᾗ καὶ τὴν <lb n="1799,001,,,,13,15,1"></lb> |
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16525 πυραμίδα, καὶ δεῖξαι, ὅτι ἡ τῆς σφαίρας διάμετρος δυνάμει <lb n="1799,001,,,,13,15,2"></lb> |
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16526 τριπλασίων ἐστὶ τῆς τοῦ κύβου πλευρᾶς. <pb></pb> <lb n="1799,001,,,,13,15,3"></lb> |
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16527 Ἐκκείσθω ἡ τῆς δοθείσης σφαίρας διάμετρος ἡ ΑΒ <lb n="1799,001,,,,13,15,4"></lb> |
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16528 καὶ τετμήσθω κατὰ τὸ Γ ὥστε διπλῆν εἶναι τὴν ΑΓ τῆς <lb n="1799,001,,,,13,15,5"></lb> |
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16529 ΓΒ, καὶ γεγράφθω ἐπὶ τῆς ΑΒ ἡμικύκλιον τὸ ΑΔΒ, καὶ <lb n="1799,001,,,,13,15,6"></lb> |
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16530 ἀπὸ τοῦ Γ τῇ *αΒ πρὸς ὀρθὰς ἤχθω ἡ ΓΔ, καὶ ἐπεζεύχθω <lb n="1799,001,,,,13,15,7"></lb> |
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16531 ἡ ΔΒ, καὶ ἐκκείσθω τετράγωνον τὸ ΕΖΗΘ ἴσην ἔχον <lb n="1799,001,,,,13,15,8"></lb> |
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16532 τὴν πλευρὰν τῇ ΔΒ, καὶ ἀπὸ τῶν Ε, Ζ, Η, Θ τῷ τοῦ <lb n="1799,001,,,,13,15,9"></lb> |
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16533 ΕΖΗΘ τετραγώνου ἐπιπέδῳ πρὸς ὀρθὰς ἤχθωσαν αἱ <lb n="1799,001,,,,13,15,10"></lb> |
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16534 ΕΚ, ΖΛ, ΗΜ, ΘΝ, καὶ ἀφῃρήσθω ἀπὸ ἑκάστης τῶν <lb n="1799,001,,,,13,15,11"></lb> |
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16535 ΕΚ, ΖΛ, ΗΜ, ΘΝ μιᾷ τῶν ΕΖ, ΖΗ, ΗΘ, ΘΕ ἴση ἑκά-<lb n="1799,001,,,,13,15,12"></lb> |
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16536 στη τῶν ΕΚ, ΖΛ, ΗΜ, ΘΝ, καὶ ἐπεζεύχθωσαν αἱ ΚΛ, <lb n="1799,001,,,,13,15,13"></lb> |
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16537 ΛΜ, ΜΝ, ΝΚ· κύβος ἄρα συνέσταται ὁ ΖΝ ὑπὸ ἓξ <lb n="1799,001,,,,13,15,14"></lb> |
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16538 τετραγώνων ἴσων περιεχόμενος.</s> <s>δεῖ δὴ αὐτὸν καὶ σφαίρᾳ <lb n="1799,001,,,,13,15,15"></lb> |
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16539 περιλαβεῖν τῇ δοθείσῃ καὶ δεῖξαι, ὅτι ἡ τῆς σφαίρας διά-<lb n="1799,001,,,,13,15,16"></lb> |
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16540 μετρος δυνάμει τριπλασία ἐστὶ τῆς πλευρᾶς τοῦ κύβου.</s> <lb n="1799,001,,,,13,15,17"></lb> |
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16541 <s>Ἐπεζεύχθωσαν γὰρ αἱ ΚΗ, ΕΗ.</s> <s>καὶ ἐπεὶ ὀρθή ἐστιν <lb n="1799,001,,,,13,15,18"></lb> |
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16542 ἡ ὑπὸ ΚΕΗ γωνία διὰ τὸ καὶ τὴν ΚΕ ὀρθὴν εἶναι πρὸς <lb n="1799,001,,,,13,15,19"></lb> |
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16543 τὸ ΕΗ ἐπίπεδον δηλαδὴ καὶ πρὸς τὴν ΕΗ εὐθεῖαν, τὸ <lb n="1799,001,,,,13,15,20"></lb> |
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16544 ἄρα ἐπὶ τῆς ΚΗ γραφόμενον ἡμικύκλιον ἥξει καὶ διὰ τοῦ <lb n="1799,001,,,,13,15,21"></lb> |
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16545 Ε σημείου.</s> <s>πάλιν, ἐπεὶ ἡ ΗΖ ὀρθή ἐστι πρὸς ἑκατέραν <lb n="1799,001,,,,13,15,22"></lb> |
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16546 τῶν ΖΛ, ΖΕ, καὶ πρὸς τὸ ΖΚ ἄρα ἐπίπεδον ὀρθή ἐστιν <lb n="1799,001,,,,13,15,23"></lb> |
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16547 ἡ ΗΖ· ὥστε καὶ ἐὰν ἐπιζεύξωμεν τὴν ΖΚ, ἡ ΗΖ ὀρθὴ <lb n="1799,001,,,,13,15,24"></lb> |
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16548 ἔσται καὶ πρὸς τὴν ΖΚ· καὶ διὰ τοῦτο πάλιν τὸ ἐπὶ τῆς <lb n="1799,001,,,,13,15,25"></lb> |
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16549 ΗΚ γραφόμενον ἡμικύκλιον ἥξει καὶ διὰ τοῦ Ζ.</s> <s>ὁμοίως <lb n="1799,001,,,,13,15,26"></lb> |
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16550 καὶ διὰ τῶν λοιπῶν τοῦ κύβου σημείων ἥξει.</s> <s>ἐὰν δὴ μενού-<lb n="1799,001,,,,13,15,27"></lb> |
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16551 σης τῆς ΚΗ περιενεχθὲν τὸ ἡμικύκλιον εἰς τὸ αὐτὸ ἀπο-<lb n="1799,001,,,,13,15,28"></lb> |
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16552 κατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, ἔσται σφαίρᾳ περι- <pb></pb><lb n="1799,001,,,,13,15,29"></lb> |
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16553 ειλημμένος ὁ κύβος.</s> <s>λέγω δή, ὅτι καὶ τῇ δοθείσῃ.</s> <s>ἐπεὶ <lb n="1799,001,,,,13,15,30"></lb> |
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16554 γὰρ ἴση ἐστὶν ἡ ΗΖ τῇ ΖΕ, καί ἐστιν ὀρθὴ ἡ πρὸς τῷ Ζ <lb n="1799,001,,,,13,15,31"></lb> |
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16555 γωνία, τὸ ἄρα ἀπὸ τῆς ΕΗ διπλάσιόν ἐστι τοῦ ἀπὸ τῆς <lb n="1799,001,,,,13,15,32"></lb> |
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16556 ΕΖ.</s> <s>ἴση δὲ ἡ ΕΖ τῇ ΕΚ· τὸ ἄρα ἀπὸ τῆς ΕΗ διπλά-<lb n="1799,001,,,,13,15,33"></lb> |
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16557 σιόν ἐστι τοῦ ἀπὸ τῆς ΕΚ· ὥστε τὰ ἀπὸ τῶν ΗΕ, ΕΚ, <lb n="1799,001,,,,13,15,34"></lb> |
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16558 τουτέστι τὸ ἀπὸ τῆς ΗΚ, τριπλάσιόν ἐστι τοῦ ἀπὸ τῆς <lb n="1799,001,,,,13,15,35"></lb> |
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16559 ΕΚ.</s> <s>καὶ ἐπεὶ τριπλασίων ἐστὶν ἡ ΑΒ τῆς ΒΓ, ὡς δὲ ἡ ΑΒ <lb n="1799,001,,,,13,15,36"></lb> |
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16560 πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΔ, <lb n="1799,001,,,,13,15,37"></lb> |
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16561 τριπλάσιον ἄρα τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΒΔ.</s> <s>ἐδείχθη <lb n="1799,001,,,,13,15,38"></lb> |
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16562 δὲ καὶ τὸ ἀπὸ τῆς ΗΚ τοῦ ἀπὸ τῆς ΚΕ τριπλάσιον.</s> <s>καὶ <lb n="1799,001,,,,13,15,39"></lb> |
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16563 κεῖται ἴση ἡ ΚΕ τῇ ΔΒ· ἴση ἄρα καὶ ἡ ΚΗ τῇ ΑΒ.</s> <s>καί <lb n="1799,001,,,,13,15,40"></lb> |
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16564 ἐστιν ἡ ΑΒ τῆς δοθείσης σφαίρας διάμετρος· καὶ ἡ ΚΗ <lb n="1799,001,,,,13,15,41"></lb> |
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16565 ἄρα ἴση ἐστὶ τῇ τῆς δοθείσης σφαίρας διαμέτρῳ.</s> <lb n="1799,001,,,,13,15,42"></lb> |
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16566 <s>Τῇ δοθείσῃ ἄρα σφαίρᾳ περιείληπται ὁ κύβος· καὶ <lb n="1799,001,,,,13,15,43"></lb> |
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16567 συναποδέδεικται, ὅτι ἡ τῆς σφαίρας διάμετρος δυνάμει <lb n="1799,001,,,,13,15,44"></lb> |
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16568 τριπλασίων ἐστὶ τῆς τοῦ κύβου πλευρᾶς· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,15,45"></lb> |
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16569 <s>Εἰκοσάεδρον συστήσασθαι καὶ σφαίρᾳ περιλαβεῖν, ᾗ <lb n="1799,001,,,,13,16,1"></lb> |
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16570 καὶ τὰ προειρημένα σχήματα, καὶ δεῖξαι, ὅτι ἡ τοῦ εἰκο-<lb n="1799,001,,,,13,16,2"></lb> |
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16571 σαέδρου πλευρὰ ἄλογός ἐστιν ἡ καλουμένη ἐλάττων.</s> <lb n="1799,001,,,,13,16,3"></lb> |
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16572 <s>Ἐκκείσθω ἡ τῆς δοθείσης σφαίρας διάμετρος ἡ ΑΒ καὶ <lb n="1799,001,,,,13,16,4"></lb> |
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16573 τετμήσθω κατὰ τὸ Γ ὥστε τετραπλῆν εἶναι τὴν ΑΓ τῆς <lb n="1799,001,,,,13,16,5"></lb> |
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16574 ΓΒ, καὶ γεγράφθω ἐπὶ τῆς ΑΒ ἡμικύκλιον τὸ ΑΔΒ, καὶ <pb></pb> <lb n="1799,001,,,,13,16,6"></lb> |
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16575 ἤχθω ἀπὸ τοῦ Γ τῇ ΑΒ πρὸς ὀρθὰς γωνίας εὐθεῖα γραμ-<lb n="1799,001,,,,13,16,7"></lb> |
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16576 μὴ ἡ ΓΔ, καὶ ἐπεζεύχθω ἡ ΔΒ, καὶ ἐκκείσθω κύκλος <lb n="1799,001,,,,13,16,8"></lb> |
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16577 ὁ ΕΖΗΘΚ, οὗ ἡ ἐκ τοῦ κέντρου ἴση ἔστω τῇ ΔΒ, καὶ <lb n="1799,001,,,,13,16,9"></lb> |
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16578 ἐγγεγράφθω εἰς τὸν ΕΖΗΘΚ κύκλον πεντάγωνον ἰσό-<lb n="1799,001,,,,13,16,10"></lb> |
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16579 πλευρόν τε καὶ ἰσογώνιον τὸ ΕΖΗΘΚ, καὶ τετμήσθωσαν <lb n="1799,001,,,,13,16,11"></lb> |
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16580 αἱ ΕΖ, ΖΗ, ΗΘ, ΘΚ, ΚΕ περιφέρειαι δίχα κατὰ τὰ Λ, <lb n="1799,001,,,,13,16,12"></lb> |
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16581 Μ, Ν, Ξ, Ο σημεῖα, καὶ ἐπεζεύχθωσαν αἱ ΛΜ, ΜΝ, ΝΞ, <lb n="1799,001,,,,13,16,13"></lb> |
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16582 ΞΟ, ΟΛ, ΕΟ.</s> <s>ἰσόπλευρον ἄρα ἐστὶ καὶ τὸ ΛΜΝΞΟ <lb n="1799,001,,,,13,16,14"></lb> |
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16583 πεντάγωνον, καὶ δεκαγώνου ἡ ΕΟ εὐθεῖα.</s> <s>καὶ ἀνεστά-<lb n="1799,001,,,,13,16,15"></lb> |
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16584 τωσαν ἀπὸ τῶν Ε, Ζ, Η, Θ, Κ σημείων τῷ τοῦ κύκλου <lb n="1799,001,,,,13,16,16"></lb> |
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16585 ἐπιπέδῳ πρὸς ὀρθὰς γωνίας εὐθεῖαι αἱ ΕΠ, ΖΡ, ΗΣ, <lb n="1799,001,,,,13,16,17"></lb> |
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16586 ΘΤ, ΚΥ ἴσαι οὖσαι τῇ ἐκ τοῦ κέντρου τοῦ ΕΖΗΘΚ <lb n="1799,001,,,,13,16,18"></lb> |
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16587 κύκλου, καὶ ἐπεζεύχθωσαν αἱ ΠΡ, ΡΣ, ΣΤ, ΤΥ, ΥΠ, <lb n="1799,001,,,,13,16,19"></lb> |
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16588 ΠΛ, ΛΡ, ΡΜ, ΜΣ, ΣΝ, ΝΤ, ΤΞ, ΞΥ, ΥΟ, ΟΠ.</s> <s>καὶ <lb n="1799,001,,,,13,16,20"></lb> |
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16589 ἐπεὶ ἑκατέρα τῶν ΕΠ, ΚΥ τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθάς <lb n="1799,001,,,,13,16,21"></lb> |
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16590 ἐστιν, παράλληλος ἄρα ἐστὶν ἡ ΕΠ τῇ ΚΥ.</s> <s>ἔστι δὲ αὐτῇ <lb n="1799,001,,,,13,16,22"></lb> |
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16591 καὶ ἴση· αἱ δὲ τὰς ἴσας τε καὶ παραλλήλους ἐπιζευγνύουσαι <lb n="1799,001,,,,13,16,23"></lb> |
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16592 ἐπὶ τὰ αὐτὰ μέρη εὐθεῖαι ἴσαι τε καὶ παράλληλοί εἰσιν.</s> <s>ἡ <lb n="1799,001,,,,13,16,24"></lb> |
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16593 ΠΥ ἄρα τῇ ΕΚ ἴση τε καὶ παράλληλός ἐστιν.</s> <s>πενταγώνου <lb n="1799,001,,,,13,16,25"></lb> |
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16594 δὲ ἰσοπλεύρου ἡ ΕΚ· πενταγώνου ἄρα ἰσοπλεύρου καὶ ἡ <lb n="1799,001,,,,13,16,26"></lb> |
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16595 ΠΥ τοῦ εἰς τὸν ΕΖΗΘΚ κύκλον ἐγγραφομένου.</s> <s>διὰ <lb n="1799,001,,,,13,16,27"></lb> |
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16596 τὰ αὐτὰ δὴ καὶ ἑκάστη τῶν ΠΡ, ΡΣ, ΣΤ, ΤΥ πεντα-<lb n="1799,001,,,,13,16,28"></lb> |
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16597 γώνου ἐστὶν ἰσοπλεύρου τοῦ εἰς τὸν ΕΖΗΘΚ κύκλον <lb n="1799,001,,,,13,16,29"></lb> |
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16598 ἐγγραφομένου· ἰσόπλευρον ἄρα τὸ ΠΡΣΤΥ πεντάγωνον.</s> <lb n="1799,001,,,,13,16,30"></lb> |
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16599 <s>καὶ ἐπεὶ ἑξαγώνου μέν ἐστιν ἡ ΠΕ, δεκαγώνου δὲ ἡ ΕΟ, <lb n="1799,001,,,,13,16,31"></lb> |
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16600 καί ἐστιν ὀρθὴ ἡ ὑπὸ ΠΕΟ, πενταγώνου ἄρα ἐστὶν ἡ ΠΟ· <pb></pb> <lb n="1799,001,,,,13,16,32"></lb> |
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16601 ἡ γὰρ τοῦ πενταγώνου πλευρὰ δύναται τήν τε τοῦ ἑξαγώνου <lb n="1799,001,,,,13,16,33"></lb> |
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16602 καὶ τὴν τοῦ δεκαγώνου τῶν εἰς τὸν αὐτὸν κύκλον ἐγγρα-<lb n="1799,001,,,,13,16,34"></lb> |
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16603 φομένων.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἡ ΟΥ πενταγώνου ἐστὶ <lb n="1799,001,,,,13,16,35"></lb> |
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16604 πλευρά.</s> <s>ἔστι δὲ καὶ ἡ ΠΥ πενταγώνου· ἰσόπλευρον ἄρα ἐστὶ <lb n="1799,001,,,,13,16,36"></lb> |
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16605 τὸ ΠΟΥ τρίγωνον.</s> <s>διὰ τὰ αὐτὰ δὴ καὶ ἕκαστον τῶν ΠΛΡ, <lb n="1799,001,,,,13,16,37"></lb> |
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16606 ΡΜΣ, ΣΝΤ, ΤΞΥ ἰσόπλευρόν ἐστιν.</s> <s>καὶ ἐπεὶ πενταγώ-<lb n="1799,001,,,,13,16,38"></lb> |
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16607 νου ἐδείχθη ἑκατέρα τῶν ΠΛ, ΠΟ, ἔστι δὲ καὶ ἡ ΛΟ <lb n="1799,001,,,,13,16,39"></lb> |
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16608 πενταγώνου, ἰσόπλευρον ἄρα ἐστὶ τὸ ΠΛΟ τρίγωνον.</s> <s>διὰ <lb n="1799,001,,,,13,16,40"></lb> |
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16609 τὰ αὐτὰ δὴ καὶ ἕκαστον τῶν ΛΡΜ, ΜΣΝ, ΝΤΞ, ΞΥΟ <lb n="1799,001,,,,13,16,41"></lb> |
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16610 τριγώνων ἰσόπλευρόν ἐστιν.</s> <s>εἰλήφθω τὸ κέντρον τοῦ ΕΖΗ <lb n="1799,001,,,,13,16,42"></lb> |
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16611 ΘΚ κύκλου τὸ Φ σημεῖον· καὶ ἀπὸ τοῦ Φ τῷ τοῦ κύκλου <lb n="1799,001,,,,13,16,43"></lb> |
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16612 ἐπιπέδῳ πρὸς ὀρθὰς ἀνεστάτω ἡ ΦΩ, καὶ ἐκβεβλήσθω <lb n="1799,001,,,,13,16,44"></lb> |
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16613 ἐπὶ τὰ ἕτερα μέρη ὡς ἡ ΦΨ, καὶ ἀφῃρήσθω ἑξαγώνου μὲν <lb n="1799,001,,,,13,16,45"></lb> |
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16614 ἡ ΦΧ, δεκαγώνου δὲ ἑκατέρα τῶν ΦΨ, ΧΩ, καὶ ἐπεζεύχ-<lb n="1799,001,,,,13,16,46"></lb> |
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16615 θωσαν αἱ ΠΩ, ΠΧ, ΥΩ, ΕΦ, ΛΦ, ΛΨ, ΨΜ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,13,16,47"></lb> |
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16616 ἑκατέρα τῶν ΦΧ, ΠΕ τῷ τοῦ κύκλου ἐπιπέδῳ πρὸς ὀρθάς <lb n="1799,001,,,,13,16,48"></lb> |
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16617 ἐστιν, παράλληλος ἄρα ἐστὶν ἡ ΦΧ τῇ ΠΕ.</s> <s>εἰσὶ δὲ καὶ ἴσαι· <lb n="1799,001,,,,13,16,49"></lb> |
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16618 καὶ αἱ ΕΦ, ΠΧ ἄρα ἴσαι τε καὶ παράλληλοί εἰσιν.</s> <s>ἑξα-<lb n="1799,001,,,,13,16,50"></lb> |
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16619 γώνου δὲ ἡ ΕΦ· ἑξαγώνου ἄρα καὶ ἡ ΠΧ.</s> <s>καὶ ἐπεὶ ἑξαγώ-<lb n="1799,001,,,,13,16,51"></lb> |
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16620 νου μέν ἐστιν ἡ ΠΧ, δεκαγώνου δὲ ἡ ΧΩ, καὶ ὀρθή ἐστιν <lb n="1799,001,,,,13,16,52"></lb> |
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16621 ἡ ὑπὸ ΠΧΩ γωνία, πενταγώνου ἄρα ἐστὶν ἡ ΠΩ.</s> <s>διὰ <lb n="1799,001,,,,13,16,53"></lb> |
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16622 τὰ αὐτὰ δὴ καὶ ἡ ΥΩ πενταγώνου ἐστίν, ἐπειδήπερ, ἐὰν <lb n="1799,001,,,,13,16,54"></lb> |
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16623 ἐπιζεύξωμεν τὰς ΦΚ, ΧΥ, ἴσαι καὶ ἀπεναντίον ἔσονται, <lb n="1799,001,,,,13,16,55"></lb> |
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16624 καί ἐστιν ἡ ΦΚ ἐκ τοῦ κέντρου οὖσα ἑξαγώνου· ἑξαγώ-<lb n="1799,001,,,,13,16,56"></lb> |
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16625 νου ἄρα καὶ ἡ ΧΥ.</s> <s>δεκαγώνου δὲ ἡ ΧΩ, καὶ ὀρθὴ ἡ ὑπὸ <pb></pb> <lb n="1799,001,,,,13,16,57"></lb> |
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16626 ΥΧΩ· πενταγώνου ἄρα ἡ ΥΩ.</s> <s>ἔστι δὲ καὶ ἡ ΠΥ πεντα-<lb n="1799,001,,,,13,16,58"></lb> |
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16627 γώνου· ἰσόπλευρον ἄρα ἐστὶ τὸ ΠΥΩ τρίγωνον.</s> <s>διὰ τὰ αὐτὰ <lb n="1799,001,,,,13,16,59"></lb> |
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16628 δὴ καὶ ἕκαστον τῶν λοιπῶν τριγώνων, ὧν βάσεις μέν <lb n="1799,001,,,,13,16,60"></lb> |
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16629 εἰσιν αἱ ΠΡ, ΡΣ, ΣΤ, ΤΥ εὐθεῖαι, κορυφὴ δὲ τὸ Ω ση-<lb n="1799,001,,,,13,16,61"></lb> |
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16630 μεῖον, ἰσόπλευρόν ἐστιν.</s> <s>πάλιν, ἐπεὶ ἑξαγώνου μὲν ἡ ΦΛ, <lb n="1799,001,,,,13,16,62"></lb> |
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16631 δεκαγώνου δὲ ἡ ΦΨ, καὶ ὀρθή ἐστιν ἡ ὑπὸ ΛΦΨ γωνία, <lb n="1799,001,,,,13,16,63"></lb> |
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16632 πενταγώνου ἄρα ἐστὶν ἡ ΛΨ.</s> <s>διὰ τὰ αὐτὰ δὴ ἐὰν ἐπι-<lb n="1799,001,,,,13,16,64"></lb> |
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16633 ζεύξωμεν τὴν ΜΦ οὖσαν ἑξαγώνου, συνάγεται καὶ ἡ ΜΨ <lb n="1799,001,,,,13,16,65"></lb> |
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16634 πενταγώνου.</s> <s>ἔστι δὲ καὶ ἡ ΛΜ πενταγώνου· ἰσόπλευρον <lb n="1799,001,,,,13,16,66"></lb> |
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16635 ἄρα ἐστὶ τὸ ΛΜΨ τρίγωνον.</s> <s>ὁμοίως δὴ δειχθήσεται, ὅτι <lb n="1799,001,,,,13,16,67"></lb> |
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16636 καὶ ἕκαστον τῶν λοιπῶν τριγώνων, ὧν βάσεις μέν εἰσιν <lb n="1799,001,,,,13,16,68"></lb> |
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16637 αἱ ΜΝ, ΝΞ, ΞΟ, ΟΛ, κορυφὴ δὲ τὸ Ψ σημεῖον, ἰσόπλευ-<lb n="1799,001,,,,13,16,69"></lb> |
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16638 ρόν ἐστιν.</s> <s>συνέσταται ἄρα εἰκοσάεδρον ὑπὸ εἴκοσι τρι-<lb n="1799,001,,,,13,16,70"></lb> |
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16639 γώνων ἰσοπλεύρων περιεχόμενον.</s> <lb n="1799,001,,,,13,16,71"></lb> |
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16640 <s>Δεῖ δὴ αὐτὸ καὶ σφαίρᾳ περιλαβεῖν τῇ δοθείσῃ καὶ δεῖ-<lb n="1799,001,,,,13,16,72"></lb> |
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16641 ξαι, ὅτι ἡ τοῦ εἰκοσαέδρου πλευρὰ ἄλογός ἐστιν ἡ καλου-<lb n="1799,001,,,,13,16,73"></lb> |
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16642 μένη ἐλάσσων.</s> <lb n="1799,001,,,,13,16,74"></lb> |
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16643 <s>Ἐπεὶ γὰρ ἑξαγώνου ἐστὶν ἡ ΦΧ, δεκαγώνου δὲ ἡ ΧΩ, <lb n="1799,001,,,,13,16,75"></lb> |
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16644 ἡ ΦΩ ἄρα ἄκρον καὶ μέσον λόγον τέμηται κατὰ τὸ Χ, καὶ <lb n="1799,001,,,,13,16,76"></lb> |
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16645 τὸ μεῖζον αὐτῆς τμῆμά ἐστιν ἡ ΦΧ· ἔστιν ἄρα ὡς ἡ ΩΦ <lb n="1799,001,,,,13,16,77"></lb> |
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16646 πρὸς τὴν ΦΧ, οὕτως ἡ ΦΧ πρὸς τὴν ΧΩ.</s> <s>ἴση δὲ ἡ μὲν ΦΧ <lb n="1799,001,,,,13,16,78"></lb> |
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16647 τῇ ΦΕ, ἡ δὲ ΧΩ τῇ ΦΨ· ἔστιν ἄρα ὡς ἡ ΩΦ πρὸς τὴν <lb n="1799,001,,,,13,16,79"></lb> |
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16648 ΦΕ, οὕτως ἡ ΕΦ πρὸς τὴν ΦΨ.</s> <s>καί εἰσιν ὀρθαὶ αἱ ὑπὸ <lb n="1799,001,,,,13,16,80"></lb> |
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16649 ΩΦΕ, ΕΦΨ γωνίαι· ἐὰν ἄρα ἐπιζεύξωμεν τὴν ΕΩ εὐθεῖαν, <lb n="1799,001,,,,13,16,81"></lb> |
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16650 ὀρθὴ ἔσται ἡ ὑπὸ ΨΕΩ γωνία διὰ τὴν ὁμοιότητα τῶν ΨΕΩ, <pb></pb> <lb n="1799,001,,,,13,16,82"></lb> |
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16651 ΦΕΩ τριγώνων.</s> <s>διὰ τὰ αὐτὰ δὴ ἐπεί ἐστιν ὡς ἡ ΩΦ πρὸς <lb n="1799,001,,,,13,16,83"></lb> |
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16652 τὴν ΦΧ, οὕτως ἡ ΦΧ πρὸς τὴν ΧΩ, ἴση δὲ ἡ μὲν ΩΦ τῇ <lb n="1799,001,,,,13,16,84"></lb> |
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16653 ΨΧ, ἡ δὲ ΦΧ τῇ ΧΠ, ἔστιν ἄρα ὡς ἡ ΨΧ πρὸς τὴν ΧΠ, <lb n="1799,001,,,,13,16,85"></lb> |
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16654 οὕτως ἡ ΠΧ πρὸς τὴν ΧΩ.</s> <s>καὶ διὰ τοῦτο πάλιν ἐὰν ἐπιζεύ-<lb n="1799,001,,,,13,16,86"></lb> |
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16655 ξωμεν τὴν ΠΨ, ὀρθὴ ἔσται ἡ πρὸς τῷ Π γωνία· τὸ ἄρα ἐπὶ <lb n="1799,001,,,,13,16,87"></lb> |
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16656 τῆς ΨΩ γραφόμενον ἡμικύκλιον ἥξει καὶ διὰ τοῦ Π.</s> <s>καὶ ἐὰν <lb n="1799,001,,,,13,16,88"></lb> |
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16657 μενούσης τῆς ΨΩ περιενεχθὲν τὸ ἡμικύκλιον εἰς τὸ αὐτὸ <lb n="1799,001,,,,13,16,89"></lb> |
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16658 πάλιν ἀποκατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, ἥξει καὶ διὰ <lb n="1799,001,,,,13,16,90"></lb> |
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16659 τοῦ Π καὶ τῶν λοιπῶν σημείων τοῦ εἰκοσαέδρου, καὶ ἔσται <lb n="1799,001,,,,13,16,91"></lb> |
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16660 σφαίρᾳ περιειλημμένον τὸ εἰκοσάεδρον.</s> <s>λέγω δή, ὅτι καὶ <lb n="1799,001,,,,13,16,92"></lb> |
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16661 τῇ δοθείσῃ.</s> <s>τετμήσθω γὰρ ἡ ΦΧ δίχα κατὰ τὸ Α#.</s> <s>καὶ <lb n="1799,001,,,,13,16,93"></lb> |
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16662 ἐπεὶ εὐθεῖα γραμμὴ ἡ ΦΩ ἄκρον καὶ μέσον λόγον τέτμηται <lb n="1799,001,,,,13,16,94"></lb> |
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16663 κατὰ τὸ Χ, καὶ τὸ ἔλασσον αὐτῆς τμῆμά ἐστιν ἡ ΩΧ, ἡ <lb n="1799,001,,,,13,16,95"></lb> |
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16664 ἄρα ΩΧ προσλαβοῦσα τὴν ἡμίσειαν τοῦ μείζονος τμήμα-<lb n="1799,001,,,,13,16,96"></lb> |
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16665 τος τὴν ΧΑ# πενταπλάσιον δύναται τοῦ ἀπὸ τῆς ἡμισείας <lb n="1799,001,,,,13,16,97"></lb> |
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16666 τοῦ μείζονος τμήματος· πενταπλάσιον ἄρα ἐστὶ τὸ ἀπὸ τῆς <lb n="1799,001,,,,13,16,98"></lb> |
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16667 ΩΑ# τοῦ ἀπὸ τῆς Α#Χ.</s> <s>καί ἐστι τῆς μὲν ΩΑ# διπλῆ ἡ ΩΨ, <lb n="1799,001,,,,13,16,99"></lb> |
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16668 τῆς δὲ Α#Χ διπλῆ ἡ ΦΧ· πενταπλάσιον ἄρα ἐστὶ τὸ ἀπὸ <lb n="1799,001,,,,13,16,100"></lb> |
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16669 τῆς ΩΨ τοῦ ἀπὸ τῆς ΧΦ.</s> <s>καὶ ἐπεὶ τετραπλῆ ἐστιν ἡ ΑΓ <lb n="1799,001,,,,13,16,101"></lb> |
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16670 τῆς ΓΒ, πενταπλῆ ἄρα ἐστὶν ἡ ΑΒ τῆς ΒΓ.</s> <s>ὡς δὲ ἡ ΑΒ <lb n="1799,001,,,,13,16,102"></lb> |
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16671 πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΔ· <pb></pb> <lb n="1799,001,,,,13,16,103"></lb> |
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16672 πενταπλάσιον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΒΔ.</s> <lb n="1799,001,,,,13,16,104"></lb> |
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16673 <s>ἐδείχθη δὲ καὶ τὸ ἀπὸ τῆς ΩΨ πενταπλάσιον τοῦ ἀπὸ τῆς <lb n="1799,001,,,,13,16,105"></lb> |
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16674 ΦΧ.</s> <s>καί ἐστιν ἴση ἡ ΔΒ τῇ ΦΧ· ἑκατέρα γὰρ αὐτῶν ἴση <lb n="1799,001,,,,13,16,106"></lb> |
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16675 ἐστὶ τῇ ἐκ τοῦ κέντρου τοῦ ΕΖΗΘΚ κύκλου· ἴση ἄρα <lb n="1799,001,,,,13,16,107"></lb> |
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16676 καὶ ἡ ΑΒ τῇ ΨΩ.</s> <s>καί ἐστιν ἡ ΑΒ ἡ τῆς δοθείσης σφαίρας <lb n="1799,001,,,,13,16,108"></lb> |
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16677 διάμετρος· καὶ ἡ ΨΩ ἄρα ἴση ἐστὶ τῇ τῆς δοθείσης σφαί-<lb n="1799,001,,,,13,16,109"></lb> |
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16678 ρας διαμέτρῳ.</s> <s>τῇ ἄρα δοθείσῃ σφαίρᾳ περιείληπται τὸ <lb n="1799,001,,,,13,16,110"></lb> |
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16679 εἰκοσάεδρον.</s> <lb n="1799,001,,,,13,16,111"></lb> |
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16680 <s>Λέγω δή, ὅτι ἡ τοῦ εἰκοσαέδρου πλευρὰ ἄλογός ἐστιν <lb n="1799,001,,,,13,16,112"></lb> |
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16681 ἡ καλουμένη ἐλάττων.</s> <s>ἐπεὶ γὰρ ῥητή ἐστιν ἡ τῆς σφαί- <pb></pb><lb n="1799,001,,,,13,16,113"></lb> |
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16682 ρας διάμετρος, καί ἐστι δυνάμει <lb n="1799,001,,,,13,16,114"></lb> |
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16683 πενταπλασίων τῆς ἐκ τοῦ κέν-<lb n="1799,001,,,,13,16,115"></lb> |
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16684 τρου τοῦ ΕΖΗΘΚ κύκλου, <lb n="1799,001,,,,13,16,116"></lb> |
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16685 ῥητὴ ἄρα ἐστὶ καὶ ἡ ἐκ τοῦ <lb n="1799,001,,,,13,16,117"></lb> |
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16686 κέντρου τοῦ ΕΖΗΘΚ κύκλου· <lb n="1799,001,,,,13,16,118"></lb> |
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16687 ὥστε καὶ ἡ διάμετρος αὐτοῦ ῥητή <lb n="1799,001,,,,13,16,119"></lb> |
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16688 ἐστιν.</s> <s>ἐὰν δὲ εἰς κύκλον ῥητὴν <lb n="1799,001,,,,13,16,120"></lb> |
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16689 ἔχοντα τὴν διάμετρον πεντάγωνον <lb n="1799,001,,,,13,16,121"></lb> |
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16690 ἰσόπλευρον ἐγγραφῇ, ἡ τοῦ πεν-<lb n="1799,001,,,,13,16,122"></lb> |
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16691 ταγώνου πλευρὰ ἄλογός ἐστιν <lb n="1799,001,,,,13,16,123"></lb> |
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16692 ἡ καλουμένη ἐλάττων.</s> <s>ἡ δὲ τοῦ <lb n="1799,001,,,,13,16,124"></lb> |
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16693 ΕΖΗΘΚ πενταγώνου πλευρὰ <lb n="1799,001,,,,13,16,125"></lb> |
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16694 ἡ τοῦ εἰκοσαέδρου ἐστίν.</s> <s>ἡ ἄρα <lb n="1799,001,,,,13,16,126"></lb> |
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16695 τοῦ εἰκοσαέδρου πλευρὰ ἄλογός <lb n="1799,001,,,,13,16,127"></lb> |
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16696 ἐστιν ἡ καλουμένη ἐλάττων.</s> <lb n="1799,001,,,,13,16,128"></lb> |
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16697 <s>Πόρισμα <lb n="1799,001,,,,13,16,129"></lb> |
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16698 Ἐκ δὴ τούτου φανερόν, ὅτι ἡ τῆς σφαίρας διάμετρος <lb n="1799,001,,,,13,16,130"></lb> |
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16699 δυνάμει πενταπλασίων ἐστὶ τῆς ἐκ τοῦ κέντρου τοῦ κύκλου, <lb n="1799,001,,,,13,16,131"></lb> |
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16700 ἀφ' οὗ τὸ εἰκοσάεδρον ἀναγέγραπται, καὶ ὅτι ἡ τῆς σφαί-<lb n="1799,001,,,,13,16,132"></lb> |
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16701 ρας διάμετρος σύγκειται ἔκ τε τῆς τοῦ ἑξαγώνου καὶ δύο <lb n="1799,001,,,,13,16,133"></lb> |
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16702 τῶν τοῦ δεκαγώνου τῶν εἰς τὸν αὐτὸν κύκλον ἐγγραφο-<lb n="1799,001,,,,13,16,134"></lb> |
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16703 μένων.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,16,135"></lb> |
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16704 <s>Δωδεκάεδρον συστήσασθαι καὶ σφαίρᾳ περιλαβεῖν, ᾗ <lb n="1799,001,,,,13,17,1"></lb> |
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16705 καὶ τὰ προειρημένα σχήματα, καὶ δεῖξαι, ὅτι ἡ τοῦ δωδε-<lb n="1799,001,,,,13,17,2"></lb> |
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16706 καέδρου πλευρὰ ἄλογός ἐστιν ἡ καλουμένη ἀποτομή.</s> <s><pb></pb> <lb n="1799,001,,,,13,17,3"></lb> |
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16707 Ἐκκείσθωσαν τοῦ προειρημένου κύβου δύο ἐπίπεδα <lb n="1799,001,,,,13,17,4"></lb> |
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16708 πρὸς ὀρθὰς ἀλλήλοις τὰ ΑΒΓΔ, ΓΒΕΖ, καὶ τετμήσθω <lb n="1799,001,,,,13,17,5"></lb> |
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16709 ἑκάστη τῶν ΑΒ, ΒΓ, ΓΔ, ΔΑ, ΕΖ, ΕΒ, ΖΓ πλευρῶν <lb n="1799,001,,,,13,17,6"></lb> |
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16710 δίχα κατὰ τὰ Η, Θ, Κ, Λ, Μ, Ν, Ξ, καὶ ἐπεζεύχθωσαν αἱ <lb n="1799,001,,,,13,17,7"></lb> |
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16711 ΗΚ, ΘΛ, ΜΘ, ΝΞ, καὶ τετμήσθω ἑκάστη τῶν ΝΟ, <lb n="1799,001,,,,13,17,8"></lb> |
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16712 ΟΞ, ΘΠ ἄκρον καὶ μέσον λόγον κατὰ τὰ Ρ, Σ, Τ σημεῖα, <lb n="1799,001,,,,13,17,9"></lb> |
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16713 καὶ ἔστω αὐτῶν μείζονα τμήματα τὰ ΡΟ, ΟΣ, ΤΠ, καὶ <lb n="1799,001,,,,13,17,10"></lb> |
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16714 ἀνεστάτωσαν ἀπὸ τῶν Ρ, Σ, Τ σημείων τοῖς τοῦ κύβου <lb n="1799,001,,,,13,17,11"></lb> |
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16715 ἐπιπέδοις πρὸς ὀρθὰς ἐπὶ τὰ ἐκτὸς μέρη τοῦ κύβου αἱ ΡΥ, <lb n="1799,001,,,,13,17,12"></lb> |
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16716 ΣΦ, ΤΧ, καὶ κείσθωσαν ἴσαι ταῖς ΡΟ, ΟΣ, ΤΠ, καὶ <lb n="1799,001,,,,13,17,13"></lb> |
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16717 ἐπεζεύχθωσαν αἱ ΥΒ, ΒΧ, ΧΓ, ΓΦ, ΦΥ.</s> <s>λέγω, ὅτι τὸ <lb n="1799,001,,,,13,17,14"></lb> |
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16718 ΥΒΧΓΦ πεντάγωνον ἰσόπλευρόν τε καὶ ἐν ἑνὶ ἐπιπέδῳ <lb n="1799,001,,,,13,17,15"></lb> |
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16719 καὶ ἔτι ἰσογώνιόν ἐστιν.</s> <s>ἐπεζεύχθωσαν γὰρ αἱ ΡΒ, ΣΒ, <lb n="1799,001,,,,13,17,16"></lb> |
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16720 ΦΒ.</s> <s>καὶ ἐπεὶ εὐθεῖα ἡ Ν*ο ἄκρον καὶ μέσον λόγον τέτμη-<lb n="1799,001,,,,13,17,17"></lb> |
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16721 ται κατὰ τὸ Ρ, καὶ τὸ μεῖζον τμῆμά ἐστιν ἡ ΡΟ, τὰ ἄρα <lb n="1799,001,,,,13,17,18"></lb> |
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16722 ἀπὸ τῶν ΟΝ, ΝΡ τριπλάσιά ἐστι τοῦ ἀπὸ τῆς ΡΟ.</s> <s>ἴση <lb n="1799,001,,,,13,17,19"></lb> |
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16723 δὲ ἡ μὲν ΟΝ τῇ ΝΒ, ἡ δὲ ΟΡ τῇ ΡΥ· τὰ ἄρα ἀπὸ τῶν <lb n="1799,001,,,,13,17,20"></lb> |
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16724 ΒΝ, ΝΡ τριπλάσιά ἐστι τοῦ ἀπὸ τῆς ΡΥ.</s> <s>τοῖς δὲ ἀπὸ τῶν <lb n="1799,001,,,,13,17,21"></lb> |
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16725 ΒΝ, ΝΡ τὸ ἀπὸ τῆς ΒΡ ἐστιν ἴσον· τὸ ἄρα ἀπὸ τῆς ΒΡ <lb n="1799,001,,,,13,17,22"></lb> |
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16726 τριπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΡΥ· ὥστε τὰ ἀπὸ τῶν ΒΡ, <lb n="1799,001,,,,13,17,23"></lb> |
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16727 ΡΥ τετραπλάσιά ἐστι τοῦ ἀπὸ τῆς ΡΥ.</s> <s>τοῖς δὲ ἀπὸ τῶν <lb n="1799,001,,,,13,17,24"></lb> |
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16728 ΒΡ, ΡΥ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΒΥ· τὸ ἄρα ἀπὸ τῆς ΒΥ <lb n="1799,001,,,,13,17,25"></lb> |
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16729 τετραπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΥΡ· διπλῆ ἄρα ἐστὶν ἡ <lb n="1799,001,,,,13,17,26"></lb> |
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16730 ΒΥ τῆς ΡΥ.</s> <s>ἔστι δὲ καὶ ἡ ΦΥ τῆς ΥΡ διπλῆ, ἐπειδήπερ <pb></pb> <lb n="1799,001,,,,13,17,27"></lb> |
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16731 καὶ ἡ ΣΡ τῆς ΟΡ, τουτέστι τῆς ΡΥ, ἐστι διπλῆ· ἴση ἄρα <lb n="1799,001,,,,13,17,28"></lb> |
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16732 ἡ ΒΥ τῇ ΥΦ.</s> <s>ὁμοίως δὴ δειχθήσεται, ὅτι καὶ ἑκάστη τῶν <lb n="1799,001,,,,13,17,29"></lb> |
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16733 ΒΧ, ΧΓ, ΓΦ ἑκατέρᾳ τῶν ΒΥ, ΥΦ ἐστιν ἴση.</s> <s>ἰσόπλευ-<lb n="1799,001,,,,13,17,30"></lb> |
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16734 ρον ἄρα ἐστὶ τὸ ΒΥΦΓΧ πεντάγωνον.</s> <s>λέγω δή, ὅτι καὶ <lb n="1799,001,,,,13,17,31"></lb> |
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16735 ἐν ἑνί ἐστιν ἐπιπέδῳ.</s> <s>ἤχθω γὰρ ἀπὸ τοῦ Ο ἑκατέρᾳ τῶν <lb n="1799,001,,,,13,17,32"></lb> |
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16736 ΡΥ, ΣΦ παράλληλος ἐπὶ τὰ ἐκτὸς τοῦ κύβου μέρη ἡ ΟΨ, <lb n="1799,001,,,,13,17,33"></lb> |
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16737 καὶ ἐπεζεύχθωσαν αἱ ΨΘ, ΘΧ· λέγω, ὅτι ἡ ΨΘΧ εὐθεῖά <lb n="1799,001,,,,13,17,34"></lb> |
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16738 ἐστιν.</s> <s>ἐπεὶ γὰρ ἡ ΘΠ ἄκρον καὶ μέσον λόγον τέτμηται κατὰ <lb n="1799,001,,,,13,17,35"></lb> |
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16739 τὸ Τ, καὶ τὸ μεῖζον αὐτῆς τμῆμά ἐστιν ἡ ΠΤ, ἔστιν ἄρα ὡς <lb n="1799,001,,,,13,17,36"></lb> |
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16740 ἡ ΘΠ πρὸς τὴν ΠΤ, οὕτως ἡ ΠΤ πρὸς τὴν ΤΘ.</s> <s>ἴση δὲ ἡ <lb n="1799,001,,,,13,17,37"></lb> |
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16741 μὲν ΘΠ τῇ ΘΟ, ἡ δὲ ΠΤ ἑκατέρᾳ τῶν ΤΧ, ΟΨ· ἔστιν <lb n="1799,001,,,,13,17,38"></lb> |
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16742 ἄρα ὡς ἡ ΘΟ πρὸς τὴν ΟΨ, οὕτως ἡ ΧΤ πρὸς τὴν ΤΘ.</s> <lb n="1799,001,,,,13,17,39"></lb> |
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16743 <s>καί ἐστι παράλληλος ἡ μὲν ΘΟ τῇ ΤΧ· ἑκατέρα γὰρ αὐτῶν <lb n="1799,001,,,,13,17,40"></lb> |
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16744 τῷ ΒΔ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν· ἡ δὲ ΤΘ τῇ ΟΨ· ἑκα-<lb n="1799,001,,,,13,17,41"></lb> |
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16745 τέρα γὰρ αὐτῶν τῷ ΒΖ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν.</s> <s>ἐὰν δὲ <lb n="1799,001,,,,13,17,42"></lb> |
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16746 δύο τρίγωνα συντεθῇ κατὰ μίαν γωνίαν, ὡς τὰ ΨΟΘ, <lb n="1799,001,,,,13,17,43"></lb> |
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16747 ΘΤΧ, τὰς δύο πλευρὰς ταῖς δυσὶν ἀνάλογον ἔχοντα, ὥστε <lb n="1799,001,,,,13,17,44"></lb> |
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16748 τὰς ὁμολόγους αὐτῶν πλευρὰς καὶ παραλλήλους εἶναι, αἱ <lb n="1799,001,,,,13,17,45"></lb> |
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16749 λοιπαὶ εὐθεῖαι ἐπ' εὐθείας ἔσονται· ἐπ' εὐθείας ἄρα ἐστὶν <lb n="1799,001,,,,13,17,46"></lb> |
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16750 ἡ ΨΘ τῇ ΘΧ.</s> <s>πᾶσα δὲ εὐθεῖα ἐν ἑνί ἐστιν ἐπιπέδῳ· ἐν <lb n="1799,001,,,,13,17,47"></lb> |
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16751 ἑνὶ ἄρα ἐπιπέδῳ ἐστὶ τὸ ΥΒΧΓΦ πεντάγωνον.</s> <lb n="1799,001,,,,13,17,48"></lb> |
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16752 <s>Λέγω δή, ὅτι καὶ ἰσογώνιόν ἐστιν.</s> <lb n="1799,001,,,,13,17,49"></lb> |
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16753 <s>Ἐπεὶ γὰρ εὐθεῖα γραμμὴ ἡ ΝΟ ἄκρον καὶ μέσον λόγον <lb n="1799,001,,,,13,17,50"></lb> |
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16754 τέτμηται κατὰ τὸ Ρ, καὶ τὸ μεῖζον τμῆμά ἐστιν ἡ ΟΡ <lb n="1799,001,,,,13,17,51"></lb> |
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16755 [ἔστιν ἄρα ὡς συναμφότερος ἡ ΝΟ, ΟΡ πρὸς τὴν ΟΝ, <pb></pb> <lb n="1799,001,,,,13,17,52"></lb> |
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16756 οὕτως ἡ ΝΟ πρὸς τὴν ΟΡ], ἴση δὲ ἡ ΟΡ τῇ ΟΣ [ἔστιν <lb n="1799,001,,,,13,17,53"></lb> |
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16757 ἄρα ὡς ἡ ΣΝ πρὸς τὴν ΝΟ, οὕτως ἡ ΝΟ πρὸς τὴν ΟΣ], <lb n="1799,001,,,,13,17,54"></lb> |
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16758 ἡ ΝΣ ἄρα ἄκρον καὶ μέσον λόγον τέτμηται κατὰ τὸ Ο, <lb n="1799,001,,,,13,17,55"></lb> |
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16759 καὶ τὸ μεῖζον τμῆμά ἐστιν ἡ ΝΟ· τὰ ἄρα ἀπὸ τῶν ΝΣ, <lb n="1799,001,,,,13,17,56"></lb> |
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16760 ΣΟ τριπλάσιά ἐστι τοῦ ἀπὸ τῆς ΝΟ.</s> <s>ἴση δὲ ἡ μὲν ΝΟ <lb n="1799,001,,,,13,17,57"></lb> |
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16761 τῇ ΝΒ, ἡ δὲ ΟΣ τῇ ΣΦ· τὰ ἄρα ἀπὸ τῶν ΝΣ, ΣΦ τετρά-<lb n="1799,001,,,,13,17,58"></lb> |
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16762 γωνα τριπλάσιά ἐστι τοῦ ἀπὸ τῆς ΝΒ· ὥστε τὰ ἀπὸ τῶν <lb n="1799,001,,,,13,17,59"></lb> |
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16763 ΦΣ, ΣΝ, ΝΒ τετραπλάσιά ἐστι τοῦ ἀπὸ τῆς ΝΒ.</s> <s>τοῖς <lb n="1799,001,,,,13,17,60"></lb> |
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16764 δὲ ἀπὸ τῶν ΣΝ, ΝΒ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΣΒ· τὰ ἄρα <lb n="1799,001,,,,13,17,61"></lb> |
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16765 ἀπὸ τῶν ΒΣ, ΣΦ, τουτέστι τὸ ἀπὸ τῆς ΒΦ ̔̓ρθὴ γὰρ ἡ <lb n="1799,001,,,,13,17,62"></lb> |
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16766 ὑπὸ ΦΣΒ γωνίἀ τετρα- <lb n="1799,001,,,,13,17,63"></lb> |
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16767 πλάσιόν ἐστι τοῦ ἀπὸ τῆς <lb n="1799,001,,,,13,17,64"></lb> |
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16768 ΝΒ· διπλῆ ἄρα ἐστὶν ἡ ΦΒ <lb n="1799,001,,,,13,17,65"></lb> |
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16769 τῆς ΒΝ.</s> <s>ἔστι δὲ καὶ ἡ ΒΓ <lb n="1799,001,,,,13,17,66"></lb> |
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16770 τῆς ΒΝ διπλῆ· ἴση ἄρα <lb n="1799,001,,,,13,17,67"></lb> |
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16771 ἐστὶν ἡ ΒΦ τῇ ΒΓ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,13,17,68"></lb> |
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16772 δύο αἱ ΒΥ, ΥΦ δυσὶ ταῖς <lb n="1799,001,,,,13,17,69"></lb> |
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16773 ΒΧ, ΧΓ ἴσαι εἰσίν, καὶ <lb n="1799,001,,,,13,17,70"></lb> |
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16774 βάσις ἡ ΒΦ βάσει τῇ ΒΓ <lb n="1799,001,,,,13,17,71"></lb> |
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16775 ἴση, γωνία ἄρα ἡ ὑπὸ ΒΥΦ <lb n="1799,001,,,,13,17,72"></lb> |
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16776 γωνίᾳ τῇ ὑπὸ ΒΧΓ ἐστιν <lb n="1799,001,,,,13,17,73"></lb> |
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16777 ἴση.</s> <s>ὁμοίως δὴ δείξομεν, ὅτι <lb n="1799,001,,,,13,17,74"></lb> |
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16778 καὶ ἡ ὑπὸ ΥΦΓ γωνία ἴση <lb n="1799,001,,,,13,17,75"></lb> |
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16779 ἐστὶ τῇ ὑπὸ ΒΧΓ· αἱ ἄρα <lb n="1799,001,,,,13,17,76"></lb> |
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16780 ὑπὸ ΒΧΓ, ΒΥΦ, ΥΦΓ <pb></pb> <lb n="1799,001,,,,13,17,77"></lb> |
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16781 τρεῖς γωνίαι ἴσαι ἀλλήλαις εἰσίν.</s> <s>ἐὰν δὲ πενταγώνου ἰσο-<lb n="1799,001,,,,13,17,78"></lb> |
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16782 πλεύρου αἱ τρεῖς γωνίαι ἴσαι ἀλλήλαις ὦσιν, ἰσογώνιον <lb n="1799,001,,,,13,17,79"></lb> |
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16783 ἔσται τὸ πεντάγωνον· ἰσογώνιον ἄρα ἐστὶ τὸ ΒΥΦΓΧ <lb n="1799,001,,,,13,17,80"></lb> |
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16784 πεντάγωνον.</s> <s>ἐδείχθη δὲ καὶ ἰσόπλευρον· τὸ ἄρα ΒΥΦΓΧ <lb n="1799,001,,,,13,17,81"></lb> |
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16785 πεντάγωνον ἰσόπλευρόν ἐστι καὶ ἰσογώνιον, καί ἐστιν ἐπὶ <lb n="1799,001,,,,13,17,82"></lb> |
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16786 μιᾶς τοῦ κύβου πλευρᾶς τῆς ΒΓ.</s> <s>ἐὰν ἄρα ἐφ' ἑκάστης τῶν <lb n="1799,001,,,,13,17,83"></lb> |
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16787 τοῦ κύβου δώδεκα πλευρῶν τὰ αὐτὰ κατασκευάσωμεν, <lb n="1799,001,,,,13,17,84"></lb> |
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16788 συσταθήσεταί τι σχῆμα στερεὸν ὑπὸ δώδεκα πενταγώνων <lb n="1799,001,,,,13,17,85"></lb> |
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16789 ἰσοπλεύρων τε καὶ ἰσογωνίων περιεχόμενον, ὃ καλεῖται <lb n="1799,001,,,,13,17,86"></lb> |
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16790 δωδεκάεδρον.</s> <lb n="1799,001,,,,13,17,87"></lb> |
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16791 <s>Δεῖ δὴ αὐτὸ καὶ σφαίρᾳ περιλαβεῖν τῇ δοθείσῃ καὶ δεῖ-<lb n="1799,001,,,,13,17,88"></lb> |
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16792 ξαι, ὅτι ἡ τοῦ δωδεκαέδρου πλευρὰ ἄλογός ἐστιν ἡ καλου-<lb n="1799,001,,,,13,17,89"></lb> |
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16793 μένη ἀποτομή.</s> <lb n="1799,001,,,,13,17,90"></lb> |
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16794 <s>Ἐκβεβλήσθω γὰρ ἡ ΨΟ, καὶ ἔστω ἡ ΨΩ· συμβάλλει <lb n="1799,001,,,,13,17,91"></lb> |
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16795 ἄρα ἡ ΟΩ τῇ τοῦ κύβου διαμέτρῳ, καὶ δίχα τέμνουσιν <lb n="1799,001,,,,13,17,92"></lb> |
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16796 ἀλλήλας· τοῦτο γὰρ δέδεικται ἐν τῷ παρατελεύτῳ θεωρή-<lb n="1799,001,,,,13,17,93"></lb> |
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16797 ματι τοῦ ἑνδεκάτου βιβλίου.</s> <s>τεμνέτωσαν κατὰ τὸ Ω· τὸ Ω <lb n="1799,001,,,,13,17,94"></lb> |
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16798 ἄρα κέντρον ἐστὶ τῆς σφαίρας τῆς περιλαμβανούσης τὸν <lb n="1799,001,,,,13,17,95"></lb> |
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16799 κύβον, καὶ ἡ ΩΟ ἡμίσεια τῆς πλευρᾶς τοῦ κύβου.</s> <s>ἐπεζεύχ-<lb n="1799,001,,,,13,17,96"></lb> |
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16800 θω δὴ ἡ ΥΩ.</s> <s>καὶ ἐπεὶ εὐθεῖα γραμμὴ ἡ ΝΣ ἄκρον καὶ <lb n="1799,001,,,,13,17,97"></lb> |
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16801 μέσον λόγον τέτμηται κατὰ τὸ Ο, καὶ τὸ μεῖζον αὐτῆς <lb n="1799,001,,,,13,17,98"></lb> |
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16802 τμῆμά ἐστιν ἡ ΝΟ, τὰ ἄρα ἀπὸ τῶν ΝΣ, ΣΟ τριπλάσιά <lb n="1799,001,,,,13,17,99"></lb> |
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16803 ἐστι τοῦ ἀπὸ τῆς ΝΟ.</s> <s>ἴση δὲ ἡ μὲν ΝΣ τῇ ΨΩ, ἐπειδήπερ <lb n="1799,001,,,,13,17,100"></lb> |
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16804 καὶ ἡ μὲν ΝΟ τῇ ΟΩ ἐστιν ἴση, ἡ δὲ ΨΟ τῇ ΟΣ.</s> <s>ἀλλὰ <lb n="1799,001,,,,13,17,101"></lb> |
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16805 μὴν καὶ ἡ ΟΣ τῇ ΨΥ, ἐπεὶ καὶ τῇ ΡΟ· τὰ ἄρα ἀπὸ τῶν <lb n="1799,001,,,,13,17,102"></lb> |
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16806 ΩΨ, ΨΥ τριπλάσιά ἐστι τοῦ ἀπὸ τῆς ΝΟ.</s> <s>τοῖς δὲ ἀπὸ <pb></pb> <lb n="1799,001,,,,13,17,103"></lb> |
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16807 τῶν ΩΨ, ΨΥ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΥΩ· τὸ ἄρα ἀπὸ τῆς <lb n="1799,001,,,,13,17,104"></lb> |
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16808 ΥΩ τριπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΝΟ.</s> <s>ἔστι δὲ καὶ ἡ ἐκ <lb n="1799,001,,,,13,17,105"></lb> |
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16809 τοῦ κέντρου τῆς σφαίρας τῆς περιλαμβανούσης τὸν κύβον <lb n="1799,001,,,,13,17,106"></lb> |
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16810 δυνάμει τριπλασίων τῆς ἡμισείας τῆς τοῦ κύβου πλευρᾶς· <lb n="1799,001,,,,13,17,107"></lb> |
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16811 προδέδεικται γὰρ κύβον συστήσασθαι καὶ σφαίρᾳ περι-<lb n="1799,001,,,,13,17,108"></lb> |
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16812 λαβεῖν καὶ δεῖξαι, ὅτι ἡ τῆς σφαίρας διάμετρος δυνάμει <lb n="1799,001,,,,13,17,109"></lb> |
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16813 τριπλασίων ἐστὶ τῆς πλευρᾶς τοῦ κύβου.</s> <s>εἰ δὲ ὅλη τῆς ὅλης, <lb n="1799,001,,,,13,17,110"></lb> |
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16814 καὶ [ἡ] ἡμίσεια τῆς ἡμισείας· καί ἐστιν ἡ ΝΟ ἡμίσεια τῆς <lb n="1799,001,,,,13,17,111"></lb> |
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16815 τοῦ κύβου πλευρᾶς· ἡ ἄρα ΥΩ ἴση ἐστὶ τῇ ἐκ τοῦ κέντρου <lb n="1799,001,,,,13,17,112"></lb> |
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16816 τῆς σφαίρας τῆς περιλαμβανούσης τὸν κύβον.</s> <s>καί ἐστι τὸ <lb n="1799,001,,,,13,17,113"></lb> |
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16817 Ω κέντρον τῆς σφαίρας τῆς περιλαμβανούσης τὸν κύβον· <lb n="1799,001,,,,13,17,114"></lb> |
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16818 τὸ Υ ἄρα σημεῖον πρὸς τῇ ἐπιφανείᾳ ἐστὶ τῆς σφαίρας.</s> <lb n="1799,001,,,,13,17,115"></lb> |
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16819 <s>ὁμοίως δὴ δείξομεν, ὅτι καὶ ἑκάστη τῶν λοιπῶν γωνιῶν <lb n="1799,001,,,,13,17,116"></lb> |
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16820 τοῦ δωδεκαέδρου πρὸς τῇ ἐπιφανείᾳ ἐστὶ τῆς σφαίρας· <lb n="1799,001,,,,13,17,117"></lb> |
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16821 περιείληπται ἄρα τὸ δωδεκάεδρον τῇ δοθείσῃ σφαίρᾳ.</s> <lb n="1799,001,,,,13,17,118"></lb> |
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16822 <s>Λέγω δή, ὅτι ἡ τοῦ δωδεκαέδρου πλευρὰ ἄλογός ἐστιν <lb n="1799,001,,,,13,17,119"></lb> |
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16823 ἡ καλουμένη ἀποτομή.</s> <lb n="1799,001,,,,13,17,120"></lb> |
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16824 <s>Ἐπεὶ γὰρ τῆς ΝΟ ἄκρον καὶ μέσον λόγον τετμημένης <lb n="1799,001,,,,13,17,121"></lb> |
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16825 τὸ μεῖζον τμῆμά ἐστιν ἡ ΡΟ, τῆς δὲ ΟΞ ἄκρον καὶ μέσον <lb n="1799,001,,,,13,17,122"></lb> |
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16826 λόγον τετμημένης τὸ μεῖζον τμῆμά ἐστιν ἡ ΟΣ, ὅλης ἄρα <lb n="1799,001,,,,13,17,123"></lb> |
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16827 τῆς ΝΞ ἄκρον καὶ μέσον λόγον τεμνομένης τὸ μεῖζον <lb n="1799,001,,,,13,17,124"></lb> |
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16828 τμῆμά ἐστιν ἡ Ρ*ς.</s> <s>οἷον ἐπεί ἐστιν ὡς ἡ ΝΟ πρὸς τὴν ΟΡ, <lb n="1799,001,,,,13,17,125"></lb> |
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16829 ἡ ΟΡ πρὸς τὴν ΡΝ, καὶ τὰ διπλάσια· τὰ γὰρ μέρη τοῖς <pb></pb> <lb n="1799,001,,,,13,17,126"></lb> |
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16830 ἰσάκις πολλαπλασίοις τὸν αὐτὸν ἔχει λόγον· ὡς ἄρα ἡ ΝΞ <lb n="1799,001,,,,13,17,127"></lb> |
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16831 πρὸς τὴν ΡΣ, οὕτως ἡ ΡΣ πρὸς συναμφότερον τὴν ΝΡ, <lb n="1799,001,,,,13,17,128"></lb> |
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16832 ΣΞ.</s> <s>μείζων δὲ ἡ ΝΞ τῆς ΡΣ· μείζων ἄρα καὶ ἡ ΡΣ <lb n="1799,001,,,,13,17,129"></lb> |
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16833 συναμφοτέρου τῆς ΝΡ, ΣΞ· ἡ ΝΞ ἄρα ἄκρον καὶ μέσον <lb n="1799,001,,,,13,17,130"></lb> |
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16834 λόγον τέτμηται, καὶ τὸ μεῖζον αὐτῆς τμῆμά ἐστιν ἡ ΡΣ.</s> <lb n="1799,001,,,,13,17,131"></lb> |
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16835 <s>ἴση δὲ ἡ ΡΣ τῇ ΥΦ· τῆς ἄρα ΝΞ ἄκρον καὶ μέσον λόγον <lb n="1799,001,,,,13,17,132"></lb> |
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16836 τεμνομένης τὸ μεῖζον τμῆμά ἐστιν ἡ ΥΦ.</s> <s>καὶ ἐπεὶ ῥητή <lb n="1799,001,,,,13,17,133"></lb> |
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16837 ἐστιν ἡ τῆς σφαίρας διάμετρος καί ἐστι δυνάμει τριπλα-<lb n="1799,001,,,,13,17,134"></lb> |
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16838 σίων τῆς τοῦ κύβου πλευρᾶς, ῥητὴ ἄρα ἐστὶν ἡ ΝΞ πλευρὰ <lb n="1799,001,,,,13,17,135"></lb> |
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16839 οὖσα τοῦ κύβου.</s> <s>ἐὰν δὲ ῥητὴ γραμμὴ ἄκρον καὶ μέσον λό-<lb n="1799,001,,,,13,17,136"></lb> |
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16840 γον τμηθῇ, ἑκάτερον τῶν τμημάτων ἄλογός ἐστιν ἀποτομή.</s> <lb n="1799,001,,,,13,17,137"></lb> |
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16841 <s>Ἡ ΥΦ ἄρα πλευρὰ οὖσα τοῦ δωδεκαέδρου ἄλογός ἐστιν <lb n="1799,001,,,,13,17,138"></lb> |
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16842 ἀποτομή.</s> <lb n="1799,001,,,,13,17,139"></lb> |
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16843 <s>Πόρισμα <lb n="1799,001,,,,13,17,140"></lb> |
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16844 Ἐκ δὴ τούτου φανερόν, ὅτι τῆς τοῦ κύβου πλευρᾶς ἄκρον <lb n="1799,001,,,,13,17,141"></lb> |
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16845 καὶ μέσον λόγον τεμνομένης τὸ μεῖζον τμῆμά ἐστιν ἡ τοῦ <lb n="1799,001,,,,13,17,142"></lb> |
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16846 δωδεκαέδρου πλευρά.</s> <s>ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,17,143"></lb> |
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16847 <s>Τὰς πλευρὰς τῶν πέντε σχημάτων ἐκθέσθαι καὶ συγκρῖ-<lb n="1799,001,,,,13,18,1"></lb> |
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16848 ναι πρὸς ἀλλήλας.</s> <lb n="1799,001,,,,13,18,2"></lb> |
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16849 <s>Ἐκκείσθω ἡ τῆς δοθείσης σφαίρας διάμετρος ἡ ΑΒ, <lb n="1799,001,,,,13,18,3"></lb> |
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16850 καὶ τετμήσθω κατὰ τὸ Γ ὥστε ἴσην εἶναι τὴν ΑΓ τῇ ΓΒ, <lb n="1799,001,,,,13,18,4"></lb> |
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16851 κατὰ δὲ τὸ Δ ὥστε διπλασίονα εἶναι τὴν ΑΔ τῆς ΔΒ, καὶ <lb n="1799,001,,,,13,18,5"></lb> |
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16852 γεγράφθω ἐπὶ τῆς ΑΒ ἡμικύκλιον τὸ ΑΕΒ, καὶ ἀπὸ τῶν <pb></pb> <lb n="1799,001,,,,13,18,6"></lb> |
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16853 Γ, Δ τῇ ΑΒ πρὸς ὀρθὰς <lb n="1799,001,,,,13,18,7"></lb> |
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16854 ἤχθωσαν αἱ ΓΕ, ΔΖ, καὶ <lb n="1799,001,,,,13,18,8"></lb> |
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16855 ἐπεζεύχθωσαν αἱ ΑΖ, ΖΒ, <lb n="1799,001,,,,13,18,9"></lb> |
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16856 ΕΒ.</s> <s>καὶ ἐπεὶ διπλῆ ἐστιν ἡ <lb n="1799,001,,,,13,18,10"></lb> |
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16857 ΑΔ τῆς ΔΒ, τριπλῆ ἄρα <lb n="1799,001,,,,13,18,11"></lb> |
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16858 ἐστὶν ἡ ΑΒ τῆς ΒΔ.</s> <s>ἀνα-<lb n="1799,001,,,,13,18,12"></lb> |
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16859 στρέψαντι ἡμιολία ἄρα ἐστὶν <lb n="1799,001,,,,13,18,13"></lb> |
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16860 ἡ ΒΑ τῆς ΑΔ.</s> <s>ὡς δὲ ἡ ΒΑ <lb n="1799,001,,,,13,18,14"></lb> |
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16861 πρὸς τὴν ΑΔ, οὕτως τὸ ἀπὸ <lb n="1799,001,,,,13,18,15"></lb> |
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16862 τῆς ΒΑ πρὸς τὸ ἀπὸ τῆς ΑΖ· ἰσογώνιον γάρ ἐστι τὸ <lb n="1799,001,,,,13,18,16"></lb> |
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16863 ΑΖΒ τρίγωνον τῷ ΑΖΔ τριγώνῳ· ἡμιόλιον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,13,18,17"></lb> |
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16864 ἀπὸ τῆς ΒΑ τοῦ ἀπὸ τῆς ΑΖ.</s> <s>ἔστι δὲ καὶ ἡ τῆς σφαίρας <lb n="1799,001,,,,13,18,18"></lb> |
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16865 διάμετρος δυνάμει ἡμιολία τῆς πλευρᾶς τῆς πυραμίδος.</s> <lb n="1799,001,,,,13,18,19"></lb> |
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16866 <s>καί ἐστιν ἡ ΑΒ ἡ τῆς σφαίρας διάμετρος· ἡ ΑΖ ἄρα ἴση <lb n="1799,001,,,,13,18,20"></lb> |
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16867 ἐστὶ τῇ πλευρᾷ τῆς πυραμίδος.</s> <lb n="1799,001,,,,13,18,21"></lb> |
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16868 <s>Πάλιν, ἐπεὶ διπλασίων ἐστὶν ἡ ΑΔ τῆς ΔΒ, τριπλῆ <lb n="1799,001,,,,13,18,22"></lb> |
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16869 ἄρα ἐστὶν ἡ ΑΒ τῆς ΒΔ.</s> <s>ὡς δὲ ἡ ΑΒ πρὸς τὴν ΒΔ, οὕτως <lb n="1799,001,,,,13,18,23"></lb> |
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16870 τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΖ· τριπλάσιον ἄρα ἐστὶ <lb n="1799,001,,,,13,18,24"></lb> |
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16871 τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΒΖ.</s> <s>ἔστι δὲ καὶ ἡ τῆς σφαί-<lb n="1799,001,,,,13,18,25"></lb> |
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16872 ρας διάμετρος δυνάμει τριπλασίων τῆς τοῦ κύβου πλευ-<lb n="1799,001,,,,13,18,26"></lb> |
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16873 ρᾶς.</s> <s>καί ἐστιν ἡ ΑΒ ἡ τῆς σφαίρας διάμετρος· ἡ ΒΖ ἄρα <lb n="1799,001,,,,13,18,27"></lb> |
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16874 τοῦ κύβου ἐστὶ πλευρά.</s> <lb n="1799,001,,,,13,18,28"></lb> |
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16875 <s>Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΓ τῇ ΓΒ, διπλῆ ἄρα ἐστὶν ἡ <lb n="1799,001,,,,13,18,29"></lb> |
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16876 ΑΒ τῆς ΒΓ.</s> <s>ὡς δὲ ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ τῆς <lb n="1799,001,,,,13,18,30"></lb> |
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16877 ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΕ· διπλάσιον ἄρα ἐστὶ τὸ ἀπὸ τῆς <lb n="1799,001,,,,13,18,31"></lb> |
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16878 ΑΒ τοῦ ἀπὸ τῆς ΒΕ.</s> <s>ἔστι δὲ καὶ ἡ τῆς σφαίρας διάμετρος <pb></pb> <lb n="1799,001,,,,13,18,32"></lb> |
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16879 δυνάμει διπλασίων τῆς τοῦ ὀκταέδρου πλευρᾶς.</s> <s>καί ἐστιν <lb n="1799,001,,,,13,18,33"></lb> |
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16880 ἡ ΑΒ ἡ τῆς δοθείσης σφαίρας διάμετρος· ἡ ΒΕ ἄρα τοῦ <lb n="1799,001,,,,13,18,34"></lb> |
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16881 ὀκταέδρου ἐστὶ πλευρά.</s> <lb n="1799,001,,,,13,18,35"></lb> |
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16882 <s>Ἤχθω δὴ ἀπὸ τοῦ Α σημείου τῇ ΑΒ εὐθείᾳ πρὸς <lb n="1799,001,,,,13,18,36"></lb> |
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16883 ὀρθὰς ἡ ΑΗ, καὶ κείσθω ἡ ΑΗ ἴση τῇ ΑΒ, καὶ ἐπεζεύχθω <lb n="1799,001,,,,13,18,37"></lb> |
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16884 ἡ ΗΓ, καὶ ἀπὸ τοῦ Θ ἐπὶ τὴν ΑΒ κάθετος ἤχθω ἡ ΘΚ.</s> <lb n="1799,001,,,,13,18,38"></lb> |
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16885 <s>καὶ ἐπεὶ διπλῆ ἐστιν ἡ ΗΑ τῆς ΑΓ· ἴση γὰρ ἡ ΗΑ τῇ <lb n="1799,001,,,,13,18,39"></lb> |
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16886 ΑΒ· ὡς δὲ ἡ ΗΑ πρὸς τὴν ΑΓ, οὕτως ἡ ΘΚ πρὸς τὴν <lb n="1799,001,,,,13,18,40"></lb> |
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16887 ΚΓ, διπλῆ ἄρα καὶ ἡ ΘΚ τῆς ΚΓ.</s> <s>τετραπλάσιον ἄρα ἐστὶ <lb n="1799,001,,,,13,18,41"></lb> |
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16888 τὸ ἀπὸ τῆς ΘΚ τοῦ ἀπὸ τῆς ΚΓ· τὰ ἄρα ἀπὸ τῶν ΘΚ, <lb n="1799,001,,,,13,18,42"></lb> |
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16889 ΚΓ, ὅπερ ἐστὶ τὸ ἀπὸ τῆς ΘΓ, πενταπλάσιόν ἐστι τοῦ ἀπὸ <lb n="1799,001,,,,13,18,43"></lb> |
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16890 τῆς ΚΓ.</s> <s>ἴση δὲ ἡ ΘΓ τῇ ΓΒ· πενταπλάσιον ἄρα ἐστὶ τὸ <lb n="1799,001,,,,13,18,44"></lb> |
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16891 ἀπὸ τῆς ΒΓ τοῦ ἀπὸ τῆς ΓΚ.</s> <s>καὶ ἐπεὶ διπλῆ ἐστιν ἡ ΑΒ <lb n="1799,001,,,,13,18,45"></lb> |
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16892 τῆς ΓΒ, ὧν ἡ ΑΔ τῆς ΔΒ ἐστι διπλῆ, λοιπὴ ἄρα ἡ ΒΔ <lb n="1799,001,,,,13,18,46"></lb> |
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16893 λοιπῆς τῆς ΔΓ ἐστι διπλῆ.</s> <s>τριπλῆ ἄρα ἡ ΒΓ τῆς ΓΔ· <lb n="1799,001,,,,13,18,47"></lb> |
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16894 ἐνναπλάσιον ἄρα τὸ ἀπὸ τῆς ΒΓ τοῦ ἀπὸ τῆς ΓΔ.</s> <s>πεντα-<lb n="1799,001,,,,13,18,48"></lb> |
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16895 πλάσιον δὲ τὸ ἀπὸ τῆς ΒΓ τοῦ ἀπὸ τῆς ΓΚ· μεῖζον ἄρα <lb n="1799,001,,,,13,18,49"></lb> |
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16896 τὸ ἀπὸ τῆς ΓΚ τοῦ ἀπὸ τῆς ΓΔ.</s> <s>μείζων ἄρα ἐστὶν ἡ ΓΚ <lb n="1799,001,,,,13,18,50"></lb> |
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16897 τῆς ΓΔ.</s> <s>κείσθω τῇ ΓΚ ἴση ἡ ΓΛ, καὶ ἀπὸ τοῦ Λ τῇ ΑΒ <lb n="1799,001,,,,13,18,51"></lb> |
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16898 πρὸς ὀρθὰς ἤχθω ἡ ΛΜ, καὶ ἐπεζεύχθω ἡ ΜΒ.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,13,18,52"></lb> |
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16899 πενταπλάσιόν ἐστι τὸ ἀπὸ τῆς ΒΓ τοῦ ἀπὸ τῆς ΓΚ, καί <lb n="1799,001,,,,13,18,53"></lb> |
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16900 ἐστι τῆς μὲν ΒΓ διπλῆ ἡ ΑΒ, τῆς δὲ ΓΚ διπλῆ ἡ ΚΛ, <lb n="1799,001,,,,13,18,54"></lb> |
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16901 πενταπλάσιον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΚΛ.</s> <lb n="1799,001,,,,13,18,55"></lb> |
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16902 <s>ἔστι δὲ καὶ ἡ τῆς σφαίρας διάμετρος δυνάμει πενταπλασίων <lb n="1799,001,,,,13,18,56"></lb> |
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16903 τῆς ἐκ τοῦ κέντρου τοῦ κύκλου, ἀφ' οὗ τὸ εἰκοσάεδρον ἀνα-<lb n="1799,001,,,,13,18,57"></lb> |
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16904 γέγραπται.</s> <s>καί ἐστιν ἡ ΑΒ ἡ τῆς σφαίρας διάμετρος· ἡ ΚΛ <pb></pb> <lb n="1799,001,,,,13,18,58"></lb> |
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16905 ἄρα ἐκ τοῦ κέντρου ἐστὶ τοῦ κύκλου, ἀφ' οὗ τὸ εἰκο-<lb n="1799,001,,,,13,18,59"></lb> |
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16906 σάεδρον ἀναγέγραπται· ἡ ΚΛ ἄρα ἑξαγώνου ἐστὶ πλευρὰ <lb n="1799,001,,,,13,18,60"></lb> |
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16907 τοῦ εἰρημένου κύκλου.</s> <s>καὶ ἐπεὶ ἡ τῆς σφαίρας διάμετρος <lb n="1799,001,,,,13,18,61"></lb> |
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16908 σύγκειται ἔκ τε τῆς τοῦ ἑξαγώνου καὶ δύο τῶν τοῦ δεκα-<lb n="1799,001,,,,13,18,62"></lb> |
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16909 γώνου τῶν εἰς τὸν εἰρημένον κύκλον ἐγγραφομένων, καί <lb n="1799,001,,,,13,18,63"></lb> |
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16910 ἐστιν ἡ μὲν ΑΒ ἡ τῆς σφαίρας διάμετρος, ἡ δὲ ΚΛ ἑξα-<lb n="1799,001,,,,13,18,64"></lb> |
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16911 γώνου πλευρά, καὶ ἴση ἡ ΑΚ τῇ ΛΒ, ἑκατέρα ἄρα τῶν <lb n="1799,001,,,,13,18,65"></lb> |
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16912 ΑΚ, ΛΒ δεκαγώνου ἐστὶ πλευρὰ τοῦ ἐγγραφομένου εἰς τὸν <lb n="1799,001,,,,13,18,66"></lb> |
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16913 κύκλον, ἀφ' οὗ τὸ εἰκοσάεδρον ἀναγέγραπται.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,13,18,67"></lb> |
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16914 δεκαγώνου μὲν ἡ ΛΒ, ἑξαγώνου δὲ ἡ Μ*λ· ἴση γάρ ἐστι <lb n="1799,001,,,,13,18,68"></lb> |
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16915 τῇ ΚΛ, ἐπεὶ καὶ τῇ ΘΚ· ἴσον γὰρ ἀπέχουσιν ἀπὸ τοῦ <lb n="1799,001,,,,13,18,69"></lb> |
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16916 κέντρου· καί ἐστιν ἑκατέρα τῶν ΘΚ, ΚΛ διπλασίων τῆς <lb n="1799,001,,,,13,18,70"></lb> |
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16917 ΚΓ· πενταγώνου ἄρα ἐστὶν ἡ ΜΒ.</s> <s>ἡ δὲ τοῦ πενταγώνου <lb n="1799,001,,,,13,18,71"></lb> |
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16918 ἐστὶν ἡ τοῦ εἰκοσαέδρου· εἰκοσαέδρου ἄρα ἐστὶν ἡ ΜΒ.</s> <lb n="1799,001,,,,13,18,72"></lb> |
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16919 <s>Καὶ ἐπεὶ ἡ ΖΒ κύβου ἐστὶ πλευρά, τετμήσθω ἄκρον καὶ <lb n="1799,001,,,,13,18,73"></lb> |
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16920 μέσον λόγον κατὰ τὸ Ν, καὶ ἔστω μεῖζον τμῆμα τὸ ΝΒ· <lb n="1799,001,,,,13,18,74"></lb> |
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16921 ἡ ΝΒ ἄρα δωδεκαέδρου ἐστὶ πλευρά.</s> <lb n="1799,001,,,,13,18,75"></lb> |
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16922 <s>Καὶ ἐπεὶ ἡ τῆς σφαίρας διάμετρος ἐδείχθη τῆς μὲν <lb n="1799,001,,,,13,18,76"></lb> |
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16923 ΑΖ πλευρᾶς τῆς πυραμίδος δυνάμει ἡμιολία, τῆς δὲ τοῦ <lb n="1799,001,,,,13,18,77"></lb> |
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16924 ὀκταέδρου τῆς ΒΕ δυνάμει διπλασίων, τῆς δὲ τοῦ κύβου <lb n="1799,001,,,,13,18,78"></lb> |
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16925 τῆς ΖΒ δυνάμει τριπλασίων, οἵων ἄρα ἡ τῆς σφαίρας διά-<lb n="1799,001,,,,13,18,79"></lb> |
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16926 μετρος δυνάμει ἕξ, τοιούτων ἡ μὲν τῆς πυραμίδος τεσσά-<lb n="1799,001,,,,13,18,80"></lb> |
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16927 ρων, ἡ δὲ τοῦ ὀκταέδρου τριῶν, ἡ δὲ τοῦ κύβου δύο.</s> <s>ἡ μὲν <lb n="1799,001,,,,13,18,81"></lb> |
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16928 ἄρα τῆς πυραμίδος πλευρὰ τῆς μὲν τοῦ ὀκταέδρου πλευ-<lb n="1799,001,,,,13,18,82"></lb> |
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16929 ρᾶς δυνάμει ἐστὶν ἐπίτριτος, τῆς δὲ τοῦ κύβου δυνάμει <lb n="1799,001,,,,13,18,83"></lb> |
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16930 διπλῆ, ἡ δὲ τοῦ ὀκταέδρου τῆς τοῦ κύβου δυνάμει ἡμιολία.</s> <lb n="1799,001,,,,13,18,84"></lb> |
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16931 <s>αἱ μὲν οὖν εἰρημέναι τῶν τριῶν σχημάτων πλευραί, λέγω <pb></pb> <lb n="1799,001,,,,13,18,85"></lb> |
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16932 δὴ πυραμίδος καὶ ὀκταέδρου καὶ κύβου, πρὸς ἀλλήλας <lb n="1799,001,,,,13,18,86"></lb> |
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16933 εἰσὶν ἐν λόγοις ῥητοῖς.</s> <s>αἱ δὲ λοιπαὶ δύο, λέγω δὴ ἥ τε τοῦ <lb n="1799,001,,,,13,18,87"></lb> |
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16934 εἰκοσαέδρου καὶ ἡ τοῦ δωδεκαέδρου, οὔτε πρὸς ἀλλήλας <lb n="1799,001,,,,13,18,88"></lb> |
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16935 οὔτε πρὸς τὰς προειρημένας εἰσὶν ἐν λόγοις ῥητοῖς· ἄλο-<lb n="1799,001,,,,13,18,89"></lb> |
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16936 γοι γάρ εἰσιν, ἡ μὲν ἐλάττων, ἡ δὲ ἀποτομή.</s> <lb n="1799,001,,,,13,18,90"></lb> |
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16937 <s>Ὅτι μείζων ἐστὶν ἡ τοῦ εἰκοσαέδρου πλευρὰ ἡ ΜΒ τῆς <lb n="1799,001,,,,13,18,91"></lb> |
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16938 τοῦ δωδεκαέδρου τῆς ΝΒ, δείξομεν οὕτως.</s> <lb n="1799,001,,,,13,18,92"></lb> |
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16939 <s>Ἐπεὶ γὰρ ἰσογώνιόν ἐστι τὸ ΖΔΒ τρίγωνον τῷ ΖΑΒ <lb n="1799,001,,,,13,18,93"></lb> |
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16940 τριγώνῳ, ἀνάλογόν ἐστιν ὡς ἡ ΔΒ πρὸς τὴν ΒΖ, οὕτως <lb n="1799,001,,,,13,18,94"></lb> |
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16941 ἡ ΒΖ πρὸς τὴν ΒΑ.</s> <s>καὶ ἐπεὶ τρεῖς εὐθεῖαι ἀνάλογόν εἰσιν, <lb n="1799,001,,,,13,18,95"></lb> |
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16942 ἔστιν ὡς ἡ πρώτη πρὸς τὴν τρίτην, οὕτως τὸ ἀπὸ τῆς πρώ-<lb n="1799,001,,,,13,18,96"></lb> |
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16943 της πρὸς τὸ ἀπὸ τῆς δευτέρας· ἔστιν ἄρα ὡς ἡ ΔΒ πρὸς <lb n="1799,001,,,,13,18,97"></lb> |
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16944 τὴν ΒΑ, οὕτως τὸ ἀπὸ τῆς ΔΒ πρὸς τὸ ἀπὸ τῆς ΒΖ· ἀνά-<lb n="1799,001,,,,13,18,98"></lb> |
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16945 παλιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΒΔ, οὕτως τὸ ἀπὸ τῆς ΖΒ <lb n="1799,001,,,,13,18,99"></lb> |
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16946 πρὸς τὸ ἀπὸ τῆς ΒΔ.</s> <s>τριπλῆ δὲ ἡ ΑΒ τῆς ΒΔ· τριπλά-<lb n="1799,001,,,,13,18,100"></lb> |
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16947 σιον ἄρα τὸ ἀπὸ τῆς ΖΒ τοῦ ἀπὸ τῆς ΒΔ.</s> <s>ἔστι δὲ καὶ τὸ <lb n="1799,001,,,,13,18,101"></lb> |
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16948 ἀπὸ τῆς ΑΔ τοῦ ἀπὸ τῆς ΔΒ τετραπλάσιον· διπλῆ γὰρ <lb n="1799,001,,,,13,18,102"></lb> |
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16949 ἡ ΑΔ τῆς ΔΒ· μεῖζον ἄρα τὸ ἀπὸ τῆς ΑΔ τοῦ ἀπὸ τῆς <lb n="1799,001,,,,13,18,103"></lb> |
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16950 ΖΒ· μείζων ἄρα ἡ ΑΔ τῆς ΖΒ· πολλῷ ἄρα ἡ ΑΛ τῆς ΖΒ <lb n="1799,001,,,,13,18,104"></lb> |
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16951 μείζων ἐστίν.</s> <s>καὶ τῆς μὲν ΑΛ ἄκρον καὶ μέσον λόγον <lb n="1799,001,,,,13,18,105"></lb> |
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16952 τεμνομένης τὸ μεῖζον τμῆμά ἐστιν ἡ ΚΛ, ἐπειδήπερ ἡ μὲν <lb n="1799,001,,,,13,18,106"></lb> |
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16953 ΛΚ ἑξαγώνου ἐστίν, ἡ δὲ ΚΑ δεκαγώνου· τῆς δὲ ΖΒ <lb n="1799,001,,,,13,18,107"></lb> |
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16954 ἄκρον καὶ μέσον λόγον τεμνομένης τὸ μεῖζον τμῆμά ἐστιν ἡ <lb n="1799,001,,,,13,18,108"></lb> |
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16955 ΝΒ· μείζων ἄρα ἡ ΚΛ τῆς ΝΒ.</s> <s>ἴση δὲ ἡ ΚΛ τῇ ΛΜ· <pb></pb> <lb n="1799,001,,,,13,18,109"></lb> |
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16956 μείζων ἄρα ἡ ΛΜ τῆς ΝΒ [τῆς δὲ ΛΜ μείζων ἐστὶν ἡ <lb n="1799,001,,,,13,18,110"></lb> |
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16957 ΜΒ].</s> <s>πολλῷ ἄρα ἡ ΜΒ πλευρὰ οὖσα τοῦ εἰκοσαέδρου <lb n="1799,001,,,,13,18,111"></lb> |
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16958 μείζων ἐστὶ τῆς ΝΒ πλευρᾶς οὔσης τοῦ δωδεκαέδρου· <lb n="1799,001,,,,13,18,112"></lb> |
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16959 ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,18,113"></lb> |
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16960 <s>Λέγω δή, ὅτι παρὰ τὰ εἰρημένα πέντε σχήματα οὐ <lb n="1799,001,,,,13,18,114"></lb> |
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16961 συσταθήσεται ἕτερον σχῆμα περιεχόμενον ὑπὸ ἰσοπλεύ-<lb n="1799,001,,,,13,18,115"></lb> |
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16962 ρων τε καὶ ἰσογωνίων ἴσων ἀλλήλοις.</s> <lb n="1799,001,,,,13,18,116"></lb> |
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16963 <s>Ὑπὸ μὲν γὰρ δύο τριγώνων ἢ ὅλως ἐπιπέδων στερεὰ <lb n="1799,001,,,,13,18,117"></lb> |
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16964 γωνία οὐ συνίσταται.</s> <s>ὑπὸ δὲ τριῶν τριγώνων ἡ τῆς πυρα-<lb n="1799,001,,,,13,18,118"></lb> |
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16965 μίδος, ὑπὸ δὲ τεσσάρων ἡ τοῦ ὀκταέδρου, ὑπὸ δὲ πέντε ἡ <lb n="1799,001,,,,13,18,119"></lb> |
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16966 τοῦ εἰκοσαέδρου· ὑπὸ δὲ ἓξ τριγώνων ἰσοπλεύρων τε καὶ <lb n="1799,001,,,,13,18,120"></lb> |
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16967 ἰσογωνίων πρὸς ἑνὶ σημείῳ συνισταμένων οὐκ ἔσται στε-<lb n="1799,001,,,,13,18,121"></lb> |
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16968 ρεὰ γωνία· οὔσης γὰρ τῆς τοῦ ἰσοπλεύρου τριγώνου γωνίας <lb n="1799,001,,,,13,18,122"></lb> |
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16969 διμοίρου ὀρθῆς ἔσονται αἱ ἓξ τέσσαρσιν ὀρθαῖς ἴσαι· ὅπερ <lb n="1799,001,,,,13,18,123"></lb> |
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16970 ἀδύνατον· ἅπασα γὰρ στερεὰ γωνία ὑπὸ ἐλασσόνων ἢ τεσ-<lb n="1799,001,,,,13,18,124"></lb> |
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16971 σάρων ὀρθῶν περιέχεται.</s> <s>διὰ τὰ αὐτὰ δὴ οὐδὲ ὑπὸ πλει-<lb n="1799,001,,,,13,18,125"></lb> |
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16972 όνων ἢ ἓξ γωνιῶν ἐπιπέδων στερεὰ γωνία συνίσταται.</s> <s>ὑπὸ <lb n="1799,001,,,,13,18,126"></lb> |
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16973 δὲ τετραγώνων τριῶν ἡ τοῦ κύβου γωνία περιέχεται· ὑπὸ <lb n="1799,001,,,,13,18,127"></lb> |
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16974 δὲ τεσσάρων ἀδύνατον· ἔσονται γὰρ πάλιν τέσσαρες ὀρθαί.</s> <lb n="1799,001,,,,13,18,128"></lb> |
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16975 <s>ὑπὸ δὲ πενταγώνων ἰσοπλεύρων καὶ ἰσογωνίων, ὑπὸ μὲν <lb n="1799,001,,,,13,18,129"></lb> |
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16976 τριῶν ἡ τοῦ δωδεκαέδρου· ὑπὸ δὲ τεσσάρων ἀδύνατον· <lb n="1799,001,,,,13,18,130"></lb> |
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16977 οὔσησ γὰρ τῆς τοῦ πενταγώνου ἰσοπλεύρου γωνίας ὀρθῆς <lb n="1799,001,,,,13,18,131"></lb> |
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16978 καὶ πέμπτου, ἔσονται αἱ τέσσαρες γωνίαι τεσσάρων ὀρθῶν <pb></pb> <lb n="1799,001,,,,13,18,132"></lb> |
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16979 μείζους· ὅπερ ἀδύνατον.</s> <s>οὐδὲ μὴν ὑπὸ πολυγώνων ἑτέρων <lb n="1799,001,,,,13,18,133"></lb> |
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16980 σχημάτων περισχεθήσεται στερεὰ γωνία διὰ τὸ αὐτὸ ἄτοπον.</s> <lb n="1799,001,,,,13,18,134"></lb> |
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16981 <s>Οὐκ ἄρα παρὰ τὰ εἰρημένα πέντε σχήματα ἕτερον σχῆμα <lb n="1799,001,,,,13,18,135"></lb> |
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16982 στερεὸν συσταθήσεται ὑπὸ ἰσοπλεύρων τε καὶ ἰσογωνίων <lb n="1799,001,,,,13,18,136"></lb> |
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16983 περιεχόμενον· ὅπερ ἔδει δεῖξαι.</s> <lb n="1799,001,,,,13,18,137"></lb> |
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16984 <s>Λῆμμα <lb n="1799,001,,,,13,18,138"></lb> |
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16985 Ὅτι δὲ ἡ τοῦ ἰσοπλεύρου καὶ ἰσογωνίου πενταγώνου <lb n="1799,001,,,,13,18,139"></lb> |
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16986 γωνία ὀρθή ἐστι καὶ πέμπτου, οὕτω δεικτέον.</s> <lb n="1799,001,,,,13,18,140"></lb> |
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16987 <s>Ἔστω γὰρ πεντάγωνον ἰσόπλευρον καὶ ἰσογώνιον τὸ <lb n="1799,001,,,,13,18,141"></lb> |
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16988 ΑΒΓΔΕ, καὶ περιγεγράφθω περὶ αὐτὸ κύκλος ὁ ΑΒΓ <lb n="1799,001,,,,13,18,142"></lb> |
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16989 ΔΕ, καὶ εἰλήφθω αὐτοῦ τὸ κέντρον τὸ Ζ, καὶ ἐπεζεύχθω-<lb n="1799,001,,,,13,18,143"></lb> |
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16990 σαν αἱ ΖΑ, ΖΒ, ΖΓ, ΖΔ, ΖΕ.</s> <s>δίχα ἄρα τέμνουσι τὰς <lb n="1799,001,,,,13,18,144"></lb> |
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16991 πρὸς τοῖς Α, Β, Γ, Δ, Ε τοῦ πενταγώνου γωνίας.</s> <s>καὶ ἐπεὶ <lb n="1799,001,,,,13,18,145"></lb> |
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16992 αἱ πρὸς τῷ Ζ πέντε γωνίαι τέσσαρσιν ὀρθαῖς ἴσαι εἰσὶ καί <lb n="1799,001,,,,13,18,146"></lb> |
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16993 εἰσιν ἴσαι, μία ἄρα αὐ-<lb n="1799,001,,,,13,18,147"></lb> |
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16994 τῶν, ὡς ἡ ὑπὸ ΑΖΒ, <lb n="1799,001,,,,13,18,148"></lb> |
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16995 μιᾶς ὀρθῆς ἐστι παρὰ <lb n="1799,001,,,,13,18,149"></lb> |
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16996 πέμπτον· λοιπαὶ ἄρα αἱ <lb n="1799,001,,,,13,18,150"></lb> |
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16997 ὑπὸ ΖΑΒ, ΑΒΖ μιᾶς <lb n="1799,001,,,,13,18,151"></lb> |
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16998 εἰσιν ὀρθῆς καὶ πέμπτου.</s> <lb n="1799,001,,,,13,18,152"></lb> |
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16999 <s>ἴση δὲ ἡ ὑπὸ ΖΑΒ τῇ <lb n="1799,001,,,,13,18,153"></lb> |
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17000 ὑπὸ ΖΒΓ· καὶ ὅλη ἄρα <lb n="1799,001,,,,13,18,154"></lb> |
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17001 ἡ ὑπὸ ΑΒΓ τοῦ πεντα-<lb n="1799,001,,,,13,18,155"></lb> |
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17002 γώνου γωνία μιᾶς ἐστιν <lb n="1799,001,,,,13,18,156"></lb> |
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17003 ὀρθῆς καὶ πέμπτου· ὅπερ <lb n="1799,001,,,,13,18,157"></lb> |
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17004 ἔδει δεῖξαι.</s> <s><pb></pb> <lb n="1799,001,,,,13,18,158"></lb> |
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17005 </s></p></chap></body><back></back></text></archimedes> |