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a very large, delightful bath for the city; and finding the adjacent place dedicated to the Dioscuri, he made this a hippodrome, decorating it with platforms and porticoes, and cutting down a grove subject to the ownership of two brothers and transforming this to its present beauty. That the Romans called two-yoked chariots *bigas*, from which also came *bigarii*. 1.13 Aeneas, having sailed from there after very many wanderings, landed in a city of Italy called Laurentia, which they say was once named Oppica, from which the Italians say *oppicizare*, and as the multitude says, *officizare*, meaning to speak barbarously. Then, having become the son-in-law of Latinus, who was ruling the country, and having ruled himself for three years, he departed. Therefore, since so many foreigners settled in Italy, as has been shown, they called the natives Latins and those who spoke Greek Graeci, from Latinus, whom we just mentioned, and Graecus, the brothers, as Hesiod says in the *Catalogues*: Agrius and Latinus. And the daughter in the halls of noble Deucalion, Pandora, mixed in love with father Zeus, leader of all the gods, bore Graecus, staunch in battle. 1.14 Romulus founded Rome on the eleventh day before the Kalends of May, in the third year of the sixth Olympiad, or as others say, in the second year of the eighth. The hour of the city was the second hour before the third, as Tarrutius the mathematician established, with Jupiter in Pisces, Saturn and Venus and Mars and Mercury in Scorpio, the Sun in Taurus and the Moon in Libra; or as others say, the Sun in Taurus, the Moon in Virgo, Saturn in Libra, Jupiter in Leo, Mars in Libra, Venus in Taurus, Mercury in Aries. Romulus makes March the beginning of the year in honor of Mars; for the story is that he was born of him .... with only ten months. 1.15 The decad is a complete number, for which reason it is also called *panteleia* (all-perfect), containing all the forms of the other numbers and ratios and proportions and harmonies. For the decad is the rule among existing things, characterizing everything, and in particular defining the infinite in each thing, and being the collector and assembler and finisher of all things that both the intelligible nature and the sublunary nature contain. For thus Parmenides hands down to us: first, the intelligible; second, the things in numbers; third, the cohesive; fourth, the perfecting; fifth, the divisive; sixth, the life-giving; seventh, the demiurgic; eighth, the assimilative; ninth, the absolute; tenth, the worldly. Rightly, therefore, did Philolaus call it a decad, as receptive of the infinite, and Orpheus called it the branch-bearer, from which, as if they were branches, all numbers grow. 1.16 The year remained, observed according to the course of the moon, from the time of Saturn until the founding of Rome, but in the time of Romulus it was defined, as we said before, as consisting of ten months, with some months allotted more than thirty days, and others fewer; for the interval of time was not yet reckoned by them from the motion of the sun. 1.17 Pompilius Numa, in whose time Pythagoras lived, defined the year to be numbered by twelve months, according to Socrates in the *Phaedrus*, who says that the orders of all things are comprehended by the dodecad; for god used this number for the universe, designing it, as Plato says. For this form is proper to the idea of the universe; for it is also cyclical, since the nature of the dodecad is also wonderful, both for other reasons and because it is composed of the most elementary and eldest of the forms taken among essences, as those from the mathematical sciences say, the right-angled triangle; for its sides, being of three and four and five, complete the number of the twelve, the paradigm of the zodiacal circle; and by the doubling of the hexad, the most fertile, which is the beginning of perfection, being completed from its own parts .... But Numa arranges the year of twelve months for the completion of the

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πόλεως ἐπιτερπὲς λουτρὸν παμμέγεθες· εὑρὼν δὲ καὶ τὸν παρακείμενον τόπον τοῖς ∆ιοσκόροις ἀνακείμενον ἐποίησε τοῦτον ἱπποδρόμιον, ἰκρίοις καὶ στοαῖς διακοσμήσας αὐτόν, καὶ ἄλσος ἐκκόψας δύο τινῶν ἀδελφῶν ὑποκείμενον δεσποτείᾳ καὶ εἰς τὸ νῦν ὁρώμενον κάλλος τοῦτο μεταγαγών. ὅτι Ῥωμαῖοι τὰ δίζυγα ἅρματα βίγας ἐκάλουν, ἐξ ὧν καὶ βιγάριοι. 1.13 Πλεύσας ἐκεῖθεν Αἰνείας μετὰ πολλὰς ὅσας πλάνας κατάγεται ἐν πόλει τῆς Ἰταλίας λεγομένῃ Λαυρεντίᾳ, ἣν καὶ Ὀππικήν φασιν ὀνομασθῆναί ποτε, ἐξ ἧς καὶ ὀππικίζειν, καὶ ὡς τὸ πλῆθος, ὀφφικίζειν τὸ βαρβαρίζειν Ἰταλοὶ λέγουσιν. εἶτα ἐπιγαμβρεύσας Λατίνῳ βασιλεύοντι τῆς χώρας αὐτός τε βασιλεύσας τρισὶν ἐνιαυτοῖς οἴχεται. τοσούτων οὖν ἐπιξενωθέντων τῆς Ἰταλίας, ὥσπερ ἐδείχθη, Λατίνους μὲν τοὺς ἐπιχωριάζοντας, Γραικοὺς δὲ τοὺς ἑλληνίζοντας ἐκάλουν, ἀπὸ Λατίνου τοῦ ἄρτι ἡμῖν ῥηθέντος καὶ Γραικοῦ, τῶν ἀδελφῶν, ὥς φησιν Ἡσίοδος ἐν Καταλόγοις· Ἄγριον ἠδὲ Λατῖνον. Κούρη δ' ἐν μεγάροισιν ἀγαυοῦ ∆ευκαλίωνος Πανδώρη ∆ιὶ πατρί, θεῶν σημάντορι πάντων μιχθεῖσ' ἐν φιλότητι τέκε Γραικὸν μενεχάρμην. 1.14 Ῥωμύλος δὲ κτίζει τὴν Ῥώμην τῇ πρὸ δεκαμιᾶς Καλενδῶν Μαΐου, κατὰ τὸ τρίτον τῆς ἕκτης, ἢ ὡς ἕτεροι κατὰ τὸ δεύτερον τῆς ὀγδόης· ἡ δὲ ὥρα τῆς πόλεως ὥρα δευτέρα πρὸ τρίτης, ὡς Ταρρούτιος ὁ μαθηματικὸς κατεστήριξεν, ∆ιὸς μὲν ἰχθῦσι, Κρόνου δὲ καὶ Ἀφροδίτης καὶ Ἄρεος καὶ Ἑρμοῦ Σκορπίῳ,Ἡλίου δὲ Ταύρῳ καὶ Σελήνης Ζυγῷ· ἢ ὡς ἕτεροι Ἠλίου μὲν Ταύρῳ, Σελήνης δὲ Παρθένῳ, Κρόνου δὲ Ζυγῷ, ∆ιὸς δὲ Λέοντι, Ἄρεος Ζυγῷ, Ἀφροδίτης Ταύρῳ,Ἑρμοῦ Κριῷ. ἀρχὴν δὲ ἐνιαυτοῦ δίδωσι Ῥωμύλος τὸν Μάρτιον πρὸς τιμὴν Ἄρεος· ἐξ αὐτοῦ γὰρ αὐτὸν γενέσθαι λόγος .... δέκα μόνοις μησίν. 1.15 Ἡ δεκὰς πλήρης ἀριθμός ἐστιν, ὅθεν καὶ παντέλεια καλεῖται, πάσας τὰς ἰδέας τῶν ἄλλων ἀριθμῶν καὶ λόγων καὶ ἀναλογιῶν καὶ συμφωνιῶν περιέχουσα· γνώμων γὰρ ἐν τοῖς οὖσίν ἐστιν ἡ δεκὰς πάντα χαρακτηρίζουσα, καὶ ἰδίως τὸ ἐν ἑκάστῳ ἄπειρον ὁρίζουσα, καὶ πάντων οὖσα συναγωγός τε καὶ συνακτικὴ καὶ ἀποτελεστικὴ τῶν ὅσα ἥ τε νοητὴ περιέχει φύσις ἥ τε ὑπὸ Σελήνην. οὕτως γὰρ ἡμῖν παραδίδωσιν ὁ Παρμενίδης· πρώτιστα μὲν τὰ νοητά, δεύτερα τὰ ἐν τοῖς ἀριθμοῖς, τρίτα τὰ συνεκτικά, τέταρτα τὰ τελεσιουργά, πέμπτα τὰ διαιρετικά, ἕκτα τὰ ζωογονικά, ἕβδομα τὰ δημιουργικά, ὄγδοα τὰ ἀφομοιωματικά, ἔννατα τὰ ἀπόλυτα, δέκατα τὰ ἐγκόσμια. ὀρθῶς οὖν αὐτὴν ὁ Φιλόλαος δεκάδα προσηγόρευσεν, ὡς δεκτικὴν τοῦ ἀπείρου, Ὀρφεὺς δὲ κλαδοῦχον, ἐξ ἧς ὡσεὶ κλάδοι τινὲς πάντες οἱ ἀριθμοὶ φύονται. 1.16 ∆ιέμεινεν ἀπὸ Κρόνου ἕως κτίσεως Ῥώμης κατὰ τὸν ἐν Σελήνῃ δρόμον παραφυλαχθεὶς ὁ ἐνιαυτός, ἐπὶ δὲ Ῥωμύλου ὁρίζεται, ὡς ἐλέγομεν ἔμπροσθεν, δεκαμηνιαῖος, τῶν μὲν ὑπὲρ τριακάδος ἡμέρας πλείους, τῶν δὲ ἐλάττονας λαχόντων μηνῶν· οὔπω γὰρ ἀπὸ τῆς Ἡλίου κινήσεως ἐφυσιολογεῖτο αὐτοῖς τὸ τοῦ χρόνου διάστημα. 1.17 Ὁ Πομπίλιος Νουμᾶς, ἐφ' οὗ Πυθαγόρας ἦν, δυοκαίδεκα μησὶ τὸν ἐνιαυτὸν ἀριθμεῖσθαι διώρισε κατὰ τὸν ἐν Φαίδρῳ Σωκράτην, ὅς φησι τὰς τῶν ὅλων τάξεις τῇ δωδεκάδι περιειλῆφθαι· ἐπὶ τὸ πᾶν γὰρ ὁ θεὸς τῷ ἀριθμῷ τούτῳ κατεχρήσατο διαζωγραφῶν αὐτό, ὥς φησιν ὁ Πλάτων. οἰκεῖον γὰρ τὸ σχῆμα τοῦτο τῇ τοῦ παντὸς ἰδέᾳ· καὶ γὰρ κυκλικόν, ἐπεὶ καὶ θαυμαστή ἐστιν ἡ τῆς δωδεκάδος φύσις, διά τε ἄλλα καὶ ἐπειδὴ συνέστηκεν ἐκ τοῦ στοιχειωδεστάτου καὶ πρεσβυτάτου τῶν ἐν οὐσίαις εἰδῶν παραλαμβανομένων, ὥς φασιν οἱ ἀπὸ τῶν μαθημάτων, ὀρθογωνίου τριγώνου· αἱ γὰρ τοῦδε πλευραὶ ἐκ τριῶν οὖσαι καὶ τεττάρων καὶ πέντε συμπληροῦσι τὸν ἀριθμὸν τῶν δώδεκα τοῦ ζωοφόρου κύκλου τὸ παράδειγμα· διπλασιασθείσης δὲ ἑξάδος τῆς γονιμωτάτης, ἥτις ἐστὶν ἀρχὴν τελειότητος ἐκ τῶν ἰδίων συμπληρουμένη μερῶν .... διατάττει δὲ Νουμᾶς δυοκαίδεκα μηνῶν τὸν ἐνιαυτὸν πρὸς συντέλειαν τῶν