§5#1Φέρε γὰρ, ἵνα σαφηνείας ἕνεκεν τῆς ἀρχιτεκτονίας τι λέξω παράδειγμα πόλεως κτιζομένης, προκείσθω τοῖς οἰκήσουσιν αὐτὴν ἐπίστασθαι βούλεσθαι, μὴ στοχαστικῶς, ἀλλ’ ἀκριβῶς ἐφ’ ἑκάστης ἡμέρας, ὁπόσον τε παρελήλυθεν ἤδη τοῦ χρόνου τοῦ κατ’ αὐτὴν, ὁπόσον τε ὑπόλοιπόν ἐστιν ἄχρι δύσεως ἡλίου.
come, in order that I may state for the sake of clarity some example from architecture of a city being founded: let it be proposed for those who will inhabit it that they wish to know, not conjecturally, but accurately on each day, both how much of the time belonging to it has already passed, and how much is remaining until the setting of the sun.
τοῦτο τὸ πρόβλημα κατὰ τὴν ἀναλυτικὴν μέθοδον ἐπὶ τὸ πρῶτον ἀναχθῆναι χρὴ κριτήριον, εἰ μέλλοι τις εὑρήσειν αὐτὸ, καθ’ ὃν ἐμάθομεν τρόπον ἐν τῇ γνωμονικῇ πραγματείᾳ, πάλιν ἐξ ἐκείνων τὴν αὐτὴν ὁδὸν ὑπεναντίως ἰόντων συντεθῆναι, καθότι καὶ τοῦτ’ ἐμάθομεν ἐπὶ τῆς αὐτῆς πραγματείας.
This problem must be reduced to the first criterion according to the analytical method, if one is to find it, in the manner we learned in the treatise on gnomonics, and again from those first principles to be synthesized by going back along the same path in reverse, just as we also learned this in the same treatise.
εὑρεθείσης δ’ οὕτω καθόλου τε καὶ κοινῆς ὁδοῦ, καὶ γνόντων ἡμῶν, ὅτι διὰ γραμμῶν τοιῶνδε τὴν μέτρησιν τῶν μερῶν τοῦ χρόνου ἀναγκαῖόν ἐστι ποιεῖσθαι, μεταβῆναι προσήκει ἐπὶ τὰ μέλλοντα σώματα δέξασθαι εἰς τὴν τῶν γραμμῶν καταγραφὴν καὶ τὸν γνώμονα, καὶ πρῶτον μὲν ζητῆσαι, τίνα σχήματα σωμάτων ἐπιτήδεια πρὸς τὴν εὑρημένην καταγραφήν ἐστιν, εἶτα καθ’ ἕκαστον αὐτῶν ἐξ ἀναλύσεώς τε καὶ συνθέσεως εὑρεῖν, ὁποῖα ἂν αὕτη γίγνοιτο, κᾀπειδὰν ἡ λογικὴ μέθοδος ἐναργῆ τὴν πίστιν ἡμῖν ἐνδείξηται τῆς τῶν προκειμένων εὑρέσεως, ἑξῆς ἐπὶ τὸ πράττειν ἔργῳ τὰ δι’ αὐτῆς εὑρεθέντα παραγενομένους ἐπισκέψασθαι πάλιν, ὅπως ἂν ἐπιφάνειαν ὁμαλὴν ἐργασώμεθα τοῦ μέλλοντος καταγράφεσθαι σώματος.
And when such a universaland common path has been found, and we have known that it is necessary to make the measurement of the parts of time through such lines, it is fitting to proceed to the bodies that are to receive the drawing of the lines and the gnomon, and first to seek what shapes of bodies are suitable for the found drawing, and then for each of them, from analysis and synthesis, to find what sort ofshape this would become, and when the logical method shows us clear proof of the discovery of the proposed things, next, having come to the actual execution of the things discovered through it, to consider again how we might work a flat surface on the body that is to be drawn upon.
εὑρόντας δὲ τοῦτο ἐξ ἀναλύσεώς τε καὶ συνθέσεως, εἶτα κατασκευάσαντάς τι τοιοῦτον σῶμα, διὰ τῶν ὀργάνων αὐτὸ δεῖ καταγράψαι ζητῆσαι, καὶ πάλιν ἐξ ἀναλύσεώς τε καὶ συνθέσεως εὑρεθέντος αὐτοῦ πειρᾶσαι κατασκευάσαι τοιαῦτα τὸ εἶδος, οἷά περ ἡ μέθοδος ἐδίδαξεν, εἶτα ποιήσαντας ἐφεξῆς καταγραφὴν ἐπὶ πολλῶν σχημάτων, καὶ δοῦναι τοῖς ἀνθρώποις ἔργῳ πειραθῆναι γεγονὸς ἤδη τὸ προβληθέν.
And having found this from analysis and synthesis, then having constructed some such body, one must seek to draw upon it by means of instruments, and again when this has been found from analysis and synthesis, to try to construct such things in form as the method taught, then subsequently making the drawing upon many shapes, and to give tomen to experience in deed what had already been proposed.
ὅταν γὰρ ἥ τε πρώτη γραμμὴ τὴν ἀκτῖνα δέχηται τοῦ ἡλίου, κατὰ ταῦτα δὲ καὶ τὴν τελευταίαν ἡ ὑστάτη, καὶ τοῦτ’ ἐπὶ πάντων φαίνηται τῶν καταγεγραμμένων ὡρολογίων τῶν ἡλιακῶν, ἤδη μέν που ἕν τι γνώρισμα ἕξομεν ἐναργὲς τοῦτο προβληθέν.
For when the first line receives the ray of the sun, and according to this also the last line receives the final ray, and this is seen in all the constructed sundials, we shall already have this as one clear sign of the proposed task.
εὑρῆσαι δὲ πάλιν δεύτερον ἄλλο, ὅταν ἀλλήλοις καταγραφέντα πάντα συμφωνῇ, καὶ τρίτον, ὅταν ὕδατος ὁμαλοῦ ῥύσις αὐτοῖς μαρτυρῇ.
And again to find a second, another sign, when all the drawings agree with one another, and a third, when the flow of uniform water bears witness to them.
καὶ γὰρ τοῦθ’ ὁ λόγος εὑρίσκει κριτήριον ἐσόμενον τῆς τῶν καταγεγραμμένων ὡρολογίων ἀληθείας. ὃ δὲ λέγω, τοιοῦτόν ἐστιν.
For reason finds that this too will be a criterion of the truth of the constructed sundials.
ἀγγεῖον τρήσας ἐξ ἧς ἂν ὕλης ἐθέλῃς γεγονὸς, ἐπίθες ὕδατι καθαρῷ τὴν πρώτην ἀκτῖνα τοῦ ἡλίου θεασάμενος· εἷθ’ ὅταν σοι τὸ καταγεγραμμένον ὡρολόγιον ἀγγείλῃ τὴν πρώτην ὥραν ἠνύσθαι, σημηνάμενος ὅσον ἐπληρώθη τοῦ ἀγγείου μέρος ὑπὸ τοῦ ὕδατος, ἔτι ἐκκενώσας εὐθέως αὐτὸ πάλιν, ἐπίθες ἐπὶ ταὐτὸν ὕδωρ· ὅταν δὲ τὴν δευτέραν ὥραν ἀγγείλῃ τὸ ἡλιακὸν ὡρολόγιον, ἐπίσκεψαι τὸ ἀγγεῖον· εἶθ’ ὅταν εὕρῃς ἐν αὐτῷ τὸ ὕδωρ ἐπὶ τὸ αὐτὸ μέρος ἀφιγμένον κατὰ τὴν πρώτην ὥραν, ἐν σήμανον ταχέως, καὶ ἐκχέας ἔνθες κατ’ αὐτοῦ τοῦ ὕδατος, καὶ ἐπισκέπτου πάλιν, εἰ μέχρι τῆς τρίτης ὥρας ἐνδείκνυσι τὸ ὡρολόγιον, ἐπὶ τὴν αὐτὴν ὥραν ἀφικνεῖσθαι τοῦ ἀγγείου τὸ ὕδωρ, ἐφ’ ἣν κατὰ τὴν πρώτην, ἢ κατὰ τὴν δευτέραν· ὅταν οὖν εὕρῃς οὕτω γεγονὸς, ἐκχέας αὖθις ἐπίθες μέχρι τῆς τετάρτης ὥρας, ἰδών τε πάλιν ἐπὶ τὸ αὐτὸ μέρος τοῦ ἀγγείου τὸ ὕδωρ ἀναβεβηκὸς, ἐκχέας αὐτὸ παραχρῆμα, πάλιν ἐπιθεὶς ὁμοίως ἐξέταζε κατὰ τὴν πέμπτην ὥραν.
And what I mean is as follows: having pierced a vessel made of whatever material you wish, place it on clean water when you have observed the first ray of the sun; then, when the constructed sundial announces to you that the first hour has been completed, having marked how large a part of the vessel was filled by the water, and having immediately emptied it again, place it on the same water; and when the sundial announces the second hour, inspect the vessel; then, when you find in itthe water having reached the same point as during the first hour, mark it quickly, and having poured it out, place it into the same water, and inspect again if, until the third hour the sundial indicates, the water of the vessel reaches the same mark as it did during the first or the second hour; so when you find it having happened thus, having poured it out, place it again until the fourth hour, and having seen again the water having risen to the same part of the vessel, having poured it out immediately, having placed it again similarly examine it at the fifth hour.
ὅταν δὲ καὶ ταύτης εὕρῃς ἐπὶ τὴν αὐτὴν χώραν ἀφιγμένον τὸ ὕδωρ, εἶτα κατὰ τὴν ἕκτην τε καὶ κατὰ τὰς ἐφεξῆς μέχρι τῆς δωδεκάτης, εἰ παντελῶς γνώσῃ, πεισθήσῃ καλῶς καταγεγράφθαι τὸ ὡρολόγιον, εἴ γε τὸ προκείμενον ἐπεδείξατο· προκείμενον δ’ ἦν εἰς δώδεκα μοίρας ἴσας νεμηθῆναι τὸν χρόνον τῆς ὅλης ἡμέρας, ὅτι, ἀλλὰ τοῦτον μὲν τὸν ἀριθμὸν ὡς χρησιμώτατον προείλοντο, καὶ γὰρ ἥμισυ ἔχει, καὶ δίμοιρον, καὶ τέταρτον, ἕκτον τε καὶ δωδέκατον, ἅπερ οὐδεὶς ἄλλος ἔχει μετ’ αὐτὸν ἄχρι τοῦ εἰκοστοῦ τετάρτου.
And when you find the water having reached the same point for this too, and then during the sixth and the subsequent ones up to the twelfth, if you know this completely, you will be persuaded that the sundial has been constructed well, if indeed it showed what was proposed; and what was proposed was for the time of the whole day to be divided into twelve equal parts, because, indeed, they chose this number as the most useful, for it has a half, two-thirds, a fourth, a sixth, and a twelfth,which no other number has after it until twenty-four.
τοῦτον μὲν ὡς μακρὸν παρῃτήσαντο, σύμμετρον δ’ εἶναι κρίναντες τὸν δωδέκατον, εἰς τοσαῦτα μέρη τὸν χρόνον τῆς ὅλης ἡμέρας διέτεμον.
They declined the latter as too large, but judging twelve to be commensurate, they cut the time of the whole day into so many parts.
ὅτι δὲ χρήσιμός ἐστιν ἡ τοιαύτη τομὴ, τῇ πείρᾳ βασανίσαντες αὐτὴν ἄλλοι τε πολλοὶ καὶ Ῥωμαῖοι χρῶνται τὴν οὐσίαν ἅπασαν, ὅταν διατίθωνται, διαιροῦντες εἰς δώδεκα μέρη, καὶ τῶν ἐν τῷ βίῳ σταθμῶν τε καὶ μέτρων εἰς δώδεκα μοίρας τέμνονται τὰ πλεῖστα.
And that such a division is useful, having tested it by experience, many others and also the Romans use it, dividing their whole property when they make their wills into twelve parts, and most of the weights and measures in life are cut into twelve parts.
σὺ δ’ εἰ κελεύεις, καταγράψομαί σοι κατὰ τὴν μέθοδον ὡρολόγιον, εἴτ’ εἰς δώδεκα τὴν ὅλην ἡμέραν διαιρεῖν ἐθελήσεις, εἴτ’ εἰς ἄλλον τινὰ τὸν ἐφεξῆς ἀριθμὸν, εὑρήσεις γὰρ ἐπ’ ἐκείνου πάλιν ὃ προὔβαλες γεγονὸς ἔκ τε τοῦ συμφωνεῖν τῇ διὰ τῶν τετρημένων ἀγγείων ἐκμετρήσει κᾀκ τοῦ πάντοτε ἀλλήλοις ὁμολογεῖν καταγεγραμμένα, πρός τε τοῦ τὰς τελευταίας ἐν αὐτοῖς γραμμὰς ὁρίζειν τὰ πέρατα τῆς ἡμέρας.
And if you command, I will draw a sundial for you according to the method, whether you wish to divide the whole day into twelve or into some other subsequent number, for you will find in that case too what you proposed having been achieved, both from its agreeing with the measurement by means of pierced vessels and from the constructed sundials always agreeing with one another, and besides from their last lines defining the limits of the day.