§1## PROBLEMA BOVINUM
Πρόβλημα, ὅπερ Ἀρχιμήδης ἐν ἐπιγράμμασιν εὑρὼν τοῖς ἐν Ἀλεξανδρείᾳ περὶ ταῦτα πραγματευομένοις ζητεῖν ἀπέστειλεν ἐν τῇ πρὸς Ἐρατοσθένην τὸν Κυρηναῖον ἐπιστολῇ.
## PROBLEMA BOVINUM A problem which Archimedes, having found it in epigrams, sent to those who busy themselves with such matters in Alexandria, to be solved in a letter to Eratosthenes of Cyrene.
Πληθὺν Ἠελίοιο βοῶν, ὦ ξεῖνε, μέτρησον φροντίδʼ ἐπιστήσας, εἰ μετέχεις σοφίης, πόσση ἄῤ ἐν πεδίοις Σικελῆς ποτʼ ἐβόσκετο νήσου Θρινακίης τετραχῇ στίφεα δασσαμένη χροιὴν ἀλάσσοντα· τὸ μὲν λευκοῖο γάλακτος, κυανέῳ δʼ ἕτερον χρώματι λαμπόμενον, ἄλλο γε μὲν ξανθόν, τὸ δὲ ποικίλον.
O stranger, if you partake of wisdom, measure the multitude of the cattle of Helios, applying your mind, how many of them once grazed in the plains of the Sicilian island of Thrinacia, divided into four herds of different colors: one white as milk, another shining with dark-blue color, a third yellow, and the fourth variegated.
Ἐν δὲ ἑκάστῳ στίφει ἔσαν ταῦροι πλήθεσι βριθόμενοι συμμετρίης τοιῆσδε τετευχότες· ἀργότριχας μὲν κυανέων ταύρων ἡμίσει ἠδὲ τρίτῳ καὶ ξανθοῖς σύμπασιν ἴσους, ὧ ξεῖνε, νόησον, αὐτὰρ κυανέους τῷ τετράτῳ τε μέρει μικτοχρόων καὶ πέμπτῳ, ἔτι ξανθοῖσί τε πᾶσιν.
In each herd there were bulls in great abundance, having the following proportions: consider, O stranger, that the white-haired bulls were equal to a half and a third of the dark-blue bulls, plus the whole of the yellow ones; while the dark-blue were equal to a fourth and a fifth part of the variegated, plus the whole of the yellow ones.
Τοὺς δʼ ὑπολειπομένους ποικιλόχρωτας ἄθρει ἀργεννῶν ταύρων ἕκτῳ μέρει ἑβδομάτῳ τε καὶ ξανθοῖς αὐτοὺς πᾶσιν ἰσαζομένους.
And consider that the remaining variegated ones were equal to a sixth and a seventh part of the white bulls, plus the whole of the yellow ones.
Θηλείαισι δὲ βουσὶ τάδʼ ἔπλετο· λευκότριχες μὲν ἧσαν συμπάσης κυανέης ἀγέλης τῷ τριτάτῳ τε μέρει καὶ τετράτῳ ἀτρεκὲς ἶσαι· αὐτὰρ κυάνεαι τῷ τετράτῳ τε πάλιν μικτοχρόων καὶ πέμπτῳ ὁμοῦ μέρει ἰσάζοντο σὺν ταύροις πάσαις εἰς νομὸν ἐρχομέναις.
And for the cows, the following was the case: the white-haired ones were exactly equal to a third and a fourth part of the whole dark-blue herd; while the dark-blue ones were equal to a fourth and a fifth part of the whole variegated herd, coming to the pasture together with all the cows.
Ξανθοτρίχων δʼ ἀγέλης πέμπτῳ μέρει ἠδὲ καὶ ἕκτῳ ποικίλαι ἰσάριθμον πλῆθος ἔχον τετραχῇ.
And the variegated ones, in the fifth and sixth part of the yellow herd, had an equal number in four ways.
Ξανθαὶ δʼ ἠριθμεῦντο μέρους τρίτου ἡμίσει ἶσαι ἀργεννῆς ἀγέλης ἑβδομάτῳ τε μέρει.
And the yellow ones were numbered equal to a half of a third part of the white herd, plus a seventh part of it.
Ξεῖνε, σὺ δʼ Ἠελίοιο βόες πόσαι ἀτρεκὲς εἰπών, χωρὶς μὲν ταύρων ζατρεφέων ἀριθμόν, χωρὶς δʼ αὖ θήλειαι ὅσαι κατὰ χροιὰν ἕκασται, οὐκ ἄιδρίς κα λέγοι οὐδʼ ἀριθμῶν ἀδαής, οὐ μὴν πώ γε σοφοῖς ἐναρίθμιος.
O stranger, if you can tell exactly how many were the cattle of Helios, stating the number of well-fed bulls separately, and separately how many cows there were of each color, you would not be called ignorant nor unskilled in numbers, yet you would not yet be numbered among the wise.
Ἀλλʼ ἴθι φράζευ καὶ τάδε πάντα βοῶν Ἠελίοιο πάθη.
But come, consider also all these conditions of the cattle of Helios.
Ἀργότριχες ταῦροι μὲν ἐπεὶ μιξαίατο πληθὺν κυανέοις, ἵσταντ᾿ ἔμπεδον ἰσόμετροι εἰς βάθος εἰς εὖρός τε, τὰ δʼ αὖ περιμήκεα πάντη πίμπλαντο πλίνθου Θρινακίης πεδία.
When the white-haired bulls mixed their multitude with the dark-blue ones, they stood firmly, equal in depth and width, and the far-stretching plains of the Thrinacian brick were filled.
Ξανθοὶ δʼ αὖτʼ εἰς ἓν καὶ ποικίλοι ἀθροισθέντες ἵσταντ᾿ ἀμβολάδην ἐξ ἑνὸς ἀρχόμενοι σχῆμα τελειοῦντες τὸ τρικράσπεδον οὔτε προσόντων ἀλλοχρόων ταύρων οὔτ᾿ ἐπιλειπομένων.
And when the yellow and variegated ones were gathered into one, they stood in a pile starting from one, completing a triangular shape, with no bulls of other colors added nor any lacking.
Ταῦτα συνεξευρὼν καὶ ἐνὶ πραπίδεσσιν ἀθροίσας καὶ πληθέων ἀποδούς, ξεῖνε, τὰ πάντα μέτρα ἔρχεο κυδιόων νικηφόρος ἴσθι τε πάντως κεκριμένος ταύτῃ γʼ ὄμπνιος ἐν σοφίῃ.
If you find all these things together and gather them in your mind, and give all the measures of the multitudes, O stranger, go, exulting in victory, and know that you are judged by all means as highly accomplished in this wisdom.
§2## Σχόλιον
Τὸ μὲν οὖν πρόβλημα διὰ τοῦ ποιήματος ὁ Ἀρχιμήδης ἐδήλωσε σαφῶς· ἰστέον δὲ λεγόμενον, ὅτι τέσσαρας ἀγέλας εἶναι δεῖ βοῶν, λευκοτρίχων μὲν μίαν ταύρων καὶ θηλειῶν, ὧν τὸ πλῆθος ὁμοῦ συνάγει μυριάδας διπλᾶς ιδ΄ καὶ ἁπλᾶς φπβ΄ καὶ μονάδας ζτξ΄, κυανοχρόων δʼ ἄλλην ὁμοῦ ταύρων καὶ θηλειῶν, ὧν τὸ πλῆθός ἐστι μυριάδων διπλῶν ἐννέα καὶ ἁπλῶν ηωλ΄ καὶ μονάδων ω΄, μιξοτρίχων δʼ ἄλλην ταύρων καὶ θηλειῶν, ὧν τὸ πλῆθός ἐστι μυριάδων διπλῶν η΄ καὶ ἁπλῶν ςϠ(??)α΄ καὶ μονάδων υ΄· τῆς δὲ λοιπῆς ἀγέλης ξανθοχρόων συνάγει τὸ πλῆθος διπλᾶς μυριάδας ζ΄ καὶ ἁπλᾶς ςψη΄, μονάδας δὲ η· ὥστε συνάγεσθαι ὁμοῦ τὸ πλῆθος τῶν δʼ ἀγελῶν μυριάδας διπλᾶς μ΄ καὶ ἁπλᾶς γριβ΄ καὶ μονάδας ςφξ΄.
## SCHOLION The commentary: Archimedes, therefore, clearly set forth the problem through the poem; but one must know what is said, namely, that there must be four herds of cattle, one of white-haired bulls and cows, whose multitude together amounts to 14 double myriads, 582 single myriads, and 7360 units; another of dark-blue bulls and cows together, whose multitude is 9 double myriads, 8830 single myriads, and 800 units; another of mixed-haired bulls and cows, whose multitude is 8 double myriads, 6991(??) single myriads, and 400 units; and of the remaining herd of yellow-colored, the multitude amounts to 7 double myriads, 698 single myriads, and 8 units; so that the total multitude of the four herds together amounts to 40 double myriads, 3112 single myriads, and 6560 units.
Καὶ ἡ μὲν ἀγέλη τῶν λευκοτρίχων ταύρων ἔχει μυριάδας διπλᾶς η΄ καὶ ἁπλᾶς βϠλα΄ καὶ μονάδας ηφξ΄, θηλειῶν δὲ μυριάδας διπλᾶς ε΄ καὶ ἁπλᾶς ζχν΄ καὶ μονάδας ηω΄, ἡ δὲ ἀγέλη τῶν κυανοχρόων ταύρων ἔχει μὲν μυριάδας διπλᾶς ε΄ καὶ ἁπλᾶς θχπδ΄ καὶ μονάδας αρκ΄, θηλειῶν δὲ μυριάδας διπλᾶς γ΄ καὶ ἁπλᾶς θρμε΄ καὶ μονάδας θχπ΄, ἡ δʼ ἀγέλη τῶν ποικιλοτρίχων ταύρων ἔχει μὲν μυριάδας διπλᾶς ε΄ καὶ ἁπλᾶς ηωξδ᾿ καὶ μονάδας δω΄, θηλειῶν δὲ μυριάδας διπλᾶς β΄ καὶ ἁπλᾶς ηρκς΄ καὶ μονάδας εχ᾿, ἡ δʼ ἀγέλη τῶν ξανθοχρωμάτων ταύρων ἔχει μὲν μυριάδας διπλᾶς γ΄ καὶ ἁπλᾶς γρ(??)ε΄ καὶ μονάδας Ϡξ΄, θηλειῶν δὲ μυριάδας διπλᾶς δ΄ καὶ ἁπλᾶς γφιγ΄ καὶ μονάδας ζμ΄.
And the herd of white-haired bulls has 8 double myriads, 2931 single myriads, and 8560 units, and of cows 5 double myriads, 7650 single myriads, and 8800 units; and the herd of dark-blue bulls has 5 double myriads, 9684 single myriads, and 1120 units, and of cows 3 double myriads, 9145 single myriads, and 9680 units; and the herd of variegated bulls has 5 double myriads, 8864 single myriads, and 4800 units, and of cows 2 double myriads, 8126 single myriads, and 5600 units; and the herd of yellow-colored bulls has 3 double myriads, 31(??)5 single myriads, and 960 units, and of cows 4 double myriads, 3513 single myriads, and 7040 units.
Καί ἐστι τὸ πλῆθος τῶν λευκοτρίχων ταύρων ἴσον τῷ ἡμίσει καὶ τρίτῳ μέρει τοῦ πλήθους τῶν κυανοχρόων ταύρων καὶ ἔτι ὅλῃ τῇ τῶν ξανθοχρωμάτων ἀγέλῃ, τὸ δὲ πλῆθος τῶν κυανοχρωμάτων ἴσον τῷ τετάρτῳ καὶ πέμπτῳ μέρει τῶν ποικιλοτρίχων ταύρων καὶ ὅλῳ τῷ πλήθει τῶν ξανθοχρωμάτων, τὸ δὲ πλῆθος τῶν ποικιλοτρίχων ταύρων ἴσον τῷ ἕκτῳ καὶ ἑβδόμῳ μέρει τῶν λευκοτρίχων ταύρων καὶ ἔτι τῷ πλήθει ὅλῳ τῶν ξανθοχρωμάτων ταύρων, καὶ πάλιν τὸ πλῆθος τῶν λευκῶν θηλειῶν ἴσον τῷ τρίτῳ καὶ τετάρτῳ μέρει ὅλης τῆς ἀγέλης τῶν κυανοχρόων, τὸ δὲ τῶν κυανοχρόων ἴσον τῷ τετάρτῳ καὶ πέμπτῳ μέρει τῆς ὅλης ἀγέλης τῶν ποικιλοτρίχων, τὸ δὲ τῶν ποικιλοτρίχων ἴσον τῷ πέμπτῳ καὶ ἕκτῳ μέρει τῆς ὅλης τῶν ξανθῶν βοῶν.
And the multitude of white-haired bulls is equal to a half and a third part of the multitude of dark-blue bulls plus the whole yellow-colored herd; and the multitude of dark-blue is equal to a fourth and a fifth part of the variegated bulls plus the whole multitude of the yellow-colored; and the multitude of variegated bulls is equal to a sixth and a seventh part of the white-haired bulls plus the whole multitude of the yellow-colored bulls; and again, the multitude of white cows is equal to a third and a fourth part of the whole herd of dark-blue; and that of the dark-blue is equal to a fourth and a fifth part of the whole herd of variegated; and that of the variegated is equal to a fifth and a sixth part of the whole of the yellow cattle.
Πάλιν δὲ τὸ τῶν ξανθῶν θηλειῶν πλῆθος ἦν ἴσον τῷ ἕκτῳ τε καὶ ἑβδόμῳ μέρει τῆς ὅλης ἀγέλης τῶν λευκῶν βοῶν.
And again, the multitude of yellow cows was equal to a sixth and a seventh part of the whole herd of white cattle.
Καὶ ἡ μὲν ἀγέλη τῶν λευκοτρίχων ταύρων καὶ ἡ τῶν κυανοχρόων ταύρων συντεθεῖσα ποιεῖ τετράγωνον ἀριθμόν, ἡ δʼ ἀγέλη τῶν ξανθοτρίχων ταύρων μετὰ τῆς ἀγέλης τῶν ποικιλοχρόων συντεθεῖσα ποιεῖ τρίγωνον, ὡς ἔχει τὰ τῶν ὑποκειμένων κανόνων καθʼ ἕκαστον χρῶμα.
And the herd of white-haired bulls and that of the dark-blue bulls, when added together, make a square number, and the herd of yellow-haired bulls, when added together with the herd of variegated-colored, makes a triangular number, as shown in the tables set down for each color.