§15.1Διὰ τί διάμετρος καλεῖται μόνη τῶν δίχα διαιρουσῶν τὰ εὐθύγραμμα ἡ ἐκ γωνίας εἰς γωνίαν ἀχθεῖσα γραμμή;
Why is the line drawn from corner to corner alone of those dividing rectilinear figures in two called the diagonal?
ἢ ὅτι διάμετρος δίχα διαιρεῖ, καθάπερ τοὔνομα ὑποσημαίνει, οὐ φθείρουσα τὸ μετρούμενον;
Is it because the diagonal (diameter) divides in two, as its name implies, without destroying the thing measured?
ἡ μὲν οὖν κατὰ τὰς συνθέσεις διαιροῦσα (λέγω δὲ τὰς γωνίας) διάμετρος ἔσται· οὐ γὰρ φθείρει, ἀλλὰ διαιρεῖ, καθάπερ οἱ τὰ στρατιωτικὰ σκεύη διαιροῦντες.
Therefore, the line that divides at the joints (I mean the corners) will be the diagonal; for it does not destroy but divides, just like those who distribute military equipment.
ἡ δὲ κατὰ τὰς γραμμὰς σύνθετα τέμνουσα φθείρει· σύγκειται γὰρ τὸ εὐθύγραμμον κατὰ τὰς γωνίας.
On the other hand, the line that cuts the composite parts along the sides destroys them; for the rectilinear figure is composed at the corners.
§15.2Διὰ τί διάμετρος καλεῖται;
Why is it called the diagonal (diameter)?
ἢ ὅτι δίχα μόνη διαιρεῖ;
Is it because it alone divides in two?
ὥσπερ οὖν εἴ τις εἴποι διχάμετρός ἐστιν.
As if one were to say it is a "dichameter" (two-measurer).
καὶ διότι μόνη τῶν δίχα τοῦτο καλεῖται.
And because it alone of those that divide in two is called this.
ἢ ὅτι κατὰ μέλη ᾗ κέκαμπται μόνη διαιρεῖ, αἱ δὲ ἄλλαι κατὰ πλευράς;
Or is it because it alone divides at the joints where it is bent, while the others divide along the sides?
§15.3Διὰ τί πάντες ἄνθρωποι, καὶ βάρβαροι καὶ Ἕλληνες, εἰς τὰ δέκα καταριθμοῦσι, καὶ οὐκ εἰς ἄλλον ἀριθμόν, οἷον β′, γ′, δ′, ε′, εἶτα πάλιν ἐπαναδιπλοῦσιν, ἓν πέντε, δύο πέντε, ὥσπερ ἕνδεκα, δώδεκα;
Why do all men, both barbarians and Greeks, count up to ten, and not to some other number, such as two, three, four, or five, and then repeat again, saying "one-five", "two-five", as they say "eleven", "twelve"?
οὐδ’ αὖ ἐξωτέρω παυσάμενοι τῶν δέκα, εἶτα ἐκεῖθεν ἐπαναδιπλοῦσιν;
Nor, again, do they stop beyond ten and then repeat from there?
ἔστι μὲν γὰρ ἕκαστος τῶν ἀριθμῶν ὁ ἔμπροσθεν καὶ ἓν ἢ δύο, καὶ οὕτως ἄλλος τις, ἀριθμοῦσι δ’ ὅμως ὁρίσαντες ἄχρι τῶν δέκα.
For indeed, each of the numbers is the preceding one plus one or two, and so on for any other, but they nevertheless count by defining the limit up to ten.
οὐ γὰρ δὴ ἀπὸ τύχης γε αὐτὸ ποιοῦντες φαίνονται καὶ ἀεί· τὸ δὲ ἀεὶ καὶ ἐπὶ πάντων οὐκ ἀπὸ τύχης, ἀλλὰ φυσικόν.
For they certainly do not appear to do this by chance, and always; and what happens always and in all cases is not by chance, but is natural.
πότερον ὅτι τὰ δέκα τέλειος ἀριθμός;
Is it because ten is a perfect number?
ἔχων γὰρ πάντα τὰ τοῦ ἀριθμοῦ εἴδη, ἄρτιον, περιττόν, τετράγωνον, κύβον, μῆκος ἐπίπεδον, πρῶτον σύνθετον.
For it contains all species of number: even, odd, square, cube, plane-oblong, and the first composite number.
ἢ ὅτι ἀρχὴ ἡ δεκάς;
Or is it because the decade is a principle?
ἓν γὰρ καὶ δύο καὶ τρία καὶ τέτταρα γίνεται δεκάς.
For one, two, three, and four make a decade.
ἢ ὅτι τὰ φερόμενα σώματα ἐννέα;
Or is it because the moving bodies are nine?
ἢ ὅτι ἐν δέκα ἀναλογίαις τέτταρες κυβικοὶ ἀριθμοὶ ἀποτελοῦνται, ἐξ ὧν φασὶν ἀριθμῶν οἱ Πυθαγόρειοι τὸ πᾶν συνεστάναι;
Or because in ten proportions four cubic numbers are completed, from which numbers the Pythagoreans say the universe is composed?
ἢ ὅτι πάντες ὑπῆρξαν ἄνθρωποι ἔχοντες δέκα δακτύλους;
Or is it because all men naturally had ten fingers?
οἷον οὖν ψήφους ἔχοντες τοῦ οἰκείουἀριθμοῦ, τούτῳ τῷ πλήθει καὶ τὰ ἄλλα ἀριθμοῦσιν.
So, as if having counters of their own number, they count other things by this quantity.
μόνοι δὲ ἀριθμοῦσι τῶν Θρᾳκῶν γένος τι εἰς τέτταρα, διὰ τὸ ὥσπερ τὰ παιδία μὴ δύνασθαι μνημονεύειν ἐπὶ πολύ, μηδὲ χρῆσιν μηδενὸς εἶναι πολλοῦ αὐτοῖς.
But a certain tribe of Thracians alone counts up to four, because, like children, they cannot remember very far, nor do they have any use for anything in large quantities.
§15.4† ὅτι ἡ γῆ κέντρον· ἀεὶ γὰρ ὅμοια τὰ φαινόμενα ἡμῖν σχήματα.
† That the earth is the center; for the shapes appearing to us are always similar.
δοκεῖ τοῦτο εἶναι, ἐὰν μὴ ἀπὸ τοῦ μέσου τις θεωρῇ, ἀλλ’ ὁτὲ μὲν τρίγωνα, ὁτὲ δὲ τραπέζια, ὁτὲ δὲ ἀλλοῖα ἐδόκει ἡ γῆ μέσον ἡμῖν, εἰ ἀπὸ τούτων ἔνι ἡμᾶς θεωρεῖν.
This seems to be the case; if one does not observe from the center, but sometimes triangles, sometimes trapeziums, and sometimes other shapes would appear, the earth would seem to us to be in the center, if it is possible for us to observe from these.
οὔσης γὰρ σφαιροειδοῦς τῆς γῆς ταὐτὸ κέντρον τούτου καὶ τῆς γῆς ἔσται.
For since the earth is spherical, the center of this (the universe) and of the earth will be the same.
ἡμεῖς δὲ ἐπάνω τῆς γῆς οἰκοῦμεν, ὥστε οὐκ ἀπὸ τούτου, ἀλλὰ τὸ ἥμισυ τῆς διαμέτρου ἀφεστῶσιν ἡμῖν τοιαῦτα φαίνεται.
But we live on top of the earth, so that not from this (the center), but to us who are at a distance of half the diameter, they appear in this way.
τί οὖν κωλύει πλέονος γινομένου τοῦ διαστήματος διαμένειν τὴν τῶν σχημάτων φαντασίαν;
What then prevents the appearance of the shapes from remaining the same when the distance becomes greater?