§7.def.1μονάς ἐστιν, καθʼ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται.
A unit is that by virtue of which each of the things that exist is called one.
§7.def.2ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος.
A number is a multitude composed of units.
§7.def.3μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τὸν μείζονα.
A number is a part of a number, the less of the greater, when it measures the greater.
§7.def.4μέρη δέ, ὅταν μὴ καταμετρῇ.
and parts, when it does not measure it.
§7.def.5πολλαπλάσιος δὲ ὁ μείζων τοῦ ἐλάσσονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάσσονος.
and the greater is a multiple of the less, when it is measured by the less.
§7.def.6ἄρτιος ἀριθμός ἐστιν ὁ δίχα διαιρούμενος.
An even number is that which is divisible into two equal parts.
§7.def.7περισσὸς δὲ ὁ μὴ διαιρούμενος δίχα ἢ μονάδι διαφέρων ἀρτίου ἀριθμοῦ.
and an odd number is that which is not divisible into two equal parts, or which differs from an even number by a unit.
§7.def.8ἀρτιάκις ἄρτιος ἀριθμός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ ἄρτιον ἀριθμόν.
An even-times even number is that which is measured by an even number according to an even number.
§7.def.9ἀρτιάκις δὲ περισσός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν.
and an even-times odd number is that which is measured by an even number according to an odd number.
. §7.def.10περισσάκις δὲ περισσὸς ἀριθμός ἐστιν ὁ ὑπὸ περισσοῦ ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν.
and an odd-times odd number is that which is measured by an odd number according to an odd number.
§7.def.11πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος.
A prime number is that which is measured by a unit alone.
§7.def.12πρῶτοι πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ μονάδι μόνῃ μετρούμενοι κοινῷ μέτρῳ.
Numbers prime to one another are those which are measured by a unit alone as a common measure.
§7.def.13σύνθετος ἀριθμός ἐστιν ὁ ἀριθμῷ τινι μετρούμενος.
A composite number is that which is measured by some number.
§7.def.14σύνθετοι δὲ πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ ἀριθμῷ τινι μετρούμενοι κοινῷ μέτρῳ.
and numbers composite to one another are those which are measured by some number as a common measure.
§7.def.15ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί τις.
A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.
§7.def.16ὅταν δὲ δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος ἐπίπεδος καλεῖται, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.
And when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.
§7.def.17ὅταν δὲ τρεῖς ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος στερεός ἐστιν, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.
And when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.
§7.def.18τετράγωνος ἀριθμός ἐστιν ὁ ἰσάκις ἴσος ἢ ὑπὸ δύο ἴσων ἀριθμῶν περιεχόμενος.
A square number is that which is equal multiplied by equal, or that which is contained by two equal numbers.
§7.def.19κύβος δὲ ὁ ἰσάκις ἴσος ἰσάκις ἢ ὑπὸ τριῶν ἴσων ἀριθμῶν περιεχόμενος.
and a cube is that which is equal multiplied by equal and again by equal, or that which is contained by three equal numbers.
§7.def.20ἀριθμοὶ ἀνάλογόν εἰσιν, ὅταν ὁ πρῶτος τοῦ δευτέρου καὶ ὁ τρίτος τοῦ τετάρτου ἰσάκις ᾖ πολλαπλάσιος ἢ τὸ αὐτὸ μέρος ἢ τὰ αὐτὰ μέρη ὦσιν.
Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.
§7.def.21ὅμοιοι ἐπίπεδοι καὶ στερεοὶ ἀριθμοί εἰσιν οἱ ἀνάλογον ἔχοντες τὰς πλευράς.
Similar plane and solid numbers are those which have their sides proportional.
§7.def.22τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος ὤν.
A perfect number is that which is equal to its own parts.