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