---
title: "Elements · Στοιχεῖα"
author: "Euclid"
translator: "Thomas Little Heath"
languages: [grc, en]
shelf: "Classical Greek"
section: 24 of 49
url: https://scriptura.si/works/euclid-elements/24/
cite: "Euclid, tr. Thomas Little Heath, Elements · Στοιχεῖα, section 24. Scriptura (scriptura.si), https://scriptura.si/works/euclid-elements/24/"
---

# Elements · Στοιχεῖα

*Euclid, tr. Thomas Little Heath*

The thirteen books of geometry and number theory that taught mathematics for two thousand years, from the definition of a point to the five regular solids.

> Greek: ed. Johan Ludvig Heiberg, from the Perseus Digital Library (canonical-greekLit, tlg1799.tlg001, CC BY-SA 4.0).

> English: Thomas Little Heath, from the Perseus Digital Library (tlg1799.tlg001.perseus-eng2). Notes are not included.

> Rows follow the references the two texts share (books, chapters and sections), cut where both reach the end of a sentence.

## 24. Book VII · Definitions

Source: https://scriptura.si/works/euclid-elements/24/ · cite a paragraph as https://scriptura.si/works/euclid-elements/24/#p-N

[¶0] 1. An unit is that by virtue of which each of the things that exist is called one.

> Greek: 1. μονάς ἐστιν, καθʼ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται.

[¶1] 2. A number is a multitude composed of units.

> Greek: 2. ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος.

[¶2] 3. A number is a part of a number, the less of the greater, when it measures the greater;

> Greek: 3. μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τὸν μείζονα.

[¶3] 4. but parts when it does not measure it.

> Greek: 4. μέρη δέ, ὅταν μὴ καταμετρῇ.

[¶4] 5. The greater number is a multiple of the less when it is measured by the less.

> Greek: 5. πολλαπλάσιος δὲ ὁ μείζων τοῦ ἐλάσσονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάσσονος.

[¶5] 6. An even number is that which is divisible into two equal parts.

> Greek: 6. ἄρτιος ἀριθμός ἐστιν ὁ δίχα διαιρούμενος.

[¶6] 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.

> Greek: 7. περισσὸς δὲ ὁ μὴ διαιρούμενος δίχα ἢ [ὁ] μονάδι διαφέρων ἀρτίου ἀριθμοῦ.

[¶7] 8. An even-times even number is that which is measured by an even number according to an even number.

> Greek: 8. ἀρτιάκις ἄρτιος ἀριθμός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ ἄρτιον ἀριθμόν.

[¶8] 9. An even-times odd number is that which is measured by an even number according to an odd number.

> Greek: 9. ἀρτιάκις δὲ περισσός ἐστιν ὁ ὑπὸ ἀρτίου ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν. [Περισσάκις ἀρτιός ἐστιν ὁ ὑπὸ περισσοῦ ἀριθμοῦ μετρούμενος κατὰ ἄρτιον ἀριθμόν].

[¶9] 10. An odd-times odd number is that which is measured by an odd number according to an odd number.

> Greek: 10. περισσάκις δὲ περισσὸς ἀριθμός ἐστιν ὁ ὑπὸ περισσοῦ ἀριθμοῦ μετρούμενος κατὰ περισσὸν ἀριθμόν.

[¶10] 11. A prime number is that which is measured by an unit alone.

> Greek: 11. πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος.

[¶11] 12. Numbers prime to one another are those which are measured by an unit alone as a common measure.

> Greek: 12. πρῶτοι πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ μονάδι μόνῃ μετρούμενοι κοινῷ μέτρῳ.

[¶12] 13. A composite number is that which is measured by some number.

> Greek: 13. σύνθετος ἀριθμός ἐστιν ὁ ἀριθμῷ τινι μετρούμενος.

[¶13] 14. Numbers composite to one another are those which are measured by some number as a common measure.

> Greek: 14. σύνθετοι δὲ πρὸς ἀλλήλους ἀριθμοί εἰσιν οἱ ἀριθμῷ τινι μετρούμενοι κοινῷ μέτρῳ.

[¶14] 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.

> Greek: 15. ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί τις.

[¶15] 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.

> Greek: 16. ὅταν δὲ δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος ἐπίπεδος καλεῖται, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.

[¶16] 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.

> Greek: 17. ὅταν δὲ τρεῖς ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, ὁ γενόμενος στερεός ἐστιν, πλευραὶ δὲ αὐτοῦ οἱ πολλαπλασιάσαντες ἀλλήλους ἀριθμοί.

[¶17] 18. A square number is equal multiplied by equal, or a number which is contained by two equal numbers.

> Greek: 18. τετράγωνος ἀριθμός ἐστιν ὁ ἰσάκις ἴσος ἢ [ὁ] ὑπὸ δύο ἴσων ἀριθμῶν περιεχόμενος.

[¶18] 19. And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.

> Greek: 19. κύβος δὲ ὁ ἰσάκις ἴσος ἰσάκις ἢ [ὁ] ὑπὸ τριῶν ἴσων ἀριθμῶν περιεχόμενος.

[¶19] 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.

> Greek: 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.

[¶20] 21. Similar plane and solid numbers are those which have their sides proportional.

> Greek: 21. ὅμοιοι ἐπίπεδοι καὶ στερεοὶ ἀριθμοί εἰσιν οἱ ἀνάλογον ἔχοντες τὰς πλευράς.

[¶21] 22. A perfect number is that which is equal to its own parts.

> Greek: 22. τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος ὤν.