---
title: "Elements · Στοιχεῖα"
author: "Euclid"
translator: "Thomas Little Heath"
languages: [grc, en]
shelf: "Classical Greek"
section: 14 of 49
url: https://scriptura.si/works/euclid-elements/14/
cite: "Euclid, tr. Thomas Little Heath, Elements · Στοιχεῖα, section 14. Scriptura (scriptura.si), https://scriptura.si/works/euclid-elements/14/"
---

# 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.

## 14. Book IV · Definitions

Source: https://scriptura.si/works/euclid-elements/14/ · cite a paragraph as https://scriptura.si/works/euclid-elements/14/#p-N

[¶0] 1. A rectilineal figure is said to be inscribed in a rectilineal figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.

> Greek: 1. σχῆμα εὐθύγραμμον εἰς σχῆμα εὐθύγραμμον ἐγγράφεσθαι λέγεται, ὅταν ἑκάστη τῶν τοῦ ἐγγραφομένου σχήματος γωνιῶν ἑκάστης πλευρᾶς τοῦ, εἰς ὃ ἐγγράφεται, ἅπτηται.

[¶1] 2. Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.

> Greek: 2. σχῆμα δὲ ὁμοίως περὶ σχῆμα περιγράφεσθαι λέγεται, ὅταν ἑκάστη πλευρὰ τοῦ περιγραφομένου ἑκάστης γωνίας τοῦ, περὶ ὃ περιγράφεται, ἅπτηται.

[¶2] 3. A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.

> Greek: 3. σχῆμα εὐθύγραμμον εἰς κύκλον ἐγγράφεσθαι λέγεται, ὅταν ἑκάστη γωνία τοῦ ἐγγραφομένου ἅπτηται τῆς τοῦ κύκλου περιφερείας.

[¶3] 4. A rectilineal figure is said to be circumscribed about a circle, when each side of the circumscribed figure touches the circumference of the circle.

> Greek: 4. σχῆμα δὲ εὐθύγραμμον περὶ κύκλον περιγράφεσθαι λέγεται, ὅταν ἑκάστη πλευρὰ τοῦ περιγραφομένου ἐφάπτηται τῆς τοῦ κύκλου περιφερείας.

[¶4] 5. Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed.

> Greek: 5. κύκλος δὲ εἰς σχῆμα ὁμοίως ἐγγράφεσθαι λέγεται, ὅταν ἡ τοῦ κύκλου περιφέρεια ἑκάστης πλευρᾶς τοῦ, εἰς ὃ ἐγγράφεται, ἅπτηται.

[¶5] 6. A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed.

> Greek: 6. κύκλος δὲ περὶ σχῆμα περιγράφεσθαι λέγεται, ὅταν ἡ τοῦ κύκλου περιφέρεια ἑκάστης γωνίας τοῦ, περὶ ὃ περιγράφεται, ἅπτηται.

[¶6] 7. A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.

> Greek: 7. εὐθεῖα εἰς κύκλον ἐναρμόζεσθαι λέγεται, ὅταν τὰ πέρατα αὐτῆς ἐπὶ τῆς περιφερείας ᾖ τοῦ κύκλου.