Euclid Book 6 Definitions

Definition 1

Statement

Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional.

Original statement

ὅμοια σχήματα ϵὐθύγραμμά ἐστιν, ὅσα τάς τϵ γωνίας ἴσας ἔχϵι κατὰ μίαν καὶ τὰς πϵρὶ τὰς ἴσας γωνίας πλϵυρὰς ἀνάλογον.

Definition 2

Statement

When two sides of one figure together with two sides of another figure form antecedents and consequents in a proportion, the figures are reciprocally related.

Original statement

Ἀντιπϵπονθότα δὲ σχήματά ἐστιν, ὅταν ἐν ἑκατέρῳ τῶν σχημάτων ἡγούμϵνοί τϵ καὶ ἑπόμϵνοι λόγοι ὦσιν.

Definition 3

Statement

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

Original statement

ἄκρον καὶ μέσον λόγον ϵὐθϵῖα τϵτμῆσθαι λέγϵται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μϵῖζον τμῆμα, οὕτως τὸ μϵῖζον πρὸς τὸ ἔλαττον.

Definition 4

Statement

The height of any figure is the perpendicular drawn from the vertex to the base.

Original statement

ὕψος ἐστὶ παντὸς σχήματος ἡ ἀπὸ τῆς κορυϕῆς ἐπὶ τὴν βάσιν κάθϵτος ἀγομένη.

Definition 5

Statement

A ratio is said to be compounded of ratios when the sizes of the ratios multiplied together make something. A parallelogram not "filling" a line is a figure less than the line. If it has a surplus and the line is not enough, it is a figure greater than the line.

Original statement

Λόγος ἐκ λόγων συγκϵῖσθαι λέγϵται, ὅταν αἱ τῶν λόγων πηλικότητϵς ἐϕ᾽ ἑαυτὰς πολλαπλασιασθϵῖσαι ποιῶσί τινα.