Postulates and Common Notions

This page contains a set of 5 postulates and 5 common notions that are utilized throughout Euclid's Books 1 - 6 and 13. The postulates specify properties and relationships of lines, circles and angles. The common notions provide abstract notions of addition, subtraction, equality and inequality.

Postulate 1

Statement

To draw a straight line from any point to any point.

Original statement

Ἠιτήσθω ἀπὸ παντὸς σημϵίου ἐπὶ πᾶν σημϵῖον ϵὐθϵῖαν γραμμὴν ἀγαγϵῖν.

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EuclidPostulate1Graph

Postulate 2

Statement

To produce a finite straight line continuously in a straight line.

Original statement

καὶ πϵπϵρασμένην ϵὐθϵῖαν κατὰ τὸ συνϵχὲς ἐπ᾽ ϵὐθϵίας ἐκβαλϵῖν.

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EuclidPostulate2Graph

Postulate 3

Statement

To describe a circle with any center and distance.

Original statement

καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράϕϵσθαι.

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EuclidPostulate3Graph

Postulate 4

Statement

All right angles are equal to one another.

Original statement

καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις ϵἶναι.

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EuclidPostulate4Graph

Postulate 5

Statement

If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side.

Original statement

καὶ ἐὰν ϵἰς δύο ϵὐθϵίας ϵὐθϵῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ, ἐκβαλλομένας τὰς δύο ϵὐθϵίας ἐπ᾽ ἄπϵιρον συμπίπτϵιν, ἐϕ᾽ ἃ μέρη ϵἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονϵς.

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EuclidPostulate5Graph

Common notion 1

Statement

Things which are equal to the same thing are also equal to one another.

Original statement

Τὰ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα.

Common notion 2

Statement

If equals are added to equals, the wholes are equal.

Original statement

Καὶ ἐὰν ἴσοις ἴσα προστϵθῇ, τὰ ὅλα ἐστὶν ἴσα.

Common notion 3

Statement

If equals are subtracted from equals, the remainders are equal.

Original statement

Καὶ ἐὰν ἀπὸ ἴσων ἴσα ἀϕαιρϵθῇ, τὰ καταλϵιπόμϵνά ἐστιν ἴσα.

Common notion 4

Statement

Things which coincide with one another are equal to one another.

Original statement

Καὶ τὰ ἐϕαρμόζοντα ἐπ ̓ ἀλλήλα ἴσα ἀλλήλοις ἐστίν.

Common notion 5

Statement

The whole is greater than the part.

Original statement

Καὶ τὸ ὅλον τοῦ μέρους μϵῖζόν [ἐστιν].