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Euclid Book 1 Definitions
Statement
A point is that which has no part.
Original statement
σημϵῖόν ἐστιν, οὗ μέρος οὐθέν.
Statement
A line is breadthless length.
Original statement
γραμμὴ δὲ μῆκος ἀπλατές.
Statement
The extremities of a line are points.
Original statement
γραμμῆς δὲ πέρατα σημϵῖα.
Statement
A straight line is a line which lies evenly with the points on itself.
Original statement
ϵὐθϵῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐϕ᾽ ἑαυτῆς σημϵίοις κϵῖται.
Statement
A surface is that which has length and breadth only.
Original statement
ἐπιϕάνϵια δέ ἐστιν, ὃ μῆκος καὶ πλάτος μόνον ἔχϵι.
Statement
The extremities of a surface are lines.
Original statement
ἐπιϕανϵίας δὲ πέρατα γραμμαί.
Statement
A plane surface is a surface which lies evenly with the straight lines on itself.
Original statement
ἐπίπϵδος ἐπιϕάνϵιά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐϕ᾽ ἑαυτῆς ϵὐθϵίαις κϵῖται.
Statement
A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
Original statement
ἐπίπϵδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐπ᾽ ϵὐθϵίας κϵιμένων πρὸς ἀλλήλας τῶν γραμμῶν κλίσις.
Statement
When the lines containing the angle are straight, the angle is called rectilinear.
Original statement
ὅταν δὲ αἱ πϵριέχουσαι τὴν γωνίαν γραμμαὶ ϵὐθϵῖαι ὦσιν, ϵὐθύγραμμος καλϵῖται ἡ γωνία.
Statement
When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
Original statement
ὅταν δὲ ϵὐθϵῖα ἐπ᾽ ϵὐθϵῖαν σταθϵῖσα τὰς ἐϕϵξῆς γωνίας ἴσας ἀλλήλαις ποιῇ, ὀρθὴ ἑκατέρα τῶν ἴσων γωνιῶν ἐστι, καὶ ἡ ἐϕϵστηκυῖα ϵὐθϵῖα κάθϵτος καλϵῖται, ἐϕ᾽ ἣν ἐϕέστηκϵν.
Statement
An obtuse angle is an angle greater than a right angle.
Original statement
ἀμβλϵῖα γωνία ἐστὶν ἡ μϵίζων ὀρθῆς.
Statement
An acute angle is an angle less than a right angle.
Original statement
ὀξϵῖα δὲ ἡ ἐλάσσων ὀρθῆς.
Statement
A boundary is that which is an extremity of anything.
Original statement
ὅρος ἐστίν, ὅ τινός ἐστι πέρας.
Statement
A figure is that which is contained by any boundary or boundaries.
Original statement
σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων πϵριϵχόμϵνον.
Statement
A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.
Original statement
κύκλος ἐστὶ σχῆμα ἐπίπϵδον ὑπὸ μιᾶς γραμμῆς πϵριϵχόμϵνον ἣ καλϵῖται πϵριϕέρϵια, πρὸς ἣν ἀϕ᾽ ἑνὸς σημϵίου τῶν ἐντὸς τοῦ σχήματος κϵιμένων πᾶσαι αἱ προσπίπτουσαι ϵὐθϵῖαι πρὸς τὴν τοῦ κύκλου πϵριϕέρϵιαν ἴσαι ἀλλήλαις ϵἰσίν.
Statement
A point is called the center of a circle when from that point, all the straight lines falling upon the circle, among those lying within the circle, are equal to one another.
Original statement
κέντρον δὲ τοῦ κύκλου τὸ σημϵῖον καλϵῖται.
Statement
A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
Original statement
διάμϵτρος δὲ τοῦ κύκλου ἐστὶν ϵὐθϵῖά τις διὰ τοῦ κέντρου ἠγμένη καὶ πϵρατουμένη ἐϕ᾽ ἑκάτϵρα τὰ μέρη ὑπὸ τῆς τοῦ κύκλου πϵριϕϵρϵίας, ἥτις καὶ δίχα τέμνϵι τὸν κύκλον.
Statement
A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.
Original statement
ἡμικύκλιον δέ ἐστι τὸ πϵριϵχόμϵνον σχῆμα ὑπό τϵ τῆς διαμέτρου καὶ τῆς ἀπολαμβανομένης ὑπ᾽ αὐτῆς πϵριϕϵρϵίας. κέντρον δὲ τοῦ ἡμικυκλίου τὸ αὐτό, ὃ καὶ τοῦ κύκλου ἐστίν.
Statement
Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
Original statement
σχήματα ϵὐθύγραμμά ἐστι τὰ ὑπὸ ϵὐθϵιῶν πϵριϵχόμϵνα, τρίπλϵυρα μὲν τὰ ὑπὸ τριῶν, τϵτράπλϵυρα δὲ τὰ ὑπὸ τϵσσάρων, πολύπλϵυρα δὲ τὰ ὑπὸ πλϵιόνων ἢ τϵσσάρων ϵὐθϵιῶν πϵριϵχόμϵνα.
Statement
Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
Original statement
τῶν δὲ τριπλϵύρων σχημάτων ἰσόπλϵυρον μὲν τρίγωνόν ἐστι τὸ τὰς τρϵῖς ἴσας ἔχον πλϵυράς, ἰσοσκϵλὲς δὲ τὸ τὰς δύο μόνας ἴσας ἔχον πλϵυράς, σκαληνὸν δὲ τὸ τὰς τρϵῖς ἀνίσους ἔχον πλϵυράς.
Statement
Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.
Original statement
ἔτι δὲ τῶν τριπλϵύρων σχημάτων ὀρθογώνιον μὲν τρίγωνόν ἐστι τὸ ἔχον ὀρθὴν γωνίαν, ἀμβλυγώνιον δὲ τὸ ἔχον ἀμβλϵῖαν γωνίαν, ὀξυγώνιον δὲ τὸ τὰς τρϵῖς ὀξϵίας ἔχον γωνίας.
Statement
Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
Original statement
τῶν δὲ τϵτραπλϵύρων σχημάτων τϵτράγωνον μέν ἐστιν, ὃ ἰσόπλϵυρόν τέ ἐστι καὶ ὀρθογώνιον, ἑτϵρόμηκϵς δέ, ὃ ὀρθογώνιον μέν, οὐκ ἰσόπλϵυρον δέ, ῥόμβος δέ, ὃ ἰσόπλϵυρον μέν, οὐκ ὀρθογώνιον δέ, ῥομβοϵιδὲς δὲ τὸ τὰς ἀπϵναντίον πλϵυράς τϵ καὶ γωνίας ἴσας ἀλλήλαις ἔχον, ὃ οὔτϵ ἰσόπλϵυρόν ἐστιν οὔτϵ ὀρθογώνιον: τὰ δὲ παρὰ ταῦτα τϵτράπλϵυρα τραπέζια καλϵίσθω.
Statement
Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
Original statement
παράλληλοί ϵἰσιν ϵὐθϵῖαι, αἵτινϵς ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμϵναι ϵἰς ἄπϵιρον ἐϕ᾽ ἑκάτϵρα τὰ μέρη ἐπὶ μηδέτϵρα συμπίπτουσιν ἀλλήλαις.