§3.prop.16ἡ τῇ διαμέτρῳ τοῦ κύκλου πρὸς ὀρθὰς ἀπʼ ἄκρας ἀγομένη ἐκτὸς πεσεῖται τοῦ κύκλου, καὶ εἰς τὸν μεταξὺ τόπον τῆς τε εὐθείας καὶ τῆς περιφερείας ἑτέρα εὐθεῖα οὐ παρεμπεσεῖται, καὶ ἡ μὲν τοῦ ἡμικυκλίου γωνία ἁπάσης γωνίας ὀξείας εὐθυγράμμου μείζων ἐστίν, ἡ δὲ λοιπὴ ἐλάττων.
The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line will not intervene; and the angle of the semicircle is greater than any acute rectilineal angle, and the remaining angle is less.
ἔστω κύκλος ὁ ΑΒΓ περὶ κέντρον τὸ Δ καὶ διάμετρον τὴν ΑΒ· λέγω, ὅτι ἡ ἀπὸ τοῦ Α τῇ ΑΒ πρὸς ὀρθὰς ἀπʼ ἄκρας ἀγομένη ἐκτὸς πεσεῖται τοῦ κύκλου.
Let ΑΒΓ be a circle with center Δ and diameter ΑΒ; I say that the straight line drawn from Α at right angles to ΑΒ from its extremity will fall outside the circle.
μὴ γάρ, ἀλλʼ εἰ δυνατόν, πιπτέτω ἐντὸς ὡς ἡ ΓΑ, καὶ ἐπεζεύχθω ἡ ΔΓ.
ἐπεὶ ἴση ἐστὶν ἡ ΔΑ τῇ ΔΓ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΔΑΓ γωνίᾳ τῇ ὑπὸ ΑΓΔ. ὀρθὴ δὲ ἡ ὑπὸ ΔΑΓ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΑΓΔ· τριγώνου δὴ τοῦ ΑΓΔ αἱ δύο γωνίαι αἱ ὑπὸ ΔΑΓ, ΑΓΔ δύο ὀρθαῖς ἴσαι εἰσίν· ὅπερ ἐστὶν ἀδύνατον.
For, if not, let it fall inside, if possible, as ΓΑ, and let ΔΓ be joined. Since ΔΑ is equal to ΔΓ, the angle ΔΑΓ is also equal to the angle ΑΓΔ. And the angle ΔΑΓ is right; therefore the angle ΑΓΔ is also right; thus in the triangle ΑΓΔ the two angles ΔΑΓ, ΑΓΔ are equal to two right angles; which is impossible.
οὐκ ἄρα ἡ ἀπὸ τοῦ Α σημείου τῇ ΒΑ πρὸς ὀρθὰς ἀγομένη ἐντὸς πεσεῖται τοῦ κύκλου.
Therefore the straight line drawn from the point Α at right angles to ΒΑ will not fall inside the circle.
ὁμοίως δὴ δείξομεν, ὅτι οὐδʼ ἐπὶ τῆς περιφερείας· ἐκτὸς ἄρα.
Similarly we shall prove that neither does it fall on the circumference; therefore it falls outside.
πιπτέτω ὡς ἡ ΑΕ· λέγω δή, ὅτι εἰς τὸν μεταξὺ τόπον τῆς τε ΑΕ εὐθείας καὶ τῆς ΓΘΑ περιφερείας ἑτέρα εὐθεῖα οὐ παρεμπεσεῖται.
Let it fall as ΑΕ; I say then that into the space between the straight line ΑΕ and the circumference ΓΘΑ another straight line will not intervene.
εἰ γὰρ δυνατόν, παρεμπιπτέτω ὡς ἡ ΖΑ, καὶ ἤχθω ἀπὸ τοῦ Δ σημείου ἐπὶ τὴν ΖΑ κάθετος ἡ ΔΗ. καὶ ἐπεὶ ὀρθή ἐστιν ἡ ὑπὸ ΑΗΔ, ἐλάττων δὲ ὀρθῆς ἡ ὑπὸ ΔΑΗ, μείζων ἄρα ἡ ΑΔ τῆς ΔΗ. ἴση δὲ ἡ ΔΑ τῇ ΔΘ· μείζων ἄρα ἡ ΔΘ τῆς ΔΗ, ἡ ἐλάττων τῆς μείζονος· ὅπερ ἐστὶν ἀδύνατον.
For, if possible, let it intervene as ΖΑ, and let the perpendicular ΔΗ be drawn from the point Δ to ΖΑ. And since the angle ΑΗΔ is right, and the angle ΔΑΗ is less than a right angle, ΑΔ is therefore greater than ΔΗ. And ΔΑ is equal to ΔΘ; therefore ΔΘ is greater than ΔΗ, the less than the greater; which is impossible.
οὐκ ἄρα εἰς τὸν μεταξὺ τόπον τῆς τε εὐθείας καὶ τῆς περιφερείας ἑτέρα εὐθεῖα παρεμπεσεῖται.
Therefore into the space between the straight line and the circumference another straight line will not intervene.
λέγω, ὅτι καὶ ἡ μὲν τοῦ ἡμικυκλίου γωνία ἡ περιεχομένη ὑπό τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περιφερείας ἁπάσης γωνίας ὀξείας εὐθυγράμμου μείζων ἐστίν, ἡ δὲ λοιπὴ ἡ περιεχομένη ὑπό τε τῆς ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας ἁπάσης γωνίας ὀξείας εὐθυγράμμου ἐλάττων ἐστίν.
I say that the angle of the semicircle contained by the straight line ΒΑ and the circumference ΓΘΑ is greater than any acute rectilineal angle, and the remaining angle contained by the circumference ΓΘΑ and the straight line ΑΕ is less than any acute rectilineal angle.
εἰ γὰρ ἐστί τις γωνία εὐθύγραμμος μείζων μὲν τῆς περιεχομένης ὑπό τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περιφερείας, ἐλάττων δὲ τῆς περιεχομένης ὑπό τε τῆς ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας, εἰς τὸν μεταξὺ τόπον τῆς τε ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας εὐθεῖα περεμπεσεῖται, ἥτις ποιήσει μείζονα μὲν τῆς περιεχομένης ὑπό τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περιφερείας ὑπὸ εὐθειῶν περιεχομένην, ἐλάττονα δὲ τῆς περιεχομένης ὑπό τε τῆς ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας.
For if there is any rectilineal angle greater than that contained by the straight line ΒΑ and the circumference ΓΘΑ, or less than that contained by the circumference ΓΘΑ and the straight line ΑΕ, a straight line will intervene into the space between the circumference ΓΘΑ and the straight line ΑΕ, which will make an angle contained by straight lines greater than that contained by the straight line ΒΑ and the circumference ΓΘΑ, and less than that contained by the circumference ΓΘΑ and the straight line ΑΕ.
οὐ παρεμπίπτει δέ· οὐκ ἄρα τῆς περιεχομένης γωνίας ὑπό τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περιφερείας ἔσται μείζων ὀξεῖα ὑπὸ εὐθειῶν περιεχομένη, οὐδὲ μὴν ἐλάττων τῆς περιεχομένης ὑπό τε τῆς ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας.
But it does not intervene; therefore there will not be an acute angle contained by straight lines greater than the angle contained by the straight line ΒΑ and the circumference ΓΘΑ, nor yet less than that contained by the circumference ΓΘΑ and the straight line ΑΕ.
Πόρισμα
ἐκ δὴ τούτου φανερόν, ὅτι ἡ τῇ διαμέτρῳ τοῦ κύκλου πρὸς ὀρθὰς ἀπʼ ἄκρας ἀγομένη ἐφάπτεται τοῦ κύκλου.
Porism From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its extremity touches the circle.
ὅπερ ἔδει δεῖξαι.
Which was to be proved.