Page 87 - Euclid's Elements of Geometry

ST EW gþ.

ELEMENTS BOOK 3

to BA, will not fall inside the circle. So, similarly, we
can show that neither (will it fall) on the circumference.
Thus, (it will fall) outside (the circle).
Β B

Γ C
∆ D

Θ H
Ζ Η F G
Ε E
Α A
Πιπτέτω ὡς ἡ ΑΕ· λέγω δή, ὅτι εἰς τὸν μεταξὺ τόπον Let it fall like AE (in the figure). So, I say that another
τῆς τε ΑΕ εὐθείας καὶ τῆς ΓΘΑ περιφερείας ἑτέρα εὐθεῖα straight-line cannot be inserted into the space between
οὐ παρεμπεσεῖται. the straight-line AE and the circumference CHA.
Εἰ γὰρ δυνατόν, παρεμπιπτέτω ὡς ἡ ΖΑ, καὶ ἤχθω ἀπὸ For, if possible, let it be inserted like FA (in the fig-
τοῦ Δ σημείου ἐπὶ τῆν ΖΑ κάθετος ἡ ΔΗ. καὶ ἐπεὶ ὀρθή ure), and let DG have been drawn from point D, perpen-
ἐστιν ἡ ὑπὸ ΑΗΔ, ἐλάττων δὲ ὀρθῆς ἡ ὑπὸ ΔΑΗ, μείζων dicular to FA [Prop. 1.12]. And since AGD is a right-
ἄρα ἡ ΑΔ τῆς ΔΗ. ἴση δὲ ἡ ΔΑ τῇ ΔΘ· μείζων ἄρα ἡ ΔΘ angle, and DAG (is) less than a right-angle, AD (is)
τῆς ΔΗ, ἡ ἐλάττων τῆς μείζονος· ὅπερ ἐστὶν ἀδύνατον. οὐκ thus greater than DG [Prop. 1.19]. And DA (is) equal
ἄρα εἰς τὸν μεταξὺ τόπον τῆς τε εὐθείας καὶ τῆς περιφερείας to DH. Thus, DH (is) greater than DG, the lesser than
ἑτέρα εὐθεῖα παρεμπεσεῖται. the greater. The very thing is impossible. Thus, another
Λέγω, ὅτι καὶ ἡ μὲν τοῦ ἡμικυκλίου γωνία ἡ περιεχομένη straight-line cannot be inserted into the space between
ὑπό τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περιφερείας ἁπάσης the straight-line (AE) and the circumference.
γωνίας ὀξείας εὐθυγράμμου μείζων ἐστίν, ἡ δὲ λοιπὴ ἡ πε- And I also say that the semi-circular angle contained
ριεχομένη ὑπό τε τῆς ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας by the straight-line BA and the circumference CHA is
ἁπάσης γωνίας ὀξείας εὐθυγράμμου ἐλάττων ἐστίν. greater than any acute rectilinear angle whatsoever, and
Εἰ γὰρ ἐστί τις γωνία εὐθύγραμμος μείζων μὲν τῆς the remaining (angle) contained by the circumference
περιεχομένης ὑπό τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περι- CHA and the straight-line AE is less than any acute rec-
φερείας, ἐλάττων δὲ τῆς περιεχομένης ὑπό τε τῆς ΓΘΑ tilinear angle whatsoever.
περιφερείας καὶ τὴς ΑΕ εὐθείας, εἰς τὸν μεταξὺ τόπον τῆς For if any rectilinear angle is greater than the (an-
τε ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας εὐθεῖα παρεμ- gle) contained by the straight-line BA and the circum-
πεσεῖται, ἥτις ποιήσει μείζονα μὲν τῆς περιεχομένης ὑπὸ ference CHA, or less than the (angle) contained by the
τε τῆς ΒΑ εὐθείας καὶ τῆς ΓΘΑ περιφερείας ὑπὸ εὐθειῶν circumference CHA and the straight-line AE, then a
περιεχομένην, ἐλάττονα δὲ τῆς περιεχομένης ὑπό τε τῆς straight-line can be inserted into the space between the
ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας. οὐ παρεμπίπτει δέ· circumference CHA and the straight-line AE—anything
οὐκ ἄρα τῆς περιεχομένης γωνίας ὑπό τε τῆς ΒΑ εὐθείας which will make (an angle) contained by straight-lines
καὶ τῆς ΓΘΑ περιφερείας ἔσται μείζων ὀξεῖα ὑπὸ εὐθειῶν greater than the angle contained by the straight-line BA
περιεχομένη, οὐδὲ μὴν ἐλάττων τῆς περιεχομένης ὑπό τε and the circumference CHA, or less than the (angle)
τῆς ΓΘΑ περιφερείας καὶ τῆς ΑΕ εὐθείας. contained by the circumference CHA and the straight-
line AE. But (such a straight-line) cannot be inserted.
Thus, an acute (angle) contained by straight-lines cannot
be greater than the angle contained by the straight-line
BA and the circumference CHA, neither (can it be) less
than the (angle) contained by the circumference CHA
and the straight-line AE.

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