Name: Elements: Alternative Proofs (Books II–III)
Author: Euclid (attributed)

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Elements: Alternative Proofs (Books II–III) collects the alternative demonstrations ("otherwise" proofs) transmitted with Euclid's Elements for a few propositions in Books II and III. The first concerns the square on a line divided into two parts; the rest treat propositions of Book III on lines drawn to a circle and the angle in a semicircle. J. L. Heiberg printed them in an appendix and regarded them as largely later additions.

These short pieces show how the Elements were studied and supplemented in antiquity, when teachers and commentators added their own proofs beside Euclid's. They offer readers alternative approaches to familiar theorems of plane geometry and evidence for the growth of the text in transmission.

About this edition. The Greek text comes from the Eulogikon open corpus of public-domain editions (standard reference: Heiberg, Euclidis Elementa I (appendix)). Every word has been analyzed with odyCy, a neural language model for Ancient Greek — click or tap any word for its dictionary form, part of speech, and complete morphology, together with the full entry from the Liddell–Scott–Jones Greek–English Lexicon (LSJ). The 'aliter' proofs transmitted with the Elements are regarded by Heiberg as largely post-Euclidean interpolations.

Ancient Greek · Science & Medicine

Elements: Alternative Proofs (Books II–III)

Euclid (attributed) · every word parsed

1.(n)
Ad αʹ, prop. δʹ ΑΛΛΩΣ Λέγω, ὅτι τὸ ἀπὸ τῆς ΑΒ τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ περιεχομένῳ ὀρθογωνίῳ. 
1.(n)·2
Ἐπὶ γὰρ τῆς αὐτῆς καταγραφῆς, ἐπεὶ ἴση ἐστὶν ἡ ΒΑ τῇ ΑΔ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΑΒΔ τῇ ὑπὸ ΑΔΒ· καὶ ἐπεὶ παντὸς τριγώνου αἱ τρεῖς γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν, τοῦ ΑΔΒ ἄρα τριγώνου αἱ τρεῖς γωνίαι αἱ ὑπὸ ΑΔΒ, ΒΑΔ, ΔΒΑ δυσὶν ὀρθαῖς ἴσαι εἰσίν. ὀρθὴ δὲ ἡ ὑπὸ ΒΑΔ· λοιπαὶ ἄρα αἱ ὑπὸ ΑΒΔ, ΑΔΒ μιᾷ ὀρθῇ ἴσαι εἰσί· καί εἰσιν ἴσαι· ἑκατέρα ἄρα τῶν ὑπὸ ΑΒΔ, ΑΔΒ ἡμίσειά ἐστιν ὀρθῆς. ὀρθὴ δὲ ἡ ὑπὸ ΒΓΗ· ἴση γάρ ἐστι τῇ ἀπεναντίον τῇ πρὸς τῷ Α· λοιπὴ ἄρα ἡ ὑπὸ ΓΗΒ ἡμίσειά ἐστιν ὀρθῆς· ἴση ἄρα ἡ ὑπὸ ΓΒΗ γωνία τῇ ὑπὸ ΓΗΒ· ὥστε καὶ πλευρὰ ἡ ΒΓ τῇ ΓΗ ἐστιν ἴση. ἀλλʼ ἡ μὲν ΓΒ τῇ ΗΚ ἐστιν ἴση, ἡ δὲ ΓΗ τῇ ΒΚ· ἰσόπλευρον ἄρα ἐστὶ τὸ ΓΚ. ἔχει δὲ καὶ ὀρθὴν τὴν ὑπὸ ΓΒΚ γωνίαν· τετράγωνον ἄρα ἐστὶ τὸ ΓΚ· καί ἐστιν ἀπὸ τῆς ΓΒ. διὰ τὰ αὐτὰ δὴ καὶ τὸ ΖΘ τετράγωνόν ἐστι, καί ἐστιν ἴσον τῷ ἀπὸ τῆς ΑΓ· τὰ ἄρα ΓΚ, ΘΖ τετράγωνά ἐστι, καί ἐστιν ἴσα τοῖς ἀπὸ τῶν ΑΓ, ΓΒ. καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΑΗ τῷ ΗΕ, καί ἐστι τὸ ΑΗ τὸ ὑπὸ τῶν ΑΓ, ΓΒ· ἴση γὰρ ἡ ΓΗ τῇ ΓΒ· καὶ τὸ ΕΗ ἄρα ἴσον ἐστὶ τῷ ὑπὸ τῶν ΑΓ, ΓΒ. τὰ ἄρα ΑΗ, ΗΕ ἴσα ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ. ἔστι δὲ καὶ τὰ ΓΚ, ΘΖ ἴσα τοῖς ἀπὸ τῶν ΑΓ, ΓΒ. τὰ ἄρα ΓΚ, ΘΖ, ΑΗ, ΗΕ ἴσα ἐστὶ τοῖς τε ἀπὸ τῶν ΑΓ, ΓΒ καὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ. ἀλλὰ τὰ ΓΚ, ΘΖ καὶ τὰ ΑΗ, ΗΕ ὅλον ἐστὶ τὸ ΑΕ, ὅ ἐστιν ἀπὸ τῆς ΑΒ τετράγωνον· τὸ ἄρα ἀπὸ τῆς ΑΒ τετράγωνον ἴσον ἐστὶ τοῖς τε ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνοις καὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ περιεχομένῳ ὀρθογωνίῳ· ὅπερ ἔδει δεῖξαι. 
2.(n)
Ad γʹ, prop. ζʹ Ἢ καὶ οὕτως. ἐπεζεύχθω ἡ ΕΚ. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΗΕ τῇ ΕΚ, κοινὴ δὲ ἡ ΖΕ, καὶ βάσις ἡ ΖΗ βάσει τῇ ΖΚ ἴση, γωνία ἄρα ἡ ὑπὸ ΗΕΖ γωνίᾳ τῇ ὑπὸ ΚΕΖ ἴση ἐστίν. ἀλλὰ ἡ ὑπὸ ΗΕΖ τῇ ὑπὸ ΘΕΖ ἐστιν ἴση· καὶ ἡ ὑπὸ ΘΕΖ ἄρα τῇ ὑπὸ ΚΕΖ ἐστιν ἴση, ἡ ἐλάττων τῇ μείζονι· ὅπερ ἐστὶν ἀδύνατον. 
3.(n)
Ad γʹ, prop. ηʹ Ἢ καὶ ἄλλως. ἐπεζεύχθω ἡ ΜΝ. ἐπεὶ ἴση ἐστὶν ἡ ΚΜ τῇ ΜΝ, κοινὴ δὲ ἡ ΜΔ, καὶ βάσις ἡ ΔΚ βάσει τῇ ΔΝ ἴση, γωνία ἄρα ἡ ὑπὸ ΚΜΔ γωνίᾳ τῇ ὑπὸ ΔΜΝ ἐστιν ἴση. ἀλλʼ ἡ ὑπὸ ΚΜΔ τῇ ὑπὸ ΒΜΔ ἐστιν ἴση· καὶ ἡ ὑπὸ ΒΜΔ ἄρα τῇ ὑπὸ ΝΜΔ ἐστιν ἴση, ἡ ἐλάττων τῇ μείζονι· ὅπερ ἐστὶν ἀδύνατον. 
4.(n)
Ad γʹ, prop. θʹ ΑΛΛΩΣ Κύκλου γὰρ τοῦ ΑΒΓ εἰλήφθω τι σημεῖον ἐντὸς τὸ Δ, ἀπὸ δὲ τοῦ Δ πρὸς τὸν ΑΒΓ κύκλον προσπιπτέτωσαν πλείους ἢ δύο ἴσαι εὐθεῖαι αἱ ΑΔ, ΔΒ, ΔΓ· λέγω, ὅτι τὸ ληφθὲν σημεῖον τὸ Δ κέντρον ἐστὶ τοῦ ΑΒΓ κύκλου. Μὴ γάρ, ἀλλʼ εἰ δυνατόν, ἔστω τὸ Ε, καὶ ἐπιζευχθεῖσα ἡ ΔΕ διήχθω ἐπὶ τὰ Ζ, Η σημεῖα. ἡ ΖΗ ἄρα διάμετρός ἐστι τοῦ ΑΒΓ κύκλου. ἐπεὶ οὖν κύκλου τοῦ ΑΒΓ ἐπὶ τῆς ΖΗ δια μέτρου εἴληπταί τι σημεῖον, ὃ μή ἐστι κέντρον τοῦ κύκλου, τὸ Δ, μεγίστη μὲν ἔσται ἡ ΔΗ, μείζων δὲ ἡ μὲν ΔΓ τῆς ΔΒ, ἡ δὲ ΔΒ τῆς ΔΑ. ἀλλὰ καὶ ἴση· ὅπερ ἐστὶν ἀδύνατον· οὐκ ἄρα τὸ Ε κέντρον ἐστὶ τοῦ ΑΒΓ κύκλου. ὁμοίως δὴ δείξομεν, ὅτι οὐδʼ ἄλλο τι πλὴν τοῦ Δ· τὸ Δ ἄρα σημεῖον κέντρον ἐστὶ τοῦ ΑΒΓ κύκλου· ὅπερ ἔδει δεῖξαι. 
5.(n)
Ad γʹ, prop. ιʹ ΑΛΛΩΣ Κύκλος γὰρ πάλιν ὁ ΑΒΓ κύκλον τὸν ΔΕΖ τεμνέτω κατὰ πλείονα σημεῖα ἢ δύο τὰ Β, Η, Θ, Ζ καὶ εἰλήφθω τὸ κέντρον τοῦ ΑΒΓ κύκλου τὸ Κ, καὶ ἐπεζεύχθωσαν αἱ ΚΒ, ΚΗ, ΚΖ. Ἐπεὶ οὖν κύκλου τοῦ ΔΕΖ εἴληπταί τι σημεῖον ἐντὸς τὸ Κ, καὶ ἀπὸ τοῦ Κ πρὸς τὸν ΔΕΖ κύκλον προσπεπτώκασι πλείους ἢ δύο ἴσαι εὐθεῖαι αἱ ΚΒ, ΚΖ, ΚΗ, τὸ Κ ἄρα σημεῖον κέντρον ἐστὶ τοῦ ΔΕΖ κύκλου. ἔστι δὲ καὶ τοῦ ΑΒΓ κύκλου κέντρον τὸ Κ· δύο ἄρα κύκλων τεμνόντων ἀλλήλους τὸ αὐτὸ κέντρον ἐστὶ τὸ Κ· ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα κύκλος κύκλον τέμνει κατὰ πλείονα σημεῖα ἢ δύο· ὅπερ ἔδει δεῖξαι. 
6.(n)
Ad γʹ, prop. ιαʹ Ἀλλὰ δὴ πιπτέτω ὡς ἡ ΗΖΓ, [καὶ] ἐκβεβλήσθω ἐπʼ εὐθείας ἡ ΓΖΗ ἐπὶ τὸ Θ σημεῖον, καὶ ἐπεζεύχθωσαν αἱ ΑΗ, ΑΖ. Ἐπεὶ οὖν αἱ ΑΗ, ΗΖ μείζους εἰσὶ τῆς ΑΖ, ἀλλὰ ἡ ΖΑ ἴση [ἐστὶ] τῇ ΖΓ, τουτέστι τῇ ΖΘ, κοινὴ ἀφῃρήσθω ἡ ΖΗ· λοιπὴ ἄρα ἡ ΑΗ λοιπῆς τῆς ΗΘ μείζων ἐστίν, τουτέστιν ἡ ΗΔ τῆς ΗΘ, ἡ ἐλάττων τῆς μείζονος· ὅπερ ἐστὶν ἀδύνατον. ὁμοίως, κἂν ἐκτὸς ᾖ τοῦ μικροῦ τὸ κέντρον τοῦ μείζονος κύκλου, δείξομεν [τὸ] ἄτοπον. 
7.(n)
Ad γʹ, prop. λαʹ ΑΛΛΩΣ ἡ ἀπόδειξις τοῦ ὀρθὴν εἶναι τὴν ὑπὸ ΒΑΓ. Ἐπεὶ διπλῆ ἐστιν ἡ ὑπὸ ΑΕΓ τῆς ὑπὸ ΒΑΕ· ἴση γὰρ δυσὶ ταῖς ἐντὸς καὶ ἀπεναντίον· ἔστι δὲ καὶ ἡ ὑπὸ ΑΕΒ διπλῆ τῆς ὑπὸ ΕΑΓ. αἱ ἄρα ὑπὸ ΑΕΒ, ΑΕΓ διπλασίονές εἰσι τῆς ὑπὸ ΒΑΓ. ἀλλʼ αἱ ὑπὸ ΑΕΒ, ΑΕΓ δυσὶν ὀρθαῖς ἴσαι εἰσίν· ἡ ἄρα ὑπὸ ΒΑΓ ὀρθή ἐστιν· ὅπερ ἔδει δεῖξαι. 

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Text: Perseus Digital Library, licensed CC BY-SA. Dictionary: Liddell, Scott & Jones (Perseus, CC BY-SA). Morphology: odyCy — machine parsing, corrected against the AGDT & PROIEL treebanks and Morpheus.