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Scholia: Scholia on Euclid’s Elements

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These are the anonymous Greek scholia on the Elements of Euclid, the Alexandrian mathematician of about 300 BC, gathered from the manuscripts of the Elements. They comment on the books of the Elements in turn, beginning with the division of geometry into plane and solid, and explain definitions, postulates and proofs, supply alternative demonstrations and discuss the history of the subject. Some of the notes on the first book draw on Proclus' commentary.

The Elements was the foundation of mathematical education for two thousand years, and these scholia record how it was taught. They also preserve notable historical claims, such as the attribution of the theory of proportion in the fifth book to Eudoxus and the association of the regular solids with the Pythagoreans and Theaetetus. They are a key source for the history of Greek mathematics.

About this edition. The Greek text comes from the Eulogikon open corpus of public-domain editions (standard reference: Heiberg, Euclidis Elementa V (Teubner 1888)). 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).

Ancient Greek · Science & Medicine
Scholia on Euclid's Elements
Scholia · Book 74 · every word parsed
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1
Ὅτι μὲν ἡ τῶν πλευρῶν ἰσότης ἐφʼ ἑκάστου τῶν
τριγώνων ἴσας ἀποτελεῖ τὰς ὑπὸ τούτων ὑποτεινομένας
γωνίας, ἡ δὲ τῶν γωνιῶν ἰσότης ὡσαύτως τὰς ὑποτεινού‐
σας αὐτὰς πλευρὰς ἴσας ἀποφαίνει, μεμαθήκαμεν διά τε
5.1
τοῦ θʹ καὶ ϛʹ θεωρήματος, ὅτι δὲ καὶ ταῖς ἀνισότησιν τῶν
5.2
πλευρῶν ἡ τῶν ὑποτεινομένων γωνιῶν ἀνισότης ἀκολουθεῖ
5.3
καὶ ἀνάπαλιν, διὰ τούτων διδασκόμεθα τῶν θεωρημάτων,
5.4
τοῦ τε ὀκτωκαιδεκάτου λέγω καὶ τοῦ ιθʹ. τὸ μὲν γὰρ
5.5
δείκνυσι τὴν μείζονα πλευρὰν ὑπὸ τὴν μείζονα γωνίαν, τὸ
10.1
δὲ ὑπὸ μείζονα γωνίαν τὴν μείζονα πλευράν, ἀντιστρέφοντα
10.2
μὲν ἀλλήλοις, ἐπὶ δὲ τῶν ἐναντίων πραγμάτων τὸ αὐτὸ
10.3
θεωροῦντα συμπτώματα τῷ εʹ καὶ ϛʹ θεωρήματι. φανερὸν
10.4
δέ, ὅτι τὴν μείζονα καὶ τὴν ἐλάσσονα πλευρὰν ἀνάλογον
10.5
ληψόμεθα καὶ διαιρήσομεν τὴν μεγίστην καὶ μέσην καὶ
15.1
ἐλαχίστην καὶ τὰς γωνίας ὡσαύτως ἐπὶ τῶν σκαληνῶν
15.2
τριγώνων, ἐπὶ δὲ τῶν ἰσοπλεύρων ἀρκέσει τὸ μεῖζον καὶ
15.3
τὸ ἔλασσον· μία γάρ ἐστι ταῖς δυσὶν ἄνισος. ἢ τὸ μεῖζον ἢ
15.4
τὸ ἔλαττον ὡς ἐπὶ τῶν ἰσοπλεύρων.
1·2
Ἀπορ[εῖται], ὅτι πόθεν δῆλον, ὅτι οὐκ ἔρχεται …..
2·2
τη… ἡ ΕΒ καὶ οὐκ ἔστι …….. εἰ γὰρ δυνατόν, ἐρ‐
1·3
χέσθω. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΕΒ τῇ ΕΑ, ἀλλὰ ἡ ΑΒ τῆς
2·3
ΑΕ ἐλάττων, καὶ ἡ ΒΕ ἄρα τῆς .. ἐλάττων. ἔστι δὲ καὶ
5.1·2
μείζων· ὅπερ ἀδύνατον. ὑπερπίπτει ἄρα τὸ Α σημεῖον·
5.2·2
ὅπερ ἔδει δεῖξαι.
1·4
Μέρη τὰ ΑΗ καὶ ΔΚ· ἔστιν οὖν λόγος τοῦ ΑΗ
2·4
πρὸς τὸ ΔΚ, ὃν ἔχει τὸ ΑΒ πρὸς τὸ ΔΕ, τουτέστι τοῦ
3·2
μέρους πρὸς τὸ μέρος, ὁ αὐτός ἐστι καὶ τοῦ ὅλου πρὸς τὸ
4·2
ὅλον. οὐκοῦν καὶ τὰ ὅλα τοῖς μέρεσι τὸν αὐτὸν ἔχει λόγον.
75
Δοκεῖ ἡ ἔκθεσις τοῦ παρόντος ιεʹ θεωρήματος μὴ
1·5
Ἔστω ὁ Α μονάδωνη, ὁ δὲ Β δ καὶ ὁ Γ β. ὁ β τέταρ‐
2·5
τόν ἐστι τοῦ η, ὁμώνυμος δὲ τῷ δ· ἀπὸ γὰρ τοῦ δ ὠνόμα‐
3·3
σται ὁ β τέταρτον τοῦ η. ἔστιν οὖν τὸ τρίτον καὶ τέταρτον
4·3
καὶ πέμπτον ὁμώνυμον τῷ τρία ἀριθμῷ καὶ τῷ δ καὶ τῷ ε.
1·6
Οἷον ὁ ιβ καὶ ὁ ιϛ μήκει σύμμετροί εἰσιν, ἀλλὰ καὶ
2·6
δυνάμει· τὰ γὰρ ἀπʼ αὐτῶν τετράγωνα τὰ ρμδ καὶ σνϛ τῷ
3·4
αὐτῷ χωρίῳ τῷ δ μετροῦνται.
75·2
Οἷον ὁ ε καὶ ὁ ιε δυνάμει σύμμετροί εἰσι· τὰ γὰρ ἀπʼ
1·7
ἴση ἄρα ἐστὶν p. 129, 14] ἔστω ἴσα τμήματα ἴσων
2·7
κύκλων τὰ ΑΒΓ, ΔΕΖ, καὶ ἀπειλήφθωσαν ἴσαι περι‐
3·5
φέρειαι αἱ ΑΒ, ΔΕ, καὶ κάθετοι ἀπὸ τῶν Β, Ε αἱ ΒΗ,
4·4
ΕΘ ἐπὶ τὰς ΑΓ, ΔΖ. λέγω, ὅτι ἴση ἐστὶν ἡ μὲν ΒΗ τῇ
5.1·3
ΕΘ, ἡ δὲ ΑΗ τῇ ΔΘ. ἐπεζεύχθωσαν αἱ ΑΒ, ΔΕ. καὶ
5.2·3
ἐπεὶ ἴσαι περιφέρειαι ἀπειλημμέναι εἰσὶν αἱ ΑΒ, ΔΕ, καὶ
5.3·2
λοιπαὶ ἄρα αἱ ΒΓ, ΕΖ ἴσαι ἀλλήλαις εἰσίν. ὥστε καὶ αἱ
5.4·2
ἐπʼ αὐτῶν βεβηκυῖαι γωνίαι ἴσαι ἀλλήλαις εἰσίν. ἴση ἄρα ἡ
5.5·2
ὑπὸ ΒΑΓ γωνία τῇ ὑπὸ ΕΔΖ. ἀλλὰ καὶ ὀρθαὶ αἱ Η, Θ·
10.1·2
δύο δὴ τρίγωνα τὰ ΑΒΗ, ΔΕΘ τὰς δύο γωνίας ταῖς δύο
10.2·2
γωνίαις ἴσας ἔχει ἑκατέραν ἑκατέρᾳ καὶ μίαν πλευρὰν
10.3·2
τὴν ΒΑ μιᾷ πλευρᾷ τῇ ΔΕ ἴσην τὴν ὑποτείνουσαν ὑπὸ
10.4·2
μίαν τῶν ἴσων γωνιῶν· πάντα ἄρα πᾶσιν ἴσα ἐστίν. ἴση
10.5·2
ἄρα ἡ μὲν ΑΗ τῇ ΔΘ, ἡ δὲ ΒΗ τῇ ΕΘ· ὅπερ ἔδει δεῖξαι.
1·8
ἡ ΝΒ ἄρα δωδεκαέδρου p. 183, 17] ἡ γὰρ τοῦ κύ‐
2·8
βου πλευρὰ δωδεκαέδρου ἦν, ἀλλὰ καὶ ἄκρον καὶ μέσον
3·6
λόγον ἐτέμνετο.
Click or tap any word for its parse and dictionary entry.
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.
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