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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 26 · every word parsed
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1
Τρίτον πρόβλημα τοῦτο δεδομένας μὲν ἔχον δύο
εὐθείας κατὰ τὸ μέγεθος ἀνίσους, προστάττον δὲ ἀφελεῖν
ἀπὸ τῆς μείζονος ἴσην τῇ ἐλάσσονι. ἔστι δὲ καὶ τοῦτο πολύ‐
πτωτον· αἱ γὰρ δοθεῖσαι ἄνισοι εὐθεῖαι ἢ διεστᾶσιν ἀπʼ
5.1
ἀλλήλων ὡς παρὰ τῷ στοιχειωτῇ ἢ καθʼ ἓν πέρας συνάπ‐
5.2
τονται ἢ τέμνουσιν ἀλλήλας ἢ ἡ ἑτέρα κατὰ τὸ πέρας ἑαυ‐
5.3
τῆς τέμνει τὴν ἑτέραν καὶ τοῦτο διχῶς· ἢ γὰρ ἡ μείζων
5.4
τὴν ἐλάσσω ἢ ἡ ἐλάσσων τὴν μείζονα. ἀλλʼ εἰ μὲν καθʼ
5.5
ἓν συνάπτοιντο πέρας, δήλη ἡ ἀπόδειξις· τῷ γὰρ κοινῷ
10.1
πέρατι κέντρῳ χρησάμενος, διαστήματι δὲ τῇ ἐλάσσονι
10.2
τῶν εὐθειῶν γράψεις κύκλον καὶ τὴν μείζονα τεμεῖς καὶ
10.3
ἀφαιρήσεις ἴσην τῇ ἐλάσσονι. ὅσον γὰρ τῆς μείζονος ὁ
10.4
κύκλος ἐντὸς ἀποτέμνεται, τοσοῦτον ἴσον ἔσται τῇ ἐλάσ‐
10.5
σονι. εἰ δὲ ἡ ἑτέρα τέμνοι τὴν ἑτέραν κατὰ τὸ ἑαυτῆς πέρας,
15
ἤτοι ἡ μείζων τὴν ἐλάσσονα τεμεῖ ἢ ἀνάπαλιν, καὶ εἰ
1·2
ἀλλήλας τέμνοιεν, ἢ εἰς ἴσα τέμνονται ὑπʼ ἀλλήλων ἢ εἰς
2·2
ἄνισα ἢ ἡ μὲν εἰς ἴσα, ἡ δὲ εἰς ἄνισα, καὶ τοῦτο διχῶς.
3·2
ταῦτα γὰρ πάντα ποικιλίαν ἡμῖν καὶ θαυμαστὴν παρέχεται
4·2
γυμνασίαν.
1·3
Ἔστω ἡ εὐθεῖα μονάδων ιβ. τετμήσθω εἰς η καὶ δ.
2·3
πεπολυπλασιάσθω ἡ ὅλη ἤγουν τὰ ιβ ἐπὶ τὸ ἓν μέρος ἤγουν
3·3
τὰ δ. γίνονται μη. πεπολυπλασιάσθω καὶ τὸ ἓν τμῆμα ἐπὶ
4·3
τὸ ἕτερον τμῆμα, τουτέστι τὰ η ἐπὶ τὰ δ· γίνονται λβ. καὶ
5
τὸ ἀπὸ τοῦ δ τετράγωνον ιϛ· ὁμοῦ μη.
1·4
Ἅμα γὰρ ἐφʼ ἑκάτερα μέρη δύνανται συσταθῆναι,
2·4
τὸ μὲν ἓν ἐπὶ τοῦ ἑνὸς μέρους, τὸ δὲ ἕτερον ἐπὶ τοῦ ἑτέρου.
1·5
Ἰστέον, ὅτι οὐ δεῖ καὶ τὰ δ μεγέθη ἐξ ἀνάγκης
2·5
ἰσάκις πολυπλασιάζεσθαι· τοῦτο γὰρ ἐνέφηνεν εἰπὼν καθʼ
3·4
ὁποιονοῦν πολλαπλασιασμόν· ἀλλὰ μόνον τὸ πρῶτον καὶ
4·4
τρίτον ἰσάκις καὶ πάλιν τὸ βʹ καὶ τὸ δʹ ἰσάκις· ὥστε εἰ τὸ
5.1·2
μὲν αʹ καὶ γʹ φέρε εἰπεῖν διπλασιασθῶσι, τὸ δὲ βʹ καὶ δʹ
5.2·2
τριπλασιασθῶσιν, οὐδὲν γίνεται ἄτοπον· ἐκ γὰρ τοῦ δια‐
5.3·2
φόρως ἔχειν, ἃ δεῖ ἅμα πολυπλασιάζειν, τό τε αʹ ὁμοῦ καὶ
5.4·2
τὸ γʹ καὶ τὸ βʹ καὶ δʹ, συμβαίνει καὶ τὸ ἅμα τοὺς πολλα‐
5.5·2
πλασιασμοὺς τοῦ αʹ καὶ γʹ πρὸς τοὺς πολλαπλασιασμοὺς ἅμα
10.1·2
τοῦ βʹ καὶ δʹ ἢ ὑπεροχὴν ἔχειν ἢ ἰσότητα ἢ ἔλλειψιν. τοῦτο
10.2·2
δὲ δῆλον καὶ ἀπὸ τοῦ μετὰ τοῦτον ὅρου τοῦ λέγοντος·
10.3·2
ὅταν δὲ τῶν ἰσάκις πολλαπλασίων.
1·6
Ἔστω ἡ μὲν ΑΒ ἀριθμῶν ιβ, ἡ δὲ ΓΔ η, καὶ πάλιν
2·6
ἡ μὲν Ε ἀριθμῶν ϛ, ἡ δὲ Ζ ἀριθμῶν δ, ὡς τὰ ιβ πρὸς τὰ η,
3·5
οὕτως τὰ ϛ πρὸς τὰ δ. καὶ τὸ ὑπὸ τῶν ιβ καὶ δ περιεχόμε‐
4·5
νον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν η καὶ ϛ περιεχομέ‐
5·2
νῳ ὀρθογωνίῳ.
1·7
ἔστω ὁ Α γ, ὁ δὲ ΒΓ θ, ὁ δὲ Δ ϛ, ὁ δὲ ΕΖ ιη. τὰ
2·7
δὴ γ τοῦ θ γʹ εἰσὶ μέρος καὶ τὰ ϛ τοῦ ιη, καὶ συναμφότερα
3·6
ὁ ϛ καὶ γ ἤτοι ὁ θ συναμφοτέρων τοῦ ιη καὶ θ ἤτοι τοῦ κζ
4·6
γʹ εἰσίν.
1·8
Διὰ τὸν ὅρον τοῦ εʹ τὸν λέγοντα· ὅταν δὲ τρία με‐
2·8
γέθη ἀνάλογον ᾖ, τὸ πρῶτον πρὸς τὸ τρίτον διπλασίονα
3·7
λόγον ἔχειν λέγεται ἤπερ πρὸς τὸ βʹ.
1·9
Οἷον ἐδόθησαν ἀριθμοὶ ὁ η καὶ ὁ κζ. σκόπει, ἐὰν
2·9
ὦσι πρῶτοι πρὸς ἀλλήλους, ὥσπερ καί εἰσι. καὶ ἐπείπερ
3·8
εἰσίν, ἕτερος ἀνάλογον οὐχ εὑρίσκεται. ἀλλὰ μὴν ἐδόθη‐
4·7
σαν ἀριθμοὶ ὁ η καὶ ὁ ιβ. οὗτοι οὐκ εἰσὶ πρῶτοι πρὸς ἀλλή‐
5.1·3
λους· κοινὸν γὰρ αὐτοῖς ἐστι μέτρον ὁ δ. βούλει οὖν μαθεῖν,
5.2·3
εἰ ἕξει ὁ ιβ ἕτερον ἀνάλογον; πολλαπλασίασον τὸν ιβ· καὶ
5.3·3
ἀναβιβάζεται ρμδ. σκόπει οὖν καί, ἐὰν δύνῃ εὑρεῖν πλευρὰν
5.4·3
ἐν αὐτῷ τὸν η. εὑρήσεις καὶ τοῦ ιβ ἀνάλογον. ἔστιν οὖν·
5.5·3
ὀκτάκις γὰρ ιη ρμδ.
1·10
Ὅτι οὐκ ἔστιν ἐλάχιστον μέγεθος, ὡς οἱ Δημοκρί‐
2·10
τειοί φασιν, καὶ διὰ τούτου τοῦ θεωρήματος δείκνυται, εἴ
3·9
γε παντὸς τοῦ ἐκκειμένου μεγέθους δυνατὸν ἔλαττον λαβεῖν.
1·11
Ἀντιστρόφιον· ἐὰν ᾖ παράλληλα ἐπίπεδα, ἡ τῷ ἑνὶ
2·11
ἐπιπέδῳ πρὸς ὀρθὰς εὐθεῖα καὶ τῷ λοιπῷ πρὸς ὀρθάς
3·10
ἐστιν.
1·12
εἰς τοὺς αὐτοὺς λόγους p. 92, 4] ἐὰν γὰρ δύο εὐθεῖαι
2·12
ὑπὸ παραλλήλων ἐπιπέδων τέμνωνται, εἰς τοὺς αὐτοὺς
3·11
λόγους τμηθήσονται διὰ τὸ ιζʹ τοῦ ιαʹ.
Ἔστω λόγου ἕνεκεν τὸ Χ στερεόν τινων ϛ, ἡ δὲ
1·13
Ἔδειξε τοῦτο, ἐν οἷς ἄνωθεν ἔλεγεν ἴσην εἶναι τὴν
2·13
ὑπὸ ΒΑΓ ἤτοι τὴν ὑπὸ ΒΑΘ τῇ ὑπὸ ΑΒΕ ἤτοι τῇ ὑπὸ
3·12
ΑΒΘ.
1·14
καί εἰσι δώδεκα p. 14, 5] ἐπεὶ γὰρ ἡ μὲν ΒΘ
2·14
πενταπλασίων τῆς ΘΓ, ἡ δὲ ΒΓ τῆς ΘΓ ἑξαπλασίων,
3·13
ἑξάκις ἡ ΒΘ πεντάκις τῇ ΒΓ ἴση ἔσται, καὶ ἀναλόγως
4·8
δωδεκάκις ἡ ΒΘ δεκάκις τῇ ΒΓ ἐστιν ἴση.
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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