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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 156 · every word parsed
1
Δείκνυται τοῦτο τὸ τῆς νύμφης θεώρημα καὶ ἀριθ‐
μητικῶς οὕτως· Πλάτων τῶν ἀνισοσκελῶν ὡς δῆλον μό‐
νον ταῦτα καὶ ῥητὴν ἔχουσι τὴν πλευράν, καί ἐστιν ἐπὶ τῶν
ἀρτίων ἀριθμῶν δεικνύμενον οὕτως· λαμβάνει τὸ ἥμισυ
5.1
τοῦ προκειμένου αὐτῷ ἀριθμοῦ καὶ πολυπλασιάζει πρῶτον
5.2
ἐφʼ ἑαυτό· εἶτα ἀφαιρεῖται τούτου τὸ ἓν καὶ τὸν λοιπὸν
5.3
ἀριθμὸν τὴν ἑτέραν εἶναι λέγει πλευράν. εἶτα πάλιν προσ‐
5.4
τίθησι τῷ ἀπὸ τῆς ἡμισείας τετραγώνῳ μονάδα καὶ ταύ‐
5.5
την εἶναι τὴν ὑποτείνουσαν. ἔστω γὰρ ὡς ἐν ὑποδείγματι
10.1
τρίγωνον ἰσοσκελὲς ὀρθογώνιον τὴν μίαν ἔχον πλευρὰν δ
10.2
εἴτε σπιθαμῶν εἴτε ποδῶν,
10.3
εἴτε ὁπωσδήποτέ τις αὐτὴν
10.4
ὑποθῆται. ζητεῖται οὖν ἡ λοιπὴ
10.5
πλευρὰ καὶ ἡ ὑποτείνουσα, καὶ
15.1
λέγομεν οὕτως· δὶς δύο τέσσα‐
15.2
ρες· τοῦτο γὰρ ἦν τὸ ἥμισυ
15.3
τοῦ προκειμένου ἡμῖν ἀριθμοῦ.
15.4
εἶτα ἀφαιροῦμεν τούτου τὸ ἕν,
15.5
καὶ τοῦτό ἐστιν ἡ πλευρὰ ἤγουν
20.1
ὁ τρία. προστίθεμεν δὲ καὶ εἰς
20.2
τὸ ἀπὸ τῆς ἡμισείας τετράγωνον μονάδα, ὅπερ ἦν ὁ δ,
20.3
καί ἐστιν ἡ ὑποτείνουσα ε. δείκνυται οὖν τὸ θεώρημα
20.4
οὕτως ὡς ἐν τῷ διαγράμματι. Πυθαγόρας ἀπὸ τῶν πε‐
20.5
ρισσῶν οὕτως πολυπλασιάζει πρῶτον ὅλον τὸν προκεί‐
25.1
μενον ἀριθμόν, καὶ ἀφαιρεῖται τούτου μονάδα, καὶ τὸ
25.2
τοῦ ἀριθμοῦ τούτου ἥμισύ ἐστιν ἡ ἑτέρα πλευρά. εἶτα
25.3
προστίθησι τῷ ἡμίσει μονάδα, καί ἐστιν ἡ ὑποτείνουσα.
25.4
ἔστω γὰρ τρίγωνον ἀνισοσκελὲς ἔχον τὴν μίαν τῶν πλευ‐
25.5
ρῶν γ. ζητεῖται οὖν ἡ ἑτέρα πλευρὰ καὶ ἡ ὑποτείνουσα, καὶ
30.1
εὑρίσκει αὐτὴν οὕτως· πολυπλασιάζει τὸν ἀριθμὸν ὅλον
30.2
ἐφʼ ἑαυτὸν οὕτως· τρὶς τὰ τρία θ. εἶτα ἀφαιρεθείσης μο‐
1·2
νάδος ἐναπελείφθη ὁ ὀκτὼ ἀριθμός, καὶ τούτου τὸ ἥμισύ
2·2
ἐστιν ἡ ἑτέρα πλευρά. προστίθησι δὲ καὶ τῷ ἡμίσει τούτῳ
3·2
μονάδα, καὶ τοῦτό ἐστιν ἡ ὑποτείνουσα ἤτοι ε. δείκνυται τὸ
35
θεώρημα οὕτως ὡς ἐν τῷ διαγράμματι.
1·3
Ἔστι δὲ τὸ ὑπὸ τῶν ε, ϛ κδ ια περιεχόμενον ὀρθο‐
2·3
γώνιον λβ ο νε, καὶ ἡ δυναμένη αὐτό ἐστιν ἡ ε λθ κγ, ἥτις
3·3
ἄλογος οὖσα μέση καλεῖται.
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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