Ancient Library
The Greek and Latin classics — every word parsed, every form defined

Scholia: Scholia on Euclid’s Elements

Library › Greek: Science & Medicine › Scholia on Euclid's Elements

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 80 · every word parsed
Book123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159100160161162163164165166167168169171172173174175176177178179180181182183185186187188189191192193194195196197198199200201202204205206207208209210211212213214215216217218219220221222223224225227228229230231232233234235236237238239240241242244245246247248249250251252253254256257258259260264265266267269270271272273275276277278279281282284285290291292293294295296297298299301302303304305306307308309310311312313315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355357359361362363364365367368370371376377378385391392393394395397399400401402403405406407408410411412413414416417418420421423425428429431432438440448450451454456457458459461462463464465466467469470472474476478480481
1
Αἱ γὰρ ΑΒ, ΒΓ, ΑΓ αἱ τρεῖς ἤτοι ἴσαι ἀλλήλαις
εἰσὶν ἢ οὐ. εἰ μὲν ἴσαι ἀλλήλαις εἰσί, φανερόν, ὅτι δύο
ὁποιαιοῦν τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανό‐
μεναι. εἰ δὲ οὐ, ἔστι τις ἐν αὐταῖς μεγίστη. ἔστω ἡ ΒΓ.
5.1
ὅτι μὲν οὖν αἱ ΑΒ, ΒΓ τῆς ΑΓ μείζονές εἰσι, φανερόν·
5.2
καὶ πάλιν ὅτι αἱ ΑΓ, ΓΒ τῆς ΑΒ, καὶ τοῦτο δῆλον.
5.3
δεικτέον δή, ὅτι καὶ αἱ ΒΑ, ΑΓ τῆς ΒΓ μείζονές εἰσιν.
5.4
ἐκβεβλήσθω γὰρ ἡ ΒΑ ἐπὶ τὸ Δ σημεῖον.
1·2
Ποιοῦσι δὲ τὰ αὐτὰ πάντες οἱ ἰσάκις αὐτῶν πολλα‐
2·2
πλάσιοι.
1·3
Λόγισαι τὸ μὲν ΑΒ μέγεθος ιβ καὶ δίελε τὸ μὲν
2·3
ΑΕ εἰς η, τὸ δὲ ΕΒ εἰς δ, τὸ δὲ ΓΔ λόγισαι θ εἶναι καὶ
3·2
δίελε τὸ μὲν ΓΖ εἰς ἕξ, τὸ δὲ ΖΔ εἰς γ. ὅλον οὖν τὸ ΑΒ
4·2
ἤτοι ὁ ιβ πρὸς τὸν δ τριπλάσιος, καὶ ὁ ΓΔ ἤτοι ὁ θ πρὸς
5.1·2
τὸν ΖΔ ἤτοι τὸν γ τριπλάσιος. διπλάσιος δὲ καὶ ὁ ΑΕ
5.2·2
ἤτοι ὁ ὀκτὼ πρὸς τὸν ΕΒ τὸν δ, ὥσπερ καὶ ὁ ΓΖ ἤτοι ὁ
5.3·2
ϛ πρὸς τὸν ΖΔ τὸν γ.
1·4
Πολλῶν ἀριθμῶν ὄντων καὶ ἐχόντων τὰ αὐτὰ μέρη,
2·4
οἷον εἰ τύχοι δίδοσθαι 𐅵ʹ γʹ δʹ εʹ, εὑρεῖν τὸν ἐλάχιστον
3·3
ἀριθμὸν πάντων τῶν τὰ αὐτὰ μέρη ἐχόντων αὐτοῖς.
ὅπερ ἐστὶν ἀδύνατον p. 148, 19] κατεσκευάσθη γὰρ
81·2
ὁ Η ὑπὸ τῶν Δ, Ε, Ζ ἐλάχιστος μετρούμενος ἀριθμός.
1·5
Ἰστέον, ὅτι, ὅταν αἱ τῶν τετραγώνων πλευραὶ λόγον
2·5
ἔχωσι πρὸς ἀλλήλους, ὃν τετράγωνος ἀριθμὸς πρὸς τετρά‐
3·4
γωνον ἀριθμόν, τουτέστιν τὸν μήκει διπλασίονα, τότε καὶ
4·3
ὁ τετράγωνος πρὸς τὸν τετράγωνον τετραπλάσιός ἐστιν,
5.1·3
ὡς ἐπὶ τοῦ δ καὶ ιϛ καὶ θ καὶ λϛ. πλευρὰ γὰρ τοῦ δ ὁ β, τοῦ
5.2·3
δὲ ιϛ ὁ δ καὶ τοῦ θ ὁ γ, τοῦ λϛ ὁ ϛ. εἰσὶν οὖν αἱ τοιαῦται
5.3·3
πλευραὶ ἐν διπλασίονι λόγῳ, τουτέστιν ἐν τετραγώνου
5.4·2
ἀριθμοῦ πρὸς τετράγωνον ἀριθμὸν λόγῳ, καὶ διὰ τοῦτο τὰ
5.5
ἀπʼ αὐτῶν γεγονότα τετράγωνα χωρία ἐν τετραπλασίονι
10.1
λόγῳ θεωροῦνται κατὰ τὸ ἀξίωμα τὸ λέγον, ὅτι τὰ μήκει
10.2
διπλάσια δυνάμει εἰσὶν τετραπλάσια. ἂν δὲ ἡ πλευρὰ πρὸς
10.3
τὴν πλευρὰν ἔχῃ μέν τινα λόγον, ἡμιόλιον τυχὸν ἢ ἐπίτριτον
10.4
ἢ ἄλλον τινὰ τῶν ἐπιμορίων ἢ τῶν ἐπιμερῶν, τὰ μὲν ἀπʼ
10.5
αὐτῶν γεγονότα τετράγωνα λόγον ἔχουσι πρὸς ἄλληλα, ὃν
15.1
τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐ μὴν δὲ
15.2
τὸν τετραπλασίονα, ὡς ἐπὶ τοῦ θ καὶ τοῦ δ, ὧν αἱ πλευραὶ
15.3
λόγον μὲν ἔχουσιν, ὃν ἀριθμὸς πρὸς ἀριθμόν, οὐχ ὃν τετρά‐
1·6
γωνος πρὸς τετράγωνον· τὰ γὰρ δύο καὶ τρία, ἅπερ εἰσὶ
2·6
πλευραὶ τοῦ δ καὶ τοῦ θ, τὸν ἡμιόλιον ἔχουσι λόγον· διὸ
20.1
καὶ οὐ δύναται εἶναι ὁ θ τοῦ δ τετραπλάσιος, ὡς ὁ ιϛ τοῦ
20.2
δ καὶ ὁ λϛ τοῦ θ.
1·7
τὸ ΚΒΟΣ ἄρα τετράπλευρον p. 129, 24sq.] τετρά‐
2·7
πλευρόν ἐστιν, οὐ μὴν καὶ παραλληλόγραμμον· ὥστε οὐκ
3·5
ἀνάγκη τὴν ΣΟ ἴσην εἶναι τῇ ΚΒ.
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.
Scroll to Top