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Pseudo-Hero of Alexandria: Geometrica

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The Geometrica is a large collection of problems in practical geometry transmitted under the name of Hero of Alexandria. It is a later compilation, largely Byzantine, drawing on Heronian material, and its editor Heiberg does not treat it as Hero's own composition. After a preface on the practical use of geometry, it gives definitions and many worked problems on finding the areas of triangles, quadrilaterals, circles and other figures.

The Geometrica shows how Greek mathematics was taught for everyday use in surveying, building and administration. Its many numerical examples preserve methods of calculation handed down through the Heronian tradition, and it is an important source for applied mathematics in late antiquity and Byzantium.

About this edition. The Greek text comes from the Eulogikon open corpus of public-domain editions (standard reference: Heiberg, Heronis opera IV (Teubner 1912)). 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 Geometrica is a later (largely Byzantine) compilation of Heronian material, edited by Heiberg as 'Heronis quae feruntur' and not regarded as Hero's own composition.

Ancient Greek · Science & Medicine
Geometrica
Pseudo-Hero of Alexandria · Book 1 · every word parsed
1.(t)
GEOMETRICA Ἡ γεωμετρία αὐτὴ καθʼ ἑαυτὴν εἰ κρίνοιτο, εἰς οὐδὲν ἂν νομισθείη συντελεῖν τῷ βίῳ. ὃν τρόπον καὶ τὰ τεκτονικά [και], εἰ τύχοι, ὄργανα αὐτὰ καθʼ ἑαυτὰ σκοπούμενα ἄχρηστʼ ἂν δόξειεν εἶναι, τὴν δὲ διʼ αὐτῶν γινομένην σκοπῶν χρῆσιν οὐ μικρὰν οὐδὲ τὴν τυχοῦσαν εὑρήσεις, τὸν αὐτὸν τρόπον καὶ γεωμετρία τῶν μὲν διʼ αὐτῆς περαιουμένων γυμνωθεῖσα μάταιος εὑρίσκεται, εἰς δὲ τὴν πρὸς ἀστρονομίαν εὐεργεσίαν αὐτῆς ἀφορῶντες ὑπερθαυμάζομεν τὸ πρᾶγμα· οἷον γὰρ ὄμμα τῆς ἀστρονομίας τυγχάνει. ἐπεὶ γὰρ ἡ ἀστρονομία περὶ μεγεθῶν τε καὶ ἀριθμῶν καὶ ἀναλογιῶν διαλαμβάνει· τό τε γὰρ μέγεθος ἡλίου καὶ σελήνης πολυπραγμονεῖ καὶ τὴν τῶν ἄστρων ποσότητα καὶ τὴν πρὸς ἄλληλα τούτων ἀναλογίαν· ἐν δὲ τοῖς ἐπιπέδοις περὶ δύο διαστάσεων ἡμᾶς διδάσκει, πλάτους τε καὶ μήκους, ὧν μὴ γνωσθεισῶν οὐκ ἄν ποτε συσταίη τὰ στερεά, ἅτινα ἐκ τριῶν διαστάσεων τυγχάνει ὄντα, πλάτους τε καὶ μήκους καὶ βάθους, γνῶσιν ἡμῖν πορίζουσα τοῦ μεγέθους τὰ μέγιστα συντελεῖ πρὸς ἀστρονομίαν· ἔτι μὴν καὶ ἡ ἀπὸ τοῦ ἀριθμοῦ γνῶσις ἡ ἐν τῷ ἑβδόμῳ καὶ ὀγδόῳ καὶ ἐνάτῳ εἰρημένη. Ἄλλως. Τὰς ἀρχὰς τῆς γεωμετρίας, ὅθεν τυγχάνουσιν, ἔστιν ἐκ φιλοσοφίας δεῖξαι. ἵνα μὴ ἐξαγώνιοι γενώμεθα, εὔλογόν ἐστι τὸν ὅρον αὐτῆς εἰπεῖν. ἔστιν οὖν ἡ γεωμετρία ἐπιστήμη σχημάτων καὶ μεγεθῶν καὶ τῶν περὶ ταῦτα παθῶν, ὁ δὲ σκοπὸς αὐτῆς περὶ τούτων διαλαμβάνειν, ὁ δὲ τρόπος τῆς διδασκαλίας ἐστὶ συνθετικός· ἀρξάμενος γὰρ ἀπὸ σημείου ἀδιαστάτου ὄντος διὰ μέσης γραμμῆς καὶ ἐπιφανείας καταντᾷ ἐπὶ τὸ στερεόν. τὸ δὲ χρήσιμον αὐτῆς ἄντικρυς εἰς φιλοσοφίαν συντελεῖ· τοῦτο γὰρ καὶ τῷ θείῳ Πλάτωνι δοκεῖ, ἔνθα φησί· ταῦτα τὰ μαθήματα εἴτε χαλεπὰ εἴτε ῥᾴδια, ταύτῃ ἰτέον. ἐπιγέγραπται δὲ στοιχεῖα, διότι ὁ μὴ διὰ τούτων πρότερον ἀχθεὶς οὐχ οἷός τέ ἐστι συνιέναι τι τῶν γεωμετρικῶν θεωρημάτων. ἡ δὲ γεωμετρία ἐξ ἀφαιρέσεως τὴν διδασκαλίαν ἐποιήσατο· λαβοῦσα γὰρ φυσικὸν σῶμα, ὅ ἐστι τριχῆ διαστατὸν μετὰ ἀντιτυπίας, καὶ χωρίσασα τούτου τὴν ἀντιτυπίαν ἐποιήσατο τὸ μαθηματικὸν σῶμα, ὅ ἐστι στερεόν, καὶ ἀφαιροῦσα κατήντησεν ἐπὶ τὸ σημεῖον. Ἥρωνος ἀρχὴ τῶν γεωμετρουμένων. [40]
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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.
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