Figure 92. A diagram of the geometrical proof offered in the Greek fragment of Heron's Catoptrica. Taken from the critical edition Herons von Alexandria Mechanik und Katoptrik (Teuber, 1900), pg. 371. Schmidt himself notes his printed diagram is taken from Aldina (1551).
This is likely a reference to the imprint belonging to the Venetian humanist Aldo Manuzio, as he published many Greek and Latin editions. However, it is not clear in which volume one would find this diagram, particularly because the imprint only published between 1495 and Manuzio's death in 1515, which is a few years short of 1551 (unless Schmidt is guilty of a numerical typo).
As is my habit to release posts early and return to them often, the reader should be warned this one is still a "work in progress".
Introduction
An introduction. Heron of Alexandria was perhaps one of the most ingenious inventors of Western antiquity, if not all history. Living in the first century, his catalogue of inventions includes a prototype windmill, mechanical theater technology, a water fountain, syringes, and the first coin-based vending machine. More than this, Heron had detailed design notes for the first steam engine, a technology which would not resurface for another 1500 years. It is remarkable a man living in the same century as Augustus and Jesus would designed such technologies.
Heron is a remarkable demonstration of the fact that historical anachronism can be found in real life as well.
Many of Heron's works do not survive to this day, as is the case for much literature from antiquity. Some works such as the Mechanica come to us through Arabic translation. Not all of his extant works have been translated into English either. It was only in 2019, that a PhD thesis translated his Automata into English.
My interest in him comes from his original contributions to classical optics and thus the history of science.
His Catoptrica, one of the most significant works in the history of optics, has not yet made it into English. I will attempt to remedy this by starting with the single extant Greek fragment, and then proceeding through a translation of the entire Latin work into English in a subsequent post.
Comments and feedback are always appreciated, so long as they do not steer too deeply into the pedantic.
Let us begin.
The Text
The text is taken W. Schmidt's critical edition Herons von Alexandria Mechanik und Katoptrik published by Teuber in 1900, pages 368-373. I referred to the German translation in a few spots but will offer more analysis of it and textual variants in the forthcoming Commentary section.
ἐπειδὴ γὰρ τοῦτο ὡμολογημένον ἐστὶ παρὰ πᾶσιν, ὃτι οὐδὲν μάτην ἐργάζεται ἡ φύσις οὐδὲ ματαιοπονεῖ, ἐὰν μὴ δώσωμεν πρὸς ἲσας γωνὶας γὶνεσθαι τὴν ἀνάκλασιν, πρὸς ἀνὶσους ματαιοπονεῖ ἡ φύσις, καὶ ἀντὶ τοῦ διὰ βραχείας περιόδου φθάσαι τὸ ὁρώμενον τὴν ὂψιν, διὰ μακρᾶς περιόδου τοῦτο φανήσεται τὴν ὂψιν, διὰ μακρᾶς περιόδου τοῦτο φανήσεται καταλαμβάνουσα. εὑρεθήσονται γὰρ αἱ τὰς ἀνίσους γωνίας περιέχουσαι εὐθεῖαι, αἳτινες ἀπὸ τῆς ἀνὶσους γωνίας περιέχουσαι εὐθεῖαι, αἳτινες ἀπὸ τῆς ὂψεως [περιέχουσαι] φερομένας πρὸς τὸ κάτοπτρον κἀκεῖθεν πρὸς τὸ δρώμενον, μείζοντες οὖσαι τῶν τὰς ἲσας γωνίας περιεχουσῶν εὐθειῶν. καὶ ὃτι τοῦτο ἀληθές, δῆλον ἐντεῦθεν.
Ὑποκείσθω γὰρ τὸ κάτοπτρον εὐθεῖά τις ἡ ΑΒ, καὶ ἒστω τὸ μὲν ὁρῶν Γ, τὸ δ' ὁρώμενον τὸ Δ, τὸ δὲ Ε σημεῖον τοῦ κατόπτρου, ᾧ προςπίπτουσα ἡ ὂψις ἀνακλᾶται πρὸς τὸ ὁρώμενον, ἒστω, καὶ ἐπεζεύχθω ἡ ΓΕ, ΕΔ. λέγω ὃτι ἡ ὑπὸ ΑΕΓ γωνία ἲση ἐστὶ τῇ ὑπὸ ΔΕΒ. εἰ γὰρ μὴ ἒστιν ἲση, ἒστω ἓτερον σημεῑον τοῦ κατόπτρου, ἐν ᾧ προσπίπτουσα, ἡ ὂψις πρὸς ἀνίσους γωνίας περιέχουσιν ὑποκειμένης τῆς ΑΒ εὐθείας, μείζονές εἰσι τῶν ΓΕ, ΕΔ εὐθειῶν, αἳτινες τὰς ἲσας γωνἰας περιέρχουσι μετὰ τῆς ΑΒ. ἢχθω γὰρ κάθετος ἀπὸ τοῦ Δ ἐπὶ τὴν ΑΒ κατὰ τὸ Η σημεῖον (213) | καὶ ἐκβεβλήσθω ἐπ' εὐθείας ὡς ἐπὶ τὸ Θ. φανερὸν δὴ ὃτι αἱ πρὸς τῷ Η γωνίαι ἲσαι εἰσίν. ὀρθαὶ γάρ εἰσι. καὶ ἒστω ἡ ΔΗ τῇ ΗΘ ἲση, καὶ ἐπεζεύχθω ἡ ΘΖ καὶ ἡ ΘΕ. αὓτη μὲν ἡ κατασκευή. ἐπεὶ οὖν ἲση ἒστὶν ἡ ΔΗ τῇ ΗΘ, ἀλλὰ καὶ ἡ ὑπὸ ΔΗΕ γωνία τῇ ὑπὸ ΘΗΕ γωνίᾳ ἲση ἐστί, κοινὴ δὲ πλευρὰ τῶν δύο τριγώνων ἡ ΗΕ, καὶ βάσις ἡ ΘΕ βἀσει τῇ ΕΔ ἲση ἐστί, καὶ τὸ ΗΘΕ τρίγωνον τῷ ΔΗΕ τριγώνῳ ἲσον ἐστί, καὶ <αἱ> λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις εἰσὶν ἲσαι, ὑφ' ἃς αἱ ἲσαι πλευραὶ ὑποτείνουσιν. ἲση ἂρα ἡ ΘΕ τῇ ΕΔ. πάλιν ἐπειδὴ τῇ ΗΘ ἲση ἐστίν ἡ ΗΔ καὶ γωνία ἡ ὑπὸ ΔΗΖ γωνίᾳ τῇ ὑπὸ ΘΗΖ ἲση ἐστί, κοινὴ δὲ ἡ ΗΖ τῶν δύο τριγώνων τῶν ΔΗΖ καὶ ΘΗΖ, καὶ βὰσις ἂρα ἡ ΘΖ βἀσει τῇ ΖΔ ἲση ἒστί, καὶ τὸ ΖΗΔ τρίγωνον τῷ ΘΗΖ τριγὠνῳ ἲσον ἐστίν. ἲση ἂρα ἐστὶν ἡ ΘΖ τῇ ΖΔ. καὶ ἐπεὶ ἲση ἐστὶν ἡ ΘΕ τῇ ΕΔ, κοινὴ προσκείσθω ἡ ΕΓ. δύο ἂρα αἱ ΓΕ, ΕΔ δυσὶ ταῖς ΓΕ, ΕΘ ἲσαι εἰσίν. ὃλη ἂρα ἡ ΓΘ δυσὶ ταῖς ΓΕ, ΕΔ ἲση ἐστί. καὶ ἐπεὶ παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, τριγώνου ἂρα τοῦ ΘΖΓ αἱ δύο πλευραὶ αἱ ΘΖ, ΖΓ μιᾶς τῆς ΓΘ μείζονές εἰσιν. ἀλλ'ἡ ΓΘ ἲση ἐστὶ ταῖς ΓΕ, ΕΔ. αἱ ΘΖ, ΖΓ ἂρα μείζονές εἰσι τῶν ΓΕ, ΕΔ. ἀλλ' ἡ ΘΖ τῇ ΖΔ ἐστὶν ἲση. αἱ ΖΓ, ΖΔ ἂρα τῶν ΓΕ, ΕΔ μείζονές εἰσι. καὶ εἰσιν αἱ ΓΖ, ΖΔ αἱ τὰς ἀνίσους γωνιάς περιέρχουσαι· αἱ ἂρα τὰς ἀνίσους γωνίας περιέχουσαι μείζονές εἰσι τῶν τὰς ἲσας γωνίας περιεχουσῶν.
The Translation
(This is a first draft of the translation. The English rendering will be tidied on the next passthrough.)
For since this is agreed upon by all, that in no way is nature employed fruitlessly nor does it do anything in vain, if we do not grant that the reflection occurs at equal angles, nature does work in vain at unequal ones, and instead of the ray striking the object of vision by a short line, it will appear to the vision to go by a long line, this detection appears through a long line. For it will be found that the [rays] of the straight lines contain unequal angles, whichever ones the straight lines contain from the unequal angles, whichever ones are bearing [them] to the mirror [and] from there to the object, being larger than the straight lines containing the equal angles. And seeing that this is true, the following is clear.
Let the mirror be a certain straight line (AB), and on the one hand let the viewer be Γ, and on the other hand Δ as the object of view, while E is the point of the mirror, which upon striking, the vision is bent back to the object of view, and let ΓΕ connect ΕΔ. I say that the angle under ΑΕΓ is equal to ΔΕΒ. For if it is not equal, let it then be the other point of the mirror, upon striking it, the vision contains the stretching of the line AB toward unequal angles, they are larger than the straight lines ΓΕ and ΕΔ, whichever ones contain the equal angles beyond AB. For let the perpendicular be drawn from Δ on AB according to the point Η.
(213) And let it fall on the straight line just as on Θ. It is no doubt apparent that these angles are equal to H. For they are straight. And let ΔΗ be equal to ΗΘ, and let ΘΖ and ΘΕ be connected. This is the reasoning. Seeing that is the case therefore ΔΗ is equal to ΗΘ, but also the angle ΔΗΕ is equal to the angle ΘΗΕ, and the line of the triangle ΗΕ is shared between the two triangles, and the base ΘΕ is equal to the base ΕΔ, and the triangle ΗΘΕ is equal to the triangle ΔΗΕ, and these remaining angles are equal to the other remaining angles, the equal lines extend to those. Therefore ΘΕ is equal to ΕΔ. Again therefore ΗΘ is equal to ΗΔ and the angle along ΔΗΖ is equal to ΘΗΖ, and ΗΖ is shared between the two triangles ΔΗΖ and ΘΗΖ, and therefore ΘΖ is equal to ΖΔ. And consequently ΘΕ is equal to ΕΔ, let (the shared line) ΕΓ be attached. Therefore the two lines ΓΕ and ΕΔ are equal to the two lines ΓΕ and ΕΘ. Therefore the whole ΓΘ is equal to the two ΓΕ and ΕΔ. And therefore the two larger sides of the remaining one of every triangle are shared on every side, consequently the two sides ΘΖ and ΖΓ of triangle ΘΖΓ are larger than the one [side] ΓΘ. But ΓΘ is equal to ΓΕ and ΕΔ. Therefore the two sides ΘΖ and ΖΓ are larger than ΓΕ and ΕΔ. But ΘΖ is equal to ΖΔ. Consequently, ΖΓ and ΖΔ are larger than ΓΕ and ΕΔ. Also, ΓΖ and ΖΔ contain unequal angles; therefore the contained unequal angles are larger than unequal angles containing them.
The Commentary
Coming soon...
Bibliography
For more reading, see:
Drachmann, Aage Gerhardt. The Mechnical Technology of Greek and Roman Antiquity: A Study of the Literary Sources. Madison, WI: University of Wisconsin Press. 1963.
Heath, Thomas. "Mensuration: Heron of Alexandria" in A Greek History of Mathematics, Vol. 2, 298-354. Oxford Clarendon Press, 1921.