The same ratio for the river, then for the wall
From K let KΘ be drawn parallel to AI, and from Θ let ΘΡ be drawn parallel to IΥ. Since then of right-angled triangle AIΥ, IΥ has been bisected at K, and ΘΚ is parallel to AI and ΘΡ to IΥ, AI too is bisected at Ρ. Measure off the interval from I to Ρ. The interval from Ρ to A is therefore also given. Subtracting from this the interval from Ρ to Φ, we shall have the rest: that is, the breadth of the river.
If it seems laborious to someone to take a larger interval standing off on our own land, the sight there being of necessity disturbed and the result confused, we may, standing on the same bank of the river, easily take the magnitude of the breadth in this way. Again let point A have been taken on the opposite part. On the part toward us let point B have been taken, so that AB is at right angles to the line through the bank, BΓ. Some point Δ has been taken on BΓ, on which let rod ΔΕ lie. At the end of the rod let gnomon E be raised, so that if rod ΔΕ should touch the surface of the water, the gnomon is at the surface. Let the rod be carried, at right angles to ΔΕ, along BΓ, until from some point Γ on line BΓ, through a dioptra, points E and A are seen. It will be analogon: as BΓ is to ΓΔ, so AB is to EΔ. The ratio of BΓ to ΓΔ has been given. The ratio of AB to ΔΕ has therefore also been given. And ΔΕ is given. AB is therefore also given.
By the same ratio the height of a wall will also be taken, on the same diagram set upright. Let the top of the battlement be A, the base B, and the line from the wall to us, outside missile-range, BΓ. A dioptra hangs from a pole (which is called a “lampstand”), planted at right angles at Γ. Let the pole be line ΔΓ. Tilting the dioptra, I sight the top of the wall, which is A. Going to the other vessel, on the same straight I take point E. Triangle AEB it will be, and parallel to one of the sides, AB, is ΓΔ. The ratio that ΕΓ has to ΓΔ, this EB has to BA. The ratio of ΕΓ to ΓΔ has been given, for each of them has been given. The ratio of EB to AB has therefore also been given. EB has also been given, as was shown on the river. BA is therefore also given, which was to be shown.
Greek
Καὶ ἀπὸ τοῦ Κ, τῇ ΑΙ <ἤχθω> παράλληλος ἡ ΚΘ, ἀπὸ δὲ τοῦ Θ, τῇ ΙΥ παράλληλος ἡ ΘΡ. Ἐπεὶ οὖν τριγώνου τοῦ ΑΙΥ ὀρθογωνίου ἡ ΙΥ δίχα τέτμηται τῷ Κ καὶ ἔστι παράλληλος ἡ ΘΚ τῇ ΑΙ <καὶ ἡ ΘΡ τῇ ΙΥ>, καὶ ἡ ΑΙ ἄρα δίχα τέτμηται κατὰ τὸ Ρ. Ἀποτιμᾶν δὴ τὸ ἀπὸ τοῦ Ι ἐπὶ τὸ Ρ διάστημα. ∆έδοται ἄρα καὶ τὸ ἀπὸ τοῦ Ρ ἐπὶ τὸ Α. Τούτου δ' ἀφελόντες τὸ ἀπὸ τοῦ Ρ ἐπὶ τὸ Φ καὶ τὸ λοιπὸν ἕξομεν, τοῦτ' ἔστι τὸ τοῦ ποταμοῦ πλάτος. Εἰ δέ τῳ ἐργῶδες εἶναι δόξει τὸ πλέον ἀποστάντα ἐπὶ τῆς ἡμεδαπῆς διάστημα λαβεῖν, ἀνάγκης ἐκεῖ τούτου γινομένης τὴν ὄψιν ἐπιταράττεσθαι <καὶ> συγχεῖσθαι τὸ γιγνόμενον, λάβοιμεν ἄν, ἐπὶ τῆς αὐτῆς ὄχθης ἑστῶτες τοῦ ποταμοῦ, ῥᾳδίως τὸ μέγεθος τοῦ πλάτους τοῦτον τὸν τρόπον. Ἔστω γὰρ πάλιν ἐπὶ τοῦ καταντικρὺ μέρους εἰλημμένον σημεῖον τὸ Α. Ἐπὶ δὲ τοῦ πρὸς ἡμᾶς μέρους εἰλήφθω σημεῖον τὸ Β, ὥστε εἶναι τὴν ΑΒ πρὸς ὀρθὰς τῇ διὰ τῆς ὄχθης γραμμῇ, τῇ ΒΓ. Εἴληπται δέ τι σημεῖον ἐπὶ τῆς ΒΓ, τὸ ∆, ἐφ' οὗ κανὼν κείσθω ὁ ∆Ε. Ἐπὶ δὲ τοῦ ἄκρου τοῦ κανόνος μετέωρος ἔστω γνώμων ὁ Ε, ὥστε, εἰ ὁ ∆Ε κανὼν τῆς τοῦ ὕδατος ἐπιφανείας ἅπτοιτο, ἐπιπολῆς εἶναι τὸν γνώμονα. Καὶ μέχρι τούτου ὁ κανών, πρὸς ὀρθὰς <τῇ ∆Ε>, τῇ ΒΓ παραφερέσθω, μέχρις οὗ ἀπό τινος <σημείου, τοῦ Γ>, ἐπὶ τῆς ΒΓ γραμμῆς, διὰ διόπτρας θεωρηθῇ σημεῖα τὰ ΕΑ. Καὶ ἔσται ἀνά λογον ὡς ΒΓ πρὸς Γ∆ οὕτως ἡ ΑΒ πρὸς Ε∆. ∆έδοται δὲ ὁ τῆς ΒΓ πρὸς Γ∆ λόγος. ∆έδοται ἄρα καὶ ὁ τῆς ΑΒ πρὸς ∆Ε. Καὶ ἔστιν δοθεῖσα ἡ ∆Ε. ∆οθεῖσα ἄρα καὶ ἡ ΑΒ. Τῷ δὲ αὐτῷ λόγῳ καὶ τείχους ὕψος ληφθήσεται ἐπὶ τοῦ αὐτοῦ διαγράμματος ὀρθουμένου. Ἔστω τὸ μὲν ἄκρον τοῦ προμαχῶνος τὸ Α, βάσις δὲ τὸ Β, ἡ δὲ ἀπὸ τοῦ τείχους εἰς ἡμᾶς ἔξω βέλους γραμμὴ ΒΓ. Κρέμαται διόπτρα ἀπὸ κάμακος (ὃ δὴ «λυχνία» καλεῖται) πηγνυμένη πρὸς ὀρθὰς κατὰ τὸ Γ. Ἔστω δὴ γραμμὴ ὁ κάμαξ ∆Γ. Τὴν δὴ διόπτραν ἐπικλίνας, διοπτεύω τοῦ τείχους τὸ ἄκρον, ὅ ἐστιν Α. Καὶ μετελθὼν ἐπὶ τὸ ἕτερον ἀγγεῖον, ἐπὶ τῆς αὐτῆς εὐθείας λαμβάνω ση μεῖον <τὸ Ε. Καὶ ἔσται τρίγωνον> τὸ ΑΕΒ, καὶ παρὰ μίαν τῶν πλευρῶν τὴν ΑΒ παράλληλος ἡ Γ∆. Ὃν ἄρα λόγον ἔχει ἡ ΕΓ πρὸς Γ∆, τοῦτον ἡ ΕΒ πρὸς ΒΑ. ∆έδοται δὲ ὁ τῆς ΕΓ πρὸς Γ∆ λόγος· δέδοται γὰρ αὐτῶν ἑκατέρα. ∆έδοται ἄρα καὶ ὁ τῆς ΕΒ πρὸς ΑΒ <λόγος. ∆έδοται δὲ καὶ ΕΒ>, ὡς ἐπὶ τοῦ ποταμοῦ δέδεικται. ∆οθεῖσα ἄρα καὶ ἡ ΒΑ, ὅπερ ἔδει δεῖξαι.
About this text
Julius Africanus, Cesti (Embroidered Girdles): a miscellany of history, science, and craft, in the surviving fragments — books 7, 2, and 3 in full, with parts of books 4, 8, 9, and 13, and the colophon of Cestus 18. The older English translates his letters only. The book-2 table of contents and a book-7 appendix are not in this volume.
New English from locked PG 10 / Khazarzar Greek of the Cesti fragmenta through 9.5. Earlier lemma-led scaffold English was discarded. Book-2 pinax and book-7 appendix held. This lock is exhausted.
Catalogue & scope
Khazarzar scan
Witnesses
- Copy-text PG 10 Cesti fragmenta (Khazarzar) (Greek)
This is an AI-assisted study translation. Source fidelity and completeness have not been independently certified. Open Greek on each section for the source text. This is not a complete critical edition.