A Euclid cut to measure a river you cannot cross
To find the breadth of a river and the height of a wall. Those moderately possessed of encyclical education have, as is likely, touched Euclid's Elements to some extent. It is not hard, then, through the first book, to establish these things too: to measure the breadth of a river, the other bank being untrodden because of the enemies stationed on it, so as to bring on a matching span to bridge it; and by the same ratio to take the height of a wall from a distance, so as to bring up helepolis engines of equal standing.
For ease of learning the proof, this theorem will lead: “If one of the sides about the right angle of a right-angled triangle is bisected, and from the cut a straight line is raised at right angles, and through the point where it cuts the remaining side a parallel is drawn, the remaining sides of the triangle are also bisected.”
Let ABΓ be a right-angled triangle, having the right angle at B. Let AB be bisected at Δ. Let ΔΕ be drawn at right angles. Through E let EZ be drawn parallel. I say that the remaining sides of the triangle are also bisected: AΓ at E, BΓ at Z. Join ΔZ. Since AΔ is equal to ΔB, and ΔB to EZ, AΔ is therefore equal to EZ and parallel. And the lines joining equals and parallels on the same parts are equal and parallel. ΔΕ and ZΓ also make parallelogram ΓΕΔZ. ΔZ is therefore equal to ΕΓ. But it was also equal to AE. ΕΓ is therefore equal to AE. Again, since each of BΔ EZ and ΓΕ ΔZ is a parallelogram, ΔΕ is therefore equal to each of BZ and ZΓ, for they are opposite. So BZ and ZΓ are also equal. The proofs hold also for every triangle.
Following these things, the breadth of a river will be measured from a distance. Let there be banks: opposite, the enemies' bank, on which is point A; toward us, ΦΗ. A split dioptra is planted in a place toward us, at I, so that the interval from I to our bank of the river is greater than the river. This is easy to guess. Two points are sighted at right angles: one on the opposite bank — a stone or a bush or some other easy mark — and let it be A; the other, the point toward us, from the other line of the cross, Υ. Moving the dioptra onto Υ I sight A and make a right-angled triangle. Let IΥ be bisected at K.
Greek
1.15 Ποταμοῦ πλάτος εὑρεῖν καὶ τείχους <ὕψος> Οἱ τῆς ἐγκυκλίου μετρίως ἐπήβολοι παιδείας τῶν Εὐκλείδου «Στοιχείων» ἐπὶ ποσόν, ὡς εἰκός, ἐφήψαντο. Οὐ δὴ χαλεπὸν διὰ τοῦ πρώτου συστῆσαι καὶ τάδε· ποταμοῦ πλάτος ἐκμετρῆσαι, τῆς ἑτέρας ὄχθης ἀβάτου διὰ τοὺς ἐφεστῶτας αὐτῇ πολεμίους, πρὸς τὸ γεφυρῶσαι σύμμετρον ἐπαγαγόντας ζεῦγμα, τῷ τ' αὐτῷ λόγῳ τείχους ὕψος ἐκ διαστήματος λαβεῖν εἰς τὸ τὰς ἑλεπόλεις μηχανὰς ἰσοστασίους ἐπενεγκεῖν. Εἰς εὐμαθίαν δὲ τῆς ἀποδείξεως ἡγήσεται θεώρημα τόδε· «Ἐὰν ὀρθογωνίου τριγώνου μία τῶν περὶ τὴν ὀρθὴν γωνίαν <πλευρῶν> δίχα τμηθῇ, ἀπὸ δὲ τῆς τομῆς πρὸς ὀρθὰς <εὐθεῖα> ἀναταθῇ, καὶ διὰ τοῦ σημείου καθὸ τέμνει τὴν λοιπὴν πλευρὰν παράλληλος ἀχθῇ, καὶ αἱ λοιπαὶ τοῦ τριγώνου δίχα τέμνονται πλευραί». Ἔστω γὰρ τρίγωνον ὀρθογώνιον τὸ ΑΒΓ, ὀρθὴν ἔχον τὴν Β γωνίαν. Καὶ τετμήσθω δίχα ἡ ΑΒ, τῷ ∆. Καὶ πρὸς ὀρθὰς ἤχθω ἡ ∆Ε. Καὶ διὰ τοῦ Ε, παράλληλος ἤχθω ἡ ΕΖ. Λέγω ὅτι καὶ <αἱ> λοιπαὶ τοῦ τριγώνου πλευραὶ δίχα τέμνονται, ἡ μὲν ΑΓ κατὰ τὸ Ε, ἡ δὲ ΒΓ κατὰ τὸ Ζ. Ἐπεζεύχθω γὰρ ἡ ∆Ζ. Ἐπεὶ ἴση ἐστὶν ἡ Α∆ τῇ ∆Β, <ἡ δὲ ∆Β τῇ ΕΖ>, ἄρα ἡ Α∆ τῇ ΕΖ ἴση καὶ παράλληλος. Αἱ δὲ ἴσας τε καὶ παραλλήλους ἐπὶ τὰ αὐτὰ μέρη ἐπιζευγνύουσαι ἴσαι τε καὶ παράλληλοί εἰσιν. Ἀλλὰ καὶ αἱ ∆Ε ΖΓ παραλληλόγραμμον τὸ ΓΕ∆Ζ. Ἴση ἄρα ἡ ∆Ζ τῇ ΕΓ. Ἀλλὰ καὶ τῇ ΑΕ ἦν ἴση. <Ἡ ΕΓ ἄρα τῇ ΑΕ ἐστιν ἴση>. Πάλιν, ἐπεὶ ἑκάτερον τῶν Β∆ ΕΖ, ΓΕ ∆Ζ παραλληλόγραμμον, ἡ ∆Ε ἄρα ἴση ἐστὶν ἑκατέρᾳ τῶν ΒΖ ΖΓ· ἀπεναντίον γάρ. Ὥστε καὶ αἱ ΒΖ ΖΓ ἴσαι εἰσίν. Εἰσὶν αἱ ἀποδείξεις καὶ κατὰ παντὸς τριγώνου. Ἀκολούθως δὴ τοῖσδε ποταμοῦ πλάτος ἐκ διαστήματος μετρηθήσεται. Ἔστωσαν ὄχθαι, καταντικρὺ μὲν ἣ τῶν πολεμίων ἐφ' ᾗ σημεῖον τὸ Α, ἣ δὲ πρὸς ἡμᾶς ἡ ΦΗ. Πήγνυται διόπτρα ἐν χώρᾳ τῇ πρὸς ἡμᾶς ἡ σχιστή, κατὰ τὸ Ι, οὕτως ὥστε τὸ διάστημα τὸ τοῦ Ι μέχρι τῆς πρὸς ἡμᾶς ὄχθης τοῦ ποταμοῦ μεῖζον εἶναι τοῦ ποταμοῦ. Τοῦτο δὲ ῥᾴδιον στοχάσασθαι. Καὶ πρὸς ὀρθὰς δύο σημεῖα κατοπτεύεται, ἓν μὲν ἐπὶ τῇ ὄχθῃ καταντικρύ, ἢ λίθος ἢ θάμνος ἤ τις ἄλλος εὐκάτοπτος σκοπός, καὶ ἔστω τὸ Α, τὸ δὲ ἕτερον τὸ πρὸς ἡμᾶς σημεῖον, ἐκ τῆς ἑτέρας τοῦ χιασμοῦ γραμμῆς, τὸ Υ. Τὴν δὲ διόπτραν μεταγαγὼν ἐπὶ τὸ Υ κατοπτεύω τὸ Α καὶ ποιῶ τρίγωνον ὀρθογώνιον. Τετμήσθω ἡ ΙΥ δίχα κατὰ τὸ Κ.
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.