Next, by joining BG, CH, we prove in like manner that the rectangles AR, AQ are equal. Now the square on BC is equal to the sum of the rectangles BP, CP, i.e. to the sum of the rectangles BR, CQ, i.e. to the sum of the squares BH, CG diminished by the rectangles AR, AQ. But the rectangles AR, AQ are equal, and they are respectively the rectangles contained by BA, AN and by CA, AM. Therefore the square on BC is less than the squares on BA, AC by twice the rectangle BA, AN or CA, AM. Next suppose that we have to prove the theorem in the case where the triangle has an obtuse angle at A. Take B as the acute angle under consideration, so that AC is the side opposite to it. Now the square on CA is equal to the difference of the rectangles CQ, AQ, i.e. to the difference between CP and i.e. to the difference between the square i.e. to the difference between the square i.e. to the difference between the sum of (since AR is the difference between BR and BH). R M F But BP, BR are equal, and they are respectively the rectangles CB, BL and AB, BN. Therefore the square on CA is less than the squares on AB, BC by twice the rectangle CB, BL or AB, BN. Heron's proof of his converse proposition (an-Nairizi, ed. Curtze, p. 110), which is also given by the Greek scholiast above quoted, is of course simple. For let ABC be a triangle in which the square on AC is less than the squares on AB, BC. Draw BD at right angles to BC and of length equal to BA. Join DC. Then, since the angle CBD is right, the square on DC is equal to the squares on DB, BC, i.e. to the squares on AB, BC. [1. 47] But the square on AC is less than the squares on AB, BC. A Therefore the square on AC is less than the square on DC. Hence in the two triangles DBC, ABC the sides about the angles DBC, ABC are respectively equal, but the base DC is greater than the base AC. Therefore the angle DBC (a right angle) is greater than the angle ABC [1. 25], which latter is therefore acute. It may be noted, lastly, that 11. 12, 13 are supplementary to I. 47 and complete the theory of the relations between the squares on the sides of any triangle, whether right-angled or not. PROPOSITION 14. To construct a square equal to a given rectilineal figure. thus it is required to construct a square equal to the rectilineal figure A. A B C H 5 For let there be constructed the rectangular parallelogram BD equal to the rectilineal figure A. [I. 45] Then, if BE is equal to ED, that which was enjoined will have been done; for a square BD has been constructed equal to the rectilineal figure A. But, if not, one of the straight lines BE, ED is greater. let EF be made equal to ED, and let BF be bisected at G. With centre G and distance one of the straight lines GB, GF let the semicircle BHF be described; let DE be produced 15 to H, and let GH be joined. 20 Then, since the straight line BF has been cut into equal segments at G, and into unequal segments at E, the rectangle contained by BE, EF together with the square on EG is equal to the square on GF. But GF is equal to GH; [11. 5] therefore the rectangle BE, EF together with the square on GE is equal to the square on GH. But the squares on HE, EG are equal to the square on GH; [1. 47] 25 therefore the rectangle BE, EF together with the square on GE is equal to the squares on HE, EG. Let the square on GE be subtracted from each; 30 therefore the rectangle contained by BE, EF which remains is equal to the square on EH. But the rectangle BE, EF is BD, for EF is equal to ED; therefore the parallelogram BD is equal to the square on HE. And BD is equal to the rectilineal figure A. Therefore the rectilineal figure A is also equal to the square 35 which can be described on EH. Therefore a square, namely that which can be described on EH, has been constructed equal to the given rectilineal figure A. Q. E. F. 7. that which was enjoined will have been done, literally “would have been done,” γεγονὸς ἂν εἴη τὸ ἐπιταχθέν. 35, 36. which can be described, expressed by the future passive participle, dvaypaønσομένῳ, ἀναγραφησόμενον. Heiberg (Mathematisches zu Aristoteles, p. 20) quotes as bearing on this proposition Aristotle's remark (De anima II. 2, 413 a 19: cf. Metaph. 996 b 21) that "squaring” (Tetpaywvioμós) is better defined as the "finding of the mean (proportional)" than as "the making of an equilateral rectangle equal to a given oblong," because the former definition states the cause, the latter the conclusion only. This, Heiberg thinks, implies that in the text-books which were in Aristotle's hands the problem of 11. 14 was solved by means of proportions. As a matter of fact, the actual construction is the same in 11. 14 as in VI. 13; and the change made by Euclid must have been confined to substituting in the proof of the correctness of the construction an argument based on the principles of Books 1. and II. instead of Book vi. As II. 12, 13 are supplementary to I. 47, so II. 14 completes the theory of transformation of areas so far as it can be carried without the use of proportions. As we have seen, the propositions 1. 42, 44, 45 enable us to construct a parallelogram having a given side and angle, and equal to any given rectilineal figure. The parallelogram can also be transformed into an equal triangle with the same given side and angle by making the other side about the angle twice the length. Thus we can, as a particular case, construct a rectangle on a given base (or a right-angled triangle with one of the sides about the right angle of given length) equal to a given square. Further, 1. 47 enables us to make a square equal to the sum of any number of squares or to the difference between any two squares. The problem still remaining unsolved is to transform any rectangle (as representing an area equal to that of any rectilineal figure) into a square of equal area. The solution of this problem, given in II. 14, is of course the equivalent of the extraction of the square root, or of the solution of the pure quadratic equation x2= ab. Simson pointed out that, in the construction given by Euclid in this case, it was not necessary to put in the words "Let BE be greater," since the construction is not affected by the question whether BE or ED is the greater. This is true, but after all the words do little harm, and perhaps Euclid may have regarded it as conducive to clearness to have the points B, G, E, F in the same relative positions as the corresponding points A, C, D, B in the figure of 11. 5 which he quotes in the proof. dyúviov, angle-less (figure) 187 ἀδύνατον: ἡ εἰς τὸ ἀδ. ἀπαγωγή, ἡ διὰ τοῦ ἀδ. dußreîa (ywvia), obtuse (angle) 181 ȧμephs, indivisible 41, 268 aupikoiλos (of curvilineal angles) 178 ἀναγράφειν ἀπό to describe on contrasted with ȧvaσтpopixos (species of locus) 330 ἀπαγωγή, reduction 135: εἰς τὸ ἀδύνατον 136 ȧwbdeikis, proof (one of necessary divisions of a proposition) 129, 130 arTeobal, to meet (occasionally touch) 57 douμBaros, incompatible 129 doveros, incomposite: (of lines) 160, 161: &TAKTOS, unordered: (of problems) 128: (of ἄτομοι γραμμαί, indivisible lines” 268 ayls segment of circle less than semicircle = 187 Bálos, depth 158-9 Báois, base 248-9 γεγράφθω 242 γνώμων, see gnomon papun, line (or curve) q.v. Yраμμкŵs, graphically 400 dedouévos given, different senses 132-3: delypara, illustrations, of Stoics 329 = didypapua proposition (Aristotle) 252 170, 171: method of division (exhaustion) Keivos Euclid 400 = EKOEOLS, setting-out, one of formal divisions of ἐκτός, κατὰ τὸ (of an exterior angle in sense EXKoelons, spiral-shaped 159 Neps, falling-short (with reference to évaλáž, alternately or (adjectivally) alternate Evvola, notion, use of, 221 ἐντός, κατὰ τὸ οἱ ἡ ἐντὸς (γωνία) of an interior εὐθύ, τό, the straight 159: εὐθεῖα (γραμμή), evoúyраμμos, rectilineal 187: neuter as sub- épárтeσbal, to touch 57 ἐφαρμόζειν, to coincide, ἐφαρμόζεσθαι, to be applied to 168, 224-5, 249 épeKTIKOS (of a class of loci) 330 epens, "in order" 181: of adjacent angles Bewpnua, theorem q.v. Oupeós (shield) = ellipse 165 ITTOV Téon (horse-fetter), name for a certain ισομέτρων σχημάτων, περί, on isometric figures κάθετος εὐθεῖα γραμμή, perpendicular 181-2, KATATOμŃ KAVÓνOs, Sectio canonis, of Euclid 17 412 INDEX OF GREEK WORDS AND FORMS Kelow, "let it be made" 269 κέντρον, centre 183, 184, 199: ἡ ἐκ τοῦ κέντρου KEраTOELONS (ywvla), horn-like (angle) 177, κλᾶν, to infect or defect, κεκλάσθαι, κεκλασ KOLλoyúviov, hollow-angled figure (Zenodorus) Koival Evvoia, Common Notions (= axioms) κοινὴ προσκείσθω, ἀφῃρήσθω 276 Kopuń, vertex: kaтà kоopʊþýν, vertical (angles) xplxos, ring (Heron) 163 λñμμa, lemma (=something assumed, Xaμ- λοιπός : λοιπὴ ἡ ΑΛ λαπῇ τῇ ΒΗ ἴση ἐστίν 245 μκos, length, 158-9 unvocions, lune-like (of angle) 26, 201: Tò MKTÓS, "mixed" (of lines or curves) 161, 162: μονὰς προσλαβοῦσα θέσιν, definition of a point μονόστροφος έλιξ "single-turn spiral” 122- vevoels, inclinations, a class of problems 8uotos, similar" (of numbers) 357: (of angles) ὀξεία (γωνία), acute (angle) 181 ὅπερ ἔδει δεῖξαι (or ποιῆσαι) Q. E.D. (or F.) 57 8pos, opioμos, definition 143: original mean- πάντῃ μεταλαμβανόμεναι, " taken together in Tapaßoλǹ Tŵv xwplwv, application of areas 36, παράδοξος τόπος, ό, “the Treasury of Para- wapaλλάTTш, "fall beside" or "awry" 262 πέρας, extremity 165, 182: πέρας συγκλείον poßλnua, problem q.v. ponyoúμevos, leading: (of conversion) = com- Tр@ros, prime, two senses of, 146 ῥητός, rational 137: ῥητὴ διάμετρος τῆς πεμ- oтáoμn, a mathematical instrument 371 OTOXEîov, element 114-6 OTPOYYÚλOV, TÓ, round (circular), in Plato Túveros, composite: (of lines or curves) 160: σúvevais, convergence 282 ovvioτaobai, construct: special connotation σχηματογραφεῖν, σχηματογραφία, represent- 50, 410 TETрáуwvov, square: sometimes (but not in Toμh, section, =point of section 170, 171, 278 place (where things may be found), thus τόπος ἀναλυόμενος 8, 10 : παράδοξος τόπος 329 of application of areas 36, 343-5, 386-7 ipio pevn rapuh, determinate line (curve), |