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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
AQ,

i.e. to the difference between the square
BE and the sum of the rectangles
BP, AQ,

i.e. to the difference between the square
BE and the sum of the rectangles
BP, AR,

i.e. to the difference between the sum of
the squares BE, BH and the sum
of the rectangles BP, BR

(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.
Therefore AC is less than 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.
Let A be the given rectilineal figure ;

thus it is required to construct a square equal to the rectilineal figure A.

A

B

C

H

5

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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 BE be greater, and let it be produced to F;

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.
δεῖξις, ἡ εἰς τὸ ἀδ. ἄγουσα ἀπόδειξις 136
dkidocions, barb-like 188

dußreîa (ywvia), obtuse (angle) 181
außλvywvios, obtuse-angled 187

ȧμephs, indivisible 41, 268

aupikoiλos (of curvilineal angles) 178
ἀμφίκυρτος 178

ἀναγράφειν ἀπό to describe on contrasted with
to construct (συστήσασθαι) 348
ἀναλυόμενος (τόπος), Treasury of Analysis 8,
10, 11, 138

ȧvaσтpopixos (species of locus) 330
dvoμocoμephs, non-uniform 40, 161-2
ȧvτισтρоý, conversion 256-7: leading variety,
ἡ προηγουμένη or ή κυρίως, ibid.
ȧvýжаρктоs, non-existent 129
dópioтos, indeterminate: (of lines or curves)
160: (of problems) 129

ἀπαγωγή, reduction 135: εἰς τὸ ἀδύνατον 136
ἄπειρος, infinite: ἡ ἐπ ̓ ἄπ. ἐκβαλλομένη οι
line or curve extending without limit and
not "forming a figure" 160-1: éπ' áπ. or
εἰς ἄπ. adverbial 190: ἐπ ̓ ἄπ. διαιρεῖσθαι
268: Aristotle on Tò deɩpov 232-4
drλous, simple: (of lines or curves) 161-2:
(of surfaces) 170

ȧwbdeikis, proof (one of necessary divisions of

a proposition) 129, 130

arTeobal, to meet (occasionally touch) 57
appηros, irrational: of Xoyos 137: of diameter
(diagonal) 399

douμBaros, incompatible 129
dσúμжTWтоs, not-meeting, non-secant, asymp
totic 40, 161, 203

doveros, incomposite: (of lines) 160, 161:
(of surfaces) 170

&TAKTOS, unordered: (of problems) 128: (of
irrationals) 115

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
Euclid's dedouéva or Data, q.v.

132-3:

delypara, illustrations, of Stoics 329
dei dh, "thus it is required," introducing
διορισμός 293

=

didypapua proposition (Aristotle) 252
dualpeois: point of division (Aristotle) 165,

170, 171: method of division (exhaustion)
285: Euclid's περὶ διαιρέσεων 8, 9, 18, 87, 110
diaσráσeis, almost = dimensions 157, 158
διαστατόν extended, ἐφ' ἓν one way, ἐπὶ δύο
two ways, en rpia three ways (of lines,
surfaces and solids respectively) 158, 170

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Keivos Euclid 400

=

EKOEOLS, setting-out, one of formal divisions of
proposition 129: may be omitted some-
times 130

ἐκτός, κατὰ τὸ (of an exterior angle in sense
of re-entrant) 263: ǹ EKTOS ywvla, the
exterior angle 280

EXKoelons, spiral-shaped 159

Neps, falling-short (with reference to
application of areas) 36, 343-5, 383-4
EXλès рóßλnua, a deficient (= indeter-
minate) problem 129

évaλáž, alternately or (adjectivally) alternate
308

Evvola, notion, use of, 221
EvoTaois, objection 135

ἐντός, κατὰ τὸ οἱ ἡ ἐντὸς (γωνία) of an interior
angle 263, 280: ἡ ἐντὸς καὶ ἀπεναντίον
ywvla, the interior and opposite angle 280
ἐπεζεύχθωσαν (επιζεύγνυμι, join) 242
érlredov, plane in Euclid, used for surface
also in Plato and Aristotle 169
ἐπιπροσθεῖν, ἐπίπροσθεν εἶναι, to stand in
front of (hiding from view) 165, 166
émipáveia, surface (Euclid) 169
repoμnkes, oblong 151, 188

εὐθύ, τό, the straight 159: εὐθεῖα (γραμμή),
straight line 165-9

evoúyраμμos, rectilineal 187: neuter as sub-
stantive 346

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
181, 278

Bewpnua, theorem q.v.

Oupeós (shield) = ellipse 165

ITTOV Téon (horse-fetter), name for a certain
curve 162-3, 176

ισομέτρων σχημάτων, περί, on isometric figures
(Zenodorus) 26, 27, 333

κάθετος εὐθεῖα γραμμή, perpendicular 181-2,
271: "plane" and "solid" 272
κаμяúλos, curved (of lines) 159
KaTaσKEVÝ, construction, or machinery, one of
divisions of a proposition 129: sometimes
unnecessary 130

KATATOμŃ KAVÓνOs, Sectio canonis, of Euclid 17

412

INDEX OF GREEK WORDS AND FORMS

Kelow, "let it be made" 269
kekaμμévŋ, bent (of lines) 159, 176

κέντρον, centre 183, 184, 199: ἡ ἐκ τοῦ κέντρου
=radius 199

KEраTOELONS (ywvla), horn-like (angle) 177,
178, 182

κλᾶν, to infect or defect, κεκλάσθαι, κεκλασ
μένη, κλάσις 118, 150, 159, 176, 178
Kλois, inclination, 176

KOLλoyúviov, hollow-angled figure (Zenodorus)
27, 188

Koival Evvoia, Common Notions (= axioms)
221-2: called also τὰ κοινά, κοιναὶ δόξαι
(Aristotle) 120, 221

κοινὴ προσκείσθω, ἀφῃρήσθω 276

Kopuń, vertex: kaтà kоopʊþýν, vertical (angles)
278

xplxos, ring (Heron) 163

λñμμa, lemma (=something assumed, Xaμ-
βανόμενον) 133-4

λοιπός : λοιπὴ ἡ ΑΛ λαπῇ τῇ ΒΗ ἴση ἐστίν 245
μépn, parts (=direction) 190, 308, 323:
(=side) 271

μκos, length, 158-9

unvocions, lune-like (of angle) 26, 201: Tò
μηνοειδές (σχήμα), lune 187

MKTÓS, "mixed" (of lines or curves) 161, 162:
(of surfaces) 170

μονὰς προσλαβοῦσα θέσιν, definition of a point
155

μονόστροφος έλιξ "single-turn spiral” 122-
3., 164-5: in Pappus cylindrical helix
165

vevoels, inclinations, a class of problems
150-1: vevel, to verge 118, 150
EvoTpOeidhs, scraper-like (of angle) 178
duoeidhs, "of the same form" 250

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8uotos, similar" (of numbers) 357: (of angles)
= equal (Thales, Aristotle) 252
opoloμephs, uniform (of lines or curves) 40,
161-2

ὀξεία (γωνία), acute (angle) 181
¿žvy❝vios, acute-angled 187

ὅπερ ἔδει δεῖξαι (or ποιῆσαι) Q. E.D. (or F.) 57
op@oywvios, right-angled: as used of quadri-
laterals = rectangular 188-9

8pos, opioμos, definition 143: original mean-
ing of opos 143: boundary, limit 182
os, visual ray 166

πάντῃ μεταλαμβανόμεναι, " taken together in
any manner 282

Tapaßoλǹ Tŵv xwplwv, application of areas 36,
343-5: contrasted with repẞoλń (exceed.
ing) and Xes (falling-short) 343: wаpa-
βολή contrasted with σύστασις (construction)
343: application of terms to conics by
Apollonius 344-5

παράδοξος τόπος, ό, “the Treasury of Para-
doxes" 329

wapaλλάTTш, "fall beside" or "awry" 262
жараж\hpwμа, complement, q.v.

πέρας, extremity 165, 182: πέρας συγκλείον
(Posidonius' definition of figure) 183
περιεχομένη (of angle), περιεχόμενον (of rect-
angle), contained 370: Tò dis repιexbμevov,

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poßλnua, problem q.v.

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ponyoúμevos, leading: (of conversion) = com-
plete 256-7 : προηγούμενον (θεώρημα) leading
(theorem) contrasted with converse 257
pós, in geometry, various meanings 277
póraσis, enunciation 129-30
Tротεivw, "propound" 128

Tр@ros, prime, two senses of, 146
πτῶσις, case 134

ῥητός, rational 137: ῥητὴ διάμετρος τῆς πεμ-
rádos ("rational diameter of 5") 399
onμetov, point 155-6

oтáoμn, a mathematical instrument 371
σTLYμn, point 156

OTOXEîov, element 114-6

OTPOYYÚλOV, TÓ, round (circular), in Plato
159, 184: στρογγυλότης, roundness 182
Tνμπéρаσμа, conclusion (of a proposition)
129, 130

Túveros, composite: (of lines or curves) 160:
(of surfaces) 170

σúvevais, convergence 282

ovvioτaobai, construct: special connotation
259, 289: with evrós 289: contrasted with
παραβάλλειν (αρρίν) 343

σχηματογραφεῖν, σχηματογραφία, represent-
ing (numbers) by figures of like shape 359
σχηματοποιοῦσα οι σχῆμα ποιοῦσα, forming a
figure (of a line or curve) 160-1
TETαYμévov (of a problem), "ordered" 128
TETPAYWVIOμÓS, squaring, definitions of 149–

50, 410

TETрáуwvov, square: sometimes (but not in
Euclid) any four-angled figure 188
Tетράжλεuрov, quadrilateral 187

Toμh, section, =point of section 170, 171, 278
TожιKÒV Оεúρημa, locus-theorem 329
TOTOS: locus 329-31: room or space 23 n.:

place (where things may be found), thus

τόπος ἀναλυόμενος 8, 10 : παράδοξος τόπος 329
Tóρvos, instrument for drawing a circle 371
Tрinλεupov, three-sided figure 187
TUXÒν onμeîov, a point at random 252
UrEpBoth, exceeding, with reference to method

of application of areas 36, 343-5, 386-7
ὑπό, in expressions for an angle (ἡ ὑπὸ ΒΑΓ
ywvia) 249, and a rectangle 370
vwóкeιtαι, “is by hypothesis" 303, 312
VTOTELVE, subtend, with acc. or vrò and acc.
249, 283, 350

ipio pevn rapuh, determinate line (curve),
"forming a figure" 160

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