534 σχέσις, “relation”: ποιὰ σχέσις, “ a sort of relation" (in def. of ratio) II. 116-7 σχηματογραφεῖν, σχηματογραφία, representing (numbers) by figures of like shape I. 359 σχηματοποιοῦσα οι σχῆμα ποιοῦσα, “ forming a figure" (of a line or curve) I. 160-1
GENERAL INDEX OF GREEK WORDS AND FORMS
TaνтоμÝкns, of square number (Nicomachus) II. 293
ταυτότης λόγων, sameness of ratios" II. 119 TéλELOS, perfect (of a class of numbers) 11. 293-4
τεταγμένος, " ordered” : τεταγμένον πρόβλημα,
"ordered" problem I. 128: TETAYμÉVN ávaλoyla, "ordered" proportion 11. 137 τεταραγμένη ἀναλογία, perturbed proportion II. 136
TEтраyшvioμós, squaring, definitions of, 1. 149-
TEтрáуwνov, square: sometimes (but not in Euclid) any four-angled figure 1. 188 тетрάπλενроv, quadrilateral I. 187: not a "polygon" II. 239
τμῆμα κύκλου, segment of a circle: τμήματος γωνία, angle of a segment II. 4: ἐν τμήματι ywvia, angle in a segment II. 4. τομεὺς (κύκλου), sector (of a circle): σκυτοτο- μKÒS TOμεÚS, "shoemaker's knife" II. 5 TOμn, section, = point of section 1. 170, 171, 278: кown тоuń, "common section " III. 263
TOμоEidns (of figure), sector-like 11. 5 тожιкÒν Oεúρημa, locus-theorem 1. 329 TÓTOS, locus I. 329-31: room or space I. 23 n. place (where things may be found), thus Tóπos ávaλvóμevos, Treasury of Analysis I. 8, 10, παράδοξος τόπος, Treasury of Paradoxes, I. 329
Tópvos, instrument for drawing a circle 1. 371 TоσаνTATÁσιO, "the same multiple" II. 146 τρίγωνον, triangle: τὸ τριπλοῦν, τὸ δι' ἀλλή Awv, triple, interwoven triangle, =penta- gram II. 99
τριπλάσιος, triple, τριπλασίων, triplicate (of ratios) II. 133
Tрlλeupov, three-sided figure 1. 187 τυγχάνειν, happen: τυχὸν σημεῖον, any point at random I. 252: τυχοῦσα γωνία, σε any angle" II. 212 : ἄλλα, ἃ ἔτυχεν, ισάκις πολ λanλária, “other, chance, equimultiples " II. 143-4
UTEPẞoh, exceeding, with reference to method of application of areas I. 36, 343-5, 386-7
ὑπερτελής οι υπερτέλειος, “over-perfect” (of a class of numbers) II. 293-4
ὑπό, in expressions for an angle (ἡ ὑπὸ ΒΑΓ ywvía) 1. 249, and a rectangle I. 370 Úπodiяλáσios, sub-duplicate, = half (Nico- machus) II. 280
ὑποκείμενος, laid down or assumed : τὸ ὑπο- κείμενον ἐπίπεδον, the plane of reference
ὑπόκειται, “is by hypothesis I. 303, 312 Úжожоλαπλáσιos, submultiple (Nicomachus) II. 280
VæоTelve, subtend, with acc. or væò and acc. I. 249, 283, 350 yos, height 11. 189
wρioμévη ypaμμh, determinate line (curve), "forming a figure" 1. 160
[The references are to volumes and pages.]
al-'Abbas b. Sa'id al-Jauhari 1. 85 "Abthiniathus" (or "Anthisathus") I. 203 Abū 'l 'Abbās al-Fadl b. Hatim, see an- Nairizi
Abū 'Abdallah Muḥ. b. Mu'adh al-Jayyānī
Abu Ali al-Başri 1. 88
Abū 'Ali al-Hasan b. al-Hasan b. al-Haitham 1. 88, 89
Abū Da'úd Sulaiman b. 'Uqba 1. 85, 90 Abu Ja'far al-Khazin 1. 77, 85 Abu Ja'far Muh. b. Muh. b. al-Hasan Naşiraddin at-Tusi, see Naṣiraddin Abu Muḥ. b. Abdalbāqi al-Baģdādi al-Faraḍī I. 8 n., 90
Abu Muḥ. al-Hasan b. 'Ubaidallah b. Sulai- man b. Wahb 1. 87
Abū Naṣr Gars al-Na'ma 1. 90
Abu Naṣr Mansur b. 'Ali b. Iraq I. 90 Abu Nasr Muh. b. Muḥ. b. Tarkhan b. Uzlag al-Farabi 1. 88
Abu Sahl Wijan b. Rustam al-Kūhi 1. 88 Abū Sa'id Sinan b. Thābit b. Qurra 1. 88 Abū 'Uthman ad-Dimashqi 1. 25, 77 Abū 'I Wafa al-Būzjānī 1. 77, 85, 86 Abū Yusuf Ya'qūb b. Isḥāq b. aṣ-Ṣabbāḥ al- Kindi 1. 86
Abu Yusuf Ya'qub b. Muḥ. ar-Razi 1. 86 Adjacent (épens), meaning 1. 181
Aenaeas (or Aigeias) of Hierapolis 1. 28, 311 Aganis 1. 27-8, 191
Aḥmad b. al-Husain al-Ahwāzī al-Kātib 1. 89 Aḥmad b. 'Umar al-Karābisi 1. 85 al-Ahwazi I. 89
Aigeias (? Aenaeas) of Hierapolis 1. 28, 311 Alcinous II. 98
Alexander Aphrodisiensis I. 7 n., 29, II. 120 Algebra, geometrical 1. 372-4: classical
method was that of Eucl. 11. (cf. Apol- lonius) I. 373: preferable to semi-alge- braical method 1. 377-8: semi-algebraical method due to Heron 1. 373, and favoured by Pappus 1. 373: geometrical equivalents of algebraical operations 1. 374: algebraical equivalents of propositions in Book II., I. 372-3: equivalents in Book x. of pro- positions in algebra, √ √ cannot be
equal to k', III. 58-60: if a±√b=x±√y, then a=x, b=y, III. 93-4, 167-8
'Ali b. Aḥmad Abū 'l Qāsim al-Antaki 1. 86 Allman, G. J. 1. 135 n., 318, 352, III. 18- 9, 439
Alternate: (of angles) 1. 308: (of ratios), alternately II. 134
Alternative proofs, interpolated 1. 58, 59: cf. III. 9 and following II. 22: that in III. 10 claimed by Heron II. 23-4 Amaldi, Ugo I. 175, 179-80, 193, 201, 313, 328, 11. 30, 126
Ambiguous case 1. 306-7: in VI. 7, II. 208-9 Amphinomus I. 125, 128, 150 n.
Amyclas of Heraclea 1. 117
Analysis (and synthesis) 1. 18: definitions of, interpolated, I. 138, III. 442: described by Pappus I. 138-9: mystery of Greek analysis III. 246: modern studies of Greek analysis 1. 139: theoretical and problem- atical analysis 1. 138: Treasury of Analy sis (τόπος ἀναλυόμενος) 1. 8, 10, 11, 138: method of analysis and precautions neces- sary to, I. 139-40: analysis and synthesis of problems 1. 140-2: two parts of analysis (a) transformation, (b) resolution, and two parts of synthesis, (a) construction, (b) demonstration I. 141: example from Pappus 1. 141-2: analysis should also reveal dioptoubs (conditions of possibility) I. 142: interpolated alternative proofs of XIII. 1-5 by analysis and synthesis I. 137, III. 442-3
Analytical method 1. 36: supposed discovery of, by Plato 1. 134, 137 Anaximander I. 370, II. III Anaximenes II. III
Euclid I. 178: Syrianus' compromise 1. 178: treatise on the Angle by Eudemus I. 34, 38, 177-8: classification of angles (Geminus) 1. 178-9: curvilineal and "mixed" angles 1. 26, 178-9, horn-like (KEрATOELDS) I. 177, 178, 182, 265, II. 4, 39, 40, lune-like (unvoeidńs) 1. 26, 178–9, scraper-like (Evoтpoeldńs) I. 178: angle of a segment 1. 253, II. 4: angle of a semi- circle 1. 182, 253, II. 4: controversies about
angle of semicircle" and hornlike angle II. 39-42: definitions of angle classified I. 179: recent Italian views I. 179-81: angle as cluster of straight lines or rays 1. 180-1, defined by Veronese 1. 180: as part of a plane ("angular sector") 1. 179- 80: flat angle (Veronese etc.) 1. 180-1, 269: three kinds of angles, which is prior (Aristotle)? 1. 181-2: angles not less than two right angles not recognised as angles (cf. Heron, Proclus, Zenodorus) II. 47-9: did Euclid extend "angle" to angles greater than two right angles in vi. 33? 11. 275-6: adjacent angles 1. 181: alternate 1. 308: similar (= equal) 1. 178, 182, 252: vertical 1. 278: exterior and interior (to a figure) 1. 263, 280: exterior when re-entrant 1. 263, in which case we have a hollow-angled figure 1. 27, 188, 11. 48: interior and opposite 1. 280: construction by Apollonius of angle equal to angle 1. 296: angle in a semicircle, theorem of, I. 317-9 trisection of angle, by con- choid of Nicomedes 1. 265-6, by quadratrix of Hippias 1. 266, by spiral of Archimedes 1. 267: dihedral angle III. 264-5: solid angle III. 261, 267-8
Annex (πpooаpubfovoa) = the straight line which, when added to a compound ir- rational straight line formed by subtraction, makes up the greater term, "i.e. the negative """term III. 159
Antecedents (leading terms in proportion) II. 134
"Anthisathus" (or "Abthiniathus") 1. 203 Antiparallels: may be used for construction of VI. 12, II. 215 Antiphon I. 7., 35
Apastamba-Sulba-Sutra 1. 352: evidence in, as to early discovery of Eucl. 1. 47 and use of gnomon 1. 360-4: Bürk's claim that Indians had discovered the irrational 1. 363-4: approximation to /2 and Thibaut's explanation 1. 361, 363-4: inaccurate values of in, 1. 364 Apollodorus "Logisticus I. 37, 319, 351 Apollonius: disparaged by Pappus in com- parison with Euclid 1. 3: supposed by some Arabians to be author of the Ele- ments 1. 5: a "carpenter" I. 5: on ele- mentary geometry I. 42: on the line 1. 159: on the angle 1. 176: general defini- tion of diameter 1. 325: tried to prove axioms I. 42, 62, 222-3: his "general
treatise" I. 42: constructions by, for bisection of straight line 1. 268, for a perpendicular I. 270, for an angle equal to an angle 1. 296: on parallel-axiom (?) 1. 42-3 adaptation to conics of theory of application of areas 1. 344-5: geometrical algebra in, 1.373: Plane Loci, I. 14, 259, 330, theorem from (arising out of Eucl. vi. 3), also found in Aristotle 11. 198-200: Plane veúσeis 1. 151, problem from, 11. 81, lemma by Pappus on, II. 64-5: comparison of do- decahedron and icosahedron I. 6, III. 439, 512, 513 on the cochlias 1. 34, 42, 162: on "unordered" irrationals 1. 42, 115, III. 3, 10, 246, 255-9: general definition of ob- lique (circular) cone III. 270: 1. 138, 188, 221, 222, 246, 259, 370, 373, II. 75, 190, 258, 111. 264, 267
A potome: compound irrational straight line (difference between two "terms ") 111. 7: defined III. 158-9: connected by Theae- tetus with harmonic mean III. 3, 4: biquadratic from which it arises III. 7: uniquely formed 111. 167-8: first, second, third, fourth, fifth and sixth apotomes, quadratics from which arising 111. 5-6, defined III. 177, and found respectively (x. 85-90) III. 178-90: apotome equivalent to square root of first apotome 111. 190-4: first, second, third, fourth, fifth and sixth apotomes equivalent to squares of apotome, first apotome of a medial etc. III. 212-29: apotome cannot be binomial also III. 240-2: different from medial (straight line) and from other irrationals of same series with itself III. 242: used to rationalise binomial with proportional terms III. 243-8, 252-4, Apotome of a medial (straight line): first and second, and biquadratics of which they are roots III. 7: first apotome of a medial defined III. 159-60, uniquely formed III. 168-9, equivalent to square root of second apotome III. 194-8: second apotome of a medial, defined III. 161-2, uniquely formed III. 170-2, equivalent to square root of third apotome III. 199-202 Application of areas 1. 36, 343-5: contrasted with exceeding and falling-short 1. 343: complete method equivalent to geometrical solution of mixed quadratic equation 1. 344-5, 383-5, 386-8, 11. 187, 258-60, 263-5, 266-7: adaptation to conics (Apol- lonius) I. 344-5: application contrasted with construction (Proclus) I. 343 Approximations: 7/5 as approximation to√2 (Pythagoreans and Plato) II. 119: approxi- mations to 3 in Archimedes and (in sexagesimal fractions) in Ptolemy II. 119: to T (Archimedes) 11. 119: to √4500 (Theon of Alexandria) 11. 119: remarkably close approximations (stated in sexagesimal fractions) in scholia to Book X., III. 523 Aqaton I. 88
Arabian editors and commentators I. 75- 90
a proportion between commensurables to cover incommensurables II. 193: "Axiom" of (called however "lemma," assumption, by A. himself) 1. 234: relation of "Axiom" to X. I, III. 15-6: "Axiom" already used by Eudoxus and mentioned by Aristotle III. 16: proved by means of Dedekind's Postulate (Stolz) III. 16: on discovery by Eudoxus of method of ex- haustion III. 365-6, 374: new fragment of," method (ěpodos) of Archimedes about mechanical theorems," or épóôtov, dis- covered by Heiberg and published and annotated by him and Zeuthen II. 40, III. 366-8, adds new chapter to history of integral calculus, which the method actually is, III. 366-7: application to area of para- bolic segment, ibid.: spiral of Archimedes I. 26, 267: I. 116, 142, 225, 370, II. 136, 190, III. 246, 270, 375, 521
Archytas 1. 20: proof that there is no numerical geometric mean between ʼn and n+1 II. 295
Areskong, M. E. 1. 113
Arethas, Bishop of Caesarea 1. 48: owned Bodleian MS. (B) 1. 47-8: had famous Plato Ms. of Patmos (Cod. Clarkianus) written I. 48
Aristaeus 1. 138: on conics 1. 3: Solid Loci 1. 16, 329: comparison of five (regular solid) figures 1. 6, 111. 438-9, 513 Aristotelian Problems 1. 166, 182, 187 Aristotle: on nature of elements 1. 116: on first principles 1. 177 sqq.: on definitions 1. 117, 119-20, 143-4, 146-50: on distinc- tion between hypotheses and definitions I. 119, 120, between hypotheses and postulates 1. 118, 119, between hypotheses and axioms I. 120: on axioms 1. 119-21: axioms indemonstrable 1. 121: on defini- tion by negation 1. 156-7: on points 1. 155-6, 165: on lines, definitions of, I. 158-9, classification of, 1. 159-60: quotes Plato's definition of straight line 1. 166: on definitions of surface 1. 170: definition of "body" as that which has three dimensions or as "depth" III. 262: body "bounded by surfaces" (émɩmédois) III. 263: speaks of six "dimensions" III. 263: definition of sphere III. 269: on the angle 1. 176-8: on priority as between right and acute angles 1. 181-2: on figure and definition of, 1. 182-3: definitions of "squaring" I. 149-50, 410: on parallels 1. 190-2, 308-9: on gnomon 1. 351, 355, 359: on attributes κατὰ παντός and πρῶτον
that angle in semicircle is right 11. 63: on sum of angles of triangle 1. 319-21: on sum of exterior angles of polygon 1. 322: on def. of same ratio (= same ávтavalpeois) II. 120-1: on proportion as "equality of ratios" 11. 119: on theorem in proportion (alternando) not proved generally till his time II. 113: on proportion in three terms (ovvexns, continuous), and in four terms (diŋpnuévn, discrete) II. 131, 293: on alternate ratios II. 134: on inverse ratio II. 134, 149: on similar rectilineal figures 11. 188 has locus-theorem (arising out of Eucl. VI. 3) also given in Apollonius' Piane Loci II. 198-200: on unit 11. 279: on number II. 280: on non-applicability of arithmetical proofs to magnitudes if these are not numbers II. 113: on definitions of odd and even by one another II. 281: on prime numbers II. 284-5 on composite numbers as plane and solid II. 286, 288, 290: on representation of numbers by pebbles forming figures II. 288: gives proof (no doubt Pythagorean) of incom- mensurability of 2, III. 2: I. 38, 45, 117, 150 n., 181, 184, 185, 187, 188, 195, 202, 203, 221, 222, 223, 226, 259, 262-3, 283, II. 2, 4, 22, 79, 112, 135, 149, 159, 160, 165, 184, 188, 189, III. 4 Arithmetic, Elements of, anterior to Euclid II. 295
al-Arjānī, Ibn Rāhawaihi 1. 86 Ashkal at-ta'sis I. 5 n.
Ashraf Shamsaddin as-Samarqandi, Muḥ. b. I. 5 n., 89
Asymptotic (non-secant): of lines 1. 40, 161, 203: of parallel planes 111. 265 Atheihard of Bath 1. 78, 93-6 Athenaeus of Cyzicus 1. 117
August, E. F. 1. 103, 11. 23, 25, 149, 238, 256, 412, III. 2, 48
Austin, W. I. 103, 111, II. 172, 188, 211, 259 Autolycus, On the moving sphere, 1. 17 Avicenna, I. 77, 89
"Axiom of Archimedes" III. 15-6: already used by Eudoxus, III. 15, and mentioned by Aristotle, III. 16: relation of, to Eucl. x. I, III. 15-6 Axioms, distinguished from postulates by Aristotle 1. 118-9, by Proclus (Geminus and "others") I. 40, 121-3: Proclus on difficulties in distinctions I. 123-4: distin- guished from hypotheses, by Aristotle 1. 120-1, by Proclus 1. 121-2: indemonstrable 1. 121: attempt by Apollonius to prove I. 222-3: "common (things)" or "common
opinions" in Aristotle 1. 120, 221: com- mon to all sciences I. 119, 120: called 66 common notions" in Euclid 1. 121, 221: which are genuine? 1. 221 sqq.: Proclus recognises five 1. 222, Heron three 1. 222: interpolated axioms 1. 224, 232: Pappus' additions to axioms 1. 25, 223, 224, 232: axioms of congruence, (1) Euclid's Common Notion 4, 1. 224-7, (2) modern systems (Pasch, Veronese and Hilbert) I. 228-31:
axiom" with Stoics every simple declaratory statement 1. 41, 221: axioms tacitly assumed, in Book v., II. 137, in Book VI., II. 294
Axis of sphere III. 261, 269: of cone III. 261, 271: of cylinder 111. 262, 271
Babylonians: knowledge of triangle 3, 4, 5, 1. 352: supposed discoverers of "harmonic proportion
Bacon, Roger 1. 94 Baermann, G. F. II. 213 Balbus, de mensuris 1. 91 Baltzer, R. 11. 30
Barbarin I. 219
Barlaam, arithmetical commentary on Eucl. II., I. 74
Barrow on Eucl. v. Def. 3, II. 117: on v. Def. 5, 11. 121: I. 103, 105, 110, 111, II. 56, 186, 238
Base: meaning 1. 248-9: of cone III. 262: of cylinder III. 262
Basel editio princeps of Eucl., 1. 100–1 Basilides of Tyre 1. 5, 6, III. 512 Baudhayana Sulba-Sutra 1. 360° Bayfius (Baïf, Lazare) 1. 100 Becker, J. K. 1. 174
Beez 1. 176
Beltrami, E. I. 219
Benjamin of Lesbos I. 113
Bergh, P. I. 400-1
Bernard, Edward I. 102
Besthorn and Heiberg, edition of al-Hajjaj's translation and an-Nairizi's commentary
I. 22, 27 n., 79 n. Bhaskara 1. 355
Billingsley, Sir Henry, 1. 109-10, 11. 56, 238, III. 48
Bimedial (straight line): first and second, and biquadratic equations of which they are roots III. 7: first bimedial defined III. 84-5, equivalent to square root of second binomial III. 84, 120-3, uniquely divided III. 94-5 second bimedial defined III. 85-7, equivalent to square root of third binomial III. 84, 124-5, uniquely divided III. 95-7
Binomial (straight line): compound ir- rational straight line (sum of two "terms") III. 7 defined III. 83, 84: connected by Theaetetus with arithmetic mean III. 3, 4: biquadratic of which binomial is a positive root III. 7: first, second, third, fourth, fifth and sixth binomials, quadratics from which arising III. 5-6, defined III. 101-2,
and found respectively (x. 48-53) III. 102- 15, are equivalent to squares of binomial, first bimedial etc. III. 132-45: binomial equivalent to square root of first binomial III. 116-20: binomial uniquely divided, and algebraical equivalent of this fact III. 92-4: cannot be apotome also III. 240-2: different from medial (straight line) and from other irrationals (first bimedial etc.) of same series with itself III. 242: used to rationalise apotome with proportional terms III. 248-52, 252-4 al-Biruni I. 90
Björnbo, Axel Anthon I. 17 n., 93 Boccaccio 1. 96
Bodleian MS. (B) 1. 47, 48, III. 521 Boeckh 1. 351, 371
Boethius 1. 92, 95, 184, II. 295 Bologna Ms. (b) I. 49 Bolyai, I. 1. 219
Bolyai, W. 1. 174-5, 219, 328 Bolzano I. 167
Boncompagni 1. 93 n., 104 n. Bonola, R. I. 202, 219, 237
Borelli, Giacomo Alfonso 1. 106, 194, II. 2, 84 Boundary (8pos) 1. 182, 183 Bråkenhjelm, P. R. 1. 113
Breadth (of numbers) = second dimension or factor II. 288
Breitkopf, Joh. Gottlieb Immanuel 1. 97 Bretschneider 1. 136 n., 137, 295, 304, 344, 354, 358, III. 439, 442 Briconnet, François I. 100
Briggs, Henry 1. 102, II. 143
Brit. Mus. palimpsest, 7th-8th c., I. 50 Bryson, I. 8 n.
Bürk, A. 1. 352, 360-4
Bürklen 1. 179
Buteo (Borrel), Johannes 1. 104.
Cabasilas, Nicolaus and Theodorus 1. 72 Caiani, Angelo I. 101
Camerarius, Joachim 1. 101, III. 523 Camerer, J. G. I. 103, 293, II. 22, 25, 28, 33, 34, 40, 67, 121, 131, 189, 213, 244 Çamorano, Rodrigo, 1. 112 Campanus, Johannes 1. 3, 78, 94-6, 104, 106, 110, 407, 11. 28, 41, 56, 90, 116, 119, 121, 146, 189, 211, 234, 235, 253, 275, 320, 322, 328 Candalla, Franciscus Flussates (François de Foix, Comte de Candale) 1. 3, 104, 110, 11. 189
Cantor, Moritz I. 7 n., 20, 272, 304, 318, 320, 333, 352, 355, 357-8, 360, 401, II. 5, 40, 97, III. 8, 15, 438
Cardano, Hieronimo II. 41, III. 8 Carduchi, L. I. 112
Carpus, on Astronomy, I. 34, 43: 45, 127, 128, 177
Case, technical term 1. 134: cases inter- polated 1. 58, 59: Greeks did not infer limiting cases but proved them separately
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