ENGLISH INDEX.¶
al-‘Abbās b. Sa’īd al-Jauharī 85
“Abthiniathus” (or “Anthisathus”) 203
Abū ‘l ‘Abbās al-Faḍl b. Ḥātim, see an-Nairīzī
Abū ‘Abdallāh Muḥ. b. Mu’ādh al-Jayyānī 90
Abū ‘Alī al-Baṣrī 88
Abū ‘Alī al-Ḥasan b. al-Ḥasan b. al-Haitham 88, 89
Abū Dā’ūd Sulaimān b. ‘Uqba 85, 90
Abū Ja’far al-Khāzin 77, 85
Abū Ja’far Muḥ. b. Muḥ. b. al-Ḥasan Naṣīraddīn at-Ṭūsī, see Naṣīraddīn
Abū Muḥ. b. Abdalbāqī al-Baġdādī al-Faraḍī 8 n., 90
Abū Muḥ. al-Ḥasan b. ‘Ubaidallāh b. Sulaimān b. Wahb 87
Abū Naṣr Ġars al-Na’ma 90
Abū Naṣr Manṣūr b. ‘Alī b. ‘Irāq 90
Abū Naṣr Muḥ. b. Muḥ. b. Tarkhān b. Uzlaġ al-Fārābī 88
Abū Sahl Wījan b. Rustam al-Kūhī 88
Abū Sa’īd Sinān b. Thābit b. Qurra 88
Abū ‘Uthmān ad-Dimashqī 25, 77
Abū ‘l Wafā al-Būzjānī 77, 85, 86
Abū Yūsuf Ya’qūb b. Isḥāq b. aṣ-Ṣabbāḥ al-Kindī 86
Abū Yūsuf Ya’qūb b. Muḥ. ar-Rāzī 86
Adjacent (ἐφεξῆς), meaning 181
Aenaeas (or Aigeias) of Hierapolis 28, 311
Aganis 27–8, 191
Aḥmad b. al-Ḥusain al-Ahwāzī al-Kātib 89
Aḥmad b. ‘Umar al-Karābīsī 85
al-Ahwāzī 89
Aigeias (? Aenaeas) of Hierapolis 28, 311
Alexander Aphrodisiensis 7 n., 29
Algebra, geometrical, 372–4: classical method was that of Eucl. II. (cf. Apollonius) 373: preferable to semi-algebraical method 377–8: semi-algebraical method due to Heron 373, and favoured by Pappus 373: geometrical equivalents of algebraical operations 374: algebraical equivalents of propositions in Book II. 372–3
Alī b. Aḥmad Abū ‘l Qāsim al-Anṭākī 86
Allman, G. J. 135 n., 318, 352
Alternate (angles) 308
Alternative proofs, interpolated, 58, 59
Amaldi 175, 179–80, 193, 201, 313, 328
Ambiguous case 306–7
Amphinomus 125, 128, 150 n.
Amyclas of Heraclea 117
Analysis (and synthesis) 18: alternative proofs of XIII. 1–5 by, 137: definitions of, interpolated, 138: described by Pappus 138–9: modern studies of Greek analysis 139: theoretical and problematical analysis 138: Treasury of analysis (τόπος ἀναλυόμενος) 8, 10, 11, 138: method of analysis and precautions necessary to 139–40: analysis and synthesis of problems 140–2: two parts of analysis (a) transformation, (b) resolution, and two parts of synthesis, (a) construction, (b) demonstration 141: example from Pappus 141–2: analysis should also reveal διορισμός (conditions of possibility) 142
Analytical method 36: supposed discovery of, by Plato 134, 137
Anaximander 370
Anchor-ring 163
Andron 126
Angle. Curvilineal and rectilineal, Euclid’s definition of, 176 sq.: definition criticised by Syrianus 176: Aristotle’s notion of angle as κλάσις 176: Apollonius’ view of, as contraction 176, 177: Plutarch and Carpus on, 177: to which category does it belong? quantum, Plutarch, Carpus, “Aganis” 177, Euclid 178; quale, Aristotle and Eudemus 177–8: relation, Euclid 178: Syrianus’ compromise 178: treatise on the Angle by Eudemus 34, 38, 177–8: classification of angles (Geminus) 178–9: curvilineal and “mixed” angles 26, 178–9, horn-like (κερατοειδής) 177, 178, 182, 265, lune-like (μηνοειδής) 26, 178–9, scraper-like (ξυστροειδής) 178: angle of a segment 253: angle of a semicircle 182, 253: definitions of angle classified 179: recent Italian views 179–81: angle as cluster of straight lines or rays 180–1, defined by Veronese 180: as part of a plane (“angular sector”) 179–80: flat angle (Veronese etc.) 180–1, 269: three kinds of angles, which is prior (Aristotle)? 181–2: adjacent angles 181: alternate 308: similar (= equal) 178, 182, 252: vertical 278: exterior and interior (to a figure) 263, 280: exterior when re-entrant 263: interior and opposite 280: construction by Apollonius of angle equal to angle 296: angle in a semicircle, theorem of, 317–19: trisection of angle, by conchoid of Nicomedes 265–6, by quadratrix of Hippias 266, by spiral of Archimedes 267
al-Anṭākī 86
Antiphon 7 n., 35
“Anthisathus” (or “Abthiniathus”) 203
Āpastamba-Śulba-Sūtra 352: evidence in, as to early discovery of Eucl. I. 47 and use of gnomon 360–4: Bürk’s claim that Indians had discovered the irrational 363–4: approximation to $\sqrt{2}$ and Thibaut’s explanation 361, 363–4: inaccurate values of $\pi$ in, 364
Apollodorus “Logisticus” 37, 319, 351
Apollonius: disparaged by Pappus in comparison with Euclid 3: supposed by some Arabians to be author of the Elements 5: a “carpenter” 5: on elementary geometry 42: on the line 159: on the angle 176: general definition of diameter 325: tried to prove axioms 42, 62, 222–3: his “general treatise” 42: constructions by, for bisection of straight line 268, for a perpendicular 270, for an angle equal to an angle 296: on parallel-axiom (?) 42–3: adaptation to conics of theory of application of areas 344–5: geometrical algebra in, 373: Plane Loci 14, 259, 330: Plane νεύσεις 151: comparison of dodecahedron and icosahedron 6: on the cochlias 34, 42, 162: on unordered irrationals 42, 115: 138, 188, 221, 222, 246, 249, 259, 370, 373
Application of areas 36, 343–5: contrasted with exceeding and falling-short 343: complete method equivalent to geometric solution of mixed quadratic equation 344–5, 383–5, 386–8: adaptation to conics (Apollonius) 344–5: application contrasted with construction (Proclus) 343
“Aqaton” 88
Arabian editors and commentators 75–90
Arabic numerals in scholia to Book X., 12th c., 71
Archibald, R. C. 9 n., 10
Archimedes 20, 21, 116, 142: “postulates” in, 120, 123: famous “lemma” (assumption) known as Postulate of Archimedes 234: “Porisms” in, 11 n., 13: spiral of, 26, 267: on straight line 166: on plane 171–2: 225, 249, 370
Archytas 20
Areskong, M. E. 113
Arethas, Bishop of Caesarea 48: owned Bodleian MS. (B) 47–8: had famous Plato MS. of Patmos (Cod. Clarkianus) written 48
Argyrus, Isaak 74
Aristaeus 138: on conics 3: Solid Loci 16, 329: comparison of five (regular solid) figures 6
Aristotelian Problems 166, 182, 187
Aristotle: on nature of elements 116: on first principles 117 sqq.: on definitions 117, 119–20, 143–4, 146–50: on distinction between hypotheses and definitions 119, 120, between hypotheses and postulates 118, 119, between hypotheses and axioms 120: on axioms 119–21: axioms indemonstrable 121: on definition by negation 156–7: on points 155–6, 165: on lines, definitions of 158–9, classification of 159–60: quotes Plato’s definition of straight line 166: on definitions of surface 170: on the angle 176–8: on priority as between right and acute angles 181–2: on figure and definition of 182–3: definitions of “squaring” 149–50, 410: on parallels 190–2, 308–9: on gnomon 351, 355, 359: on attributes κατὰ παντός and πρῶτον καθόλου 319, 320, 325: on the objection 135: on reduction 135: on reductio ad absurdum 136: on the infinite 232–4: supposed postulate or axiom about divergent lines taken by Proclus from, 45, 207: gives pre-Euclidean proof of I. 5 252–3: on theorem of angle in a semicircle 149: on sum of angles of triangle 319–21: on sum of exterior angles of polygon 322: 38, 45, 117, 150 n., 181, 184, 185, 187, 188, 195, 202, 203, 221, 222, 223, 226, 259, 262–3, 283, 411
Arithmetical calculations in scholia to Bk. X. 71, 74
al-Arjānī, Ibn Rāhwaihī 86
Ashkāl at-ta’sīs 5 n.
Ashraf Shamsaddīn as-Samarqandī, Muḥ. b. 5 n., 89
Astaroff, Ivan 113
Asymptotic (non-secant) lines 40, 161, 203
Athelhard of Bath 78, 93–6
Athenaeus 20, 351
Athenaeus of Cyzicus 117
August, E. F. 103
Austin, W. 103, 111
Autolycus, On the moving sphere 17
Avicenna 77, 89
Axioms, distinguished from postulates by Aristotle 118–9, by Proclus (Geminus and “others”) 40, 121–3: Proclus on difficulties in distinctions 123–4: distinguished from hypotheses, by Aristotle 120–1, by Proclus 121–2: indemonstrable 121: attempt by Apollonius to prove 222–3: = “common (things)” or “common opinions” in Aristotle 120, 221: common to all sciences 119, 120: called “common notions” in Euclid 121, 221: which are genuine? 221 sqq.: Proclus recognises five 222, Heron three 222: interpolated axioms 224, 232: Pappus’ additions to axioms 25, 223, 224, 232: axioms of congruence, (1) Euclid’s Common Notion 4, 224–7, (2) modern systems (Pasch, Veronese and Hilbert) 228–31: “axiom” with Stoics = every simple declaratory statement 41, 221
Bacon, Roger 94, 416
Balbus, de mensuris 91
Barbarin 219
Barlaam, arithmetical commentary on Eucl. II. 74
Barrow, 103, 105, 110, 111
Base, meaning 248–9
Basel, editio princeps of Eucl. 100–1
Basilides of Tyre 5, 6
Bāudhāyana Śulba-Sūtra 360
Bayfius (Baïf, Lazare) 100
Becker, J. K. 174
Beez 176
Behā-ad-dīn 417
Beltrami, E. 219
Benjamin of Lesbos 113
Bergh, P. 400–1
Bernard, Edward, 102
Besthorn and Heiberg, edition of al-Ḥajjāj’s translation and an-Nairīzī’s commentary 22, 27 n., 79 n.
Bhāskara 355, 418
Billingsley, Sir Henry 109–10, 418
al-Bīrūnī 90
Björnbo, Axel Anthon 17 n., 93
Boccaccio 96
Bodleian MS. (B) 47, 48
Boeckh 351, 371
Boethius 92, 95, 184
Bologna MS. (b) 49
Bolyai, J. 219
Bolyai, W. 174–5, 219, 328
Bolzano 167
Boncompagni 93 n., 104 n.
Bonola, R. 202, 219
Borelli, Giovanni Alfonso 106, 194
Boundary (ὅρος) 182, 183
Bråkenhjelm, P. R. 113
Breitkopf, Joh. Gottlieb Immanuel 97
Bretschneider 136 n., 137, 295, 304, 344, 354, 358
Briconnet, François 100
“Bride, Theorem of the,” = Eucl. I. 47, 417–8
“Bride’s Chair,” name for I. 47, 417–8
Briggs, Henry 103
Brit. Mus. palimpsest, 7th–8th c., 50
Bryson 8 n.
Bürk, A. 352, 360–4
Bürklen 179
Buteo (Borrel), Johannes 104
Cabasilas, Nicolaus and Theodorus 72
Caiani, Angelo 101
Camerarius, Joachim 101
Camerer, J. G. 103, 293
Çamorano, Rodrigo 112
Campanus, Johannes, 3, 78, 94–96, 104, 106, 110, 407
Candalla, Franciscus Flussates (François de Foix, Comte de Candale) 3, 104, 110
Cantor, Moritz 272, 304, 318, 320, 333, 352, 355, 357–8, 360, 401
Carduchi, L. 112
Carpus, on Astronomy 34, 43: 45, 127, 128, 177
Case, technical term 134: cases interpolated 58, 59
Casiri 4 n., 17 n.
Cassiodorus, Magnus Aurelius 92
Cataldi, Pietro Antonio 106
Catoptrica, attributed to Euclid, probably Theon’s 17: Catoptrica of Heron 21, 253
“Cause”: consideration of, omitted by commentators 19, 45: definition should state cause (Aristotle) 149: cause = middle term (Aristotle) 149: question whether geometry should investigate cause (Geminus), 45, 150 n.
Censorinus 91
Centre, κέντρον 184–5
Ceria Aristotelica 35
Chasles on Porisms of Euclid 10, 11, 14, 15
Chaucer: Dulcarnon in 416–7, 418
Chinese, knowledge of triangle 3, 4, 5, 352: “Chóu-peï” 355
Chou Kung 362
Chrysippus 330
Cicero 91, 351
Circle: definition of, 183–5: = round, στρογγύλον (Plato) 184: = περιφερόγραμμον (Aristotle) 184: a plane figure 183–4: centre of, 184–5: pole of, 185: bisected by diameter (Thales) 185, (Saccheri) 185–6: intersections with straight line 237–8, 272–4, with another circle 238–40, 242–3, 293–4
Circumference, περιφέρεια 184
Cissoid 161, 164, 176, 330
Clairaut 328
Claymundus, Joan. 101
Clavius (Christoph Klau?) 103, 105, 194, 232, 381, 391, 407
Cleonides, Introduction to Harmony 17
Cochlias or cochlion (cylindrical helix) 162
Codex Leidensis 399, 1: 22, 27 n., 79 n.
Coets, Hendrik 109
Commandinus 4, 102, 103, 104–5, 106, 110, 111, 407: scholia included in translation of Elements 73: edited (with Dee) De divisionibus 8, 9, 110
Commentators on Eucl. criticised by Proclus 19, 26, 45
Common Notions: = axioms 62, 120–1, 221–2: which are genuine? 221 sq.: meaning and appropriation of term 221: called “axioms” by Proclus 221
Complement, παραπλήρωμα: meaning of, 341: “about diameter” 341: not necessarily parallelograms 341: use for application of areas 342–3
Composite, σύνθετος, (of lines) 160, (of surfaces) 170
Conchoids 160–1, 265–6, 330
Conclusion, συμπέρασμα: necessary part of a proposition 129–30: particular conclusion immediately made general 131: definition merely stating conclusion 149
Congruence-Axioms or Postulates: Common Notion 4 in Euclid 224–5: modern systems of (Pasch, Veronese, Hilbert), 228–31
Congruence theorems for triangles, recapitulation of, 305–6
Conics, of Euclid 3, 16: of Aristaeus 3, 16: of Apollonius 3, 16: fundamental property as proved by Apollonius equivalent to Cartesian equation 344–5: focus-directrix property proved by Pappus 15
Constantinus Lascaris 3
Construct (συνίστασθαι) contrasted with describe on 348, with apply to 343: special connotation 259, 289
Construction, κατασκευή, one of formal divisions of a proposition 129: sometimes unnecessary 130: turns nominal into real definition 146: mechanical 151, 387
Continuity, Principle of, 234 sq., 242, 272, 294
Conversion, geometrical: distinct from logical 256: “leading” and partial varieties 256–7, 337
Copernicus 101
Cordonis, Mattheus 97
Cosmic figures (= five regular solids) 413–4
Cratistus 133
Crelle, on the plane 172–4
Ctesibius 20, 21, 59 n.
Cunn, Samuel 111
Curtze, Maximilian, editor of an-Nairīzī 22, 78, 92, 94, 96, 97 n.
Curves, classification of: see line
Cylindrical helix 161, 162, 329, 330
Czecha, Jo. 113
Dasypodius (Rauchfuss), Conrad 73, 102
Data of Euclid 8, 132, 141, 385, 391
Deahna 174
Dechales, Claude François Milliet 106, 107, 108, 110
Dedekind’s Postulate, and applications 235–40
Dee, John 109, 110: discovered De divisionibus 8, 9
Definition, in sense of “closer statement” (διορισμός), one of formal divisions of a proposition 129: may be unnecessary 130
Definitions: Aristotle on, 117, 119, 120, 143: a class of thesis (Aristotle) 120: distinguished from hypotheses 119, but confused therewith by Proclus 121–2: must be assumed 117–9, but say nothing about existence (except in the case of a few primary things) 119, 143: terms for, ὅρος and ὁρισμός 143: real and nominal definitions (real = nominal plus postulate or proof), Mill anticipated by Aristotle, Saccheri and Leibniz 143–5: Aristotle’s requirements in, 146–50, exceptions 148: should state cause or middle term and be genetic 149–50: Aristotle on unscientific definitions (ἐκ μὴ προτέρων) 148–9: Euclid’s definitions agree generally with Aristotle’s doctrine 146: interpolated definitions 61, 62: definitions of technical terms in Aristotle and Heron, not in Euclid 150
De levi et ponderoso, tract 18
Demetrius Cydonius 72
Democritus 38: treatise on irrationals 413
De Morgan 246, 260, 269, 284, 291, 298, 300, 309, 313, 314, 315, 369, 376
Desargues 193
Describe on (ἀναγράφειν ἀπό) contrasted with construct 348
De Zolt 328
Diagonal (διαγώνιος) 185
“Diagonal” numbers: see “Side-” and “diagonal-” numbers
Diameter (διάμετρος), of circle or parallelogram 185: as applied to figures generally 325: “rational” and “irrational” diameter of 5 (Plato) 399, taken from Pythagoreans 399–400, 413
Diels, H. 412
Dimensions (cf. διαστάσεις) 157, 158: Aristotle’s view of, 158–9
Dinostratus 117, 266
Diocles 164
Diodorus 203
Diogenes Laertius 37, 305, 317, 351
Dionysius, friend of Heron, 21
Diophantus 86
Diorismus (διορισμός) = (a) “definition” or “specification,” a formal division of a proposition 129: (b) condition of possibility 128, determines how far solution possible and in how many ways 130–1, 243: diorismi said to have been discovered by Leon 116: revealed by analysis 142: introduced by δεῖ δή 293: first instances in Elements 234, 293
Dippe 108
Direction, as primary notion, discussed 179: direction-theory of parallels 191–2
Distance, διάστημα: = radius 199: in Aristotle has usual general sense and = dimension 199
Division (method of), Plato’s 134
Divisions (of figures) by Euclid 8, 9: translated by Muḥammad al-Baġdādī 8: found (by Woepcke) in Arabic 9, and (by Dee) in Latin translation 8, 9: 110
Dodecahedron in sphere 411
Dodgson, C. L. 194, 254, 261, 313
Dou, Jan Pieterszoon 108
Duhamel 139, 328
Dulcarnon, name for Eucl. I. 47, 416, 418
Egyptians, knowledge of $3^2 + 4^2 = 5^2$, 352
Eibe, Thyra 113
Elefuga, name for Eucl. I. 5, 416–7
Elements: pre-Euclidean Elements, by Hippocrates of Chios, Leon 116, Theudius 117: contributions to, by Eudoxus 1, 37, Theaetetus 1, 37, Hermotimus of Colophon 117: Euclid’s Elements, ultimate aims of 2, 115–6: commentators on 19–45, Proclus 19, 29–45 and passim, Heron 20–24, an-Nairīzī 21–24, Porphyry 24, Pappus 24–27, Simplicius 28, Aenaeas (Aigeias) 28: MSS. of 46–51: Theon’s changes in text 54–58: means of comparing Theonine with ante-Theonine text 51–53: interpolations before Theon’s time 58–63: scholia 64–74: external sources throwing light on text, Heron, Taurus, Sextus Empiricus, Proclus, Iamblichus 62–3: Arabic translations (1) by al-Ḥajjāj 75, 76, 79, 80, 83–4, (2) by Isḥāq and Thābit b. Qurra 75–80, 83–4, (3) Naṣīraddīn at-Ṭūsī 77–80, 84: Hebrew translation by Moses b. Tibbon or Jakob b. Machir 76: Arabian versions compared with Greek text 79–83, with one another 83, 84: translation by Boethius 92: old translation of 10th c., 92: translation by Athelhard 93–6, Gherard of Cremona 93–4,
ENGLISH INDEX¶
Campanus 94–6, 97–110 etc., Zamberti 98–100, Commandinus 104–5: introduction into England, 10th c., 95: translation by Billingsley 109–10: Greek texts, editio princeps 100–1, Gregory’s 102–3, Peyrard’s 103, August’s 103, Heiberg’s passim: translations and editions generally 97–113: on the nature of elements (Proclus) 114–6, (Menaechmus) 114, (Aristotle) 116: Proclus on advantages of Euclid’s Elements 115: immediate recognition of, 116: first principles of, definitions, postulates, common notions (axioms) 117–24: technical terms in connexion with, 125–42: no definitions of such technical terms 150: sections of Book I. 308
Elinuam 95
Engel and Stäckel 219, 321
Enriques, F. 113, 157, 175, 193, 195, 201, 313
Enunciation (πρότασις), one of formal divisions of a proposition 129–30
Epicureans, objection to I. 20 41, 287: Savile on, 287
Equality, in sense different from that of congruence (=”equivalent,” Legendre) 327–8: two senses of equal (1) “divisibly-equal” (Hilbert) or “equivalent by sum” (Amaldi), (2) “equal in content” (Hilbert) or “equivalent by difference” (Amaldi) 328: modern definition of, 228
Eratosthenes 1: contemporary with Archimedes 1, 2
Errard, Jean, de Bar-le-Duc 108
Erycinus 27, 290, 329
Euclid: account of, in Proclus’ summary 1; date 1–2: allusions to in Archimedes 1: (according to Proclus) a Platonist 2: taught at Alexandria 2: Pappus on personality of, 3: story of (in Stobaeus) 3: not “of Megara” 3, 4: supposed to have been born at Gela 4: Arabian traditions about, 4, 5: “of Tyre” 4–6: “of Ṭūs” 4, 5 n.: Arabian derivation of name (“key of geometry”) 6: Elements, ultimate aim of, 2, 115–6: other works, Conics 16, Pseudaria 7, Data 8, 132, 141, 385, 391, On divisions (of figures) 8, 9, Porisms 10–15, Surface-loci 15, 16, Phaenomena 16, 17, Optics 17, Elements of Music or Sectio Canonis 17: on “three- and four-line locus” 3: Arabian list of works 17, 18: bibliography 91–113
Eudemus 29: On the Angle 34, 38, 177–8: History of Geometry 34, 35–8, 278, 295, 304, 317, 320, 387, 412
Eudoxus 1, 37, 74, 116: discoverer of theory of proportion as expounded generally in Bks. V., VI. 137, 351: on the golden section 137: founder of method of exhaustion 234: inventor of a certain curve, the hippopede, horse-fetter 163: possibly wrote Sphaerica 17
Euler, Leonhard 401
Eutocius 25, 35, 39, 142, 161, 164, 259, 317, 329, 330, 373
Exterior and interior (of angles) 263, 280
Extremity, πέρας 182, 183
Falk, H. 113
al-Faraḍī 8 n., 90
Figure, as viewed by Plato 182, by Aristotle 182–3, by Euclid 183: according to Posidonius is confining boundary only 41, 183: figures bounded by two lines classified 187: angle-less (ἀγώνιον) figure 187
Figures, printing of, 97
Fihrist 4 n., 5 n., 17, 21, 24, 25, 27: list of Euclid’s works in 17, 18
Finaeus, Orontius (Oronce Fine) 101, 104
Flauti, Vincenzo 107
Florence MS. Laurent. XXVIII. 3, (F) 47
Flussates, see Candalla
Forcadel, Pierre 108
Fourier 173–4
Francisci tunica, “Franciscan’s cowl,” name for Eucl. I. 47, 418
Frankland, W. B. 173, 199
Frischauf 174
Gartz 17 n.
Gauss 172, 193, 194, 202, 219, 321
Geminus: name 38–9: title of work (φιλοκαλία) quoted from by Proclus 39: elements of astronomy 38: comm. on Posidonius 39: Proclus’ obligations to, 39–42: on postulates and axioms 122–3: on theorems and problems 128: two classifications of lines (or curves) 160–2: on homoeomeric (uniform) lines 162: on “mixed” lines (curves) and surfaces 162: classification of surfaces 170, of angles 178–9: on parallels 191: on Postulate 4, 200: on stages of proof of theorem of I. 32, 317–20: 27–8, 37, 44, 45, 74, 133 n., 203, 265, 330
Geometrical algebra 372–4: Euclid’s method in Book II. evidently the classical method 373: preferable to semi-algebraical method 377–8
Georgius Pachymeres 417
Gherard of Cremona, translator of Elements 93–4: of an-Nairīzī’s commentary 22, 27 n., 94: of tract De divisionibus 9, 10 n.
Giordano, Vitale 106, 176
Given, δεδομένος, different senses, 132–3
Gnomon: literally “that enabling (something) to be known” 64, 370: successive senses of, (1) upright marker of sundial 181, 185, 271–2, introduced into Greece by Anaximander 370, (2) carpenter’s square for drawing right angles 371, (3) figure placed round square to make larger square 351, 371, Indian use of gnomon in this sense 362, (4) use extended by Euclid to parallelograms 371, (5) by Heron and Theon to any figures 371–2: Euclid’s method of denoting in figure 383: arithmetical use of, 358–60, 371
“Gnomon-wise” (κατὰ γνώμονα), old name for perpendicular (κάθετος) 36, 181, 272
ENGLISH INDEX¶
Görland, A. 233, 234
“Golden section” = section in extreme and mean ratio 137: connexion with theory of irrationals 137
“Goose’s foot” (pes anseris), name for Eucl. III. 7, 99, 418
Gow, James 135 n.
Gracilis, Stephanus 101–2
Grandi, Guido 107
Greenhill, Sir George 415, 418
Gregory, David 102–3
Gregory of St Vincent 401, 404
Gromatici 91 n., 95
Grynaeus 100–1
al-Haitham 88, 89
al-Ḥajjāj b. Yūsuf b. Maṭar, translator of the Elements 22, 75, 76, 79, 80, 83, 84
Halifax, William 108, 110
Halliwell 95 n.
Hankel, H. 139, 141, 232, 234, 344, 354
Harmonica of Ptolemy, Comm. on, 17
Harmony, Introduction to, not by Euclid 17
Hārūn ar-Rashīd 75
al-Ḥasan b. ʿUbaidallāh b. Sulaimān b. Wahb 87
Hauff, J. K. F. 108
“Heavy and Light,” tract on, 18
Heiberg, J. L. passim
Helix, cylindrical 161, 162, 329, 330
Helmholtz 226, 227
Henrici and Treutlein 313, 404
Henrion, Denis 108
Hérigone, Pierre 108
Herlin, Christian 100
Hermotimus of Colophon 1, 117
Herodotus 37 n., 370
“Heromides” 158
Heron of Alexandria, mechanicus, date of 20–1: Heron and Vitruvius 20–1: commentary on Euclid’s Elements 20–4: direct proof of I. 25, 301: comparison of areas of triangles in I. 24, 334–5: addition to I. 47, 366–8: apparently originated semi-algebraical method of proving theorems of Book II. 373, 378: 137 n., 159, 163, 168, 170, 171–2, 176, 183, 184, 185, 188, 189, 222, 223, 243, 253, 285, 287, 299, 351, 369, 371, 405, 407, 408
Heron, Proclus’ instructor 29
“Herundes” 156
Hieronymus of Rhodes 305
Hilbert 157, 193, 201, 228–31, 249, 313, 328
Hipparchus 4 n., 20, 30 n., 74 n.
Hippasus 411
Hippias of Elis 42, 265–6, 413
Hippocrates of Chios 8 n., 29, 35, 38, 116, 135, 136 n., 386–7, 413
Hippopede (ἵππου πέδη), a certain curve used by Eudoxus 162–3, 176
Hoffmann, Heinrich 107
Hoffmann, John Jos. Ign. 108, 365
Holtzmann, Wilhelm (Xylander) 107
Homoeomeric (uniform) lines 40, 161, 162
Hornlike (angle), κερατοειδής 177, 178, 182, 265
Horsley, Samuel 106
Hoüel, J. 219
Hudson, John 102
Hultsch, F. 17 n., 74, 329, 400
Ḥunain b. Isḥāq al-ʿIbādī 75
Hypotheses, in Plato 122: in Aristotle 118–20: confused by Proclus with definitions 121–2: geometer’s hypotheses not false (Aristotle) 119
Hypothetical construction 199
Hypsicles 5: author of Book XIV. 5, 6
Iamblichus 63, 83, 417
Ibn al-ʿAmīd 86
Ibn al-Haitham 88, 89
Ibn al-Lubūdī 90
Ibn Rāhawaihi al-Arjānī 86
Ibn Sīnā (Avicenna) 77, 89
“Iflaton” 88
Incomposite (of lines) 160–1, (of surfaces) 170
Indivisible lines (ἄτομοι γραμμαί), theory of, rebutted 268
Infinite, Aristotle on the, 232–4: infinite division not assumed, but proved, by geometers 268
Infinity, parallels meeting at, 192–3
Ingrami, G. 175, 193, 195, 201, 227–8
Interior and exterior (of angles) 263, 280: interior and opposite angle 280
Interpolations in the Elements before Theon’s time 58–63: by Theon 46, 55–6: I. 40 interpolated 338
Irrational: discovered with reference to $\sqrt{2}$ 351, 411, 412–3: claim of India to priority of discovery 363–4: “irrational diameter of 5” (Pythagoreans and Plato) 399–400, 413: approximation to $\sqrt{2}$ by means of “side-” and “diagonal-” numbers 399–401, to $\sqrt{2}$ and $\sqrt{3}$ in sexagesimal fractions 74 n.: Indian approximation to $\sqrt{2}$ 361, 363–4: unordered irrationals (Apollonius) 42, 115: irrational ratio (ἄρρητος λόγος) 137
Isaacus Monachus (or Argyrus) 73–4, 407
Isḥāq b. Ḥunain b. Isḥāq al-ʿIbādī, Abū Yaʿqūb, translation of Elements by, 75–80, 83–4
Ismaʿīl b. Bulbul 88
Isoperimetric (or isometric) figures: Pappus and Zenodorus on, 26, 27, 333
Isosceles (ἰσοσκελής) 187: of numbers (=even) 188: isosceles right-angled triangle 352
Jakob b. Machir 76
Jan, C. 17
al-Jauharī, al-ʿAbbās b. Saʿīd 85
al-Jayyānī 90
Joannes Pediasimus 72–3
Junge, G., on attribution of theorem of I. 47 and discovery of irrationals to Pythagoras 351, 411, 413
Kästner, A. G. 78, 97, 101
al-Karābīsī 85
ENGLISH INDEX¶
Kātyāyana Śulba-Sūtra 360
Keill, John 105, 110–11
Kepler 193
al-Khāzin, Abū Jaʿfar 77, 85
Killing, W. 194, 219, 225–6, 235, 242, 272
al-Kindī 5 n., 86
Klamroth, M. 75–84
Klau (?), Christoph = Clavius 105
Klügel, G. S. 212
Knesa, Jakob 112
Knoche 32 n., 33 n., 73
Kroll, W. 399–400
al-Kūhī 88
Lambert, J. H. 212–3
Lardner, Dionysius 112, 246, 250, 298, 404
Lascaris, Constantinus 3
Leading theorems (as distinct from converse) 257: leading variety of conversion 256–7
Leeke, John 110
Lefèvre, Jacques 100
Legendre, Adrien Marie 112, 169, 213–9
Leibniz 145, 169, 176, 194
Leiden MS. 399, 1 of al-Ḥajjāj and an-Nairīzī 22
Lemma 114; meaning 133–4: lemmas interpolated 59–60, especially from Pappus 67
Leodamas of Thasos 36, 134
Leon 116
Leonardo of Pisa 9 n., 10
Leucippus 413
Linderup, H. C. 113
Line: Platonic definition 158: objection of Aristotle 158: “magnitude extended one way” (Aristotle, “Heromides”) 158: “divisible or continuous one way” (Aristotle) 158–9: “flux of point” 159: Apollonius on, 159: classification of lines, Plato and Aristotle 159–60, Heron 159–60, Geminus, first classification 160–1, second 161: straight (εὐθεῖα), curved (καμπύλη), circular (περιφερής), spiral-shaped (ἑλικοειδής), bent (κεκαμμένη), broken (κεκλασμένη), round (στρογγύλος) 159, composite (σύνθετος), incomposite (ἀσύνθετος), “forming a figure” (σχηματοποιοῦσα), determinate (ὡρισμένη), indeterminate (ἀόριστος) 160: “asymptotic” or non-secant (ἀσύμπτωτος), secant (συμπτωτός) 161: simple, “mixed” 161–2: homoeomeric (uniform) 161–2: Proclus on lines without extremities 165: loci on lines 329, 330
Linear, loci 330: problems 330
Lionardo da Vinci, proof of I. 47 365–6
Lippert 88 n.
Lobachewsky, N. I. 174–5, 213, 219
Locus-theorems (τοπικὰ θεωρήματα) and loci (τόποι): locus defined by Proclus 329: loci likened by Chrysippus to Platonic ideas 330–1: locus-theorems and loci (1) on lines (a) plane loci (straight lines and circles) (b) solid loci (conics), (2) on surfaces 329: corresponding distinction between plane and solid problems, to which Pappus adds linear problems 330: further distinction in Pappus between (1) ἐφεκτικοί (2) διεξοδικοί (3) ἀναστροφικοὶ τόποι 330: Proclus regards locus in I. 35, III. 21, 31 as an area which is locus of area (parallelogram or triangle) 330
Logical conversion, distinct from geometrical 256
Logical deductions 256, 284–5, 300: logical equivalents 309, 314–5
Lorenz, J. F. 107–8
Loria, Gino 7 n., 10 n., 11 n., 12 n.
Luca Paciuolo 98–9, 100, 418
Lucas, Edouard 418
Lundgren, F. A. A. 113
Machir, Jakob b. 76
Magni, Domenico 106
Magnitude: common definition vicious 148
al-Māhānī 85
al-Maʾmūn, Caliph 75
Manitius, C. 38
Mansion, P. 219
al-Manṣūr, Caliph 75
Manuscripts of Elements 46–51
“Marriage, Figure of” (Plutarch), name for Pythagorean triangle (3, 4, 5), 417
Martianus Capella 91, 155
Martin, T. H. 20, 29 n., 30 n.
Masʿūd b. al-Qass al-Baġdādī 90
Maximus Planudes, scholia and lectures on Elements 72
meguar = axis 93
Mehler, F. G. 404
Menaechmus: story of M. and Alexander 1: on elements 114: 117, 125, 133 n.
Menelaus 21, 23: direct proof of I. 25 300
Menge, H. 16 n., 17
Middle term, or cause, in geometry, illustrated by III. 31 149
Mill, J. S. 144
“Mixed” (lines) 161–2: (surfaces) 162, 170: different meanings of “mixed” 162
Mocenigo, Prince 97–8
Mollweide, C. B. 108
Mondoré (Montaureus), Pierre 102
Moses b. Tibbon 76
Motion, in mathematics 226: motion without deformation considered by Helmholtz necessary to geometry 226–7, but shown by Veronese to be petitio principii 226–7
Müller, J. H. T. 189
Müller, J. W. 365
Muḥammad (b. ʿAbdalbāqī) al-Baġdādī, translator of De divisionibus 8 n., 90, 110
Muḥ. b. Aḥmad Abū ʾr-Raiḥān al-Bīrūnī 90
Muḥ. b. Ashraf Shamsaddīn as-Samarqandī 89
Muḥ. b. ʿĪsā Abū ʿAbdallāh al-Māhānī 85
Munich MS. of enunciations (R) 94–5
Mūsā b. Muḥ. b. Maḥmūd Qāḍīzāde ar-Rūmī 5 n., 90
Music, Elements of (Sectio Canonis), by Euclid 17
al-Mustaʿṣim, Caliph 90
ENGLISH INDEX¶
al-Mutawakkil, Caliph 75
an-Nairīzī, Abū ʾl ʿAbbās al-Faḍl b. Ḥātim, 21–24, 85, 184, 190, 191, 195, 223, 232, 258, 270, 285, 299, 303, 326, 364, 367, 369, 373, 405, 408
Napoleon 103
Naṣīraddīn aṭ-Ṭūsī 4, 5 n., 77, 84, 89, 208–10
Naẓīf b. Yumn (Yaman) al-Qass 76, 77, 87
Neide, J. G. C. 103
Nicomachus 92, 417
Nicomedes 42, 160–1, 265–6
Nipsus, Marcus Junius 305
Nominal and real definitions: see Definitions
“Nuptial Number” = Plato’s Geometrical Number in Republic 417
Objection (ἔνστασις), technical term, in geometry 135, 257, 260, 265: in logic (Aristotle) 135
Oblong 62, 151, 188
Oenopides of Chios 34, 36, 126, 271, 295, 371, 414
Ofterdinger, L. F. 9
Olympiodorus 29
Oppermann 151
Optics of Euclid 17
Oresme, N. 97
Orontius Finaeus (Oronce Fine) 101, 104
Ozanam, Jaques 107, 108
Paciuolo, Luca 98–9, 100, 418
Pamphile 317, 319
Pappus: contrasts Euclid and Apollonius 3: on Euclid’s Porisms 10–14, Surface-loci 15, 16, Data 8: on Treasury of Analysis 8, 10, 11, 138: commentary on Elements 24–7, partly preserved in scholia 66: evidence of scholia as to Pappus’ text 66–7: lemmas in Book X. interpolated from, 67: on Analysis and Synthesis 138–9, 141–2: additional axioms by, 25, 223, 224, 232: on converse of Post. 4 25, 201: proof of I. 5 by, 254: extension of I. 47 366: semi-algebraical methods in 373, 378: on loci 329, 330: on conchoids 161, 266: on quadratrix 266: on isoperimetric figures 26, 27, 333: on paradoxes of Erycinus 27, 290: 20, 39, 133 n., 137, 151, 225, 388, 391, 401
Papyrus, Herculanensis No. 1061 50, 184: Oxyrhynchus 50: Fayūm 51, 337, 338: Rhind 304, 352
Paradoxes, in geometry 188: of Erycinus 27, 290, 329: an ancient “Budget of Paradoxes” 329
Parallelogram (= parallelogrammic area), first introduced 325: rectangular parallelogram 370
Parallels: Aristotle on, 190, 191–2: definitions, by “Aganis” 191, by Geminus 191, Posidonius 190, Simplicius 190: as equidistants 190–1, 194: direction-theory of, 191–2, 194: definitions classified 192–4: Veronese’s definition and postulate 194: Parallel Postulate, see Postulate 5: Legendre’s attempt to establish theory of 213–9
Paris MSS. of Elements, (p) 49: (q) 50
Pasch, M. 157, 228, 250
“Peacock’s tail,” name for III. 8 99, 418
Pediasimus, Joannes 72–3
Peet, T. Eric 352
Peithon 203
Peletarius (Jacques Peletier) 103, 104, 249, 407
Pena 104
Perpendicular (κάθετος): definition 181: “plane” and “solid” 272: perpendicular and obliques 291
Perseus 42, 162–3
Pesch, J. G. van, De Procli fontibus 23 sqq., 29 n.
Petrus Montaureus (Pierre Mondoré) 102
Peyrard and Vatican MS. 190 (P) 46, 47, 103: 108
Pfleiderer, C. F. 168, 298
Phaenomena of Euclid 16, 17
Philippus of Medma 1, 116
Phillips, George 112
Philo of Byzantium 20, 23: proof of I. 8 263–4
Philolaus 34, 351, 371, 399, 411
Philoponus 45, 191–2
Pirckenstein, A. E. Burkh. von 107
Plane (or plane surface): Plato’s definition of, 171: Proclus’ and Simplicius’ interpretation of Euclid’s def. 171: possible origin of Euclid’s def. 171: Archimedes’ assumption 171, 172: other ancient definitions of, in Proclus, Heron, Theon of Smyrna, an-Nairīzī 171–2: “Simson’s” definition and Gauss on 172–3: Crelle’s tract on, 172–4: other definitions by Fourier 173, Deahna 174, J. K. Becker 174, Leibniz 176, Beez 176: evolution of, by Bolyai and Lobachewsky 174–5: Enriques and Amaldi, Ingrami, Veronese and Hilbert on, 175
“Plane loci” 329–30: Plane Loci of Apollonius 14, 259, 330
“Plane problems” 329
Planudes, Maximus 72
Plato: 1, 2, 3, 137, 155–6, 159, 184, 187, 203, 221, 411, 417: supposed invention of Analysis by, 134: def. of straight line 165–6: def. of plane surface 171: generation of cosmic figures by putting together triangles 226, 413: rule for rational right-angled triangles 356, 357, 359, 360, 385: “rational diameter of 5” 399, 413
“Platonic” figures 2, 413–4
Playfair, John 103, 111: “Playfair’s” Axiom 220: used to prove I. 29, 312, and Eucl. Post. 5, 313: comparison of Axiom with Post. 5, 313–4
Pliny 333
Plutarch 29, 37, 177, 343, 351, 417
Point: Pythagorean definition of, 155:
ENGLISH INDEX¶
Interpretation of Euclid’s definition 155: Plato’s view of, and Aristotle’s criticism 155–6: attributes of, according to Aristotle 156: terms for (στιγμή, σημεῖον) 156: other definitions by “Herundes,” Posidonius 156, Simplicius 157: negative character of Euclid’s def. 156: is it sufficient? 156: motion of, produces line 157: an-Nairīzī on, 157: modern explanations by abstraction 157
Polybius 331
Polygon: sum of interior angles (Proclus’ proof) 322: sum of exterior angles 322
“Pons asinorum” 415–6: “Pont aux ânes” ibid.
Porism: two senses 13: (1) = corollary 134, 278–9: interpolated Porisms (corollaries) 60–1, 381: (2) as used in Porisms of Euclid, distinguished from theorems and problems 10, 11: account of the Porisms given by Pappus 10–13: modern restorations by Simson and Chasles 14: views of Heiberg 11, 14, and Zeuthen 15
Porphyry 17: commentary on Euclid 24: Symmikta 24, 34, 44: 136, 277, 283, 287
Posidonius, the Stoic 20, 27, 28 n., 39, 189, 197: book directed against the Epicurean Zeno 34, 43: on parallels 40, 190: definition of figure 41, 183
Postulate, distinguished from axiom, by Aristotle 118–9, by Proclus (Geminus and “others”) 121–3: from hypothesis, by Aristotle 120–1, by Proclus 121–2: postulates in Archimedes 120, 123: Euclid’s view of, reconcileable with Aristotle’s 119–20, 124: postulates do not confine us to ruler and compasses 124: Postulates 1, 2, significance of, 195–6: famous “Postulate of Archimedes” 234
Postulate 4: significance of, 200: proofs of, resting on other postulates 200–1, 231: converse true only when angles rectilineal (Pappus) 201
Postulate 5: due to Euclid himself 202: Proclus on, 202–3: attempts to prove, Ptolemy 204–6, Proclus 206–8, Naṣīraddīn aṭ-Ṭūsī 208–10, Wallis 210–1, Saccheri 211–2, Lambert 212–3: substitutes for, “Playfair’s” axiom (in Proclus) 220, others by Proclus 207, 220, Posidonius and Geminus 220, Legendre 213, 214, 220, Wallis 220, Carnot, Laplace, Lorenz, W. Bolyai, Gauss, Worpitzky, Clairaut, Veronese, Ingrami 220: Post. 5 proved from, and compared with, “Playfair’s” axiom 313–4: I. 30 is logical equivalent of, 220
Potts, Robert 112, 246
Prime (of numbers): two senses of, 146
Principles, First 117–124
Problem, distinguished from theorem 124–8: problems classified according to number of solutions (a) one solution, ordered (τεταγμένα) (b) a definite number, intermediate (μέσα) (c) an infinite number of solutions, unordered (ἄτακτα) 128: in widest sense anything propounded (possible or not) but generally a construction which is possible 128–9: another classification (1) problem in excess (πλεονάζον), asking too much 129, (2) deficient problem (ἐλλιπὲς πρόβλημα), giving too little 129
Proclus: details of career 29–30: remarks on earlier commentators 19, 33, 45: commentary on Eucl. I, sources of, 29–45, object and character of, 31–2: commentary probably not continued, though continuation intended 32–3: books quoted by name in, 34: famous “summary” 37–8: list of writers quoted 44: his own contributions 44–5: character of MS. used by, 62, 63: on the nature of elements and things elementary 114–6: on advantages of Euclid’s Elements, and their object 115–6: on first principles, hypotheses, postulates, axioms 121–4: on difficulties in three distinctions between postulates and axioms 123: on theorems and problems 124–9: attempt to prove Postulate 5 206–8: on Eucl. I. 47, 350, 412: on discovery of five regular solids 413: commentary on Plato’s Republic, allusion in, to “side-” and “diagonal-” numbers in connexion with Eucl. II. 9, 10 399–400
Proof (ἀπόδειξις), necessary part of proposition 129–30
Proposition, formal divisions of, 129–131
Protarchus 5
Psellus, Michael, scholia by, 70, 71
Pseudaria of Euclid 7: Pseudographemata 7 n.
Pseudoboethius 92
Ptolemy I.: 1, 2: story of Euclid and Ptolemy 1
Ptolemy II. Philadelphus 20
Ptolemy VII. (Euergetes II.), Physcon 20
Ptolemy, Claudius 21, 30 n.: Harmonica of, and commentary on 17: on Parallel-Postulate 28 n., 34, 43, 45: attempt to prove it 204–6
Punch on “Pons Asinorum” 416
Pythagoras 4 n., 36: supposed discoverer of the irrational 351, 411, 412, of application of areas 343–4, of theorem of I. 47 343–4, 350–4, 411, 412, of construction of five regular solids 413–4; story of sacrifice 37, 343, 350: probable method of discovery of I. 47 and proof of, 352–5: suggestions by Bretschneider and Hankel 354, by Zeuthen 355–6: rule for forming right-angled triangles in rational numbers 351, 356–9, 385
Pythagoreans 19, 36, 155, 188, 279, 411–414: term for surface (χροιά) 169: angles of triangle equal to two right angles, theorem and proof 317–20: three polygons which in contact fill space round point 318: method of application of areas (including exceeding and falling short) 343, 384, 403: gnomon Pythagorean 351: “rational” and “irrational diameter of 5” 399–400, 413
ENGLISH INDEX¶
Qāḍīzāde ar-Rūmī 5 n., 90
Q.E.D. (or F.) 57
al-Qifṭī 4 n., 94
Quadratic equation, geometrical solution of, 383–5, 386–8: solution assumed by Hippocrates 386–7
Quadratrix 265–6, 330
Quadrature (τετραγωνισμός), definitions of, 149
Quadrilaterals, varieties of, 188–90
Quintilian 333
Qusṭā b. Lūqā al-Baʿlabakkī, translator of “Books XIV, XV” 76, 87, 88
Radius, no Greek word for, 199
Ramus, Petrus (Pierre de la Ramée) 104
Ratdolt, Erhard 78, 97
Rational (ῥητός): (of ratios) 137: “rational diameter of 5” 399–400, 413: rational right-angled triangles, see right-angled triangles
Rauchfuss, see Dasypodius
Rausenberger, O. 157, 175, 313
ar-Rāzī, Abū Yūsuf Yaʿqūb b. Muḥ. 86
Rectangle: = rectangular parallelogram 370: “rectangle contained by” 370
Rectilineal angle: definitions classified 179–81: rectilineal figure 187: “rectilineal segment” 196
Reductio ad absurdum 134: described by Aristotle and Proclus 136: synonyms for, in Aristotle 136: a variety of Analysis 140: by exhaustion 285, 293: nominal avoidance of 309
Reduction (ἀπαγωγή), technical term, explained by Aristotle and Proclus 135: first “reduction” of a difficult construction due to Hippocrates 135
Regiomontanus (Johannes Müller of Königsberg) 93, 96, 100
Reyher, Samuel 107
Rhaeticus 101
Rhind Papyrus 304, 352
Rhomboid 62, 151, 189
Rhombus 62, 151, meaning and derivation 189
Riccardi, P. 96, 112, 202
Riemann, B. 219, 273, 274, 280
Right angle: definition 181: drawing straight line at right angles to another, Apollonius’ construction for, 270: construction when drawn at extremity of second line (Heron) 270
Right-angled triangles, rational: rule for finding, by Pythagoras 356–9, by Plato 356, 357, 359, 360, 385: discovery of rules by means of gnomons 358–60: connexion of rules with Eucl. II. 4, 8, 360: rational right-angled triangles in Āpastamba 361, 363
Röth 357–8
Rouché and de Comberousse 313
Rudd, Capt. Thos. 110
Ruellius, Joan. (Jean Ruel) 100
Russell, Bertrand 227, 249
Saccheri, Gerolamo 106, 144–5, 167–8, 185–6, 194, 197–8, 200–1
Ṣaʿīd b. Masʿūd b. al-Qass 90
Śathapatha-Brāhmaṇa 362
Savile, Henry 105, 166, 245, 250, 262
Scalene (σκαληνός or σκαληνής) 187–8: of numbers (=odd) 188: of cone (Apollonius) 188
Schessler, Chr. 107
Scheubel, Joan. 101, 107
Schiaparelli, G. V. 163
Schmidt, Max C. P. 304, 319
Scholia to Elements and MSS. of, 64–74: historical information in, 64: evidence in, as to text 64–5, 66–7: sometimes interpolated in text 67: classes of, “Schol. Vat.” 65–9, “Schol. Vind.” 69–70: miscellaneous 71–4: “Schol. Vat.” partly derived from Pappus’ commentary 66: many scholia partly extracted from Proclus on Bk. I. 66, 69, 72: numerical illustrations in, in Greek and Arabic numerals 71: scholia by Psellus 70–1, by Maximus Planudes 72, Joannes Pediasimus 72–3: scholia in Latin published by G. Valla, Commandinus, Conrad Dasypodius 73: scholia on Eucl. II. 13 407
Schooten, Franz van 108
Schopenhauer 227, 354
Schotten, H. 167, 174, 179, 192–3, 202
Schumacher 321
Schur, F. 328
Schweikart, F. K. 219
Scipio Vegius 99
Sectio Canonis attributed to Euclid 17
Section (τομή): = point of section 170, 171, 383: “the section” = “golden section” q.v.
Segment of circle, angle of, 253: segment less than semicircle called ἁψίς 187
Semicircle 186: centre of, 186: angle of, 182, 253
Seqt 304
Serenus of Antinoeia 203
Serle, George 110
Setting-out (ἔκθεσις), one of formal divisions of a proposition 129: may be omitted 130
Sexagesimal fractions in scholia to Book X. 74
Sextus Empiricus 62, 63, 184
Shamsaddīn as-Samarqandī 5 n., 89
“Side-” and “diagonal-” numbers, described 398–400: due to Pythagoreans 400: connexion with Eucl. II. 9, 10 398–400: use for approximation to $\sqrt{2}$ 399
Sigboto 94
“Similar” (=equal) angles 182, 252: “similar” numbers 357
Simon, Max 108, 155, 157–8, 167, 202, 328
Simplicius: commentary on Euclid 27–8: on lunes of Hippocrates 29, 35, 386–7: on Eudemus’ style 35, 38: on parallels 190–1: 22, 167, 171, 184, 185, 197, 203, 223, 224, 413
Simson, Robert: on Euclid’s Porisms 14:
ENGLISH INDEX¶
on “vitiations” in Elements due to Theon 46, 103, 104, 106, 111, 148: definition of plane 172–3: 185, 186, 255, 259, 287, 293, 296, 322, 328, 384, 387, 403
Sind b. ʿAlī Abū ʾṭ-Ṭaiyib 86
Smith, D. E. 362, 417
Smith and Bryant 404
“Solid loci” 329, 330: Solid Loci of Aristaeus 16, 329
“Solid problems” 329, 330
Speusippus 125
Sphaerica, early treatise on, 17
Spiral, “single-turn” 122–3 n., 164–5: in Pappus = cylindrical helix 165
Spiral of Archimedes 26, 267
Spire (tore) or Spiric surface 163, 170; varieties of 163
Spiric curves or sections, discovered by Perseus 161, 162–4
Steenstra, Pybo 109
Steiner, Jakob 193
Steinmetz, Moritz 101
Steinschneider, M. 8 n., 76 sqq.
Stephanus Gracilis 101–2
Stephen Clericus 47
Stobaeus 3
Stoic “axioms” 41, 221: illustrations (δείγματα) 329
Stolz, O. 328
Stone, E. 105
Straight line: pre-Euclidean (Platonic) definition 165–6: Archimedes’ assumption respecting, 166: Euclid’s definition, interpreted by Proclus and Simplicius 166–7: language and construction of, 167, and conjecture as to origin 168: other definitions 168–9, in Heron 168, by Leibniz 169, by Legendre 169: two straight lines cannot enclose a space 195–6, cannot have a common segment 196–9: one or two cannot make a figure 169, 183: division of straight line into any number of equal parts (an-Nairīzī) 326
Strömer, Mårten 113
Studemund, W. 92 n.
St Vincent, Gregory of, 401, 404
Subtend, meaning and construction 249, 283, 350
Suidas 370
Sulaimān b. ʿUsma (or Oqba) 85, 90
Superposition: Euclid’s dislike of method of, 225, 249: apparently assumed by Aristotle as legitimate 226: used by Archimedes 225: objected to by Peletarius 249: no use theoretically, but merely furnishes practical test of equality 227: Bertrand Russell on, 227, 249
Surface: Pythagorean term for, χροιά (= colour, or skin) 169: terms for, in Plato and Aristotle 169: ἐπιφάνεια in Euclid (not ἐπίπεδον) 169: alternative definition of, in Aristotle 170: produced by motion of line 170: divisions or sections of solids are surfaces 170, 171: classifications of surfaces by Heron and Geminus 170: composite, incomposite, simple, mixed 170: spiric surfaces 163, 170: homoeomeric (uniform) 170: spheroids 170: plane surface, see plane: loci on surfaces 329, 330
Surface-loci of Euclid 15, 16, 330: Pappus’ lemmas on, 15, 16
Suter, H. 8 n., 17 n., 18 n., 25 n., 78 n., 85–90
Suvoroff, Pr. 113
Swinden, J. H. van 169
Synthesis, see Analysis and Synthesis
Syrianus 30, 44, 176, 178
Tacquet, André 103, 105, 111
Tāittirīya-Saṁhitā 362
Tannery, P. 7 n., 37–40, 44, 160, 163, 221, 223, 224, 225, 232, 305, 353, 412, 417
Taʾrīkh al-Ḥukamā 4 n.
Tartaglia, Niccolò 3, 103, 106
Taurinus, F. A. 219
Taurus 62, 184
Taylor, H. M. 248, 377–8, 404
Taylor, Th. 259
Thābit b. Qurra, translator of Elements 42, 75–80, 82, 84, 87, 94: proof of I. 47 364–5
Thales 36, 37, 185, 252, 253, 278, 317, 318, 319: on distance of ship from shore 304–5
Theaetetus 1, 37
Theodorus Antiochita 71
Theodorus Cabasilas 72
Theodorus Metochita 3
Theodorus of Cyrene 411, 412–3
Theognis 371
Theon of Alexandria: edition of Elements 46: changes made by, 46: Simson on “vitiations” by, 46: principles for detecting his alterations, by comparison of P, ancient papyri and “Theonine” MSS. 51–3: character of changes by, 54–8
Theon of Smyrna 172, 357, 358, 371, 398
Theorem and problem, distinguished by Speusippus 125, Amphinomus 125, 128, Menaechmus 125, Zenodotus, Posidonius 126, Euclid 126, Carpus 127, 128: views of Proclus 127–8, and of Geminus 128: “general” and “not-general” (or partial) theorems (Proclus) 325
Theudius of Magnesia 117
Thibaut, B. F. 321
Thibaut, C.: On Śulvasūtras 360, 363–4
Thompson, Thomas Perronet 112
Thucydides 333
Tibbon, Moses b. 76
Tiraboschi 94 n.
Tittel, K. 38, 39
Todhunter, I. 112, 189, 246, 258, 277, 283, 293, 298, 307
Tonstall, Cuthbert 100
Tore 163
Transformation of areas 346–7, 410
Trapezium: Euclid’s definition his own 189: further division into trapezia and trapezoids (Posidonius, Heron) 189–90: a
ENGLISH INDEX¶
theorem on area of parallel-trapezium 338–9
Treasury of Analysis (ἀναλυόμενος τόπος) 8, 10, 11, 138
Trendelenburg 146 n., 148, 149
Treutlein, P. 358–60
Triangle: seven species of, 188: “four-sided” triangle, called also “barb-like” (ἀκιδοειδές) and (by Zenodorus) κοιλογώνιον 27, 188: construction of isosceles and scalene triangles 243
Trisection of an angle 265–7
aṭ-Ṭūsī, see Naṣīraddīn
Unger, E. S. 108, 169
Vacca, Giovanni 113
Vachtchenko-Zakhartchenko 113
Vailati, G. 144 n., 145 n.
Valerius Maximus 3
Valla, G., De expetendis et fugiendis rebus 73, 98
Van Swinden 169
Vatican MS. 190 (P) 46, 47
Vaux, Carra de 20
Verona palimpsest 91
Veronese, G. 157, 168, 175, 180, 193–4, 195, 201, 226–7, 228, 249, 328
Vertical (angles) 278
Viennese MS. (V) 48, 49
Vinci, Lionardo da 365–6
Vitali, G. 237
Vitruvius 352: Vitruvius and Heron 20, 21
Viviani, Vincenzo 107, 401
Vogt, Heinrich 360, 364, 411–414
Vooght, C. J. 108
Wachsmuth, C. 32 n., 73
Wallis, John 103: edited Comm. on Ptolemy’s Harmonica 17: attempt to prove Post. 5 210–1
Weber (H.) and Wellstein (J.) 157
Weissenborn, H. 78 n., 92 n., 94 n., 95, 96, 97 n., 418
Whiston, W. 111
Williamson, James 111, 293
Witt, H. A. 113
Woepcke, F., discovered De divisionibus in Arabic and published translation 9: on Pappus’ commentary on Elements 25, 66, 77: 85 n., 86, 87
Xenocrates 268, 413
Ximenes, Leonardo 107
Xylander 107
Yaḥyā b. Khālid b. Barmak 75
Yaḥyā b. Muḥ. b. ʿAbdān b. ʿAbdalwāḥid (Ibn al-Lubūdī) 90
Yrinus = Heron 22
Yūḥannā b. Yūsuf b. al-Ḥārith b. al-Biṭrīq al-Qass 76, 87
Zamberti, Bartolomeo 98–100, 101, 104, 106
Zeno the Epicurean 34, 196, 197, 199, 242
Zenodorus 26, 27, 188, 333
Zenodotus 126
Zeuthen, H. G. 15, 113, 139, 141, 146 n., 151, 355–6, 360, 363, 387, 398, 399