GENERAL INDEX.

[The references are to volumes and pages.]

al-’Abbās b. Sa‘īd al-Jauharī I. 85
“Abthiniathus” (or “Anthisathus”) I. 203
Abū ’l ’Abbās al-Faḍl b. Ḥātim, see an-Nairīzī
Abū ’Abdallāh Muḥ. b. Mu‘ādh al-Jayyānī I. 90
Abū ’Alī al-Baṣrī I. 88
Abū ’Alī al-Ḥasan b. al-Ḥasan b. al-Haitham I. 88, 89
Abū Dā’ūd Sulaimān b. ’Uqba I. 85, 90
Abū Ja‘far al-Khāzin I. 77, 85
Abū Ja‘far Muḥ. b. Muḥ. b. al-Ḥasan Naṣīraddīn aṭ-Ṭūsī, see Naṣīraddīn
Abū Muḥ. b. Abdalbāqī al-Baġdādī al-Faraḍī I. 8 n., 90
Abū Muḥ. al-Ḥasan b. ’Ubaidallāh b. Sulaimān b. Wahb I. 87
Abū Naṣr Gars al-Na‘ma I. 90
Abū Naṣr Manṣūr b. ’Alī b. ’Irāq I. 90
Abū Naṣr Muḥ. b. Muḥ. b. Tarkhān b. Uzlaġ al-Fārābī I. 88
Abū Sahl Wījan b. Rustam al-Kūhī I. 88
Abū Sa‘īd Sinān b. Thābit b. Qurra, I. 88
Abū ’Uthmān ad-Dimashqī I. 25, 77
Abū ’l Wafā al-Būzjānī I. 77, 85, 86
Abū Yūsuf Ya‘qūb b. Isḥāq b. aṣ-Ṣabbāḥ al-Kindī I. 86
Abū Yūsuf Ya‘qūb b. Muḥ. ar-Rāzī I. 86
Adjacent (ἐφεξῆς), meaning I. 181
Adrastus II. 292
Aenaeas (or Aigeias) of Hierapolis I. 28, 311
Aganis I. 27–8, 191
Aḥmad b. al-Ḥusain al-Ahwāzī al-Kātib I. 89
Aḥmad b. ’Umar al-Karābīsī I. 85
al-Ahwāzī I. 89
Aigeias (? Aenaeas) of Hierapolis I. 28, 311
Alcinous II. 98
Alexander Aphrodisiensis I. 7 n., 29, II. 120
Algebra, geometrical I. 372–4: classical method was that of Eucl. II. (cf. Apollonius) I. 373: preferable to semi-algebraical method I. 377–8: semi-algebraical method due to Heron I. 373, and favoured by Pappus I. 373: geometrical equivalents of algebraical operations I. 374: algebraical equivalents of propositions in Book II., I. 372–3: equivalents in Book X. of propositions in algebra, $\sqrt{k} - \sqrt{\lambda}$ cannot be equal to $k'$, III. 58–60: if $a \pm \sqrt{b} = x \pm \sqrt{y}$, then $a = x$, $b = y$, III. 93–4, 167–8
‘Alī b. Aḥmad Abū ’l Qāsim al-Antakī I. 86
Allman, G. J. I. 135 n., 318, 352, III. 18–9, 439
Alternate: (of angles) I. 308: (of ratios), alternately II. 134
Alternative proofs, interpolated I. 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, II. 30, 126
Ambiguous case I. 306–7: in VI. 7, II. 208–9
Amphinomus I. 125, 128, 150 n.
Amyclas of Heraclea I. 117
Analysis (and synthesis) I. 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 I. 139: theoretical and problematical analysis I. 138: Treasury of Analysis (τόπος ἀναλυόμενος) I. 8, 10, 11, 138: method of analysis and precautions necessary to, I. 139–40: analysis and synthesis of problems I. 140–2: two parts of analysis (a) transformation, (b) resolution, and two parts of synthesis, (a) construction, (b) demonstration I. 141: example from Pappus I. 141–2: analysis should also reveal διορισμός (conditions of possibility) I. 142: interpolated alternative proofs of XIII. 1–5 by analysis and synthesis I. 137, III. 442–3
Analytical method I. 36: supposed discovery of, by Plato I. 134, 137
Anaximander I. 370, II. 111
Anaximenes II. 111
Anchor-ring I. 163
Andron I. 126
Angle: curvilineal and rectilineal, Euclid’s definition of, I. 176 sq.: definition criticised by Syrianus I. 176: Aristotle’s notion of angle as κλάσις I. 176: Apollonius’ view of, as contraction I. 176, 177: Plutarch and Carpus on, I. 177: to which category does it belong? quantum, Plutarch, Carpus, “Aganis” I. 177, Euclid I. 178; quale, Aristotle and Eudemus I. 177–8: relation,
Euclid I. 178: Syrianus’ compromise I. 178: treatise on the Angle by Eudemus I. 34, 38, 177–8: classification of angles (Geminus) I. 178–9: curvilineal and “mixed” angles I. 26, 178–9, horn-like (κερατοειδής) I. 177, 178, 182, 265, II. 4, 39, 40, lune-like (μηνοειδής) I. 26, 178–9, scraper-like (ξυστροειδής) I. 178: angle of a segment I. 253, II. 4: angle of a semicircle I. 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 I. 180–1, defined by Veronese I. 180: as part of a plane (“angular sector”) I. 179–80: flat angle (Veronese etc.) I. 180–1, 269: three kinds of angles, which is prior (Aristotle)? I. 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? II. 275–6: adjacent angles I. 181: alternate I. 308: similar (=equal) I. 178, 182, 252: vertical I. 278: exterior and interior (to a figure) I. 263, 280: exterior when re-entrant I. 263, in which case we have a hollow-angled figure I. 27, 188, II. 48: interior and opposite I. 280: construction by Apollonius of angle equal to angle I. 296: angle in a semicircle, theorem of, I. 317–9: trisection of angle, by conchoid of Nicomedes I. 265–6, by quadratrix of Hippias I. 266, by spiral of Archimedes I. 267: dihedral angle III. 264–5: solid angle III. 261, 267–8
Annex (προσαρμόζουσα) = the straight line which, when added to a compound irrational straight line formed by subtraction, makes up the greater “term,” i.e. the negative “term” III. 159
al-Antākī I. 86
Antecedents (leading terms in proportion) II. 134
“Anthisathus” (or “Abthiniathus”) I. 203
Antiparallels: may be used for construction of VI. 12, II. 215
Antiphon I. 7 n., 35
Āpastamba-Śulba-Sūtra I. 352: evidence in, as to early discovery of Eucl. I. 47 and use of gnomon I. 360–4: Bürk’s claim that Indians had discovered the irrational I. 363–4: approximation to $\sqrt{2}$ and Thibaut’s explanation I. 361, 363–4: inaccurate values of $\pi$ in, I. 364
Apollodorus “logisticus” I. 37, 319, 351
Apollonius: disparaged by Pappus in comparison with Euclid I. 3: supposed by some Arabians to be author of the Elements I. 5: a “carpenter” I. 5: on elementary geometry I. 42: on the line I. 159: on the angle I. 176: general definition of diameter I. 325: tried to prove axioms I. 42, 62, 222–3: his “general treatise” I. 42: constructions by, for bisection of straight line I. 268, for a perpendicular I. 270, for an angle equal to an angle I. 296: on parallel-axiom (?) I. 42–3: adaptation to conics of theory of application of areas I. 344–5: geometrical algebra in, I. 373: Plane Loci, I. 14, 259, 330, theorem from (arising out of Eucl. VI. 3), also found in Aristotle II. 198–200: Plane νεύσεις I. 151, problem from, II. 81, lemma by Pappus on, II. 64–5: comparison of dodecahedron and icosahedron I. 6, III. 439, 512, 513: on the cochlias I. 34, 42, 162: on “unordered” irrationals I. 42, 115, III. 3, 10, 246, 255–9: general definition of oblique (circular) cone III. 270: I. 138, 188, 221, 222, 246, 259, 370, 373, II. 75, 190, 258, III. 264, 267
Apotome: compound irrational straight line (difference between two “terms”) III. 7: defined III. 158–9: connected by Theaetetus with harmonic mean III. 3, 4: biquadratic from which it arises III. 7: uniquely formed III. 167–8: first, second, third, fourth, fifth and sixth apotomes, quadratics from which arising III. 5–6, defined III. 177, and found respectively (X. 85–90) III. 178–90: apotome equivalent to square root of first apotome III. 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 I. 36, 343–5: contrasted with exceeding and falling-short I. 343: complete method equivalent to geometrical solution of mixed quadratic equation I. 344–5, 383–5, 386–8, II. 187, 258–60, 263–5, 266–7: adaptation to conics (Apollonius) I. 344–5: application contrasted with construction (Proclus) I. 343
Approximations: $7/5$ as approximation to $\sqrt{2}$ (Pythagoreans and Plato) II. 119: approximations to $\sqrt{3}$ in Archimedes and (in sexagesimal fractions) in Ptolemy II. 119: to $\pi$ (Archimedes) II. 119: to $\sqrt{4500}$ (Theon of Alexandria) II. 119: remarkably close approximations (stated in sexagesimal fractions) in scholia to Book X., I. 74 n.
“Aqaṭon” I. 88
Arabian editors and commentators I. 75–90
Arabic numerals in scholia to Book X., 11th c., I. 71
Archibald, R. C. I. 9 n., 10
Archimedes: “postulates” in, I. 120, 123: “porisms” in, I. 11 n., 13: on straight line I. 166: on plane I. 171–2: Liber assumptorum, proposition from, II. 65: approximations to $\sqrt{3}$, square roots of large numbers and to $\pi$, II. 119: extension of a proportion between commensurables to cover incommensurables II. 193: “Axiom” of (called however “lemma,” assumption, by A. himself) I. 234: relation of “Axiom” to X. 1, 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 exhaustion III. 365–6, 374: new fragment of, “Method (ἔφοδος) of Archimedes about mechanical theorems,” or ἐφόδιον, discovered by Heiberg and published by him 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 parabolic segment, ibid.: spiral of Archimedes I. 26, 267: I. 20, 21, 116, 142, 225, 249, 370, II. 136, 190, III. 246, 270, 375
Archytas I. 20: proof that there is no numerical geometric mean between $n$ and $n + 1$ II. 295
Areskong, M. E. I. 113
Arethas, Bishop of Caesarea I. 48: owned Bodleian MS. (B) I. 47–8: had famous Plato MS. of Patmos (Cod. Clarkianus) written I. 48
Argyrus, Isaak I. 74
Aristaeus I. 138: on conics I. 3: Solid Loci I. 16, 329: comparison of five (regular solid) figures I. 6, III. 438–9, 513
*Aristotelian Problems* I. 166, 182, 187
Aristotle: on nature of elements I. 116: on first principles I. 117 sqq.: on definitions I. 117, 119–20, 143–4, 146–50: on distinction between hypotheses and definitions I. 119, 120, between hypotheses and postulates I. 118, 119, between hypotheses and axioms I. 120: on axioms I. 119–21: axioms indemonstrable I. 121: on definition by negation I. 156–7: on points I. 155–6, 165: on lines, definitions of, I. 158–9, classification of, I. 159–60: quotes Plato’s definition of straight line I. 166: on definitions of surface I. 170: definition of “body” as that which has three dimensions or as “depth” III. 262: body “bounded by surfaces” (ἐπιπέδοις) III. 263: speaks of six “dimensions” III. 263: definition of sphere III. 269: on the angle I. 176–8: on priority as between right and acute angles I. 181–2: on figure and definition of, I. 182–3: definitions of “squaring” I. 149–50, 410: on parallels I. 190–2, 308–9: on gnomon I. 351, 355, 359: on attributes κατὰ παντός and πρῶτον καθόλου I. 319, 320, 325: on the objection I. 135: on reduction I. 135: on reductio aa absurdum I. 136: on the infinite I. 232–4: supposed postulate or axiom about divergent lines taken by Proclus from, I. 45, 207: gives pre-Euclidean proof of Eucl. I. 5, I. 252–3: on theorem of angle in a semicircle I. 149; has proof (pre-Euclidean) that angle in semicircle is right II. 63: on sum of angles of triangle I. 319–21: on sum of exterior angles of polygon I. 322: on def. of same ratio (=*same ἀνταναίρεσις*) II. 120–1: on proportion as “equality of ratios” II. 119: on theorem in proportion (alternando) not proved generally till his time II. 113: on proportion in three terms (συνεχής, continuous), and in four terms (διῃρημένη, discrete) II. 131, 293: on alternate ratios II. 134: on inverse ratio II. 134, 149: on similar rectilinear figures II. 188: has locus-theorem (arising out of Eucl. VI. 3) also given in Apollonius’ Plane Loci II. 198–200: on unit II. 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 incommensurability of $\sqrt{2}$, III. 2: I. 38, 45, 117, 150 n., 181, 184, 185, 187, 188, 195, 202, 203, 221, 222, 223, 226, 259, 262–3, 283, 411, II. 2, 4, 22, 79, 112, 135, 149, 159, 160, 165, 184, 188, 189, III. 4
Arithmetic, Elements of, anterior to Euclid II. 295
Arithmetical calculations in scholia to Book X., I. 71, 74
al-Arjānī, Ibn Rāhawaihi I. 86
Ashkāl at-ta’sīs I. 5 n.
Ashraf Shamsaddīn as-Samarqandī, Muḥ. b. I. 5 n., 89
Astaroff, Ivan I. 113
Asymptotic (non-secant): of lines I. 40, 161, 203: of parallel planes III. 265
Athelhard of Bath I. 78, 93–6
Athenaeus of Cyzicus I. 117
Athenaeus I. 20
August, E. F. I. 103, II. 23, 25, 149, 238, 250, 412, III. 2, 48
Austin, W. I. 103, 111, II. 172, 188, 211, 259
Autolycus, On the moving sphere, I. 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. 1, III. 15–6
Axioms, distinguished from postulates by Aristotle I. 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 I. 120–1, by Proclus I. 121–2: indemonstrable I. 121: attempt by Apollonius to prove I. 223–3: = “common (things)” or “common opinions” in Aristotle I. 120, 221: common to all sciences I. 119, 120: called “common notions” in Euclid I. 121, 221: which are genuine? I. 221 sqq.: Proclus recognises five I. 222, Heron three I. 222: interpolated axioms I. 224, 232: Pappus’ additions to axioms I. 25, 223, 224, 232: axioms of congruence, (1) Euclid’s Common Notion 4, I. 224–7, (2) modern systems (Pasch, Veronese and Hilbert) I. 228–31: “axiom” with Stoics = every simple declaratory statement I. 41, 221: axioms tacitly assumed, in Book V., II. 137, in Book VII., II. 294
Axis: of sphere III. 261, 269: of cone III. 261, 271: of cylinder III. 262, 271
Babylonians: supposed discoverers of “harmonic proportion” II. 112
Bacon, Roger I. 94, 416
Baermann, G. F. II. 213
Balbus, de mensuris I. 91
Baltzer, R. II. 30
Barbarin I. 219
Barlaam, arithmetical commentary on Eucl. II., I. 74
Barrow: on Eucl. V. Def. 3, II. 117: on V. Def. 5, II. 121: I. 103, 105, 110, 111, II. 56, 186, 238
*Base*: meaning I. 248–9: of cone III. 262: of cylinder III. 262
Basel editio princeps of Eucl., I. 100–1
Basilides of Tyre I. 5, 6, III. 512
Bāudhāyana Śulba-Sūtra I. 360
Bayfius (Baïf, Lazare) I. 100
Becker, J. K. I. 174
Beez I. 176
Behā-ad-dīn I. 417
Beltrami, E. I. 219
Benjamin of Lesbos I. 113
Bergh, P. I. 400–1
Bernard, Edward I. 102
Besthorn and Heiberg, edition of al Hajjāj’s translation and an-Nairīzī’s commentary I. 22, 27 n., 79 n.
Bhāskara I. 355, 418
Billingsley, Sir Henry, I. 109–10, 418, II. 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 irrational 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-Bīrūnī I. 90
Björnbo, Axel Anthon I. 17 n., 93
Boccaccio I. 96
Bodleian MS. (B) I. 47, 48
Boeckh I. 351, 371
Boethius I. 92, 95, 184, II. 295
Bologna MS. (b) I. 49
Bolyai, J. I. 219
Bolyai, W. I. 174–5, 219, 328
Bolzano I. 167
Boncompagni I. 93 n., 104 n.
Bonola, R. I. 202, 219
Borelli, Giovanni Alfonso I. 106, 194, II. 2, 84
Boundary (ὅρος) I. 182, 183
Bråkenhjelm, P. R. I. 113
Breadth (of numbers) = second dimension or factor II. 288
Breitkopf, Joh. Gottlieb Immanuel I. 97
Bretschneider I. 136 n., 137, 295, 304, 344, 354, 358, III. 439, 442
Briconnet, François I. 100
“Bride, Theorem of the” = Eucl. I. 47, I. 417–8
“Bride’s Chair,” name for Eucl. I. 47, I. 417–8
Briggs, Henry I. 102, II. 143
Brit. Mus. palimpsest, 7th–8th c., I. 50
Bryson, I. 8 n.
Bürk, A. I. 352, 360–4
Bürklen I. 179
Buteo (Borrel), Johannes I. 104
Cabasilas, Nicolaus and Theodorus I. 72
Caiani, Angelo I. 101
Camerarius, Joachim I. 101
Camerer, J. G. I. 103, 293, II. 22, 25, 28, 33, 34, 40, 67, 121, 131, 189, 213, 244
Camorano, Rodrigo, I. 112
Campanus, Johannes I. 3, 78, 94–6, 104, 106, 110, 407, II. 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) I. 3, 104, 110, II. 189
Cantor, Moritz I. 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: I. 45, 127, 128, 177
Case, technical term I. 134: cases interpolated I. 58, 59: Greeks did not infer limiting cases but proved them separately II. 75
Casey, J. II. 227
Casiri, I. 4 n., 17 n.
Cassiodorus, Magnus Aurelius I. 92
Catalan, II. 426
Cataldi, Pietro Antonio, I. 106
Catoptrica**, attributed to Euclid**, probably Theon’s I. 17: Catoptrica of Heron I. 21, 253
Cauchy III. 267: proof of Eucl. XI. 4, III. 280
“Cause”: consideration of, omitted by commentators I. 19, 45: definition should state cause (Aristotle) I. 149: cause = middle term (Aristotle) I. 149: question whether geometry should investigate cause (Geminus), I. 45, 150 n.
Censorinus I. 91
Centre, κέντρον I. 184–5
*Ceria Aristotelica* I. 35
Cesàro, E. II. 426
“Chance equimultiples” in phrase “other, chance, equimultiples” II. 143–4
Chasles on Porisms of Euclid I. 10, 11, 14, 15
Chaucer, Dulcarnon in, I. 416–7, 418
Chinese, knowledge of triangle 3, 4, 5, I. 352, 362: “Chóu pei” I. 355
Chou Kung I. 362
Christensen III. 8
Chrysippus I. 330
Chrystal, G. III. 19
Cicero I. 91, 351
Circle: definition of, I. 183–5: =round, στρογγύλον (Plato), I. 184: =περιφερόγραμμον (Aristotle) I. 184: a plane figure I. 183–4: exceptionally in sense of “circumference” II. 23: centre of, I. 184–5: pole of, I. 185: bisected by diameter (Thales) I. 185, (Saccheri) I. 185–6: intersections with straight line I. 237–8, 272–4, with another circle I. 238–40, 242–3, 293–4: definition of “equal circles” II. 2: circles touching, meaning of definition, II. 3: circles intersecting and touching, difficulties in Euclid’s treatment of, II. 25–7, 28–9, modern treatment of, II. 30–2
Circumference, περιφέρεια, I. 184
Cissoid, I. 161, 164, 176, 330
Clairaut I. 328
Clavius (Christoph Klau?) I. 103, 105, 194, 232, 381, 391, 407, II. 2, 41, 42, 47, 49, 53, 56, 67, 70, 73, 130, 170, 190, 231, 238, 244, 271, III. 273, 331, 341, 350, 359, 433
Claymundus, Joan. I. 101
Cleonides, Introduction to Harmony, I. 17
Cochlias or cochlion (cylindrical helix) I. 162
Codex Leidensis 399, 1: I. 22, 27 n., 79 n.
Coets, Hendrik, I. 109
Commandinus I. 4, 102, 103, 104–5, 106, 110, 111, 407, II. 47, 130, 190: scholia included in translation of Elements I. 73: edited (with Dee) De divisionibus I. 8, 9, 110
Commensurable: defined III. 10: commensurable in length, commensurable in square, and commensurable in square only defined III. 10, 11: symbols used in notes for these terms III. 34
Commentators on Eucl. criticised by Proclus I. 19, 26, 45
Common Notions: =axioms I. 62, 120–1, 221–2: which are genuine? I. 221 sq.: meaning and appropriation of term I. 221: called “axioms” by Proclus I. 221
Complement, παραπλήρωμα: meaning of, I. 341: “about diameter” I. 341: not necessarily parallelograms I. 341: use for application of areas I. 342–3
Componendo (συνθέντι), denoting “composition” of ratios q.v.: componendo and separando used relatively to each other II. 168, 170
Composite, σύνθετος: (of lines) I. 160: (of surfaces) I. 170: (of numbers) II. 286: with Eucl. and Theon of Smyrna composite numbers may be even, but with Nicom. and Iamblichus are a subdivision of odd II. 286, plane and solid numbers are species of, II. 286
“Composite to one another” (of numbers) II. 286–7
Composition of ratio (σύνθεσις λόγου), denoted by componendo (συνθέντι), distinct from compounding ratios II. 134–5
Compound ratio: explanation of, II. 132–3: (interpolated?) definition of, II. 189–90: compounded ratios in V. 20–3, II. 176–8
Conchoids I. 160–1, 265–6, 330
Conclusion, συμπέρασμα: necessary part of a proposition I. 129–30: particular conclusion immediately made general I. 131: definition merely stating conclusion I. 149
Cone: definitions of, by Euclid III. 262, 270, by Apollonius III. 270: distinction between right-angled, obtuse-angled and acute-angled cones a relic of old theory of conics III. 270: similar cones, definition of, III. 262, 271
Congruence-Axioms or Postulates: Common Notion 4 in Euclid I. 224–5: modern systems of (Pasch, Veronese, Hilbert) I. 228–31
Congruence theorems for triangles, recapitulation of, I. 305–6
Conics, of Euclid, I. 3, 16: of Aristaeus, I. 3, 16: of Apollonius I. 3, 16: fundamental property as proved by Apollonius equivalent to Cartesian equation I. 344–5: focus-directrix property proved by Pappus I. 15
Consequents (“following” terms in a proportion) II. 134, 238
Constantinus Lascaris I. 3
Construct (συνίστασθαι) contrasted with describe on I. 348, with apply to I. 343: special connotation I. 259, 289
Construction, κατασκευή, one of formal divisions of a proposition I. 129: sometimes unnecessary I. 130: turns nominal into real definition I. 146: mechanical constructions I. 151, 387
Continuity, Principle of, I. 234 sq., 242, 272, 294
Continuous proportion (συνεχὴς or συνημμένη ἀναλογία) in three terms II. 131
Conversion, geometrical: distinct from logical I. 256: “leading” and partial varieties of, I. 256–7, 337
Conversion of ratio (ἀναστροφὴ λόγου), denoted by convertendo (ἀναστρέψαντι) II. 135: convertendo theorem not established by V. 19, Por. II. 174–5, but proved by Simson’s Prop. E. II. 175, III. 38: Euclid’s roundabout substitute III. 38
Convertendo denoting “conversion” of ratios, q.v.
Copernicus I. 101
Cordonis, Mattheus I. 97
Corresponding magnitudes II. 134
Cosmic figures (=five regular solids) I. 413–4
Cossali III. 8
Cratistus I. 133
Crelle, on the plane I. 172–4, III. 263
Ctesibius I. 20, 21, 39 n.
Cube: defined III. 262: problem of inscribing in sphere, Euclid’s solution III. 478–80, Pappus’ solution III. 480: duplication of cube reduced by Hippocrates of Chios to problem of two mean proportionals I. 135, II. 133: cube number, defined II. 291: two mean proportionals between two cube numbers II. 294, 364–5
Cunn, Samuel I. 111
Curtze, Maximilian, editor of an-Nairīzī I. 22, 78, 92, 94, 96, 97 n., II. 426
Curves, classification of: see line
Cyclic, of a particular kind of square number II. 291
Cyclomathia of Leotaud II. 42
Cylinder: definition of, III. 262: similar cylinders defined III. 262
Cylindrical helix I. 161, 162, 329, 330
Czecha, Jo. I. 113
Dasypodius (Rauchfuss) Conrad I. 73, 102
Data of Euclid: I. 8, 132, 141, 385, 391: Def. 2, II. 248: Prop. 8, II. 249–50: Prop. 24, II. 246–7: Prop. 55, II. 254: Props. 56 and 68, II. 249: Prop. 58, II. 263–5: Props. 59 and 84, II. 266–7: Prop. 67 assumes part of converse of Simson’s Prop. B (Book VI.) II. 224: Prop. 70, II. 250: Prop. 85, II. 264: Prop. 87, II. 228: Prop. 93, II. 227
Deahna I. 174
Dechales, Claude François Milliet I. 106, 107, 108, 110, II. 259
Dedekind’s theory of irrational numbers corresponds exactly to Eucl. V. Def. 5, II. 124–6; Dedekind’s Postulate and applications of, I. 235–40, III. 16
Dee, John I. 109, 110; discovered De divisionibus I. 8, 9
Definition, in sense of “closer statement” (διορισμός), one of formal divisions of a proposition I. 129: may be unnecessary I. 130
Definitions: Aristotle on, I. 117, 119, 120, 143: a class of thesis (Aristotle) I. 120: distinguished from hypotheses I. 119, but confused therewith by Proclus I. 121–2: must be assumed I. 117–9, but say nothing about existence (except in the case of a few primary things) I. 119, 143: terms for, ὅρος and ὁρισμός I. 143: real and nominal definitions (real = nominal plus postulate or proof), Mill anticipated by Aristotle, Saccheri and Leibniz I. 143–5: Aristotle’s requirements in, I. 146–50, exceptions I. 148: should state cause or middle term and be genetic I. 149–50: Aristotle on unscientific definitions (ἐκ μὴ προτέρων) I. 148–9: Euclid’s definitions agree generally with Aristotle’s doctrine I. 146: interpolated definitions I. 61, 62: definitions of technical terms in Aristotle and Heron, not in Euclid I. 150
De levi et ponderoso**, tract** I. 18
Demetrius Cydonius I. 72
Democritus I. 38: “On difference of gnomon” etc. (? on “angle of contact”) II. 40: on parallel and infinitely near sections of cone, II. 40, III. 368: stated, without proving, propositions about volumes of cone and pyramid, II. 40, III. 366: was evidently on the track of the infinitesimal calculus III. 368: treatise on irrationals (περὶ ἀλόγων γραμμῶν καὶ ναστῶν βʹ) I. 413, III. 4
De Morgan, A. I. 246, 260, 269, 284, 291, 298, 300, 309, 313, 314, 315, 369, 376, II. 5, 7, 9–10, 11, 15, 20, 22, 29, 56, 76–7, 83, 101, 104, 116–9, 120, 130, 139, 145, 197, 202, 217–8, 232, 233, 234, 272, 275: on definition of ratio II. 116–7: on extension of meaning of ratio to cover incommensurables II. 118: means of expressing ratios between incommensurables by approximation to any extent II. 118–9: defence and explanation of V. Def. 5, II. 121–4: on necessity of proof that tests for greater and less, or greater and equal, ratios cannot coexist II. 130–1, 157: on compound ratio II. 132–3, 234: sketch of proof of existence of fourth proportional (assumed in V. 18) II. 171; proposed lemma about duplicate ratios as alternative means of proving VI. 22, II. 246–7: on Book X., III. 8
Dercyllides II. 111
Desargues I. 193
Describe on (ἀναγράφειν ἀπό) contrasted with *construct* I. 348
De Zolt I. 328
Diagonal (διαγώνιος) I. 185
“Diagonal” numbers: see “Side-” and “diagonal-” numbers
Diameter (διάμετρος), of circle or parallelo-
gram I. 185: of sphere III. 261, 269, 270: as applied to figures generally I. 325; “rational” and “irrational” diameter of 5 (Plato) I. 399, 413, taken from Pythagoreans I. 399–400, III. 12
Dickson, L. E. II. 426
Diels, H. I. 412
Dihedral angle = inclination of plane to plane, measured by a plane angle III. 264–5
Dimensions (cf. διαστάσεις), I. 157, 158: Aristotle’s view of, I. 158–9, III. 262–3, speaks of six III. 263
Dinostratus I. 117, 266
Diocles I. 164
Diodorus I. 203
Diogenes Laertius I. 37, 305, 317, 351, III. 4
Dionysius, friend of Heron, I. 21
Diophantus I. 86
Diorismus (διορισμός) = (a) “definition” or “specification,” a formal division of a proposition I. 129: (b) condition of possibility I. 128, determines how far solution possible and in how many ways I. 130–1, 243: diorismi said to have been discovered by Leon I. 116; revealed by analysis I. 142: introduced by δεῖ δή I. 293: first instances in Elements I. 234, 293: for solution of quadratic II. 259
Dippe I. 108
Direction, as primary notion, discussed I. 179: direction-theory of parallels I. 191–2
Discrete proportion, διῃρημένη or διεζευγμένη ἀναλογία, in four terms, II. 131, 293
“Dissimilarly ordered” proportion (ἀνομοίως τεταγμένων τῶν λόγων) in Archimedes = “perturbed proportion” II. 136
Distance, διάστημα: =radius I. 199: in Aristotle has usual general sense and =dimension I. 199
Dividendo (of ratios): see Separation, separando
Division (method of), Plato’s I. 134
Divisions (of figures)**, treatise by Euclid**, I. 8, 9: translated by Muḥammad al-Baġdādī I. 8: found by Woepcke in Arabic I. 9, and by Dee in Latin translation I. 8, 9: I. 110: proposition from, II. 5
Dodecahedron: in sphere I. 411: decomposition of faces into elementary triangles II. 98: definition of, III. 262: dodecahedra found, apparently dating from centuries before Pythagoras III. 438, though said to have been discovered by Pythagoreans ibid.: problem of inscribing in sphere, Euclid’s solution III. 493, Pappus’ solution III. 501–3
Dodgson, C. L. I. 194, 254, 261, 313, II. 48, 275
Dou, Jan Pieterszoon I. 108
Duhamel, J. M. C. I. 139, 328
Dulcarnon, name for Eucl. I. 47, I. 416, 418
Duplicate ratio II. 133; διπλασίων, duplicate, distinct from διπλάσιος, double (=ratio 2 : 1), though use of terms not uniform II. 133: “duplicate” of given ratio found by VI. 11, II. 214: lemma on duplicate ratio as alternative to method of VI. 22 (De Morgan and others) II. 242–7
Duplication of cube; reduction of, by Hippocrates, to problem of finding two mean proportionals I. 135, II. 133: wrongly supposed to be alluded to in Timaeus 32 A, B, II. 294–5 n.
Egyptians II. 112: knowledge of $3^2 + 4^2 = 5^2$, I. 352: view of number II. 280
Eibe, Thyra I. 113
Elefuga**, name for Eucl. I. 5**, I. 416–7
Elements: pre-Euclidean Elements, by Hippocrates of Chios, Leon I. 116, Theudius I. 117: contributions to, by Eudoxus I. 1, 37, II. 112, III. 15, 365–6, 374, 441, Theaetetus I. 1, 37, III. 3, 438, Hermotimus of Colophon I. 117; Euclid’s Elements, ultimate aims of, I. 2, 115–6; commentators on, I. 19–45, Proclus I. 19, 29–45 and passim, Heron I. 20–4, an-Nairīzī I. 21–4, Porphyry I. 24, Pappus I. 24–7, Simplicius I. 28, Aenaeas (Aigeias) I. 28: MSS. of, I. 46–51: Theon’s changes in text I. 54–8: means of comparing Theonine with ante-Theonine text I. 51–3: interpolations before Theon’s time I. 58–63: scholia I. 64–74: external sources throwing light on text, Heron, Taurus, Sextus Empiricus, Proclus, Iamblichus I. 62–3: Arabic translations (1) by al-Ḥajjāj I. 75, 76, 79, 80, 83–4, (2) by Isḥāq and Thābit b. Qurra I. 75–80, 83–4, (3) Naṣīraddīn aṭ-Ṭūsī I. 77–80, 84: Hebrew translation by Moses b. Tibbon or Jakob b. Machir I. 76: Arabian versions compared with Greek text I. 79–83, with one another I. 83, 84: translation by Boethius I. 92: old translation of 10th c. I. 92; translations by Athelhard I. 93–6, Gherard of Cremona I. 93–4, Campanus I. 94–6, 97–100 etc., Zamberti I. 98–100, Commandinus I. 104–5: introduction into England, 10th c., I. 95: translation by Billingsley I. 109–10: Greek texts, editio princeps I. 100–1, Gregory’s I. 102–3, Peyrard’s I. 103, August’s I. 103, Heiberg’s passim: translations and editions generally I. 97–113: writers on Book X., III. 8–9: on the nature of elements (Proclus) I. 114–6, (Menaechmus) I. 114, (Aristotle) I. 116: Proclus on advantages of Euclid’s Elements I. 115: immediate recognition of, I. 116: first principles of, definitions, postulates, common notions (axioms) I. 117–24: technical terms in connexion with, I. 125–42: no definitions of such technical terms I. 150: sections of Book I., I. 308
Elinuam I. 95
Engel and Stäckel I. 219, 321
Enriques, F. I. 113, 157, 175, 193, 195, 201, 313, II. 30, 126
Enunciation (πρότασις), one of formal divisions of a proposition I. 129–30
Epicureans, objection to Eucl. I. 20, I. 41, 287: Savile on, I. 287
Equality, in sense different from that of congruence (= “equivalent,” Legendre) I. 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) I. 328: modern definition of, I. 228
Equimultiples: “any equimultiples whatever,” ἰσάκις πολλαπλάσια καθ’ ὁποιονοῦν πολλαπλασιασμόν II. 120: stereotyped phrase “other, chance, equimultiples” II. 143–4: should include once each magnitude II. 145
Eratosthenes: I. 1: contemporary with Archimedes I. 1, 2: Archimedes’ “Method” addressed to, III. 366: measurement of obliquity of ecliptic ($23^\circ\ 51'\ 20''$) II. 111
Errard, Jean, de Bar-le-Duc I. 108
Erycinus I. 27, 290, 329
Escribed circles of triangle II. 85, 86–7
Euclid: account of, in Proclus’ summary I. 1: date I. 1–2: allusions to, in Archimedes I. 1: (according to Proclus) a Platonist I. 2: taught at Alexandria I. 2: Pappus on personality of, I. 3: story of (in Stobaeus) I. 3: not “of Megara” I. 3, 4: supposed to have been born at Gela I. 4: Arabian traditions about, I. 4, 5: “of Tyre” I. 4–6: “of Ṭūs” I. 4, 5 n.: Arabian derivation of name (“key of geometry”) I. 6: Elements, ultimate aim of, I. 2, 115–6: other works, Conics I. 16, Pseudaria I. 7, Data I. 8, 132, 141, 385, 391, On divisions (of figures) I. 8, 9, Porisms I. 10–15, Surface-loci I. 15, 16, Phaenomena I. 16, 17, Optics I. 17, Elements of Music or Sectio Canonis I. 17, II. 295: on “three- and four-line locus” I. 3: Arabian list of works I. 17, 18: bibliography I. 91–113
Eudemus I. 29: On the Angle I. 34, 38, 177–8: History of Geometry I. 34, 35–8, 278, 295, 304, 317, 320, 387, 412, II. 99, 111, III. 3, 366
Eudoxus I. 1, 37, 74, 116, II. 40, 99, 280, 295: discoverer of theory of proportion covering incommensurables as expounded generally in Bks. V., VI., I. 137, 351, II. 112: on the golden section I. 137: discoverer of method of exhaustion I. 234, III. 365–6, 374: used “Axiom of Archimedes” III. 15: first to prove theorems about volume of pyramid (Eucl. XII. 7 Por.) and cone (Eucl. XII. 10), also theorem of Eucl. XII. 2, III. 15: theorems of Eucl. XIII. 1–5 probably due to, III. 441: inventor of a certain curve, the hippopede, horse-fetter I. 163: possibly wrote Sphaerica I. 17: III. 442
Euler, Leonhard I. 401, II. 426
Eutocius: I. 25, 35, 39, 142, 161, 164, 259, 317, 329, 330, 373: on “VI. Def. 5” and meaning of πηλικότης II. 116, 132, 189–90: gives locus-theorem from Apollonius’ Plane Loci II. 198–200
Even (number): definitions by Pythagoreans and in Nicomachus II. 281: definitions of odd and even by one another unscientific (Aristotle) I. 148–9, II. 281: Nicom. divides even into three classes (1) even-times even and (2) even-times odd as extremes, and (3) odd-times even as intermediate II. 282–3
Even-times even: Euclid’s use differs from use by Nicomachus, Theon of Smyrna and Iamblichus II. 281–2
Even-times odd in Euclid different from even-odd of Nicomachus and the rest II. 282–4
Ex aequali**, of ratios**, II. 136: ex aequali propositions (V. 20, 22), and ex aequali “in perturbed proportion” (V. 21, 23) II. 176–8
Exhaustion, method of: discovered by Eudoxus I. 234, III. 365–6: evidence of Archimedes III. 365–6: III. 374–7
Exterior and interior (of angles) I. 263, 280
Extreme and mean ratio (line cut in): defined, II. 188: known to Pythagoreans I. 403, II. 99, III. 19: irrationality of segments of (apotomes) III. 19, 449–51
Extremity, πέρας, I. 182, 183
Faifofer II. 126
Falk, H. I. 113
al-Fārābī I. 8 n., 90
Fauquembergue, E. II. 426
Fermat II. 425, 426
Figure, as viewed by Plato I. 182, by Aristotle I. 182–3, by Euclid I. 183: according to Posidonius is confining boundary only I. 41, 183: figures bounded by two lines classified I. 187: angle-less (ἀγώνιον) figure I. 187
Figures, printing of, I. 97
Fihrist I. 4 n., 5 n., 17, 21, 24, 25, 27; list of Euclid’s works in, I. 17, 18
Finaeus, Orontius (Oronce Fine) I. 101, 104
Flauti, Vincenzo I. 107
Florence MS. Laurent. XXVIII. 3 (F) I. 47
Flussates, see Candalla
Forcadel, Pierre I. 108
Fourier: definition of plane based on Eucl. XI. 4, I. 173–4, III. 263
Fourth proportional: assumption of existence of, in V. 18, and alternative methods for avoiding (Saccheri, De Morgan, Simson, Smith and Bryant) II. 170–4: Clavius made the assumption an axiom II. 170: sketch of proof of assumption by De Morgan II. 171: condition for existence of number which is a fourth proportional to three numbers II. 409–11
Francisci tunica, “Franciscan’s cowl,” name for Eucl. I. 47, I. 418
Frankland, W. B. I. 173, 199
Frischauf, J. I. 174
Galileo Galilei: on angle of contact II. 42
Gartz I. 17
Gauss I. 172, 193, 194, 202, 219, 321
Geminus: name Latin or Greek? I. 38–9: title of work (Φιλοκαλία) quoted from by
Proclus I. 39, and by Schol., I. 74: elements of astronomy I. 38: comm. on Posidonius I. 39: Proclus’ obligations to, I. 39–42: on postulates and axioms I. 122–3: on theorems and problems I. 128: two classifications of lines (or curves) I. 160–2: on homoeomeric (uniform) lines I. 162: on “mixed” lines (curves) and surfaces I. 162: classification of surfaces I. 170, of angles I. 178–9: on parallels I. 191: on Postulate 4, I. 200: on stages of proof of theorem of I. 32, I. 317–20: I. 27–8, 37, 44, 45, 74, 133 n., 203, 265, 330
Geometrical algebra I. 372–4: Euclid’s method in Book II. evidently the classical method I. 373: preferable to semi-algebraical method I. 377–8
Geometrical progression II. 346 sqq.: summation of $n$ terms of (IX. 35) II. 420–1
Geometric means II. 357 sqq.: one mean between square numbers II. 294, 363, or between similar plane numbers II. 371–2: two means between cube numbers II. 294, 364–5, or between similar solid numbers II. 373–5
Georgius Pachymeres I. 417
Gherard of Cremona, translator of Elements I. 93–4: of an-Nairīzī’s commentary I. 22, 27 n., 94, II. 47: of tract De divisionibus I. 9, 10 n.
Giordano, Vitale I. 106, 176
Given**, δεδομένος**, different senses, I. 132–3
Gnomon**: literally “that enabling (something) to be known**” I. 64, 370: successive senses of, (1) upright marker of sundial, I. 181, 185, 271–2, introduced into Greece by Anaximander I. 370, (2) carpenter’s square for drawing right angles I. 371, (3) figure placed round square to make larger square I. 351, 371, Indian use of gnomon in this sense I. 362, (4) use extended by Euclid to parallelograms I. 371, (5) by Heron and Theon to any figures I. 371–2: Euclid’s method of denoting in figure I. 383: arithmetical use of, I. 358–60, 371, II. 289
“Gnomon-wise” (κατὰ γνώμονα), old name for perpendicular (κάθετος) I. 36, 181, 272
Görland, A. I. 233, 234
Golden section (section in extreme and mean ratio), discovered by Pythagoreans I. 137, 403, II. 99: connexion with theory of irrationals I. 137, III. 19: theory carried further by Plato and Eudoxus II. 99: theorems of Eucl. XIII. 1–5 on, probably due to Eudoxus III. 441
“Goose’s foot” (pes anseris), name for Eucl. III. 7, I. 99, 418
Gow, James I. 135 n.
Gracilis, Stephanus I. 101–2
Grandi, Guido I. 107
Greater ratio: Euclid’s criterion not the only one II. 130: arguments from greater to less ratios etc. unsafe unless they go back to original definitions (Simson on V. 10) II. 156–7: test for, cannot coexist with test for equal or less ratio II. 130–1
Greatest common measure: Euclid’s method of finding corresponds exactly to ours II. 118, 299, III. 18, 21–2: Nicomachus gives the same method II. 300: method used to prove incommensurability III. 18–9; for this purpose often unnecessary to carry it far (cases of extreme and mean ratio and of $\sqrt{2}$) III. 18–9
Greenhill, Sir George, I. 415, 418
Gregory, David I. 102–3, II. 116, 143, III. 32
Gregory of St Vincent I. 401, 404
Gromatici I. 91 n., 95
Grynaeus I. 100–1
Häbler, Th. II. 294 n.
al-Haitham I. 88, 89
al-Ḥajjāj b. Yūsuf b. Maṭar, translator of the Elements I. 22, 75, 76, 79, 80, 83, 84
Halifax, William I. 108, 110
Halliwell (-Phillipps) I. 95 n.
Hankel, H. I. 139, 141, 232, 234, 344, 354, II. 116, 117, III. 8
Harmonica of Ptolemy, Comm. on, I. 17
Harmony, Introduction to**, not by Euclid**, I. 17
Hārūn ar-Rashīd I. 75
al-Ḥasan b. ’Ubaidallāh b. Sulaimān b. Wahb I. 87
Hauber, C. F. II. 244
Hauff, J. K. F. I. 108
“Heavy and Light,” tract on, I. 18
Heiberg, J. L. passim
Helix, cylindrical I. 161, 162, 329, 330
Helmholtz I. 226, 227
Henrici and Treutlein I. 313, 404, II. 30
Henrion, Denis I. 108
Hérigone, Pierre I. 108
Herlin, Christian I. 100
Hermotimus of Colophon I. 1, 117
Herodotus I. 37 n., 370
“Heromides” I. 158
Heron of Alexandria, mechanicus, date of, I. 20–1: Heron and Vitruvius I. 20–1; commentary on Euclid’s Elements I. 20–4: direct proof of I. 25, I. 301: comparison of areas of triangles in I. 24, I. 334–5: addition to I. 47, I. 366–8: apparently originated semi-algebraical method of proving theorems of Book II., I. 373, 378: Eucl. III. 12 interpolated from, II. 28: extends III. 20, 21 to angles in segments less than semicircles II. 47–8: does not recognise angles equal to or greater than two right angles II. 47–8: proof of formula for area of triangle, $\Delta = \sqrt{s(s-a)(s-b)(s-c)}$, II. 87–8: I. 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, II. 5, 16–7, 24, 28, 33, 36, 44, 47, 48, 116, 189, 302, 320, 383, 395, III. 24, 263, 265, 267, 268, 269, 270, 366, 404, 442
Heron, Proclus’ instructor I. 29
“Herundes” I. 156
Hieronymus of Rhodes I. 305
Hilbert, D. I. 157, 193, 201, 228–31, 249, 313, 328
Hipparchus I. 4 n., 20, 30 n., 74 n.
Hippasus I. 411, II. 97, III. 438
Hippias of Elis I. 42, 265–6, 413
Hippocrates of Chios I. 8 n., 29, 35, 38, 116, 135, 136 n., 386–7, 413, II. 133: first proved that circles (and similar segments of circles) are to one another as the squares on their diameters III. 366, 374
Hippopede (ἵππου πέδη), a certain curve used by Eudoxus I. 162–3, 176
Hoffmann, Heinrich I. 107
Hoffmann, Joh. Jos. Ign. I. 108, 365
Holgate, T. F. III. 284, 303, 331
Holtzmann, Wilhelm (Xylander) I. 107
Homoeomeric (uniform) lines I. 40, 161, 162
Hornlike angle (κερατοειδὴς γωνία) I. 177, 178, 182, 265, II. 4, 39, 40: hornlike angle and angle of semicircle, controversies on, II. 39–42: Proclus on, II. 39–40: Democritus may have written on hornlike angle II. 40: view of Campanus (“not angles in same sense”) II. 41: of Cardano (quantities of different orders or kinds): of Peletier (hornlike angle no angle, no quantity, nothing; angles of all semicircles right angles and equal) II. 41: of Clavius II. 42: of Vieta and Galileo (“angle of contact no angle”) II. 42: of Wallis (angle of contact not inclination at all but degree of curvature) II. 42
Horsley, Samuel I. 106
Hoüel, J. I. 219
Hudson, John I. 102
Hultsch, F. I. 17 n., 74, 329, 400, II. 133, 190, III. 4
Ḥunain b. Isḥāq al-’Ibādī I. 75
Hypotheses, in Plato I. 122: in Aristotle I. 118–20: confused by Proclus with definitions I. 121–2: geometer’s hypotheses not false (Aristotle) I. 119
Hypothetical construction I. 199
Hypsicles I. 5: author of Book XIV. I. 5, 6, III. 438–9, 512
Iamblichus I. 63, 83, 417, II. 97, 116, 279, 280, 281, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 419, 425, 426
Ibn al-’Amīd I. 86
Ibn al-Haitham I. 88, 89
Ibn al-Lubūdī I. 90
Ibn Rāhawaihi al-Arjānī I. 86
Ibn Sīnā (Avicenna) I. 77, 89
Icosahedron II. 98: defined III. 262: discovery of, attributed to Theaetetus III. 438: problem of inscribing in sphere, Euclid’s solution III. 481–9, Pappus’ solution III. 489–91: Mr H. M. Taylor’s construction III. 491–2
“Iflaton” I. 88
Inclination (κλίσις) of straight line to plane, defined III. 260, 263–4: of plane to plane (=dihedral angle) III. 260, 264
Incommensurables: discovered by Pythagoreans III. 1, 2, 3, and with reference to $\sqrt{2}$, I. 351, III. 1, 2, 19: incommensurable a natural kind, unlike irrational which depends on convention or assumption (Pythagoreans) III. 1: proof of incommensurability of $\sqrt{2}$ no doubt Pythagorean III. 2, proof in Chrystal’s Algebra III. 19–20: incommensurable in length and incommensurable in square defined III. 10, 11: symbols for, used in notes III. 34: method of testing incommensurability (process of finding G.C.M.) II. 118, III. 18–9: means of expression consist in power of approximation without limit (De Morgan) II. 119: approximations to $\sqrt{2}$ by means of side- and diagonal-numbers I. 399–401, II. 119, by means of sexagesimal fractions I. 74 n., to $\sqrt{3}$, I. 74 n., II. 119: to $\sqrt{4500}$ by means of sexagesimal fractions II. 119: to $\pi$, II. 119
Incomposite: (of lines) I. 160–1, (of surfaces) I. 170: (of number) = prime II. 284
Indivisible lines (ἄτομοι γραμμαί), theory of, rebutted I. 268
Infinite, Aristotle on the, I. 232–4: infinite division not assumed, but proved, by geometers I. 268
Infinity, parallels meeting at, I. 192–3
Ingrami, G. I. 175, 193, 195, 201, 227–8, II. 30, 126
Integral calculus, in new fragment of Archimedes III. 366–7
Interior and exterior (of angles) I. 263, 280: interior and opposite angle I. 280
Interpolations in the Elements before Theon’s time I. 58–63: by Theon I. 46, 55–6: Eucl. I. 40 interpolated I. 338: other propositions interpolated, (III. 12) II. 28, (proposition after XI. 37) III. 360, (XIII. 6) III. 449–51: cases in XI. 23, III. 319–21: defs. of analysis and synthesis, and proofs of XIII. 1–5 by, III. 442–3
Inverse (ratio), inversely (ἀνάπαλιν) II. 134: inversion is subject of V. 4, Por. (Theon) II. 144, and of V. 7, Por. II. 149, but is not properly put in either place II. 149: Simson’s Prop. B on, directly deducible from V. Def. 5, II. 144
Irrational: discovered by Pythagoreans I. 351, 411, 412–3, III. 1–2, 3, and with reference to $\sqrt{2}$, I. 351, III. 1, 2, 19: depends on assumption or convention, unlike incommensurable which is a natural kind (Pythagoreans) III. 1: claim of India to priority of discovery I. 363–4; “irrational diameter of 5” (Pythagoreans and Plato) I. 399–400, 413, III. 12: approximation to $\sqrt{2}$ by means of “side-” and “diagonal-” numbers I. 399–401, II. 119, to $\sqrt{2}$ and $\sqrt{3}$ in sexagesimal fractions I. 74 n.: Indian approximation to $\sqrt{2}$, I. 361, 363–4: unordered irrationals (Apollonius) I. 42, 115, III. 3, 10, 246, 255–9: irrational ratio (ἄρρητος λόγος) I. 137: an irrational straight line is so relatively to any straight

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line taken as rational III. 10, 11 : irrational area incommensurable with rational area or square on rational straight line III. 10, 12 : Euclid’s irrationals, object of classification of, III. 4, 5 : Book X. a repository of results of solution of different types of quadratic and biquadratic equations III. 5 : types of equations of which Euclid’s irrationals are positive roots III. 5-7 : actual use of Euclid’s irrationals in Greek geometry III. 9-10 : compound irrationals in Book X. all different III. 242-3

Isaacus Monachus (or Argyrus) I. 73-4, 407 Isḥāq b. Ḥunain b. Isḥāq al-‘Ibādī, Abū Ya’qūb, translation of Elements by, I. 75-80, 83-4 Isidorus of Miletus III. 520 Ismā’īl b. Bulbul I. 88 Isoperimetric (or isometric) figures: Pappus and Zenodorus on, I. 26, 27, 333 Isosceles (ἰσοσκελής) I. 187 : of numbers (= even) I. 188 : isosceles right-angled triangle I. 352 : isosceles triangle of IV. 10, construction of, due to Pythagoreans I. 414, II. 97-9

Jacobi, C. F. A. II. 188 Jakob b. Machir I. 76 Jan, C. I. 17 al-Jauharī, al-‘Abbās b. Sa’īd I. 85 al-Jayyānī I. 90 Joannes Pediasimus I. 72-3 Johannes of Palermo III. 8 Junge, G., on attribution of theorem of I. 47 and discovery of irrationals to Pythagoras I. 351, 411, 413

Kästner, A. G. I. 78, 97, 101 al-Karābīsī I. 85 Kātyāyana Śulba-Sūtra I. 360 Keill, John I. 105, 110-11 Kepler I. 193 al-Khāzin, Abū Ja’far I. 77, 85 Killing, W. I. 194, 219, 225-6, 235, 242, 272, III. 276 al-Kindī I. 5 n., 86 Klamroth, M. I. 75-84 Klau (?), Christoph = Clavius q.v. Klügel, G. S. I. 212 Kluge III. 520 Knesa, Jakob I. 112 Knoche I. 32 n., 33 n., 73 Kroll, W. I. 399-400 al-Kūhī I. 88

Lachlan, R. II. 226, 227, 245-6, 247, 256, 272 Lambert, J. H. I. 212-3 Lardner, Dionysius I. 112, 246, 250, 298, 404, II. 58, 259, 271 Lascaris, Constantinus 3 Leading theorems (as distinct from converse) I. 257 : leading variety of conversion I. 256-7 Least common multiple II. 336-41 Leeke, John I. 110 Lefèvre, Jacques I. 100

Legendre, Adrien Marie I. 112, 169, 213-9, II. 30, III. 263, 264, 265, 266, 267, 268, 273, 275, 298, 309, 356, 436 : proves VI. 1 and similar propositions in two parts (1) for commensurables, (2) for incommensurables II. 193-4 : proof of Eucl. XI. 4, III. 280, of XI. 6, 8, III. 284, 289, of XI. 15, III. 299, of XI. 19, III. 305 : definition of planes at right angles III. 303 : alternative proofs of theorems relating to prisms III. 331-3 : on equivalent parallelepipeds III. 335-6 : proof of Eucl. XII. 2, III. 377-8 : propositions on volumes of pyramids III. 389-91, of cylinders and cones III. 422-3 Leibniz I. 145, 169, 176, 194 Leiden MS. 399, 1 of al-Ḥajjāj and an-Nairīzī I. 22, 27 n., 79 n. Lemma I. 114 : meaning (=assumption) I. 133-4 : lemmas interpolated I. 59-60, especially from Pappus I. 67 : lemma assumed in VI. 22, II. 242-3 : alternative propositions on duplicate ratios and ratios of which they are duplicate (De Morgan and others) II. 242-7 : lemmas interpolated, (after X. 9) III. 30-1, (after X. 59) III. 97, 131-2 : lemmas suspected, (those added to X. 18, 23) III. 48, (that after XII. 2) III. 375, (that after XIII. 2) III. 444-5 Length, μῆκος (of numbers in one dimension) II. 287 : Plato restricts term to side of integral square number II. 287 Leodamas of Thasos I. 36, 134 Leon I. 116 Leonardo of Pisa I. 9 n., 10, III. 8 Leotaud, Vincent II. 42 Leucippus I. 413 Linderup, H. C. I. 113 Line : Platonic definition I. 158 : objection of Aristotle I. 158 : “magnitude extended one way” (Aristotle, “Heromides”) I. 158 : “divisible or continuous one way” (Aristotle) I. 158-9 : “flux of point” I. 159 : Apollonius on, I. 159 : classification of lines, Plato and Aristotle I. 159-60, Heron I. 159-60, Geminus, first classification I. 160-1, second I. 161 : straight (εὐθεῖα), curved (καμπύλη), circular (περιφερής), spiral-shaped (ἑλικοειδής), bent (κεκαμμένη), broken (κεκλασμένη), round (τὸ στρογγύλον) I. 159, composite (σύνθετος), incomposite (ἀσύνθετος), “forming a figure” (σχηματοποιοῦσα), determinate (ὡρισμένη), indeterminate (ἀόριστος) I. 160 : “asymptotic” or non-secant (ἀσύμπτωτος), secant (συμπτωτός) I. 161 : simple, “mixed” I. 161-2 : homoeomeric (uniform) I. 161-2 : Proclus on lines without extremities I. 165 : loci on lines I. 329, 330 Linear, loci I. 330 : problems I. 330 : numbers=(1) in one dimension II. 287, (2) prime II. 285 Lionardo da Vinci, proof of I. 47, I. 365-6 Lippert I. 88 n. Lobachewsky, N. I. 1. 174-5, 213, 219 Locus-theorems (τοπικὰ θεωρήματα) and loci <!– Page 538 –>

(τόποι) : locus defined by Proclus I. 329 : loci likened by Chrysippus to Platonic ideas I. 330-1 : locus-theorems and loci (1) on lines (a) plane loci (straight lines and circles) (b) solid loci (conics), (2) on surfaces I. 329 : corresponding distinction between plane and solid problems, to which Pappus adds linear problems I. 330 : further distinction in Pappus between (1) ἐφεκτικοί (2) διεξοδικοί (3) ἀναστροφοικοὶ τόποι I. 330 : Proclus regards locus in I. 35, III. 21, 31 as an area which is locus of area (parallelogram or triangle) I. 330 Logical conversion, distinct from geometrical I. 256 Logical deductions I. 256, 284-5, 300 : not made by Euclid II. 22, 29 : logical equivalents I. 309, 314-5 Lorenz, J. F. I. 107-8, III. 34 Loria, Gino I. 7 n., 10 n., 11 n., 12 n., II. 425, III. 8, 9 Luca Paciuolo I. 98-9, 100, 418, III. 8 Lucas, E. I. 418, II. 426 Lucian II. 99 Lundgren, F. A. I. 113

Machir, Jakob b. I. 76 Magni, Domenico I. 106 Magnitude : common definition vicious I. 148 al-Māhānī I. 85 Major (irrational) straight line : biquadratic of which it is positive root III. 7 : defined III. 87-8 : equivalent to square root of fourth binomial III. 84, 125-7 : uniquely divided III. 98 : extension of meaning to irrational straight line of three terms III. 258 al-Ma’mūn, Caliph I. 75 Manitius, C. I. 38 Mansion, P. I. 219 al-Manṣūr, Caliph I. 75 Manuscripts of Elements I. 46-51 “Marriage, Figure of” (Plutarch), name for Pythagorean triangle (3, 4, 5), I. 417 Martianus Capella I. 91, 155 Martin, T. H. I. 20, 29 n., 30 n. Mas’ūd b. al-Qass al-Baġdādī I. 90 Maximus Planudes, scholia and lectures on Elements I. 72 Means : three kinds, arithmetic, geometric and harmonic II. 292-3 : geometric mean is “proportion par excellence” (κυρίως) II. 292-3 : one geometric mean between two square numbers, two between two cube numbers (Plato) II. 294, 363-5 : one geometric mean between similar plane numbers, two between similar solid numbers II. 371-5 : no numerical geometric mean between n and n + 1 (Archytas and Euclid) II. 295 Medial (straight line) : connected by Theaetetus with geometric mean III. 3, 4 : defined III. 49, 50 : medial area III. 54-5 : an unlimited number of irrationals can be derived from medial straight line, III. 254-5 Meguar = axis I. 93 Mehler, F. G. I. 404, III. 268, 284-5 Menaechmus : story of M. and Alexander I. 1 : on elements I. 114 : I. 117, 125, 133 n. Menelaus I. 21, 23 : direct proof of I. 25, I. 300 Menge, H. I. 16 n., 17 Middle term, or cause, in geometry, illustrated by Eucl. III. 31, I. 149 Mill, J. S. I. 144 Minor (irrational) straight line : biquadratic of which it is root III. 7 : defined III. 163-4 : uniquely formed III. 172-3 : equivalent to square root of fourth apotome III. 203-6 “Mixed” (lines) I. 161-2 : (surfaces) I. 162, 170 : different meanings of “mixed” I. 162 Mocenigo, Prince I. 97-8 Moderatus, a Pythagorean II. 280 Mollweide, C. B. I. 108 Mondoré (Montaureus), Pierre I. 102 Moses b. Tibbon I. 76 Motion, in mathematics I. 226 : motion without deformation considered by Helmholtz necessary to geometry I. 226-7, but shown by Veronese to be petitio principii I. 226-7 Müller, J. H. T. I. 189 Müller, J. W. I. 365 Muḥammad (b. ‘Abdalbāqī) al-Baġdādī, translator of De divisionibus I. 8 n., 90, 110 Muḥ. b. Aḥmad Abū ‘r-Raiḥān al-Bīrūnī I. 90 Muḥ. b. Ashraf Shamsaddīn as-Samarqandī I. 89 Muḥ. b. ‘Īsā Abū ‘Abdallāh al-Māhānī I. 85 Multinomial (straight line) : an extension from binomial, probably investigated by Apollonius, III. 256 Multiplication, definition of, II. 287 Munich MS. of enunciations (R) I. 94-5 Mūsā b. Muḥ. b. Maḥmūd Qāḍīzāde ar-Rūmī I. 5 n., 90 Music, Elements of (Sectio Canonis), attributed to Euclid, I. 17, II. 295, III. 33 al-Musta’ṣim, Caliph I. 90 al-Mutawakkil, Caliph I. 75

an-Nairīzī, Abū ‘l-‘Abbās al-Faḍl b. Ḥātim, I. 21-4, 85, 184, 190, 191, 195, 223, 232, 258, 270, 285, 299, 303, 326, 364, 367, 369, 373, 405, 408, II. 5, 16, 28, 34, 36, 44, 47, 302, 320, 383 Napoleon I. 103 Naṣīraddīn aṭ-Ṭūsī I. 4, 5 n., 77, 84, 89, 208-10, II. 28 Nazīf b. Yumn (Yaman) al-Qass I. 76, 77, 87 Neide, J. G. C. I. 103 Nesselmann, G. H. F. II. 287, 293, III. 8 Nicomachus I. 92, 417, II. 116, 119, 131, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 300, 363, 425 Nicomedes I. 42, 160-1, 265-6 Nipsus, Marcus Junius I. 305 Nixon, R. C. J. II. 16 Nominal and real definitions : see Definitions Number : defined by Thales, Eudoxus, Moderatus, Aristotle, Euclid II. 280 :

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Nicomachus and Iamblichus on, II. 280 : represented by lines II. 287, and by points or dots II. 288-9 “Nuptial Number” = Plato’s Geometrical Number in Republic, I. 417

Objection (ἔνστασις), technical term, in geometry I. 135, 257, 260, 265 : in logic (Aristotle) I. 135 Oblong : (of geometrical figure) I. 62, 151, 188 : (of number) in Plato either προμήκης or ἑτερομήκης II. 288 : but these terms denote two distinct divisions of plane numbers in Nicomachus, Theon of Smyrna and Iamblichus II. 289-90 Octahedron II. 98 : definition of, III. 262 : discovery of, attributed to Theaetetus III. 438 : problem of inscribing in sphere, Euclid’s solution III. 474-7, Pappus’ solution III. 477 Odd (number) : defs. of in Nicomachus II. 281 : Pythagorean definition II. 281 : def. of odd and even by one another unscientific (Aristotle) I. 148-9, II. 281 : Nicom. and Iambl. distinguish three classes of odd numbers (1) prime and incomposite, (2) secondary and composite, as extremes, (3) secondary and composite in themselves but prime and incomposite to one another, which is intermediate II. 287 Odd-times even (number) : definition in Eucl. spurious II. 283-4, and differs from definitions by Nicomachus etc. ibid. Odd-times odd (number) : defined in Eucl. but not in Nicom. and Iambl. II. 284 : Theon of Smyrna applies term to prime numbers II. 284 Oenopides of Chios I. 34, 36, 126, 271, 295, 371, 414, II. 111 Ofterdinger, L. F. I. 9 Olympiodorus I. 29 Oppermann I. 151 Optics of Euclid I. 17 “Ordered” proportion (τεταγμένη ἀναλογία), interpolated definition of, II. 137 Oresme, N. I. 97 Orontius Finaeus (Oronce Fine) I. 101, 104 Ozanam, Jaques I. 107, 108

Paciuolo, Luca I. 98-9, 100, 418, III. 8 Pamphile I. 317, 319 Pappus : contrasts Euclid and Apollonius I. 3 : on Euclid’s Porisms I. 10-14, Surface-loci I. 15, 16, Data I. 8 : on Treasury of Analysis I. 8, 10, 11, 138 : commentary on Elements I. 24-7, partly preserved in scholia I. 66 : evidence of scholia as to Pappus’ text I. 66-7 : commentary on Book X. survives in Arabic III. 3 : quotations from it, III. 3-4, 255-9 : lemmas in Book X. interpolated from, I. 67 : on Analysis and Synthesis I. 138-9, 141-2 : additional axioms by, I. 25, 223, 224, 232 : on converse of Post. 4, I. 25, 201 : proof of I. 5 by, I. 254 : extension of I. 47, I. 366 : semi-algebraical methods in, I. 373, 378 : on loci I. 329, 330 : on conchoids I. 161, 266 : on quadratrix I. 266 : on isoperimetric figures I. 26, 27, 333 : on paradoxes of Erycinus I. 27, 290 : lemma on Apollonius’ Plane νεύσεις II. 64-5 : problem from same work II. 81 : assumes case of VI. 3 where external angle bisected (Simson’s VI. Prop. A) II. 197 : theorem from Apollonius’ Plane Loci II. 198 : theorem that ratio compounded of ratios of sides (of equiangular parallelograms) is equal to ratio of rectangles contained by sides II. 250 : use of Euclid’s irrationals III. 9, 10 : methods of inscribing regular solids in sphere, tetrahedron III. 472-3, octahedron III. 477, cube III. 480, icosahedron III. 489-91, dodecahedron III. 501-3 : I. 20, 39, 133 n., 137, 151, 225, 388, 391, 401 : II. 4, 27, 29, 67, 79, 81, 113, 133, 211, 250, 251, 292 Papyrus, Herculanensis No. 1061, I. 50, 184 : Oxyrhynchus I. 50 : Fayūm I. 51, 337, 338 : Rhind I. 304, 352 Paradoxes, in geometry I. 188 : of Erycinus I. 27, 290, 329 : an ancient “Budget of Paradoxes” I. 329 “Parallelepipedal” = with parallel planes or faces : “parallelepipedal solid” (not “solid parallelepiped”) or “parallelepiped” III. 326 : generally has six faces but sometimes more (“parallelepipedal prism”) III. 401, 404 : “parallelepipedal” (solid) numbers in Nicomachus have two of sides differing by unity II. 290 Parallelogram (=parallelogrammic area), first introduced I. 325 : rectangular parallelogram I. 370 Parallels : Aristotle, on I. 190, 191-2 : definitions, by “Aganis” I. 191, Geminus I. 191, Posidonius I. 190, Simplicius I. 190 : as equi-distants I. 190-1, 194 : direction-theory of, I. 191-2, 194 : definitions classified I. 192-4 : Veronese’s definition and postulate I. 194 : Parallel Postulate, see Postulate 5 : Legendre’s attempt to establish theory of, I. 213-9 : parallel planes, definition of, III. 260, 265 Paris MSS. of Elements, (p) I. 49, (q) I. 50 Pasch, M. I. 157, 228, 250 “Peacock’s tail,” name for Eucl. III. 8, I. 99, 418 Pediasimus, Joannes I. 72-3 Peet, T. Eric I. 352 Peithon I. 203 Peletarius (Jacques Peletier) I. 103, 104, 249, 407, II. 47, 56, 84, 146, 190 : on angle of contact and angle of semicircle II. 41 Pena I. 104 Pentagon : decomposition of regular pentagon into 30 elementary triangles II. 98 : relation to pentagram II. 99 Pentagonal numbers II. 289 “Perfect” (of a class of numbers) II. 293-4, 421-6 : Pythagoreans applied term to 10, <!– Page 540 –>

  1. 294 : 3 also called “perfect” I. 417, II. 294

Perpendicular (κάθετος) : definition I. 181 : “plane” and “solid” I. 272 : perpendicular and obliques I. 291 : perpendicular to plane, III. 260, 263 : perpendicular to two straight lines not in one plane III. 306-7 Perseus I. 42, 162-3 Perturbed proportion (τεταραγμένη ἀναλογία) II. 136, 176-7 Pesch, J. G. van, De Procli fontibus I. 23 sqq., 29 n. Petrus Montaureus (Pierre Mondoré) I. 102 Peyrard and Vatican MS. 190 (P) I. 46, 47, 103 : I. 108 Pfleiderer, C. F. I. 168, 298, II. 2 Phaenomena of Euclid I. 16, 17 Philippus of Medma I. 1, 116 Phillips, George I. 112 Philo of Byzantium I. 20, 23 : proof of I. 8, I. 263-4 Philolaus I. 34, 351, 371, 399, 411 Philoponus I. 45, 191-2, II. 234, 282 Pirckenstein, A. E. Burkh. von, I. 107 Plane (or plane surface) : Plato’s definition of, I. 171, III. 272-3 : Proclus’ and Simplicius’ interpretation of Euclid’s def. I. 171 : possible origin of Euclid’s def. I. 171 : Archimedes’ assumption I. 171, 172 : other ancient definitions of, in Proclus, Heron, Theon of Smyrna, an-Nairīzī I. 171-2 : “Simson’s” definition (“axiom of the plane”) I. 172-3, III. 273, and Gauss on, I. 172-3 : Crelle’s tract on, I. 172-4 : other definitions by Fourier I. 173, Deahna I. 174, J. K. Becker I. 174, Leibniz I. 176, Beez I. 176 : evolution of, by Bolyai and Lobachewsky I. 174-5 : Enriques and Amaldi, Ingrami, Veronese and Hilbert on, I. 175 : plane at right angles to plane, Euclid’s definition of, III. 260, 263, and alternative definition making it a particular case of “inclination” III. 303-4 : parallel planes defined III. 260, 265 “Plane loci” I. 329-30 : Plane Loci of Apollonius I. 14, 259, 330, II. 198-200 Plane numbers, product of two factors (“sides” or “length” and “breadth”) II. 287-8 : in Plato either square or oblong II. 287-8 : similar plane numbers II. 293 : one mean proportional between similar plane numbers II. 371-2 “Plane problems” I. 329 Planudes, Maximus I. 72 Plato : I. 1, 2, 3, 137, 155-6, 159, 184, 187, 203, 221, 411, 417, III. 1, 3 : supposed invention of Analysis by, I. 134 : def. of straight line I. 165-6 : def. of plane surface I. 171 : on golden section II. 99 : on art of stereometry (length, breadth, and depth) as one of three μαθήματα, next to geometry but commonly put after astronomy because little advanced III. 262 : generation of cosmic figures by putting together triangles, I. 226, 413-4, II. 97-8, III. 267 : rule for rational right-angled triangles I. 356, 357, 359, 360, 385 : “rational diameter of 5” I. 399, 413, gives 7/5 as approximation to √2, II. 119 : passage of Theaetetus on δυνάμεις (square roots or surds) I. 411, 412-3, II. 288, 290, III. 1-3 : on square and oblong numbers II. 288, 290 : theorem that between square numbers one mean suffices, between cube numbers two means are necessary II. 294, 364 “Platonic” figures I. 2, 413-4 : scholium on, III. 438 Playfair, John I. 103, 111 : “Playfair’s” Axiom I. 220 : used to prove Eucl. I. 29, I. 312, and Eucl. Post. 5, I. 313 : comparison of Axiom with Post. 5, I. 313-4 : II. 2 Pliny I. 333 Plutarch I. 29, 37, 177, 343, 351, 417, II. 98, 254, III. 368 Point : Pythagorean definition of, I. 155 : interpretation of Euclid’s definition I. 155 : Plato’s view of, and Aristotle’s criticism I. 155-6 : attributes of, according to Aristotle I. 156 : terms for (στιγμή, σημεῖον) I. 156 : other definitions by “Herundes,” Posidonius I. 156, Simplicius I. 157 : negative character of Euclid’s def. I. 156 : is it sufficient? I. 156 : motion of, produces line I. 157 : an-Nairīzī on, I. 157 : modern explanations by abstraction I. 157 Polybius I. 331 Polygon : sum of interior angles (Proclus’ proof) I. 322 : sum of exterior angles I. 322 Polygonal numbers II. 289 Polyhedral angles, extension of XI. 21 to, III. 310-1 “Pons Asinorum” I. 415-6 : Pont aux ânes, ibid. Porism : two senses I. 13 : (1)=corollary I. 134, 278-9 : as corollary to proposition precedes “Q.E.D.” or “Q.E.F.” II. 8, 64 : Porism to IV. 15 mentioned by Proclus II. 109 : Porism to VI. 19, II. 234 : interpolated Porisms (corollaries) I. 60-1, 381 : (2) as used in Porisms of Euclid, distinguished from theorems and problems I. 10, 11 : account of the Porisms given by Pappus I. 10-13 : modern restorations by Simson and Chasles I. 14 : views of Heiberg I. 11, 14, and of Zeuthen I. 15 Porphyry I. 17 : commentary on Euclid I. 24 : Symmikta I. 24, 34, 44 : I. 136, 277, 283, 287 Poselger III. 8 Posidonius, the Stoic I. 20, 27, 28 n., 39, 189, 197 : book directed against the Epicurean Zeno I. 34, 43 : on parallels I. 40, 190 : definition of figure I. 41, 183 Postulate, distinguished from axiom, by Aristotle I. 118-9, by Proclus (Geminus and “others”) I. 121-3 : from hypothesis, by Aristotle I. 120-1, by Proclus I. 121-2 : postulates in Archimedes I. 120, 123 : Euclid’s view of, reconcileable with Aristotle’s I. 119-20, 124 : postulates do not confine us to ruler and compasses I. 124 :

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Postulates 1, 2, significance of, I. 195-6 : famous “Postulate” or “Axiom of Archimedes” I. 234, III. 15-6 Postulate 4 : significance of, I. 200 : proofs of, resting on other postulates I. 200-1, 231 : converse true only when angles rectilineal (Pappus) I. 201 Postulate 5 : probably due to Euclid himself I. 202 : Proclus on, I. 202-3 : attempts to prove, Ptolemy I. 204-6, Proclus I. 206-8, Naṣīraddīn aṭ-Ṭūsī I. 208-10, Wallis I. 210-1, Saccheri I. 211-2, Lambert I. 212-3 : substitutes for, “Playfair’s” axiom (in Proclus) I. 220, others by Proclus I. 207, 220, Posidonius and Geminus I. 220, Legendre I. 213, 214, 220, Wallis I. 220, Carnot, Laplace, Lorenz, W. Bolyai, Gauss, Worpitzky, Clairaut, Veronese, Ingrami I. 220 : Post. 5 proved from, and compared with, “Playfair’s” Axiom I. 313-4 : I. 30 is logical equivalent of, I. 220 Potts, Robert I. 112, 246 Powers, R. E. II. 426 Prestet, Jean II. 426 Prime (number) : definitions of, II. 284-5 : Aristotle on two senses of “prime” I. 146, II. 285 : 2 admitted as prime by Eucl. and Aristotle, but excluded by Nicomachus, Theon of Smyrna and Iamblichus, who make prime a subdivision of odd II. 284-5 : “prime and incomposite (ἀσύνθετος)” II. 284 : different names for prime, “odd-times odd” (Theon), “linear” (Theon), “rectilinear” (Thymaridas), “euthymetric” (Iamblichus) II. 285 : prime absolutely or in themselves as distinct from prime to one another (Theon) II. 285 : definitions of “prime to one another” II. 285-6 Principles, First, I. 117-24 Prism, definition of, by Euclid III. 261, by others III. 268-9 : “parallelepipedal prisms” III. 404 Problem, distinguished from theorem I. 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) I. 128 : in widest sense anything propounded (possible or not) but generally a construction which is possible I. 128-9 : another classification (1) problem in excess (πλεονάζον), asking too much I. 129, (2) deficient problem (ἐλλιπὲς πρόβλημα), giving too little I. 129 Proclus : details of career I. 29-30 : remarks on earlier commentators I. 19, 33, 45 : commentary on Eucl. I, sources of, I. 29-45, object and character of, I. 31-2 : commentary probably not continued, though continuation intended I. 32-3 : books quoted by name in, I. 34 : famous “summary” I. 37-8 : list of writers quoted I. 44 : his own contributions I. 44-5 : character of MS. used by, I. 62, 63 : on the nature of elements and things elementary I. 114-6 : on advantages of Euclid’s Elements, and their object I. 115-6 : on first principles, hypotheses, postulates, axioms I. 121-4 : on difficulties in three distinctions between postulates and axioms I. 123 : on theorems and problems I. 124-9 : on formal divisions of proposition I. 129-31, II. 100 : attempt to prove Postulate 5, I. 206-8 : commentary on Plato’s Republic, allusion in, to “side-” and “diagonal-” numbers in connexion with Eucl. II. 9, 10, I. 399-400 : on use of “quindecagon” for astronomy II. 111 : I. 412, II. 4, 39, 40, 193, 247, 269, III. 10, 264, 267, 273, 310, 441 Proof (ἀπόδειξις), necessary part of proposition I. 129-30 Proportion : complete theory applicable to incommensurables as well as commensurables is due to Eudoxus I. 137, 351, II. 112 : old (Pythagorean) theory practically represented by arithmetical theory of Eucl. VII., II. 113 : in giving older theory as well Euclid simply followed tradition II. 113 : Aristotle on general proof (new in his time) of theorem (alternando) in proportion II. 113 : X. 5 as connecting two theories II. 113 : De Morgan on extension of meaning of ratio to cover incommensurables II. 118 : power of expressing incommensurable ratio is power of approximation without limit II. 119 : interpolated definitions of proportion as “sameness” or “similarity of ratios” II. 119 : definition in V. Def. 5 substituted for that of VII. Def. 20 because latter found inadequate, not vice versa II. 121 : De Morgan’s defence of V. Def. 5 as necessary and sufficient II. 122-4 : V. Def. 5 corresponds to Weierstrass’ conception of number in general and to Dedekind’s theory of irrationals II. 124-6 : alternatives for V. Def. 5 by a geometer-friend of Saccheri, by Faifofer, Ingrami, Veronese, Enriques and Amaldi II. 126 : proportionals of VII. Def. 20 (numbers) a particular case of those of V. Def. 5 (Simson’s Props. C, D and notes) II. 126-9, III. 25 : proportion in three terms (Aristotle makes it four) the “least” II. 131 : “continuous” proportion (συνεχὴς or συνημμένη ἀναλογία, in Euclid ἑξῆς ἀνάλογον) II. 131, 293 : three “proportions” II. 292, but proportion par excellence or primary is continuous or geometric II. 292-3 : “discrete” or “disjoined” (διῃρημένη, διεζευγμένη) II. 131, 293 : “ordered” proportion (τεταγμένη), interpolated definition of, II. 137 : “perturbed” proportion (τεταραγμένη) II. 136, 176-7 : extensive use of proportions in Greek geometry II. 187 : proportions enable any quadratic equation with real roots to be solved II. 187 : supposed use of propositions of Book V. in arithmetical Books II. 314, 320 Proposition, formal divisions of, I. 129-31 Protarchus I. 5, III. 512 <!– Page 542 –>

Psellus, Michael, scholia by, I. 70, 71, II. 234 Pseudaria of Euclid I. 7 : Pseudographemata I. 7 n. Pseudoboethius I. 92 Ptolemy I. : I. 1, 2 : story of Euclid and Ptolemy I. 1 Ptolemy II. Philadelphus I. 20 Ptolemy VII. (Euergetes II.), Physcon, I. 20 Ptolemy, Claudius I. 21, 30 n. : Harmonica of, and commentary on, I. 17 : on Parallel-Postulate I. 28 n., 34, 43, 45 : attempt to prove it I. 204-6 : lemma about quadrilateral in circle (Simson’s VI. Prop. D) II. 225-7 : II. 111, 117, 119 Punch on Pons Asinorum I. 416 Pyramid, definitions of, by Euclid III. 261, by others III. 268 Pyramidal numbers II. 290 : pyramids truncated, twice-truncated etc. II. 291 Pythagoras I. 4 n., 36 : supposed discoverer of the irrational I. 351, 411, 412, of application of areas I. 343-4, of theorem of I. 47, I. 343-4, 350-4, 411, 412, of construction of five regular solids I. 413-4, II. 97 : story of sacrifice I. 37, 343, 350 : probable method of discovery of I. 47 and proof of, I. 352-5 ; suggestions by Bretschneider and Hankel I. 354, by Zeuthen I. 355-6 : rule for forming right-angled triangles in rational numbers I. 351, 356-9, 385 : construction of figure equal to one and similar to another rectilineal figure II. 254 : introduced “the most perfect proportion in four terms and specially called ‘harmonic’” into Greece II. 112 Pythagoreans I. 19, 36, 155, 188, 279, 411-4 : term for surface (χροιά) I. 169 : angles of triangle equal to two right angles, theorem and proof I. 317-20 : three polygons which in contact fill space round point I. 318, II. 98 : method of application of areas (including exceeding and falling-short) I. 343, 384, 403, II. 187, 258-60, 263-5, 266-7 : gnomon Pythagorean I. 351 : “rational” and “irrational diameter of 5” I. 399-400, 413 : story of Pythagorean who, having divulged the irrational, perished by shipwreck I. 411, III. 1 : 7/5 as approximation to √2, II. 119 : approximation to √2 by “side-” and “diagonal-” numbers I. 398-400, III. 2, 20 : proof of incommensurability of √2, III. 2 : construction of isosceles triangle of Eucl. IV. 10, and of regular pentagon, I. 414, II. 97-8 : possible method of discovery of latter II. 97-9 : distinguished three sorts of means, arithmetic, geometric and harmonic II. 112 : had theory of proportion applicable to commensurables only II. 112 : construction of dodecahedron in sphere II. 97, and of other regular solids I. 413-4, III. 438 : definitions of unit II. 279, of even and odd II. 281 : called 10 “perfect” II. 294

Qāḍīzāde ar-Rūmī I. 5 n., 90 Q.E.D. (or F.) I. 57 al-Qifṭī I. 4 n., 94 Quadratic equations : solution assumed by Hippocrates I. 386-7 : geometrical solution of particular quadratics I. 383-5, 386-8 : solution of general quadratic by means of proportions II. 187, 263-5, 266-7 : διορισμός or condition of possibility of solving equation of Eucl. VI. 28, II. 259 : one solution only given, for obvious reasons II. 260, 264, 267 : but method gives both roots if real II. 258 : exact correspondence of geometrical to algebraical solution, II. 263-4, 266-7 : indication that Greeks solved them numerically III. 43-4 Quadratrix I. 265-6, 330, 413 Quadrature (τετραγωνισμός), definitions of, I. 149 Quadrilateral : varieties of, I. 188-90 : inscribing in circle of quadrilateral equiangular to another II. 91-2 : condition for inscribing circle in, II. 93, 95 : quadrilateral in circle, Ptolemy’s lemma (Simson’s VI. Prop. D) II. 225-7 : quadrilateral not a “polygon” II. 239 Quadrinomial (straight line), compound irrational (extension from binomial) III. 256 “Quindecagon” (fifteen-angled figure) : useful for astronomy II. 111 Quintilian I. 333 Qusṭā b. Lūqā al-Ba’labakkī, translator of “Books XIV, XV” I. 76, 87, 88

Radius, no Greek word for, I. 199, II. 2 Ramus, Petrus (Pierre de la Ramée) I. 104, II. 121 Ratdolt, Erhard I. 78, 97 Ratio : definition of, II. 116-9, no sufficient ground for regarding it as spurious II. 117, Barrow’s defence of it II. 117 : method of transition from arithmetical to more general sense covering incommensurables II. 118 : means of expressing ratio of incommensurables is by approximation to any degree of accuracy II. 119 : def. of greater ratio only one criterion (there are others) II. 130 : tests for greater, equal and less ratios mutually exclusive II. 130-1 : test for greater ratio easier to apply than that for equal ratio II. 129-30 : arguments about greater and less ratios unsafe unless they go back to original definitions (Simson on V. 10) II. 156-7 : compound ratio II. 132-3, 189-90, 234 : operation of compounding ratios II. 234 : “ratio compounded of their sides” (careless expression) II. 248 : duplicate, triplicate etc. ratio as distinct from double, triple etc. II. 133 : alternate ratio, alternando II. 134 : inverse ratio, inversely II. 134 : composition of ratio, componendo, different from compounding ratios II. 134-5 : separation of ratio, separando (commonly dividendo) II. 135 : conversion of ratio, convertendo II. 135 : ratio ex aequali II. 136, ex aequali in perturbed proportion II. 136 : division of ratios used in Data as <!– Page 543 –>

general method alternative to compounding II. 249-50 : names for particular arithmetical ratios II. 292 Rational (ῥητός) : (of ratios) I. 137 : “rational diameter of 5” I. 399-400, 413 : rational right-angled triangles, see right-angled triangles : any straight line may be taken as rational and the irrational is irrational in relation thereto III. 10 : rational straight line is still rational if commensurable with rational straight line in square only (extension of meaning by Euclid) III. 10, 11-12 Rationalisation of fractions with denominator of form a ± √B or √A ± √B, III. 243-52 Rauchfuss, see Dasypodius Rausenberger, O. I. 157, 175, 313, III. 307, 309 ar-Rāzī, Abū Yūsuf Ya’qūb b. Muḥ. I. 86 Reciprocal or reciprocally-related figures : definition spurious II. 189 Rectangle : =rectangular parallelogram I. 370 : “rectangle contained by” I. 370 Rectilineal angle : definitions classified I. 179-81 : rectilineal figure I. 187 : “rectilineal segment” I. 196 Reductio ad absurdum I. 134 : described by Aristotle and Proclus I. 136 : synonyms for, in Aristotle I. 136 : a variety of Analysis I. 140 : by exhaustion I. 285, 293 : nominal avoidance of, I. 369 : the only possible method of proving Eucl. III. 1, II. 8 Reduction (ἀπαγωγή), technical term, explained by Aristotle and Proclus I. 135 : first “reduction” of a difficult construction due to Hippocrates I. 135, II. 133 Regiomontanus (Johannes Müller of Königsberg) I. 93, 96, 100 Reyher, Samuel I. 107 Rhaeticus I. 101 Rhind Papyrus I. 304, 352 Rhomboid I. 62, 151, 189 Rhombus I. 62, 151, meaning and derivation I. 189 Riccardi, P. I. 96, 112, 202 Riemann, B. I. 219, 273, 274, 280 Right angle : definition I. 181 : drawing straight line at right angles to another, Apollonius’ construction for, I. 270 : construction when drawn at extremity of second line (Heron) I. 270 Right-angled triangles, rational : rule for finding, by Pythagoras I. 356-9, by Plato I. 356, 357, 359, 360, 385, by Euclid III. 63-4 : discovery of rules by means of gnomons I. 358-60 : connexion of rules with Eucl. II. 4, 8, I. 360 : rational right-angled triangles in Āpastamba I. 361, 363 Röth I. 357-8 Rouché and de Comberousse I. 313 Rudd, Capt. Thos. I. 110 Ruellius, Joan. (Jean Ruel) I. 100 “Rule of three” : Eucl. VI. 12 equivalent to, II. 215 Russell, Bertrand I. 227, 249

Saccheri, Gerolamo I. 106, 144-5, 167-8, 185-6, 194, 197-8, 200-1, II. 126, 130 : proof of existence of fourth proportional by Eucl. VI. 1, 2 and 12, II. 170 Ṣā’id b. Mas’ūd b. al-Qass I. 90 Śatapatha-Brāhmaṇa I. 362 Savile, Henry I. 105, 166, 245, 250, 262, II. 190 Scalene (σκαληνός or σκαληνής) I. 187-8 : of numbers (= odd) I. 188 : a class of solid numbers II. 290 : of cone (Apollonius) I. 188 Schessler, Chr. I. 107 Scheubel, Joan. I. 101, 107 Schiaparelli, G. V. I. 163 Schmidt, Max C. P. I. 304, 319 Scholia to Elements and MSS. of, I. 64-74 : historical information in, I. 64 : evidence in, as to text I. 64-5, 66-7 : sometimes interpolated in text I. 67 : classes of, “Schol. Vat.” I. 65-9, “Schol. Vind.” I. 69-70, miscellaneous I. 71-4 : “Schol. Vat.” partly derived from Pappus’ commentary I. 66 : many scholia partly extracted from Proclus on Bk. I., I. 66, 69, 72 : many from Geminus solely I. 74 : numerical illustrations in, in Greek and Arabic numerals I. 71, 74 n. : scholia by Psellus I. 70-1, by Maximus Planudes I. 72, Joannes Pediasimus I. 72-3 : scholia in Latin published by G. Valla, Commandinus, Conrad Dasypodius I. 73 : scholia on Eucl. II. 13, I. 407 : Scholium IV. No. 2 ascribes Book IV. to Pythagoreans I. 414, II. 97 : Scholium V. No. 1 attributes Book V. to Eudoxus II. 112 : Scholium X. No. 1 attributes discovery of irrational and incommensurable to Pythagoreans III. 1 : scholium published later by Heiberg attributes Scholium X. No. 62 to Proclus I. 32 n. Scholiast to Clouds of Aristophanes II. 99 Schooten, Franz van I. 108 Schopenhauer I. 227, 354 Schotten, H. I. 167, 174, 179, 192-3, 202 Schultze, A. and Sevenoak, F. L. III. 284, 303, 331 Schumacher I. 321 Schur, F. I. 328 Schweikart, F. K. I. 219 Scipio Vegius I. 99 Sectio Canonis attributed to Euclid I. 17, II. 295, III. 33 Section (τομή) : =point of section I. 170, 171, 383 : “the section”=”golden section” q.v. Sector (of circle) : explanation of name : two kinds (1) with vertex at centre, (2) with vertex at circumference II. 5 Sector-like (figure) II. 5 : bisection of such a figure by straight line II. 5 Seelhoff, P. II. 426 Segment of circle : angle of, I. 253, II. 4 : similar segments II. 5 : segment less than semicircle called ἁψίς I. 187 Semicircle : I. 186 : centre of, I. 186 : angle of, I. 182, 253, II. 4, 39-41 (see Angle) : angle in semicircle a right angle, pre-Euclidean proof II. 63 <!– Page 544 –>

Separation of ratio, διαίρεσις λόγου, and separando (διελόντι) II. 135 : separando and componendo used relatively to one another, not to original ratio II. 168, 170 Seqt I. 304 Serenus of Antinoeia I. 203 Serle, George I. 110 Servais, C. II. 426 Setting-out (ἔκθεσις), one of formal divisions of a proposition I. 129 : may be omitted I. 130 Sexagesimal fractions in scholia I. 74 Sextus Empiricus I. 62, 63, 184 Shamsaddīn as-Samarqandī I. 5 n., 89 “Side-” and “diagonal-” numbers, described I. 398-400 : due to Pythagoreans I. 400, III. 2, 20 : connexion with Eucl. II. 9, 10, I. 398-400 : use for approximation to √2, I. 399 “Side of a medial minus a medial area” (in Euclid “that which produces with a medial area a medial whole”), a compound irrational straight line : biquadratic of which it is a root III. 7 : defined III. 165-6 : uniquely formed III. 174-7 : equivalenc to square root of sixth apotome III. 209-11 “Side of a medial minus a rational area” (in Euclid “that which produces with a rational area a medial whole”), a compound irrational straight line : biquadratic of which it is a root III. 7 : defined III. 164 : uniquely formed III. 173-4 : equivalent to square root of fifth apotome III. 206-8 “Side of a rational plus a medial area,” a compound irrational straight line : biquadratic equation of which it is a root III. 7 : defined III. 88-9 : uniquely divided III. 99 : equivalent to square root of fifth binomial III. 84, 128-9 “Side of the sum of two medial areas,” a compound irrational straight line : biquadratic of which it is a root III. 7 : defined III. 89-91 : uniquely divided III. 99-101 : equivalent to square root of sixth binomial III. 84, 130-1 “Side” used in translation of Book X. for ἡ δυναμένη (τὸ χωρίον), “side of a square equal to (the area)” III. 13, 119 Sides of plane and solid numbers, II. 287-8 Sigboto I. 94 “Similar” (=equal) angles I. 182, 252 “Similarly inclined” (of planes) III. 260, 265 Similar plane and solid numbers I. 357, II. 293 : one mean between two similar plane numbers II. 294, 371-2, two means between two similar solid numbers II. 294, 373-5 Similar rectilineal figures : def. of, given in Aristotle II. 188 : def. gives at once too little and too much II. 188 : similar figures on straight lines which are proportional are themselves proportional and conversely (VI. 22), alternatives for proposition II. 242-7 Similar segments of circles II. 5 Similar solids : definitions of, III. 261, 265-7

Simon, Max I. 108, 155, 157-8, 167, 202, 328, II. 124, 134 Simplicius : commentary on Euclid I. 27-8 : on lunes of Hippocrates I. 29, 35, 386-7 : on Eudemus’ style I. 35, 38 : on parallels I. 190-1 : I. 22, 167, 171, 184, 185, 197, 203, 223, 224, 413, III. 366 Simpson, Thomas, II. 121, III. 274 Simson, Robert : on Euclid’s Porisms I. 14 : on “vitiations” in Elements due to Theon I. 46, 103, 104, 106, 111, 148 : definition of plane I. 172-3 : Props. C, D (Bk. V.) connecting proportionals of VII. Def. 20 as particular case with those of V. Def. 5, II. 126-9, III. 25 : Axioms to Bk. V., II. 137 : Prop. B (inversion) II. 144 : Prop. E (convertendo) II. 175 : shortens V. 8 by compressing two cases into one II. 152-3 : important note showing flaw in V. 10 and giving alternative II. 156-7 : Bk. VI. Prop. A extending VI. 3 to case where external angle bisected II. 197 : Props. B, C, D II. 222-7 : remarks on VI. 27-9, II. 258-9 : Prop. D, Book XI., III. 345 : I. 185, 186, 255, 259, 287, 293, 296, 322, 328, 384, 387, 403, II. 2, 3, 8, 22, 23, 33, 34, 37, 43, 49, 53, 70, 73, 79, 90, 117, 131, 132, 140, 143-4, 145, 146, 148, 154, 161, 162, 163, 165, 170-2, 177, 179, 180, 182, 183, 184, 185, 186, 189, 193, 195, 209, 211, 212, 230-1, 238, 252, 269, 270, 272-3, III. 265, 266, 273-4, 275, 276, 286-7, 289, 295, 301, 309, 314, 321, 324, 327, 331, 334, 340, 341, 349, 351, 359, 362, 375, 433, 434 Sind b. ‘Alī Abū ‘t-Ṭaiyib I. 86 Size, proper translation of πηλικότης in V. Def. 3, II. 116-7, 189-90 Smith and Bryant, alternative proofs of V. 16, 17, 18 by means of VI. 1, where magnitudes are straight lines or rectilineal areas II. 165-6, 169, 173-4 : I. 404, III. 268, 275, 284, 303, 307 Smith, D. E. I. 362, 417 Solid : definition of, III. 260, 262-3 : similar solids, definitions of, III. 261, 265-7 : equal and similar solids, ibid. Solid angle : definitions of, III. 261, 267-8 : solid “angle” of “quarter of sphere,” of cone, or of half-cone III. 268 “Solid loci” I. 329, 330 : Solid Loci of Aristaeus I. 16, 329 Solid numbers, three varieties according to relative lengths of sides II. 290-1 “Solid problems” I. 329, 330 Speusippus I. 125 Sphaerica, early treatise on, I. 17 Sphere : definitions of, by Euclid III. 261, 269, by others III. 269 Spherical number, a particular species of cube number II. 291 Spiral, “single-turn,” I. 122-3 n., 164-5 : in Pappus = cylindrical helix I. 165 Spiral of Archimedes I. 26, 267 Spire (tore) or Spiric surface I. 163, 170 : varieties of, I. 163 <!– Page 545 –>

Spiric curves or sections, discovered by Perseus I. 161, 162-4 Square number, product of equal numbers II. 289, 291 : one mean between square numbers II. 294, 363-4 Staudt, Ch. von III. 276 Steenstra, Pybo I. 109 Steiner, Jakob I. 193 Steinmetz, Moritz I. 101 Steinschneider, M. I. 8 n., 76 sqq. Stephanus Gracilis I. 101-2 Stephen Clericus I. 47 Stevin, Simon III. 8 Stifel, Michael III. 8 Stobaeus I. 3, II. 280 Stoic “axioms” I. 41, 221 : illustrations (δείγματα) I. 329 Stolz, O. I. 328, III. 16 Stone, E. I. 105 Straight line : pre-Euclidean (Platonic) definition I. 165-6 : Archimedes’ assumption respecting, I. 166 : Euclid’s definition, interpreted by Proclus and Simplicius I. 166-7 : language and construction of, I. 167, and conjecture as to origin I. 168 : other definitions I. 168-9, in Heron I. 168, by Leibniz I. 169, by Legendre I. 169 : two straight lines cannot enclose a space I. 195-6, cannot have a common segment I. 196-9, III. 273 : one or two cannot make a figure I. 169, 183 : division of straight line into any number of equal parts (an-Nairīzī) I. 326 : straight line at right angles to plane, definition of, III. 260, alternative constructions for, III. 293-4 Strömer, Mårten I. 113 Studemund, W. I. 92 n. St Vincent, Gregory of, I. 401, 404 Subduplicate of any ratio found by Eucl. VI. 13, II. 216 Subtend, meaning and construction I. 249, 283, 350 Suidas I. 370, 413, III. 360, 438 Sulaimān b. ‘Uṣma (or ‘Uqba) I. 85, 90 Superposition : Euclid’s dislike of method of, I. 225, 249 : apparently assumed by Aristotle as legitimate I. 226 : used by Archimedes I. 225 : objected to by Peletarius I. 249 : no use theoretically, but merely furnishes practical test of equality I. 227 : Bertrand Russell on, I. 227, 249 Surface : Pythagorean term for, χροιά (=colour, or skin) I. 169 : terms for, in Plato and Aristotle I. 169 : ἐπιφάνεια in Euclid (not ἐπίπεδον) I. 169 : alternative definition of, in Aristotle I. 170 : produced by motion of line I. 170 : divisions or sections of solids are surfaces I. 170, 171 : classifications of surfaces by Heron and Geminus I. 170 : composite, incomposite, simple, mixed I. 170 : spiric surfaces I. 163, 170 : homoeomeric (uniform) surfaces I. 170 : spheroids I. 170 : plane surface, see plane : loci on surfaces I. 329, 330

Surface-loci of Euclid I. 15, 16, 330 : Pappus’ lemmas on, I. 15, 16 Suter, H. I. 8 n., 17 n., 18 n., 25 n., 78 n., 85-90, III. 3 Suvoroff, Pr. I. 113 Swinden, J. H. van I. 169, II. 188 Sylvester, J. II. 426 Synthesis, see Analysis and Synthesis Syrianus I. 30, 44, 176, 178

Tacquet, André I. 103, 105, 111, II. 121, 258 Tāittirīya-Saṁhitā I. 362 Tannery, P. I. 7 n., 37-40, 44, 160, 163, 221, 223, 224, 225, 232, 305, 353, 412, 417, II. 112, 113, III. 1, 5 Ta’rīkh al-Hukamā I. 4 n. Tartaglia, Niccolò I. 3, 103, 106, II. 2, 47 Taurinus, F. A. I. 219 Taurus I. 62, 184 Taylor, H. M. I. 248, 377-8, 404, II. 16, 22, 29, 56, 75, 102, 227, 244, 247, 272, III. 268, 275, 303, 491-2, 498 Taylor, Th. I. 259 Tetrahedron, regular : II. 98 : problem of inscribing in given sphere, Euclid’s solution III. 467-72, Pappus’ solution III. 472-3 Thābit b. Qurra, translator of Elements I. 42, 75-80, 82, 84, 87, 94 : proof of I. 47, I. 364-5 Thales I. 36, 37, 185, 252, 253, 278, 317, 318, 319, II. 111, 280 : on distance of ship from shore I. 304-5 Theaetetus I. 1, 37 : contributions to theory of incommensurables III. 3 : Eucl. X. 9 attributed to, III. 3, 30 : supposed to have discovered octahedron and icosahedron III. 438 : was the first to write a treatise on regular solids III. 438 : III. 442 Theodorus Antiochita I. 71 Theodorus Cabasilas I. 72 Theodorus of Cyrene : proved incommensurability of √3, √5 etc. up to √17, I. 411, 412-3, III. 1, 2 Theodorus Metochita, I. 3 Theodosius II. 37, III. 269, 366, 472 Theognis I. 371 Theon of Alexandria : edition of Elements I. 46 : changes made by, I. 46 : Simson on “vitiations” by, I. 46 : principles for detecting his alterations, by comparison of P, ancient papyri and “Theonine” MSS. I. 51-3 : character of changes by, I. 54-8 : interpolation in V. 13 and Porism II. 144 : interpolated Porism to VI. 20, II. 239 : additions to VI. 33 (about sectors) II. 274-6 : II. 43, 109, 117, 119, 149, 152, 161, 186, 190, 234, 235, 240, 242, 256, 262, 311, 322, 412 Theon of Smyrna : I. 172, 357, 358, 371, 398, II. 111, 119, 279, 280, 281, 284, 285, 286, 288, 289, 290, 291, 292, 293, 294, 425, III. 2, 263, 273 Theorem and problem, distinguished by Speusippus I. 125, Amphinomus I. 125, 128, Menaechmus I. 125, Zenodotus, Posi- <!– Page 546 –>

donius I. 126, Euclid I. 126, Carpus I. 127, 128 : views of Proclus I. 127-8, and of Geminus I. 128 : “general” and “not-general” (or partial) theorems (Proclus) I. 325 Theudius of Magnesia I. 117 Thibaut, B. F. I. 321 Thibaut, C. : On Śulvasūtras I. 360, 363-4 Thompson, Thomas Perronet I. 112 Thrasyllus II. 292 Thucydides I. 333 Thymaridas II. 279, 285 Tibbon, Moses b. I. 76 Timaeus of Plato I. 413-4, II. 97-8, 294-5, 263 Tiraboschi I. 94 n. Tittel, K. I. .38, 39 Todhunter, I. I. 112, 189, 246, 258, 277, 283, 293, 298, 307, II. 3, 7, 22, 49, 51, 52, 67, 73, 90, 99, 172, 195, 202, 204, 208, 259, 271, 272, 300 Tonstall, Cuthbert I. 100 Tore I. 163 Transformation of areas I. 346-7, 410 Trapezium : Euclid’s definition his own I. 189 : further division into trapezia and trapezoids (Posidonius, Heron) I. 189-90 : a theorem on area of parallel-trapezium I. 338-9 : name applied to truncated pyramidal numbers (Theon of Smyrna) II. 291 Treasury of Analysis (ἀναλυόμενος τόπος) I. 8, 10, 11, 138 Trendelenburg, F. A. I. 146 n., 148, 149 Treutlein, P. I. 358-60 Triangle : seven species of, I. 188 : “four-sided” triangle, called also “barb-like” (ἀκιδοειδές) and (by Zenodorus) κοιλογώνιον I. 27, 188 : construction of isosceles and scalene triangles I. 243 : Heron’s proof of expression for area in terms of sides, √s (s - a) (s - b) (s - c), II. 87-8 : right-angled triangle which is half of equilateral triangle used for construction of tetrahedron, octahedron and icosahedron (Timaeus of Plato) II. 98 Triangular numbers II. 289 Trihedral angles : conditions of equality III. 311-2 : symmetrical trihedral angles III. 312 Trimedial (straight line), first and second, corresponding to bimedial III. 257-8 Trinomial (straight line), extension from binomial III. 256 Triplicate, distinct from triple, ratio II. 133 Trisection of an angle I. 265-7 aṭ-Ṭūsī, see Naṣīraddīn

Unger, E. S. I. 108, 169 Unit : definitions of, by Thymaridas, “some Pythagoreans,” Chrysippus, Aristotle and others II. 279 : Euclid’s definition that of the “more recent” writers II. 279 : μονάς connected etymologically by Theon of Smyrna and Nicomachus with μόνος (solitary) or μονή (rest) II. 279

Vacca, Giovanni I. 113 Vachtchenko-Zakhartchenko I. 113 Vailati, G. I. 144 n., 145 n. Valerius Maximus I. 3 Valla, G., De expetendis et fugiendis rebus I. 73, 98 Van Swinden, J. H. I. 169, II. 188 Vatican MS. 190 (P) I. 46, 47 Vaux, Carra de, I. 20 Verona palimpsest I. 91 Veronese, G. I. 157, 168, 175, 180, 193-4, 195, 201, 226-7, 228, 249, 328, II. 30, 126 Vertical (angles) I. 278 Vettius Valens III. 3 Viennese MS. (V) I. 48, 49 Vieta : on angle of contact II. 42 Vinci, Lionardo da I. 365-6 Vitali, G. I. 237 Vitruvius I. 352 : Vitruvius and Heron I. 20, 21 Viviani, Vincenzo I. 107, 401 Vogt, Heinrich, I. 360, 364, 411-4 Vooght, C. J. I. 108

Wachsmuth, C. I. 32 n., 73 Walker, John II. 204, 208, 259 Wallis, John I. 103 : edited Comm. on Ptolemy’s Harmonica I. 17 : attempt to prove Post. 5, I. 210-11 : on angle of contact (“degree of curvature”) II. 42 Weber (H.) and Wellstein (J.) I. 157 Weierstrass II. 124 Weissenborn, H. I. 78 n., 92 n., 94 n., 95, 96, 97 n., 418 Whiston, W. I. 111 Williamson, James I. 111, 293 Witt, H. A. I. 113 Woepcke, F., discovered De divisionibus in Arabic and published translation I. 9 : on Pappus’ commentary on Elements I. 25, 66, 77 : I. 85 n., 86, 87, II. 5, III. 3, 255, 256, 258, 259

Xenocrates I. 268, 413 Ximenes, Leonardo I. 107 Xylander I. 107

Yaḥyā b. Khālid b. Barmak I. 75 Yaḥyā b. Muḥ. b. ‘Abdān b. ‘Abdalwāḥid (Ibn al-Lubūdī) I. 90 Yrinus = Heron I. 22 Yūḥannā b. Yūsuf b. al-Ḥārith b. al-Biṭrīq al-Qass I. 76, 87

Zamberti, Bartolomeo I. 98-100, 101, 104, 106 Zeno the Epicurean I. 34, 196, 197, 199, 242 Zenodorus I. 26, 27, 188, 333, II. 276 Zenodotus I. 126 Zeuthen, H. G. I. 15, 113, 139, 141, 146 n., 151, 355-6, 360, 363, 387, 398, 399, III. 4

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