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Heath T.L. (ed.) The Thirteen Books of Euclid's Elements, Vol. 2
Heath T.L. (ed.)  The Thirteen Books of Euclid's Elements, Vol. 2









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: The Thirteen Books of Euclid's Elements, Vol. 2

: Heath T.L. (ed.)

:

Volume 2 of three-volume set containing complete English text of all 13 books of the Elements plus critical apparatus analyzing each definition, postulate and proposition in great detail. Covers textual and linguistic matters; mathematical analyses of Euclid's ideas; classical, medieval, Renaissance and modern commentators; refutations, supports, extrapolations, reinterpretations and historical notes. Total in set: 995 figures.


: en

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ed2k: ed2k stats

: 1908

: 436

: 02.04.2008

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Adrastus      292
Alcinous      98
Alternate and alternately (of ratios)      134
Alternative proofs that in III. 10 claimed by Heron      2324
Alternative proofs, interpolated (cf. 111. 9 and following)      22
Amaldi, Ugo      30 126
Ambiguous case of VI. 7      208209
an-Nairizi      5 16 28 34 36 44 47 302 320
Anaximander      111
Anaximenes      111
Angle, angles not less than two right angles not recognised as angles (cf. Heron, Proclus, Zenodorus)      4749
Angle, controversies about angle of semicircle and hornlike angle      3942 (see also Hornlike)
Angle, did Euclid extend angle to angles greater than two right angles in VI. 33?      275276
Angle, hollow-angled figure (the re-entrant angle was exterior)      48
Angle, hornlike angle      4 39 40
Angle, angle of semicircle and of segment      4
Antecedents (leading terms in proportion)      134
Antiparallels, may be used for construction of VI. 12      215
Apollonius      75 190 258
Apollonius lemma by Pappus on      6465
Apollonius Plane $\nu\epsilon\upsilon\sigma\epsilon\iota\varsigma$, problem from      81
Apollonius Plane Loci, theorem from (arising out of Eucl. VI. 3), also found in Aristotle      198200
Application of areas (including exceeding and falling short) corresponding to solution of quadratic equations      187 258260 263265 266267
Approximations to $\pi$ (Archimedes)      119
Approximations to \sqrt{4500} (Theon of Alexandria)      119
Approximations, 7/5 as approximation to $\sqrt{2}$ (Pythagoreans and Plato)      119
Approximations, approximations to $\sqrt{3}$ in Archimedes and (in sexagesimal fractions) in Ptolemy      119
Archimedes      136 190
Archimedes approximations to $\sqrt{3}$, square roots of large numbers and to $\pi$      119
Archimedes extension of a proportion in commensurables to cover incommensurables      193
Archimedes, Liber assumptorum, proposition from      65
Archimedes, new fragment of      40
Archytas proof that there is no numerical geometric mean between n and n+1      295
Aristotle has locus-theorem (arising out of Eucl. VI. 3) also given in Apollonius Plane Loci      198200
Aristotle indicates proof (pre-Euclidean) that angle in semicircle is right      63
Aristotle on alternate ratios      134
Aristotle on composite numbers as plane and solid      286 288 290
Aristotle on def. of same ratio (=same $\alpha\nu\tau\alpha\nu\alpha\iota\rho\epsilon\sigma\iota\varsigma$)      120121
Aristotle on definions of odd and even by one another      281
Aristotle on inverse ratio      134 149n
Aristotle on non-applicability of arithmetical proofs to magnitudes if these are not numbers      113
Aristotle on number      280;
Aristotle on prime numbers      284285
Aristotle on proportion as equality of ratios      119
Aristotle on proportion in three terms ($\sigma\iota\nu\epsilon\chi\eta\varsigma$, continuous) and in four terms ($\delta\iota\eta\mu\epsilon\nu\eta$, discrete)      131 293
Aristotle on representation of numbers by pebbles forming figures      288
Aristotle on theorem in proportionnot proved generally till his time      113
Aristotle on unit      279
Aristotle, similar rectilineal figures      188
Arithmetic, Elements of, anterior to Euclid      295
at-Tusi      see Nasiraddln
August, E. F.      23 25 149 238 256 412
Austin, W.      172 188 211 259
Axioms tacitly assumed in Book V.      137
Axioms tacitly assumed in Book VII.      294
Babylonians      112
Baermann, G.F.      213
Baltzer, R.      30
Barrow      56 186 238
Barrow on Eucl. V. Def. 3      117
Barrow on V. Def. 5      121
Billingsley, H.      56 238
Boethius      295
Borelli, G.A.      2 84
Breadth (of numbers) = second dimension or factor      288
Briggs, H.      143
Camerer, J.G.      22 25 28 33 34 40 67 121 131 189 213 244
Campanus      28 41 56 90 116 119 121 146 189 211 234 235 253 275 320 322 328
Candalla      189
Cantor, Moritz      5 40 97
Cardano, Hieronimo      41
Case, Greeks did not infer limiting cases, but proved them separately      75
Casey, J.      227
Circle, circles intersecting and touching, difficulties in Euclids treatment of      2527 2829
Circle, circles touching,meaning of definition      3
Circle, definition of equal circles      2
Circle, modern treatment of      3032
Circle, circle in sense of circumference      23
Clavius      2 41 42 47 49 53 56 67 70 73 130 170 190 231 238 244 271
Commandinus      47 130 190
Componendo ($\sigma\upsilon\nu\theta\epsilon\nu\tau\iota$), componendo and separando used relatively to each other      168 170
Componendo ($\sigma\upsilon\nu\theta\epsilon\nu\tau\iota$), denoting composition of ratios      q.v.
Composite numbers in Euclid      286
Composite numbers in Euclid, plane and solid numbers species of      286
Composite numbers in Euclid, with Eucl. and Theon of Smyrna may be even, but with Nicom. and Iambl, are a subdivision of odd      286
Composition of ratios ($\sigma\upsilon\nu\theta\epsilon\sigma\iota\varsigma \lambdao\gammao\upsilon$), denoted by componendo ($\sigma\upsilon\nu\theta\epsilon\nu\tau\iota$), distinct from compounding ratios      134135
Compound ratio, compounded ratios in V. 2023      176178
Compound ratio, explanation of      132133
Compound ratio, interpolated definition of      189190
Consequents (following terms in proportion)      134 238
Continuous proportion ($\sigma\upsilon\nu\epsilon\chi\eta\varsigma$ or $\sigma\upsilon\nu\eta\mu\mu\epsilon\nu\eta \alpha\nu\alpha\lambdao\gamma\iota\alpha$) in three terms      131
Conversion of ratio ($\alpha\nu\alpha\sigma\tau\rho\phi\eta \lambdao\gammao\upsilon$), denoted by convertendo ($\alpha\nu\alpha\sigma\tau\rho\epsilon\psi\alpha\nu\tau\iota$)      135
Conversion of ratio ($\alpha\nu\alpha\sigma\tau\rho\phi\eta \lambdao\gammao\upsilon$), denoted by convertendo ($\alpha\nu\alpha\sigma\tau\rho\epsilon\psi\alpha\nu\tau\iota$), convertendo theorem not established by Por.      174175
Conversion of ratio ($\alpha\nu\alpha\sigma\tau\rho\phi\eta \lambdao\gammao\upsilon$), denoted by convertendo ($\alpha\nu\alpha\sigma\tau\rho\epsilon\psi\alpha\nu\tau\iota$), convertendo theorem not established by V.      19
Conversion of ratio ($\alpha\nu\alpha\sigma\tau\rho\phi\eta \lambdao\gammao\upsilon$), denoted by convertendo ($\alpha\nu\alpha\sigma\tau\rho\epsilon\psi\alpha\nu\tau\iota$), convertendo theorem proved by Simsons Prop. E      175
Convertendo denoting conversion of ratios      q.v.
Corresponding magnitudes      134
Cube, cube number, def. of      291
Cube, duplication of, reduced by Hippocrates to problem of two mean proportionals      133
Cube, two mean proportionals between two cube numbers      294 364365
Cyclic of a particular kind of square number      291
Cyclomathia of Leotaud      42
Data of Euclid, Def. 2      248
Data of Euclid, Prop. 24      246247
Data of Euclid, Prop. 55      254
Data of Euclid, Prop. 58      263 265
Data of Euclid, Prop. 67 assumes part of converse of Simsons Prop. (Book VI.)      224
Data of Euclid, Prop. 70      250
Data of Euclid, Prop. 8      249250
Data of Euclid, Prop. 85      264
Data of Euclid, Prop. 87      228
Data of Euclid, Prop. 93      227
Data of Euclid, Props. 56 and 68      249
Data of Euclid, Props. 59 and 84      266267
De Morgan, A.      5 7 910 11 15 20 22 29 56 7677 83 101 104 116119 120 130 139 145 197 202 217218 232 233 234 272 275
De Morgan, A. means of expressing ratios between incommensurables by approximation to any extent      118119
De Morgan, A. on compound ratio      132133 234
De Morgan, A. on definition of ratio      116117
De Morgan, A. on extension of meaning of ratio to cover incommensurables      118
De Morgan, A. on nessity of proof that tests for greater and less, or greater and equal, ratios cannot coexist      130131 157
De Morgan, A., defence and explanation of V. Def. 5      122124
De Morgan, A., proposed lemma about duplicate ratios asalternative means of proving VI. 22      246247
De Morgan, A., sketch of proof of existence of fourth proportional (assumed in V. 18)      171
Dechales      259
Dedekinds theory of irrational numbers corresponds exactly to Eucl. V. Def. 5      124126
Democritus, On difference of gnomon etc. (? on angle of contact)      40
Democritus, On difference of gnomon etc., on parallel and infinitely near sections of cone      40
Democritus, On difference of gnomon etc., stated, without proving, propositions about volumes of cone and pyramid      40
Dercyllides      111
Diorismns for solution of a quadratic      259
Discrete proportion, $\delta\iota\eta\rho\eta\mu\epsilon\nu\eta$ or $\delta\iota\epsilon\zeta\epsilon\upsilon\gamma\mu\epsilon\nu\eta \alpha\nu\alpha\lambdao\gamma\iota\alpha$, in four terms      131 293
Dividendo (of ratios)      see Separation Separando
Divisions (of figures), On, treatise by Euclid, proposition from      5
Dodecahedron, decomposition of faces into elementary triangles      98
Dodgson, C.L.      48 275
Duplicate ratio      133
Duplicate ratio, $\delta\iota\pi\lambda\alpha\sigma\iota\omega\nu$, duplicate, distinct from $\delta\iota\pi\lambda\alpha\sigma\iotao\varsigma$, double (=ratio 2:1), though use of terms not uniform      133
Duplicate ratio, lemma on duplicate ratio as alternative to method of VI. 22 (De Morgan and others)      242247
Duplicate ratio, duplicate of given ratio found by VI. 11      214
Duplication of cube, reduction of, by Hippocrates, to problem of finding two mean proportionals      133
Duplication of cube, wrongly supposed to be alluded to in Timaeus 32 A, B      294295n.
Egyptians      112
Egyptians, Egyptian view of number      280
Enriques (F.) and Amaldi (U.)      30 126
Equimultiples should include once each magnitude      145
Equimultiples, stereotyped phrase other, chance, equimultiples      143144
Equimultiples, any equimultiples whatever, $\iota\sigma\alpha\kappa\iota\varsigma \pio\lambda\lambda\alpha\pi\lambda\alpha\sigma\iota\alpha \kappa\alpha\theta o\pio\iotao\nuo\upsilon\nu \pio\lambda\lambda\alpha\pi\lambda\alpha\sigma\iota\alpha\sigma\muo\nu$      120
Eratosthenes measurement of obliquity of ecliptic ($23^{\circ} 51' 20''$)      111
Escribed circles of triangle      85 8687
Eudemus      99 111
Eudoxus      99 280 295
Eudoxus, discovered general theory of proportionals covering incommensurables      112113
Eudoxus, was first to prove scientifically the propositions about volumes of cone and pyramid      40
Eutocius gives locustheorem from Apollonius Plane Loci      198200
Eutocius on VI. Def. 5 and meaning of $\pi\eta\lambda\iota\kappao\tau\eta\varsigma$      116 132 189190
Even (number), definitions by Pythagoreans and in Xicomachus      251
Even (number), definitions of odd and even by one another unscientific (Aristotle)      281
Even (number), Nicom. divides even into three classes (1) even-times even and (2) eventimes odd as extremes, and (3) odd-times even as intermediate      282283
Even-times even, Euclids use differs from use by Nicomachus, Theon of Smyrna and Iamblichus      281282
Even-times odd in Euclid different from even-odd of Nicomachus and the rest      282284
Ex aequali of ratios      136
Ex aequali of ratios, ex aequoii propositions (V. 20, 22), and ex aequali in perturbed proportion (V. 21, 23)      176178
Faifofer      126
Fourth proportional, assumption of existence of, in V. 18, and alternative methods for avoiding (Saccheri, De Morgan, Simson, Smith and Bryant)      170174
Fourth proportional, Clavius made the assumption an axiom      170
Fourth proportional, condition forexistence of number which is fourth proportional to thiee numbers      409411
Fourth proportional, sketch of proof of assumption by De Morgan      171
Galileo Galilei, on angle of contact      42
Geometric means      357sqq.
Geometric means, one mean between square numbers      294 363
Geometric means, one mean between square numbers or similar plane numbers      371372
Geometric means, two means between cube numbers      294 364365
Geometric means, two means between cube numbers or between similar solid numbers      373375
Geometrical progression      346sqq.
Geometrical progression, summation of n terms of (IX. 35)      420421
Gherard of Cremona      47
Gnomon (of numbers)      289
Golden section (section in extreme and mean ratio), discovered by Pythagoreans      99
Golden section (section in extreme and mean ratio), discovered by Pythagoreans, theory carried further by Plato and Eudoxus      99
Greater ratio, arguments from greater to less ratios etc. unsafe unless they go back to original definitions (Simson on V. 10)      156157
Greater ratio, Euclids criterion not the only one      130
Greater ratio, test for, cannot coexist with test for equal or less ratio      130131
Greatest common measure, Euclids method of finding corresponds exactly to ours      118 299
Greatest common measure, Nicomachus gives the same method      300
Gregory, D.      116 143
Habler, Th.      294n.
Hankel, H.      116 117
Hauber, C.F.      244
Heiberg, J.L.      passim
Henrici and Treutlein      30
Heron of Alexandria      5 1617 24 28 34 36 44 116 189 302 320 383 395
Heron of Alexandria does not recognise angles equal to or greater than two right angles      4748
Heron of Alexandria proof of formula for area of triangle, $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$      8788
Heron of Alexandria, Eucl. III. 12, interpolated from      28
Heron of Alexandria, extends III. 20, 21, to angles in segments less than semicircles      4748
Hippasus      97
Hippocrates of Chios      133
Hornlike angle ($\kappa\epsilon\rho\alpha\tauo\epsilon\iota\delta\eta\varsigma \gamma\omega\nu\iota\alpha$)      4 39 40
Hornlike angle ($\kappa\epsilon\rho\alpha\tauo\epsilon\iota\delta\eta\varsigma \gamma\omega\nu\iota\alpha$), Campanus (not angles in same sense)      41
Hornlike angle ($\kappa\epsilon\rho\alpha\tauo\epsilon\iota\delta\eta\varsigma \gamma\omega\nu\iota\alpha$), Cardano (quantities of different orders or kinds), Peletier (hornlike angle no angle, no quantity, nothing; angles of all semicircles right angles and equal)      41
Hornlike angle ($\kappa\epsilon\rho\alpha\tauo\epsilon\iota\delta\eta\varsigma \gamma\omega\nu\iota\alpha$), Clavius      42
Hornlike angle ($\kappa\epsilon\rho\alpha\tauo\epsilon\iota\delta\eta\varsigma \gamma\omega\nu\iota\alpha$), Democritus may have written on hornlike angle      40
Hornlike angle ($\kappa\epsilon\rho\alpha\tauo\epsilon\iota\delta\eta\varsigma \gamma\omega\nu\iota\alpha$), hornlike angle and angle of semicircle, controversies on      3942
Hornlike angle ($\kappa\epsilon\rho\alpha\tauo\epsilon\iota\delta\eta\varsigma \gamma\omega\nu\iota\alpha$), Proclus on      3940
Hornlike angle ($\kappa\epsilon\rho\alpha\tauo\epsilon\iota\delta\eta\varsigma \gamma\omega\nu\iota\alpha$), Vieta and Galileo (angle of contact no angle)      42
Hornlike angle ($\kappa\epsilon\rho\alpha\tauo\epsilon\iota\delta\eta\varsigma \gamma\omega\nu\iota\alpha$), Waliis (angle of contact not inclination at all but degree if curvature)      42
Hultsch, F.      133
Iamblichus      97 116 279 280 281 283 284 285 286 287 288 289 290 291 292 293 419
icosahedron      98
Incommensurables approximations to $\sqrt{2}$ (by means of side- and diagonal-numbers      119
Incommensurables approximations to $\sqrt{3}$ and to $\pi$, tig; to \sqrt{4500} by means of sexagesimal fractions      119
Incommensurables means of expression consist in power of approximation without limit (De Morgan)      119
Incommensurables method of testing incommensurability (process of finding G.C.M.)      118
Incomposite (of number) = prime      284
Ingrami, G.      30 126
Inverse (ratio), inversely ($\alpha\nu\alpha\pi\alpha\lambda\iota\nu$)      134
Inverse (ratio), inversely ($\alpha\nu\alpha\pi\alpha\lambda\iota\nu$), inversion is subject of V. 4, Por. (Theuii)      144
Inverse (ratio), inversely ($\alpha\nu\alpha\pi\alpha\lambda\iota\nu$), inversion is subject of V. 7, Por      149
Inverse (ratio), inversely ($\alpha\nu\alpha\pi\alpha\lambda\iota\nu$), is not properly put in either place      149
Inverse (ratio), inversely ($\alpha\nu\alpha\pi\alpha\lambda\iota\nu$), Simsons Prop. B on directly deducibie from V. Def. 5      144
Isosceles triangle of IV. 10, construction of, by Pythagoreans      9799
Jacobi, C.F.A.      1S8
Lachlan, R.      226 227 245246 247 256 272
Lardner, D.      58 259 271
Least common multiple      336341
legendre      30
Legendre proves VI. 1 and similar propositions in two parts: (1) for commensurables, (2) for incommensurables      193194
Lemma assumed in VI. 22      242243
Lemma assumed in VI., alternative propositions on duplicate ratios and ratios of which they are duplicate (De Morgan and others)      242247
Length, $\mu\eta\kappao\varsigma$ (of numbers in one dimension)      287
Length, $\mu\eta\kappao\varsigma$ (of numbers in one dimension), Plato restricts term to side of complete square      287
Leotaud, Vincent      42
Linear (of numbers) = (1) in one dimension      287
Linear (of numbers) = (2) prime      285
Logical inferences, not made by Euclid      22 29
Lucian      99
Means, geometric mean is proportion par excellence ($\kappa\upsilon\rho\iota\omega\varsigma$)      292293
Means, no numerical geometric mean between n and n+1 (Archytas and Euclid)      295
Means, one geometric mean between similar plane numbers, two between similar solid numbers      371375
Means, one geometric mean between two square numbers, two between two cube numbers (Plato)      294 363365
Means, three kinds, arithmetic, geometric and harmonic      292293
Moderatus, a Pythagorean      280
Multiplication, definition of      287
Musici Scriptores Graeci      295
Nasiraddin at-Tusi      28
Nesseimann, G.H.F.      287 293
Nicomachus      116 119 131 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 300 363
Nixon, R.C.J.      16
Number, defined by Nicomachus and Iamblichus on      280
Number, defined by Thales, Eudoxus, Moderatus, Aristotle, Euclid      280
Number, represented by lines      288
Number, represented by lines and by points or dots      288289
Oblong (of number) in Plato either $\pi\rhoo\mu\eta\kappa\eta\varsigma$ or $\epsilon\tau\epsilon\rhoo\mu\eta\kappa\eta\varsigma$      288
Oblong (of number), but these terms denote two distinct divisions of plane numbers in Nicomachus, Theon of Smyrna and Iamblichus      289290
octahedron      98
Odd (number), def. of odd and even by one another unscientific (Aristotle)      281
Odd (number), defs. of in Nicomachus      281
Odd (number), Nicom. and Iambl, distinguish three classes of odd numbers (1) prime and incomposite, (2) secondary and composite, as etremes, (3) secondary and composite in itself but prime and incomposite to one another, which is intermediate      257
Odd (number), Pythagorean definition      281
Odd-times even (number), definition in Eucl. spurious      283284
Odd-times even (number), definition in Eucl. spurious and differs from definitions by Nicomachus etc.      ibid.
Odd-times odd (number), defined in Eucl. but not in Nicom. and Iambi      284
Odd-times odd (number), Theon of Smyrna applies term to prime numbers      284
Oenopides of Chios      111
Pappus      4 27 29 67 79 81 113 133 211 250 251 292
Pappus assumes case of VI. 3 where external angle bisected (Simsons VI. Prop. A)      197
Pappus lemma on Apollonius Plane $\nu\epsilon\upsilon\sigma\epsilon\iota\varsigma$      64-65
Pappus problem from same work      81
Pappus theorem from Apoilonius Plane Loci      198
Pappus theorem that ratio compounded of ratios of sides is equal to ratio of rectangles contained by sides      250
Peletarius      47 56 84 146 190
Peletarius (Peletier) on angle of contact and angle of semicircle      41
Pentagon decomposition of regular pentagon into 30 elementary triangles      98
Pentagon relation to pentagram      99
Pentagonal numbers      289
Perturbed proportion ($\tau\epsilon\tau\alpha\rho\alpha\gamma\mu\epsilon\nu\eta \alpha\nu\alpha\lambdao\gamma\iota\alpha$)      136 176177
Pfleiderer, C.F.      2
Philolaus      97
Philoponus      234 282
Plane numbers, product of two factors (sides or length and breadth)      287-288
Plane numbers, product of two factors (sides or length and breadth), in Plato either square or oblong      257-258
Plane numbers, product of two factors (sides or length and breadth), one mean proportional between similar plane numbers      371372
Plane numbers, product of two factors (sides or length and breadth), similar plane numbers      293
Plato 7/5 as approximation to \sqrt{2}      119
Plato construction of regular solids from triangles      9798
Plato on $\delta\upsilon\nu\alpha\mu\epsilon\iota\varsigma$ (square roots or surds)      288 290
Plato on golden section      99
Plato on square and oblong numbers      288 293
Plato theorem that between square numbers one mean suffices, between cube numbers two means necessary      294 364
Playfair, John      2
Plutarch      98 254
Polygonal numbers      289
Porism (corollary) to proposition precedes Q.E.D. or Q.E.F.      8 64
Porism (corollary) to proposition precedes Q.E.D. or Q.E.F., Porism to IV. 15 mentioned by Proclus      109
Porism (corollary) to proposition precedes Q.E.D. or Q.E.F., Porism to VI. 19      234
Prime (number)      4 39 40 193 247 269
Prime (number), 2 admitted as prime by Eucl. and Aristotle, but excluded by Nicomachus, Theon of Smyrna and Iamblichus, who make prime a subdivision of odd      284285
Prime (number), Aristotle on two senses of prime      285:
Prime (number), definitions of      284285
Prime (number), definitions of prime to one another      285282
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