Heath T.L. (ed.) — The Thirteen Books of Euclid's Elements, Vol. 2 :: Электронная библиотека попечительского совета мехмата МГУ

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

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

Автор: Heath T.L. (ed.)

Аннотация:

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.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1908

Количество страниц: 436

Добавлена в каталог: 02.04.2008

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Предметный указатель

Prime (number), different names for prime, “odd-times odd” (Theon), “linear” (Theon), “rectilinear” (Thymaridas), “euthymetric” (Iamblichus)      285
Prime (number), prime absolutely or in themselves as distinct from prime to one another (Theon)      285
Prime (number), “prime and incomposite ( $\alpha\sigma\upsilon\nu\theta\epsilon\tauo\varsigma$ )”      284
Proclus on absence of formal divisions of proposition in certain cases, e.g. IV. 10      100
Proclus on use of “quindecagon” for astronomy III      4 39 40 193 247 269
Proportion in giving older theory as well Euclid simply followed tradition      113
Proportion, alternatives for v. Def. 5 by a geometer-friend of Saccheri, by Faifofer, Ingrami, Veronese, Enriques and Amaldi      126
Proportion, Aristotle on general proof (new in his time) of theorem (alternando) in proportion      113
Proportion, complete theory applicable to incommensurables as well as commensurables is due to Eudoxus      112
Proportion, De Morgan on extension of meaning of ratio to cover incommensurables      118
Proportion, De Morgan’s defence of V. Def. 5 as necessary and sufficient      122—124
Proportion, definition in V. Def. 5 substituted for that of VII. Def. 20 because latter found inadequate, not vice versa      121
Proportion, extensive use of proportions in Greek geometry      187
Proportion, interpolated definitions of proportion as “sameness” or “similarity of ratios”      119
Proportion, old (Pythagorean) theory practically represented by arithmetical theory of Eucl. VII      113
Proportion, power of expressing incommensurable ratio is power of approximation without limit      119
Proportion, proportion in three terms (Aristotle makes it four) the “least”      131
Proportion, proportionals of VII. Def. 20 (numbers) a particular case of those of V. Def. 5 (Simson’s Props. C, D and notes)      126—129
Proportion, proportions enable any quadratic equation with real roots to be solved      187
Proportion, supposed use of propositions of Book V. in arithmetiPsellus      234
Proportion, three “proportions” 292, but proportion par excellence or primary is continuous or geometric      292—293
Proportion, V. Def. 5 corresponds to Weierstrass’ conception of number in general and to Dedekind’s theory of irrationals      124—126
Proportion, X. 5 as connecting two theories      113
Proportion, “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 Euclid $\epsilon\xi\eta\varsigma \alpha\nu\alpha\lambdao\gammao\nu$ )      131 293
Proportion, “discrete” or “disjoined” ( $\delta\iota\eta\rho\eta\mu\epsilon\nu\eta, \delta\iota\epsilon\zeta\epsilon\upsilon\gamma\mu\epsilon\nu\eta$ )      131 293
Proportion, “ordered” proportion ( $\tau\epsilon\tau\alpha\gamma\mu\epsilon\nu\eta$ ), interpolated definition of      137
Proportion, “perturbed” proportion ( $\tau\epsilon\tau\alpha\rho\alpha\gamma\mu\epsilon\nu\eta$ )      136 176—177
Ptolemy, Claudius      111 117 119
Ptolemy, Claudius, lemma about quadrilateral in circle (Simson’s VI. Prop. D)      225—227
Pyramidal numbers      290
Pyramidal numbers, pyramids truncated, twice-truncated etc.      291
Pythagoras construction of figure equal to one and similar to another rectilineal figure      254
Pythagoras introduced “the most perfect proportion in four terms and specially called “harmonic”” into Greece      112
Pythagoras, reputed discoverer of construction of five regular solids      97
Pythagoreans called 10 “perfect”      294
Pythagoreans distinguished three sorts of means, arithmetic, geometric, harmonic      112
Pythagoreans had theory of proportion applicable to commensurables only      112
Pythagoreans of even and odd      281
Pythagoreans theorem about only three regular polygons filling space round a point      98
Pythagoreans, 7/5 as approximation to $\sqrt{2}$       119
Pythagoreans, construction of dodecahedron in sphere      97
Pythagoreans, construction of isosceles triangle of IV. 10 and of regular pentagon due to      97—98
Pythagoreans, definitions of unit      279
Pythagoreans, possible method of discovery of latter      97—99
Quadratic equations exact correspondence of geometrical to algebraical solution      263—264 266—267
Quadratic equations, $\delta\iotao\rho\iota\sigma\muo\varsigma$ or condition of possibility of solving equation of Eucl. VI. 28      259
Quadratic equations, but method gives both roots if real      258
Quadratic equations, one solution only given, for obvious reasons      260 264 267
Quadratic equations, solution by means of proportions      187 263—265 266—267:
Quadrilateral condition for inscribing circle in      93 95
Quadrilateral in circle, Ptolemy’s lemma on (Simson’s VI. Prop. D)      225—227
Quadrilateral inscribing in circle of quadrilateral equiangular to another      91—92
quadrilateral not a “polygon”      239
Radius, no Greek word for      2
Ratio ex aequali      136
Ratio ex aequali in perturbed proportion      136
Ratio, alternate ratio, alternando      134
Ratio, arguments about greater and less ratios unsafe unless they go back to original definitions (Simson on V. 10)      156—157
Ratio, Barrow’s defence of it      117
Ratio, composition of ratio, componendo, different from compounding ratios      134—135
Ratio, compound ratio      132—133 189—190 234
Ratio, conversion of ratio, convertendo      135
Ratio, def. of greater ratio only one criterion (there are others)      130
Ratio, definition of      116—119 110
Ratio, division of ratios used in Data as general method alternative to compounding      249—250
Ratio, duplicate, triplicate etc. ratio as distinct from double, triple etc      133
Ratio, inverse ratio, inversely      134
Ratio, means of expressing ratio of incommensurables is by approximation to any degree of accuracy      119
Ratio, method of transition from arithmetical to more general sense covering incommensurables      118
Ratio, names for particular arithmetReciprocal or reciprocally related figures: definition spurious      189
Ratio, operation of compounding ratios      234
Ratio, separation of ratio, separando (commonly dividendo)      135
Ratio, sufficient ground for regarding it as spurious      117
Ratio, test for greater ratio easier to apply than that for equal ratio      129—130
Ratio, tests for greater equal and less ratios mutually exclusive      130—131
Ratio, “ratio compounded of their sides” (careless expression)      248
Reductio ad absurdum, the only possible method of proving III. 1      8
Saccheri, Gerolamo      126 130
Saccheri, Gerolamo, proof of existence of fourth proportional by VI. 1, 2, 12      170
Savile, H.      190
Scalene, a class of solid numbers      290
Scholia, IV. No. 2 ascribes Book IV. to Pythagoreans      97

Scholia, V. No. 1 attributes Book V. to Eudoxus      112
Scholiast to Clouds of Aristophanes      99
Sectio canonis of Euclid      295
Sector (of circle), explanation of name, two kinds (1) with vertex at centre, (2) with vertex at circumference      5
Sector-like (figure)      5
Sector-like (figure), bisection of such a figure by straight line      5
Segment of circle, angle of      4
Segment of circle, similar segments      5
Semicircle, angle of      4 39—41 see
Semicircle, angle of, angle in semicircle a right angle, pre-Euclidean proof      63
Separation of ratio, $\delta\iota\alpha\iota\rho\epsilon\sigma\iota\varsigma \lambdao\gammao\upsilon$ , and separando ( $\delta\iota\epsilon\lambdao\nu\tau\iota$ )      135
Separation of ratio, $\delta\iota\alpha\iota\rho\epsilon\sigma\iota\varsigma \lambdao\gammao\upsilon$ , and separando ( $\delta\iota\epsilon\lambdao\nu\tau\iota$ ), separando and componendo used relatively to one another, not to original ratio      168 170
Sides of plane and solid numbers      287—288
Similar plane and solid numbers      293
Similar plane and solid numbers, one mean between two similar plane numbers      371—372
Similar plane and solid numbers, two means between two similar solid numbers      294 373—375
Similar rectilineal figures, def. gives at once too little and too much      188
Similar rectilineal figures, def. of, given in Aristotle      188
Similar rectilineal figures, similar figures on straight lines which are proportional are themselves proportional and conversely (VI. 22), alternatives for proposition      242—247
Similar segments of circles      5
Simon, Max      124 134
SIMPSON, THOMAS      121
Simson, R.      2 3 8 22 23 33 34 37 43 49 53 70 73 79 90 ll7 131 132 140 143—144 145 146 148 154 161 162 163 165 170—172 177 179 180 182 183 184 185 l86 189 I93 I95 209 211 212 230—231 238 252 269 270 272-273
Simson, R., Axioms to Book V.      137
Simson, R., Book VI. Prop. A extending VI. 3 to case where external angle bisected      197
Simson, R., important note showing flaw in V. 10 and giving alternative      156—157
Simson, R., Prop. B (inversion)      144
Simson, R., Prop. E (convertendo)      175
Simson, R., Props. B, C, D      222—227
Simson, R., remarks, on VI. 27—29      258—259
Simson, R., shortens V. 8 by compressing two cases into one      152—153
Simson, R.: Props. C, D (Book V.) connecting proportionals of VII. Def. 20 as particular case with those of V. Def. 5      126—129
Size, proper translation of $\pi\eta\lambda\iota\kappao\tau\eta\varsigma$ in V. Def. 3      116—117 189—190
Smith and Bryant, alternative proofs of V, 16, 17, 18 by means of VI. 1, where magnitudes are straight lines or rectilineal areas      165—166 169 173—174
Solid numbers, three varieties according to relative length of sides      290—291
Spherical number, a particular kind of cube number      291
Square number, product of equal numbers      289 291
Square number, product of equal numbers, one mean between square numbers      294 363—364
Stobaeus      280
Subduplicate of any ratio found by VI. 13      216
Swinden, J.H. van      188
Tacquet, A.      121 258
Tannery, P.      112 113
Tartaglia, Niccolo      2 47
Taylor, H.M.      16 22 29 56 75 102 227 244 247 272
tetrahedron      98
Thales      111 280
Theodosius      37
Theon of Alexandria      43 109 117 119 149 152 161 186 190 234 235 240 242 250 262 311 322 412
Theon of Alexandria interpolation in V. 13 and Porism      144
Theon of Alexandria, additions to VI. 33 (about sectors)      274—276
Theon of Alexandria, interpolated Porism to VI. 20      239
Theon of Smyrna, in      119 279 280 281 284 285 286 288 289 290 291 292 293 294
Thrasyllus      292
Thymaridas      279 285
Timaeus of Plato      97—98 294—295 363
Todhunter, I.      3 7 22 49 51 52 67 73 90 99 172 195 202 204 208 259 271 272 300
Trapezium, name applied to truncated pyramidal numbers (Theon of Smyrna)      291
Triangle, Heron’s proof of expression for area in terms of sides, $\sqrt{s(s-a)(s-b)(s-c)}$       87—88
Triangle, right-angled triangle which is half of equilateral triangle used for construction of tetrahedron, octahedron and icosahedron (Timaeus of Plato)      98
Triangular numbers      289
Triplicate, distinct from triple, ratio      133
Unit, $\muo\nu\alpha\varsigma$ connected etymologically by Theon of Smyrna and Nicomachus with $\muo\nuo\varsigma$ (solitary) or $\muo\nu\eta$ (rest)      279
Unit, definitions of, by Thymaridas, “some Pythagoreans,” Chrysippus, Aristotle and others      279
Unit, Euclid’s definition was that of the “more recent” writers      279
Veronese, G.      30 126
Vieta, on angle of contact      42
Walker      204 208 259
Wallis, John, on angle of contact (“degree of curvature”)      42
Weierstrass      124
Woepcke      5
Zenodorus      276
“Chance equimultiples” in phrase “other, chance, equimultiples”      143—144
“Composite to one another” (of numbers)      286—287
“Dissimilarly ordered” proportion ( $\alpha\nuo\muo\iota\omega\varsigma \tau\epsilon\tau\alpha\gamma\mu\epsilon\nu\omega\nu \tau\omega\nu \lambdao\gamma\omega\nu$ ) in Archimedes = “perturbed proportion”      136
“Division (of ratios)”      see “Separation”
“Ordered” proportion ( $\tau\epsilon\tau\alpha\gamma\mu\epsilon\nu\eta \alpha\nu\alpha\lambdao\gamma\iota\alpha$ ), interpolated definition of      137
“Parallelepipedal” (solid) numbers, two of the three factors differ by unity (Nicomachus)      290
“Perfect” (of a class of numbers)      293—294 421—425
“Perfect” (of a class of numbers), 3 also called “perfect”      294
“Perfect” (of a class of numbers), Pythagoreans applied term to      10 294:
“Quindecagon” (fifteen-angled figure), useful for astronomy      111
“Rule of three”, VI. 12 equivalent to      215

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