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| Heath T.L. (ed.) — Thirteen Books of Euclid's Elements, Vol. 3 |
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| Ïðåäìåòíûé óêàçàòåëü |
Proportion, proportion in three terms (Aristotle makes it four) the "least" II. 131
Proportion, proportionals of vII. Def. 20 (numbers) a particular case of those of V. Def. 5 (Simson's Props. C, D and notes) II. 126—129 III.
Proportion, proportions enable any quadratic equation with real roots to be solved II. 187
Proportion, supposed use of propositions of Book V. in arithmetical Books II. 314 320
Proportion, three "proportions" II. 292
Proportion, three "proportions", but proportion par excellence or primary is continuous or geometric II. 292—293
Proportion, V. Def. 5 corresponds to Weierstrass' conception of number in general and to Dedekind's theory of irrationals II. 124—126
Proportion, X. 5 as connecting two theories II. 113
Proposition, formal divisions of I. 129—131
Protarchus I. 5 III.
Psellus, Michael, scholia by I. 70 71 II.
Pseudana of Euclid I. 7
Pseudana of Euclid, Pseudographemata I. 7n.
Pseudoboethius I. 92
Ptolemy I. I. 1 2
Ptolemy I., story of Euclid and Ptolemy I. 1
Ptolemy, Claudius I. 30n. II. 117 119 III.
Ptolemy, Claudius on Parallel-Postulate I. 28n. 34 43 45
Ptolemy, Claudius, attempt to prove it I. 204—206
Ptolemy, Claudius, Harmonica of, and commentary on I. 17
Ptolemy, Claudius, lemma about quadrilateral in circle (Simson's VI. Prop. D) II. 225—227
Pyramid, definitions of, by Euclid III. 261
Pyramid, definitions of, by others III. 268
Pyramidal numbers II. 290
Pyramidal numbers, pyramids truncated, twice-truncated etc. II. 291
Pythagoras I. 4n. 36
Pythagoras, construction of figure equal to one and similar to another rectilineal figure II. 254
Pythagoras, introduced "the most perfect proportion in four terms and specially called 'harmonic'" into Greece II. 112
Pythagoras, probable method of discovery of I. 47 and proof of I. 352—355
Pythagoras, rule for forming right-angled triangles in rational numbers I. 351 356—359 385
Pythagoras, story of sacrifice I. 37 343 350
Pythagoras, suggestions by Bretschneider and Hankel I. 354
Pythagoras, suggestions by Zeuthen I. 355—356
Pythagoras, supposed discoverer of application of areas I. 343—344 III.
Pythagoras, supposed discoverer of construction of five regular solids II. 97 III.
Pythagoras, supposed discoverer of the irrational I. 351 III. 524—525
Pythagoras, supposed discoverer of theorem of I. 47 I. 343—344 350—354 III.
Pythagoreans I. 19 36 155 188 279
Pythagoreans, "rational" and "irrational diameter of 5" I. 399—400 III.
Pythagoreans, 7/5 as approximation to  II. 119
Pythagoreans, and of other regular solids III. 438 525
Pythagoreans, angles of triangle equal to two right angles, theorem and proof I. 317—320
Pythagoreans, approximation to by "side-" and "diagonal-" numbers I. 398—400 III. 20
Pythagoreans, called 10 "perfect" II. 294
Pythagoreans, construction of dodecahedron in sphere II. 97
Pythagoreans, construction of isosceles triangle of Eucl. IV. 10, and of regular pentagon II. 97—98 III.
Pythagoreans, definitions of even and odd II. 281
Pythagoreans, definitions of unit II. 279
Pythagoreans, distinguished three sorts of means, arithmetic, geometric and harmonic II. 112
Pythagoreans, gnomon Pythagorean I. 351
Pythagoreans, had theory of proportion applicable to commensurables only II. 112
Pythagoreans, method of application of areas (including exceeding and falling-short) I. 343 384 403 II. 258—260 263—265 266—267
Pythagoreans, possible method of discovery of latter II. 97—99
Pythagoreans, proof of incommensurability of III. 2
Pythagoreans, story of Pythagorean who, having divulged the irrational, perished by shipwreck III. 1
Pythagoreans, term for surface I. 169
Pythagoreans, three polygons which in contact fill space round point I. 318 II.
Q.E.D. (or F.) I. 57
Qadizade ar-Rumi I. 5n. 90
Quadratic equations, but method gives both roots if real II. 258
Quadratic equations, condition of possibility of solving equation of Eucl. VI. 28 II. 259
Quadratic equations, exact correspondence of geometrical to algebraical solution II. 263—264 266—267
Quadratic equations, geometrical solution of particular quadratics I. 383—385 386—388
Quadratic equations, indication that Greeks solved them numerically III. 43—44
Quadratic equations, one solution only given, for obvious reasons II. 260 264 267
Quadratic equations, solution assumed by Hippocrates I. 386—387
Quadratic equations, solution of general quadratic by means of proportions II. 187 263—265 266—267
Quadratrix I. 265—266 330
Quadrature definitions of I. 149
Quadrilateral, condition for inscribing circle in II. 93 95
Quadrilateral, inscribing in circle of quadrilateral equiangular to another II. 91—92
Quadrilateral, quadrilateral in circle, Ptolemy's lemma (Simson's VI. Prop. D) II. 225—227
Quadrilateral, quadrilateral not a "polygon" II. 239
Quadrilateral, varieties of I. 188—190
Quadrinomial (straight line), compound irrational (extension from binomial) III. 256
Quintilian I. 333
Qusta b. Luqa al-Ba'labakki, translator of "Books XIV, XV" I. 76 87 88
Radius, no Greek word for I. 199 II.
Ramus, Petrus (Pierre de la Ramee) I. 104
Ratdolt, Erhard I. 78 97
Ratio, "ratio compounded of their sides" (careless expression) II. 248
Ratio, alternate ratio, alternando II. 134
Ratio, arguments about greater and less ratios unsafe unless they go back to original definitions (Simson on V. 10) II. 156—157
Ratio, composition of ratio, componendo, different from compounding ratios II. 134—135
Ratio, compound ratio II. 132—133 189—190 234
Ratio, conversion of ratio, convertendo II. 135
Ratio, def. of greater ratio only one criterion (there are others) II. 130
Ratio, definition of II. 116—119
Ratio, definition of, Barrow's defence of it II. 117
Ratio, definition of, no sufficient ground for regarding it as spurious II. 117
Ratio, division of ratios used in Data as general method alternative to compounding II. 249—250
Ratio, duplicate, triplicate etc. ratio as distinct from double, triple etc. II. 133
Ratio, ex aequali in perturbed proportion II. 136
Ratio, inverse ratio, inversely II. 134
Ratio, means of expressing ratio of incommensurables is by approximation to any degree of accuracy II. 119
Ratio, method of transition from arithmetical to more general sense covering incommensurables II. 118
Ratio, names for particular arithmetical ratios II. 292
Ratio, operation of compounding ratios II. 234
Ratio, ratio ex aequali II. 136
Ratio, separation of ratio, separando (commonly dividendo) II. 135
Ratio, test for greater ratio easier to apply than that for equal ratio II. 129—130
Ratio, tests for greater, equal and less ratios mutually exclusive II. 130—131
Rational, "rational diameter of 5" I. 399—400
Rational, (of ratios) I. 137
Rational, any straight line may lie taken as rational and the irrational is irrational in relation thereto III. 10
Rational, rational right-angled triangles see "Right-angled triangles"
Rational, 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  or  III. 243—252
Rauchfuss see "Dasypodius"
Rausenberger, O. I. 157 175 313 III. 309
Reciprocal or reciprocally-related figures, definition spurious II. 189
Rectangle = rectangular parallelogram I. 370
Rectangle, "rectangle contained by" I. 370
Rectilineal angle, "rectilineal segment" I. 196
Rectilineal angle, definitions classified I. 179—181
Rectilineal angle, rectilineal figure I. 187
Reductio ad absurdum I. 134
Reductio ad absurdum, a variety of Analysis I. 140
Reductio ad absurdum, by exhaustion I. 285 293
Reductio ad absurdum, described by Aristotle and Proclus I. 136
Reductio ad absurdum, nominal avoidance of I. 369
Reductio ad absurdum, synonyms for, in Aristotle I. 136
Reductio ad absurdum, the only possible method of proving Eucl. III. 1 II. 8
Reduction, technical term, explained by Aristotle and Proclus I. 135
Reduction, technical term, explained by Aristotle and Proclus, first "reduction" of a difficult construction due to Hippocrates I. 135 II.
Regiomontanus (Johannes Mueller of Koenigsberg) I. 93 96 100
Reyher, Samuel I. 107
Rhaeticus I. 10 III.
Rhomboid I. 189
Rhombus, meaning and derivation I. 189
Riccardi, P. I. 96 112 202
Riemann, B. I. 219 273 274 280
Right angle, construction when drawn at extremity of second line (Heron) I. 270
Right angle, definition I. 181
Right angle, drawing straight line at right angles to another, Apollonius' construction for I. 270
Right-angled triangles, rational, connexion of rules with Eucl. II. 4, 8 I. 360
Right-angled triangles, rational, discovery of rules by means of gnomons I. 358—360
Right-angled triangles, rational, rational right-angled triangles in Apastamba I. 361 363
Right-angled triangles, rational, rule for finding, by Euclid 63—64
Right-angled triangles, rational, rule for finding, by Plato I. 356 357 359 360 385
Right-angled triangles, rational, rule for finding, by Pythagoras I. 356—359
Roeth I. 357—358
Rouche and de Comberousse I. 313
Rudd, Capt. Thos. I. 110
Ruellius, Joan. (Jean Ruel) I. 100
Russell, Bertrand I. 227 249
Sa'id b. Mas'ud b. al-Qass I. 90
Saccheri, Gerolamo I. 106 144—145 167—168 185—186 194 197—198 200—201 II. 130
Saccheri, Gerolamo, proof of existence of fourth proportional by Eucl. VI. 1, 2 and 12 II. 170
Sathapatha-Brahmana I. 362
| Savile, Henry I. 105 166 245 250 262 II.
Scalene I. 187—198
Scalene of cone (Apollonius) I. 188
Scalene of numbers (=odd) I. 188
Scalene, a class of solid numbers II. 290
Schessler, Chr. I. 107
Scheubel, Joan. I. 101 107
Schiaparelli, G.V. I. 163
Schluessel, Christoph see "Clavius"
Schmidt, Max C.P. I. 304 319
Schmidt, W., editor of Heron, on Heron's date I. 20—21
Scholia to Elements and MSS. of I. 64—74
Scholia to Elements and MSS. of, "Schol. Vat." partly derived from Pappus' commentary I. 66
Scholia to Elements and MSS. of, classes of, "Schol. Vat." I. 65—69
Scholia to Elements and MSS. of, classes of, "Schol. Vind." I. 69—70
Scholia to Elements and MSS. of, classes of, miscellaneous I. 71—74
Scholia to Elements and MSS. of, evidence in, as to text I. 64—65 66—67
Scholia to Elements and MSS. of, historical information in I. 64
Scholia to Elements and MSS. of, many from Geminus solely III. 522
Scholia to Elements and MSS. of, many scholia partly extracted from Proclus on Bk. I. I. 60 69 72
Scholia to Elements and MSS. of, numerical illustrations in, in Greek and Arabic numerals I. 71 III.
Scholia to Elements and MSS. of, scholia by Joannes Pediasimus I. 72—73
Scholia to Elements and MSS. of, scholia by Maximus Planudes I. 72
Scholia to Elements and MSS. of, scholia by Psellus I. 70—71
Scholia to Elements and MSS. of, scholia in Latin published by G. Valla, Commandinus, Conrad Dasypodius I. 73
Scholia to Elements and MSS. of, scholia on Eucl. II. 13 I. 407
Scholia to Elements and MSS. of, Scholium IV. No. 2 ascribes Book IV. to Pythagoreans II. 97 III.
Scholia to Elements and MSS. of, scholium published later by Heiberg attributes Scholium X. No. 62 to Proclus III. 521—522
Scholia to Elements and MSS. of, Scholium V. No. 1 attributes Book V. to Eudoxus II. 112
Scholia to Elements and MSS. of, Scholium X. No. 1 attributes discovery of irrational and incommensurable to Pythagoreans III. 1
Scholia to Elements and MSS. of, sometimes interpolated in text I. 67
Scholia to Elements and MSS. of, sources go back as far as Theodorus III. 522
Scholiast to Clouds of Aristophanes II. 99
Schopenhauer I. 227 354
Schotten, H. I. 167 174 179 192—193 202
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 by Euclid I. 17 II. III.
Section = point of section I. 170 171 383
Section, "the section" see "Golden section"
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
Sector-like (figure), bisection of such a figure by straight line II. 5
Seelhoff, P. III. 527
Segment of circle, angle of I. 253 II.
Segment of circle, segment less than semicircle I. 187
Segment of circle, similar segments II. 5
Semicircle I. 186
Semicircle, angle in semicircle a right angle, pre-Euclidean proof II. 63
Semicircle, angle of I. 182 253 II. 39—41 see
Semicircle, centre of I. 186
Separation of ratio and separando II. 135
Separation of ratio and separando, 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. III. 527
Setting-out, one of formal divisions of a proposition I. 129
Setting-out, one of formal divisions of a proposition, may be omitted I. 130
Sexagesimal fractions in scholia III. 523
Sextus Empiricus I. 62 63 184
Shamsaddin as-Samarqandi I. 5n. 89
Sides of plane and solid numbers II. 287—288
Sigboto I. 94
Similar plane and solid numbers I. 357 II.
Similar plane and solid numbers, one mean between two similar plane numbers II. 294 371—272
Similar plane and solid numbers, two means between two similar solid numbers II. 294 373—375
Similar rectilineal figures, def. gives at once too little and too much II. 188
Similar rectilineal figures, def. of, given in Aristotle II. 188
Similar rectilineal figures, similar figures on straight lines which are proportional are themselves proportional and conversely (VI. 22), alternatives for proposition II. 242—247
Similar segments of circles II. 5
Similar solids, definitions of III. 261 265—267
Simon, Max I. 108 155 157—158 167 202 328 II. 134
Simplicius I. 22 167 171 184 185 197 203 223 224 III.
Simplicius on Eudemus' style I. 35 38
Simplicius on lunes of Hippocrates I. 29 35 386—387
Simplicius on parallels I. 190—191
Simplicius, commentary on Euclid I. 27—28
SIMPSON, THOMAS II. 121 III.
Simson, Robert I. 185 186 255 259 287 293 296 322 328 384 387 403 II. 3 8 22 23 33 34 37 43 49 53 70 73 79 90 117 131 132 140 143—144 145 146 148 154 161 162 163 165 170—172 177 179 180 182 183 184 185 186 189 193 195 209 211 212 230—231 238 252 269 270 272—273 III. 266 273—274 275 276 286—287 289 295 301 309 314 321 324 327 331 334 340 341 349 351 359 362 375 433 434
Simson, Robert on "vitiations" in Elements due to Theon I. 46 103 104 106 111 148
Simson, Robert on Euclid's Porisms I. 14
Simson, Robert, Axioms to Bk. V. II. 137
Simson, Robert, Bk. VI. Prop. A extending VI. 3 to case where external angle bisected II. 197
Simson, Robert, definition of plane I. 172—173
Simson, Robert, important note showing flaw in V. 10 and giving alternative II. 156—157
Simson, Robert, Prop. B (inversion) II. 144
Simson, Robert, Prop. D, Book XI. III. 345
Simson, Robert, Prop. E (convertendo) II. 175
Simson, Robert, Props. B, C, D II. 222—227
Simson, Robert, Props. C, D (Bk. V.) connecting proportionals of VII. Def. 20 as particular case with those of V. Def. 5 II. 126—129 III.
Simson, Robert, remarks on VI. 27-29 II. 258—259
Simson, Robert, shortens V. 8 by compressing two cases into one II. 152—153
Sind b. 'Ali Abu 't-Taiyib I. 86
Size, proper translation in V. Def. 3 II. 116—167 189—190
Smith and Bryant, alternative proofs of V. 16, 17, 18 by means of VI. 1 I. 404 III. 275 284 303 307
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—166 169 173—174
Solid angle, definitions of III. 261 267—268
Solid angle, solid "angle" of "quarter of sphere", of cone, or of half-cone III. 268
Solid numbers, three varieties according to relative lengths of sides II. 290—291
Solid, definition of III. 260 262—263
Solid, equal and similar solids III. 261 265—267
Solid, similar solids, definitions of III. 261 265—267
Speusippus I. 125
Sphaerica, early treatise on I. 17
Sphere, definitions of, by Euclid III. 261 269
Sphere, definitions of, by others III. 269
Spherical number, a particular species of cube number II. 291
Spiral of Archimedes I. 26 267
Spiral, "single-turn" I. 122—123n. 164—165
Spiral, "single-turn" in Pappus = cylindrical helix I. 165
Spire (tore) or Spiric surface I. 163 170
Spire (tore) or Spiric surface, varieties of I. 163
Spiric curves or sections, discovered by Perseus I. 161 162—164
Square number, product of equal numbers II. 289 291
Square number, product of equal numbers, one mean between square numbers II. 294 363—264
St Vincent, Gregory of I. 401 404
Steenstra, Pybo I. 109
Steiner, Jakob I. 193
Steinmann, Johann III. 523
Steinmetz, Moritz I. 101 III.
Steinschneider, M. I. 8n. 76sqq.
Stephanus Gracilis I. 101—102
Stephen Clericus I. 47
Stevin, Simon III. 8
Stifel, Michael III. 8
Stobaeus I. 3 II.
Stoic "axioms" I. 41 221
Stoic "axioms", illustrations I. 329
Stolz, O. I. 328 III.
Stone, E. I. 105
Straight line, Archimedes' assumption respecting I. 166
Straight line, division of straight line into any number of equal parts (an-Nairizi) I. 326
Straight line, Euclid's definition, interpreted by Proclus and Simplicius I. 166—167
Straight line, language and construction of I. 167
Straight line, language and construction of, and conjecture as to origin I. 168
Straight line, one or two cannot make a figure I. 169 183
Straight line, other definitions I. 168—169
Straight line, other definitions in Heron I. 168
Straight line, other definitions, by Legendre I. 169
Straight line, other definitions, by Leibniz I. 169
Straight line, pre-Euclidean (Platonic) definition I. 165—166
Straight line, straight line at right angles to plane, definition of III. 260
Straight line, straight line at right angles to plane, definition of, alternative constructions for III. 293—294
Straight line, two straight lines cannot enclose a space I. 195—196
Straight line, two straight lines cannot have a common segment I. 196—199 III.
Stroemer, Marten I. 113
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