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Veblen O., Young J.W. — Projective Geometry. Vol 1
Veblen O., Young J.W. — Projective Geometry. Vol 1

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Название: Projective Geometry. Vol 1

Авторы: Veblen O., Young J.W.


The present work, which is to consist of two volumes and is intended to be available as a text in courses offered in American universities to upper-class and graduate students, seeks to avoid this difficulty by deferring the study of order and continuity to the second volume. The more elementary part of the subject rests on a very simple set of assumptions which characterize what may be called "general projective geometry." It will be found that the theorems selected on this basis of logical simplicity are also elementary in the sense of being easily comprehended and often used.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1946

Количество страниц: 350

Добавлена в каталог: 05.07.2008

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Предметный указатель
$H_0$, assumption      45
$H_0$, role of      81 261
Abelian group      67
Abscissa      170
Abstract science      2
Addition, of points      142 231
Addition, other definitions of      167 exs.3 4
Addition, theorems on      142—144
Adjacent sides or vertices of simple n-line      37
Algebraic curve      259
Algebraic problem      238
Algebraic surface      259
Alignment, assumptions of      16
Alignment, consistency of assumptions of      17
Alignment, theorems of, for 3-space      20—24
Alignment, theorems of, for 4-space      25 ex.
Alignment, theorems of, for n-space      29—33
Alignment, theorems of, for the plane      17—20
Amodeo, F.      120 294
Anharmonic ratio      159
Apollonius      286
Associative law, for addition of points      143
Associative law, for correspondences      66
Associative law, for multiplication of points      146
Assumption      298
Assumption, $H_0$      45
Assumption, $H_0$, role of      81 261
Assumption, of projectivity      95
Assumption, of projectivity, alternative forms of      105 106 exs.10—12
Assumptions, are necessary      2
Assumptions, categoricalness of      6
Assumptions, consistency of      3
Assumptions, examples of, for a mathematical science      2
Assumptions, for an n-space      33
Assumptions, independence of      6
Assumptions, of alignment      16
Assumptions, of alignment, consistency of      17
Assumptions, of closure      24
Assumptions, of extension      18 24
Axial pencil      55
Axial perspectivity      57
Axis, of coordinates      169 191
Axis, of homology      104
Axis, of pencil      55
Axis, of perspective collineation      72
Axis, of perspectivity      36
Axis, of projectivity on conic      218
Base, of pencil of complexes      332
Base, of plane of points or lines      55
Bilinear equation, binary, represents projectivity on a line      156
Bilinear equation, ternary, represents correlation in a plane      267
Binary form      251 252 254
Bocher, M.      156 272 289 330
Braikenridge      119
Brianchon point      111
Brianchon's theorem      111
Bundle, of complexes      334 ex.
Bundle, of conies      297 exs.9—12
Bundle, of planes or lines      27 55
Bundle, of quadrics      311
Burnside, W.      150
Bussey, W.H.      202
Canonical forms, of collineations in plane      274—276
Canonical forms, of correlations in a plane      281
Canonical forms, of pencils of conies      287—293
Castelnuovo, G.      139 140 237 297
Categorical set of assumptions      6
Cayley, A.      52 140
Center, of bundle      55
Center, of coordinates      170
Center, of flat pencil      55
Center, of homology      104
Center, of perspective collineation in plane      72
Center, of perspective collineation in space      75
Center, of perspectivity      36
Center, of projectivity on conic      218
Central perspectivity      57
Characteristic constant of parabolic projectivity      207
Characteristic equation of matrix      165
Characteristic throw and cross ratio      212 exs.5 7
Characteristic throw and cross ratio, of involution      206
Characteristic throw and cross ratio, of one-dimensional projectivity      205 211
Characteristic throw and cross ratio, of parabolic projectivity      206
Chasles      125
Class, elements of      2
Class, notation for      57
Class, notion of      2
Class, relation of belonging to a      2
Class, subclass of a      2
Class, undefined      15
Clebseh, A.      289
Cogredient n-line      84 ex.
Cogredient triangle      84 exs.7 10
Collineation, between two planes, analytic form of      190
Collineation, characteristic equation of      272
Collineation, defined      71
Collineation, double elements of, in plane      271
Collineation, in plane, analytic form of      189 190 268
Collineation, in space, analytic form of      200
Collineation, invariant figure of, is self-dual      272
Collineation, is the product of two polarities      265
Collineation, leaving conic invariant      214 220 235 ex.
Collineation, perspective, in plane      72
Collineation, perspective, in space      75
Collineation, transforming a conic into a conic      132
Collineation, transforming a five-point into a five-point      77
Collineation, transforming a quadrangle into a quadrangle      74
Collineation, which is the product of two reflections      282 ex.
Collineations      135 ex. 136 ex.
Collineations, associated with two conies of a pencil      131 exs.2 4 6
Collineations, canonical forms of      274—276
Collineations, group of, in plane      268
Collineations, represented by matrices      268—270
Collineations, two, not in general commutative      268
Collineations, types of, in plane      106 273
Commutative correspondence      66
Commutative group      67 70 ex. 228
Commutative law of multiplication      148
Commutative projectivities      70 210 228
Compass, constructions with      246
Complete n-line, in plane      37
Complete n-line, on point      38
Complete n-plane, in space      37
Complete n-plane, on point      38
Complete n-point, in plane      37
Complete n-point, in space      36
Complete quadrangle and quadrilateral      44
Complex, determined by correspondence between points and planes of space      324
Complex, determined by five independent lines      324
Complex, determined by skew pentagon      319
Complex, determined by two projective fiat pencils      323
Complex, equation of      329 331
Complex, general and special      320
Complex, generated by involution on regulus      326 ex.
Complex, linear      312
Complex, null system of      324
Complexes, bundle of      334 ex.
Complexes, in involution      333
Complexes, pencil of      332
Con well, G.M.      204
Concrete representation or application of an abstract science      2
concurrent      16
Cone      118
Cone, as degenerate case of quadric      308
Cone, of lines      109
Cone, of planes      109
Cone, section of, by plane, is conic      109
Configuration      38
Configuration, of Desargues      40 51
Configuration, of Mobius      326 ex.
Configuration, of Pappus      98 249
Configuration, quadrangle-quadrilateral      44
Configuration, symbol of      38
Congruence, determined by four independent lines      317
Congruence, determined by projective planes      317
Congruence, determined by two complexes      325
Congruence, elliptic, hyperbolic, parabolic, degenerate      315
Congruence, equation of      329 330
Congruence, linear      312
conic      109 118 294 exs.3 4
Conic section      118
Conic, determined by conjugate points      293 ex.
Conic, equation of      185 245
Conic, intersection of line with      240 242 246
Conic, polar system of      120—124
Conic, projectivity on      217
Conic, theorems on      109—140
Conic, through four points and meeting given line in two points harmonic with two given points      250 ex.
Conic, through four points and tangent to line      250 ex.
Conic, through three points and tangent to two lines      250 ex.
Conies, pencils and ranges of      128—136 287—293
Conies, projective      212 304
Conjugate groups      209
Conjugate pair of involution      102
Conjugate points (lines)      140 ex. 293 ex.
Conjugate points (lines), on line (point), form involution      124
Conjugate points (lines), with regard to a pencil of conies      136 ex.
Conjugate points (lines), with regard to conic      122
Conjugate projectivities      208
Conjugate projectivities, conditions for      208 209
Conjugate subgroups      211
Consistency, of a set of assumptions      3
Consistency, of assumptions of alignment      17
Consistency, of notion of elements at infinity      9
Construct      45
Constructions, linear (first degree)      236
Constructions, of second degree      245 249—250
Constructions, of third and fourth degrees      294—296
Contact, of second order between two conies      134
Contact, of third order between two conies      136
Contact, point of, of line of line conic      112
Coordinates, homogeneous, in space      194
Coordinates, homogeneous, of points and lines in plane      174
Coordinates, homogeneous, of points on line      163
Coordinates, in a bundle      179 ex.
Coordinates, Klein's line      333
Coordinates, nonhomogeneous, in space      190
Coordinates, nonhomogeneous, of lines in plane      170
Coordinates, nonhomogeneous, of points in plane      169
Coordinates, nonhomogeneous, of points on line      152
Coordinates, of quadrangle-quadrilateral configuration      181 ex.
Coordinates, Pliicker's line      327
Coplanar      24
Copunctal      16
Correlation, between two planes, analytic representation of      266 267
Correlation, between two-dimensional forms      262 263
Correlation, between two-dimensional forms determined by four pairs of homologous elements      264
Correlation, double pairs of a      278—281
Correlation, induced      262
Correlation, represented by matrices      270
Correlation, represented by ternary bilinear form      267
Correlation, which interchanges vertices and sides of triangle is polarity      264
Correlations and duality      268
Correspondence      293 ex.
Correspondence, (1, 1) of two figures      35
Correspondence, as a logical term      5
Correspondence, general theory of      64—66
Correspondence, identical      65
Correspondence, inverse of      65
Correspondence, involutoric or reflexive      66
Correspondence, period of      66
Correspondence, periodic or cyclic      66
Correspondence, perspective      12
Correspondence, perspective between two planes      71
Correspondence, quadratic      139 exs.22 24
Correspondences, associative law for      66
Correspondences, commutative      66
Correspondences, groups of      67
Correspondences, leaving a figure invariant form a group      68
Correspondences, resultant or product of two      65
Corresponding elements      35
Corresponding elements, doubly      102
Covariant      257
Covariant, example of      258
Cremona, L.      137 138
Cross ratio      159
Cross ratio, as an invariant of two quadratic binary forms      254 ex.
Cross ratio, characteristic, of involution      206
Cross ratio, characteristic, of projectivity      205
Cross ratio, definition of      160
Cross ratio, expression for      160
Cross ratio, in homogeneous coordinates      165
Cross ratio, of four complexes      332
Cross ratio, of harmonic set      159 161
Cross ratio, theorems on      167 168
Cross ratios, the six, defined by four elements      161
Curve, algebraic      259
Curve, of third order      217 exs.7 8 9
Cyclic correspondence      66
Darboux, G.      95
Degenerate conies      126
Degenerate regulus      311
Degree of geometric problem      236
Derivative      255
Desargues, configuration of      40 51
Desargues, theorem on conies      127 128
Desargues, theorem on perspective triangles      41 180
Descartes, R.      285
Diagonal point (line), of complete n-point (n-line) in plane      44
Diagonal point (line), of complete quadrangle (quadrilateral)      44
Diagonal triangle of quadrangle (quadrilateral)      44
Dickson, L.E.      66
Difference of two points      148
Differential operators      256
Dimensions, assumptions for space of n      33
Dimensions, space of five      331
Dimensions, space of n      30
Dimensions, space of three      20
Directrices, of a congruence      315
Directrices, of a regulus      299
Directrices, of a special complex      324
Distributive law for multiplication with respect to addition      147
Division of points      149
Domain of rationality      238
Double element (point, line, plane) of correspondence      68
Double pairs of a correlation      97
Double points, and lines of collineation in plane      271 295
Double points, of a projectivity on a line satisfy a quadratic equation      156
Double points, of involution determined by covariant      258
Double points, of projectivity always exist in extended space      242
Double points, of projectivity on a line, construction of      246
Double points, of projectivity on a line, homogeneous coordinates of      164
Double ratio      159
Doubly parabolic point      274
Duality, a consequence of existence of correlations      268
Duality, at a point      29
Duality, in four-space      29
Duality, in plane      29
Duality, in three-space      28
Edge of n-point or n-plane      36 37
Elation, in plane      72
Elation, in space      75
Element, fundamental      1
Element, ideal      7
Element, invariant, or double, or fixed      68
Element, lineal      107
Element, of a figure      1
Element, simple, of space      34
Element, undefined      1
Eleven-point, plane section of      53 ex.
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