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Pasch's theorem

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in 1902 that the axiom is redundant, and revised editions now list it as a theorem. Thus Pasch's theorem is also known as
291: 43: 406: 197:, p. 179) states the result in 12.274 but does not refer to it specifically as Pasch's theorem. 139: 233: 130:
originally included Pasch's theorem as an axiom in his modern treatment of Euclidean geometry in
309: 266: 276: 223: 163: 158: 145: 20: 280: 347: 259: 39: 148:, a separate statement, is also included and remains an axiom in Hilbert's treatment. 390: 127: 312: 287: 35: 16:
Result about 4 points on a line which cannot be derived from Euclid's postulates
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on a line, if it is known that the points are ordered as (
351: 258: 296:(in German) (2nd ed.), Leipzig: B.G. Teubner 216:Transactions of the American Mathematical Society 371: 34:, stated in 1882 by the German mathematician 8: 58: 378: 364: 227: 194: 176: 209:"On the projective axioms of geometry" 122: 265:(2nd ed.), John Wiley and Sons, 183: 7: 332: 330: 23:regarding a line through a triangle. 14: 293:Vorlesungen uber neuere Geometrie 134:(1899). However, it was found by 334: 123:Hilbert's use of Pasch's theorem 104:), then it is also true that ( 1: 54:The statement is as follows: 42:which cannot be derived from 350:. You can help Knowledge by 132:The Foundations of Geometry 433: 412:Theorems in plane geometry 329: 18: 417:Elementary geometry stubs 140:Hilbert's discarded axiom 397:Euclidean plane geometry 261:Introduction to geometry 19:Not to be confused with 402:Foundations of geometry 346:-related article is a 207:Moore, E.H. (1902), 344:elementary geometry 62: —  44:Euclid's postulates 310:Weisstein, Eric W. 60: 359: 358: 313:"Pasch's Theorem" 272:978-0-471-18283-2 38:, is a result in 424: 380: 373: 366: 338: 331: 323: 322: 297: 283: 264: 241: 240: 231: 213: 204: 198: 192: 186: 181: 159:Ordered geometry 115: 111: 107: 103: 99: 95: 91: 87: 83: 79: 75: 71: 67: 63: 432: 431: 427: 426: 425: 423: 422: 421: 387: 386: 385: 384: 327: 308: 307: 304: 286: 273: 255:Coxeter, H.S.M. 253: 250: 245: 244: 229:10.2307/1986321 211: 206: 205: 201: 193: 189: 182: 178: 173: 168: 154: 125: 118: 113: 109: 105: 101: 97: 93: 89: 85: 81: 77: 73: 69: 65: 61: 59:Pasch's theorem 52: 32:Pasch's theorem 24: 17: 12: 11: 5: 430: 428: 420: 419: 414: 409: 404: 399: 389: 388: 383: 382: 375: 368: 360: 357: 356: 339: 325: 324: 303: 302:External links 300: 299: 298: 284: 271: 249: 246: 243: 242: 222:(1): 142–158, 199: 187: 175: 174: 172: 169: 167: 166: 161: 155: 153: 150: 124: 121: 56: 51: 48: 40:plane geometry 15: 13: 10: 9: 6: 4: 3: 2: 429: 418: 415: 413: 410: 408: 405: 403: 400: 398: 395: 394: 392: 381: 376: 374: 369: 367: 362: 361: 355: 353: 349: 345: 340: 337: 333: 328: 320: 319: 314: 311: 306: 305: 301: 295: 294: 289: 288:Pasch, Moritz 285: 282: 278: 274: 268: 263: 262: 256: 252: 251: 247: 239: 235: 230: 225: 221: 217: 210: 203: 200: 196: 195:Coxeter (1969 191: 188: 185: 180: 177: 170: 165: 164:Pasch's axiom 162: 160: 157: 156: 151: 149: 147: 146:Pasch's axiom 143: 141: 137: 133: 129: 128:David Hilbert 120: 117: 64:Given points 55: 49: 47: 45: 41: 37: 33: 29: 22: 21:Pasch's axiom 407:Order theory 352:expanding it 341: 326: 316: 292: 260: 219: 215: 202: 190: 179: 144: 131: 126: 119: 57: 53: 36:Moritz Pasch 31: 25: 391:Categories 281:0181.48101 248:References 184:Pasch 1912 136:E.H. Moore 318:MathWorld 290:(1912) , 50:Statement 257:(1969), 152:See also 28:geometry 238:1986321 92:) and ( 279:  269:  236:  76:, and 342:This 234:JSTOR 212:(PDF) 171:Notes 348:stub 267:ISBN 277:Zbl 224:doi 116:). 26:In 393:: 315:. 275:, 232:, 218:, 214:, 142:. 112:, 108:, 100:, 96:, 88:, 84:, 72:, 68:, 46:. 30:, 379:e 372:t 365:v 354:. 321:. 226:: 220:3 114:d 110:b 106:a 102:d 98:c 94:b 90:c 86:b 82:a 78:d 74:c 70:b 66:a

Index

Pasch's axiom
geometry
Moritz Pasch
plane geometry
Euclid's postulates
David Hilbert
E.H. Moore
Hilbert's discarded axiom
Pasch's axiom
Ordered geometry
Pasch's axiom
Pasch 1912
Coxeter (1969
"On the projective axioms of geometry"
doi
10.2307/1986321
JSTOR
1986321
Coxeter, H.S.M.
Introduction to geometry
ISBN
978-0-471-18283-2
Zbl
0181.48101
Pasch, Moritz
Vorlesungen uber neuere Geometrie
Weisstein, Eric W.
"Pasch's Theorem"
MathWorld
Stub icon

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