240:
31:
414:
The classical equivalence between
Playfair's axiom and Euclid's fifth postulate collapses in the absence of triangle congruence. This is shown by constructing a geometry that redefines angles in a way that respects Hilbert's axioms of incidence, order, and congruence, except for the Side-Angle-Side
405:
Given that
Playfair's postulate implies that only the perpendicular to the perpendicular is a parallel, the lines of the Euclid construction will have to cut each other in a point. It is also necessary to prove that they will do it in the side where the angles sum to less than two right angles, but
202:
Playfair acknowledged Ludlam and others for simplifying the
Euclidean assertion. In later developments the point of intersection of the two lines came first, and the denial of two parallels became expressed as a unique parallel through the given point.
20:
218:
My own view is that Euclid's
Twelfth Axiom in Playfair's form of it, does not need demonstration, but is part of our notion of space, of the physical space of our experience, which is the representation lying at the bottom of all external
156:
where the concept of parallelism is central. In the affine geometry setting, the stronger form of
Playfair's axiom (where "at most one" is replaced by "one and only one") is needed since the axioms of
267:
The complexity of this statement when compared to
Playfair's formulation is certainly a leading contribution to the popularity of quoting Playfair's axiom in discussions of the parallel postulate.
274:
the two statements are equivalent, meaning that each can be proved by assuming the other in the presence of the remaining axioms of the geometry. This is not to say that the statements are
297:
The easiest way to show this is using the
Euclidean theorem (equivalent to the fifth postulate) that states that the angles of a triangle sum to two right angles. Given a line
231:(1899), providing a new set of axioms for Euclidean geometry, he used Playfair's form of the axiom instead of the original Euclidean version for discussing parallel lines.
790:
This argument assumes more than is needed to prove the result. There are proofs of the existence of parallels which do not assume an equivalent of the fifth postulate.
395:
351:
315:
286:
one statement is true and the other isn't. Logically equivalent statements have the same truth value in all models in which they have interpretations.
90:. The "at most" clause is all that is needed since it can be proved from the first four axioms that at least one parallel line exists given a line
160:
are not present to provide a proof of existence. Playfair's version of the axiom has become so popular that it is often referred to as
1001:
993:
985:
950:
776:
727:
1017:
901:
278:(i.e., one can be proved from the other using only formal manipulations of logic), since, for example, when interpreted in the
243:
If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.
423:
Proposition 30 of Euclid reads, "Two lines, each parallel to a third line, are parallel to each other." It was noted by
263:, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
825:
Brown, Elizabeth T.; Castner, Emily; Davis, Stephen; O’Shea, Edwin; Seryozhenkov, Edouard; Vargas, A. J. (2019-08-01).
722:, translated by Leo Unger from the 10th German edition (2nd English ed.), La Salle, IL: Open Court Publishing,
703:
149:, two lines are said to be parallel if they never meet and other characterizations of parallel lines are not used.
175:(410–485 A.D.) clearly makes the statement in his commentary on Euclid I.31 (Book I, Proposition 31).
484:
415:(SAS) congruence. This geometry models the classical Playfair's axiom but not Euclid's fifth postulate.
109:: Using the axioms and previously established theorems, you can construct a line perpendicular to line
539:
428:
354:
275:
146:
603:
555:
283:
228:
211:
71:
67:
971:
838:
424:
279:
256:
198:
Two straight lines which intersect one another cannot be both parallel to the same straight line.
80:
58:
23:
997:
989:
981:
946:
856:
772:
723:
591:
271:
373:(since it is not the perpendicular) and the hypothesis of the fifth postulate holds, and so,
848:
157:
380:
336:
300:
535:
527:
153:
443:
the set of distinct pairs of lines each of which is parallel to a single common line. If
145:
The statement is often written with the phrase, "there is one and only one parallel". In
922:
754:
658:
635:
543:
531:
190:
This brief expression of
Euclidean parallelism was adopted by Playfair in his textbook
179:
138:
289:
The proofs below assume that all the axioms of absolute (neutral) geometry are valid.
239:
1011:
964:
918:
715:
224:
207:
87:
84:
37:
of
Playfair's axiom: a second line, parallel to the first, passing through the point
627:
252:
152:
This axiom is used not only in
Euclidean geometry but also in the broader study of
30:
959:
432:
260:
562:
in common. It follows that they are the same line, which is Playfair's axiom."
852:
186:
Two straight lines, meeting at a point, are not both parallel to a third line.
34:
860:
123:: A second perpendicular line is drawn to the first one, starting from point
826:
771:(3rd ed.), Upper Saddle River, NJ: Pearson Prentice Hall, p. 139,
19:
526:
More recently the implication has been phrased differently in terms of the
42:
259:
forming two interior angles on the same side that sum to less than two
172:
141:(i.e the alternate interior angles are congruent as per the 4th axiom).
550:
be a point not on line 2. Suppose both line 1 and line 3 pass through
54:
843:
827:"On the equivalence of Playfair's axiom to the parallel postulate"
238:
79:
It is equivalent to Euclid's parallel postulate in the context of
50:
29:
18:
874:
Supplementary Remarks on the first six Books of Euclid's Elements
558:, they are parallel to each other, and hence cannot have exactly
353:
and form a triangle, which is stated in Book 1 Proposition 27 in
769:
Experiencing Geometry: Euclidean and Non-Euclidean with History
755:
Elements of Geometry (containing the first six books of Euclid)
943:
Euclidean and Non-Euclidean Geometries/Development and History
329:, and then a perpendicular to this perpendicular at the point
214:
and expressed this opinion in his address to the Association:
447:
represents a pair of distinct lines, then the statement,
292:
164:, even though it was not Euclid's version of the axiom.
357:. Now it can be seen that no other parallels exist. If
70:, given a line and a point not on it, at most one line
26:
of Playfair's axiom: a line and a point not on the line
400:
439:
be the set of pairs of distinct lines which meet and
383:
339:
303:
133:: This second perpendicular line will be parallel to
515:
is Euclid I.30, the transitivity of parallelism (No
435:in 1908. De Morgan’s argument runs as follows: Let
325:, perpendicular to the given one through the point
53:that can be used instead of the fifth postulate of
963:
431:to Playfair’s axiom. This notice was recounted by
389:
345:
309:
401:Playfair's axiom implies Euclid's fifth postulate
293:Euclid's fifth postulate implies Playfair's axiom
74:to the given line can be drawn through the point.
16:Modern formulation of Euclid's parallel postulate
333:. This line is parallel because it cannot meet
542:, which means that a line is considered to be
475:is Playfair's axiom (in De Morgan's terms, No
8:
767:Henderson, David W.; Taimiņa, Daina (2005),
700:Theories of Parallelism: A Historic Critique
194:(1795) that was republished often. He wrote
842:
799:
382:
338:
302:
182:expressed the parallel axiom as follows:
686:
674:
604:Euclid's elements, Book I, definition 23
578:
966:The Thirteen Books of Euclid's Elements
571:
235:Relation with Euclid's fifth postulate
889:
812:
646:
614:
7:
741:
321:not on that line, construct a line,
247:Euclid's parallel postulate states:
590:more precisely, in the context of
14:
934:A Survey of Geometry (Volume One)
757:, p. 3, Baldwin, Cradock, and Joy
698:William Barrett Frankland (1910)
410:Importance of triangle congruence
83:and was named after the Scottish
945:, San Francisco: W.H. Freeman,
902:The College Mathematics Journal
554:and are parallel to line 2. By
538:the relation is taken to be an
483:) and its logically equivalent
234:
121:Construct another perpendicular
941:Greenberg, Marvin Jay (1974),
1:
663:The Rudiments of Mathematics
970:( 2nd ed.). New York:
419:Transitivity of parallelism
1034:
811:The proof may be found in
704:Cambridge University Press
369:makes an acute angle with
361:was a second line through
936:, Boston: Allyn and Bacon
853:10.1007/s00022-019-0496-9
427:that this proposition is
107:Construct a perpendicular
878:Companion to the Almanac
720:Foundations of Geometry
406:this is more difficult.
255:intersects two straight
1018:Foundations of geometry
753:George Phillips (1826)
546:. Andy Liu wrote, "Let
229:Foundations of Geometry
162:Euclid's parallel axiom
391:
347:
311:
270:Within the context of
265:
244:
77:
38:
27:
932:Eves, Howard (1963),
392:
390:{\displaystyle \ell }
348:
346:{\displaystyle \ell }
312:
310:{\displaystyle \ell }
249:
242:
210:was president of the
137:by the definition of
63:
33:
22:
924:Elements of Geometry
540:equivalence relation
429:logically equivalent
381:
337:
301:
276:logically equivalent
192:Elements of Geometry
113:that passes through
831:Journal of Geometry
665:, p. 145, Cambridge
284:elliptical geometry
212:British Association
972:Dover Publications
544:parallel to itself
425:Augustus De Morgan
387:
343:
307:
245:
81:Euclidean geometry
59:parallel postulate
39:
28:
592:absolute geometry
355:Euclid's Elements
272:absolute geometry
147:Euclid's Elements
1025:
975:
969:
960:Heath, Thomas L.
955:
937:
928:
905:
899:
893:
892:, Vol. 1, p. 314
887:
881:
871:
865:
864:
846:
822:
816:
815:, Vol. 1, p. 313
809:
803:
797:
791:
788:
782:
781:
764:
758:
751:
745:
739:
733:
732:
712:
706:
696:
690:
684:
678:
672:
666:
656:
650:
649:, Vol. 1, p. 220
644:
638:
624:
618:
617:, Vol. 1, p. 190
612:
606:
601:
595:
588:
582:
576:
396:
394:
393:
388:
352:
350:
349:
344:
316:
314:
313:
308:
227:wrote his book,
158:neutral geometry
47:Playfair's axiom
1033:
1032:
1028:
1027:
1026:
1024:
1023:
1022:
1008:
1007:
958:
953:
940:
931:
917:
914:
909:
908:
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896:
888:
884:
872:
868:
824:
823:
819:
810:
806:
798:
794:
789:
785:
779:
766:
765:
761:
752:
748:
740:
736:
730:
714:
713:
709:
697:
693:
685:
681:
673:
669:
657:
653:
645:
641:
632:Linear Geometry
625:
621:
613:
609:
602:
598:
589:
585:
577:
573:
568:
536:affine geometry
528:binary relation
421:
412:
403:
379:
378:
335:
334:
299:
298:
295:
280:spherical model
237:
170:
154:affine geometry
102:, as follows:
17:
12:
11:
5:
1031:
1029:
1021:
1020:
1010:
1009:
1006:
1005:
977:
976:
956:
951:
938:
929:
919:Playfair, John
913:
910:
907:
906:
894:
882:
866:
817:
804:
800:Greenberg 1974
792:
783:
777:
759:
746:
734:
728:
716:Hilbert, David
707:
691:
679:
667:
659:William Ludlam
651:
639:
636:Addison-Wesley
626:for instance,
619:
607:
596:
583:
570:
569:
567:
564:
532:parallel lines
513:
512:
485:contrapositive
473:
472:
420:
417:
411:
408:
402:
399:
386:
342:
306:
294:
291:
236:
233:
221:
220:
200:
199:
188:
187:
180:William Ludlam
169:
166:
143:
142:
139:parallel lines
128:
118:
15:
13:
10:
9:
6:
4:
3:
2:
1030:
1019:
1016:
1015:
1013:
1003:
1002:0-486-60090-4
999:
995:
994:0-486-60089-0
991:
987:
986:0-486-60088-2
983:
979:
978:
973:
968:
967:
961:
957:
954:
952:0-7167-0454-4
948:
944:
939:
935:
930:
927:. W. E. Dean.
926:
925:
920:
916:
915:
911:
903:
898:
895:
891:
886:
883:
879:
875:
870:
867:
862:
858:
854:
850:
845:
840:
836:
832:
828:
821:
818:
814:
808:
805:
801:
796:
793:
787:
784:
780:
778:0-13-143748-8
774:
770:
763:
760:
756:
750:
747:
743:
738:
735:
731:
729:0-87548-164-7
725:
721:
717:
711:
708:
705:
701:
695:
692:
689:, p. 291
688:
687:Playfair 1846
683:
680:
676:
675:Playfair 1846
671:
668:
664:
660:
655:
652:
648:
643:
640:
637:
633:
629:
623:
620:
616:
611:
608:
605:
600:
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587:
584:
580:
579:Playfair 1846
575:
572:
565:
563:
561:
557:
553:
549:
545:
541:
537:
533:
530:expressed by
529:
524:
522:
518:
510:
506:
502:
498:
494:
490:
489:
488:
486:
482:
478:
470:
466:
462:
458:
454:
450:
449:
448:
446:
442:
438:
434:
430:
426:
418:
416:
409:
407:
398:
384:
376:
372:
368:
364:
360:
356:
340:
332:
328:
324:
320:
304:
290:
287:
285:
281:
277:
273:
268:
264:
262:
258:
254:
248:
241:
232:
230:
226:
225:David Hilbert
217:
216:
215:
213:
209:
208:Arthur Cayley
204:
197:
196:
195:
193:
185:
184:
183:
181:
176:
174:
167:
165:
163:
159:
155:
150:
148:
140:
136:
132:
131:Parallel Line
129:
126:
122:
119:
116:
112:
108:
105:
104:
103:
101:
97:
93:
89:
88:John Playfair
86:
85:mathematician
82:
76:
75:
73:
69:
62:
60:
56:
52:
48:
44:
36:
32:
25:
21:
965:
942:
933:
923:
897:
885:
877:
873:
869:
834:
830:
820:
807:
795:
786:
768:
762:
749:
737:
719:
710:
699:
694:
682:
677:, p. 11
670:
662:
654:
642:
634:, page 202,
631:
628:Rafael Artzy
622:
610:
599:
586:
581:, p. 29
574:
559:
556:transitivity
551:
547:
525:
520:
516:
514:
508:
504:
500:
496:
492:
480:
476:
474:
468:
464:
460:
456:
452:
444:
440:
436:
422:
413:
404:
374:
370:
366:
362:
358:
330:
326:
322:
318:
317:and a point
296:
288:
269:
266:
261:right angles
253:line segment
250:
246:
222:
205:
201:
191:
189:
177:
171:
161:
151:
144:
134:
130:
124:
120:
114:
110:
106:
99:
95:
94:and a point
91:
78:
65:
64:
46:
40:
980:(3 vols.):
744:, pp. 385-7
702:, page 31,
433:T. L. Heath
219:experience.
996:(vol. 2),
988:(vol. 1),
912:References
890:Heath 1956
844:1903.05233
813:Heath 1956
647:Heath 1956
615:Heath 1956
507:is not in
467:is not in
35:Consequent
24:Antecedent
1004:(vol. 3).
904:42(5):372
861:1420-8997
837:(2): 42.
742:Eves 1963
718:(1990) ,
385:ℓ
341:ℓ
305:ℓ
1012:Category
962:(1956).
921:(1846).
802:, p. 107
491:For all
451:For all
206:In 1883
178:In 1785
72:parallel
43:geometry
880:, 1849.
876:in the
661:(1785)
630:(1965)
365:, then
173:Proclus
168:History
98:not on
1000:
992:
984:
949:
859:
775:
726:
499:is in
459:is in
377:meets
55:Euclid
49:is an
839:arXiv
566:Notes
534:: In
503:then
495:, if
463:then
455:, if
257:lines
251:If a
223:When
68:plane
66:In a
57:(the
51:axiom
998:ISBN
990:ISBN
982:ISBN
947:ISBN
857:ISSN
773:ISBN
724:ISBN
849:doi
835:110
523:).
519:is
479:is
282:of
61:):
41:In
1014::
855:.
847:.
833:.
829:.
487:,
397:.
45:,
974:.
863:.
851::
841::
594:.
560:P
552:P
548:P
521:X
517:Y
511:,
509:X
505:z
501:Y
497:z
493:z
481:Y
477:X
471:,
469:Y
465:z
461:X
457:z
453:z
445:z
441:Y
437:X
375:n
371:t
367:n
363:P
359:n
331:P
327:P
323:t
319:P
135:L
127:.
125:P
117:.
115:P
111:L
100:L
96:P
92:L
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