288:
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47:
1939:
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2082:) the three Euclidean properties mentioned above are not equivalent and only the second one (Line m is in the same plane as line l but does not intersect l) is useful in non-Euclidean geometries, since it involves no measurements. In general geometry the three properties above give three different types of curves,
461:, so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson (a.k.a.
520:
Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better. The main difficulty, as pointed out by
Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of
525:
suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line. This can not be proved and must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860
516:
reviewed this text and declared it a failure, primarily on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson also devotes a large section of his play (Act II, Scene VI § 1) to denouncing Wilson's treatment of parallels. Wilson edited this concept
412:
Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in
Euclidean space, but the first and third properties involve measurement, and so, are "more complicated" than the second. Thus, the second property is the one usually chosen as the defining
2236:
542:
1245:
1080:
1230:
492:. Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, "Two straight lines that meet one another have different directions, and the difference of their directions is the
2368:
742:
the distance between the two lines can be found by locating two points (one on each line) that lie on a common perpendicular to the parallel lines and calculating the distance between them. Since the lines have slope
452:
At the end of the nineteenth century, in
England, Euclid's Elements was still the standard textbook in secondary schools. The traditional treatment of geometry was being pressured to change by the new developments in
100:
each other or intersect and keep a fixed minimum distance. In three-dimensional
Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called
2628:, Kambly, and the writers of the last fifty years who have been already quoted in connection with other pictographs. Before about 1875 it does not occur as often Hall and Stevens use "par or ∥" for parallel
1794:
1584:
932:
845:
530:
requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles then all transversals must do so. Again, a new axiom is needed to justify this statement.
1489:{\displaystyle d={\sqrt {\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}}={\sqrt {\left({\frac {b_{1}m-b_{2}m}{m^{2}+1}}\right)^{2}+\left({\frac {b_{2}-b_{1}}{m^{2}+1}}\right)^{2}}}\,,}
2139:
While in
Euclidean geometry two geodesics can either intersect or be parallel, in hyperbolic geometry, there are three possibilities. Two geodesics belonging to the same plane can either be:
221:
character set, the "parallel" and "not parallel" signs have codepoints U+2225 (∥) and U+2226 (∦), respectively. In addition, U+22D5 (⋕) represents the relation "equal and parallel to".
943:
1091:
1894:
Similar to the fact that parallel lines must be located in the same plane, parallel planes must be situated in the same three-dimensional space and contain no point in common.
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and all great circles intersect each other. Thus, there are no parallel geodesics to a given geodesic, as all geodesics intersect. Equidistant curves on the sphere are called
1698:
1642:
2435:
parallel to itself so that the reflexive and transitive properties belong to this type of parallelism, creating an equivalence relation on the set of lines. In the study of
204:
737:
690:
2865:
172:
321:
261:
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lines on a globe. Parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center of the sphere.
580:
2135:
just off the image. This is just an artifact of the visualisation. On a real hyperbolic plane the lines will get closer to each other and 'meet' in infinity.
608:
2306:
2657:
2837:
Only the third is a straightedge and compass construction, the first two are infinitary processes (they require an "infinite number of steps".)
2431:(1957) adopted a definition of parallelism where two lines are parallel if they have all or none of their points in common. Then a line
1709:
1808:
that do not intersect need not be parallel. Only if they are in a common plane are they called parallel; otherwise they are called
1505:
646:
there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines,
856:
769:
560:
2682:
2612:'s sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in
2011:
425:
The definition of parallel lines as a pair of straight lines in a plane which do not meet appears as
Definition 23 in Book I of
2523:
Although this postulate only refers to when lines meet, it is needed to prove the uniqueness of parallel lines in the sense of
1983:
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3011:(vol. 3). Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.
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1990:
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When the lines are given by the general form of the equation of a line (horizontal and vertical lines are included):
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in three-dimensional space, the line not lying in that plane, are parallel if and only if they do not intersect.
38:
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1979:
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2621:
2567:
1075:{\displaystyle \left(x_{1},y_{1}\right)\ =\left({\frac {-b_{1}m}{m^{2}+1}},{\frac {b_{1}}{m^{2}+1}}\right)\,}
120:
1225:{\displaystyle \left(x_{2},y_{2}\right)\ =\left({\frac {-b_{2}m}{m^{2}+1}},{\frac {b_{2}}{m^{2}+1}}\right).}
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1235:
These formulas still give the correct point coordinates even if the parallel lines are horizontal (i.e.,
637:
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489:
31:
3065:
La théorie des parallèles de Johann
Heinrich Lambert : Présentation, traduction et commentaires
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1949:
696:
429:. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the
652:
500:, p. 2) In definition 15 he introduces parallel lines in this way; "Straight lines which have the
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of line l. They separate the lines intersecting line l and those that are ultra parallel to line
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414:
83:
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property of parallel lines in
Euclidean geometry. The other properties are then consequences of
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2004:
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397:
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538:
The three properties above lead to three different methods of construction of parallel lines.
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2613:
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446:
405:
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also mentions
Posidonius' definition as well as its modification by the philosopher Aganis.
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2409:
2152:, if they do not intersect in the plane, but converge to a common limit point at infinity (
426:
401:
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116:
8:
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to get the coordinates of the points. The solutions to the linear systems are the points
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1925:. This will never hold if the two planes are not in the same three-dimensional space.
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in two antipodal points. They share two common perpendiculars (one shown in blue).
323:, shown here both to the right of the transversal, one above and adjacent to line
142:
is a special instance of this type of geometry. In some other geometries, such as
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344:
135:
67:
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2609:
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2071:, a four-dimensional manifold with 3 spatial dimensions and 1 time dimension.
2047:, the concept of a straight line is replaced by the more general concept of a
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2276:
2056:
462:
224:
The same symbol is used for a binary function in electrical engineering (the
2566:. Vol. 1 (two volumes in one unaltered reprint ed.). Chicago, US:
2363:{\displaystyle l\parallel m\ \land \ m\parallel n\ \implies \ l\parallel n.}
2617:
2457:
2248:
2067:, particles not under the influence of external forces follow geodesics in
229:
146:, lines can have analogous properties that are referred to as parallelism.
108:
3067:, Paris: Collection Sciences dans l'histoire, Librairie Albert Blanchard,
2608:(1677) , a posthumous work (§ 184) §368. Signs for parallel lines. when
2974:
2564:
A History of
Mathematical Notations - Notations in Elementary Mathematics
2200:
2153:
2063:, a surface (or higher-dimensional space) which may itself be curved. In
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Equivalently, they are parallel if and only if the distance from a point
643:
361:
37:"Parallel lines" and "Parallel line" redirect here. For other uses, see
2947:
2428:
1809:
438:
287:
103:
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2675:, 43. edition (Breslau, 1876), p. 8. H. S. Hall and F. H. Stevens,
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1938:
2288:
2048:
382:
63:
55:
3022:
Mathematical Visions: The Pursuit of Geometry in Victorian England
2424:
in the set of lines where parallelism is an equivalence relation.
437:
attributes a definition of parallel lines as equidistant lines to
2235:
442:
434:
218:
97:
46:
2589:
2251:, the equivalent of a straight line in spherical geometry. Line
2131:
in the hyperbolic plane. The parallel lines appear to intersect
2108:
30:
This article is about the geometry concept. For other uses, see
2280:
2240:
1789:{\displaystyle d={\frac {|c_{2}-c_{1}|}{\sqrt {a^{2}+b^{2}}}}.}
127:
2259:
but is not a great circle. It is a parallel of latitude. Line
521:
Posidonius, expounded by Francis Cuthbertson in his 1874 text
476:
of 1868. Wilson based his definition of parallel lines on the
1579:{\displaystyle d={\frac {|b_{2}-b_{1}|}{\sqrt {m^{2}+1}}}\,.}
472:
One of the early reform textbooks was James Maurice Wilson's
93:
927:{\displaystyle {\begin{cases}y=mx+b_{2}\\y=-x/m\end{cases}}}
840:{\displaystyle {\begin{cases}y=mx+b_{1}\\y=-x/m\end{cases}}}
355:
is located at exactly the same (minimum) distance from line
2934:
Andy Liu (2011) "Is parallelism an equivalence relation?",
2214:), and diverge on both sides of this common perpendicular.
920:
833:
541:
2745:, Chap. 4: Euclid and the English Schoolchild. pp. 161–200
504:, but are not parts of the same straight line, are called
299:
are parallel. This can be proved because the transversal
2210:
Ultra parallel lines have single common perpendicular (
2162:, if they do not have a common limit point at infinity.
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1905:
are parallel if and only if the distance from a point
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772:
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243:
180:
160:
2849:"A Not So Gentle Introduction to General Relativity"
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as a common perpendicular. Solve the linear systems
2146:, if they intersect in a common point in the plane,
3063:Papadopoulos, Athanase; Théret, Guillaume (2014),
3062:
2536:
2362:
1788:
1692:
1636:
1578:
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528:The Elements of Geometry, simplified and explained
517:out of the third and higher editions of his text.
315:
255:
198:
166:
2981:(2nd ed. ed.), New York: Dover Publications
2500:Handbook of mathematics and computational science
2220:
396:are both intersected by a third straight line (a
3087:
2243:there is no such thing as a parallel line. Line
642:Because parallel lines in a Euclidean plane are
2396:. According to Euclid's tenets, parallelism is
2388:considered parallel in Euclidean geometry. The
2279:. Great circles divide the sphere in two equal
50:Line art drawing of parallel lines and curves.
2790:Einführung in die Grundlagen der Geometrie, I
2694:"Mathematical Operators – Unicode Consortium"
2496:
2439:, this variant of parallelism is used in the
747:, a common perpendicular would have slope −1/
282:
1928:
250:
244:
1967:. Unsourced material may be challenged and
347:, the following properties are equivalent:
2648:(Edinburgh, 1714), characters explained.
2548:
2530:
2344:
2340:
3042:(1st ed.), London: Macmillan and Co.
2544:. Vol. Book IV. London. p. 177.
2031:Learn how and when to remove this message
1689:
1633:
1572:
1482:
1239:= 0). The distance between the points is
1071:
725:
681:
570:has everywhere the same distance to line
404:of intersection with the transversal are
327:and the other above and adjacent to line
265:as well as from the logical or operator (
134:. Parallelism is primarily a property of
3016:
2813:
2742:
2726:
2724:
2679:, Parts I and II (London, 1889), p. 10.
2497:Harris, John W.; Stöcker, Horst (1998).
2234:
2107:
2092:geodesics sharing a common perpendicular
1823:in three-dimensional space are parallel
540:
303:produces congruent corresponding angles
286:
272:
45:
2979:The Thirteen Books of Euclid's Elements
2754:
2187:As in the illustration through a point
751:and we can take the line with equation
586:Property 2: Take a random line through
277:
14:
3088:
3037:
2866:"5.3: Theorems of Hyperbolic Geometry"
2846:
2778:
2554:
2392:between parallel lines is evidently a
2097:
509:
497:
469:, in which these texts are lambasted.
232:brackets, U+2016 (‖), that indicate a
3046:
2973:
2825:
2801:
2730:
2721:
2715:
2606:Opuscula mathematica hactenus inedita
2263:is another geodesic which intersects
1850:
2294:
1965:adding citations to reliable sources
1932:
1800:Two lines in three-dimensional space
269:) in several programming languages.
2490:
1847:. This never holds for skew lines.
1703:their distance can be expressed as
632:Distance between two parallel lines
123:(not necessarily the same length).
115:are parallel if they have the same
24:
3056:
2636:(London, 1673), Book IV, p. 177.
2176:Geodesics intersecting at infinity
1917:is independent of the location of
1878:is independent of the location of
1839:is independent of the location of
291:As shown by the tick marks, lines
126:Parallel lines are the subject of
25:
3117:
3081:
2913:From Affine to Euclidean Geometry
2503:. Birkhäuser. Chapter 6, p. 332.
622:share a transversal line through
545:The problem: Draw a line through
2600:§359. ∥ for parallel occurs in
1937:
1693:{\displaystyle ax+by+c_{2}=0,\,}
607:
579:
559:
2941:
2936:The College Mathematics Journal
2928:
2902:
2883:
2858:
2847:Church, Benjamin (2022-12-03).
2840:
2831:
2819:
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2772:
2372:In this case, parallelism is a
2303:are three distinct lines, then
1637:{\displaystyle ax+by+c_{1}=0\,}
533:
488:the idea may be traced back to
39:Parallel lines (disambiguation)
3038:Wilson, James Maurice (1868),
2748:
2736:
2709:
2686:
2644:(London, 1706). John Wilson,
2517:
2341:
2221:Spherical or elliptic geometry
2199:lines, one for each direction
2059:(definition of distance) on a
1913:to the nearest point in plane
1874:to the nearest point in plane
1835:to the nearest point on line
1751:
1723:
1547:
1519:
335:Given parallel straight lines
199:{\displaystyle AB\parallel CD}
13:
1:
2967:
2642:Synopsis palmarioum matheseos
2568:Open court publishing company
2384:, the superimposed lines are
2055:straight with respect to the
1889:
732:{\displaystyle y=mx+b_{2}\,,}
381:(recall that lines extend to
373:is in the same plane as line
2757:Euclid and His Modern Rivals
685:{\displaystyle y=mx+b_{1}\,}
467:Euclid and His Modern Rivals
7:
2446:
2170:geodesics are often called
2074:In non-Euclidean geometry (
626:that intersect them at 90°.
415:Euclid's Parallel Postulate
228:). It is distinct from the
10:
3122:
3024:, Boston: Academic Press,
2224:
2101:
1827:the distance from a point
638:Distance between two lines
635:
420:
400:) in the same plane, the
283:Conditions for parallelism
167:{\displaystyle \parallel }
36:
29:
3047:Wylie, C. R. Jr. (1964),
1929:In non-Euclidean geometry
210:is parallel to line
149:
32:Parallel (disambiguation)
27:Relation used in geometry
2894:Introduction to Geometry
2755:Carroll, Lewis (2009) ,
2669:Die Elementar-Mathematik
2538:Kersey (the elder), John
2483:
2418:pencil of parallel lines
1980:"Parallel" geometry
3049:Foundations of Geometry
2656:(London, 1763), p. 4.
2255:is equidistant to line
1806:three-dimensional space
377:but does not intersect
316:{\displaystyle \theta }
154:The parallel symbol is
84:three-dimensional space
3106:Orientation (geometry)
2870:Mathematics LibreTexts
2759:, Barnes & Noble,
2364:
2268:
2136:
2045:non-Euclidean geometry
1804:Two lines in the same
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459:non-Euclidean geometry
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2898:John Wiley & Sons
2572:193, 402–403, 411–412
2560:"§ 184, § 359, § 368"
2478:Ultraparallel theorem
2365:
2285:parallels of latitude
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2212:ultraparallel theorem
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385:in either direction).
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273:Euclidean parallelism
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256:{\displaystyle \|x\|}
201:
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49:
2654:Elements of Geometry
2410:equivalence relation
2307:
2275:, all geodesics are
1961:improve this section
1897:Two distinct planes
1710:
1649:
1596:
1506:
1246:
1092:
944:
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402:corresponding angles
351:Every point on line
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278:Two lines in a plane
241:
230:double-vertical-line
206:indicates that line
178:
158:
3096:Elementary geometry
3040:Elementary Geometry
2412:. Nevertheless, in
2376:. However, in case
2374:transitive relation
2104:hyperbolic geometry
2098:Hyperbolic geometry
2080:hyperbolic geometry
2061:Riemannian manifold
2051:, a curve which is
1815:Two distinct lines
474:Elementary Geometry
455:projective geometry
445:in a similar vein.
144:hyperbolic geometry
2437:incidence geometry
2402:reflexive relation
2394:symmetric relation
2360:
2273:spherical geometry
2269:
2227:Spherical geometry
2166:In the literature
2137:
2088:parallel geodesics
2084:equidistant curves
2065:general relativity
1851:A line and a plane
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431:parallel postulate
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140:Euclidean geometry
132:parallel postulate
121:opposite direction
66:infinite straight
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3074:978-2-85367-266-5
3018:Richards, Joan L.
2953:Geometric Algebra
2766:978-1-4351-2348-9
2677:Euclid's Elements
2468:Limiting parallel
2453:Clifford parallel
2422:equivalence class
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2295:Reflexive variant
2287:analogous to the
2231:Elliptic geometry
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2181:limiting parallel
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136:affine geometries
113:Euclidean vectors
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16:(Redirected from
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1005:
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980:
979:
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974:
973:
961:
960:
933:
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891:
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826:
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804:
738:
736:
735:
730:
724:
723:
691:
689:
688:
683:
680:
679:
611:
590:that intersects
583:
563:
478:primitive notion
322:
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205:
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165:
21:
3121:
3120:
3116:
3115:
3114:
3112:
3111:
3110:
3101:Affine geometry
3086:
3085:
3084:
3075:
3059:
3057:Further reading
3032:
2970:
2965:
2964:
2946:
2942:
2933:
2929:
2907:
2903:
2888:
2884:
2875:
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2714:
2710:
2701:
2699:
2696:
2692:
2691:
2687:
2660:
2594:
2592:
2582:
2556:Cajori, Florian
2553:
2549:
2535:
2531:
2522:
2518:
2511:
2495:
2491:
2486:
2449:
2420:is taken as an
2414:affine geometry
2390:binary relation
2308:
2305:
2304:
2297:
2233:
2223:
2217:
2106:
2100:
2037:
2026:
2020:
2017:
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1051:
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1034:
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854:
831:
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822:
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719:
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675:
671:
654:
651:
650:
640:
634:
627:
612:
603:
584:
575:
564:
536:
496:between them."
486:Wilhelm Killing
484:. According to
423:
345:Euclidean space
308:
305:
304:
285:
280:
275:
266:
242:
239:
238:
237:
179:
176:
175:
174:. For example,
159:
156:
155:
152:
89:Parallel curves
76:Parallel planes
42:
35:
28:
23:
22:
15:
12:
11:
5:
3119:
3109:
3108:
3103:
3098:
3083:
3082:External links
3080:
3079:
3078:
3073:
3058:
3055:
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3044:
3035:
3030:
3013:
3012:
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2909:Wanda Szmielew
2901:
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2475:
2473:Parallel curve
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2293:
2222:
2219:
2195:there are two
2168:ultra parallel
2164:
2163:
2160:ultra parallel
2157:
2147:
2123:lines through
2121:ultra parallel
2099:
2096:
2039:
2038:
1945:
1943:
1936:
1930:
1927:
1891:
1888:
1852:
1849:
1825:if and only if
1801:
1798:
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718:
714:
711:
708:
705:
702:
692:
678:
674:
670:
667:
664:
661:
658:
636:Main article:
633:
630:
629:
628:
613:
606:
604:
585:
578:
576:
565:
558:
535:
532:
506:parallel lines
502:same direction
422:
419:
410:
409:
386:
367:
312:
284:
281:
279:
276:
274:
271:
252:
249:
246:
195:
192:
189:
186:
183:
163:
151:
148:
74:at any point.
60:parallel lines
26:
18:Parallel lines
9:
6:
4:
3:
2:
3118:
3107:
3104:
3102:
3099:
3097:
3094:
3093:
3091:
3076:
3070:
3066:
3061:
3060:
3051:, McGraw–Hill
3050:
3045:
3041:
3036:
3033:
3031:0-12-587445-6
3027:
3023:
3019:
3015:
3014:
3010:
3009:0-486-60090-4
3006:
3002:
3001:0-486-60089-0
2998:
2994:
2993:0-486-60088-2
2990:
2986:
2985:
2980:
2976:
2972:
2971:
2960:
2956:
2954:
2949:
2944:
2937:
2931:
2925:
2924:90-277-1243-3
2921:
2918:
2914:
2910:
2905:
2899:
2895:
2891:
2886:
2871:
2867:
2861:
2850:
2843:
2834:
2827:
2822:
2816:, pp. 180–184
2815:
2814:Richards 1988
2810:
2803:
2798:
2791:
2786:
2780:
2775:
2768:
2762:
2758:
2751:
2744:
2743:Richards 1988
2739:
2733:, pp. 190–194
2732:
2727:
2725:
2717:
2712:
2695:
2689:
2683:
2680:
2678:
2674:
2670:
2664:
2659:
2655:
2651:
2647:
2643:
2639:
2635:
2631:
2627:
2623:
2619:
2615:
2611:
2607:
2603:
2591:
2587:
2583:
2581:0-486-67766-4
2577:
2573:
2569:
2565:
2561:
2557:
2551:
2543:
2539:
2533:
2526:
2520:
2512:
2510:0-387-94746-9
2506:
2502:
2501:
2493:
2489:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2454:
2451:
2450:
2444:
2442:
2438:
2434:
2430:
2427:To this end,
2425:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2391:
2387:
2383:
2379:
2375:
2370:
2357:
2354:
2351:
2348:
2334:
2331:
2328:
2322:
2316:
2313:
2310:
2302:
2292:
2290:
2286:
2282:
2278:
2277:great circles
2274:
2266:
2262:
2258:
2254:
2250:
2246:
2242:
2237:
2232:
2228:
2218:
2215:
2213:
2208:
2206:
2202:
2198:
2194:
2190:
2185:
2183:
2182:
2177:
2173:
2169:
2161:
2158:
2155:
2151:
2148:
2145:
2142:
2141:
2140:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2105:
2095:
2093:
2089:
2085:
2081:
2077:
2072:
2070:
2066:
2062:
2058:
2054:
2050:
2046:
2035:
2032:
2024:
2013:
2010:
2006:
2003:
1999:
1996:
1992:
1989:
1985:
1982: –
1981:
1977:
1976:Find sources:
1970:
1966:
1962:
1956:
1955:
1951:
1946:This section
1944:
1940:
1935:
1934:
1926:
1924:
1920:
1916:
1912:
1908:
1904:
1900:
1895:
1887:
1885:
1881:
1877:
1873:
1869:
1864:
1862:
1858:
1848:
1846:
1842:
1838:
1834:
1830:
1826:
1822:
1818:
1813:
1811:
1807:
1783:
1775:
1771:
1767:
1762:
1758:
1745:
1741:
1737:
1732:
1728:
1716:
1713:
1706:
1705:
1704:
1686:
1683:
1680:
1675:
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1655:
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1524:
1512:
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1452:
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1436:
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1427:
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1399:
1396:
1391:
1387:
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1365:
1360:
1356:
1349:
1342:
1335:
1330:
1324:
1320:
1316:
1311:
1307:
1302:
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1281:
1277:
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1238:
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1200:
1196:
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1161:
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1146:
1142:
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1114:
1110:
1105:
1101:
1096:
1088:
1087:
1086:
1067:
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1057:
1052:
1048:
1041:
1037:
1031:
1025:
1022:
1017:
1013:
1007:
1002:
998:
994:
987:
983:
976:
970:
966:
962:
957:
953:
948:
940:
939:
938:
914:
910:
906:
903:
900:
897:
888:
884:
880:
877:
874:
871:
868:
862:
853:
852:
851:
827:
823:
819:
816:
813:
810:
801:
797:
793:
790:
787:
784:
781:
775:
766:
765:
764:
762:
758:
754:
750:
746:
726:
720:
716:
712:
709:
706:
703:
700:
693:
676:
672:
668:
665:
662:
659:
656:
649:
648:
647:
645:
639:
625:
621:
617:
610:
605:
601:
598:. Move point
597:
593:
589:
582:
577:
573:
569:
562:
557:
556:
552:
548:
543:
539:
531:
529:
524:
518:
515:
511:
507:
503:
499:
495:
491:
487:
483:
479:
475:
470:
468:
464:
463:Lewis Carroll
460:
456:
450:
448:
444:
440:
436:
432:
428:
418:
416:
407:
403:
399:
395:
391:
387:
384:
380:
376:
372:
368:
365:
363:
358:
354:
350:
349:
348:
346:
342:
338:
330:
326:
310:
302:
298:
294:
289:
270:
247:
235:
231:
227:
222:
220:
215:
213:
209:
193:
190:
187:
184:
181:
161:
147:
145:
141:
137:
133:
129:
124:
122:
118:
114:
110:
109:Line segments
106:
105:
99:
95:
91:
90:
85:
81:
77:
73:
69:
65:
61:
57:
48:
44:
40:
33:
19:
3064:
3048:
3039:
3021:
2978:
2952:
2943:
2930:
2912:
2904:
2893:
2885:
2874:. Retrieved
2872:. 2021-10-30
2869:
2860:
2842:
2833:
2821:
2809:
2797:
2789:
2785:
2774:
2756:
2750:
2738:
2711:
2700:. Retrieved
2688:
2676:
2672:
2668:
2653:
2646:Trigonometry
2645:
2641:
2633:
2605:
2599:
2593:. Retrieved
2563:
2550:
2541:
2532:
2519:
2499:
2492:
2458:Collinearity
2441:affine plane
2432:
2426:
2405:
2397:
2385:
2381:
2377:
2371:
2300:
2298:
2284:
2270:
2264:
2260:
2256:
2252:
2249:great circle
2244:
2216:
2209:
2204:
2192:
2191:not on line
2188:
2186:
2179:
2175:
2171:
2167:
2165:
2159:
2149:
2144:intersecting
2143:
2138:
2132:
2128:
2124:
2120:
2116:
2113:Intersecting
2112:
2091:
2087:
2083:
2073:
2042:
2027:
2018:
2008:
2001:
1994:
1987:
1975:
1959:Please help
1947:
1922:
1918:
1914:
1910:
1906:
1902:
1898:
1896:
1893:
1883:
1879:
1875:
1871:
1867:
1865:
1860:
1859:and a plane
1856:
1854:
1844:
1840:
1836:
1832:
1828:
1820:
1816:
1814:
1803:
1702:
1588:
1498:
1236:
1234:
1084:
936:
849:
760:
756:
752:
748:
744:
741:
641:
623:
619:
615:
602:to infinity.
599:
595:
591:
587:
571:
567:
550:
549:parallel to
546:
537:
534:Construction
527:
522:
519:
510:Wilson (1868
505:
501:
498:Wilson (1868
493:
481:
473:
471:
466:
451:
424:
411:
393:
389:
378:
374:
370:
360:
356:
352:
340:
336:
334:
328:
324:
300:
296:
292:
223:
216:
211:
207:
153:
125:
102:
96:that do not
87:
82:in the same
75:
70:that do not
59:
53:
43:
2987:(3 vols.):
2779:Wilson 1868
2718:, pp. 92—94
2673:Planimetrie
2661: [
2630:John Kersey
2570:. pp.
2281:hemispheres
2201:ideal point
2178:are called
2154:ideal point
644:equidistant
441:and quotes
398:transversal
388:When lines
362:equidistant
3090:Categories
3003:(vol. 2),
2995:(vol. 1),
2968:References
2948:Emil Artin
2876:2024-08-22
2826:Heath 1956
2802:Heath 1956
2731:Heath 1956
2716:Wylie 1964
2702:2013-04-21
2671:, Part 2:
2650:W. Emerson
2624:, Wilson,
2595:2019-07-22
2429:Emil Artin
2225:See also:
2102:See also:
1991:newspapers
1890:Two planes
1810:skew lines
447:Simplicius
439:Posidonius
104:skew lines
2955:, page 52
2938:42(5):372
2917:D. Reidel
2896:, p 192,
2658:L. Kambly
2558:(1993) .
2408:to be an
2404:and thus
2352:∥
2342:⟹
2332:∥
2323:∧
2314:∥
2069:spacetime
1948:does not
1921:in plane
1909:in plane
1738:−
1534:−
1437:−
1369:−
1317:−
1274:−
1143:−
995:−
904:−
817:−
512:, p. 12)
482:direction
406:congruent
311:θ
251:‖
245:‖
188:∥
162:∥
117:direction
72:intersect
3020:(1988),
2977:(1956),
2915:, p 17,
2828:, p. 194
2804:, p. 194
2638:W. Jones
2602:Oughtred
2590:93-29211
2540:(1673).
2447:See also
2289:latitude
2150:parallel
2117:parallel
2076:elliptic
2049:geodesic
2021:May 2017
1882:on line
1870:on line
1843:on line
1831:on line
383:infinity
64:coplanar
56:geometry
2950:(1957)
2911:(1983)
2892:(1961)
2634:Algebra
2626:Emerson
2618:Caswell
2610:Recorde
2542:Algebra
2301:l, m, n
2239:On the
2053:locally
2005:scholar
1969:removed
1954:sources
1855:A line
490:Leibniz
443:Geminus
435:Proclus
421:History
219:Unicode
217:In the
3071:
3028:
3007:
2999:
2991:
2922:
2792:, p. 5
2763:
2614:Kersey
2588:
2578:
2507:
2346:
2338:
2326:
2320:
2241:sphere
2057:metric
2007:
2000:
1993:
1986:
1978:
1129:
981:
526:text,
236:(e.g.
150:Symbol
128:Euclid
94:curves
80:planes
2852:(PDF)
2697:(PDF)
2665:]
2622:Jones
2484:Notes
2406:fails
2247:is a
2156:), or
2012:JSTOR
1998:books
494:angle
369:Line
364:lines
98:touch
68:lines
3069:ISBN
3026:ISBN
3005:ISBN
2997:ISBN
2989:ISBN
2957:via
2920:ISBN
2761:ISBN
2586:LCCN
2576:ISBN
2505:ISBN
2229:and
2119:and
2090:and
1984:news
1952:any
1950:cite
1901:and
1819:and
1085:and
850:and
618:and
457:and
392:and
339:and
295:and
234:norm
138:and
111:and
92:are
78:are
62:are
2604:'s
2398:not
2386:not
2299:If
2271:In
2174:.
2078:or
2043:In
1963:by
755:= −
594:in
508:."
480:of
343:in
130:'s
119:or
54:In
3092::
2868:.
2723:^
2667:,
2663:de
2652:,
2640:,
2632:,
2620:,
2616:,
2598:.
2584:.
2574:.
2562:.
2443:.
2433:is
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2207:.
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2115:,
2086:,
1886:.
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433:.
417:.
366:).
267:||
263:),
214:.
212:CD
208:AB
107:.
58:,
2879:.
2854:.
2705:.
2527:.
2513:.
2382:n
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2133:l
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2034:)
2028:(
2023:)
2019:(
2009:·
2002:·
1995:·
1988:·
1971:.
1957:.
1923:q
1919:P
1915:r
1911:q
1907:P
1903:r
1899:q
1884:m
1880:P
1876:q
1872:m
1868:P
1861:q
1857:m
1845:m
1841:P
1837:l
1833:m
1829:P
1821:m
1817:l
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1776:2
1772:b
1768:+
1763:2
1759:a
1752:|
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1724:|
1717:=
1714:d
1687:,
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1672:c
1668:+
1665:y
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1659:+
1656:x
1653:a
1631:0
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1623:1
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1615:+
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1609:b
1606:+
1603:x
1600:a
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1068:)
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1026:1
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988:(
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863:{
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776:{
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759:/
757:x
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745:m
727:,
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713:+
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677:1
673:b
669:+
666:x
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359:(
357:l
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325:a
301:t
297:b
293:a
248:x
194:D
191:C
185:B
182:A
41:.
34:.
20:)
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