Axiom:Euclid's Fifth Postulate
== {{AuthorRef|Euclid}}'s Statement ==  {{EuclidPostulateStatement|Fifth|If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.}}  There are many equivalent ways to state this postulate.  See below for a selection of them. == Parallel Postulate == {{:Axiom:Parallel Postulate}} == Playfair's Axiom == {{:Axiom:Playfair's Axiom}} == Other equivalent statements == Many further attempts have been made to formulate equivalent definitions of this axiom, often with a view to finding a proof which relies on the other four axioms. (Such attempts have been universally doomed to failure.) Here are a few examples, in approximate chronological order: === {{AuthorRef|Proclus Lycaeus|Proclus}} === :If a straight line intersects one of two parallels, it will intersect the other also. Straight lines parallel to the same straight line are parallel to one another. === {{AuthorRef|Posidonius of Apameia|Posidonius}} and {{AuthorRef|Geminus of Rhodes|Geminus}} === :There exist straight lines everywhere equidistant from one another. (This can be compared with {{AuthorRef|Proclus Lycaeus|Proclus}}' tacit assumption that "Parallels remain, throughout their length, at a finite distance from one another.") === {{AuthorRef|Adrien-Marie Legendre|Legendre}} === :There exists a triangle in which the sum of the three angles is equal to two right angles. === {{AuthorRef|John Wallis|Wallis}}, {{AuthorRef|Lazare Nicolas Marguerite Carnot|Carnot}}, {{AuthorRef|Pierre-Simon de Laplace|Laplace}} === :Given any figure, there exists a figure similar to it of any size we please. {{AuthorRef|Giovanni Girolamo Saccheri|Saccheri}} points out that it is necessary only to postulate that: :There exist two unequal triangles with equal angles. === {{AuthorRef|Adrien-Marie Legendre|Legendre}} (again) === :Through any point within an angle less than two-thirds of a right angle a straight line can always be drawn which meets both sides of the angle. === {{AuthorRef|Johann Friedrich Lorenz|Lorenz}} === :Every straight line through a point within an angle must meet one of the sides of the angle.<ref>{{AuthorRef|Johann Friedrich Lorenz}}: ''Grundriss der reinen und angewandten Mathematik'', 1791.</ref> === {{AuthorRef|Adrien-Marie Legendre|Legendre}} and {{AuthorRef|Farkas Wolfgang Bolyai|W. Bolyai}} === :Given any three points not in a straight line, there exists a circle passing through them. === {{AuthorRef|Carl Friedrich Gauss|Gauss}} === :If I could prove that a rectilineal triangle is possible the content of which is greater than any given area, I am in a position to prove perfectly rigorously the whole of geometry.<ref>{{AuthorRef|Carl Friedrich Gauss|Gauss}}, in a letter to {{AuthorRef|Farkas Wolfgang Bolyai|W. Bolyai}}, 1799.</ref> === {{AuthorRef|Julius Worpitzky|Worpitzky}} === :There exists no triangle in which every angle is as small as we please. === {{AuthorRef|Alexis Claude Clairaut|Clairault}} === :If in a quadrilateral three angles are right angles, the fourth angle is a right angle also. (1741) === {{AuthorRef|Giuseppe Veronese|Veronese}} === :If two straight lines are parallel, they are figures opposite to (or the reflex of) one another with respect to the middle points of all their transversal segments.<ref>{{AuthorRef|Giuseppe Veronese|Veronese}}: ''Elementi'', 1904.</ref> === {{AuthorRef|G. Ingrami|Ingrami}} === :Two parallel straight lines intercept, on every transversal which passes through the middle point of a segment included within them, another segment the middle point of which is the middle point of the first.<ref>{{AuthorRef|G. Ingrami|Ingrami}}: ''Elementi'', 1904.</ref> == Historical Note == {{:Axiom:Euclid's Fifth Postulate/Historical Note}} == Also see == * {{TarskiAxiomLink|axiom = Euclid's Axiom|pageName = Euclid's Fifth Postulate}} == References == <references/> == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Mathematician:Dionysius Lardner|next = Mathematician:Proclus Lycaeus}}: Preface to the Second Edition * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 1|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Axiom:Euclid's Fourth Postulate|next = Axiom:Euclid's Common Notion 1}}: Book $\text{I}$. Postulates * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Axiom:Euclid's Fourth Postulate|next = Axiom:Playfair's Axiom|entry = Euclid's axioms}} Euclid Postulate 5 cmj46mqk4dtm5m3ls5f1eonb6lckqmo