This article offers an indirect proof of an implication of Euclid’s fifth postulate: that through any given point off a line, there exists one and only one other line parallel to the first. We will show this by assuming the contrary and exposing the contradiction in such an assumption.

            In “The Universal Validity of Euclidean Geometry,” I had irrefutably demonstrated that two parallel lines do not by definition intersect. As in that demonstration, let us assume that L₁ and L₂ have slope m. At the y-axis, L₁ has coordinates (0,y₁), and L₂ has coordinates (0,y₂). The vertical distance between L₁ and L₂ along the y-axis is (y₂-y₁).

            Let us assume, for the purposes of this proof, that L₁ has a second line parallel to it through the point (0,y₂). We will call the line L₃. If L₃ is to be a unique line, it cannot be equal to L₂ and must thus differ from L₂, which is y=mx+y₂. Because we defined L₃ as passing through (0,y₂), it cannot differ from L₂ in that respect. It can only differ in its slope, which would have to be unequal to m. Yet m is also the slope of L₁, and mutually parallel lines are defined to have the same slope. Thus, if L₃ is y=nx+y₂, where n≠m, it cannot by definition be parallel to L₁. The second line we assumed to be parallel to L₁ is shown not to be parallel to it by the implications of that very assumption.

            Also, L₃ will eventually intersect L₁. L₁ is y=mx+y₁ and L₃ is y=nx+y₂, where n≠m. To demonstrate my contention, we can try setting the two equal to each other and find the x-value at which they intersect:

mx+y₁=nx+y₂

mx-nx=y_2- y₁

(m-n)x=y_2- y₁

x = (y_2-y₁)/(m-n)

            Because, by our conditions, n≠m and y₂≠y₁, both the numerator and denominator of the above expression have nonzero real values. Dividing them produces a real value for x, which hence must occur somewhere along the progression of L₁ and L₃. Of course, if L₃ intersects L_1, the two lines cannot be parallel.  L₁ has only one parallel line to it at a distance of separation of (y₂-y₁). That parallel line is L₂, which will never intersect L₁.

            Thus, we have proved that through any given point off a line, there exists one and only one other line parallel to the first, by showing any contrary instance of it to be false and contradictory.

G. Stolyarov II is a science fiction novelist, independent filosofical essayist, poet, amateur mathematician, composer, contributor to Enter Stage Right, The Autonomist, Le Quebecois Libre, and the Ludwig von Mises Institute, Senior Writer for The Liberal Institute, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress. His newest science fiction novel is Eden against the Colossus. His latest non-fiction treatise is A Rational Cosmology. Mr. Stolyarov can be contacted at gennadystolyarovii@yahoo.com.