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  "body": "\n## Statement\n\nLet $\\ell$ be a [[line|0MP9BCF6DP]] in a [[plane|0V3XAMF3C1]] and let $P$ be a point of the plane not on $\\ell$.\nThen there is at most one line in the plane through $P$ that does not meet $\\ell$.\n\n## Remarks\n\nIn [[absolute geometry|VKFPCSRNDH]], at least one such line always exists: drop the perpendicular from $P$ to $\\ell$ and take the perpendicular to it at $P$.\nSo, over those axioms, Playfair's axiom says that the parallel through $P$ exists and is unique.\nIt is the parallel axiom in Hilbert's system, and it is equivalent to Euclid's fifth postulate over the other axioms.\nIn the hyperbolic plane it fails: through $P$ there are infinitely many lines that do not meet $\\ell$.\nIn the real plane $\\R^2$ with $\\ell = \\set{(x, 0) : x \\in \\R}$ and $P = (0, 1)$, the only line through $P$ missing $\\ell$ is $y = 1$.\n\n## Explanations\n\n### L1 human\n\nThrough a point off a line, you can draw exactly one line that never meets it.\nThis is the parallel axiom in its most common form.\nIt is named after John Playfair, who used it in his edition of Euclid.\n\n### L2 human\n\nPlayfair's axiom is the form of the parallel postulate that one uses in practice.\nIt speaks only of incidence, and not of angles, so it also makes sense in an affine plane, where it is one of the defining axioms.\n",
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