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  "body": "\n## Summary\n\nBirkhoff's axioms are four postulates for plane [[Euclidean geometry|KRZD96GSVC]] that George David Birkhoff gave in 1932.\nThey take the [[real numbers|08J2M1JYAG]] as known and measure distances with a ruler and angles with a protractor.\nBecause the real numbers do much of the work, the list is very short, and it became the basis of many school geometry courses.\n\n## Explanations\n\n### L1 human\n\nHilbert's axioms build geometry from scratch, which makes the list long.\nBirkhoff instead allows a ruler and a protractor: every line is a copy of the number line, and every angle has a size in degrees.\nWith these tools and one axiom about similar triangles, all of plane geometry follows.\nFor example, the angle sum of a triangle is a short consequence of the similarity axiom.\n",
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