The book everyone skips
115 propositions sorting irrational lengths into named kinds. It seems impenetrable because it is written about things nobody could calculate with.
Some lengths a straightedge and compass produce are whole numbers or fractions. Most are not. Euclid gives thirteen names to the ones that are not — medial, binomial, apotome, and so on — each defined by whether certain sums and products come out rational. Every one of those conditions is a yes-or-no question the exact arithmetic can answer, so the whole classification becomes a single function.
| Length | Value | Euclid's name for it | Degree |
|---|---|---|---|
| sqrt(sqrt(2)) | 1.189207115 | medial | 4 |
| 5 + sqrt(24) | 9.898979486 | binomial | 2 |
| (1 + sqrt(2))*sqrt(sqrt(2)) | 2.870999946 | first bimedial | 4 |
| sqrt(sqrt(2)) + ((1/4*sqrt(2))*sqrt(sqrt(2)))*sqrt(24) | 3.248974259 | second bimedial | 8 |
| (sqrt(2))*sqrt(1/2 + 1/4*sqrt(2)) | 1.306562965 | major | 4 |
| (-1 + sqrt(5))*sqrt(1 + 1/2*sqrt(5)) | 1.798907440 | the side of a rational plus a medial area | 4 |
| (1 + (-1/6 + (1/12*sqrt(2))*sqrt(5))*sqrt(24))*sqrt(1/2*sqrt(2) + 1/2*sqrt(5)) | 1.992013751 | the side of the sum of two medial areas | 8 |
| 3 - sqrt(2) | 1.585786438 | apotome | 2 |
| (-1 + sqrt(2))*sqrt(sqrt(2)) | 0.492585716 | first apotome of a medial line | 4 |
| -sqrt(sqrt(2)) + ((1/4*sqrt(2))*sqrt(sqrt(2)))*sqrt(24) | 0.870560029 | second apotome of a medial line | 8 |
| (2 - sqrt(2))*sqrt(1/2 + 1/4*sqrt(2)) | 0.541196100 | minor | 4 |
| (3 - sqrt(5))*sqrt(1 + 1/2*sqrt(5)) | 1.111785941 | that which produces with a rational area a medial whole | 4 |
| (1 + (1/6 + (-1/12*sqrt(2))*sqrt(5))*sqrt(24))*sqrt(1/2*sqrt(2) + 1/2*sqrt(5)) | 0.709941667 | that which produces with a medial area a medial whole | 8 |
The six apotome species are the awkward ones: their two terms are the roots of a single quadratic, so neither can be written without the other and the sum shows no seam to split it at. Squaring puts the seam back — the square is the sum of the squares plus twice the rectangle, and those come apart — after which the recovered pair is checked by adding it up again.
The simplest example outside the thirteen is 1 + sqrt(2) + sqrt(3). It is constructible with a straightedge and compass, but Euclid has no name for it. His classification comes out of applying areas, which produces sums and differences of two terms; this one resolves into three, and falls outside. It is the simplest such number the search finds. The finding →
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