The book everyone skips

Book X, running

One hundred and fifteen propositions sorting irrational lengths into named kinds. It reads as impenetrable because it is written about things nobody could calculate with. This project can.

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(5)	2.236067977	rational (commensurable in square only)	2
sqrt(sqrt(2))	1.189207115	medial	4
5 + sqrt(24)	9.898979486	first binomial	2
3 + sqrt(2)	4.414213562	fourth binomial	2
1 + sqrt(5)	3.236067977	fifth binomial	2
1/2 + 1/2*sqrt(5)	1.618033989	fifth binomial	2
sqrt(2) + (1/4sqrt(2))sqrt(24)	3.146264370	sixth binomial	4
3 - sqrt(2)	1.585786438	fourth apotome	2
-1 + sqrt(5)	1.236067977	fifth apotome	2
-sqrt(2) + (1/4sqrt(2))sqrt(24)	0.317837245	sixth apotome	4
(1 + sqrt(2))*sqrt(sqrt(2))	2.870999946	first bimedial	4
(-1 + sqrt(2))*sqrt(sqrt(2))	0.492585716	first apotome of a medial line	4
sqrt(sqrt(2)) + ((1/4sqrt(2))sqrt(sqrt(2)))*sqrt(24)	3.248974259	second bimedial	8
-sqrt(sqrt(2)) + ((1/4sqrt(2))sqrt(sqrt(2)))*sqrt(24)	0.870560029	second apotome of a medial line	8
4*sqrt(2)	5.656854249	rational (commensurable in square only)	2
1/2sqrt(2) + (1/2sqrt(2))*sqrt(7)	2.577935475	sixth binomial	4
(sqrt(2))sqrt(1/2 + 1/4sqrt(2))	1.306562965	major	4
(2 - sqrt(2))sqrt(1/2 + 1/4sqrt(2))	0.541196100	minor	4
(-1 + sqrt(5))sqrt(1 + 1/2sqrt(5))	1.798907440	the side of a rational plus a medial area	4
(3 - sqrt(5))sqrt(1 + 1/2sqrt(5))	1.111785941	that which produces with a rational area a medial whole	4
(1 + (-1/6 + (1/12sqrt(2))sqrt(5))sqrt(24))sqrt(1/2sqrt(2) + 1/2sqrt(5))	1.992013751	the side of the sum of two medial areas	8
(1 + (1/6 + (-1/12sqrt(2))sqrt(5))sqrt(24))sqrt(1/2sqrt(2) + 1/2sqrt(5))	0.709941667	that which produces with a medial area a medial whole	8
1 + sqrt(2) + (1/4sqrt(2))sqrt(24)	4.146264370	an irrational outside Euclid's thirteen species	4

All thirteen names are above. The last six 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.

sqrt(18) + sqrt(2) is not a binomial: the two parts are multiples of each other, so it collapses to 4·sqrt(2), and the classifier says so. sqrt(4 + sqrt 7) untangles itself before being named. And the second bimedial turns on its rectangle being a medial area rather than a medial line — one square shallower, and a distinction easy to lose.

Where the thirteen run out

The last row. 1 + sqrt(2) + sqrt(3) is constructible with a straightedge and compass like everything above it, and 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 →

Try it

euclid classify "sqrt(3) + sqrt(5)"
euclid classify "(1+sqrt(5))/2"
euclid gap