by Luis Eduardo Muñoz Guerrero
ABSTRACT
This study measures how close the order of the propositions in Euclid’s Elements comes to minimising the total distance between each proposition and those its proof uses. The 465 propositions and 1,676 dependencies marked in Heath’s translation define, book by book, a single-machine scheduling problem with precedence constraints, which is NP-hard. Dynamic programming over order ideals and Sidney’s decomposition give exactly the expected length under uniformly random admissible orders and the minimum; an optimality index maps them to zero and one. The analysis plan was frozen beforehand. The text is shorter than all 100,000 sampled admissible orders (p = 1.0 × 10−5), but its index is 0.364, below the pre-registered threshold of 0.5 under every reading of the citations (0.31–0.37), although an objective restricted to dependencies inside each scheduling problem raises it to 0.49–0.54. Placebo citations lower it to 0.069. Spinoza’s Ethics, the preregistered replication, reaches 0.353. The largest departures involve widely cited results, such as the triangle’s angle sum, stated with their subject, not near their uses. The order is economical, yet it recovers just over a third of the reduction the logic allows, so minimising total citation distance does not alone account for it.
