We develop a new approach to the linear ordering of the braid group B_n,
based on investigating its restriction to the set \Div(\Delta\_n^d) of all
divisors of Δ_nd in the monoid B_∞+, i.e., to positive
n-braids whose normal form has length at most d. In the general case, we
compute several numerical parameters attached with the finite orders
(\Div(\Delta\_n^d), <). In the case of 3 strands, we moreover give a complete
description of the increasing enumeration of (\Div(\Delta\_3^d), <). We
deduce a new and specially direct construction of the ordering on B_3, and a
new proof of the result that its restriction to B_3+ is a well-ordering of
ordinal type ωω