  A classical theory of Desarguesian geometry, originating with D. Hilbert in
his 1899 treatise, Grundlagen der Geometrie, leads from axioms to the
construction of a division ring from which coordinates may be assigned to
points, and equations to lines; this theory is highly nonconstructive. The
present paper develops this coordinatization theory constructively, in
accordance with the principles introduced by Errett Bishop in his 1967 book,
Foundations of Constructive Analysis. The traditional geometric axioms are
adopted, together with two supplementary axioms which are constructively
stronger versions of portions of the usual axioms. Stronger definitions, with
enhanced constructive meaning, are also selected; these are based on a single
primitive notion, and are classically equivalent to the traditional
definitions. Brouwerian counterexamples are included; these point out specific
nonconstructivities in the classical theory, and the consequent need for
strengthened definitions and results in a constructive theory. All the major
results of the classical theory are established, in their original form,
revealing their hidden constructive content.
