  During the early 1830's Bernard Bolzano, working in Prague, wrote a
manuscript giving a foundational account of numbers and their properties. In
the final section of his work he described what he called `infinite number
expressions' and `measurable numbers'. This work was evidently an attempt to
provide an improved proof of the sufficiency of the criterion usually known as
the `Cauchy criterion' for the convergence of an infinite sequence. Bolzano had
in fact published this criterion four years earlier than Cauchy who, in his
work of 1821, made no attempt at a proof. Any such proof required the
construction or definition of real numbers and this, in essence, was what
Bolzano achieved in his work on measurable numbers. It therefore pre-dates the
well-known constructions of Dedekind, Cantor and many others by several
decades. Bolzano's manuscript was partially published in 1962 and more fully
published in 1976. We give an account of measurable numbers, the properties
Bolzano proved about them, and the controversial reception they have prompted
since their publication.
