  In 1969, H. Davenport and W. Schmidt established a measure of simultaneous
approximation for a real number \xi and its square by rational numbers with the
same denominator, assuming only that \xi is not rational nor quadratic over Q.
Here, we show by an example, that this measure is optimal. We also indicate
several properties of the numbers for which this measure is optimal, in
particular with respect to approximation by algebraic integers of degree at
most three.
