Some Aspects of the History of Infinite Continuous Fractions

The development of an irrational number in continued fraction is infinite. Euler explained how to determine the development for the case of square roots of non-square natural numbers, his goal being solving an indeterminate equation in natural numbers (so-called Pell’s equation). He noticed that such developments are always periodic. Lagrange proved that periodicity occurs only in the case of quadratic irrationals, and that the period begins after some number of terms. In the 19th century was discovered that the conjugate roots of a same quadratic equation have reverse periods.