If this be applied to the right-hand side of the identity m m m 2 m2 tan-=- - n n -3n-5n" it follows that the tangent of every arc **commensurable** with the radius is irrational, so that, as a particular case, an arc of 45 having its tangent rational, must be in**commensurable** with the radius; that is to say, 3r/4 is an in**commensurable** number."

If this is the case, the apsidal angle must evidently be **commensurable** with -ir, and since it cannot vary discontinuously the apsidal angle in a nearly circular orbit must be constant.

If b 2 /a 2, 3 /a 3 ..., the component fractions, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be recurring or periodic. It is obvious that every terminating continued fraction reduces to a **commensurable** number.

Any quantity, **commensurable** or in**commensurable**, can be expressed uniquely as a simple continued fraction, terminating in the case of a **commensurable** quantity, non-terminating in the case of an in**commensurable** quantity.

Since the fraction is infinite it cannot be **commensurable** and therefore its value is a quadratic surd number.