The correspondence principle requires that old (classical mechanics) and new (special relativity) theories agree in the area of overlap, where the old theory has been well-verified. If a new theory is valid, it must give the same correct results as the old theory, i.e., it must correspond, in any area where the old theory has been verified. Thus, at low speeds, special relativity and classical mechanics must give essentially identical results (and they do).
Work Step by Step
This is discussed on page 680-681.