This paper describes a novel algorithm, Follow-the-Past, for autonomous path-tracking vehicles. Common algorithms, like Pure Pursuit and Follow the Carrot, compute steering commands that make a vehicle follow approximately a predefined path. One problem with these algorithms is that they tend to cut corners, since they do not explicitly take into account the actual curvature of the path. The method presented in this paper uses recorded orientation and steering commands to overcome this problem. The algorithm is constructed within the reactive paradigm, common in modern robotics, and is divided into three separate behaviors, each responsible for one aspect of the path-tracking task. We present results from both a simulator for autonomous forest machines and experiments with a physical robot. The results are compared with the Pure-Pursuit and the Follow-the-Carrot algorithms, and show a significant improvement in performance.