Galileo's insight into the parabolic shape of projectile trajectories, which he was able to prove for the case of horizontal projection, is usually considered to be one of his major achievements finally leading to classical mechanics. Galileo failed, however, to prove his claim for the general case of oblique projection. This failure can be understood, once it is recognized that Galileo's conceptualisation of motion was not yet that of classical mechanics but was still rooted in medieval and Aristotelian concepts and thus formed a part of what can be called "preclassical mechanics." But if this assessment of Galileo's mechanics is correct, do we not have to count it as a lucky coincidence that he hit upon what later became a key insight of classical mechanics? Or does this insight, certain assumptions given, follow with some necessity in the preclassical framework? A natural way to approach questions of this kind is to explore alternative solutions possible in preclassical mechanics by studying the work of contemporaries of Galileo who were encountering the same challenges and had similar means in hand to face them. In my presentation, I want to give a brief account of the theory of projectile motion of such a contemporary, the English natural philosopher Thomas Harriot (1560-1621), as it can be reconstructed from his manuscripts. Harriot worked independently of Galileo, and still there turns out to be a broad knowledge base both scientists shared, including, for example, the medieval doctrine of the configuration of qualities used to quantify motion. In fact, Harriot  not only arrived at the conclusion that projectile trajectories are parabolas, but also encountered precisely the same difficulty as Galileo: how to combine two motions that are not perpendicular to each other, in the absence of a concept of inertia as given in classical mechanics. Harriot's resolution of this difficulty, however, differs from that of Galileo and, as a result, his parabolas are tilted in the case of oblique projection. Harriot did not publish any of his scientific achievements and hence did not leave a significant entry in the history of the emergence of classical mechanics.