What is a Particle in Classical Physics? A Wave?

The concepts of ``particle" and ``wave" play important roles in quantum mechanics in that a given microscopic system can, under the proper circumstances, display properties of each.\footnote{A Messiah, Quantum Mechanics, Vols. I and II (Wiley, 1961).} However, these categories have their genesis in classical mechanics\footnote{H. Goldstein, Classical Mechanics (Addison-Wesley, 1980, 2nd edition).} and we must turn to this area to understand what they mean. In particular, it must be clearly understood that these concepts are idealistic representations of physical phenomena and, as a consequence, each may not correspond to any actual physical system. We discuss the definition of ``particle'' from the perspective of how it has been defined in standard textbooks and carry out a similar analysis for the notion of ``wave.'' We then define ``particle'' and ``wave'' within the context of classical mechanics by means of their respective equations of motion. The general (tentative) conclusion is that within the framework of classical mechanics the concepts of ``particle'' and ``wave'' systems may be unambiguously defined.