Our exact knowledge of Rotary Motion, as of Dynamics in general, dates from the time of Newton. Euler, Laplace, Lagrange, Poisson and Poinsot are illustrious names in the development of the theory. Foucault, in 1855, demonstrated the rotation of the earth by means of the gyroscope, and gave it its name. Its practical applications date from yesterday. These began with the Griffin Grinding Mill, and have been followed by the Howell and Obry devices for keeping a torpedo on a straight course, the Schlick Stabilsator for ships, the Brennan Gyro-Monorail, the Anschuetz-Kaempfe Gyro-compass, and the end is not yet. The theory of rotary motion is not simple, nor is it yet complete. Not all inventors have understood the reason for their devices, and not all mathematicians have had a clear conception of the theory, as evidenced by the unnecessary complication of their treatments. Attempts have been made to explain gyroscopic action without mathematics, or at least without the Calculus. It is hardly necessary to say that all such attempts are futile. It is impossible to explain the actions of a gyroscope without mathematics, and it is impossible to understand them without such knowledge. Many students are afraid of what is called the higher mathematics, and are permitted to avoid them in our higher institutions of learning. Mathematics, in its broadest sense, is the science of time, space, mass and force, and the relations existing between these four quantities. It is the foundation upon which all the exact sciences are built, as it is the foundation of the universe. Everything else may and does change, but the principles of mathematics alone remain eternal. There is no doubt that mathematics are difficult: all other forms of intellectual effort are mere child's play in comparison. Hence, many who are scientifically inclined, seek a field in the inexact sciences, or in the pseudo-sciences, where these difficulties may be shirked. It is noteworthy, however, that even here, as these branches become developed, they are found to reach down to the solid bedrock of mathematics, where their cultivators, who have neglected the fundamental science of all, find themselves in an unenviable position. This is notably the case with meteorology. The day will undoubtedly come (and the sooner, the better) when mathematics will be made the foundation of every education, and no man (or woman) can be considered educated who does not know the Calculus. A virile mind will not quail before its difficulties, but will experience a joy in surmounting its obstacles - the gaudium certaminis - such as can be found in no other intellectual field. The student with an elementary knowledge of mathematics, who attempts to understand gyroscopics from a study of its scattered parts in standard treatises, and from the few monographs as yet written, will find the task tedious - probably repulsive. For this reason, it has seemed advisable to the author to write a monograph which may be easily understood by anybody possessing an elementary knowledge of mechanics and the calculus. The book is divided into two parts - the development of the theory from the Fundamental Gyroscopic Principle, and a discussion of its modern practical applications. The motions of the heavenly bodies, where gyroscopics are exhibited in their grandest and freest (frictionless) form, have been fully explained, but the engineer and cursory student, who care only for the elementary theory and an explanation of its applications, may omit the astronomical discussion without loss of continuity.