Huygens continued his mathematical studies on free fall shortly after and, in 1659, obtained a series of remarkable results. At the same time, he was aware that the periods of simple pendulums are not perfectly tautochronous, that is, they do not keep exact time but depend to some extent on their amplitude. Huygens was interested in finding a way to make the bob of a pendulum move reliably and independently of its amplitude. The breakthrough came later that same year when he discovered that the ability to keep perfect time can be achieved if the path of the pendulum bob is a cycloid.Lodder, J. (2018). The Radius of Curvature According to Christiaan Huygens, pp. 1-14. However, it was unclear what form to give the metal cheeks regulating the pendulum to lead the bob in a cycloidal path. His famous and surprising solution was that the cheeks must also have the form of a cycloid, on a scale determined by the length of the pendulum.Chareix, F. (2004). Huygens and mechanics. Proceedings of the International Conference "Titan - from discovery to encounter" (April 13–17, 2004). Noordwijk, Netherlands: ESA Publications Division, , p. 55 - 65. These and other results led Huygens to develop his theory of evolutes and provided the incentive to write a much larger work, which became the Horologium Oscillatorium.
