Horologium Oscillatorium: Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae (English: The Pendulum Clock: or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks) is a book published by Dutch mathematician and physicist Christiaan Huygens in 1673 and his major work on pendula and horology.

[1][2] It is regarded as one of the three most important works on mechanics in the 17th century, the other two being Galileo’s Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) and Newton’s Philosophiæ Naturalis Principia Mathematica (1687).

[7] The appearance of the book in 1673 was a political issue, since at that time the Dutch Republic was at war with France; Huygens was anxious to show his allegiance to his patron, which can be seen in the obsequious dedication to Louis XIV.

[8] The motivation behind Horologium Oscillatorium (1673) goes back to the idea of using a pendulum to keep time, which had already been proposed by people engaged in astronomical observations such as Galileo.

[8][12] Although Huygens’s design, published in a short tract entitled Horologium (1658), was a combination of existing ideas, it nonetheless became widely popular and many pendulum clocks by Salomon Coster and his associates were built on it.

[9][16][17] 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.

[8][13] After 1673, during his stay in the Academie des Sciences, Huygens studied harmonic oscillation more generally and continued his attempt at determining longitude at sea using his pendulum clocks, but his experiments carried on ships were not always successful.

The rest of the book is made of three, highly abstract, mathematical and mechanical parts dealing with pendular motion and a theory of curves.

This part ends with a table to adjust for the inequality of the solar day, a description on how to draw a cycloid, and a discussion of the application of pendulum clocks for the determination of longitude at sea.

The last part of the book returns to the design of a clock where the motion of the pendulum is circular, and the string unwinds from the evolute of a parabola.

[4][25] Huygens chose not to publish the majority of his results using these techniques but instead adhered as much as possible to a strictly classical presentation, in the manner of Archimedes.

The review in the Philosophical Transactions (1673) likewise praised the author for his invention but mentions other contributors to the clock design, such as William Neile, that in time would lead to a priority dispute.

Their appreciation of the text was due not exclusively on their ability to comprehend it fully but rather as a recognition of Huygens’s intellectual standing, or of his gratitude or fraternity that such gift implied.

[11] Thus, sending copies of the Horologium Oscillatorium worked in a manner similar to a gift of an actual clock, which Huygens had also sent to several people, including Louis XIV and the Grand Duke Ferdinand II.

[17] Nonetheless, the Archimedean and geometrical style of Huygens's mathematics soon fell into disuse with the advent of the calculus, making it more difficult for subsequent generations to appreciate his work.

[9] Huygens’s most lasting contribution in the Horologium Oscillatorium is his thorough application of mathematics to explain pendulum clocks, which were the first reliable timekeepers fit for scientific use.

[4] His mastery of geometry and physics to design and analyze a precision instrument arguably anticipated the advent of mechanical engineering.