He received a liberal education from his father, studying languages, music, history, geography, mathematics, logic, and rhetoric, alongside dancing, fencing and horse riding.

[17] Frans van Schooten Jr., professor at Leiden's Engineering School, became private tutor to Huygens and his elder brother, Constantijn Jr., replacing Stampioen on the advice of Descartes.

[28] Christiaan Huygens lived at the home of the jurist Johann Henryk Dauber while attending college, and had mathematics classes with the English lecturer John Pell.

The experience was bittersweet and somewhat puzzling since it became clear that Fermat had dropped out of the research mainstream, and his priority claims could probably not be made good in some cases.

[5] These results became the main reference point and the focus for further debates through correspondence and in a short article in Journal des Sçavans but would remain unknown to a larger audience until the publication of De Motu Corporum ex Percussione (Concerning the motion of colliding bodies) in 1703.

[58] His relationship with the French Académie was not always easy, and in 1670 Huygens, seriously ill, chose Francis Vernon to carry out a donation of his papers to the Royal Society in London should he die.

[5][65] The young diplomat Leibniz met Huygens while visiting Paris in 1672 on a vain mission to meet the French Foreign Minister Arnauld de Pomponne.

[72] Huygens's preferred method in his published works was that of Archimedes, though he made use of Descartes's analytic geometry and Fermat's infinitesimal techniques more extensively in his private notebooks.

[17][27] Huygens's first publication was Theoremata de Quadratura Hyperboles, Ellipsis et Circuli (Theorems on the quadrature of the hyperbola, ellipse, and circle), published by the Elzeviers in Leiden in 1651.

In this work, Huygens was able to narrow the gap between the circumscribed and inscribed polygons found in Archimedes's Measurement of the Circle, showing that the ratio of the circumference to its diameter or pi (π) must lie in the first third of that interval.

[82] His success in applying algebra to the realm of chance, which hitherto seemed inaccessible to mathematicians, demonstrated the power of combining Euclidean synthetic proofs with the symbolic reasoning found in the works of Viète and Descartes.

[83] Huygens included five challenging problems at the end of the book that became the standard test for anyone wishing to display their mathematical skill in games of chance for the next sixty years.

[8] Huygens first re-derives Archimedes's solutions for the stability of the sphere and the paraboloid by a clever application of Torricelli's principle (i.e., that bodies in a system move only if their centre of gravity descends).

[17][90] However, unlike many of his contemporaries, Huygens had no taste for grand theoretical or philosophical systems and generally avoided dealing with metaphysical issues (if pressed, he adhered to the Cartesian philosophy of his time).

[105] He derived geometrically the now standard formula for the centrifugal force, exerted on an object when viewed in a rotating frame of reference, for instance when driving around a curve.

[8] Huygens collected his results in a treatise under the title De vi Centrifuga, unpublished until 1703, where the kinematics of free fall were used to produce the first generalized conception of force prior to Newton.

In his work on pendulums Huygens came very close to the theory of simple harmonic motion; the topic, however, was covered fully for the first time by Newton in Book II of the Principia Mathematica (1687).

[117][118][119][120] Part of the incentive for inventing the pendulum clock was to create an accurate marine chronometer that could be used to find longitude by celestial navigation during sea voyages.

The first part focused on the general principles of refraction, the second dealt with spherical and chromatic aberration, while the third covered all aspects of the construction of telescopes and microscopes.

[138] Huygens also worked out practical ways to minimize the effects of spherical and chromatic aberration, such as long focal distances for the objective of a telescope, internal stops to reduce the aperture, and a new kind of ocular known as the Huygenian eyepiece.

[140][141] Huygens's lenses were known to be of superb quality and polished consistently according to his specifications; however, his telescopes did not produce very sharp images, leading some to speculate that he might have suffered from near-sightedness.

[144] There are others to whom such a lantern device has been attributed, such as Giambattista della Porta and Cornelis Drebbel, though Huygens's design used lens for better projection (Athanasius Kircher has also been credited for that).

However, Thomas Young's interference experiments in 1801, and François Arago's detection of the Poisson spot in 1819, could not be explained through Newton's or any other particle theory, reviving Huygens's ideas and wave models.

[151] The thus-named Huygens–Fresnel principle was the basis for the advancement of physical optics, explaining all aspects of light propagation until Maxwell's electromagnetic theory culminated in the development of quantum mechanics and the discovery of the photon.

Such speculations were not uncommon at the time, justified by Copernicanism or the plenitude principle, but Huygens went into greater detail, though without acknowledging Newton's laws of gravitation or the fact that planetary atmospheres are composed of different gases.

[166] He argued that extraterrestrial life is neither confirmed nor denied by the Bible, and questioned why God would create the other planets if they were not to serve a greater purpose than that of being admired from Earth.

[4][171] Huygens also helped develop the institutional frameworks for scientific research on the European continent, making him a leading actor in the establishment of modern science.

[72] Huygens brought this type of geometrical analysis to a close, as more mathematicians turned away from classical geometry to the calculus for handling infinitesimals, limit processes, and motion.

As he wrote at the end of a draft of De vi Centrifuga:[33] Whatever you will have supposed not impossible either concerning gravity or motion or any other matter, if then you prove something concerning the magnitude of a line, surface, or body, it will be true; as for instance, Archimedes on the quadrature of the parabola, where the tendency of heavy objects has been assumed to act through parallel lines.Huygens favoured axiomatic presentations of his results, which require rigorous methods of geometric demonstration: although he allowed levels of uncertainty in the selection of primary axioms and hypotheses, the proofs of theorems derived from these could never be in doubt.

[125] In demanding such mathematical tractability and precision, Huygens set an example for eighteenth-century scientists such as Johann Bernoulli, Jean le Rond d'Alembert, and Charles-Augustin de Coulomb.