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In that year Huygens, who played the harpsichord, took an interest in music, and Simon Stevin's theories on it; he showed very little concern to publish his theories on consonance, some of which were lost for centuries.
During 1663 he made what was his third visit to Paris; the Montmor Academy closed down, and Huygens took the chance to advocate a more Baconian programme in science.
In October 1646 there is the suspension bridge, and the demonstration that a catenary is not a parabola.
Some of Mersenne's concerns at the time, such as the cycloid (he sent Evangelista Torricelli's treatise on the curve), the centre of oscillation, and the gravitational constant, were matters Huygens only took seriously towards the end of the 17th century. Huygens preferred meantone temperament; he innovated in 31 equal temperament, which was not itself a new idea but known to Francisco de Salinas, using logarithms to investigate it further and show its close relation to the meantone system.
In political terms, the First Stadtholderless Period that began in 1650 meant that the House of Orange was not in power, removing Constantijn's influence.
Further, he realised that his son had no interest in such a career.
Subsequently, Huygens developed a broad range of correspondents, though picking up the threads after 1648 was hampered by the five-year Fronde in France.
Visiting Paris in 1655, Huygens called on Ismael Boulliau to introduce himself. Through Pierre de Carcavi Huygens corresponded in 1656 with Pierre de Fermat, whom he admired greatly, though this side of idolatry.
It included material discussed with Mersenne some years before, such as the fallacious nature of the squaring of the circle by Grégoire de Saint-Vincent.
After two years, from March 1647, Huygens continued his studies at the newly founded College of Orange, in Breda, where his father was a curator: the change occurred because of a duel between his brother Lodewijk and another student.
He then had a stint as a diplomat on a mission with Henry, Duke of Nassau. He took off for Denmark, visited Copenhagen and Helsingør, and hoped to cross the Øresund to visit Descartes in Stockholm. While his father Constantijn had wished his son Christiaan to be a diplomat, it also was not to be.
Mersenne wrote to Constantijn on his son's talent for mathematics, and flatteringly compared him to Archimedes (3 January 1647).
The letters show the early interests of Huygens in mathematics.