flavour: the "ancient" privileges of the guilds and civic militias—who were traditionally seen as mouthpieces of the citizenry as a whole—to curb the regent's powers were to be recognised again (as in pre-Burgundian times). The demonstrators also demanded more influence of the Calvinist preachers on the content of government policies and a roll-back of the toleration of Catholics and other dissenting denominations. The purges of the city governments were not everywhere equally thoroughgoing (and, of course, there was little mention of popular influence later on, as the new regents shared the abhorrence of the old ones of real democratic reforms). But as a whole, the new Orangist regime of the Stadtholder was well-entrenched during his following reign. Whether William had a hand in the murder of the de Witt brothers remains unanswered, like his exact role in the later Massacre of Glencoe. That he ordered the withdrawal of a federal cavalry detachment that otherwise might have prevented the lynching has always raised eyebrows. He did not prosecute the well-known ringleaders like Johan van Banchem, Cornelis Tromp, and Johan Kievit, even advancing their careers. In any case, the political turmoil did not enable the allies an opportunity to finish the Republic off. The French were effectively stymied by the water defenses. Only when the inundations froze over in the following winter was there, briefly, a chance for Marshal Luxembourg, who had taken over command of the invading army from Louis, to make an incursion with 10,000 troops on skates. This almost ended in disaster, when they were ambushed. Meanwhile, the States General managed to conclude alliances with the German emperor and Brandenburg, which helped relieve the French pressure in the East. Mathematics The kinematic description of ellipses dates from Archimedes and Proclus, as well as the contemporary Claude Mydorge. Johan de Witt describes the hyperbola with a rotating line and a sliding angle, and a parabola by means of a rotating angle and sliding line. In 1661, de Witt's work appeared in the second volume of von Schooten's Latin translation of La Géométrie. Elementa Curvarum Linearum has been described as the first textbook in analytic geometry. De Witt contributed to financial mathematics: The Worth of Life Annuities Compared to Redemption Bonds. This work combined his roles as statesman and as mathematician, and was discussed in the correspondence between Leibniz and Bernoulli concerning the use of probabilities. Ever since the Middle Ages,