On 28 May 1651, when Hendrik was about seventeen years old, his father remarried. Abraham's wife, Magdaleentje Egels, was much younger than her husband and she was carrying his child when he died on 5 November in the year following his marriage. The eighteen year old Hendrik lost his father and gained a brother soon after. However he left Haarlem around this time and went to Leiden where he entered the University on 25 May 1653 intending to study medicine. However he studied mathematics as well as medicine, studying privately under van Schooten with fellow students Huygens and Hudde. On 8 September 1655, having reached the age of twenty-one years, he inherited his father's fortune and from that time on became a man of independent means.
Little is known of van Heuraet's life after he inherited his father's money, all that is known is that he went with Hudde to the Protestant university in Saumur, a town on the river Loire in western France, in 1658. From Saumur he wrote a letter to van Schooten entitled Epistola de transmutatione curvarum linearum in rectas Ⓣ. Van Schooten published this letter in 1659 and we describe below its contents. Van Heuraet returned to Leiden in the following year and again enrolled to study medicine.
Van Schooten had established a vigorous research school in Leiden which included van Heuraet, and this school was one of the main reasons for the rapid development of Cartesian geometry in the mid 17th century. Sluze, Huygens, van Schooten, Hudde and van Heuraet corresponded regarding the properties of curves, in particular van Heuraet was interested in methods of rectification, that is methods to determine the length of a curve. Most of the correspondence was directed through van Schooten in the sense that his students would tell him of their discoveries and he would inform the others as well as publish certain of their results as appendices to his own publications.
Van Schooten edited and published a Latin translation of Descartes's La Géométrie in 1649. A second two-volume translation of the same work (1659-1661) contained appendices by de Witt, Hudde and van Heuraet. In fact there are two papers by van Heuraet which appear as appendices. One of them gives the construction of inflexion points on the conchoid (see  for details). This work was typical of that being carried out by van Schooten's research group and this work was important since attempts to discover properties of curves of this type led to methods which eventually gave rise to the differential and integral calculus. The second of the two papers by van Heuraet is the letter from Saumur we mentioned above (see  and  for details). In it he gives a rectification method which reduces, for any arbitrary algebraic curve, the rectification to a quadrature of an associated curve, that is to computing the area under an associated curve. This was particularly important since at this time mathematicians believed that it was not possible to compare the length of a curved arc with a straight line segment. Van Heuraet's breakthrough is therefore significant in the development of mathematics. In particular, in the paper, he computed the integral
∫ √(1+y' 2) dxand applied his methods to the parabola. His methods of rectification of curves became part of a more general theory by Fermat which was produced independently and at about the same time. We should note that William Neile, independently of van Heuraet, found the arc length of an algebraic curve in 1657 when he rectified the cubical parabola.
Now van Heuraet's remarkable work led to an argument with Huygens over a priority claim. Huygens wrote to van Schooten saying that he had reduced the surface area of a paraboloid to a circle. Van Schooten replied immediately that van Heuraet had made a similar discovery, indeed van Heuraet's methods were much more general. Huygens realised the significance of van Heuraet's work and tried to claim credit for being first to discover another special case. He pressed van Heuraet hard to accept his priority and van Heuraet wrote to him in February 1658 :-
If you had only known my character, it would not have been necessary to exert so much effort against me, who by no means shall seek to rob you of the pleasure and honour of the aforesaid invention, even if the same might have been found by me long ago.Van Heuraet had basically allowed Huygens his priority for the special cases to draw a line under the dispute and calm matters down. He seemed successful in this but then Huygens wrote to Wallis in June 1659 making a claim which historians believe is simply untrue. He wrote :-
When van Heuraet learned that I had measured the surface of the parabolic conoid and had determined the length of the parabola equal to a given quadrature of the hyperbola (concerning both of which I wrote you previously), he found not only both of them by his own technique but, in addition, he rectified completely all other curves of those genera that we allow in geometry.This was written despite what van Schooten wrote to Huygens on 4 February 1658 :-
... concerning your other discovery you indicated nothing to me except some obscure things which were almost completely forgotten by me when I visited Heuraet, so that I could not remember what your discovery properly comprised, much less whether his method corresponded with that of your discovery.If nothing else, Huygens strenuous attempts to gain some credit for van Heuraet's discovery tells us how significant he felt it to be. Huygens repeated his claim in a letter to Sluze a month later. When Huygens came to describe these events in Horologium Oscillatorium in 1673 he certainly modified his claims by only saying that his letter to van Schooten saying that he had reduced the surface area of a paraboloid to a circle had been shown to van Heuraet :-
... he shared that letter with van Heuraet, with whom he then associated. Indeed, it was not difficult for this man of very keen ability to deduce that the surface of that conoid is associated with the measure of the parabolic curve itself.Even if this lesser claim by Huygens is true, it does not take anything away from van Heuraet. My [EFR] guess would be that Huygens believed this version to be true, but in fact it was not true and van Heuraet's discovery was quite independent of Huygens. Certainly examining all the correspondence which survives, seeing van Heuraet's reaction to Huygens claims and also van Schooten's support for van Heuraet, all suggest that he was not directly influenced by Huygens' special cases.
There is, of course, no way now that we will ever know for certain whether or not Huygens' later claim is true. Certainly Huygens stressed the superiority of van Heuraet's methods over those of William Neile, simply, one supposes, because of his claim to have influenced van Heuraet. Wallis, who championed Neile's cause, strongly disagreed.
We do not know when van Heuraet died. All that is known is that Huygens wrote a letter in December 1659 which mentions van Heuraet who was certainly alive at the time. From the fact that no mention of his name occurs in any later correspondence, we have assumed that van Heuraet died shortly after this, probably in 1660. Given his remarkable achievements, one has to wonder at the progress he would have made had he lived as long as Huygens.
Article by: J J O'Connor and E F Robertson
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