**Hendrik van Heuraet** was a Dutch mathematician who made some contributions to integration.

- However he studied mathematics as well as medicine, studying privately under van Schooten with fellow students Huygens and Hudde.
- From Saumur he wrote a letter to van Schooten entitled Epistola de transmutatione curvarum linearum in rectas Ⓣ(Letter on rectification of a curve).
- 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' La Géométrie in 1649.
- A second two-volume translation of the same work (1659-1661) contained appendices by de Witt, Hudde and van Heuraet.
- There are two papers by van Heuraet which appear as appendices.
- 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.
- Van Heuraet's breakthrough is therefore significant in the development of mathematics.
- 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.
- Van Heuraet had basically allowed Huygens his priority for the special cases to draw a line under the dispute and calm matters down.
- If nothing else, Huygens strenuous attempts to gain some credit for van Heuraet's discovery tells us how significant he felt it to be.
- Even if this lesser claim by Huygens is true, it does not take anything away from van Heuraet.
- 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.
- 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.
- 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.

Born 8 September 1634, Haarlem, Netherlands. Died about 1660, (probably) Leiden, Netherlands.

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Origin Netherlands

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive