Stampioen taught mathematics in Rotterdam where, in 1632, he published his own treatment of spherical trigonometry appended to van Schooten's sine tables. In 1633 he challenged Descartes to a public competition by giving him a geometric problem whose solution involved the solution of a quartic equation. Descartes presented a solution but it was rejected by Stampioen as not being complete. In fact Stampioen's criticism was fair for although Descartes had taken the geometric problem and derived the correct quartic equation, he left the problem there without solving the quartic. Fair though Stampioen's criticism was, it was definitely unwise for he made an enemy of Descartes who was a very powerful figure.
In 1638 Stampioen moved from Rotterdam to The Hague on being appointed tutor to the twelve year old Prince William. This was an important position since William was first in line to succeed to stadtholder and captain general of six provinces of the Netherlands. William was the son of Frederick Henry, Prince of Orange, and he arranged William's marriage to Mary Henrietta Stuart, the eldest daughter of the English king Charles I of England. Mary was ten years old and William fifteen when they were married in May 1641. William later became Prince of Orange on the death of his father in March 1647. How successful Stampioen was in tutoring William, and whether this had any bearing on his years as Prince of Orange, we can only leave to the imagination. The move to the Hague to become tutor to William, however, set Stampioen on a new career for, while in The Hague, he opened a printing shop in which he printed his own writings on mathematics.
In 1639 Stampioen published Algebra or the New Method , a work which he had written while still teaching in Rotterdam. The problem which Stampioen was interested in came as a consequence of using the Cardan-Tartaglia formula to solve cubic equations. This formula involved taking the cube root of expressions such as a + √b, both in the case where b is positive and negative. Stampioen let this cube root be A + √B, where a, b, A and B are all natural numbers. Then cubing both sides gives
He then posed two further public challenges under the alias of John Baptista of Antwerp involving the solution of cubics. One of these problems was to find the correct position to place a siege gun so seemed of particular practical importance. Having posed the problems as if by John Baptista of Antwerp, he then proceeded to give solutions to the problems under his own name, using his methods for finding the cube root of a + √b. Clearly it was a publicity stunt to publicise his book. If this sounds like a particularly modern approach to selling, then let us simple say that human nature does not seem to have changed much over the centuries!
A young surveyor Waessenaer solved the first of John Baptista's (Stampioen's) problems, using a rule due to Descartes. Since we know that Descartes described his rule in a letter to Waessenaer written on 1 February 1640, it looks as though Descartes was probably out to discredit Stampioen and realised that it would be even more telling if it came from a relatively unknown mathematician. Stampioen rejected Waessenaer's solution which prompted Waessenaer to reply with a broadly based attack on the mathematics contained in Stampioen's Algebra or the New Method. The argument was adjudicated by van Schooten who favoured Waessenaer. Descartes also became publicly involved in the argument, although it does look as though he was always behind Waessenaer.
Was Stampioen correct? Well if we look at what he was trying to do, it was to do his arithmetic in the ring Z[√b]. Descartes (and Waessenaer's) criticisms are valid - he needs b to be square free etc. Descartes' method also starts off with an observation which we now easily see is true in Z[√b], namely a + √b has a cube root only if a2 - b is a cube. This, again in modern terms, says that the norm of a + √b must be a cube, which is true since the norm is multiplicative. Descartes then goes on to find approximate values and ends up taking integer parts. The paper  contains interesting observations on this argument and contains a French translation of Stampioen's method of finding the cube roots of a + √b as well as letters of Descartes to Mersenne on 30 September 1640 in which he criticises Stampioen's method. In this paper the author describes the origin of Descartes' rule and explains why it works.
In 1644 Stampioen was employed to tutor Huygens and his brother in mathematics. This may be seen as slightly surprising given that Huygens' father Constantijn Huygens had corresponded with Descartes and the two agreed that Stampioen's behaviour during the dispute had been bad. It may indicate that Descartes actually had a higher opinion of Stampioen than that which he showed in public.
Little else is known about Stampioen. He published a topographical map in 1650 and, were it not for the fact that he is mentioned as being a member of a panel set up to adjudicate a proposed solution to the longitude problem in 1689, we might have wrongly supposed that he died shortly after 1650. As it is, given that he was 79 in 1689, we have assumed that he did not live long after that - in other words the date of his death that we have given is a pure guess based on his age and the knowledge that he was certainly alive in 1689.
Article by: J J O'Connor and E F Robertson