# John Collins and James Gregory discuss Tschirnhaus

John Collins met Walter von Tschirnhaus in London in the summer of 1675. He told James Gregory about the meeting in a letter dated 3 August 1675:

There has been here [London] a Gent [Walter von Tschirnhaus] of a noble family in Saxony about thirteen weeks, though I have had but lately conference with him, i am apt to think (excepting yourself and Mr Newton) he is the most knowing algebraist in Europe, he is so great an admirer of Descartes that he asserts that all that has been done by Slusius, Barrow etc and the whole doctrine of quadratures, centres of gravity, straightening of curves, tangents etc are but mere corollaries of Descartes' doctrine ... this Gent is going to Paris to reside there for a year where he intends to publish a treatise of 'Algebra et de Locis' in Latin, the rough draft of which he showed me, wherein he had explained all Hudde's reductions etc, amplified the doctrine of tangents both as to geometrical and mechanical curves, affirming that Hudde never thoroughly understood the doctrine of maxima and minima. ... he affirms he can give general new methods for quadratures of curvilinear figures and straightening of curves, has much amplified the doctrine of constructions, and lastly a new method for the roots of all equations, whereby Hudde's reductions and breaking of equations are rendered useless, of which new method he gave me only one specimen in an easy case of a biquadratic equation ... I received this from him on Friday last, and then proposed this equation to be solved by his new method:
$x^{4} - 2x^{2} + 12x - 18 = 0.$
This is an equation as cannot be broke by Descartes' cubical mallet, for instead of being reduced to a cubic equation it comes to an impossible quadratic. ... He is a very worthy affable person, and I hope will prove a good correspondent at Paris ...

Collins wrote again to Gregory about Tschirnhaus one week later on 10 August:

By mine of the 3rd instant I gave you some account of a new method for finding the roots of equations etc invented by Mr Tschirnhaus, a gent of Saxony, who I told you was just upon departing for Paris; and, presuming you have that letter, I proceed. Upon the parting visit I received from him, in answer to the doubt I mentioned about that series, he said it was only fitted to the condition there proposed. I further objected that it seemed to serve only biquadratics that had two pairs of equal though different roots; in answer he affirmed it served for other cases (according to an example taken out of Descartes) wherein all the roots were unequal, and gave another rule for another easy case as follows, showing the variety thereof, affirming that he imparted only some of the rules for easy cases, reserving the universal rule to himself, but might possibly impart that when at Paris ...

Gregory must have received the letters from Collins quickly, since he replied on 20 August 1675: