## John Collins and James Gregory discuss Tschirnhaus

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 ...

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 ...

I received lately two of your letters, whereby I perceive you have fallen in acquaintance with a very learned gentleman [Walter von Tschirnhaus] and a great admirer of Descartes, whom I also admire so much that he or any other shall help him as to his solution of biquadratic and cubic equations. Descartes' method is general ... Where it fails, being fitly applied, I have not seen, neither do expect to see anything used with better success. If a man will trifle his time in particular methods, he may have enough to do perchance in the very quadratic equation, concerning which you know all the ancients busied themselves. It is true, particular methods may be, and commonly are, much easier than the general, yet are not to be preferred, since they require much time in learning, and memory in retaining. His first method ... holds only when the biquadratic equation is produced from two quadratics of such a nature ... His second method ... is only the common way of throwing off the second term; for when this determination is, the second term being put off, the second goes with it, and so the biquadratic comes to lack the alternate terms, and consequently falls to be a quadratic equation with a plane root. ... I am hardly of the gentleman's mind to think that Hudde does not thoroughly understand his method of maxima and minima ... I have as little charity for his overvaluing Descartes' method so as to think that all discovered since are but its consequences. Yet this hinders nothing the esteem I have for the gentleman, who (if I may judge) surely is a great algebraist, and, albeit probably he may be inferior to Mr Newton, is without question far beyond me, whom you are pleased too much to overvalue. I see no connection between my general method of giving the surd roots of all equations and these particular rules invented by that gentleman. Mine in the biquadratic and cubic coincides with Descartes; and in higher equations, as in Descartes' rules, there occur frequently impossible cases.

I am much obliged to you for yours of 20th August, those methods of Mr Tschirnhaus indeed are but very particular, though he asserts that he has general ones, and I told him at parting I should be more fully convinced he had if he but show me his series for finding the three roots of a cubic equation capable of so many that would express the same in cubic surds different from those of Cardan, which indeed (to my understanding) seems to imply that the said roots cannot be always (at least in that method, or any other I know of) be expressed by any manner of surds, to which he replied that his papers were packed up; otherwise he would impart the same, and there being present with him a Dane named George Mohr who lately published in low Dutch, two little books the one named Euclides Danicus where he pretends to perform all Euclid's problems with a pair of compasses only without ruler, and another entitled

*Euclides Curiosus*, wherein with a ruler and a fork (or compasses at a fixed opening) he performs the same, he [Tschirnhaus] said he would speak to the said Mohr to impart the said series, which he promised to do but as yet has not performed.

Mr Tschirnhaus whilst here, which was 13 or 14 weeks, spent most of his time in calculating the canons for the first 8 dimensions complaining much of the excessive tedium thereof, and being gone to Paris may perchance be induced to print some brief account thereof before his intended bigger book comes out ... rather in regard he promised ere long after his going home to give an account of it to Mr Oldenburg to impart to the Royal Society, and whilst he was here Dr Pell found him so learned that he declined all conferences with him, and was the rather induced to go thither in regard he heard that the Academy maintains two algebraic calculators the other being Jacques Ozanam who is a notable person often mentioned and concerned in a little book by Jacques de Billy entitled

*Diophantus Redivivus*...

JOC/EFR July 2012

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