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As I studied Cardano's formula for the cubic, Tschirnhaus transformation came up as a very important step. The more I studied it the more attractive and interesting it seemed. So I am curious about the motivation and background of the person behind it, what he had in mind when inventing it. I saw the post Motivation Behind Tschirnhaus Transformation and Cardano's Formula on Math SE, but I am interested in what motivated the idea of the Tschirnhaus transformation itself rather than its uses.

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Attempting to find substitutions that would generalize Cardano's and Ferrari's solutions to the cubic and quartic was a popular enterprise in the 17-18th centuries. Gregory, Leibniz, Euler, Bezout and Lagrange participated along with many lesser known mathematicians. So if the question is how Tschirnhaus came up with his transformation a brief answer is by algebraic experimentation inspired by Cardano and Descartes (the two he mentioned by name).

For general information on the man, his time, life and mathematical work, a good source is Kracht-Kreyszig, E. W. von Tschirnhaus: His role in early calculus and his work and impact on Algebra, section 4 is dedicated to the Tschirnhaus's Acta Eruditorum paper that introduced his transformation. Descartes left a particularly deep impression on him, he was introduced to Geometry in his youth by the brother of its editor, van Schooten, and later had Leibniz arrange for him to study Descartes's unpublished papers, see McTutor's biography.

Tschirnhaus's paper, A method for removing all intermediate terms from a given equation (1683), was essentially completed by 1677, and shared with Leibniz, but Tschirnhaus kept working to improve his results, without success, and timed the publication to his election to the French Académie des sciences. The paper has been translated into English by Green. On his motivation, Tschirnhaus says the following:

"We have learned from DesCartes’ Geometry by what method the second term might reliably be removed from a given equation; but on the question of removing multiple intermediate terms I have seen nothing hitherto in the analytic arts. On the contrary, I have encountered not a few who believed that the thing could not be done by any art. For this reason I have decided to set down here some things concerning this business, enough at least for those who have some grounding in the analytic art, since the others could scarcely be content with so brief an exposition: reserving the remainder (which they might wish to see here) for some other time... DesCartes shows [this] in removing the second term from a given equation, when he says that for removing the second term with the aid of the equation $x = y+a$, in a quadratic equation the quantity a must be equal to $\frac{p}2$ in a cubic equation $\frac{p}3$ and so forth."

Tschirnhaus proceeds to explain how to choose $a$ to remove two terms, and later comments that his solution to the cubic is "different from the Cardano expression most particularly in the fact that it includes a single cubic radical sign, where the Cardano expression contains two". After that, he gets carried away and claims that any (polynomial) equation can be reduced by his method, promising to show it elsewhere in a manner reminiscent of Fermat's famous remark on the margins:

"But finally it should be noted that by means of the equations displayed in the second paragraph not only may two, three, four, &c terms from a given equation be removed, but that from various other equations different from these I might also do the same thing. Nay, I can easily show that all possible equations may be handled by this business. But that I might be able to show by this method (according to paragraph five) all possible analytic expressions of roots of every possible equation, I shall explain more fully in its own place."

As a matter of fact, Tschirnhaus's method does not always succeed even in removing the three middle terms from the general quintic, let alone that "all possible equations may be handled by this business", as Leibniz pointed out years before the publication:

"Concerning your... method for finding the roots of an equation... I do not believe that it will be successful for equations of higher degree, except in special cases. I believe that I have a proof for this." [quoted from Kracht-Kreyszig]

Bring (1786) and Jerrard (1834) did find a way to remove the three terms (likely, independently), see Adamchik-Jeffrey, Polynomial transformations of Tschirnhaus, Bring and Jerrard. Alas, that still did not solve the quintic. Although the futility of the original quest to solve it in radicals was shown by Abel and Galois, the Tschirnhaus transformation continued to inspire. Hermite applied it to solve the quintic in terms of elliptic modular functions in 1858 (his other inspiration was Viete's trigonometric solution to the irreducible case of the cubic). For later developments, see Wolfson, Tschirnhaus transformations after Hilbert.

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    $\begingroup$ (+1) How do you do it? In the time it took me to track down the original publication and fight my way through the first paragraph, you managed to write an excellent two-page answer that wasn't there when I started. $\endgroup$
    – njuffa
    Commented Dec 7, 2023 at 10:51
  • $\begingroup$ @njuffa Mostly luck. I had a break in my chores and was better rested than usual, finding Green's translation and Kracht-Kreyszig was fortuitous. And a bit of inspiration. While reading, I became captivated by the surrounding story, not unlike the OP. How Tschirnhaus's method differed by Cardano's, what was it that he took from Descartes, Leibniz's involvement, the aftermath. So it came out naturally as I was answering my own questions. $\endgroup$
    – Conifold
    Commented Dec 8, 2023 at 4:04
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D. T., "Methodus auferendi omnes terminos intermedios ex data æquatione." Acta Eruditorum (1683), pp. 204-207.

The 'D' may stand for doctor, because the author's full name was Ehrenfried Walther von Tschirnhaus, so there is no 'D' in his name. He states his motivation in the initial paragraph of this publication: To simplify a polynomial equation by eliminating more than one of the terms.

He writes that it was known that one could always remove the second term by the method of Descartes, but that he had so far not encountered any method in the analytical art to eliminate more than one term. He therefore decided to reveal one here, albeit only briefly for those skilled in the art, with additional details reserved for the future.

Ex Geometria Dn Des Cartes notum est, qua ratione semper secundus terminus ex data æquatione possit auferri; quod plures terminos intermedios auferendos, hactenus nihil inventum vidi in Arte Analytica, imo non paucos offendi, qui crediderunt, id nulla arte perfici posse. Quapropter hic quædam circa hoc negotium aperire constitui, verum saltem pro iis, qui Artis Analyticae apprime gnari, cum aliis tam brevi explicatione vix satisfieri possit: reliqua, quæ hic desiderari possent, alii tempori reservans.

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