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Mark Kac’s First Publication: A Translation of "O nowym sposobie rozwiązywania równań stopnia trzeciego" – Background to the Kac Publication

David Derbes (University of Chicago Laboratory School, retired)

Kac related the story of how he became a mathematician in the prologue5 of his autobiography [Kac 1985]. In the spring of 1930, he was fifteen, a year from graduation, and had become obsessed with the cubic equation. His teachers dismissed his questions about the cubic as too difficult for a high school student. He decided that he would study it on his own and soon found in a textbook Cardano’s formula6 for a root of the cubic \(x^3 + px +q = 0\),

x = \sqrt[3]{-\dfrac{q}{2} + \sqrt{\left(\dfrac{q}{2}\right)^{2} + \left(\dfrac{p}{3}\right)^{3}}} +  \sqrt[3]{-\dfrac{q}{2} - \sqrt{\left(\dfrac{q}{2}\right)^{2} + \left(\dfrac{p}{3}\right)^{3}}}

 The book’s derivation began with, “Set \(x = u + v\).” The young student thought this “grossly unfair,” presupposing knowledge of the solution (that a root x would be obtained as the sum of two cube roots), and vowed to work it out for himself. He announced this lofty goal to his skeptical father, who promised to pay his son five Polish złotys (“in those days a lot of money”)7 should he succeed.

All summer he worked at the problem, “filling reams of paper with formulas before I collapsed into bed at night,” everything else put aside, even dating. At last the formulas were before him. He showed his pieced-together work to his father, who looked it over and promptly paid up. When school reopened, he presented “a neatly written manuscript” to his teacher, who submitted it on the young man’s behalf to a national journal for school mathematics, Młody Matematyk (The Young Mathematician). “That seemed to be the end of it,” writes Kac, “because the receipt was not acknowledged, and for months nothing was heard from distant Warsaw,” 487 km from Kac’s home in Krzemieniec.

In early May, a few weeks before his graduation, he was caught in the hall by the school’s principal. Fearing a bad outcome, he was surprised to be addressed with something like respect. The principal announced that “His Excellency, the Counselor of the Ministry of Education, Antoni Marian Rusiecki, who is visiting our institution, would like to see you at 2:30 this afternoon.” It took Mark Kac only a few seconds to recall that His Excellency was also the editor-in-chief of Młody Matematyk.8

Kac made himself presentable, in his Sabbath best, and met the visitor exactly at 2:30. Rusiecki told him that they had indeed received his paper and explained the lengthy delay in responding. They had been reluctant to accept his derivation, because they felt that it must have been found previously. But a careful search of the literature indicated that it was new, and so they were going to publish it. “And so they did. It appeared a few months after my graduation under the name of Katz because I thought that the German spelling was fancier than the Slavic Kac.”9 What happened next is so dramatic that only the author’s words will suffice.

Before my brief visit was over Mr. Rusiecki asked me my plans for the future. I told him that my family thought that I should study engineering. “No,” he said, “you must study mathematics; you clearly have a gift for it.” I followed this advice and it saved my life. In mathematics, as it turned out, I was good enough (and lucky enough) to win a post-doctoral fellowship to go abroad in 1938. The fellowship was endowed by the wealthy, thoroughly assimilated Polish-Jewish family Parnas, and by the terms of the endowment, one of the two yearly endowments had to be given to a Jewish applicant. I came to Johns Hopkins University in December of 1938 and the war caught me there. Had I gone into engineering, I would unquestionably have shared the fate of my family and six million others [Kac 1985, 4].

Work good enough to launch such a career and save a life should be known. But what was Mark Kac’s derivation? It was not given nor even sketched in the autobiography. A search for copies of this volume of Młody Matematyk in the United States initially proved fruitless (in part because it was listed under Parametr), nor did this publication merit either inclusion in a collection of Kac’s selected papers [Kac 1979] or mention in his bibliography [McKean 1990, 226–235]. Perhaps, I thought, it could be found.10


[5] The prologue of [Kac 1985], “How I Became a Mathematician”, originally appeared as an essay in the journal of the Weizmann Institute of Science, Israel: Rehovot, vol. 9, no. 2 (1981/82).

[6] The story of Cardano's formula is told entertainingly in [Livio 2005, 63–73]. Gerolamo Cardano (or Jerome Cardan) (1501–1576), the first to publish it, was the third to obtain it, after substantial help from the notebook of Scipione del Ferro (1465–1526), and from conversations with Niccolò Fontana (1499–1557), known to history as Niccolò Tartaglia, “the stammerer.” (Fontana acquired his nickname as a result of the 1512 French invasion of his home town, Brescia, during which he was slashed across his jaw and throat by a saber; the wound left him with a lifelong speech impediment and a perpetual beard to hide the scar.) Del Ferro never made his methods public, but he had taught a method for solving cubics to his students Annibale della Nave (ca 1500–1558), who was married to del Ferro’s daughter Filippa, and Antonio Maria del Fiore (dates unknown, fl. 1520–1535). Fiore briefly made a career out of public challenges (standard procedure for Italian Renaissance mathematicians) to solve a class of cubics. Knowing that a method existed, Tartaglia accepted Fiore’s challenge (1535) and worked out more general rules shortly before the contest, winning it easily. Cardano asked the now-famous Tartaglia to show him the method, which Tartaglia did only under a promise of secrecy, but he withheld the derivation; in those times discoveries afforded competitive advantages in contests and so were concealed like buried treasure. A rumor reached Cardano that della Nave also knew the secret. Cardano traveled with his student Lodovico de Ferrari (1522–1565) to Bologna to meet with della Nave, who allowed the visitors to read his late father-in-law’s notebook, where del Ferro had recorded his solution many years before Tartaglia. Given del Ferro’s prior claim, Cardano felt no longer bound by his promise to Tartaglia and published the formula in his landmark book on algebra, Ars Magna (The Great Art). Though Cardano gave due credit to del Ferro and Tartaglia [Cardano 1545, 8, 96], he did not mention his promise to Tartaglia.

[7] Kac cited [1985, 31] a half-stipend for a college student of 60 złotys a month, which he called “about twelve dollars,” a rate in 1931 of five złotys to 1 US dollar. That suggests five 1931 złotys would be roughly equivalent to $15–$20 in 2021 dollars.

[8] The magazine Młody Matematyk was an offshoot of the magazine Parametr (in English, Parameter), published from 1930 to 1932 and then reappearing in 1939 for one year, its publication ceasing with the German invasion of Poland. Though Polish mathematics had attained world-class status, the level of mathematics taught in the schools was in need of improvement. A. Tarski (1901–1983), at the time a secondary school teacher as well as a university professor, had promoted (in 1929) the publication of a magazine devoted to the teaching of mathematics, and Parametr was the response. The magazine was founded and edited by Antoni M. Rusiecki (1892–1956), a mathematics instructor and government official in the Ministry of Religious Denominations and Public Education (Warsaw), and Stefan Straszewicz (1889–1983), the chair of that Ministry’s mathematics committee. Straszewicz earned a PhD (in 1914) under E. Zermelo (1871–1953) at the University of Zürich. Both men taught in underground schools during the German occupation, and after the war worked to develop the Polish Mathematical Olympiad. From the beginning Parametr had included material for secondary school teachers and students, but in the 1931/1932 volume, this material was taken out of Parametr and published as a new magazine, Młody Matematyk, distributed for free with Parametr. Rusiecki and Straszewicz not only served as editors but also contributed many articles to the two magazines. As part of his governmental duties, Rusiecki visited schools throughout Poland to provide teacher training, often for twenty days a month, and the extra work of editing two magazines was unsustainable. Besides Kac, among the contributors were H. Steinhaus, (1887–1972), W. Sierpiński (1882–1969) and Tarski. See [Dabkowska 2019, 69–130] and [McFarland, McFarland, and Smith 2014, 204–213].

[9] The Polish pronunciation of “Marek Kac” and the German pronunciation of “Mark Katz” are nearly identical.

[10] As far as can be determined, the only physical copies of Młody Matematyk outside of Poland are in Brown University’s John Hay Library, Providence, RI, bound with the Library’s copies of Parametr.

David Derbes (University of Chicago Laboratory School, retired), "Mark Kac’s First Publication: A Translation of "O nowym sposobie rozwiązywania równań stopnia trzeciego" – Background to the Kac Publication," Convergence (April 2021)