Retraction of MR3437893 and MR348182

I have regretfully retracted my two reviews for Mathematical Reviews (MathSciNet) of Andrea Del Centina's article "Poncelet's porism: a long story of renewed discoveries." Without informing or consulting me, MR edited the reviews between submission and publication in ways that I believe exceeded their reasonable editorial discretion, so that the published version did not fairly represent my words and assessments. I requested some changes to MR's published version to correct this, and offered to work with MR to resolve any outstanding concerns. The Executive Editor rejected the option of correcting the review, and did not explain his decision to me. I believe the alternative of retraction was both avoidable and unfortunate. MR policy requires that the full reviews as initially published remain online in perpetuity, so I include here my proposed correction to represent what I would have preferred to appear under my name in MathSciNet.

MR3437893

Andrea Del Centina, ``Poncelet's porism: a long story of renewed discoveries, I.'' Arch. Hist. Exact Sci. (2016) 70:1-122. DOI 10.1007/s00407-015-0163-y.

This is the first installment of a two-part paper examining what Del Centina calls Poncelet's porism or, more frequently, Poncelet's closure theorem (or simply PCT). The theorem asserts that for any pair of conics in the plane there are either no polygons inscribed in one and circumscribed about the other or there are infinitely many such polygons. While inscription and circumscription problems date to antiquity, Del Centina situates in the mid-eighteenth century the first research into so-called inter-scription (simultaneous inscription and circumscription) problems beyond the straightforward case of concentric circles. Inter-scription (and related) problems became newly significant with the advent of projective geometry, and took on further meanings and interpretations with the development of modern algebraic geometry.

Owing to length, the paper is divided into two parts. The first 122 pages fill a single issue of the journal, offering a brief introduction and then covering the story through the nineteenth century. The remainder of the article, reviewed separately (MR3458182), appears in the subsequent issue and continues the story into the twentieth century, then briefly recapitulates the narrative from the beginning. Both parts are well illustrated with numerous geometrical diagrams. A more careful copyedit could have eased a number of linguistic difficulties for a piece whose scope and approach offer enough challenges of their own.

The paper aims to provide a comprehensive survey of mathematical results and developments related to Poncelet's closure theorem, including those whose relationship may be only conceptual rather than historical. The author does not advance a clear argument for how all of these sources fit together, nor for how they fit into the larger histories of geometry, algebra, and modern mathematics. Del Centina's method is primarily bibliographic, featuring detailed expositions of individual works from the past that he has identified as significant. The selection of primary sources appears reasonable, including contributions of clear importance by authors such as Carl Jacobi, Arthur Cayley, Gaston Darboux, and of course Jean-Victor Poncelet. A number of other authors, including several whose works Del Centina asserts were scarcely known by the main figures in this history, might have been either omitted or more explicitly justified.

The discussion leans heavily on direct quotation and extensive paraphrase of arguments from the primary sources in multiple languages. Checking a number of these quotations against original sources, the paper does not appear to be a reliable resource for direct quotation, though most of the errors did not substantially distort the meaning of the quoted sources. More troublesome, especially with regard to earlier sources, is Del Centina's tendency to paraphrase works anachronistically---a tendency that creeps as well into some of his translations of non-English quotations. For instance, Del Centina attributes notions and reasoning from complex projective geometry (among other areas) to Poncelet that appear to reach beyond what is evident in Poncelet's 1822 book or what would have been available to him at the time.

While he makes regular reference to other secondary scholarship on his specific topic (e.g. Bos et al. 1987, MR0917349), Del Centina does little to engage others' scholarship directly or to situate his claims with respect to the existing literature on the history of geometry or the broader history of mathematics in the periods considered. He also largely eschews putting his primary sources in historical context, mostly relegating contextual remarks to occasional footnotes. So, while Del Centina presents his primary sources in copious detail, basic questions about their meaning, motivations, and relevance remain largely unanswered. One small exception to this void of contextual information is Del Centina's attention to his authors' citations, which at least helps establish which texts may have mattered most to which later authors.

MR3458182

Andrea Del Centina, ``Poncelet's porism: a long story of renewed discoveries, II.'' Arch. Hist. Exact Sci. (2016) 70:123-173. DOI 10.1007/s00407-015-0164-x.

This is the second part of a two-part paper, covering twentieth-century developments related to Poncelet's closure theorem. My review of the first part (MR3437893) contains a brief summary and a comment on Del Centina's methods. The second part continues where the first left off (apart from an opening paragraph that repeats the start of the first installment almost word for word), in the same style and subject to the same methodological considerations. After carrying the story through the work of Griffiths and Harris and of Barth and Michel in, respectively, the 1970s and 1990s, Del Centina states the aims of the paper as a whole and then summarizes his principal observations regarding the sources considered in both parts of the paper.

Del Centina asserts that the fundamental questions of mathematical historiography are how earlier ideas and methods influenced later ones, and how later ones ``can be legitimately recognized'' in earlier ones (p. 166). Within this narrow framing, the paper leans heavily on the latter objective (the one historians would consider most dubious, but which may be more relevant to mathematicians themselves). Later, the author states that his goal was to present ``the differences and the similarities'' between key works ``so that the reader can form a personal opinion'' on the story's development (p. 171). Del Centina here appears to regard the historical literature as simply a repository of facts and mathematical assessments, and shows little interest in offering an identifiable intervention or an original argument.

As it stands, the work may be of value as a bibliographic resource for those doing their own studies in the history of geometry, or those who have an amateur interest in the subject. Yet even as bibliography, the paper, while the capacious result of an impressive effort at excavating sources, lacks the precision and explication that would make it most useful.