**Ettore Bortolotti** studied under Pincherle in Bologna, graduating in 1889. In 1886-87 the first course on Galois theory to be given in Italy was put on at Bologna. Bortolotti, at the time a 20 year old student, attended the course given by Cesare Arzelà. The notes which Bortolotti took of that course have survived and are discussed in [5]. He became an assistant there and worked at Bologna until 1891. In that year he was appointed to the Lyceum of Modica in Sicily. From 1892 he undertook postgraduate studies at Paris then, in 1893, he was appointed to the University of Rome and taught in Rome until 1900 when he became professor of infinitesimal calculus at Modena. His teaching at Modena at this time included analysis and rational mechanics.

Bortolotti was Dean of the Faculty at Modena in 1913-19, then he was appointed professor of geometry at the University of Bologna where he remained for the rest of his life, retiring in 1936. He was an extremely patriotic man; in particular he loved Bologna and it must have been a real joy to him to be able to spend the final part of his career in the University of Bologna where he had begun his studies.

Bortolotti studied topology at first but later went in the direction of analysis considering the calculus of finite differences, continued fractions, convergence of infinite algorithms, summation of series, the asymptotic behaviour of series and improper integrals.

He was always interested in the history of mathematics studying Ruffini's manuscripts while at Modena, then later editing Ruffini's *Complete Works*. Let us look at some of his achievements, and also some of his weaknesses, as a historian of mathematics. First, to put his historical work in context it is useful to see the opinion of Panza in [7] and also that of Dieudonné in his review of [7]:-

The Italian mathematician Ettore Bortolotti devoted a large part of his activity to the history of Italian mathematics from1200to1800. To him is due the discovery of a previously unknown part of Bombelli's 'Algebra', and the edition of Ruffini's complete works. But the author of the paper[7]stresses how lopsided and unconvincing were many opinions of Bortolotti on the history of mathematics. They stemmed from his powerful nationalist feelings, which led him(long before fascism)to attribute in every case priority and leadership to Italian mathematicians(and more particularly to those who worked in Bologna), with a corresponding downgrading of the work of their contemporaries. The author of[7]discusses several examples of this unfortunate tendency, usually supported by unjustified interpretations by Bortolotti of the papers he considered.

We should note that Bortolotti published Ruffini's *Complete Works* in two volumes, the first in 1915. The second volume was published in 1943 but all except three or four copies of it were destroyed by bombs during World War II. Fortunately it was possible to republish the work in 1953 using photographic methods from one of the surviving copies.

He vindicated Cataldi's claim to have discovered continued fractions, after S Günther and A Favaro in separate papers, both published in 1874, had found traces of the continued fractions algorithm long before Cataldi. For details of earlier evidence for continued fractions see the article on Bombelli's Algebra.

However, in defending the originality of Cataldi, Bortolotti does so in a manner which is not free from partiality. We also note that recent work by Fowler has added much to our understanding of the concept of continued fractions as present in ancient Greek mathematics. Nevertheless, this work of Fowler does not diminish the value of Cataldi's contibutions.

Bortolotti also studied Fibonacci, del Ferro, Tartaglia, Cardan, Ferrari. In 1915 Bortolotti wrote a paper on the National Italian Institute in the Napoleonic period while in 1929 he published the previously lost work of Books 4 and 5 of Bombelli's *Algebra* (Bortolotti's text was republished in 1966).

In 1924 Bortolotti made an important discovery. Paolo Bonasoni was a professor at the University of Bologna during the late sixteenth century. His book *Algebra geometrica*, written around 1575, was unpublished and unknown until 1924 when it was discovered in the University of Bologna archives by Bortolotti. This book was remarkable in containing a far more geometric approach to algebra than others from the same period. There is certainly a strong argument supporting the view that if the work had been widely known at the time it was written, the fusion of algebra and geometry which was achieved by Descartes might have occurred much earlier. Bortolotti's discovery was certainly an important event in the history of mathematics.

In 1937 Bortolotti attended the fourth International Congress of the History of Sciences held in Prague. He contributed three papers to the conference: *Mathematica babilonese; L'Histoire des sciences dans l'enseignement et l'enseignement de l'histoire des sciences*; and *Interpretazione storica di testi matematici babilonesi*. The papers are given in [6].

Other papers by Bortolotti on the history of mathematics written during the final ten years of his life include: *I primi algoritmi infiniti nelle opere dei matematici italiani del secolo XVII* (1939); *L'Opera geometrica di Evangelista Torricelli* (1939); *Le fonti della matematica moderna. Matematica sumerica e matematica babilonese* (1940); *Influenza del campo numerico sullo sviluppo delle teorie algebriche* (1941); *Il carteggio matematico di Giovanni Regiomontano con Giovanni Bianchini, Giacomo Speier e Cristiano Roder* (1942); *La pubblicazione delle opere e del carteggio matematico di Paolo Ruffini* (1943); *Il problema della tangente nell'opera geometrica di Evangelista Torricelli* (1943); *Le serie divergenti nel carteggio matematico di Paolo Ruffini* (1944); *Il carteggio matematico di Paolo Ruffini* (1947).

We will say just a few words about some of these papers. In the 1939 paper on Torricelli, Bortolotti gives a concise survey of Torricelli's geometrical investigations, giving his theorems and proofs in modern symbols as well as in the original form. In the 1940 paper on Babylonian mathematics, Bortolotti gives a summary of problems published by Neugebauer but argues that the fact that large series of examples for quadratic equations are made up from the same roots demonstrates that this pair of roots has an 'arcane mystic property'. Finally let us note that Neugebauer was unhappy with many aspects of the 1941 paper of Bortolotti whose title we quoted above:-

Many corrections to the first part could be made, e.g. fourth millennium should be replaced by second millennium. It is also wrong to deny the existence of approximations to irrational square roots, to assume a geometrical basis of the quadratic equations or to deny the existence of texts of this type in the Hellenistic period.

**Article by:** *J J O'Connor* and *E F Robertson*

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