**Ernesto Pascal**'s parents were Stefano Pascal and Maria Gaetana Zapegna. Of course, the name Pascal sounds French and indeed it is since Stefano Pascal's family were originally from France, coming from the town of Tarascon which is about 20 km south west of Avignon. Ernesto had a brother, Carlo Pascal (1866-1926), who was born on 21 October 1866, also in Naples. Carlo studied Latin at the University of Naples and became a professor of Latin literature at the universities of Catania, Pavia and Milan. He was an outstanding scholar, probably at least as eminent in his own field as his brother Ernesto was in the field of mathematics.

Ernesto Pascal's education was in his home town of Naples and, after attending school there, he entered the University of Naples. The mathematicians at Naples at that time included Ettore Caporali (1855-1886) and his student Pasquale Del Pezzo (1859-1936) who had graduated in 1882 and spent the rest of his career teaching at Naples. Sadly Caporali committed suicide in July 1886 at the age of 31 believing his intellectual skills were declining. Also teaching there was Gabriele Torelli (1849-1931), who was a lecturer, Federico Amodeo (1859-1946), who had been awarded his laurea from Naples in 1883, and Alfonso Del Re who had also earned a laurea from Naples and was appointed as an assistant in 1885. Pascal graduated with a laurea in mathematics in 1887 but, even before the award of the degree, he had begun to publish mathematical papers. Two of his papers, *Relazioni fra le ellissi centrali d'inerzia delle aree ed i baricentri dei volumi generati da esse* Ⓣ and *Teoremi baricentrici* Ⓣ, were published in 1886 in the journal of the Academy of Science, Physics and Mathematics of Naples. Remarkably, he published six papers in 1887, the year he was awarded his laurea, three in the same journal as his 1886 papers, and three in the *Giornale di matematiche*. This journal had been founded in Naples by Giuseppe Battaglini together with Nicola Trudi (1811-1894) and Vincenzo Janni (1819-1891) two years before Pascal had been born. His three 1887 papers in Battaglini's journal were *Costruzioni geometriche di tre poligoni regolari* Ⓣ, *Sopra una formula numerica* Ⓣ, and *Sulla risultante di una ennica e di una cubica* Ⓣ.

After graduating, Pascal left Naples and went to the University of Pisa where he continued his studies during the academic year 1887-88. There were a number of outstanding mathematicians at Pisa including Ulisse Dini, Luigi Bianchi and Vito Volterra. Pascal continued his remarkable publication record, another six papers appearing in 1888. The leading centre for mathematics in the world at this time was the University of Göttingen in Germany and this is where Pascal went after his year in Pisa. Hermann Schwarz and Felix Klein were two leading mathematicians at Göttingen at this time and Pascal particularly benefited from Klein's teaching. The year he spent in Germany had a major impact on his mathematical development. This meant that, after the year 1888-89 in Göttingen, Pascal returned to Italy in 1889, still only 24 years old, but having considerable experience which would put him in a good position in competitions for chairs in Italy. He also had by this time a remarkable publication record for someone so young with 22 papers in print by 1890. It was in that year that he entered the competition for the chair of infinitesimal calculus at the University of Pavia. Ranked in top position by the commissioners, he was appointed in 1890. He remained at Pavia until 1907 and during his time there he married and had two sons.

Both Alberto Pascal, born on 23 December 1894, and Mario Pascal, born on 31 May 1896, were born in Pavia. Alberto studied mathematics at the University of Naples (where his father was a professor) but was called up to fight in World War I in 1915 before completing his university studies. He participated in the Battle of the Three Mountains (Monte Valbella, Col del Roso and Col d'Echele) on the Asiago Plateau on 28 January 1918 between Italian forces and Austrian-Hungarian forces. He died on that day, shot down while acting as an observer in a plane. He was awarded a posthumous medal for valour by his country and an honorary mathematics degree by the University of Naples. Mario, the younger brother, also studied mathematics at the University of Naples and graduated in 1919. He became a lecturer at the Naval Institute and later at the Air Force Academy at Caserta. Let us return to discuss the career of Ernesto Pascal at Pavia.

It was during his seven years at Pavia that Pascal began some projects which he continued throughout his life. He published, in lithographed form, the first course he gave at Pavia, namely *Lezioni di calcolo infinitesimale dettate nella R. Università di Pavia nell'anno 1890-91* Ⓣ. In 1895 he published the two-volume work *Lezioni di calcolo inf“nitesimale* Ⓣ and the book of exercises *Esercizi e note critiche di calcolo infinitesimale* Ⓣ. He published *Teoria delle funzioni ellittiche* Ⓣ in 1896 and, in the same year, his lithographed notes *Introduzione alla teoria della trasformazione delle funzioni ellittiche, Lezioni dettate nella R. Università di Pavia* Ⓣ. He published *Calcolo delle variazioni e calcolo delle differenze finite* Ⓣ in 1897 and, two years later, a German translation appeared. James K Whittemore, of Harvard University, reviewing the German translation, writes:-

Pascal's production of texts, while at the same time producing a stream of high quality research papers, is quite remarkable. We certainly cannot continue to mention all his texts as this list would be far too long. Let us note, however, that in 1897 he began publication of hisThe German translation of the "Calcolo delle variazioni" published by Ernesto Pascal in1897gives to American mathematicians in convenient form the best book on the calculus of variations that has, to our knowledge, appeared up to the present time. The book consists of only150octavo pages and presents concisely the principal facts of the subject. A valuable feature of the work will certainly be found to be the very excellent and apparently complete bibliography given in connection with brief accounts of the development of the calculus of variations. No book with which we are acquainted could be better adapted to controvert the lay opinion that everything in mathematics is exact and beyond dispute. Again and again the author calls attention to errors made by writers in this field. He often calls attention, too, to gaps which remain still to be filled in the theory, and makes the reader sometimes feel that the results which we already have rest on a rather precarious foundation. The chief fault of the book, from our point of view, is that it sacrifices simple and natural discussion to the pursuit of the end so dear to Italian mathematicians, the greatest possible generality. The apparent purpose of the author is to give an account, absolutely rigorous as far as it goes, of the present condition of the science. That such an end is in the calculus of variations especially difficult to attain appears from the fact that the proofs are not always precise and that the author prefers often to tell us that the work given is not rigorous rather than to attempt to make it so.

*Repertorio di matematiche superiori*Ⓣ series of texts. The first of these,

*Repertorio di matematiche superiori. I*(

*Analisi*) Ⓣ, was reviewed by Edgar Odell Lovett (1871-1957), an assistant professor at Princeton, who writes:-

This series was translated into several languages, including German, and reviews of some of the Italian texts and German translations can be read here:In no subject is special specialization growing more imperative than in mathematics; in the midst of difficulty and demand the student should hail with delight the valuable services of a work so admirably adapted to purposes of orientation as Professor Pascal's repertorium promises to be. The author's plan, adhered to without deviation, is to present with regard to each theory of modern mathematics the fundamental definitions and notions, the characteristic necessary theorems and formulae, and citations to the principal works of its bibliography. The definitions are clear and unequivocal; the statements of the theorems, always given without demonstration, are concise and unambiguous; and the bibliographical references are sufficiently full to supply the needs of the general mathematical reader.

Reviews of Ernesto Pascal's books are at THIS LINK

Other texts by Pascal appeared in the Manuale Hoepli series. As an example of one of the many works in this series we quote from the review by Edwin Bailey Elliott of Pascal's* I Gruppi Continui di Trasformazioni* Ⓣ (1903):-

Pascal left Pavia in 1907 when he was appointed to the chair of Complementary Algebra and Higher Analysis at the University of Naples. He continued to fill this chair until his death in 1940. Let us look briefly at his views on the Italian university system which he put forward in two speeches, the first his inaugural address in Pavia and the second given in Naples in 1912. In his Pavia address he criticised the appointment system which too often led to favouring a relative or friend or friend of a relative. Too often, he argued, professors tried to increase the size of their department simply to enhance their reputation rather than for scientific or educational reasons. He also criticised those judging competitions for chairs because they tended to be older academics who often tried to perpetuate their own, perhaps outdated, areas rather than make appointments of those with newer forward-looking ideas. He made similar criticisms of the educational system in his Naples address. He argued forcibly that although great progress had been made in research, the role of teaching within the universities had not received enough attention. Appointments to science chairs, he argued, were made entirely on the grounds of research ability and the teaching ability of the candidates was not considered. As a consequence, teaching had become too specialised with the result that students were put off science. He blamed different factors for problems within the universities such as lack of discipline, disregard of the principle of authority and, perhaps most significantly, that the country had lost confidence in the educational schools and universities.The little book on Lie's theory which is before us deserves a hearty welcome. For a short time longer there is still no English book on the subject. Let those of us who know a little Italian peruse the present manual. It is all the easier to start upon because there is not room in it for the dignified style and the almost wearisome elaboration of the greater works brought out under Lie's own auspices. Few authors know so well as Sig. Pascal how to present higher mathematics in didactic form. The range of his mathematical learning is moreover cyclopedic. The rate at which one useful and up-to-date Manuale Hoepli from his pen follows another is remarkable. Signs of haste in production, though not entirely absent, are rare. The 'Manuali Hoepli' are books of size for the pocket. Two pages would go on one of an ordinary octavo. The type, which is beautifully clear, is almost extravagantly large. The purchaser for half-a-crown of the present volume might well be forgiven for expecting only a meagre sketch of first principles, and not much of the analysis, perforce abounding in triple suffixes, etc., which is formidable of aspect even on the ample page of Teubner's Lie-Engel. He will be agreeably disappointed. The work is not unambitious. Its aim is "Without lack of rigour and generality to contain in little space all that forms the basis of this advanced part of pure mathematics." The author has at any rate succeeded in making clear in their complete forms the principles and processes of the general theory. Special theories, and in particular the whole subject of contact transformations, are reserved for a promised further volume.

We get an idea of his attitude towards teaching from the above, but we get further insight into his ideas about teaching engineers by quoting from the Preface of *Lezioni di Calcolo Infinitésimale* Ⓣ (1917):-

Of course, during his years in Naples, the political scene in Italy changed markedly with the rise of Fascism in the 1920s. Pascal was strongly opposed to Fascist ideas although his colleagues described him as arrogant and authoritarian. The death of his son in 1918 had a profound impact on him and it would be fair to say that he was so badly shaken that he was never quite the same man after this tragedy. His most famous student at Naples was Renato Caccioppoli who graduated in 1925 having written a thesis guided by Pascal. We should mention one further aspect of Pascal's work which was his interest in mathematical instruments. He developed the intergraph, an instrument for the mechanical integration of differential equations. He first introduced this in a paperIt is certain that through the profound changes which the critical spirit has made in the foundations of the calculus, even a course intended for those for whom mathematics is a means rather than an aim, cannot but use the new results which have been reached . . . it would therefore exhibit a shortsighted view and little esteem for the ability of the future engineer, to believe that it would be sufficient for them, at least if they can, to learn to operate the calculus in about the way in which a workman knows how to operate a machine made by others, and of which he does not know the inner connections.

*I miei integrafi per equazioni differenziali*Ⓣ in 1913 but he surveyed his contributions in the paper

*Sull'integrazione meccanica delle equazioni differenziali, e in particolare di quella lineare di 2*Ⓣ published posthumously in 1941. Pierce W Ketchum (1903-1993), University of Illinois, writes in a review:-

^{e}ordine ausiliaria dell'altra non lineare che è fondamentale per la fisica atomicaPascal received many awards and honours for his contributions. He was elected to the Accademia dei Lincei, the Istituto Lombardo Accademia di Scienze e Lettere, and the National Academy of Sciences of Italy (the "Academy of Forty"). The Academy of XL awarded him their gold medal in 1904 and again in 1914. The Liceo Sientifico Statale Ernesto Pascal in Pompei is named after him. A road, the via Ernesto Pascal, in the city of Naples is also named for him.A discussion is given of the theory of the integraph of Abdank-Abakanowicz with various improvements and modifications which the author has made in order to enable him to solve special types of first and second order differential equations, such as the general linear first order equation, the equationy" = (y^{3}/x)^{1/2}, etc. The instruments are more compact but much less flexible than a differential analyser. The input unit for x(a two-wheeled carriage which rolls parallel to the x-axis and which carries the entire instrument), the input and output units for the dependent variables(small carriages carrying pointers and pencils, respectively, which roll on tracks attached to the main carriage), and the integrator(a friction wheel which constrains the motion of the output pencil)are retained in substantially the form used in the original integraph. The modifications consist mainly in the different types of linkages which are used to connect the integrator and the input and output units. These linkages involve curved tracks, and cams, as well as parallelograms.

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