Book Review The Volterra Chronicles Reviewed by Irwin W. Sandberg The Volterra Chronicles: The Life and Times of teens, advocated a career in a more practical area an Extraordinary Mathematician 1860–1940 such as railroad engineering or land surveying. Judith R. Goodstein Alfonso was an official at the Bank of Italy with AMS, 2007 a modest income that could not support Vito’s 310 pages, US$59.00, ISBN: 978-0821839690 aspirations. Goodstein tells us that in a letter from Alfonso to his cousin Edoardo Almagià, Alfonso This book by Judith R. Goodstein, a Caltech wrote “I make Vito come to the bank every day archivist, is an imposing tribute to Vito Volterra, where I give him some work to do [in which he an exceptional mathematician, physicist, educa- showed no interest, his uncle made clear]…God tor, statesman, and man of uncommon integrity. grant that he might overcome his disgust for It is also an interesting, carefully written, richly anything that takes him away from his beloved detailed history of his personal life and the times in which he lived. Much of the material was drawn science…[T]hat damned passion for the study of from Volterra’s personal correspondence and from pure mathematics discourages me.” interviews with his family members. The book As we know, Vito prevailed—this with the includes translations of actual correspondence, ex- encouragement and assistance of Edoardo and amples of which are given below, as well as several others, including the physicist Antonio Ròiti in photos that help to round out the biography. whom Vito found a scientific mentor and a per- Samuel Giuseppe Vito Volterra was born into a son who shared his perspective on mathematics. Jewish family in Ancona, Italy’s Jewish ghetto in Volterra’s geometry instructor at the Instituto 1860—during the period of the Italian unification Tecnico, the technical high school in Florence, and the year of the liberation of the Italian ghet- was Cesare Arzelà during the years before Arzelà tos. His family was of modest financial means, began his ascent on the academic ladder. During a and he had the misfortune that his father Abramo class meeting, young Vito solved—on the spot—a Volterra died when he was about two years old. problem put on the blackboard by an unannounced For these reasons, he and his mother Angelica inspector from the Ministry of Public Instruction, lived for most of Vito’s youth in the home of her after Arzelà, to whom the problem was addressed, brother Alfonso Almagià, first in Turin and later appeared to be confused. in Florence. Inspired by books he had read, Vito Volterra was eighteen when he left Florence to became interested in mathematics and science become a freshman at the University of Pisa. He in his preadolescent years, to the dismay of both took algebra from Cesare Finzi, a fine instructor his mother and his uncle who, during Vito’s early and a family friend. His calculus class was taught Irwin W. Sandberg is Cockrell Family Regents Chair by the brilliant and accomplished Ulisse Dini. In a Emeritus in Engineering at the University of Texas. His letter to Vito’s mother, who had asked (earnestly) email address is [email protected]. that her son write frequently, one finds: MARCH 2008 NOTICES OF THE AMS 377 Uncle asks how I like the lectures by natural and uncomplicated one. After it was intro- Finzi. They are without doubt very duced by Volterra in his 1887 paper “On functions beautiful lectures; he is a teacher who that depend on other functions,” and developed explains things with order and clarity, further by him during the succeeding three years, he never gets confused, he is suffi- it was rightfully viewed by Jacques Hadamard and ciently precise, speaks well and is also others as a highly creative idea. In a classic 1945 good and kind; one really could not ask essay, Hadamard wrote for a better person to teach algebra. But …[B]ut that “functionals” as we called I always prefer Dini, who gives lectures the new conception, could be in di- that are a bit confused, who some days rect relation with reality could not be explains and explains without reaching thought of otherwise than as mere any conclusions and at other times is absurdity. Functionals seemed to be a bit obscure. I prefer him because he an essentially and completely abstract puts his whole soul into his lectures, be- creation of mathematicians. cause the things he explains are almost always his own discoveries—at least the Now, precisely the absurd has hap- method is really all his own. He speaks pened. Hardly intelligible and con- simply and gets a little confused but in ceivable as it seems, in the ideas of the end his concepts are infinitely more contemporary physicists (in the re- clear, more concise, and, what is more cent theory of “wave mechanics”) the important, more exact, than those of new notion, the treatment of which Finzi. Dini’s lectures are always inter- is accessible only to students already esting. (p. 39) familiar with very advanced calculus, It is not surprising that it was Dini who inspired is absolutely necessary for the math- Volterra’s interest in pure mathematics. But Enrico ematical representation of any physical Betti, a highly respected professor of mathemati- phenomenon…(p. 70) cal physics and the director of the Pisa school of Today, functional analysis, as developed much mathematics, had a strong interest in applications. further by Banach, Fréchet, Hahn, and others, It was he who served as Volterra’s dissertation plays an important role in the theory of stability advisor and who provided the inspiration for Volt- of nonlinear automatic control systems, as well as erra to spend the rest of his career working, in an in other areas of engineering and science such as academic environment, on both pure mathematics system representation theory. In the stability stud- and applications in the area of physics. Volterra ies one encounters, for example, Volterra integral was twenty-two years old when he graduated from equations of the form the University of Pisa with a doctorate—not in mathematics, but in physics. By that time he had published five papers. Just one year later, in 1883 and after a competition with six other applicants, he became an associate professor at the University in which u​ and v​ are the system input and output, of Pisa—and held its Chair of Rational Mechanics. respectively, k is associated with the linear part of His promotion to full professor occurred in 1887, the system, and η represents the nonlinear part. the year in which he was awarded a gold medal for While a faculty member at Pisa, Volterra served mathematics from the Italian Society of Sciences. as an external examiner at a technical institute This medal was the first of his many awards and in Sicily where a student, upset by his failure to honors that later included membership in Italy’s pass an exam proctored by Volterra, fired a pistol prestigious Accademia dei Lincei (which translates at him. Fortunately, the student’s marksmanship to “Academy of Lynxes”, a name chosen by the was no better than his performance on the exam: academy’s founders in 1603 probably because the he missed. An interesting thread woven into mythological lynx possessed piercing eyesight). Goodstein’s lively and richly detailed account of Volterra initiated the powerful branch of math- Volterra’s life and times concerns the dispro- ematics that came to be called functional analy- portionately large number of Jews within Italy’s sis, and made important contributions in several mathematics community. She notes, for example, fields, including elasticity theory, the study of the that in 1895 Volterra received the Accademia dei movement of the earth’s axis of rotation, linear Lincei’s mathematics prize (for “various memoirs integral and differential equations, and later the on pure and applied analysis”); that the co-winner theory of ecology and biology. He left Pisa in 1893 Corrado Segre, the leader of the Italian school of to accept a professorship in rational mechanics at geometry, was also from a Jewish background; and the University of Turin. that, by the time of the collapse of the academy From the perspective of modern mathematics the under Mussolini in 1944, seven of the sixteen prize concept of a functional, while decidedly useful, is a winners in mathematics were Jewish. These Jewish 378 NOTICES OF THE AMS VOLUME 55, NUMBER 3 prize winners included Guido Castelnuovo and ment, where he par- Federigo Enriques (both known for their contribu- ticipated in many tions in the area of algebraic geometry), as well as of its deliberations. Guido Fubini (of Fubini’s theorem). The population In an early speech of Jews in Italy at that time was a tiny fraction of to the Italian Sen- one percent. ate he called for a When he was about forty, Volterra’s mother de- revamping of Italy’s cided that the time had come for him to marry, and advanced technical she proceeded to arrange a marriage. His bride was school system, actu- to be the much younger Virgina Almagià, his sec- ally arguing that too ond cousin and the daughter of Alfonso Almagià much mathemat- whose modest railroad construction business ics was taught. In had grown into a large international engineering his role as a politi- company. Virgina was educated, intellectually cian, he cited Alois brilliant, and a lovely young woman. Vito’s main Riedler, a German concern appears to have been that he did not want professor of me- to be thought of as one who married for money.