Since his father died when Betti was very young, the boy was educated by his mother. At the University of Pisa, from which he received a degree in physical and mathematical sciences, he was a disciple of O. F. Mossotti, under whose leadership he fought in the battle of Curtatone and Montanara during the first war for Italian independence.
After having taught mathematics at a Pistoia high school, in 1865 Betti was offered a professorship at the University of Pisa; he held this post for the rest of his life. He also was rector of the university and director of the teachers college in Pisa. In addition, he was a member of Parliament in 1862 and a senator from 1884. His principal aim, however, was always pure scientific research with a noble philosophical purpose.
In 1874 Betti served for a few months as undersecretary of state for public education. He longed, however, for the academic life, solitary meditation, and discussions with close friends. Among the latter was Riemann, whom Betti had met in Göttingen in 1858, and who subsequently visited him in Pisa.
In algebra, Betti penetrated the ideas of Galois by relating them to the previous research of Ruffini and Abel. He obtained fundamental results on the solubility of algebraic equations by means of radicorational operations. It should be noted that the most important results of Galois’s theory are included—without demonstration and in a very concise form—in a letter written in 1832 by Galois to his friend Chevalier on the eve of the duel in which Galois was killed. The letter was published by Liouville in 1846. When Betti was able to demonstrate—on the basis of the theory of substitutions, which he stated anew—the necessary and sufficient conditions for the solution of any algebraic equation through radicorational operations, it was still believed in high mathematical circles that the questions related to Galois’s results were obscure and sterile. Among the papers in which Betti sought to demonstrate Galois’s statements are “Sulla risoluzione delle equazioni algebriche” (1852) and “Sopra la teorica delle sostituzioni” (1855). They constitute an essential contribution to the development from classical to abstract algebra.
Figures & Tables
In this chapter, we give a brief synopsis of each of the classic papers referred to in this collection. Where relevant, we reproduce the basic equations, recast in modern notation. Supporting works also are referred to. They are listed in the “General References” section.
Table 1 is a quick outline of the key contributions of each paper reprinted in this book.
Robert Hooke, “Potentia Restitutiva, or Spring” (Oxford, 1678)
The article by Robert Hooke, “Potentia Restitutiva, or Spring,” contains the statement of the proportional relation between stress and strain universally referred to as Hooke’s law. Although the English language has evolved somewhat since 1678, the article does not require translation. Hooke describes a variety of experiments, accompanied by illustrations, confirming the stress/strain relation over a wide range of applied loads. He emphasizes the great generality of his results.
Based on his experimental work from 1660 onward, Hooke first published his law in 1676 in the form of an anagram in Latin,
which he later revealed to be “ut tensio sic vis.” Roughly translated, this means “as the force, so is the displacement” (Love, 1911; Boyce and DiPrima, 1976).
In his treatise, Hooke examined the behavior of springs, so his first casting of the equations dealt with the restoring force on a spring, for a given displacement: