Under this head we find it convenient to consider the subjects of the differential and integral calculus, the calculus of variations, infinite series, probability, and differential equations.
Prominent in the development of these subjects was Cauchy.
Augustin-Louis Cauchy[]78 (1789–1857) was born in Paris, and received his early education from his father. Lagrange and Laplace, with whom the father came in frequent contact, foretold the future greatness of the young boy. At the École Centrale du Panthéon he excelled in ancient classical studies. In 1805 he entered the Polytechnic School, and two years later the École des Ponts et Chaussées. Cauchy left for Cherbourg in 1810, in the capacity of engineer. Laplace's Mécanique Céleste and Lagrange's Fonctions Analytiques were among his book companions there. Considerations of health induced him to return to Paris after three years. Yielding to the persuasions of Lagrange and Laplace, he renounced engineering in favour of pure science. We find him next holding a professorship at the Polytechnic School. On the expulsion of Charles X., and the accession to the throne of Louis Philippe in 1830, Cauchy, being exceedingly conscientious, found himself unable to take the oath demanded of him. Being, in consequence, deprived of his positions, he went into voluntary exile. At Fribourg in Switzerland, Cauchy resumed his studies, and in 1831 was induced by the king of Piedmont to
accept the chair of mathematical physics, especially created for him at the university of Turin. In 1833 he obeyed the call of his exiled king, Charles X., to undertake the education of a grandson, the Duke of Bordeaux. This gave Cauchy an opportunity to visit various parts of Europe, and to learn how extensively his works were being read. Charles X. bestowed upon him the title of Baron. On his return to Paris in 1838, a chair in the College de France was offered to him, but the oath demanded of him prevented his acceptance. He was nominated member of the Bureau of Longitude, but declared ineligible by the ruling power. During the political events of 1848 the oath was suspended, and Cauchy at last became professor at the Polytechnic School. On the establishment of the second empire, the oath was re-instated, but Cauchy and Arago were exempt from it. Cauchy was a man of great
piety, and in two of his publications staunchly defended the Jesuits.
Cauchy was a prolific and profound mathematician. By a prompt publication of his results, and the preparation of standard text-books, he exercised a more immediate and beneficial influence upon the great mass of mathematicians than any contemporary writer. He was one of the leaders in infusing rigour into analysis. His researches extended over the field of series, of imaginaries, theory of numbers, differential equations, theory of substitutions, theory of functions, determinants, mathematical astronomy, light, elasticity, etc.,–-covering pretty much the whole realm of mathematics, pure and applied.
Encouraged by Laplace and Poisson, Cauchy published in 1821 his Cours d'Analyse de l'École Royale Polytechnique, a work of great merit. Had it been studied more diligently by writers of text-books in England and the United States, many a lax and loose method of analysis hardly as yet eradicated
from elementary text-books would have been discarded over half a century ago. Cauchy was the first to publish a rigorous proof of Taylor's theorem. He greatly improved