Jakob Steiner

Swiss mathematician

• Born: March 18, 1796
• Birthplace: Utzentorf, Canton of Bern, Switzerland
• Died: April 1, 1863
• Place of death: Bern, Switzerland

One of the greatest geometers of the first half of the nineteenth century, Steiner wrote books and dozens of articles on geometry that established him as a chief authority on isoperimetric geometry and as the founder of modern synthetic geometry in Germany.

Early Life

Jakob Steiner (SHTI-ner) was born into a family of thrifty, humble, and hardworking Swiss farmers. Though the youngest of five children, he contributed from a very early age to the family income, the family expecting nothing more than the most modest intellectual development. Consequently, he remained illiterate until he was fourteen and continued farm work until he was nineteen. According to his later recollections, before he had any formal education he developed an astounding capacity for spatial conceptualization.

Contrary to the desires of his father, Jakob entered the school of the Swiss educational reformer Johann Pestalozzi at Yverdon. Out of conformity with Swiss educational precepts, Pestalozzi continued stressing the pedagogical importance of individual training and direct experience for his students. Before Pestalozzi’s institution failed, Steiner had become a teaching assistant. Thereafter, Steiner entered Heidelberg University, where he pursued numerical perceptions in connection with imaginative spatial concepts. From 1818 until 1821, while earning a living as a teacher, Steiner worked with one of the institution’s leading geometers, whose lectures and ideas he profoundly disdained. Notwithstanding, Steiner obtained his doctorate from Heidelberg, thereafter accepting a teaching position as a tutor at a private school.

The eldest son of the famed German statesman and philologist Wilhelm von Humboldt was one of Steiner’s pupils. Steiner’s acquaintance with the distinguished Humboldt family altered his fortunes. The Humboldts introduced him to Berlin’s premier mathematicians, and Steiner was encouraged to accept a teaching post at a Berlin vocational institution during the next decade. Eventually the University of Berlin created an endowed chair, which Steiner was to fill—indeed, he had, since 1834, been a member of the Berlin Academy on the basis of his previous mathematical, or geometrical, writings.

Life’s Work

Steiner’s mathematical publications commenced in 1826, while he still tutored at his vocational school. This creative production coincided with the founding by August Leopold Crelle of what became one of the nineteenth century’s most famous mathematical publications, Journal für die reine und angewandte Mathematik (the journal for pure and applied mathematics). Professionally, Steiner expanded his reputation in 1832 with his Systematische Entwicklung der Abhängigkeit Geometrischer Gestalten (systematic evolution of the mutual dependence of geometrical forms), a planned introduction to a five-part series never to be completed.

Steiner’s work does not readily reduce to layman’s terms. It is projective geometry, built upon synthetic constructions. Geometry’s basic forms are based on planes. Projective geometry moves from the fundamental plane to lines, planar pencils of lines to pencils of planes, bundles of lines, bundles of planes—and then into space itself, steadily generating higher geometric forms. For Steiner, one form in this projective hierarchy related with the others.

It was not the originality of Steiner’s work that was dominant, although the questions he raised were then novel considering geometers’ principal preoccupations. Steiner’s own view was that “the writings of the present day have tried to reveal the organism by which the sundry phenomena of the external world are bound to one another.” What he sought to determine was how “order enters into chaos,” how all parts of the external world fit naturally into one another, and how related parts join to form well-defined groups. Specifically, it was the brilliantly stated and systematic treatment Steiner lent to his inquiries that gained for him his reputation.

The unique and justly famed French École Polytechnique, with its unparalleled training of France’s intellectual elite and special concentration of intensive mathematical training, had long before Steiner’s day divided geometry into two branches: the analytical and the synthetic, or projective. During the early seventeenth century, René Descartes had explained how numbers could be utilized to describe points in a plane or in space algebraically. Steiner, however, concentrated on the other branch: projective geometry, which did not usually resort to the measurements or lengths of angles.

Steiner learned something from Johann Pestalozzi and his eccentric preoccupation with right triangles, and as a pedagogue Steiner, like Pestalozzi, encouraged his students’ independent and rigorously logical search for learning. As might be expected, Steiner avoided figures to illustrate his lectures. His own intuitions were so much a part of his character, he sought both in teaching and writing to use them. He did not neglect his own disciplined scholarship. He read exhaustively the works of his European counterparts, staying on the cutting edge of his investigations.

Mathematical authorities agree that in midcareer Steiner still fell short of his goals by rejecting the achievements of some of his predecessors and contemporaries. For example, he lost the chance to employ signs drawn from Karl August Möbius’s synthetic geometry and therefore the opportunity for the full deployment of his imagination. It is small wonder that Steiner sometimes wrote of “the shadow land of geometry.”

Steiner’s practical ambitions, related to, but lying near or on the margin of his geometrical scholarship, were not as shadowy. Perhaps this was understandable, for in class-conscious Berlin and the German academic world, his social origins were not advantageous. His special professorship or chair created for him at Berlin University was partly an effort to avoid this implicit embarrassment. Moreover, the timing of his publications was partly calculated to advance him toward the directorship of Berlin’s planned Polytechnic Institute. Hence, in 1833 he published a short work, Die geometrischen Konstructionen, ausgeführt mittelst der geraden Linie und eines festen Kreises (Geometrical Constructions with a Ruler, Given a Fixed Circle with Its Center , 1950), which was intended for high schools and for practical purposes. Indeed, following his appointment to his Berlin chair, he never completed what he promised would be a comprehensive work.

Steiner apparently was not surprised when analytical geometricians discovered that his own results could often be verified analytically. It was not so much that Steiner disdained others’ analyses. Rather, he was headed in a different direction of inquiry, and he believed that analysis prevented geometricians from seeing things as they actually are. Like other projective geometers, he thought that because projective geometry could advance so swiftly from a few fundamental concepts to significant statements, he, like them, eschewed the formidable axiomatic studies that were the hallmark of Euclidean geometry. Most mathematicians argued against him, however, that despite Steiner’s disclaimers, there was no royal road to a new geometry. No matter how logical, clear, and intuitive Steiner’s projective geometry was, most geometricians actually wanted to see—metrically and analytically—what the projectivists were describing. Geometry was, for its nineteenth century scholars, simply too full of irrationals to make its results completely tenable.

Significance

Jakob Steiner was not the originator of projective or synthetic geometry. Nevertheless, his contributions were substantial and significant in the revival and advancement of synthetic geometry. This was the result of his clearly presented intuitions and his marvelous systematization of his projections. Before the close of his career, moreover, he had both trained others through the clarity of his lectures and writings and encouraged other geometricians such as Julius Plücker, Karl Weierstrass, and Karl von Staudt to resolve problems that had eluded or defeated him. These geometricians, through their own citations and references to his work, spread his name further throughout the European mathematical community. In addition, Steiner left a substantial body of published works.

By the 1850’s, his health declined and the eccentricities of an always contentious character increased. He journeyed from spa to spa seeking the rejuvenation of his health. He died on April 1, 1863, at such a spa in Bern, Switzerland. However, his repute and the respect of geometricians for his revitalization of synthetic geometry and its conundrums outlasted him.

Bibliography

Courant, Richard, and Herbert Robbins. What Is Mathematics? An Elementary Approach to Ideas and Methods. Revised by Ian Stewart. 2d ed. New York: Oxford University Press, 1996. This overview of mathematical history, aimed at readers with some knowledge of the subject, contains information about Steiner’s geometric constructions and “Steiner’s problem.”

Klein, Felix. Development of Mathematics in the Nineteenth Century. Translated by M. Ackerman. Brookline, Mass.: Math-Sci Press, 1979. Klein’s work is indispensable, as little of Steiner’s writing has been translated into English. Filled with technical mathematical signs, symbols, and equations, it nevertheless contains much that is understandable to lay readers. His expositions include biographical material on all mathematicians treated, with Steiner prominently among them, as well as good contextual explanations of their objectives, problems, and results. Contains ample illustrations.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. A layperson’s survey, which largely ignores Steiner but places his work in a broad comprehensible framework. Contains illustrations, a good select bibliography, and an index.

Newman, James R., ed. The World of Mathematics: A Small Library of the Literature of Mathematics from A’h-mosé, the Scribe, to Albert Einstein. 4 vols. New York: Simon & Schuster, 1956. Volume 2 of this work is pertinent to Steiner’s context and to defining aspects of his work. Illustrations help nonspecialists appreciate the nature of some synthetic, isoperimetric geometrical problems and their attempted solutions. A fine explication of certain projective geometrical investigations. There are bibliographical citations scattered throughout and a select bibliography and usable index at the end of the second volume.

Porter, Thomas Isaac. “A History of the Classical Isoperimetric Problem.” In Contributions to the Calculus of Variations, 1931-1932: Theses Submitted to the Department of Mathematics of the University of Chicago. Chicago: University of Chicago Press, 1933. Rather than a raw thesis, this essay is an excellent survey of the synthetic geometrical problems Steiner, among others, tackled. Illustrated and readily understandable for those lacking special math training. Includes a substantial, if somewhat dated, bibliography.

Torretti, Roberto. Philosophy of Geometry from Riemann to Poincaré. Boston: Reidel, 1978. This important study is critical for a sound understanding by specialists as well as nonspecialists of a creative period in the development of both German and French mathematics, once again placing Steiner in a somewhat different historical context from that of the works cited above. It has some illustrations, a select bibliography, and an index.