Jakob Steiner | Biography

Jakob Steiner, unschooled until age 18, was one of the founders of and greatest contributors to the field of projective or modern geometry. Building on the work of such greats as Gerard Desargues and Gaspard Monge, the Swiss geometer discovered both the Steiner surface and the Steiner theorem, and extrapolated the work of the French geometer, Jean Poncelet, to develop the Poncelet-Steiner theorem, all of which helped to build the developing field of synthetic or projective geometry. A firm believer in the power of geometry, he pronounced that the calculations involved in algebra and analysis simply replaced real thinking, while geometry stimulated it. Made wealthy by his lectures on geometry, at his death Steiner bequeathed a third of his fortune to the Berlin Academy to establish the Steiner Prize.

Steiner was born on March 18, 1796, in the village of Utzensdorf, near Bern, Switzerland. The last of eight children of Niklaus Steiner and Anna Barbara Weber, Steiner grew up without formal schooling, working instead on the family farm and business where his natural skills with numbers was greatly needed. He did not learn to write until age 14, and against his parents' will, he finally left home four years later to attend a school in Yverdon run by Johann Heinrich Pestalozzi, the Swiss educational reformer whose theories laid the foundation for modern elementary education. The Pestalozzi method reacts to the highly individual needs of each learner and scorns rote memorization in favor of concrete experience and critical thinking. It was a method Steiner took to eagerly, so eagerly that within a year-and-a-half of studying with Pestalozzi, Steiner himself was teaching mathematics. His studies and teaching experience at Yverdon left a lasting legacy with Steiner: the desire, as a researcher, to discover a scientific unity in his chosen field of mathematics, and the ability, as a teacher, to encourage independent thinking on the part of his students.

Establishes a Career in Berlin

Steiner left Yverdon, Switzerland, in 1818 for Heidelberg, Germany, where he studied at the university. To pay his way as a student, Steiner gave private lessons in mathematics. At the university, he attended lectures by Ferdinand Schweins in combinatorial analysis, and also studied algebra as well as differential and integral calculus. Early papers from 1821, 1824, and 1825 demonstrate the influence of his studies in Heidelberg. At the suggestion of a friend, however, Steiner decided to leave Heidelberg for the Prussian capital of Berlin in 1821.

His lack of education and stubborn will played against Steiner initially in Berlin. To receive a teaching license, he was forced to take examinations in various subjects. Steiner was not totally successful, receiving a partial license in mathematics and employment for a short time at a gymnasium, or high school. He gave private instruction to make a full living, and attended the University of Berlin from 1822 to 1824. Thereafter, he taught at a technical school in Berlin until 1834, when he was appointed a professor at the University of Berlin. He held this post until his death 29 years later.

A blunt, somewhat corrosive personality, Steiner was known for both the startling nature of his lectures and the originality of his research. The important phase of the latter began with an 1826 publication of "Einige geometrische Betrachtungen" in the new Journal fur die reine and angewandte Mathematik, founded by his friend, August Crelle. In total, Steiner contributed 62 articles to that journal. Those articles, in addition to his longer works, Systematische Entwicklung der abhangigkeit geometrischer Gestalten voneinander of 1832, and the posthumously published Vorlesungen uber synthetische Geometrie and Allgemeine Theorie uber das Beruhren und Schneiden der Kreise und der Kugeln, together dealt with Steiner's passion for reforming geometry and also made public the basics of what has come to be called projective geometry. Steiner's articles and books have been gathered in his Gesammelte Werke.

Steiner Unifies Geometrical Theory

It was Steiner's great dream to discover the commonality between seemingly unrelated theorems, providing a simple and logical way of deducing such theorems. As he noted in his Systematische Entwicklung, "Here the main thing is neither the synthetic nor the analytic method, but the discovery of the mutual dependence of the figures and of the way in which their properties are carried over from the simplest to the more complex ones." All of Steiner's work draws from one basic principle, or so contended F. Gonseth in the foreword to Steiner's posthumously published Allgemeine Theorie. That principle is the stereographic projection of the plane onto the sphere.

Steiner noted in Systematische Entwicklung the principle of duality in projective geometry is a principle normally considered a property of algebra. But Steiner showed that in projective geometry that if two operations are interchangeable or dual, then whatever results are true for one are also true for the other. He contributed a plethora of original concepts that created the scaffolding of projective geometry. Among others, he discovered the Steiner surface, containing an infinity of conic sections; the Steiner theorem, proving that the intersection points of corresponding lines of two projective pencils (sets of geometric objects) of lines form a conic section. He also built on the work of Poncelet, in his Poncelet-Steiner theorem which shows that just one given circle with its center and straight edge are needed for any Euclidian construction. His investigation of conic sections and surfaces is at the very heart of modern geometry.

Steiner the Man

Steiner never married. Over the course of a long career, many honors came his way. He received an honorary doctorate from the University of Königsberg in 1833 and was elected a member of the Prussian Academy of Science in 1834. He also became a corresponding member of the French Académie Royale des Sciences after a winter of lecturing in Paris in 1844 and 1845, and held membership in the Accademia dei Lincei as well. In the final decade of his life, a kidney ailment limited his lecturing to the winter months only, and he died in 1863, revered as both a mathematician and educator.

Over the years, Steiner was able to accumulate a relatively large estate. His lecturing proved popular and most of his savings came from this source. In addition to the bequest he left to the Berlin Academy for the establishment of a mathematics prize in his name, and to the large chunk of money left to his relatives, Steiner also left money to the school in his native village of Utzensdorf to establish prizes for students adept at mathematics. To his death, he never forgot the value of education, nor the bitter memory of the lack of it in his own youth.