Swiss mathematician Jakob Steiner (1796-1863) made groundbreaking intellectual contributions to the field of mathematics in the area of geometry.

# Jakob Steiner

Beginning his education at the age of 18, Steiner attended universities in both Berlin and Heidelberg, then worked at a Prussian school while developing the mathematical theories that caused him to be hailed by many as the most eminent geometer since Apollonius of Perga (250-220 B.C.), whose study of conic sections earned him the title "the Great Geometer." He developed the Steiner surface and the Steiner theorem, the building blocks of what we now know as projective, or modern geometry, and ended his career as an esteemed professor of mathematics at the University of Berlin.

## Demonstrated Skill with Sums

The eighth child born to farmer Niklaus Steiner and his wife, Anna, Steiner quickly found that his apptitude for mathematical calculations was useful. Born in Utzendorf, a farming village near Bern, Switzerland, on March 18, 1796, he was the youngest child in a family where hard work won out over formal education. At the age of 14 he learned to write, and the mastery of this skill Steiner saw as a way to escape the harsh life of a farmer. Despite his parents' objections, in 1814 the 18-year-old farmer's son left home and traveled to Yverdon, where educational reformer Johann Heinrich Pestalozzi (1746-1827) had established a school.

Highly influenced by the theories of Swiss enlightenment philosopher Jean Jacques Rousseau outlined in Rousseau's 1762 novel *Émile* Pestalozzi was an adherent of the philosophy that the state of nature is the best teacher. Dedicating himself to the education of the poor, he used his school to put a number of his educational theories to the test, in 1805 opening a secondary school at Yverdon. Steiner was a willing subject. His methodology, which has since become a foundation of elementary educational theory, takes into account the unique needs and talents of each student; traditional classroom repetition and memorization are replaced by hands-on learning and the resulting development of critical-thinking skills. Steiner shared the naturalistic approach to education promoted by his teacher and flourished under Pestalozzi's tutelage. Within a year and a half he was teaching math to other students and had adopted an innovative approach to his subject. Although he left Yverdon at age 22, he continued to draw on Pestalozzi's approach. With his innovative, open-minded approach to mathematics, Steiner excelled as a researcher, and during his tenure as a university professor, he also encouraged his students to think "outside the box."

## University Career in Germany

In the fall of 1818 Steiner left his native Switzerland and moved to Germany to attend the University of Heidelberg, where he fell into the company of other mathematicians, such as Norwegian-born Niels Henrik Abel (1802-1829) and German mathematician Carl Gustav Jacobi (1804-1851), both of whom independently derived elliptical functions. His teaching experience at Yverdon was sufficient to allow him to fund his education by working as a tutor, and his days were divided between attending lectures in algebra, combinatorial analysis, and differential and integral calculus by such renowned mathematicians as Ferdinand Schweins and fulfilling his own teaching duties. Three years later, in the early spring of 1821, he followed the suggestion of a friend and moved to the Prussian capital city of Berlin in hopes of finding a teaching post and then enrolling at the city's prestigious university.

The lack of a formal, structured education came back to haunt Steiner in Berlin. In order to be allowed to teach in the cosmopolitan Prussian capital, teachers were expected to be well-rounded in their education, and examinations in a variety of subjects were required. Although his ability at math was great, Steiner was less adept at history and literature, and his stubbornness, belligerent nature, and unwillingness to work out with licensing officials a way to overcome any academic deficiencies made things worse. He was only able to earn a partial teaching license, and the best job he could find was a teaching post at a *gymnasium,* the German version of the North American high school. Although he was dismissed within a few months due to his unconventional teaching methods, the resourceful young man managed to replace this job with work as a tutor, and
by November of 1822 Steiner was enrolled at the University of Berlin.

Leaving the university in August of 1824, Steiner found a post at a Berlin technical school and spent the next decade teaching mathematics there, first as assistant master and after 1929 as senior master. Although his role as educator to younger students relegated him teaching basic mathematical concepts, he challenged his intellect outside the classroom with theoretical work and published some of his most significant findings beginning in 1826 with his most notable work, "Einige geometrische Betrachtungen," which appeared in a periodical published by his friend, A. L. Crelle, the *Journal für die reine und angewandte Mathematik* (Journal for Pure and Complex Mathematics).

## Developed Projective Geometry

During the course of his career, Steiner contributed over 60 articles to Crelle's *Journal für die reine und angewandte Mathematik* as he continued his research in geometry. These articles combine with his book-length works *Systematische Entwicklung der Abhangigkeit geometrischer Gestalten voneinander* (1832), the two-volume *Vorlesungen üuber synthetische Geometrie* (1867), and *Allgemeine Theorie über das Berühren und Schneiden der Kreise und der Kugeln* (1931) to encompass Steiner's foundational work in the area of what has since become the discipline of projective, or pure geometry.

Steiner was convinced, as Euclid had been, in what he termed the "organic unity of all the objects of mathematics": that there are interrelationships between what were then considered to be unrelated geometric theorems. He desired to uncover these interrelationships, thereby allowing mathematicians to deduce such theorems by means of logical deduction. "Here the main thing is neither the synthetic nor the analytic method," he wrote in *Systematische Entwicklung,* "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." Steiner's work developed from a single principle: the stereo-graphic projection of the plane onto the sphere.

The principle underlying Steiner's projective geometry is the principle of duality, an algebraic concept that holds that if A equals B, then what holds for A holds as well for B. Algebra is the area of mathematics that uses letters or other symbols to describe the properties and relationships among complex numbers and other abstract entities. In contrast, geometry developed from ancient man's desire to measure the earth, a goal impossible to accomplish except by abstract means. Where algebra concerns itself with numerical abstractions, geometry concerns itself with the study of the properties of constant entities, such as angles, points, lines, and one-and multi-dimensional surfaces. Steiner reasoned that what held for complex numbers should hold also for complex shapes, and through his work he proved that if two geometric operations are interchangeable or dual, then whatever results are true for one are also true for the other.

Following from his original work, Steiner derived other mathematical theorems and relationships. The Steiner ellipse is an ellipse that passes through the vertices of a triangle with points A, B, and C and also shares the centroid of those points as well as a fourth point, called the Steiner point. The Steiner surface contains an infinity of conic sections; the Steiner theorem states that the points of intersection of corresponding lines of two sets of geometric objects form a conic section. The Steiner problem relates to triangle geometry: Given a plane containing three straight lines {l, m, n} and three points {P, Q, R}, construct a triangle ABC such that the ver tices A, B, and C lie on lines l, m, n and sides BC, CA, and AB pass through points P, Q, and R.

Steiner also extrapolated the work of his colleague, French geometer Jean Poncelet (1788-1837), a military engineer and professor of mechanics at the University of Paris. Like Poncelet, Steiner believed that geometry was a tool that encouraged creative thinking while algebra merely reiterated existing numerical complexities. In an example of the ability of geometry to create new relationships, the Poncelet-Steiner theorem proves that a single given circle with its center and a straight edge are needed for any Euclidian construction.

## A Dedicated Educator

Through the efforts of Jacobi as well as other noted German intellectuals, on October 8, 1834, the 38-year-old mathematician was honored by the University of Berlin, which established a chair of geometry for him. Steiner remained an extraordinary professor at the university until his death 29 years later. He never married, but dedicated much of his adult life to his students. A blunt and somewhat coarse manner combined with surprisingly liberal social attitudes to make Steiner a unique and memorable individual, particularly to students used to seeing his impassioned lectures on mathematical subjects. He was known for both the startling nature of his lectures and the originality of his research, and influenced many of his students, including the noted mathematician Georg Friedrich Bernhard Riemann. Inspired by Steiner, Riemann (1826-1866) went on to become professor of mathematics at Göttingen and developed the system of elliptical space that Albert Einstein would one day draw on in his formulation of the Theory of Relativity.

Over the course of a long career, many honors came his way. Again through the influence of Jacobi, an honorary doctorate from the University of Königsberg was conferred upon Steiner in 1833, and the following year he was elected a fellow of the Prussian Academy of Science. He also became a corresponding member of the French Acadmie Royale des Sciences after a winter of lecturing in Paris in 1844 and 1845, and held membership in the Accademia dei Lincei as well.

A successful career as a teacher and lecturer combined with the frugal nature developed during childhood to provide Steiner with a comfortable income throughout his adult lifetime. Living conservatively, he amassed a significant estate which at his death was valued at 90,000 Swiss francs. In his will Steiner bequeathed a third of his fortune to the Berlin Academy to establish the Steiner Prize. In addition, he directed that an endowment of 750 francs be presented to the public 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.

Steiner left Germany in the final ten years of his life, with a kidney ailment confining him to his home in Switzerland for much of the year. He returned to German to lecture during the winters until he became bedridden. Steiner died at age 67 in Bern, Switzerland, on April 1, 1863, mourned by students and colleagues who revered him as both a brilliant mathematician and a dedicated and inspiring teacher. Steiner's articles and books were eventually collected and published in the two-volume *Gesammelte Werke* in 1881-1882, while many of his problems and theorems continue to be included in modern texts, providing challenges for future mathematicians.

## Books

Burckhardt, *Dictionary of Scientific Biography,* edited by Charles Coulston Gillispie, Charles Scribner's Sons, 1980.

Dörrie, Heinrich, *One Hundred Great Problems of Elementary Mathematics, Their History and Solution,* translated from the German by David Antin, Dover, 1965.

Kollros, Louis, *Jakob Steiner,* Birkhauser, 1947.

*Notable Mathematicians,* Gale, 1998.

Stark, Marion Elizabeth, and Raymond Clare Archibald, *Jacob Steiner's Geometrical Constructions with a Ruler, Given a Fixed Circle with Its Center,* Scripta Mathematica, Yeshiva University, 1970.

Steiner, Jakob, *Gesammelte Werke,* two volumes, Prussian Academy of Sciences, 1881-82, reprinted, Chelsea House, 1971.

## Online

*Jakob Steiner (1796-1863) Geometer,* http://faculty.evansville.edu/ck6/bstud/steiner.html (February 12, 2003).