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Luigi Cremona and His Transformations WDS'08 Proceedings of Contributed Papers, Part I, 32–37, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS Luigi Cremona and his Transformations D. Trkovsk´a Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. The present paper is dedicated to Luigi Cremona (1830–1903), one of the leading Italian mathematicians of the second half of the 19th century. At first, it gives a brief account of his professional life. In the following text, we describe fundamental results of the theory of birational transformations, later known as Cremona transformations. At the end, we discuss Cremona’s influence on the Czech geometrical school, with main attention to Emil Weyr (1848–1894) who spent some time in Italy attending Cremona’s lectures. References to selected papers relevant to this theme are attached. Introduction Luigi Cremona had a great influence on Italian geometry, he was one of the founders of the new Italian mathematical school. He was mainly interested in projective and algebraic geometry and discovered graphical methods for solving problems in statics as well. Many propositions on synthetic geometry were revised and improved by him. Birational transformations of a projective plane or space named after Cremona play an important role in algebraic geometry. Professional Life of Luigi Cremona Luigi Cremona was born on December 7, 1830 in Pavia, Lombardy (today Italy). Having finished his study at the grammar school in Pavia, he was made by political events of the revolution of 1848 to interrupt his further studies. As an Italian nationalist he joined the Free Italy battalion and took part in the defence of Venice against the Austrian army. Although Cremona with his troops had to surrender in August 1849, their bravery had been such that the Austrian attackers allowed them to leave the city with an honour. Then, Cremona returned back to Pavia and entered the University of Pavia to study for a degree in civil engineering. During that time he was mostly influenced by his teacher Francesco Brioschi (1824–1897). By that time Camillo Benso di Cavour (1810–1861) had become the head of the government of Piemont and the unification of Italy had begun. However, the region of Lombardy was still controlled by the Austrian army. Cremona, who had been fighting against the Austrian occupation, could not obtain an official teaching post and therefore he served at first as a private teacher in several families in Pavia. At that time Cremona was also engaged in mathematical research; his first mathematical paper was published in March 1855. It had a great effect on his future because it helped him to gain a permission to teach physics on a temporary basis at the grammar school in Pavia where he had himself been educated. In May 1856, his second mathematical paper appeared and this, together with his great reputation as a teacher, helped him to gain the position of an associate professor in December 1856. Consequently, in January 1857, Cremona was appointed as a full professor at the grammar school in Pavia. He stayed there for three years and during that time he wrote a number of mathematical papers on curves using methods of projective geometry. In 1859, Italy in the alliance with France got free from the Austrian rule, Lombardy was liberated and Cremona should no longer be held back for political reasons. On November 28, 1859 he began to teach at the Lyceum of Saint Alexander in Milan and on June 10, 1860 he was appointed by the Royal Decree as an ordinary professor at the University of Bologna and served there until October 1867. During that time he published about 45 mathematical papers including his most important work on transformations of plane curves which won him 32 TRKOVSKA:´ LUIGI CREMONA AND HIS TRANSFORMATIONS the Steiner Prize1 in 1864. Also while at the University of Bologna, Cremona developed the theory of birational transformations, later known as Cremona transformations. In October 1867, on Brioschi’s recommendation, Cremona was appointed by another Royal Decree as a teacher to the Polytechnical Institute of Milan where he received the Professor title in 1872. This period is thought to be the time of Cremona’s gratest productivity. He wrote many articles on such diverse topics as conic sections, plane curves, developable surfaces, third- and fourth-degree surfaces, projective geometry and also on graphical statics. In 1873, Cremona was offered a political post of general secretary of the new Italian govern- ment. Although he was as an Italian patriot very pleased with this offer, he had to refuse it because of his serious engagement in mathematical research. Instead, on October 9, 1873, he moved to Rome where he was appointed by another Royal Decree as a director and professor of graphical statics of the newly established Polytechnical School of Engineering. However, if he had hoped that refusing of the political post would leave him able to continue his mathematical research, he soon found out that his administration and teaching duties made him very busy and he had no time for his scientific activities. In November 1877, Cremona was appointed to the Chair of higher mathematics at the University of Rome. However, political pressures made him feel that he should serve the new Italian State. On March 16, 1879 Cremona was elected as a Senator and at this point his mathematical work ended completely. In the next years he became the Minister of Public Education and ended his political career as the Vice-President of the Senate. Luigi Cremona died of a heart attack on June 10, 1903 in Rome. Figure 1. Luigi Cremona with his signature. In mathematics, Luigi Cremona was interested in projective and algebraic geometry, graphi- cal statics, differential and integral calculus and application of algebraic methods to geometry. During his professional life he published more than one hundred mathematical papers many of which have been translated into other languages. Amongst many other honours, Luigi Cremona was a corresponding member of the Royal Society of London since 1879, and a member of the Royal Society of Edinburgh since 1883. 1Jacob Steiner (1796–1863), a Swiss mathematician who was interested in synthetic and projective geometry. 33 TRKOVSKA:´ LUIGI CREMONA AND HIS TRANSFORMATIONS He was also a member of academies and scientific associations in Bologna, Napoli, G¨ottingen, Lisbon, Venice and Prague where he became the first honorary foreign member of the Union of Czech Mathematicians in 1871. After his death one of the Moon craters was named Cremona. Cremona Transformations In the second half of the 19th century, mathematicians all over the world paid attention to the study of geometric transformations. They especially drew their attention to higher degree transformations, the main stress was laid upon birational transformations. A birational transformation of a projective plane is a transformation of the form x′ = φ(x, y), y′ = ψ(x, y), where φ and ψ are rational functions of non-homogeneous coordinates x and y which can also be expressed rationally in terms of coordinates x′ and y′. In homogeneous coordinates x0, x1 and x2 every birational transformation can be expressed in the form x′i = Fi(x0, x1, x2) for i = 0, 1, 2; inverse transformations are of the form xi = Gi(x0′ , x1′ , x2′ ) for i = 0, 1, 2, where Fi and Gi are homogeneous polynomials of the corresponding variables. A Cremona transformation is subsequently defined as any birational transformation of a projective space over a field K. Cremona transformations of the given space form a group which is called the Cremona group. The simplest example of Cremona transformations is circular inversion of a plane which is defined as a transformation under which corresponding points X and X′ are collinear with the fixed centre S of inversion and the product of the distances SX and SX′ is equal to possitive constant r2, where r is the radius of the given circle. If we set up a coordinate system at the centre S of the circle, this transformation may be expressed analytically in the form r2x r2y x = , y = . ′ x2 + y2 ′ x2 + y2 Circular inversion carries a straight line or circle again into a straight line or circle. It was the first non-trivial geometric transformation which was studied in the context of birational transformations. Another examples of Cremona transformations are quadratic birational transformations of a plane. In non-homogeneous coordinates x, y they may be expressed as linear-fractional functions a1x + b1y + c1 a3x + b3y + c3 x′ = , y′ = , a2x + b2y + c2 a4x + b4y + c4 where ai, bi and ci are real coefficients for i = 1, 2, 3, 4. Among these transformations, special attention is paid to the standard quadratic transformation 1 1 x = , y = , ′ x ′ y or, in homogeneous coordinates, x0′ = x1x2, x1′ = x0x2, x2′ = x0x1. 34 TRKOVSKA:´ LUIGI CREMONA AND HIS TRANSFORMATIONS Both circular inversion and quadratic birational transformations are transformations of the second degree.2 Cremona was in his work Sulle trasformazioni geometriche delle figure piane [Geometric Transformations of Plane Figures] from 1863 and 1865 engaged in the question on how to construct general geometric transformation of an arbitrary degree, i.e. how to construct a transformation which carries straight lines into curves of an arbitrary degree. In his work Cremona derived two fundamental equations which must be held for the number of points common to all those curves. Consider two geometric figures, first in a plane P and the second one in a plane P ′ and any one-to-one transformation between them, i.e. any transformation that carries every point of the figure P into just one point of the figure P ′ and conversely. The question of principle is which curves in the second figure correspond under the transformation to straight lines in the first figure.
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