Jacob Steiner (1796–1863), "the greatest geometrician since
the time of Euclid," was born in Utzendorf in the Canton of Bern. He did not learn to write till he was fourteen. At eighteen he became a pupil of Pestalozzi. Later he studied at Heidelberg and Berlin. When Crelle started, in 1826, the celebrated mathematical journal bearing his name, Steiner and Abel became leading contributors. In 1832 Steiner published his Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander, "in which is uncovered the organism by which the most diverse phenomena (Erscheinungen) in the world of space are united to each other." Through the influence of Jacobi and others, the chair of geometry was
founded for him at Berlin in 1834. This position he occupied until his death, which occurred after years of bad health. In his Systematische Entwickelungen, for the first time, is the principle of duality introduced at the outset. This book and von Staudt's lay the foundation on which synthetic geometry
in its present form rests. Not only did he fairly complete the theory of curves and surfaces of the second degree, but he
made great advances in the theory of those of higher degrees. In his hands synthetic geometry made prodigious progress. New discoveries followed each other so rapidly that he often did not take time to record their demonstrations. In an article in Crelle's Journal on Allgemeine Eigenschaften Algebraischer
Curven he gives without proof theorems which were declared by Hesse to be "like Fermat's theorems, riddles to
the present and future generations." Analytical proofs of some of them have been given since by others, but Cremona finally proved them all by a synthetic method. Steiner discovered synthetically the two prominent properties of a surface of the third order; viz. that it contains twenty-seven straight lines and a pentahedron which has the double points for its vertices and the lines of the Hessian of the given surface
for its edges.55 The first property was discovered analytically somewhat earlier in England by Cayley and Salmon,
and the second by Sylvester. Steiner's work on this subject
was the starting-point of important researches by H. Schröter,
F. August, L. Cremona, and R. Sturm. Steiner made investigations
by synthetic methods on maxima and minima, and arrived at the solution of problems which at that time altogether surpassed the analytic power of the calculus of variations.
He generalised the hexagrammum mysticum and also
Malfatti's problem.59 Malfatti, in 1803, proposed the problem,
to cut three cylindrical holes out of a three-sided prism in such a way that the cylinders and the prism have the same altitude and that the volume of the cylinders be a maximum. This problem was reduced to another, now generally known as Malfatti's problem: to inscribe three circles in a triangle that each circle will be