**Hermann Schubert**'s father was an innkeeper. Hermann was brought up in Potsdam where he attended the Gymnasium. However he moved to the Gymnasium in Spandau where he completed his secondary education. He then entered the University of Berlin where he took his first degree in 1867, then moved to the University of Halle where he was awarded a doctorate in 1870 for his thesis on enumerative geometry *Zur Theorie der Charakteristiken*. He had published two papers on enumerative geometry before submitting his doctoral dissertation, these being on the system of sixteen spheres that touch four given spheres. This work extended that of Apollonius who proved that there are eight circles that simultaneously touch three circles in the plane.

After the award of his doctorate, Schubert qualified to become a mathematics teacher and was appointed to the Andreanum Gymnasium in Hildesheim in 1872. While a teacher at this school, Schubert married Anna Hamel in 1873; they had four daughters. He remained as a teacher in the Andreanum Gymnasium for four years before moving to Hamburg when he was appointed to the Johanneum, the renowned humanistic school where Möbius had taught twenty years earlier. In 1887 he was promoted to professor at the Johanneum and he continued to teach at the school but, in 1905, his health began to deteriorate and he started to suffer from circulatory disorders. With his health slowly deteriorating, he retired in 1908 at the age of 60. The illness was a long and difficult one which slowly took away the power in his limbs. Near the end of his life he was completely paralysed.

Schubert is famed for his work on enumerative geometry which considers those parts of algebraic geometry that involves a finite number of solutions. He won the Gold Medal of the Royal Danish Academy of Sciences (Kongelige danske videnkabernes selskab) in 1874 for solving a question posed by Zeuthen on the extension of the theory of characteristics in cubic space curves.

Using methods of Chasles, with Schröder's logical calculus as a model, he set up a system to solve such problems, he called it *the principal of conservation of the number*. Burau describes the ideas behind this in [1]:-

Algebraically, the solution of the problems of enumerative geometry amounts to finding the number of solutions for certain systems of algebraic equations with finitely many solutions. Since the direct algebraic solution of the problems is possible only in the simplest cases, mathematicians sought to transform the system of equations, by continuous variation of the constants involved, into a system for which the number of solutions could be determined more easily. Poncelet devised this process, which he called the principle of continuity; in his day, of course, the method could not be elucidated in exact terms. Schubert's achievement was to combine this procedure, which he called "the principle of conservation of number", with the Chasles correspondence principle, thus establishing the foundation of a calculus. With the aid of this calculus, which he modelled on Ernst Schröder's logical calculus, Schubert was able to solve many problems systematically.

Hilbert, in his famous Paris lecture of 1900, asked for a proof (it is Problem 15), which was given by Severi in 1912. Some remarkable counting results of Schubert were neglected for many years for their lack of rigour but recently many of them have been confirmed. In fact it appears that although the principal of conservation of the number as stated by Schubert was false, and counter-examples were given by Study, nevertheless Schubert knew exactly what he was doing and his use of the principal was always in cases where it is valid.

Schubert was editor of *Sammlung Schubert*, a series of textbooks. He wrote the first in the series *Arithmetik und Algebra* and a later book in the series on analysis *Niedere Analysis*. He edited a book of tables which contains sixteen numerical tables, including Briggsian and natural logarithms, addition and subtraction logarithms, trigonometric functions and mortality tables.

Given Schubert's outstanding mathematical contributions, it is natural to ask why he remained a school teacher and never became a university professor. Certainly it was not through lack of recognition for his contributions. Rather it was because he did not wish to move from Hamburg and turned down a number of offers of university positions. Although Hamburg had no university during the time that Schubert taught in the Andreanum, nevertheless it did have the Hamburg Mathematical Society, founded in 1690, the oldest mathematical society in the world which still exists today.

In addition to his research and textbook writing, Schubert was very interested in recreational mathematics and games, in particular chess and skat. He published the first edition of his book on recreational mathematics *Mathematische Mussestunden* in 1897. A second edition, by this time expanded to three volumes, appeared in 1900 and editions continued to be produced after his death so that the thirteenth edition (revised by J Erlebach) was published in 1967. In 1903 *Mathematical Essays and Recreations* translated into English by Thomas J McCormack was published. McCormack writes in the Translator's Note:-

The essays and recreations constituting this volume are by one of the foremost mathematicians and text-book writers of Germany. The monistic construction of arithmetic, the systematic and organic development of all its consequences from a few thoroughly established principles, is quite foreign to the general run of American and English elementary text-books, and the first three essays of Professor Schubert will, therefore, from a logical and aesthetic side, be full of suggestions for elementary mathematical teachers and students, as well as for non-mathematical readers. For the actual detailed development of the system of arithmetic here sketched, we may refer the reader to Professor Schubert's volume Arithmetik und Algebra...

The remaining essays on "Magic squares", "The fourth dimension", and "The history of squaring the circle", will be found to be the most complete generally accessible accounts in English, and to have, one and all, a distinct educational and ethical lesson.

In all these essays, which are of a simple and popular character, and designed for the general public, Professor Schubert has incorporated much of his original research.

We should note that Frege was highly critical of Schubert's approach to numbers. He wrote *Über die Zahlen des Herrn H Schubert* (1899) which is a bitter satire with little or no scientific impact although it is very witty in places. Frege's article is a negative, sarcastic and a very destructive attack on Schubert's article.

**Article by:** *J J O'Connor* and *E F Robertson*

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