##
**Jan Vilém Pexider 1874–1914.**
*(English)*
Zbl 0989.00008

Dějiny Matematiky / History of Mathematics. 5. Prague: Prometheus. 110 p., open access (1997).

The monograph is devoted to the life and work of Jan Vilém Pexider, a bit inconsistent personality, whose name is still alive in the field of mathematical analysis where it is connected to so-called Pexider equality, but who is almost unknown in the Czech and world mathematical community.

The book starts with the basic information on Pexider’s family and with the description of the story of his life, including his unsuccessful attempts on habilitation at Charles University in Prague caused by the fact that his works were sort of compilations with practically no original results, together with certain personal rifts. In this connection Pexider published at his own expenses the critique of the textbook of Eduard Weyr, the author of the disapproving habilitation opinion, which resulted in an unfortunate dispute (this quarrel is described in details in the book [Eduard Weyr (1852–1903) (History of Mathematics 2, Prometheus, Praha) (1995; Zbl 1076.01508)], where the modern and unprejudiced evaluation is also given. Later, in 1905, Pexider became a private docent at the university in Bern, where he lectured for three years (there are few information from the subsequent years; Pexider died in the psychiatric hospital).

The second chapter, written by Štefan Schwabik, discusses Pexider’s publications on mathematical analysis; these works were written in the period 1898–1909. The author outlines the contents of particular papers and gives their evaluation. Mainly they compiled formed results, nevertheless some of them were useful for public education, for they informed the Czech reader about current or less known topics (this concerns the discussion of the representation of numbers by lengths and vice versa inspired by Hilbert’s axiomatic foundation of geometry, and the work on Abel’s theorem) or about some historical aspects. Worthy of a mention is the work [Monatsch. f. Math. 14, 293–301 (1903; JFM 34.0420.01)], due to which Pexider’s name still appears in the mathematical literature. Here Pexider investigates functional equations with three functions \(f, \varphi, \psi\) and asks a question, which continuous functions satisfy them. The first of these equations bears Pexider’s name: \(f(z)+\varphi(u)=\psi(z+u).\) Similar equation with \(f=\varphi =\psi\) was already examined by A. L. Cauchy; Pexider showed that the more general problem mentioned above gives practically no new outcomes. Certain indications of the considerations contained in Pexider’s 1903 paper can be found in the work [Casopis 29, 153–195 (1900; JFM 31.0403.02)] written in the Czech language and examining the case of two unknown functions.

In the third chapter examines Tomáš Cipra three Pexider’s actuary works from 1905–1907. Although these publications don’t contain revolutionary ideas, they were probably very useful for the practise. The fourth chapter was written by Ivan Netuka and it is devoted to so-called Pexider equation in a wider context. Jiří Veselý, the author of the following two short chapters, describes two parts of a small experiment with the name of Pexider”: in the database of the journals Mathematical Reviews and Zentralblatt he searched for the expression ‘Pexider*” and found 60 and 65 items respectively, the oldest of which was dated in both cases in 1962. The discussion analyses the irregular and interesting spread of these citations and the comparison of the results obtained from both databases.

After the concluding reflection of J. Bečvář the supplement follows which contains the list of Pexider’s publications provided with the reviews from referential journals of that time, the reprints of three short Pexider’s papers that were published in the magazine Přehled (Survey) in 1905, and the collection of pictures (photographs and interesting documents).

The book starts with the basic information on Pexider’s family and with the description of the story of his life, including his unsuccessful attempts on habilitation at Charles University in Prague caused by the fact that his works were sort of compilations with practically no original results, together with certain personal rifts. In this connection Pexider published at his own expenses the critique of the textbook of Eduard Weyr, the author of the disapproving habilitation opinion, which resulted in an unfortunate dispute (this quarrel is described in details in the book [Eduard Weyr (1852–1903) (History of Mathematics 2, Prometheus, Praha) (1995; Zbl 1076.01508)], where the modern and unprejudiced evaluation is also given. Later, in 1905, Pexider became a private docent at the university in Bern, where he lectured for three years (there are few information from the subsequent years; Pexider died in the psychiatric hospital).

The second chapter, written by Štefan Schwabik, discusses Pexider’s publications on mathematical analysis; these works were written in the period 1898–1909. The author outlines the contents of particular papers and gives their evaluation. Mainly they compiled formed results, nevertheless some of them were useful for public education, for they informed the Czech reader about current or less known topics (this concerns the discussion of the representation of numbers by lengths and vice versa inspired by Hilbert’s axiomatic foundation of geometry, and the work on Abel’s theorem) or about some historical aspects. Worthy of a mention is the work [Monatsch. f. Math. 14, 293–301 (1903; JFM 34.0420.01)], due to which Pexider’s name still appears in the mathematical literature. Here Pexider investigates functional equations with three functions \(f, \varphi, \psi\) and asks a question, which continuous functions satisfy them. The first of these equations bears Pexider’s name: \(f(z)+\varphi(u)=\psi(z+u).\) Similar equation with \(f=\varphi =\psi\) was already examined by A. L. Cauchy; Pexider showed that the more general problem mentioned above gives practically no new outcomes. Certain indications of the considerations contained in Pexider’s 1903 paper can be found in the work [Casopis 29, 153–195 (1900; JFM 31.0403.02)] written in the Czech language and examining the case of two unknown functions.

In the third chapter examines Tomáš Cipra three Pexider’s actuary works from 1905–1907. Although these publications don’t contain revolutionary ideas, they were probably very useful for the practise. The fourth chapter was written by Ivan Netuka and it is devoted to so-called Pexider equation in a wider context. Jiří Veselý, the author of the following two short chapters, describes two parts of a small experiment with the name of Pexider”: in the database of the journals Mathematical Reviews and Zentralblatt he searched for the expression ‘Pexider*” and found 60 and 65 items respectively, the oldest of which was dated in both cases in 1962. The discussion analyses the irregular and interesting spread of these citations and the comparison of the results obtained from both databases.

After the concluding reflection of J. Bečvář the supplement follows which contains the list of Pexider’s publications provided with the reviews from referential journals of that time, the reprints of three short Pexider’s papers that were published in the magazine Přehled (Survey) in 1905, and the collection of pictures (photographs and interesting documents).

Reviewer: Magdalena Hyksova (Praha)

### MSC:

00B15 | Collections of articles of miscellaneous specific interest |

01-06 | Proceedings, conferences, collections, etc. pertaining to history and biography |