James T. Smith

A Contribution to the Axiomatics of Well-Ordered Sets (1921)

This chapter is devoted to an English translation of Alfred Tarski’s first published paper, Przyczynek do aksjomatyki zbioru dobrze uporządkowanego, [1921] 2014, written while he was still a student at the University of Warsaw using the surname Tajtelbaum. It appeared in volume 24 of the journal Przegląd filozoficzny.

Geometry for the Third Gimnazjum Class (1935)

This chapter is devoted to a translation of parts of the school-geometry text Geometrja dla trzeciej klasy gimnazjalnej by Zygmunt Chwialkowski, Waclaw Schayer, and Alfred Tarski. The text was published in 1935 by Panstwowe Wydawnictwo Ksiązek Szkolnych, the National Schoolbooks Publisher. Its title page is shown on page 276 of the present book. The note there under the publisher’s logo says that the price of the book, including the cost of the stamp of the Society Supporting Construction of Common Public Schools, is 1.20 zlotys. That was comparable to the retail price of three kilograms of wheat flour, one kilogram of pork, five liters of milk, or thirteen eggs.

Career and Family

This chapter describes Alfred Tarski’s work as a secondary-school mathematics teacher and teacher-trainer in Poland during 1924–1935. It also provides background for that activity, and for all of Tarski’s publications that stemmed from it. The first of those is translated in chapter 10: his [1929] 2014a report to secondary teachers about the First Congress of Mathematicians of Slavic Countries, held in 1929 in Warsaw.

Exercises Posed by Tarski

This chapter contains translations and discussions of the fourteen exercises that Alfred Tarski published for enhancing instruction in Polish gimnazjums. The translations appear in boxes in each of the numbered sections that follow. They are meant to be as faithful as possible to the originals, and follow the conventions used for the longer translations in other chapters. In particular, all [square] brackets in the translations enclose editorial comments inserted for clarification. Each individual exercise is accompanied by a discussion by the present editors: the material outside the box. These discussions include contextual information, descriptions of responses from students, teachers, and other mathematicians, and full or partial solutions by the present editors. The paragraphs immediately following this one describe general aspects of all of the exercises.

On Decomposition of Point Sets into Respectively Congruent Parts (1924)

This chapter contains an English translation of the paper Sur la decomposition des ensembles de points en parties respectivement congruents, [1924] 2014, by Stefan Banach and Alfred Tarski. It appeared in volume 6 of the journal Fundamenta Mathematicae. This is its first translation. Its best-known result is often called the Banach–Tarski paradox: any two balls with different radii can be decomposed into the same finite number of disjoint, respectively congruent parts.

On the Equivalence of Polygons (1924)

This chapter contains an English translation of Alfred Tarski’s paper O rownowazności wielokątow, [1924] 2014b, written as he was completing his doctoral studies. It appeared in volume 24 of the journal Przegląd matematyczno-fizyczny.

Circumference of a Circle (1932)

This chapter contains an English translation of Alfred Tarski’s [1932] 2014e article Teorja dlugości okregu w szkole średniej, which appeared in the October–December 1932 issue of the journal Parametr. The journal’s target audience consisted of gimnazjum teachers and their most serious students. For further information about that publication, consult section 9.7 of the present book.

Congress of Mathematicians of Slavic Countries (1929)

This chapter contains an English translation of Alfred Tarski’s [1929] 2014a article Zjazd matematykow, which appeared in the journal Ogniwo. The journal identified the author only as A.T. It is a brief account of the First Congress of Mathematicians of Slavic Countries, held in Warsaw and Poznan, 23–27 September 1929.

Area, Volume, Measure

The synthetic approach to the theory of area and volume initiated by Euclid is described in the first section of this chapter. The area a of a polygonal region P is computed by decomposing P into a finite number of polygonal components with disjoint interiors, which can be reassembled to form a rectangle R with unit base: a is then the altitude of R. The volume of a polyhedral region can be reckoned in a similar way, but for that it is necessary to use some form of Eudoxus’s method of exhaustion.

Teaching and Logic—To the New World

The preceding portions of the book contain all of the translations and background information needed to accomplish the first two goals, except for a few short works, not concerned with geometry or teaching, that had been published only in Polish. Those are translated in chapter 15, with some background information for specific items. One of those short works has to do with politics; the rest, logic. The present chapter contains some general background for chapter 15 and concludes the biographical material in this book with an account of Tarski’s 1939 voyage to the New World.