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Bernard Bolzano, Czech Pioneer of Modem

JOSEPH V. TALACKO

SUMMARY

Since the foundation of in by Emperor Charles IV in 1348 A.D., a large number of her sons have earned a permanent place in the history of our culture and sciences. Bernard Bolzano, a remarkable personality of the nineteenth century, is a shining example. He was an intellectual genius with strong convic- tions, and he made essential contributions to mathematics, , and . One hundred years after his death, some of his manuscripts and books were first published, then republished and translated and his numerous scattered papers were identified and collected. His entire writings would comprise over twenty-five volumes. Our present interest is in Bolzano, the . As a student he displayed mathematical brilliance. Because of the era in which he lived, he was destined to become a professor of philosophy instead of a professor of mathematics. He is one of pioneers of modern mathematics, an original thinker, a rationalist, and a partisan of strict, methodic inquiry. Bolzano developed the theory of function of one real variable before Cauchy and Weierstrass, and made notable contributions to the foundation of modern analysis. As a student he published and anticipated a theory of parallel lines usually known as Legendre's theory. He is recognized as a prede- cessor of Cantor in the theory of and in the of infinity, and is known as a predecessor of in modern . Other details of his colorful life and some of his work are discussed in this paper.

INTRODUCTION

Bernard Placidus Johann Nepomuk Bolzano, known to throughout the world as one of the pioneers of modern mathematics, was 1656 Joseph V. Talacho born in the old town of Prague, Bohemia, October 5, 1781, and died in Prague on December 18, 1848. He is buried in the Prague cemetery of Volsany, where a simple stone with his name marks his grave. His father was an art dealer, an Italian immigrant from Nesso in Lombardy; and his mother, born Cecilia Maurer, was of German origin. Nevertheless, born in the turbulent years after the American and French revolution, he was a true son of his adopted country in Central Europe, then governed by the Habsburg monarchy. His life, his deeds, and his destiny should place him among its greatest sons. A contemporary and a part of the awaking and restoration of the Czech nation, he proudly counted among his friends important men of the Czech enlightenment such as Josef Dobrovsky, Josef Jungmann, Norbert Vanèk, Ladislav Jandera, Praelat Pfeifer, and many others. Long is the list of his students (12), who were close to him in of trouble and persecution. His great personality, his ethical and moral attitudes, influenced young Czech intelligentsia and such Czech leaders as Palacky, Havlicek, and Celakovsky. Bernard Bolzano was an ordinary active member of the Royal Bohemian Society (Academy) of Sciences from 1815 until the end of his life. In the Proceedings (Berichte and Abhandlungen) of this Academy are records of his thirty-four readings and seven published memoirs. His influence on this formative period of the Czech nation is far-reaching and deserves much more study. In this short essay, it is almost impossible to cover all aspects of this man, who was also a great theologician, logician, and humanitarian. We must limit our discussion to Bernard Bolzano as a mathematician. It is not an exaggeration, reading about his life and his contributions to this field, to compare him to Galileo Galilei, René Descartes, or Blaise Pascal. L. Coolidge, in his book, The Mathematics of Great Amateurs (7), writes: "The distinguished Czech writer whose name stands above bears, from the point of view of this book, a certain resemblance to Blaise Pascal. Both were deeply interested in religion and philosophy, both were involved in controversy and suffered for the faith that was in them. Both did such brilliant mathematical work that they might well be classed as professionals. But I have included Pascal with the amateurs because he was more famous as a and a writer of beautiful French prose than as a mathematician, and I take up Bolzano because it seems to me interesting that a man who was a remarkable pulpit orator, only removed from his chair for his political opinions, should have taught so far into the deepest problems of a science which he never taught in a professional capacity." Bernard Bolzano, Czech Pioneer of Modern Mathematics 1657

STUDENT OF MATHEMATICS AND PROFESSOR OF THEOLOGY

Indeed, he was born a mathematician, such as a poet or a composer is born. Bolzano's formal education was that customary at that for a gifted child of a respectable, but not wealthy family. It began soon, before he was ten years old, at the Piarists Classical Gymnasium (secon- dary school), because it was near their home. As a boy, Bolzano was ailing and weak. His physical health never was robust and his eyesight was poor to the end of his life. He graduated from the Piarist Academy before he was fifteen years old. His father wanted him to be a business- man. With his mother's help and persuasion, Bernard entered the - sophical Faculty of the Charles University of Prague in 1796, studying philosophy, mathematics, and, later, theology in the Faculty of Theology. An above-average student at the secondary school, as a university student too, he soon displayed sound intellectual habits, originality, and frugal self-reliance. Bolzano's teachers were two outstanding Bohemian mathematicians: Stanislaus Wydra, a member of the Society of Jesus until its suppression, and Franz Joseph Gerstner, the founder of the Prague Polytechnik. Mathematics was the first he studied. At the beginning, he devoted himself chiefly to mathematics with excep- tional success. He made copious records of the lectures and critical notes on the textbooks by Kaestner. Carefully reading standard texts, he soon worked through memoirs of Euler and treatises of Lagrange. One of the real jewels of the Charles University is the Clementinum, the former Jesuit college, which houses the public and university library, which now has over two million books. Here, young Bolzano, already known as a brilliant student of mathematics, found the old Greek masters. As Steele (16) points out, "to cheer himself during indisposition", Bolzano took up the Elements of Euclid. Delighted by its logical depth, he swiftly dis- pelled his indisposition. Nevertheless, unsatisfied with Euclid's theory of proportion in the fifth book, he studied the , and other Greek masters' works, like Archimedes, and . Here is the key to Bolzano's conversion to logical, exact, rigorous mathematics and his future discoveries of his philosophy and life destiny. The trait of Bolzano's "Paradoxien", as well as the trait to the theory of irrational numbers, (developed later by Weierstrass and Dedekind), leads to the Eudoxus mode of almost literally. Bolzano wanted to be a university teacher. There was no opening in mathematics or in philosophy. After finishing his course work in mathe- matics, he studied theology, living, not in the Seminary, but at home, 1658 Joseph V. Talacko and always believing "that true knowledge in any sphere is attainable only by the mathematical mode of thought". After graduation from the theology faculty, he postponed his ordination, and in 1804, after eight years at the University, defended his doctor's thesis in mathematics. The Betrachtungen, published in 1804, were dedicated to his teacher Stanislaus Wydra and were so highly commended by Von Gerstner, not only for his singular pedagogical talent, but also for originality and mathematical brilliance, that they granted him prior claim to future appointments at the chair of mathematics. After the death of Wydra, Bolzano had a rival for his chair in Jandera, who already was docent there. In the meantime, the government announced the establishment of the chair of "" at the Faculties of Philosophy. Before ordained as a , Bernard Bolzano applied for this new chair at Charles University in Prague, took the examination, and won over several other older and well-known . Nevertheless, about three weeks later, he submitted an application for the open chair of mathematics to succeed Wydra, encouraged by Professor Gerstner, who recommended him strongly. On February 13, 1804, he was recom- mended for the chair of Philosophy; on April 7, 1804, was ordained as a priest, on April 17, 1804, got the Ph.D. in philosophy, and on April 19, 1804, was introduced as a professor of the philosophy of religion. The chair of mathematics went to Jandera. So, not yet twenty-four years old, Bernard Bolzano was the first ordinary professor of the new chair of the Philosophy of Religion. This establishment of the chair of Science of Religion was bitterly resented by students and the Czech public. Bolzano started his teaching career with patience and understanding. An ideal teacher, clear in language and reasoning, an outspoken advocate of the Bohemian-Catholic Enlightenment, he soon converted the distrust of his students into full devotion and admiration. In his lectures on religion and moral philosophy, he made a strong impression on his audience, and his teaching somewhat reversed the increasing religious indifference. The universities were at that time under strict supervision of the monarchy. Bolzano was expected to base his lectures on standard texts. The initiator of Bolzano's new university chair was the court chaplain in Vienna, J. Frint, the author of an expensive compulsory text not adopted by Bolzano. Over Fruit's objection, only three months after the nomination, Bolzano's professorship was confirmed on October 22, 1806. With the diploma of chair-confirmation went the duties of chaplain of the university. In some of his exhortations and university sermons, which were tinged with rationalism and , he publicly advocated Bernard Bolzano, Czech Pioneer of Modern Mathematics 1659 political and social reforms. Director of the Philosophical Faculty, Premonstratesian Abbot Milo Griin, encouraged him to write his own lectures and to publish his sermons. They made him famous in Prague circles, but led to his enforced retirement at the age of thirty-nine. It is surprising that his uneasy tenure lasted as long as fifteen years! From 1808, J. Frint gathered "evidence" against Bolzano. To make a long story short, Bolzano, through his personal friends, and through ecclesias- tical and secular authorities, was able to retain his professorship until January 20, 1820. After a long trial and harassment, by an imperial order signed on December 24, 1819, his dismissal was finally put in effect. Bolzano was reprimanded "for writing passages on war and civic obedience". It was established that the archiépiscopal curia must exam- ine his works and the notes of his students, and would forbid him the pulpit until he gave satisfaction. "As a special favour, no punishment would be imposed." As Steele (16) reports, the Consistorial Council gave his verdict in favour of Bolzano on May 27, 1820, declaring him to be an orthodox Catholic whom no one could possibly brand as a heretic. The affair ended amicably on the ecclesiastical and secular level in 1824. Vienna at last approved for Bolzano a pension from the religion fund; at the instance of Count Lev Thun, it was later even tripled. It is well known that, thanks to the powerful protection of the Archbishop of Prague, and warnings and threats of political consequences to Czech patriots (especially Josef Dobrovsky), Bernard Bolzano was rescued from a worse fate. Thus ended the direct teaching career of Bolzano. He never yielded to civil authority against his convictions, and Vienna re- newed his suspension and forbade the appointment of Bolzano to any other teaching position, even as a mathematician at the observatory. Bolzano was wounded, separated from academic youth and the public, and under constant surveillance. He stoically accepted his ordeal. He soon found that the dismissal had some advantages, that is was good for his health, and that he now had enough time for concentration on mathematics and logic, on research and writing. Great men overcome all obstacles and, on the contrary, they often do more under pressure and resistance.

Bolzano spent the remainder of his life, a most productive twenty- eight years, in studious semiretirement. From 1823 to 1830, he spent his summers on the estate of his friend, Josef Hofmann, in the village of Tëchobuz in Southern Bohemia, returning each winter to Prague. From 1830 through 1841, he lived in Tëchobuz continuously, composing his 1660 Joseph V. Talacko two main encyclopedic works: the Wissenschaftslehre and the Grössen- lehre. After 1841, he lived in the home of his brother in Prague. A pension, small but sufficient for those times, of 300 gulden a year, relieved Bolzano of all monetary worries. Those twenty years after his retirement were the happiest period in his life. His health improved and he was at the height of his creative powers, studying and writing in seclusion and peace.

BOLZANO'S MAIN PUBLICATIONS ON MATHEMATICS

Bernard Bolzano was a prolific writer. His known writings and manu- scripts comprise over twenty-five volumes. He was not only a modern mathematician, but also a theologician, a philosopher and logician, a physicist, a sociologist, a philologist, and a writer on the theory of beauty and arts. Well-educated in the best classical central European tradition, he pursued all these studies, yet doing justice to all of them. The publications about Bolzano, translations of his works, and reprints of his books would total a very large private library. It is almost im- possible to introduce all of them in this short space. Interested readers will find references in recent publications to his books, especially in Berg (2) and Steele (16). Berg cites eighty-four items of Bolzano's books, papers, contributions, letters, and particular manuscripts; the selected bibliography of Steele has sixty-four items concerning Bolzano's works. Berg's book contains over 24 pages of other references. Bolzano's writings on mathematics may be divided into four parts: his earliest publications from 1804 until 1820; publications in the years 1821 to 1848; posthumous publications (since 1848), and works not yet published, but still in manuscript. Of the last, mostly those from Bolzano's bequest to Count Leo Thun, now in the National Museum at Prague, and in the Austrian National Library (Österreichische National Bibliothek in Vienna), Jarnik (9) writes: "New discoveries can be expected from what Bolzano left behind. The large number of manu- scripts left by Bolzano have not yet been fully explored; many sketches and drafts will perhaps remain undeciphered, forever. However, a careful study of this wealth of material can be expected to throw new light on the work of this deep and original thinker." An almost complete list of his works published during his lifetime may be found in the Conference Reports (Sitzungs-Berichte) of the Vienna Academy for 1849. Nevertheless, as we know now, during the years of strict censor- ship of his publications from 1821 to 1843, Bolzano published some Bernard Bolzano, Czech Pioneer of Modem Mathematics 1661 books and papers anonymously in editions and journals not easily accessible; these may never be identified. We list here twelve publica- tions, with a free translation of titles into English. In the first period (1804-1820) of Bolzano's contributions to mathe- matics belong these five works: 1) Betrachtungen über einige Gegenstande der Elementargeometrie [Reconsideration of the Foundations of some Branches of Elementary Geometry] (Prague, 1804), 66 pp. A revised paper was prepared by Bolzano in 1844. The Betrachtungen were reprinted in Prague, 1948. 2) Beyträge zu einer Begründeteren Darstellung der Mathematik [Contributions to the Foundation of the Science of Mathematics] (Prague, 1810), pp. 152. Planned as a sequence, but only one installment ap- peared. Reprinted in 1926, Paderborn. 3) Der Binomische Lehrsatz und als Folgerung aus ihm der Poly- nomische und die Reihen, die zur Berechnung der Logarithmen und Exponential grossen dienen, genauer als bisher erwiesen [The Binomial Theorem, the Binomial Polynoms and the Binomial Infinite Series as Tools of Computation of Logarithms and the Evaluation of Exponential Functions] (Prague, 1816), pp. 144. 4) Rein Analytischer Beweis des Lehrsatzes, dass zwischen je zwey wehrten, die ein entgegengesetztes Resultat gewahren, wenigstens wine reelle Wurzel der Gleuchung liege [A pure analytic proof of the theorem that the solution of the rational polynormal equation of higher order f(x)=0, if f(a) and f(b) have opposite signs, has at least one real root between a and b] (Prague, 1817), pp. 60. Reprinted, in Wissenschaft- liche Klassiker in Faksimile-Drücken, Vol. 8 Berlin, 1894; in Ostwalds Klassiker der exact Wissenschetten, No. 153 (Leipzig, 1905), translated into Czech and published in Prague, 1881. 5) Die drey Probleme der Rectification, der Complanation und der Cubirung, ohne Betrachtung des unendlichen Kleinen, ohne die An- nahme des Archimedes und ohne irgend eine nicht streng erweisliche Voraussetzung gelöst: zugleich als Probe einer ganzlichen Umgestaltung der Raum Wissenschaft allen Mathematikern zur Prüfung vorlegt [Three problems are here solved: Finding of the length of arc of a curve, of the plane area and of the volume without any consideration of values, without Archimedes' hypothesis and without any other strong assumptions. At the same time, this essay is submitted to all mathe- maticians as a demonstration that the theory of space deserves a total reform and re-examination by them] (Leipzig, 1817), pp. 80. Reprinted in Prague, 1948. 1662 Joseph V. Talacko From the second period (1821-1848) are known only the following four publications, published before his death: 6) Review of F. X. Moth's "Theorie der Differenzial-Rechnung und ihre Anwendung zur Auflösung der Probleme der Rectification, der Complanation und der Cubirung" (Prague, 1827), published in Monat- schrift der Gesellschaft des Vaterländischen Museums in Böhmen, January 1, 1827, pp. 79-82 [F. X. Moth's Theory of the and its Applications to the Solutions of Problems of Finding the Length of Arc of a Curve, the Plane Area and the Volume] 7) Wissenschaftslehre. Versuch einer ausführlichen und grössentheils neuen Darstellung der Logik mit steter Rücksicht auf deren bisherigen Bearbeiter [Science of Knowledge] (Sulzbach, 1837), Vol. 1, pp. 571; Vol. 2, pp. 568; Vol. 3, pp. 575; Vol. 4, pp. 683. Reprinted in Vienna, 1882, second edition (Leipzig, 1929-1931). Czech translation expected soon. 8) Versuch einer objectiven Begründung der Lehre von der Zu- sammensetzung der Kräfte [An attempt at the objective foundation of science by the combination of forces or ] (Prague, by Kronberger, 1842), pp. 40. 9) Versuch einer objectiven Begründung der Lehre von den drei Dimensionen des Raumes [An attempt of the objective foundation of the theory of three-dimensional space] Betrachtungen, Proceedings of the Royal Bohemian Society of Sciences, 3 (1845), pp. 201-215. Submitted by Bolzano in 1842, but written in 1815. Privately printed, Prague, 1843. Reprinted, Prague, 1948.

From his posthumous publications on mathematics, we introduce here only three major works: 10) Paradoxien des Unendlichen [Paradoxes of the Infinite]. Published by Bolzano's student, Dr. Prihonsky, in Leipzig (1851), pp. 134. Reprinted, Berlin 1889; Hamburg, 1921 and 1955. Translated in English and published in 1950 by Steele. See Reference no. 16. 11) Bernard Bolzanos Schriften [Publications of Bernard Bolzano]. Published by the Bohemian Academy of Sciences, 1930-1931. Volumes 1-2, Edited by K. Petr. Volume 1. Functionen Lehre [Theory of Functions] by K. Rychlik. Volume 2. Zahlentheorie [Theory of Numbers], edited by K. Rychlik. 12) Ueber Haltung, Richtung, Krümmung und Schnörkelung bei Linien sowohl als Flachen sammt einigen verwandten Begriffen [On Behavior, Direction, Curvature and Turning Points of Curves and Bernard Bolzano, Czech Pioneer of Modern Mathematics 1663 Surfaces as well as Other Related Notions from our Point of View]. Edited by J. Vojtech, Prague, 1948. Written in 1843-1844. This book contains also reprints of publications (1), (5), and (9), listed in this section.

There are many other posthumous publications selected from Bolzano's original manuscripts, mainly by his students, friends, and in the last century, especially by Czech mathematicians, K. Petr, K. Rychlik, M. Jasek, V. Jarnik, and others. Interested readers will find references in the Bergs book (2).

BOLZANO'S CONTRIBUTIONS TO MODERN MATHEMATICS

Bernard Bolzano's several important discoveries in mathematics were practically overlooked during his lifetime and rediscovered only later, twenty-three years after his death. Since their first publication in 1904, Bolzano's interest in the foundation of pure mathematics has been clear. Nevertheless, some of them were unknown to the mathematical commu- nity for almost two or three generations. Bolzano returned to pure mathe- matics after 1830, when he started the Herculean task of writing five or six volumes of the encyclopedic work, Grossenlehre, of which only the first three were completed before his death in 1848. Bolzano lectured at those meetings of the Royal Bohemian Society of Sciences without any effect of impact on his contemporaries. It is known (2) that Bolzano communicated some abstracts of the Grossenlehre to his colleagues, Jandera and Kulik, as well as to Cauchy, who met Bolzano in Prague in 1834, but without any apparent effect on them. The several important "firsts" were unknown until 1871, when the German mathematician, Hermann Hankel (1839-1873), called attention to some important works of Bolzano (8). After that, two prominent pupils of (1815-1897), (1842-1905) and Herman A. Schwarz (1845-1921), during the years 1872-1881 declared Bolzano's writings remarkable, placed him on the same important level as Cauchy, and called him an innovator of that line of reasoning which made K. Weierstrass famous. Bolzano's advanced on variables, continuity, limits, infinite series, infinity, the theory of sets and mathe- matical logic made him a forerunner of Augustin L. Cauchy (1789- 1857), K .Weierstrass, Georg F. Riemann (1826-1866), and Camille Jordan (1838-1922) in modern analysis. He was ahead of Richard Dedekin (1831-1916) and (1845-1918) in the theory of 1664 Joseph V. Talacho continuum and theory of sets, and of Bertrand Russell (1872-) and other mathematical logicians. Bolzano's original about space and dimensions were not fully developed until about sixty years later. Some of his basic definitions are still modern, his ideas followed, his approach universally recognized. Some papers need modern notation and inter- pretations, but still remain a source of deep inspiration. Bernard Bolzano brought logic into mathematics and mathematics into logic. His interest in mathematics was primarily philosophical. Theoretically speaking, mathematics and logic have for a long time been two different subjects. Today, modern mathematics is more logical and modern logic is mathematized; it is really hard to find a border line between them. Bolzano's philosophy and logic were practical tools of correct thought, of abstraction, and of sound, further development of pure mathematics. In the prime of life, he declared that the science of mathematics is "the science of the general laws according to which the of all things is regulated" (16). Some of his "firsts" we summarize in this paper: a) In his first paper, published in 1804, he anticipated Legendre's well- known theory of parallel lines. b) In 1816, in his third paper on the binomial theorem, he showed the advantages of the application of this theorem and binomial series in analysis. Ten years later Abel studied the same problem. c) In his paper, (4) "Rein analytischer Beweis . ..", published in 1817, Bolzano developed the theory of one real variable before Cauchy, and made notable additions to the theory of the differentiation and integra- tion. See Jarnik (9), Steele (16). d) He proved the well-known so-called "Bolzano-Weierstrass theorem"; that "each bounded sequence of numbers has an accumulation point", in the early years after 1830, a good thirty years before Weierstrass. e) He constructed at the same time, also thirty years before Weierstrass, the well-known "Bolzano function", which is continuous everywhere in a closed interval, but passes derivatives nowhere. See Steele (16). This has been discovered in 1920 by Jasek. See Jarnik (9). f) He left remarkable writings on the theory of real numbers, dis- covered after almost 100 years. Felix Klein (1845-1925) called him the "father of the arithmetication of analysis". g) Bolzano is recognized as an inceptor of modern , fully developed at least forty years later by Cantor. The Paradoxes of the Infinite, published in 1851 by his pupil and friend, F. Pffhonsky, is generally recognized as a classical masterpiece of modern mathematics. Bernard Bolzano, Czech Pioneer of Modern Mathematics 1665 The recognition of Bolzano, the logician, took longer. Only recently, as the mathematizing tendencies in modern logic have assumed a fresh perspective, is the importance of his life work, Wissenschaftslehre, and other publications being fully recognized - see Berg (2) - and he is recognized as a father of modern .

CONCLUSION

It would be possible to add much to this short essay. Bernard Bolzano was a representative typical of the spirit of the country and era in which he was born and lived through his whole life. It was a period of great intellectual awareness and activity, characterized by questioning of old mysteries and authority, a creative interest not only in science and culture, but also in political matters. Bernard Bolzano, almost unknown during his lifetime, a genius whose career seems full of paradoxes and , remains a paradox in our times. Always a loyal son of the , he is studied at present and some of his manuscripts have been republished by those whom he would definitely oppose if he were alive. More than 130 years ago, he wrote of a social Utopia, Von dem bestem Stuate, in which he developed a plan of political and social reforms. Politically, he advocated a republican constitution, restriction on private ownership, and economic and social welfare. His fundamental ideas are not far from those of the papal encyclical, Quadragesimo Anno, because he was against revolution, oppression of personal freedom, etc. During his lifetime, he forbade all distribution or publication of this manuscript of his "Utopia". Published in German in 1932, translated into Czech in 1934, it has been republished in 1952. We may find extracts from Bolzano's Utopia in papers on pure mathematics, logic, and philosophy. It is a paradox that this man, born too soon to be recognized, is now being rediscovered. By coincidence, those other scientific manuscripts are being studied, translated, and published more than a hundred years after his death. Nevertheless, he is a man of that select group, members of which may be born once in a century - a very great man, a very great mathe- matician, who, by his original thinking, his discoveries and achievement, his imagination and versatility, by his lasting influence, will have a permanent place in the long list of famous sons of his adopted nation. 1666 Joseph V. Talacho

REFERENCES

1. Archibald, R. C., "Outline of the History of Mathematics", The American Mathematical Monthly, Vol. 56, No. 1; Memorial Paper, No. 2, pp. 1-114 (1949). 2. Berg, Jan, Bolzano's Logic (Stockholm, Almqvist & Wiksell, 1962), p. 214. 3. Bolzano, B., Bernard Bolzano's Schriften, Hrsg. von der königlichen böhm- ischen Gesellschaft der Wissenschaften, Vorrede von dr. Karel Petr (Prag, 1930), bd 1-5. 4. Bolzano, B., Rein Analytischer Beweis Des Lehrsatzes, ... (Leipzig, Engel- mann, 1905). 5. Bolzano, Bernhard, Wissenschaftslehre, Vol. 1-4 (Sulzbach, 1837). 6. Cajori, Florian, A History of Mathematics (N.Y., MacMillan, 1922). 7. Coolidge, J. L., The Mathematics of Great Amateurs (Oxford University Press, London, 1950). 8. Hankel, H., "Grenze", Allgemeine Encyklopädie der Wissenschaften und Künste, Sect. 1, Teil 90 (Leipzig 1871), pp. 185-211. 9. Jarnik, Vojtéch, "Bernard Bolzano", Czechoslovak Mathematical Journal, Vol. 11, 86, No. 4 (Prague, 1961), pp. 485-489. 10. Kline, Morris, Mathematics in Western Culture, (N.Y., Oxford University Press, 1953). 11. Langer, R. E., "René Descartes", American Mathematical Monthly, Vol. 44 (1937), pp. 495-512. 12. Ottùv slovnik naucny [Otto's Czech Encyclopedia], edited by Fr. L. Rieger (Prague, 1881), pp. 790-792. 13. Pfihonsky, Fr., Dr., Dr. Bernard Bolzano's Paradoxian des Unendlichen Budissim, Am 10. Juli, 1850, 2, unveränderte Auflage (Berlin, Mayer-Muller, 1889), p. 134. 14. Rychlik, K., 'Theorie der Reelen Zahlen im Bolzano's handschriflichen Nachlasse", Chechoslovak Math. Journal, Vol. 7, 82 (Prague, 1957), pp. 553-567. 15. Singh, J., Great Ideas of Modern Mathematics, Their and Use (Do- ver, N.Y., 1959). 16. Steele, D. A., Paradoxes of the Infinite by Dr. Bernard Bolzano. Translated from the German of the posthumous edition by Dr. Fr. Pfihonsk^ and furnished with a historical introduction (New Haven, Yale University Press, 1950). 17. Stolz, Otto, "Bolzano's Bedeutung in der Geschichte der Infinitesimalrech- nung", Math. Annalen, Vol. XVIII (1888), p. 195. 18. Struik, D. J., A Concise History of Mathematics, Vol. II (New York, Dover Pubi. Co., 1948).