## Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. [Bound with two other works by Bolzano, his doctoral thesis and his autobiography, see below].

Prague: Gottlieb Haase, 1817.

First edition, extremely rare, of this epoch-making paper in the history of mathematics, the first to provide a rigorous foundation for the calculus. “The main mathematical achievements of the paper include: (a) the formal definition of the continuity of a function of one real variable, correctly understood and applied (Preface); (b) the criterion for the (pointwise) convergence of an infinite series, although the proof of its sufficiency, prior to any definition or construction of the real numbers, is inevitably inadequate (Sect. 7); (c) the original form of the Bolzano-Weierstrass theorem (Sect. 12); (d) an analytic proof of the intermediate value theorem, now sometimes called Bolzano’s theorem (Sect. 15). The theorem in the title of the paper, where ‘equation’ is understood as ‘polynomial equation in one real variable,’ is deduced in the final paragraph (Sect. 18) from result (d)” (Russ, p. 157). It is usual to attribute (a), (b) and (d) to Augustin-Louis Cauchy’s *Cours d’analyse*, which Bolzano anticipated by four years (Cauchy’s definition of continuity actually still involved infinitesimals), and (d) was rediscovered by Karl Weierstrass half a century later (in ignorance of Bolzano’s work). Bound with this important work are his first published work, *Betrachtungen uber einige Gegenstande der Elementargeomwtrie* (Prague: Karl Barth, 1804), an attempt to axiomatise plane Euclidean geometry, and also extremely rare; and his autobiography, *Lebenschreibung des Dr. B. Bolzano* (Sulzbach: J. E. v. Seidels, 1836). “Around the turn of the nineteenth century, mathematicians in Europe were concerned with two major problems. The first was the status of Euclid’s parallel postulate, and the second was the problem of providing a solid foundation for mathematical analysis, so as to remove the so-called scandal of the infinitesimals” – this remarkable volume contains Bolzano’s responses to both of these great problems. We are aware of only one other copy of *Rein analytischer Beweis *having appeared on the market in the last 30 years; no copy is listed on ABPC/RBH. OCLC lists eight copies of each of *Rein analytischer Beweis* and *Betrachtungen*, but no copy of either in the US. No copies in auction records.

“Bolzano’s writings mark a turning-point in research on the foundations of mathematics – a transition from the mathematical style of the eighteenth century to that of the nineteenth … Bolzano was the first mathematician explicitly to reject the traditional geometric and spatial approach to foundations, calling instead, on explicitly *logical *grounds, for a 'purely analytic' grounding of the calculus – that is, a grounding in *arithmetic. *He thus stands at the head of two intertwined movements in nineteenth-century mathematics: the *arithmetization of mathematics, *a project that was to be carried forward by Cauchy, Gauss, Abel, Riemann, Dirichlet, Weierstrass, Heine, Cantor, Dedekind, and others; and the search for *logical* *foundations *that was pursued by Frege, Peirce, Peano, Russell, Brouwer, Hilbert, and Weyl” (Ewald, p. 168). “Although Bolzano’s proofs are incomplete, and although they are somewhat clumsily presented, this paper is a milestone in the history of real analysis. It was the first successful attempt to free the calculus from infinitesimals, and it is the starting point for the modern theory of the continuum; the precision of Bolzano’s definitions and the rigour of his deductions mark a break with the mathematics of the past. The project of putting the theory of the real line on a solid, arithmetical foundation was to be carried forward, largely in ignorance of Bolzano’s work, throughout the nineteenth century – most notably by Cauchy, Abel, Dirichlet, Weierstrass, Cantor, and Dedekind” (*ibid*., p. 226).

“Bolzano's most significant contribution to mathematics is his epoch-making paper on the foundations of real analysis, the *Rein analytischer Beweis *of 1817. In the years following the appearance of Berkeley’s *Analyst *(1734), mathematicians had made various attempts to put the calculus on a firmer foundation. The most common approaches were to base the calculus on one of the following ideas: on *motion *(Newton, MacLaurin); on *limits *(D’Alembert, L’Huilier); on *ratios of zeros *(Euler); on *infinitesimals *(Leibniz and – with reservations – Carnot). In perhaps the most radical proposal, James (= Jacob/Jacques) Bernoulli proposed amending the laws of logic by abandoning, for infinitesimals, the Euclidean ‘common notion’ that, if equals are subtracted from equals, the remainders are equal. Perhaps the most influential approach was that of Lagrange, who, in his *Théorie des fonctions analytiques *(1797), assumed the existence of a Taylor-series expansion for every function; the full title of his work* – Théorie des fonctions analytiques contenant les principes du calcul différentiel, dégagés de toute considération d’infiniment petits, d’évanouissans, de limites et de fluxions, et réduits a l'analyse algébrique des quantités finies – *shows the scepticism with which he, and many other mathematicians, regarded the notion of limits, and his desire to reduce the calculus to ‘l’analyse algébrique’. Bolzano’s paper was not the first to attempt to find an analytic foundation for the calculus, nor was it the first to employ the notion of limits. But, in contrast to his predecessors, Bolzano employed a limit-concept that was not based upon motion, and that was analytically defined and, more importantly, he was the first actually to use this definition to prove significant mathematical theorems.

“Bolzano’s announced aim is to prove the intermediate value theorem – in his formulation, that if *f *and *g* are continuous functions such that *f*(*a*) < *g*(*a*) while *f*(*b*) > *g*(*b*), then for some *x*, *a* < *x* < *b*, *f*(*x*) = *g*(*x*). This theorem he eventually proves in §15. But he begins with an important critique of previous proofs, and in Part II of the *Preface *he gives the first precise definition of a continuous function. His definition is essentially the same as that given by Cauchy in his *Cours d’analyse *in 1821; whether Cauchy knew of Bolzano’s work is uncertain. (Bolzano improved on his definition in his unpublished *Functionenlehre, *written in 1834; there he gives a definition of pointwise continuity and distinguishes between left and right continuity.)

“In §7, Bolzano states and attempts to prove the sufficiency of the ‘Cauchy condition’ for the convergence of an infinite series. A rigorous proof requires a precise definition of the real numbers, which Bolzano did not possess; he himself (in the *Functionenlehre*) later admitted that the §7 proof was incomplete.

“Similarly, although Bolzano had a precise definition of continuity, he did not have the modern notion and definition of *function. *Lagrange, in the *Théorie, *had indeed defined a function of one or several quantities to be ‘any mathematical expression in which those quantities appear in any manner, linked or not with some other quantities that are regarded as having given and constant values, whereas the quantities of the function may take all possible values’; but in practice he and his successors treated functions as equations. The modern conception did not enter mathematics until Dirichlet’s paper, *Ūber die Darstellung ganz willkurlicher Functionen, *in 1837.

“Having defined continuity and stated the Cauchy condition, Bolzano proceeds (§12) to prove a lemma that was eventually to become the cornerstone of the theory of real numbers. This lemma (the greatest lower bound principle) is the first published version of the Bolzano-Weierstrass theorem, which, in modern terminology, says that every bounded infinite point-set has an accumulation point …

“It is important to appreciate the role Bolzano’s objective conception of axioms, described above in the *Beiträge zu einer begründeteren Darstellung der Mathematik erste Lieferung, *1810), played in the *Rein analytischer Beweis. *Bolzano was not driven by scepticism or by fear of paradox. He makes it clear that he did not doubt the *truth *of the intermediate value theorem; and he was not attempting to place it on firmer or more obvious foundations – for the greatest lower bound principle is, if anything, less evident than the theorem he is trying to prove. Similarly, his criticism of the proofs based upon motion is not that intuitions of motion are unreliable, but the logical objection that the proofs beg the question. Bolzano’s ambition was not so much to attain some superior brand of mathematical certainty as to reveal the objective reasons for the truth of the intermediate value theorem – to uncover its true logical foundations. And *a fortiori *Bolzano’s quest for rigour in his [*Rein analytischer Beweis*] was not prompted by the ‘challenge to geometric intuition’ presented by the discovery of continuous nowhere-differentiable functions; on the contrary, it was Bolzano's rigour that made his subsequent discovery [in the *Functionenlehre*] of such counter-intuitive phenomena possible” (*ibid*., pp. 225-7).

“The seeds of Bolzano's distinctive approach to mathematics seem to have been planted early. As a student at the University of Prague he studied Abraham Gotthelf Kästner's *Anfangsgründe der Arithmetik *(1758); and in a revealing passage in his autobiography Bolzano praised Kästner because ‘he proved what is generally passed over because everyone already knows it, i.e. he sought to make the reader clearly aware of the basis (Grund) on which his judgements rest. That was what I liked most of all. My special pleasure in mathematics rested therefore particularly on its purely speculative parts, in other words, I prized only that part of mathematics which was at the same time philosophy’ (*Lebensbeschreibung*, p. 64). ‘Proving what everyone already knows’ and ‘prizing that part of mathematics which was at the same time philosophy’ are precisely the traits that were to be characteristic of Bolzano’s own mathematical work; but he was to pursue both far more deeply than anything in Kästner.

“Bolzano’s new and more rigorous approach to axiomatics can already be detected in his earliest mathematical writings … Bolzano’s predecessors, in contrast, had taken a more relaxed approach. In their view, the essential requirement for an axiom was that it be certain – an immediate and obvious truth on which the calculus of fluxions or Euclidean geometry could be founded. [The *Betrachtungen*] gives three reasons for rejecting this conception of axioms and for pursuing even obvious truths ‘down to their ultimate grounds’: such a procedure will be conducive to thoroughness, to the ease of learning the subject, and to the discovery of new theorems …

“Bolzano's conception of logical methodology led him to deepen his studies in the foundations of mathematics, and yielded him a rich harvest of theorems. The process begins in §§4-6 of the *Preface *to his [*Betrachtungen*], where Bolzano applies his general methodological principles to the particular case of the foundations of geometry. He criticizes earlier mathematicians for importing conceptions from the theory of motion into their discussions of geometry, pointing out that the *theory of space *is logically antecedent to the *theory of the movement of objects in space, *and must therefore be developed without recourse to the latter theory. Bolzano explicitly criticizes Kant, Mercator, and Kästner on this point; but his remarks can equally well be read as a response to MacLaurin and Newton” (*ibid*., pp. 168-171).

“Bolzano's approach to mathematical problems was characterized by his ability to find new, non-traditional methods, and to use them to deal with problems that until then had withstood all attempts at solution. This approach manifests itself in geometry as well. Bolzano's first mathematical treatise ‘Betrachtungen …’ was aimed at the solution of the then popular problem of parallels. It is not essential that Bolzano solved the problem via a very general concept of similarity, but rather that already in this work he subjected to criticism the contemporary (mostly traditionally Euclidean) interpretation of elementary geometry. In Part 2 of his treatise he tried to define the straight line and the plane, starting from and studying the properties of the simplest geometric object, a pair of points. Thus he defined the notion of the direction of a pair of points, its distance, and in essence constructed geometrically the vector space, indicating also its three-dimensional analogue. In this way he arrived at a result analogous to that obtained in 1799 by C. Wessel in his geometrical interpretation of complex numbers [‘Om Directionens analytiske Betegning’], or later (1844) H. Grassmann in a much more general setting [‘Die Ausdehnungslehre’] (Folta, p. 25).

“Bolzano’s philosophical methodology thus led him to introduce powerful new *concepts *and *techniques *and *conjectures *into mathematics. This is an important aspect of his work, and sets him apart from a thinker like [Johann Heinrich] Lambert, whose methodological observations on the Axiom of Parallels (*Theorie der Parallellinien, *1786) were as shrewd as anything in Bolzano, but who was unable to put them to any actual use in the proving of new theorems. Indeed, the history of mathematics is strewn with similar examples of unexploited anticipations of great advances- recall, for example, Kant’s observations on the possibility of alternative geometries, or D’Alembert’s discussion of the concept of a *limit, *or Leibniz’s dream of a mathematical logic, or Lambert’s remarks on formal axiom systems. Such insights, unless they can be shown to perform some actual mathematical work, tend to be sterile, and are only noticed years later when somebody else has demonstrated their significance. Bolzano managed both to have a crucial insight, and to show how to develop it into new branches of mathematics; unfortunately this accomplishment was no guarantee against being ignored, and the circumstances of his life kept his work from becoming widely known” (Ewald, p. 171).

“Bolzano was born in Prague, the youngest son of an Italian father (an art dealer) and a German mother. He entered the University of Prague in 1796, where he was educated in philosophy, mathematics, and physics. In philosophy, he read the *Metaphysica *(1739) of the Wolffian philosopher, Alexander Gottlieb Baumgarten; in mathematics, he was particularly influenced by his close study of Eudoxus, Euler, and Lagrange, as well as of Kästner’s *Anfangsgründe der Arithmetik *(1758). In 1800 Bolzano took up the study of theology; he was called to the new chair of religion at the University of Prague in 1805. The chair had been established by the Emperor Franz I of Austria to shore up the position of the conservative Catholic hierarchy against the tide of freethinking and republicanism that had been rising in Central Europe since the French Revolution. From the point of view of the political and religious authorities, the appointment of Bolzano was not a happy choice. Although his appointment was confirmed in 1807, his own social, ethical, and religious sympathies inclined to the cause of Enlightenment, and he found himself in perpetual trouble with the authorities. (Among the doctrines that caused him difficulty was his publicly-expressed conviction that one day men would live without kings.) Bolzano was a popular lecturer, and in 1818 was elected head of the philosophy faculty; nevertheless, in 1819 he was dismissed from his professorship, forbidden to publish, and placed under police supervision. For the remaining decades of his life he lived in the countryside, writing on ethics, religion, politics, logic, and the foundations of science.

“Despite the clarity of his arguments, the power of his theorems, and the fruitfulness of his techniques, and although his *Paradoxien des Unendlichen *(1851) was known and admired by Peirce, Cantor, and Dedekind, Bolzano's work in real analysis – the work of an obscure theologian, most of it published by equally obscure Bohemian publishers – seems to have remained entirely unnoticed until Otto Stolz called attention to it in 1881. But by this time Bolzano’s most important results had been independently discovered by Weierstrass and his school” (*ibid*., pp. 171-2).

Parkinson, *Breakthroughs*, p. 265 (*Rein analytischer Beweis*). Ewald, *From Kant to Hilbert* (1996); Folta, ‘Life and scientific endeavor of Bernard Bolzano,’ pp. 11-31 in *Bolzano and the Foundations of Mathematical Analysis*, Jarnik et al (eds.) (1981); Russ, ‘A translation of Bolzano’s paper on the intermediate value theorem,’ *Historia Mathematica* 7 (1980), 156-185).

Three works bound in one vol., 8vo, [Rein analytischer Beweis:] pp. 60; [Betrachtungen:] pp. [xvi], 63, [1], with one folding engraved plate; [Lebensbeschreibung:] pp. lvi, 272 with engraved portrait frontispiece. Contemporary half-roan and marbled boards, paper label on spine with manuscript title, two paper labels on covers with auction numbers (?) (light edge-wear). Old Viennese library stamps on titles and elsewhere with their release on front free endpaper (some light browning and scattered foxing).

Item #4362

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Price:
$35,000.00
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