1. A theorem that a picture should not prove

A pamphlet published in Prague in 1817 announced a purely analytic proof of a familiar result: between two arguments at which a continuous expression takes opposite signs, its equation has a real root. The striking part of the title was not the theorem but the demand for a purely analytic proof. Draw a continuous curve below the horizontal axis at one end and above it at the other. Surely it must cross the axis somewhere between.

Bernard Bolzano was dissatisfied with that surely. A picture can make the conclusion persuasive without establishing which properties of numbers and functions guarantee it. If the result is to support other arguments, including arguments involving objects that cannot usefully be pictured, it needs a justification that does not borrow its decisive step from the reader's visual expectations.

This resembles the difficulty we encountered at the beginning of Euclid. A construction uses the intersection of two circles, while the stated assumptions do not fully justify why those circles must meet. The drawing carries an unannounced burden. Bolzano was not challenging the intermediate value theorem. He was challenging the adequacy of a particular way of defending it.

By the nineteenth century, analysis increasingly dealt with infinite sums, complicated functions, and successive limiting operations. A finite drawing could not display all the behavior under discussion. This did not make pictures useless. It made their evidential limits impossible to ignore. The problem was to say what continuity meant and what properties of the number system permitted a proof to proceed.

Bolzano was not employed as a mathematics professor. A Catholic priest, he held a chair concerned with religious doctrine in the philosophical faculty at Prague. Mathematics was a sustained intellectual commitment rather than the title of his post. In 1819 he was dismissed because of religious and political positions considered dangerously liberal. Investigation, surveillance, restrictions on correspondence, and limits on publication burdened his subsequent life.

Those circumstances help explain his work's limited influence, but they do not supply a complete single-cause explanation. Nor was the pamphlet literally unread. It was reviewed in the German-speaking world, known to mathematicians in Prague, and related Bolzano publications were known to Abel. The missing thing was a continuing network that made these arguments normal starting points in teaching and research.

Later nineteenth-century writers recognized his anticipation of important ideas about convergence and series; an 1881 study helped establish his historical position. A bibliographical trace and a canonical origin are different things, as the preceding chapter showed. Bolzano's pamphlet contained techniques later seen as foundational, but its eventual importance should not be projected backward as an already functioning school.

One can reasonably imagine that wider circulation might have reduced the need for later rediscovery. That is a counterfactual, not an observable event. The firmer claim is that important arguments existed before the institutions and textbooks that would make related arguments standard. The history of rigor therefore includes a history of access, transmission, and professional location, not only a sequence of definitions.

2. Cauchy did not expel infinitesimals

The chapter's title needs quotation marks. A familiar story says that eighteenth-century calculus rested on mysterious quantities smaller than anything imaginable but not zero; Cauchy replaced them with limits; Weierstrass then formalized everything; rigor arrived. The first transition in that story is already misleading. Cauchy did not remove infinitesimals from his mathematics.

His Cours d'analyse of 1821 explicitly treated their properties as fundamental. A variable whose successive numerical values decrease indefinitely toward zero becomes, in his terminology, infinitesimal. In 1823 he published a summary of his lectures on infinitesimal calculus. The word appears in the course's very title. Whatever his achievement was, it cannot accurately be described as refusing to use the concept. The Bibliothèque nationale de France records the 1821 edition.

Cauchy's limit language describes a variable taking successive values that approach a fixed value so closely that their difference can become as small as desired. Continuity is expressed through small changes in the independent variable producing small changes in the function, including the formulation that an infinitesimal increment produces an infinitesimal change. Series convergence is explained through partial sums approaching a finite value.

Much of the material of later analysis is already here: arbitrarily small differences, control through estimates, and limiting behavior. What is not uniformly present is the later standardized syntax that makes the order and dependence of choices explicit. When those dependencies remain partly implicit, the reader must supply them. Sometimes the intended arrangement is correct; sometimes an unnoticed change of order turns a true claim into a false one.

Cauchy also used highly precise inequalities in proofs. It would be as misleading to describe him as merely intuitive as to credit him with having completed every later foundational clarification. His work combines methods that do not fit a simple division between an informal past and a formal present.

Historians disagree about how to understand his infinitesimals. One approach reads them principally as variables tending to zero and translates the arguments into later limit language. Another stresses operations in which infinitesimal quantities seem to play a more substantial role: comparison of orders, substitution into functions, and the use of inputs depending on an indefinitely increasing index. From that perspective, treating every infinitesimal as removable verbal shorthand loses something important.

The counterwarning is equally necessary. A twentieth-century structure may provide an illuminating interpretation of a nineteenth-century calculation without establishing that its author possessed that structure or its ontology. Cauchy did not have modern hyperreal fields or a formal transfer principle. A reconstruction can make a historical argument intelligible without becoming the historical argument itself. A research discussion of the competing readings makes the interpretive dispute explicit.

The safe common ground is substantial: Cauchy used infinitesimals and made them important to his exposition. The disagreement concerns exactly what work they did and how faithfully different later languages reproduce it. As elsewhere in this series, the ability to restate an older author elegantly in our own terms does not show that those were the terms in which the author originally thought.

3. The theorem has exceptions

The 1821 textbook contained a theorem about sums of continuous functions. In the ordinary modern pointwise reading, its conditions are insufficient: a convergent series of continuous functions can have a discontinuous sum. Abel drew attention to this in 1826, saying that the theorem admitted exceptions. His example involved a trigonometric series, the kind of expansion made central by Fourier's work on heat. It was not the power-series example later classrooms often substitute for it.

Every individual term may behave continuously while the infinite limiting object fails to do so. There is no contradiction in that statement. Continuity at each finite stage controls that stage; it does not automatically provide one control that survives passage through infinitely many stages. The missing distinction concerns how convergence depends on the point at which the functions are evaluated.

For a simple modern illustration, xⁿ is continuous on the interval from zero to one. At each point below one it tends to zero; at one it remains one. The limiting function jumps at the endpoint. This is a sequence rather than Abel's historical trigonometric example, and it should be labeled as an illustration, not silently substituted into his paper. It isolates the phenomenon with very little calculation.

Pointwise convergence allows the stage required for a given accuracy to depend on the chosen point. Uniform convergence requires one sufficiently late stage to work at every point in the specified domain. Near one, xⁿ can remain large however late the stage if the point is chosen sufficiently close to one. That is why the common control is absent. The order of choices is mathematical content, not verbal decoration.

Cauchy returned to the series theorem in 1853. He strengthened his condition on the remainder, using language that it should always become infinitesimal, and examined a trigonometric example at an input changing with the index. Standard historical readings relate this stronger condition to uniform convergence. Other readings interpret the universal requirement as including infinitesimal inputs and reconstruct it in an extended number system.

The latter approach can establish a coherent modern interpretation. It cannot by itself show that Cauchy's 1821 readers shared precisely that semantics. Conversely, reading the earlier statement exclusively through later pointwise terminology risks overlooking differences in the earlier conceptual setting. The disagreement is not resolved simply by announcing that Cauchy was either obviously wrong or secretly possessed a later theory in complete form.

Around 1847 Seidel and Stokes published conditions arising from this cluster of problems. Cauchy's return followed in 1853; a standard, widely usable concept developed through further work. The decades involved show a mathematical community clarifying what a statement should mean and which hypothesis makes its intended conclusion valid. They do not imply that a logically precise proposition was literally neither true nor false for thirty years.

Nor are modern mathematical claims always immediately proved or refuted. Open problems remain, and some questions are independent of specified axioms. The historical contrast is between unstable or incompletely articulated conditions and the later discipline of stating those conditions explicitly. Rigor did not abolish uncertainty; it made a particular source of uncertainty easier to identify.

The eventual remedy was not to catalogue every misbehaving example. It was to formulate a sufficient condition whose scope could be checked. Under uniform convergence, the limit of continuous functions is continuous. Without it, continuity may still hold, but it no longer follows from the same theorem. The price of precision is often a longer hypothesis. Its reward is knowing exactly what has and has not been promised.

4. A schoolteacher's route

Weierstrass did not follow the familiar trajectory of a precocious university prodigy. His studies at Bonn, undertaken under his father's influence, did not lead to a successful degree. He later trained as a teacher and spent much of the 1840s and early 1850s teaching in Prussian secondary schools. Some of his serious research appeared in school annual publications with limited visibility in the main European mathematical networks.

The turn came in 1854 when work on Abelian functions attracted attention and brought an honorary doctorate from Königsberg. In 1856 he moved to Berlin, first through a technical teaching appointment and then into the university and Academy. A full professorship followed in 1864. He was around forty when he emerged from provincial school teaching into metropolitan mathematical prominence.

It is tempting to connect his long teaching experience with the step-by-step form of his later foundational lectures. That is a plausible interpretation, not a documented psychological law. What can be observed is that his mature presentation was a continuing teaching project, repeatedly revised in Berlin rather than a single finished youthful system. The precision associated with his name was developed through courses as well as papers.

His relative reluctance to publish a comprehensive account of those foundations is a material fact of the story. He certainly published research—the paper that changed his career is an obvious example—but his influence far exceeded the reach of a definitive textbook written by himself. Students' lecture notes became major evidence for the development of his ideas from the 1860s through the 1880s.

Hermann Amandus Schwarz's notes from the 1861 differential calculus lectures are especially important. Schwarz himself described them as compressed and imperfect. Consequently, a statement that Weierstrass defined a concept in 1861 may rest on a student's manuscript rather than on a book that Weierstrass issued that year. The distinction does not disqualify the evidence; it tells us what kind of evidence it is.

As his health worsened, a recognizable classroom mechanism developed. He lectured seated, selected students wrote dictated material on the board, and others copied it. Notes were recopied and circulated beyond Berlin and Germany. Some changed hands through purchase as well as exchange. Weierstrass himself could consult or borrow students' records when preparing later teaching.

A sparsely publishing scholar was therefore not necessarily concealing his ideas. The chain ran from the chair to spoken exposition, from speech to blackboard, from blackboard to notebooks, and from notebooks to distant readers. Formal editorial publication could come at the end of a long period of active circulation. The history cannot be reconstructed by counting printed books alone.

Earlier chapters followed a scraped parchment and private working papers never intended for publication. Lecture notes supply a third material setting, but this one was not merely an accidental survival. It was an operating institution of transmission. Someone had to listen, write, copy, exchange, and sometimes pay. How far an unpublished idea traveled depended on that labor as well as on its mathematical quality.

5. Name an error

There is no single publication in which Weierstrass suddenly handed the world the finished modern definition of a limit. Nevertheless, the central change can be explained without drowning in notation. Instead of asking the reader to imagine a quantity moving indefinitely closer to another, specify a challenge and a response. Name any positive error tolerance. The proof must supply a restriction that forces the error below that tolerance.

For a function limit, the distinction between input and output is essential. To say that f(x) tends to L as x tends to a means this: for every positive ε, there is a positive δ such that whenever x lies in the domain and 0 < |x − a| < δ, one has |f(x) − L| < ε. The δ controls how close the input is to a; the ε controls how close the output is to L. Mixing those roles produces something other than the intended definition.

The order matters. The challenger names ε first. The response δ may depend on ε, and possibly on the fixed point a and the function, but cannot be chosen separately after seeing every particular x inside the permitted neighborhood. Once the neighborhood has been chosen, the guarantee must cover all eligible inputs there. A precise statement records both what may vary and what must remain fixed.

For a sequence, the response is a stage N rather than an input neighborhood: after N, every term lies within ε of the proposed limit. The picture of motion remains a helpful intuition, but the definition no longer needs a literal temporal journey. It states an arrangement of quantifiers and inequalities that another mathematician can examine independently.

This is not the claim that checking infinitely many tolerances means testing them one by one. A proof establishes a general method or argument covering arbitrary positive tolerances. Nor does the language remove every demand on understanding. It relocates the burden from an imagined process to explicit dependencies. Someone who disagrees with the intuition can still ask whether the inequalities follow.

The documentary history is gradual. Student records from 1861 provide early clear evidence; the Berlin course cycle developed from the 1860s onward; later notes show continuing refinement. Uniform convergence makes the difficulty of assigning a unique birthday particularly clear. Gudermann used language about equal rates of convergence in 1838, but the relation of that wording to the full later concept needs interpretation.

A Weierstrass paper dated 1841 was not printed until 1894. It has often been treated as decisive evidence for an already complete early concept. Modern research cautions that the printed text underwent revisions and that its exact formulation cannot simply be carried back to the manuscript's assigned date. Seidel and Stokes's public work around 1847 and Cauchy's 1853 reformulation belong to related but not identical paths. Klaus Viertel's study of the lecture notes and editions specifically examines this dating problem.

Who first noticed the phenomenon? Who first wrote an adequate condition? Who first published it? Who made it a general organizing principle in analysis? These are different questions. The answers need not be the same person. A history that collapses them into one claim of invention gives a cleaner anniversary and a less accurate account of how a concept became usable.

6. The convergence of 1872

On July 18, 1872, Weierstrass presented a continuous function without a finite derivative at any point. Continuity and differentiability had often been associated through the image of a smooth curve interrupted by an occasional corner. The new example forced them apart everywhere. A continuous graph need not possess the local linear approximation expressed by a derivative.

It is better not to call this a curve with a corner at every point. A corner normally has two well-defined one-sided directions; nowhere differentiability need not arise that way. Nor are derivative and every possible geometric notion of tangent simply synonymous. The exact result concerns the failure of the difference quotient to approach a finite limit at every point.

The shock was not just that another strange function had joined a collection. Visual continuity could no longer authorize a general assumption of local smoothness. If a theorem required differentiability, its hypothesis had to secure differentiability. The absence of breaks in a picture did not do that work. Related earlier constructions existed, some unpublished, so the episode should not become a claim that no one before Weierstrass had imagined severe irregularity.

The dates of presentation and public circulation also differ. Du Bois-Reymond's 1875 publication helped bring the example to a wider audience. Hermite's famous language of horror and aversion belongs to an 1893 letter to Stieltjes. Poincaré's remarks about logical monsters come later in the century, around 1899. They are evidence of subsequent responses, not a transcript of what the Berlin Academy audience said in July 1872.

That year also brought several influential constructions of the real numbers. Dedekind published Continuity and Irrational Numbers, developed from a difficulty encountered in teaching in 1858. A numerical theory of continuity should not perpetually borrow the geometrical line to guarantee that the necessary numbers exist. His cuts divide the rationals into ordered parts and supply numbers corresponding to divisions not represented by any rational. Dedekind's essay is available in English translation.

Cantor's 1872 work constructed numbers through suitable sequences of rationals, identifying sequences whose difference tends to zero. Heine published a related treatment the same year. These were not three wholly independent inventions of irrational numbers, which mathematicians had used for centuries. Heine and Cantor belonged to connected discussions, and other approaches also existed.

The year is better understood as a meeting point. Pressure accumulated around limits, irrational quantities, functions, and continuity produced several public constructions that could be compared. A nonempty bounded-above set of real numbers has a least upper bound; a Cauchy sequence of real numbers has a real limit. Those carefully qualified properties can now be established within the construction instead of being borrowed from the imagined completeness of a drawn line.

Not every bounded set has a greatest member, and not every sequence converges. Precision must survive even in a summary of precision's history. Cuts and equivalence classes of sequences are different constructions of structures satisfying the required properties. Their later isomorphism explains why they can serve the same mathematics; it does not make them two excavations of an identical ready-made object hidden in nature.

7. The word arithmetization

Arithmetization of analysis is now an easy historical label. Its own history is less straightforward. An early traceable use occurs in Kronecker's writing in 1887, but what he wanted under that heading was not simply what later histories grouped together under the names Weierstrass, Dedekind, and Cantor. He emphasized integers and finite constructive procedures, and objected to important parts of the treatment of completed infinite objects associated with his contemporaries.

It would therefore be anachronistic to make arithmetization the common public slogan of a united 1870s movement. Klein's 1895 essay using the word helped it become a broader retrospective description. A term can migrate from a particular program into a historian's category encompassing programs its early user opposed. That does not make the category useless, but it requires care about whose language is being reported.

The mathematical development involved several connected projects, not one replacement operation. Limits needed explicit control. Continuity required separation from differentiability. Infinite series required conditions for interchanging limiting operations. The real number system required a treatment of completeness. Integration, too, had to be examined beyond the safe territory of smooth curves and continuous functions.

Riemann's 1854 habilitation dissertation on trigonometric series investigated the definite integral and the range of functions to which it could apply. It was published posthumously in 1867, not to be confused with the 1868 publication of his geometrical lecture discussed in the preceding chapter. The distinction matters because the two works share an early date and a later posthumous appearance while addressing different questions. The Göttingen library catalogue records the 1867 edition.

For a bounded function on a finite interval, the issue is not simply whether the graph looks continuous. Discontinuities can be allowed under appropriate conditions. Darboux's 1875 treatment organized the problem through upper and lower sums: partition the interval and use upper and lower bounds of the function on each part. Their relation provides a systematic way of describing when the integral exists.

This does not mean that every wildly discontinuous function becomes integrable. It means that the boundary can be stated mathematically rather than by an informal appeal to a drawable curve. The construction invites extreme examples because they expose exactly which hypothesis is doing the work. What looks pathological in one framework becomes a diagnostic instrument in another.

The scope of functions, the meaning of convergence, the completeness of numbers, and the definition of integration thus moved together. Their connections explain why the rebuilding took decades and why no single definition could complete it. A correct rule about limits still needs a number system in which the relevant limits exist, and a definition of function still leaves open which functions satisfy the assumptions of a particular theorem.

The historical label should not flatten these distinctions. Arithmetization names a family of attempts to put analysis on explicitly numerical foundations. It does not name a unanimous party, a single date, or an argument that made every other rigorous approach impossible. Its later success can be recognized without rewriting its internal disputes out of existence.

8. What entered when the fence moved

There was a cost, but we must not invent contemporary complaints to dramatize it. The evidence assembled here does not establish that, before the 1880s, observers were already denouncing the new formal language itself as impossible to read or teach. Hermite's 1893 letter and Poincaré's later comments are strong evidence for a different claim: discomfort with continuous functions lacking derivatives. Their targets were mathematical objects, not simply unfamiliar notation.

Poincaré also distinguished a logically secure order of exposition from the order in which a learner most naturally comes to understand. A foundation can arrange dependencies impeccably without supplying the best first encounter with a subject. That educational issue remains different from whether a theorem is true or a definition adequate. Historical quotations should not be made to answer all three questions at once.

At least three consequences are visible. Geometry ceased to have the last word in many analytic proofs. Explicit definitions admitted objects excluded by familiar pictures. And the order most suitable for justification could diverge from the order most suitable for learning. The middle consequence is especially striking because it reverses a common expectation about rigor.

The project seems at first to be one of exclusion: remove what cannot be explained and allow only secure objects to remain. Yet the new definitions opened the door wider. An inherited picture suggested that continuous curves should usually have tangents. An explicit definition of continuity contains no such promise. If no theorem derives differentiability from the stated conditions, a continuous nowhere-differentiable function has every right to exist within that theory.

The standard did not fail because it admitted unexpected objects. It clarified what it actually required. Once admissibility rests on specified properties rather than resemblance to familiar examples, unfamiliar constructions can qualify. We encountered a related pattern when symbolic algebra loosened the demand that every sign already possess a direct interpretation as an ordinary magnitude. The analogy is illuminating, though it should not be mistaken for a universal law of all mathematical change.

Moving the fence from what can comfortably be pictured to what can be precisely defined changes the territory inside it. Definitions still exclude many things, and their consistency and usefulness remain questions. But they may exclude less than an older intuition because they make fewer unacknowledged promises. The objects once dismissed as monsters can reveal those promises by failing to keep them.

Finally, modern nonstandard analysis, developed by Abraham Robinson in the 1960s, gave a rigorous setting containing nonzero infinitesimals. Appropriate extensions of the real numbers support differential and integral reasoning through precise structural principles. This does not show that nineteenth-century limit analysis was a mistake. It shows that infinitesimal reasoning and lack of rigor are not identical.

The later dominance of one foundational presentation is a historical fact, not a proof that logic allows no alternative. Cauchy's work should not be assigned Robinson's machinery in advance, but Robinson's achievement prevents us from treating the old vocabulary as inherently disreputable. What matters is what the objects are, which rules govern them, and why those rules justify the conclusions.

The infinitesimal was not simply driven out. Its place in explanation and proof was reorganized, while other methods came to dominate instruction. The quotation marks around the title belong there to the end. Mathematics continued to calculate, but the authority to decide whether an argument stood securely had changed hands—and the resulting mathematics was larger than the pictures it left behind.