Solution
= Solution
{c}
Standards of <mathematical proof> changed between the seventeenth and nineteenth centuries because both the objects studied and the operations performed on them changed. A finite geometrical construction, an infinite expansion, and an abstract <group> raise different questions about justification. There was no universal seventeenth-century indifference to rigour followed by its sudden nineteenth-century invention. The more defensible thesis is that successful methods repeatedly exposed unstated assumptions; later work made some of those assumptions into explicit definitions, hypotheses and existence arguments.
<Pierre de Fermat>'s <adequality> illustrates the distinction between an effective method and its justification. For $F(a)=a(L-a)$, comparing neighbouring expressions gives
$$
F(a+e)-F(a)=e(L-2a)-e^2.
$$
Dividing by $e$ and suppressing the remaining term in $e$ produces $a=L/2$. There is an independent finite verification:
$$
\boxed{F(a)=\frac{L^2}{4}-\left(a-\frac L2\right)^2\leq\frac{L^2}{4}.}
$$
Thus the result can be secure even when the general procedure raises a question: why is the increment first nonzero for division and then discarded? In the <parabola> construction of question 2, the same issue appears in replacing a strict geometrical comparison by <adequality> and extracting the <subtangent>. A historical explanation must describe this operation on its own terms rather than silently equate it with a modern <derivative>.<Fermat's rule for maxima and minima by the Garden of Archimedes>[Procedure of adequality and tangent example]{ref}
<Isaac Newton> and <Gottfried Wilhelm Leibniz> supplied different languages for the new <calculus>. <Isaac Newton>'s <Introduction to the Quadrature of Curves by Isaac Newton>, first published with the 1704 Opticks, presents quantities generated by motion and considers their rates of change. Its mature language of first and last ratios seeks exact relations as increments vanish; it should not be confused with simply asserting that a small error is harmless. <Gottfried Wilhelm Leibniz>'s 1684 <Nova methodus pro maximis et minimis by Gottfried Wilhelm Leibniz> presents a calculus of differentials and operational rules. Their common problem was to explain why procedures using vanishing increments establish exact results, while their chosen objects and modes of explanation differed.<Introduction to the Quadrature of Curves by Isaac Newton>[Opening discussion of generated quantities and fluxions; 1704 publication distinguished from the archived 1711 Latin and 1710 English texts]{ref}<Nova methodus pro maximis et minimis by Gottfried Wilhelm Leibniz>[Acta Eruditorum, October 1684, 467–473, original pages reproduced by the Mathematical Association of America]{ref}
<George Berkeley>'s <The Analyst by George Berkeley> (1734) exposes the logical pressure point. For the cube,
$$
\frac{(x+h)^3-x^3}{h}=3x^2+3xh+h^2\qquad(h\ne0).
$$
The finite identity does not become a legal division at $h=0$. The modern repair is to keep $h\ne0$ throughout the identity and prove that its right-hand side tends to $3x^2$. That <limit> is different from a quotient evaluated at zero. <George Berkeley>'s criticism therefore concerns the grounds of inference, not whether the established rules often produce correct answers. His theological polemic also reminds us that standards of <mathematical proof> could be contested outside specialist calculation.<The Analyst by George Berkeley>[1734, §§13–18, critique of evanescent increments and changes of hypothesis]{ref}
The infinite introduced another burden. <Joseph Fourier>'s 1822 use of <trigonometric series> to represent arbitrary <functions> greatly extended what calculations in <mathematical physics> could express. But deriving <Fourier coefficients> by <orthogonality> only shows what those <coefficients> must be if the proposed operations are legitimate. It does not prove <convergence>, equality with the initial data, or interchange of an <integral> and an infinite <sum>. The example of a constant expanded in a <Fourier sine series> makes a hidden qualification visible: the terms all vanish at the endpoints, even when the assigned data do not. The correct theorem must specify the domain and mode of representation.<Théorie analytique de la chaleur by Joseph Fourier/Propagation de la chaleur dans un solide rectangle infini>[§233, broad representation claim and integral coefficients]{ref}
<Peter Gustav Lejeune Dirichlet>'s 1829 memoir supplies a particularly precise example of changing proof obligations. It criticises a convergence argument of <Augustin-Louis Cauchy>'s which used complex substitution into an arbitrary <function> and an insufficient comparison between signed <series>. <Peter Gustav Lejeune Dirichlet> supplies the instructive pair
$$
a_n=\frac{(-1)^n}{\sqrt n},\qquad
b_n=\frac{(-1)^n}{\sqrt n}+\frac1n.
$$
The first <series> converges by the <alternating series test>; the second diverges because their difference is the <harmonic series>. Nevertheless $b_n/a_n=1+(-1)^n/\sqrt n\to1$. Thus asymptotic equality of terms does not transfer convergence between conditionally convergent signed <series>. A comparison principle valid for positive terms needs additional justification when cancellation is involved. The example refutes a mode of inference rather than merely correcting a numerical answer.<Sur la convergence des séries trigonométriques by Peter Gustav Lejeune Dirichlet>[1829, opening objections and explicit counterexample]{ref}
<Augustin-Louis Cauchy>'s own foundations programme is part of this change. The 1821 <Cours d'analyse by Augustin-Louis Cauchy> makes <limits>, <continuity> and <series convergence> central to elementary reasoning. Yet its <Sum theorem of Cauchy> also shows why announcing rigour does not make every proof adequate. In modern terminology, continuous <functions> can have a discontinuous <pointwise limit>. An explicit <series> version is
$$
\sum_{n=1}^{\infty}(x^n-x^{n+1})=
\begin{cases}x&0\leq x<1,\\0&x=1.
\end{cases}
$$
Each term is a <polynomial> and the <partial sum> is $x-x^{N+1}$. The convergence does not hold in <uniform convergence>: with $x_N=1-1/N$, the error $x_N^{N+1}$ tends to $e^{-1}$ rather than zero. The modern <uniform limit theorem> repairs the inference by requiring one bound on the error to work throughout the domain. This example diagnoses the missing control under a pointwise reading; it does not settle every historical interpretation of <Augustin-Louis Cauchy>'s infinitesimal language.<Cours d'analyse by Augustin-Louis Cauchy>[Chapter VI, §I, sum theorem and preceding proof]{ref}
The issue is the order of dependence in quantifiers. <Pointwise convergence> permits the threshold $N$ to depend on both the error tolerance and the point. <Uniform convergence> requires a threshold depending only on the tolerance. If $f_N$ is continuous near $x_0$ and the convergence is uniform, choose $N$ so that $|f-f_N|<\varepsilon/3$ nearby, then use the continuity of this fixed $f_N$ to make $|f_N(x)-f_N(x_0)|<\varepsilon/3$. The <triangle inequality> proves $|f(x)-f(x_0)|<\varepsilon$. Writing out this proof reveals what a vague statement that the tail is small leaves unproved.
Another nineteenth-century development concerns the foundations of the <real numbers> themselves. <Richard Dedekind>'s <Continuity and Irrational Numbers by Richard Dedekind> (1872) asks for an arithmetical account of the continuity often borrowed from a drawn line. A <Dedekind cut> separates the <rational numbers> into lower and upper classes. For instance,
$$
A=\{q\in\mathbb Q:q<0\text{ or }q^2<2\},\qquad B=\mathbb Q\setminus A
$$
defines a gap in the <rational numbers> corresponding to $\sqrt2$. The cut is determined without assuming that a rational number fills it. The extended number system then supports existence theorems that a geometrical picture alone had suggested.<Continuity and Irrational Numbers by Richard Dedekind>[1872 preface and §§III–V; English translation in Essays on the Theory of Numbers]{ref}
This matters for apparently elementary proofs. Suppose $f$ is continuous on $[a,b]$ with $f(a)<0<f(b)$. Repeated <bisection> constructs nested <intervals> whose endpoint values have opposite signs and whose lengths tend to zero. To conclude that there is a common limiting point, one needs the <completeness of the real numbers>. Continuity then forces its value to be zero. Over the <rational numbers>, $f(q)=q^2-2$ changes sign between $1$ and $2$ but has no rational zero. Thus even a proof that avoids pictures must explain the existence of its limiting object. <Richard Dedekind>'s account makes that requirement an explicit part of the mathematical setting, rather than an intuition about a continuous line.<Continuity and Irrational Numbers by Richard Dedekind>[§VII, existence of limits and its relation to completeness]{ref}
There is a parallel change in <algebra>. <Heinrich Weber>'s 1882 <definition> in question 6 describes arbitrary elements through laws of composition. To assess such a definition one proves implications between <axioms>. The passage's <finite set> hypothesis implies that cancellative multiplication has an <identity element> and <inverse elements>; without that hypothesis, the positive <integers> under <addition> supply a counterexample. This is a new kind of proof obligation: one must establish that a proposed abstract <definition> has the intended consequences, rather than rely on their familiarity in particular numerical or permutation examples.<Beweis des Satzes über Primzahlen und quadratische Formen by Heinrich Weber>[1882 opening definition]{ref}
\b[The principal change was towards explicit control of operations, hypotheses and existence.] Effective symbolic algorithms persisted, and geometric reasoning remained valuable. What became increasingly demanding was the passage from success in examples to a general claim: an infinitesimal step needed justification, an infinite expansion needed a convergence theorem, a limiting point needed a complete number system, and an abstract definition needed its consequences proved. The examples also prevent a simple progress story: nineteenth-century mathematicians disagreed about admissible reasoning, and a programme devoted to rigour could itself become the object of a successful critique.