## History

Newton (1660s) and Leibniz (1680s) independently invented calculus, with Newton framing derivatives as "fluxions" (instantaneous rates of change) and integrals as "fluents" (accumulated quantities). Their methods produced spectacular results — planetary orbits, areas of curves, solutions to differential equations — but relied on "infinitesimals," quantities that were simultaneously nonzero and smaller than any positive number. Bishop Berkeley's 1734 critique ("ghosts of departed quantities") exposed the logical gap, but the practical success of calculus was so overwhelming that rigour was postponed for over a century.

The first steps toward rigour came from Cauchy's *Cours d'analyse* (1821), which introduced the $\varepsilon$-$N$ definition of convergence and the $\varepsilon$-$\delta$ definition of continuity. Cauchy proved the Intermediate Value Theorem and established the Cauchy criterion for convergence, though he still relied on an intuitive notion of the real numbers. His contemporary Bolzano independently proved the IVT (1817) with greater logical care, including a version of the Bolzano-Weierstrass theorem.

Weierstrass, in his Berlin lectures of the 1860s-1880s, completed the programme of rigour. He gave the fully precise $\varepsilon$-$\delta$ definitions still used today, proved the extreme value theorem, and constructed his famous continuous-but-nowhere-differentiable function (1872), shattering the belief that continuous functions are "generically" smooth. His student Heine proved that continuity on closed bounded intervals implies uniform continuity (the Heine-Cantor theorem, 1872).

The crisis of foundations reached its resolution in 1872, when Cantor, Dedekind, and Méray independently gave rigorous constructions of $\mathbb{R}$ from $\mathbb{Q}$. Cantor used equivalence classes of Cauchy sequences; Dedekind used "cuts" partitioning $\mathbb{Q}$ into two classes. Both constructions made the completeness of $\mathbb{R}$ a theorem rather than an axiom, finally providing the logical foundation for everything Cauchy had built.

Riemann's 1854 Habilitationsschrift introduced the integral bearing his name, defining it via upper and lower sums without reference to antiderivatives. This constructive approach revealed that integration is more robust than differentiation: bounded functions with countably many discontinuities are Riemann integrable, while even continuous functions can fail to be differentiable. The Fundamental Theorem of Calculus, connecting the two operations, was finally proved with full rigour.

The convergence theory for power series was developed by Abel (1826), who proved Abel's theorem on boundary convergence, and Cauchy, who introduced the root test and ratio test. The Cauchy-Hadamard formula for the radius of convergence emerged from their work, and Weierstrass's $M$-test (1880s) provided the standard tool for proving uniform convergence of series.

By 1900, the material in this course was essentially complete. The 20th century would bring Lebesgue's integral (1902), which resolved the limitations of Riemann's theory, and the abstract frameworks of topology and functional analysis. But the core ideas — $\varepsilon$-$\delta$ definitions, completeness of $\mathbb{R}$, the Mean Value Theorem, the Fundamental Theorem of Calculus — remain exactly as Cauchy, Weierstrass, and Riemann established them.