Rational and irrational numbers. When and what were the earliest proofs of irrationality of the square root of 2, pi, e? Continued fractions: what are they, what is the earliest use of them you can find, who else used them? Algebraic and transcendental numbers. Cauchy sequences. Dedekind cuts. The Peano axioms for integers. Eudoxus, Hermite, Weierstrass, Hamilton, Cantor. How many real numbers are there?

The Greek technique of exhaustion, Roberval, Tonnicelli, Fermat, D\' escartes, Barrow, Kepler, Cavalieri, Wallis, Wren, Huygens (not necessarily all of these!) and of course, Newton, Leibniz. You might prefer to concentrate on the work of Newon and/or Leibniz, perhaps give an account of the controversy about precedence, and the impact on mathematics for some time to come.

(Of course, complex numbers are not only important in Analysis.) What are the earliest uses of complex numbers that you can find. What are the latest recorded protests against the use of complex numbers that you can find? Geometrical Representaion of complex numbers (Argand, Wessel). Partial fractions, their use in integration, relation with complex functions (John Bernoulli). Exponential, cosine and sine functions (Euler, de Moivre). Use of complex functions to evalutate real integrals such as the integral of cos x divided by a polynomial. (Euler, d'Alembert, Laplace). Cauchy's fundamental work on complex analysis. (Contour integrals, Cauchy's Theorem: to what extent is it really his theorem?) holomorphic functions and Taylor's theorem.

Some key early examples: isochrone (a swinging pendulum problem): James Bernoulli, catenary/heavy string problem (John Bernoulli, Huygens, Leibniz) tractrix (James Bernoulli), Bernoulli's equation. Can you find any of the orginal works? If so, are any of the techniques at alll recongnizable, e.g. separation of variables? Who seems to have first introduced the technique? What about the Integrating factor technique? Clairaut, singular solutions. LInear equations. Series solutions of differential equations. Possibly the history of existence theorems.

A Calculus of Variations Problems are very important in the history of calculus and analysis. The isochrone, brachistochrone, heavy string/catenary (again) Geodesics. John and James Bernoulli. Dido's problem. (What is it, who was Dido anyway, and who considered the problem more recently than Dido?) Euler and Euler's differential equation. Fermat and light: Principle of Least Time. Huygens. Maupertuis: Principle of Least Action. Lagrange, mechanics. Later developments: minimal surfaces?

Where does the formula for the sum of a geometric series originate? Who said: ``divergent series are the invention of the devil ...with the exception of the geometric series, there does not exist in all of mathematics a single infinite series the sum of which has been determined rigorously''? What is the earliest known written proof of divergence of the sum of 1/k ? Is there more than one proof? What about the value of reciproals of squares of integers, or other powers? Convergence/divergence of series, Euler constant. Absolute convergence, rearrangements of series, radius of convergence. Kerala School of Astronomy and Mathematics, Euler, Abel, Cauchy, Bolzano, Gauss.

Can you find the early works of Euler and d'Alembert on the wave equation, in particular, d'Alembert's derivation of the wave equation for a vibrating string, and his solution? Connection with what are now called Fourier series and the (big) question of which functions can be represented as Fourier series. But when did Fourier first work with them, and why? Lagrange, Laplace. Connection with music, J-P. Rameau. The equation of the drum: Euler, Bessel functions.

Alternatively, it would be possible to do a project on partial differential equations with less emphasis on Fourier series. The wave equation in its various forms is obviously an important example. Another supremely important example is the Lapalace equation in its various forms. What is it? (Have you come across it?) Why is it called that? Euler, Lagrange, Gauss, Legendre, Laplace, Fourier, Poisson, Green and (especially in two dimensions) Cauchy, Riemann. Minimal surfaces and the Laplace equation. Characteristics. Monge.

What is a function? What different definitions have various people used over the past four centuries? The ``closed'' form and others. Represntation of functions (which) by Taylor series, Fourier series. Special functions: sin, cosine, hyperbolic, Bessel, elliptic. (How did some or all of these first arise, do you think?) Continuous, differentiable, nowhere differentiable, spacefilling functions. Weierstrass. (He is a possible subject of a biographical project.) Functions defined by series, uniform approximation.

9. The History of Dynamical Systems.

In some ways this will be a harder topic than the others because interest in dynamics in the twentieth century was high - and recent history tends not to have textbooks written about it. So any work on this topic would need more looking at direct mathematical sources. A good place to start might be the introduction to W. Szlenk's Dynamical Systems (QA614.8.S99) which has some limited historical comments. Key people whose bibliographical details are worth looking up include Henri Poincare, George Birkhoff, Mary Cartwright, Stephen Smale. Some precise references are given in ``suggested books''