Augustin--Louis Cauchy (1789--1857), the son of wealthy devout parents, entered the École Polytechnique, one of the famous schools founded by Napoleon to train engineers for the French Army, in 1805. He worked as an engineer on bridges and railways until returning to Paris in 1813. By this time he had already published several papers in geometry, number theory and determinants. Now he started extensive research in Complex Variables and submitted his work to the Académie Francaise. He began winning prizes for his work on hydrodynamics and PDE's. Eventually he was rewarded by membership in the Academy and appointment as Professor at the École Polytechnique.

He published prolifically and was known as a careful teacher, which he enjoyed. He submitted so much to the Journal of the Academy, that eventually they brought in special rules limiting the amount that a single author could publish in one issue.

The early teachers at the École Polytechnique set a precedent of writing textbooks and Cauchy followed the tradition. He wrote three texts on Calculus, in 1821, 1823 and 1829, and they set the pattern which nearly all calculus texts have followed to this day - definitions of limits and of continuous and differentiable functions, numerous graphical and analytic examples and exercises, simple applications, definite integration, d.e's. He didn't define real numbers or functions, but he did have dependent and independent variables. He defined derivatives as the limit of the difference quotient. Defining the definite integral and not the antiderivative enabled him to integrate non--continuous functions as we do. He proved the (Cauchy) Mean Value Theorem via Rolle's Theorem and used it to prove the Fundamental Theorem of Calculus.

Cauchy's career took a dramatic turn in 1830 when the Bourbon king Charles X was deposed and Louis--Philippe, a descendant of Napoleon, became king. Cauchy refused to take the oath of allegiance and went into exile till 1838, only returning to teaching in 1848 when the oath was no longer required. He continued to publish almost till his death. His name today is attached to several theorems in series, group theory, complex variable and differential equations, and he was regarded as one of the two foremost mathematicians of his time, the other being Gauss. However he is also known for his bad treatment as Secretary of the Academy of younger mathematicians such as Abel and Galois. His work also contained "fruitful errors", the chief one being a faulty proof that the limit of a sequence of continuous functions is continuous, which led to the definition of uniform convergence.

(a) Limits and infinitesimals

When the values successively attributed to the same variable approach a fixed. value indefinitely, in such a way as to end up by differing from it as little as one could wish, this last value is called the limit of all the others. So, for example, an irrational number is the limit of the various fractions which provide values that approximate it more and more closely. In geometry, the surface of a circle is the limit to which the surfaces of inscribed polygons converge as the number of the sides steadily increases, etc.

When the successive numerical values of the same variable decrease indefinitely in such a way as to fall below any given number, this variable becomes what one calls an infinitesimal or an infinitely small quantity. A variable of this kind has zero for its limit. When the successive numerical values of the same variables steadily increase in such a way as to exceed any given number, one says that this variable has positive infinity as its limit, indicated by the sign oo, when a positive variable is concerned, and negative infinity, indicated by the sign - oo, when a negative variable is concerned. The positive and negative infinities are jointly denoted by the name infinite quantities.

(b) Continuous functions

Among the objects which belong to the consideration of the infinitely small one must place notions relative to the continuity or discontinuity of functions. Let us first of all examine functions of a single variable from this point of view. Let f(x) be a function of the variable x and suppose that for each value of x between two given limits this function always takes a unique and finite value. If, having a value of x between these limits, one attributes to the variable x an infinitely small increase a, the function itself increases by the difference

f(x + a) - f(x),

which depends simultaneously on the new variable a and the value of x. This done, the function f(x) will be, between the two limits assigned to the variable x, a continuous function of this variable if, for each value of x intermediate between these limits the numerical value of the difference

f(x + a) - f(x)

decreases indefinitely with a. In other words, the function f(x) will remain continuous with respect to x between the given limits if, between these limits an infinitely small increase in the variable always produces an infinitely small increase in the function itself. One says furthermore that the function f(x) is, in the neighbourhood of a particular value attributed to x, a continuous function of this variable, whenever it is continuous between two limits of x, however close, which contain that value of x.

(c) Convergence

A sequence is an infinite succession of quantities u₀, u₁, u₂, u₃,... which succeed each other according to some fixed law. These quantities themselves are the different terms of the sequence considered. Let

s_n = u₀ + u₁ + u₂ + ... + u_{n - 1}

be the sum of the first n terms, where n is some integer. If the sum s_n tends to a certain limit s for increasing values of n, then the series is said to be convergent , and the limit in question is called the sum of the series. On the contrary, if the partial sum s_n approaches no fixed limit as n increases indefinitely, the series is divergent and has no sum. In either case, the term corresponding to the index n, namley u_n is called the general term . It suffices to give this general term as a function of the index n in order for the sequence to be completely determined.

One of the simplest sequences is the geometric progression 1, x, x², x³, .., whose general term is xⁿ, that is, the nth power of x. If one sums the first n terms of this sequence, one finds

1 + x + x² + .... + x^n-1 = 1/(1 - x) - xⁿ/(l - x)

and, since the magnitude of the fraction xⁿ/(l - x) either converges to zero for increasing values of n or increases beyond all limits, depending on whether one supposes the magnitude of x to be less than or greater than unity; one must conclude that under the first hypothesis the progression 1, x, x², x³, .. defines a convergent series whose sum is I/(l - x), while under the second hypothesis the same progression defines a divergent series which has no sum.

By the principles established above, for the series

(1) ... u₀ + u₁ + u₂ + ... + u_n + u_n+1

to converge it is necessary and sufficient that the sums s_n = u₀ + u₁ + u₂ + ... + u_{n - 1} converge to a fixed limit s as n increases; in other words, it is necessary and sufficient that for infinitely large values of n the sums s_n, s_n+1, s_n+2 differ from the limit s, and hence from each other, by infinitesimal quantities. Besides, the successive differences between the first sum s_n and each of the following are respectively determined by the following equations:

s_n+1 - s_n = u_n , s_n+2 - s_n = u_n + u_n+1 , s_n+3 - s_n = u_n + u_n+1 + u_n+2,

Hence., for the series (1) to converge, it is necessary that the general term u_n decrease indefinitely as n increases; but this condition is not sufficient, and it must also be true that, for increasing values of n, the different sums u_n + u_n+1 , u_n + u_n+1 + u_n+2... that is, the sums of quantities u_n , u_n+1 , u_n+2, taken in arbitrary number from the first will always have a magnitude which is less than any assignable limit. Conversely, when these various conditions are satisfied, the convergence of the series is assured.

(d) Derivatives

When the function y = f(x) is continuous between two given limits of the variable x, and one assigns a value between these limits to the variable, an infinitesimal increment Dx of the variable produces an infinitesimal increment in the function itself. Consequently, if we then set Dx = h, the two terms of the difference quotient

Dy/Dx = (f(x + h) - f(x))/h

will be infinitesimals. But whereas these terms tend to zero simultaneously, the ratio itself may converge to another limit, either positive or negative. This limit, when it exists, has a definite value for each particular value of x; but it varies with x. Thus, for example, if we take f(x) = x^m, m being a positive integer, the ratio of the infinitesimal differences will be

((x + h)^m -x^m)/h = mx^m-1 + [m(m-1)/1.2] x^m-2h + .... + h^m ,

and it will have for its limit the quantity mx^{m - 1} that is to say, a new function of the variable x. The same will hold generally; only the form of the new function which serves as the limit of the ratio [f(x + h) - f(x)]/h will depend upon the form of the given function y = f(x). In order to indicate this dependence, we give to the new function the name derivative and we designate it, using a prime, by the notation y' or f'(x).

(e) Differentials of functions of a single variable

Let y = f(x) remain a function of the independent variable x; let h be an infinitesimal and k a finite quantity. If we set h = ak, a will also be an infinitesimal quantity, and we will have identically

[f(x + h) - f(x)]/h = [f(x + ak) - f(x)]/ak,

whence one concludes that

(1).... [f(x + ak) - f(x)]/a = ([f(x + h) - f(x)]/h) k. The limit toward which the left side of equation (1) converges as the variable a tends to zero, the quantity k remaining constant, is called the differential of the function y = f(x). We indicate this differential by the symbol d, as follows:

dy or df(x).

It is easy to obtain its value when we know that of the derivative y' or f '(x). Indeed, taking the limits of the two sides of equation (1), we shall find generally

(2)... df(x) = kf'(x).

In the special case where f(x) = x, equation (2) reduces to dx = k.

(f) The definite integral

Suppose that the function y = f(x) is continuous with respect to the variable x between the two finite limits x = x₀, x = X. We designate by x₁, x₂,...,x_n-1 new values of x placed between these limits and suppose that they either always increase or always decrease between the first limit and the second. We can use these values to divide the difference X - x₀ into elements

x₁ - x₀, x₂ - x₁, x₃ - x₂, ... , X - x_n-1,

which all have the same sign. Once this has been done, let us multiply each element by the value of f(x) corresponding to the left-hand end point of that element: that is, the element x₁ - x₀ will be multiplied by f(x₀), the element x₂ - x₁ by f(x₁),. and finally, the element X- x_n-1 by f(x_n-1 ); and let

S = (x₁ - x₀)f(x₀) + (x₂ - x₁)f(x₁) + ... + (X - x_n-1)f(x_n-1)

be the sum of the products so obtained. The quantity S clearly will depend upon

lst: the number n of elements into which we have divided the difference X - x₀;

2nd: the values of these elements and therefore the mode of division adopted.

It is important to observe that if the numerical values of these elements become very small and the number n very large, the mode of division will have only an insignificant effect on the value of S. This in fact can be proved as follows....