After the rapid advances in calculus and its applications in the 18th Century by the Bernoullis, Euler, Lagrange, Lacroix and others, it became apparent through contradictions and paradoxes that there were gaps in the foundations. For example, functions that could not be expressed as power series, series which did not converge, Maclaurin series which converged but not to the function from which they were derived, series of continuous functions whose limit was not continuous, functions which were continuous on a domain but not differentiable anywhere, branching of solutions to differential equations, definition of derivative for functions with complex domains and so on.

It was clear that the most important lacks were firstly, a logically coherent definition of limit, and secondly, criteria for convergence of arbitrarily defined series. Behind these of course was a need for an axiomatic definition of real numbers with which to work.

One of the first to attack this problem was Bernhard Bolzano, a Czech priest of Italian descent who was Professor in the Philosophy of Religion at the University in Prague, then part of the Austrian empire. The Chair had been expressly established by the Emperor Franz I to counter the spread of enlightenment in Europe following the French Revolution, but he chose the wrong incumbent. Bolzano spread his own enlightened ideas in his lectures. He was eventually dismissed and even arrested on suspicion of heresy. His philosophical training attracted him to questions about the foundations of mathematics, including infinity and its paradoxes, and the properties of real numbers.

In his Paradoxes of the Infinite, Bolzano showed for example that there is a 1--1 correspondence between the intervals [1,2] and [1,3], and he suggested that there is no such correspondence between the natural numbers and the reals, anticipating later results by Cantor.

The other is: that every algebraic rational integral function of one variable quantity can be divided into real factors of first or second degree. After several unsuccessful attempts by d'Alembert, Euler, de Foncenex, Lagrange, Laplace, Kluegel, and others at proving the latter proposition, Gauss finally supplied, last year, two proofs which leave very little to be desired. Indeed, this outstanding scholar had already presented us with a proof of this proposition in 1799, but it had, as he admitted, the defect that it proved a purely analytic truth on the basis of a geometrical consideration. But his two most recent proofs are quite free of this defect; the trigonometric functions which occur in them can, and must, be understood in a purely analytical sense.

The other proposition mentioned above is not one which so far has concerned scholars to any great extent. Nevertheless, we do find mathematicians of great repute concerned with the proposition, and already different kinds of proof have been attempted. To be convinced of this one need only compare the various treatments of the proposition which have been given by, for example, Kaestner, Clairaut, Lacroix, Mettemich, Kluegel, Lagrange, Roesling, and several others.

However, a more careful examination very soon shows that none of these proofs can be viewed as adequate.

The most common kind of proof depends on a truth borrowed from geometry, namely, that every continuous line of simple curvature of which the ordinates are first positive and then negative (or conversely) must necessarily intersect the x- axis somewhere at a point that lies in between those ordinates. There is certainly no question concerning the correctness, nor indeed the obviousness, of this geometrical proposition. But it is clear that it is an intolerable offense against correct method to derive truths of pure (or general) mathematics (i.e., arithmetic, algebra, analysis) from considerations which belong to a merely applied (or special) part, namely, geometry.

No less objectionable is the proof which some have constructed from the concept of the continuity of a function with the inclusion of the concepts of time and motion. 'If two functions fx and gx', they say, 'vary according to the law of continuity, and if for x = a, fa < ga, but for x = b, fb > gb, then there must be some value u lying between a and b for which fu = gu. For if one imagines that the variable quantity x, in both these functions, gradually takes all values between a and b, and the same value is always taken by them both at the same moments, then at the beginning of this continuous change in x, fx < gx, and at the end, fx > gx. But since both functions by virtue of their continuity, must first go through all intermediate values before they can reach a higher value, there must be some intermediate moment at which they are both equal to one another.'

No one will deny that the concepts of time and motion are just as foreign to general mathematics as the concept of space. Nevertheless, if these two concepts were introduced here only for the sake for clarification, we would have nothing against them.

For we are in no way party to such an exaggerated purism, which demands, in order to keep the science free from everything alien, that in its exposition one can never use an expression borrowed from another field, even if only in a metaphorical sense and with the purpose of describing a fact more briefly and clearly than could be done by a strictly literal description; nor even if it is just to avoid the jarring of the constant repetition of the same word, or so as to remember, by the mere name given to a thing, an example which would serve to confirm the assertion. Thus it may be noted that we do not regard examples and applications as detracting in the least from the perfection of a scientific exposition. On the other hand, we strictly require only this: that examples never be put forward instead of proofs and that the essence of a deduction never be based on the merely metaphorical use of phrases or on their related ideas, so that the deduction itself would become void as soon as these were changed.

In accord with these views, the inclusion of the concept of time in the above proof may still perhaps be excused, because no conclusion is based on phrases which contain it which would not also hold without it. But in no way can the last illustration about the motion of a body be viewed as anything more than a mere example which does not prove the proposition, but rather is only to be proved by it.

So let us drop this example and examine the rest of the reasoning. Let us first note that this is based on an incorrect concept of continuity. According to a correct definition, the expression that a function fx varies according to the law of continuity for all values of x inside or outside certain limits means just that: if x is some such value, the difference f(x + w) - fx can be made smaller than any given quantity provided w can be taken as small as we please.

The following is a short summary of the method adopted.

The truth to be proved, that between any two values a and b which give results of opposite sign there always lies at least one real root, clearly rests on the more general truth that, if two continuous functions of x, fx and gx, have the property that for x = a, fa < ga, and for x = b, fx > gb, there must always be some value of x lying between a and b for which fx = gx. However, if fa < ga, then by the law of continuity it is possible that f(a + i) < g(a + i), if i is taken small enough. The property of being smaller, therefore, belongs to the function of i represented by the expression f(a + i), for all values smaller than a certain value. Nevertheless this property does not hold for all values of i without restriction, namely not for an i = b - a, for fb is already > gb.

Now the theorem holds that whenever a certain property M belongs to all values of a variable quantity i which are smaller than a given value and yet not for all values in general, then there is always some greatest value u, for which it can be asserted that all i < u possess property M.

For this value of i itself f(a + u) cannot be < g(a + u), because then, by the law of continuity, f(a + u + w) < g(a + u + w) if w were taken small enough. And consequently it would not be true that u is the greatest of the values for which the assertion holds, that all lower values of i make f(a + i) < g(a + i); for u + w would be a still greater value for which this holds.

But still less can it be true that f(a + w) > g(a + w), for then f(a + u - w) > g(a + u - w) would also be true if w were taken sufficiently small, and consequently it would not be true that for all values of i < u, f(a + i) < g(a + i). So therefore it must be that f(a + u) = g(a + u); i.e. there is a value of lying between a and b, namely a + u, for which the functions fx and gx are equal to each other.

' A function f(x) varies according to the law of continuity for all values of x inside or outside certain limits [bounds] if, when x is some such value, the difference f(x + w) - f(x) can be made smaller than any given quantity provided w can be taken as small as we please,'

and also by his criterion for convergence of a sequence:

' If a series of quantities F₁(x), F₂(x),..., F_n(x),... has the property that the difference between the n--th term F_n(x) and every later term F_n+r(x) remains smaller than any given quantity if n has been taken large enough, then there is always a certain constant quantity, indeed only one, which the terms of this series approach and to which they can come as close as desired if the series is taken far enough.'

Both results were needed to prove that every bounded set of numbers has a supremum which in turn is needed to prove the IVT: between any two values of a continuous function f which are of opposite sign, there must always lie at least one root of the equation f(x)=0.

Bolzano's version of the least upper bound property was:

' If a property M does not belong to all values of a variable x, but does belong to all values which are less than a certain u, then there is always a quantity U which is the greatest of those of which it can be asserted that all smaller x have property M.

Bolzano's proof involves the creation of a series to which his convergence result can be applied. Because M is not valid for all x, there must exist a quantity V = u + D such that M is not valid for all x smaller than V. He then constructs a sequence u, u + D/2^m, u + D/2^m + D/2^m+n,... which satisfies his convergence criterion and must therefore converge to a value U which satisfies the conditions of the theorem. (Weierstrass in 1860 simplified this result by showing that given any bounded infinite set of numbers, there is a number r in every neighbourhood of which thewre are other members of the set.)