The Development of Calculus

The study of calculus consists of three major aspects, namely, differentiation, integration and the relation between differentiation and integration. Differentiation is mainly concerned with the rate of change of a variable (with respect to time or other variables), while integration deals mainly with the calculation of the area of a figure. The relation between differentiation and integration is best illustrated by the “Fundamental Theorem of Calculus” (which is sometimes known as the “Newton-Leibniz Formula”), which says that differentiation and integration are reverse process of each other under certain conditions. We give below is a historical exposition of how calculus developed over the centuries.

1. The embryonic stage

As early as in the ancient Greeks, people already started to discuss about the concepts of “infinity”, “limits” and “infinite partitions”. These are all crucial concepts in the theory of differentiation and integration. Despite the fact that some of these discussions were absurd from a modern logical point of view, they are remembered as the first human attempt to understand the concepts of infinity and limits, which paved the way to the subsequent development of calculus.

For example, in the 5th century B.C., the Greek philosopher Democritus raised the atomic theory, saying that everything in the universe was made up of very tiny atoms. In the ancient Chinese text *Zhuangzi*, there was also the saying that “when there is a rod of length one feet and you cut away half of it per day, it can last forever” and stated that zero is an “infinitesimal quantity”. Both are examples of how early humans described concepts like limits and infinity.

Other early human descriptions of the concepts of limits and infinity include Zeno’s several paradoxes, in one of which he claimed that a man can never catch up even the slowest turtle ^{1}. He reasoned as follows: when a man arrived at the starting point of the turtle, the turtle would have already moved forward by several steps. When the man covered these several steps, the turtle would have taken another several steps forward, and so on. Zeno claimed that as a result, the man can never catch up the turtle, which is obviously absurd from a modern logical point of view. He confused the concepts of “infinity” and “infinite partitions”: the path which the man needs to take to catch up the turtle can of course be partitioned by infinitely many times; nevertheless, the length of that path is still finite, and the man can still take up the path within a finite period of time. However, these absurd arguments and debates started human discussions about the concepts of limits and infinity, which paved the way for the development of calculus in the later centuries.

It should be noted that, at the time of the ancient Greeks, Archimedes already managed to calculate certain areas by partitioning them into infinitesmally small quantities and then summing them up. This is already very similar to our modern way of calculating areas by integration; so we can see that historically men first came to master integration before they came to know differentiation, in contrast to the usual practice in the modern school curriculum, which introduces differentiation before integration.

2. The advancement in the 17th century — The contributions of Newton and Leibniz

The middle ages were a dark time for scientific developments in Europe. There was little advancement in the concepts of limits, infinity and integration. After the middle ages, following the rapid development of science and technology in Europe, the elements of calculus developed quickly in the 17th century. For integration, Kepler found the volume of a wine bucket in 1615 by regarding it to be a pile of thin horizontal circular discs. B. Cavalieri, a student of Galileo, thought of a line to be formed from infinitely many points, a plane to be formed from infinitely many lines and a solid to be formed from infinitely many planes. All these are pioneering ideas in integration.

There was also a great breakthrough in differentiation in the 17th century. Fermat mentioned in his letter to Gilles Personne Roberval how he found the maxima and minima of a function. This is essentially the way how one finds the critical points by setting the derivative to be zero in modern day. On the other hand, I. Barrow used his “differential triangle” (a triangle with sides *dx*, *dy*, *ds*) to find tangents. This is essentially the how we use derivative to find tangents today. So we see that human beings already mastered the essence of differentiation and integration by the 17th century.

However, up to the mid-17th century, people still regarded differentiation and integration as two separate concepts. It was not until then Newton and Leibniz related the two seemingly unrelated concepts of differentiation and integration by the “Fundamental Theorem of Calculus” (or the so-called “Newton-Leibniz Formula”), which clearly states that differentiation and integration are reverse processes of each other. This was a milestone in the development of the theory of calculus.

The methods of calculus proved to be extremely powerful. It solved many difficult problems which elementary mathematics failed to solve in the past. For instance, Jacob Bernoulli discovered with methods in calculus that the logarithmic spiral is invariant under various transformations ^{2}. In 1696, his brother Johann Bernoulli raised a Brachistochrone (geodesic) question: neglecting friction, along what curve will a point mass slide from a higher point to a lower point under the influence of gravity so that it takes the shortest possible time? This led to the development of “calculus of variations”, which is a branch of calculus. Other advancement in the development of calculus during the 17th century was also contained in Euler’s “*Introduction in analysin infinitorum*“, “*Institutiones calculi differentialis*” and “*Institutiones calculi integralis*“.

Although calculus proved to be so powerful, its logical foundation was under the severe challenge of the leading mathematicians of the time ^{3}. The “infinitesimal” that Newton raised was equal to zero some of the time but was non-zero under other circumstances. His thoery of limit was ambiguous as well. Leibniz’s calculus also failed to find a firm logical foundation. That is why Rolle, amongst other mathematicians of the time, opposed calculus. However, Rolle himself raised a famous theorem that had a strong calculus flavour: between any two real roots of a polynomial *f*(*x*) = *a* + *bx* + *cx*^{2} + *dx*^{3} + …, there exists a root of the derived polynomial *b* + 2*cx* + 3*dx*^{2} + …. If you know calculus, you would recognize the second expression *b* + 2*cx* + 3*dx*^{2} + … as the derivative of our original polynomial *f*(*x*) = *a* + *bx* + *cx*^{2} + *dx*^{3} + … ^{4}. Later people extended this theorem to cater for differentiable functions: between any two real roots of any differentiable function *f* (*x*), there exists a real root of the equation *f’*(*x*) = 0. This is now widely known as the Rolle’s Theorem, which has become one of the basic theorems in elementary calculus. So we see that mathematicians had already yielded fruitful results in calculus while still challenging its logical foundation.

3. The consolidation of the foundation in the 19th century

Calculus developed so quickly that people did not even care to pause and build a solid ground of the subject. It was not until the 19th century, when most applications of calculus were fully developed, that mathematicians eventually came to the establishment of a firm logical foundation of the subject, which means the first time for men to give a rigorous definition of concepts like limits, derivatives and integrals.

In 1816, for the first time in human history, B. Bolzanno gave the modern definition of a continuous function. Then in 1821, Cauchy first introduced the *e*-method in his “*Cours d’ analyse de l’ Ecole Royale Polytechnique*“. In 1823 he introduced the *d*-method in his “*Resume des lecons donnees a l’ Ecole Royale Polytechnique sur le calcul infinitesimal*“. In this way the limit process was converted into manipulations of inequalities, and this is known as the arithmetization of the limit process. Later K.T.W. Weierstrass put *e* and *d* together, which gave our modern *e*–*d* method of defining limits. This was how the first rigorous modern definition of limit emerged.

With the rigorous definition of limits in hand, mathematicians then proceeded to define derivatives and integrals in a rigorous way. Before Cauchy, differentials were usually considered to be the building blocks of calculus, and that derivatives were considered to be quotients of differentials. However, such a definition was unsatisfactory, for the concept of differentials was vague and cannot be relied on. As a result, in his “*Resume des lecons donnees a l’ Ecole Royale Polytechnique sur le calcul infinitesimal*“, Cauchy defined the derivative as a limit of difference quotients, and this became the first rigorous definition of the derivative in modern mathematics.

Cauchy was also the first one who gave a definition of the integral of a continuous function. It was stated in his “*Resume des lecons donnees a l’ Ecole Royale Polytechnique sur le calcul infinitesimal*“: Let *f* (*x*) be a continuous function on [*a*,*b*]. Take arbitrary points *a* = *x*_{0} < … < *x _{n}* =

*b*in the interval [

*a*,

*b*] and partition the interval into

*n*subintervals [

*x*

_{i-1},

*x*

_{i}] (

*i*= 1, 2, …,

*n*). If the “limit” of the sum

exists as the length of the longest subinterval tends to zero, then the limit is known as the integral of *f* (*x*) on [*a,b*]. This is consistent with the modern definition of the integral of a continuous function.

Later, Riemann generalized the definition of Cauchy. The difference between Riemann’s definition and Cauchy’s definition is in the way in which the sum *S* was defined. In Riemann’s definition the sum *S* was given by

. (note that Riemann used an arbitrary point x_{i-1} in the interval [*x*_{i-1},*x*_{i}], while Cauchy always chose the left end point *x*_{i-1} in his definition of *S*.) We say that Riemann’s definition generalized Cauchy’s definition, because the limit of *S* not only exists and agrees with Cauchy’s definition for all continuous functions on [*a*,*b*], but also exists for some functions *f* (*x*) which are not continuous on [*a*,*b*]. This is our modern definition of the Riemann integral, and the whole logical basis of the theory of calculus was then almost fully established.

After Cauchy, the most important advancement in the logical foundation of calculus was the construction of the real numbers using set theory. We say this is important for the logical foundation of calculus, because the theories calculus often makeuse of the properties of the real numbers and thus relied heavily on the theory of real numbers. The construction of the real numbers represented the joint efforts of many mathematicians; amongst them were Dedekind, Cantor and Weierstrass. In 1872, H.C.R. Méray first defined irrational numbers, and his definition was essentially the same as what Cantor suggested in the same year (which made use of convergent sequences of real numbers). The theory of real numbers, set theory and the definition of limits eventually established a sound logical foundation of the entire theory of calculus, which had been in lack of for over 300 years.