The origins of the differential and integral calculus - 1

We now sketch the origins of the differential and integral calculus, probably the most powerful technique introduced into mathematics since the golden age of Greek geometry. Archimedes, in the 3rd century BCE, had been able to calculate areas under curves and volumes of certain solids by a method of approximation, called the method of exhaustion, based on using known areas and volumes of rectangles, discs, etc. His results were usually expressed, not in absolute terms, but in terms of comparisons of volumes. For instance, suppose we have a sphere of radius r which is surrounded exactly by a circular cylinder of radius r and height 2r. Then Archimedes showed that the volume of the sphere is two thirds that of the cylinder. This is of course well known to us, as if V is the volume of the sphere and V1 is the volume of the cylinder, then

$V = \frac{4}{3} \pi r^3, \ \ V_1 = 2 \pi r^3$

and the result follows.

Each case of his area and volume calculation was worked out on its own merits, and no algorithm for handling general problems emerged. It was also important for Archimedes to maintain Greek standards of rigour in proof, for no clear idea of limiting processes or of infinity was known at the time. This made the method rather laborious by modern standards. The integral calculus eventually provided the necessary algorithm for calculating areas, volumes, centres of gravity, and so on. Some of Archimedes’s ideas were known in the Renaissance, as his work On the Sphere and Cylinder was available in Latin translations. Interestingly enough, another important contribution of Archimedes, called simply The Method, which contained further volume calculations, was not known until 1906. A palimpsest (which is a parchment that has been partly erased and then re-used), on which a copy of The Method had been written by a 10th century scribe, was discovered in Constantinople by the historian Joseph Heiberg. It is difficult to read directly, but under ultra-violet light with modern image-enhancing technology it may easily be deciphered. It is the most important Archimedes manuscript known, and is the sole source of some of his work. The manuscript was sold at auction in New York in 1998 for around 2 million dollars (probably to a computer millionaire), to the dismay of some historians of mathematics and amidst disputes about its rightful ownership.

The famous German astronomer Johann Kepler (1571-1630) knew of Archimedes’s work on volume calculation. He was able to put his knowledge to practical use when he

was asked to find the best proportions for making wine casks. His response to this practical problem was the book he wrote in 1615: Nova stereometria doliorum (New solid geometry of wine casks). In this work, Kepler considered both theoretical and applicable volume calculations. He broke away from the methods of Archimedes, abandoning the extreme rigour previously considered appropriate, and introduced new approximation techniques. One of his main contributions was the volume of the solid obtained by rotating a segment of a conic section about an axis in its own plane. His book gives the volumes of almost 100 solids. He is considered the great precursor of the infinitesimal method of the integral calculus.

A further important addition to the foundations of the integral calculus was made by the Italian priest Bonaventura Cavalieri (1598-1647). Cavalieri was a follower of the great physicist Galileo Galilei, who was himself interested in problems involving area and volume. Cavalieri’s ideas are contained in his book Geometria indivisibilibus continuorum of 1635. In this work, he expounded his method of indivisibles, an infinitesimal technique influenced by Archimedes’s method of exhaustion. It is difficult to explain exactly what indivisibles are, but we can think of them as in some sense the material out of which continous substances are constructed. Cavalieri considered a line as composed of an infinite number of points, a surface as composed of an infinite number of lines, and so on. The idea of an indivisible was not entirely new, as it had occurred already in medieval scholastic philosophy.

The main advantage of the method of indivisibles was that it was more systematic compared with the method of exhaustion, although perhaps more subject to criticism from lacking extreme rigour. The method made it easy to find the area of an ellipse or the volume of a sphere. Cavalieri found, in effect, a result equivalent to evaluating the integral

$\int_0^a x^m \ dx \ \text{as} \ \frac{a^{m + 1}}{m + 1}$

when m is a positive integer. Of course, he did not express his ideas in this form, and it should be noted that his knowledge of the new algebraic notation (as say compared with Descartes) was weak. The main thrust of his arguments was to establish inequalities for estimating the sums

1m +2m +··· +nm

when n and m are positive integers. In effect, by looking at special cases and generalizing,

he established inequalities of the form

$1^m + 2^m + \cdots + n^m > \frac{n^{m+1}}{m+1} > 1^m + 2^m + \cdots + (n-1)^m$ .

The same type of analysis had in fact already been considered by the French mathematicians Fermat and Roberval a little earlier.

The next significant contribution to the methods of the calculus that we wish to describe is that made by the English mathematician John Wallis (1616-1703). Wallis was professor of geometry at Oxford and he wrote a number of influential books. We will concentrate on his Arithmetica infinitorum, published in 1655. Wallis knew of the existence of Cavalieri’s work on indivisibles, but probably had never seen his Geometria indivisibilibus continuorum. Wallis obtained results, amounting to the calculation of definite integrals, rather similar to those of Cavalieri. His approach was, however, more arithmetical and less laborious than Cavalieri’s. Indeed, Wallis was probably more of an algebraist than a geometer, although geometric considerations still were important to him. What is more, on the basis of his preliminary findings, he was able to infer the general pattern for evaluating, in effect,

$\int_0^1 x^k \ dx$

He used, in essence, a form of induction, or more correctly, an argument by analogy, in which he did not rigorously justify all his steps, and this led to criticism from continental mathematicians. Another point to observe is that Wallis was the first mathematician to make use of fractional powers, such as x1/2 or x2/3 (recall that Descartes, twenty or so years earlier, had popularized the use of integer powers or exponents). Wallis even went so far as to integrate such fractional powers, obtaining

$\int_0^1 x^{\frac{p}{q}} \ dx = \frac{1}{\frac{p}{q} + 1}$

His argument was really by analogy with the whole number case, an idea that is still used in introductory courses in integration. Despite the lack of mathematical rigour, his results were correct and provided a powerful impetus for sweeping generalization by Newton a few years later.

Wallis also showed the relevance of infinite processes to problems of mathematical analysis. A noteworthy example is his infinite product formula

$\frac{\pi}{4} - \frac{2 \times 4 \times 4 \times 6 \times 6 \times 8 \times \cdots}{3 \times 3 \times 5 \times 5 \times 7 \times 7 \times \cdots} .$

In our modern notation, he obtained this result by considering integrals of the form

$\int_0^{\frac{\pi}{4}} \sin^m \theta \ d\theta$

His arguments in obtaining this formula were not entirely convincing but they proved to be correct, and he had thus extended the range of theorems accessible by limiting processes. The formula for $\frac{\pi}{4}$is remarkable, as р is transcendental (it is not the root of any polynomial with rational coefficients) but is shown to be a limit of simple fractions. We can say that Wallis prepared the way for the study of infinite series, rather than polynomials. Such series have been a key feature in the calculus and much of all subsequent analysis.

The origins of the differential and integral calculus - 2

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