The origins of the differential and integral calculus - 2

A person who may have played a significant role in introducing Newton to the concepts of the calculus is the English mathematician Isaac Barrow (1630-77). He was professor of mathematics at Cambridge from 1663 until 1669. His successor in the professorship was Newton. Barrow was primarily a geometer following the traditions of the ancient Greeks. However, around 1664 he became interested in the problem of finding tangents to curves and he developed an approach involving moving points and lines. In his university lectures at Cambridge, which were subsequently published, he gave his own generalization of tangent and area procedures based on his extensive reading of the works of such notable contemporary mathematicians as Descartes, Wallis, Fermat and especially the Scottish mathematician James Gregory, who is considered to be an important forerunner of Newton. The lectures contained ideas that could have been exploited but they were probably not studied outside Cambridge.

It has been a matter of much conjecture whether Newton was in any sense a student of Barrow’s. It was always assumed that he was influenced by Barrow, as he was working at Cambridge at the time of Barrow’s lectures on tangent and area problems. Furthermore, Newton’s first great advances in the foundations of the calculus date from 1664-65, which is the time when Barrow first studied the problems that underlie the calculus. It is clear that Barrow’s notion of generating curves by the motion of points was important to Newton’s foundation of the differential calculus, but on the whole Newton denied any direct influence from Barrow.

We turn now to study the work of the person generally held to be the first to develop the calculus on a systematic basis and see the connection between the differential and integral processes. Issac Newton (1642-1727) was born on Christmas Day 1642 in Lincolnshire, England. His father had already died in October 1642 and the lack of a father is thought to have had an effect on Newton’s personality. Newton lived at Woolsthorpe, near Grantham, in a house which can still be visited. He attended a grammar school in Grantham, where he learnt Latin, but little mathematics. His mother intended him to become a farmer, but he showed no aptitude for farm work. Instead, he was sent to Cambridge University in 1661, where he graduated in 1665.

As an undergraduate, he studied Descartes’s La Geometrie in a Latin translation, as well as Wallis’s Arithmetica infinitorum and Aristotle’s philosophy. He also read Euclid’s Elements but was unimpressed by it initially. In 1665, the university was closed because of the plague, and Newton spent most of the next 18 months in Lincolnshire. It was during this time that Newton made four fundamental discoveries:

•  the general binomial theorem;

•  the connection between the methods used in tangent and area problems (the fundamental theorem of the calculus);

•  the law of universal gravitation;

•  the theory of the composition of white light.

The episode of the falling apple, which is said to have occasioned the notion of gravitation, occurred at his Lincolnshire home, Newton later declared.

In 1667, he was elected a Fellow of Trinity College, Cambridge, and in 1669, at the age of 26, he became professor of mathematics at Cambridge University. Isaac Barrow, his predecessor as professor, is said to have resigned in Newton’s favour, although he was also seeking advancement in London clerical circles. In the next ten or more years, Newton gave university lectures in optics, arithmetic and algebra, and parts of what became his Principia. Much of this work was published subsequently and proved immensely influential, although it is reported that his lectures were largely unattended and unappreciated. In the 1690’s, after he had published his fundamental work, he apparently lost interest in mathematics, and eventually left Cambridge to become Master of the Mint in London.

He still, however, found time to revise and publish other parts of his work and engage in scientific correspondence and controversy.

As we already mentioned, Newton especially studied Descartes’s La Geometrie and Wallis’s Arithmetica infinitorum, which strongly influenced his work on analytic geometry and algebra, and on calculus. He also studied the work of Fermat and James Gregory. At the beginning of 1665, Newton discovered the general binomial theorem. He recalled later:

In the winter of the years 1664 and 1665 upon reading Dr Wallis’s Arithmetica Infinitorum & trying to interpole his progressions for squaring the circle I found out another infinite series for squaring the circle and then another for squaring the Hyperbola.

Here, squaring the circle means finding the area of a circle, or, in effect, calculating the integral

without assuming knowledge of р. By reading Wallis’s work on finding areas, Newton was led to understand that the integrand $\sqrt{1-x^2}$ could be expanded as an infinite series and

the general indefinite integral

could also be expressed by term by term integration as

$x - \frac{\frac{1}{2}x^3}{3} - \frac{\frac{1}{8}x^5}{5} - \frac{\frac{1}{16}x^7}{7} \cdots$

James Gregory had applied similar ideas in evaluating

by expanding (1 + x²)^-1 as an infinite geometric series, thereby obtaining an inverse tan series.

Newton also discovered the infinite series representation for the sin^-1 x function, and for $\log (1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} \cdot \cdot \cdot$

He used his formula to calculate special values of the log function, giving areas under a hyperbola, to fifty or more decimal places. The surviving manuscripts show that Newton was a formidable calculator, well able to test the accuracy of his theoretical results. Around

the same time, Newton developed a differentiation procedure based on the concept of an infinitesimally small increase o, which ultimately vanishes, of a variable x. Eventually, he settled on the notion of a fluxion of the variable, meaning a finite instantaneous speed defined with respect to an independent time variable. In our terms, the fluxion of x with respect to the time variable t is $\frac{dx}{dt}$, which is the speed in physical terms. Later, he

introduced the notation $\dot{x} \ {\bf for} \ \frac{dx}{dt}, \ddot{x} \ {\bf for} \ \frac{d^2x}{dt^2}$, and so on, which has been maintained to

some extent in dynamics. He also invented the idea of partial derivatives of a function of two independent variables in 1665, although again, his notation was very different from that which evolved later.

Between 1664 and 1666, Newton was immensely creative mathematically, but for the next two years his interest moved elsewhere. In 1669, he saw a copy of Mercator’s book Logarithmotechnia, published in late 1668, in which the infinite series for log(1 + x) was given. Newton suddenly realized that other people were discovering the method of infinite series, which he had greatly developed in 1664-65. He therefore wrote a tract, entitled De analysi per aequationes infinitas. This tract was circulated to several important mathematicians, to establish Newton’s priority of discovery. The tract was handwritten and was never printed, although a version subsequently appeared many years later, incorporated into a larger work. It was perhaps typical of Newton that he never really gave a proper presentation of his work on the differential calculus, and this was partly to blame for the famous dispute with Leibniz.

In his De analysi manuscript, Newton was able to show that the area under the curve

$y - x^{\frac{m}{n}}$

is given by the function $\frac{x^{1 + {\frac{m}{n}}}}{1 + {\frac{m}{n}}}$

by introducing an area function and considering small increments in this function. In this way, he was able to demonstrate that the area problem was the inverse of the tangent problem for curves, or, in modern terms, that integration is the inverse of differentiation. While special cases of this property were evident in the work of such mathematicians as James Gregory, Newton was the first to exploit it systematically, especially through his use of infinite series.

Newton changed his approach to the limiting processes of the calculus over the years.

However, as the limit concept is subtle, and was not properly clarified until the mid-19th century, Newton was always vulnerable to the accusation that his arguments relied on the manipulation of quantities that are ultimately set equal to 0, and that he was effectively dividing by 0. Of course, in elementary treatments of the calculus, where the limiting process is described more or less intuitively, virtually the same arguments as those of Newton are still used today.

As we mentioned earlier, Newton never published a self-contained account of his theory of the calculus. In his masterpiece Philosophiae naturalis principia mathematica (Mathematical principles of natural philosophy), published in 1687, he made use of some of his ideas of calculating derivatives, although most of the emphasis is given to geometric rather than algebraic methods. (It was once thought that Newton had an aversion to the use of algebra, especially in his published work, but existing manuscripts show that he freely used algebra in his calculations.) The Principia (as it is generally named) is mainly concerned with physics and astronomy, and its intent is to show how the laws of motion and the inverse square law of gravitation enable us to describe the workings of the solar system very accurately. It is of interest to note that, in the first two editions of the Principia, Newton acknowledged that Leibniz had independently discovered a version of the calculus, but following the acrimony of their dispute, all reference to Leibniz had been removed from the third edition of 1726.

As we have alluded in our description so far, the other person acknowledged to be the main discoverer of the calculus is Gottfried Wilhelm Leibniz (1646-1716). He was born in Leipzig, in Germany. At university, he studied a wide range of topics, including law, theology, philosophy and mathematics. Like Descartes, he was determined to find a universal procedure of reasoning, applicable to virtually all academic disciplines. (He was interested in inventing an algebra of logic, something that George Boole later introduced in his book The Laws of Thought, published in 1854.) This abstract approach to reasoning probably ensured that his version of the calculus was systematic and algorithmic, unlike Newton’s. Much of Leibniz’s career was spent in the diplomatic service of the rulers of Hanover, which enabled him to visit many of the intellectual centres of Europe, including Paris in the early 1670’s. He visited London in 1673 and 1676 and probably saw the manuscript of Newton’s De analysi during his stays.

Around 1673, Leibniz realized that the determination of the tangent to a curve depended on the ratio of the difference of the y-values (the ordinates) to the difference of

the a;-values (abscissae), as these differences become infinitesimally small. He also realized that the area under a curve is found by taking the sum of the areas of infinitely thin rectangles. Like Newton, he was led to see the role of infinite series (possibly influenced by Newton’s De analyst). By 1676, he had essentially obtained all of Newton’s slightly earlier conclusions. He saw furthermore that his procedures could deal not only with simple powers of x but also with so-called transcendental functions, such as log a; and sin a;.

We owe to Leibniz much of the familiar notation of the calculus, and it is generally agreed that Leibniz’s exposition of the calculus and its fundamental processes was much clearer and more easily used than that of Newton. He eventually decided on the use of dx and dy for smallest possible x and y increases (they are sometimes called differentials). Likewise, he introduced $\int y \ {\bf and} \ \int y \ dx$

for finding area, the J sign arising as an enlarged s, signifying sum. The calculus differ-entialis became the method for finding tangents and the calculus summatorius or calculus integralis the method for finding areas.

Leibniz was the first person to publish a complete account of the differential calculus. His paper was entitled Nova methodus pro maximis et minimis, itemque tangentibus. It appeared in an important journal, Ada Eruditorum, published in Leipzig in 1684. This journal was a major source of news in science, where reviews of such books as Newton’s Principia kept its readers up to date with the latest ideas. In his paper, Leibniz used the d notation for derivatives (or differentials). He stated quite clearly the product rule in the form

$dx v - x dv + v dx$

where the bar denotes product, and the quotient rule as

$d \frac{v}{y} = \frac{v dy - y dv}{y^2}$

He also gave the rule for differentiating powers as

dxⁿ = nxⁿ~ dx.

In 1686, in the same journal, Leibniz gave an account of the integral calculus, emphasizing the inverse nature of the procedures of integration and differentiation. Leibniz’s presentation of the differential calculus proved to be influential, as he soon found followers such as the Bernoulli brothers, who were able to extend the range of its applications significantly.

It may be of interest to examine what communication existed between Newton and Leibniz in the years when Leibniz was perfecting his differential calculus. In 1676, in answer to a request from Leibniz, Newton wrote to him to give brief details of his theory of the calculus. On August 27 of the same year, Leibniz wrote back to him to ask for fuller details on this subject. Newton replied on October 24 1676 in a long letter, describing how he had been led to some of his discoveries. He had interpolated the results of Wallis, and, working by analogy, saw how to expand functions such as

$(1 - x^2)^{\frac{1}{2}} \ {\bf and} \ (1 - x^2)^{\frac{3}{2}}$

into infinite series. He had then proceeded to deduce (or guess) the general binomial theorem. In this same letter, Newton mentioned his method of fluxions, but he gave no details. He also listed some of the functions which he could integrate, enabling him to find corresponding areas. These included

$x^m(b + cx^n)^p \ {\bf and} \ x^{mn-1}(a + bx^n + cx^{2n})^{\pm \frac{1}{2}}$

Newton concluded by expressing regret about controversy arising from his earlier research publications, with the implication that he intended to publish little in future. Leibniz replied on June 21, 1677. He described his method of drawing tangents, which proceeded not by fluxions of lines but by differences of numbers. He also introduced his notation of dx and dy for infinitesimal differences or differentials of coordinates.

While Leibniz’s differential calculus seems quite familiar to modern day students of mathematics, the same cannot be said of Newton’s fluxional calculus, which appears unnecessarily complicated. We will briefly attempt to describe how Newton’s calculus proceeded. He assumed that we may conceive of all geometric magnitudes as generated by continous motion–for example, a line is generated by the motion of a point, etc. The quantity thus generated is called the fluent or flowing quantity. The velocity of the moving magnitude is the fluxion of the fluent. There then arise two problems. The first is to find the fluxion of a given quantity, or more generally the relation of the fluents being given, to find the relation of their fluxions. This is akin to implicit differentiation. The second method is the inverse method of fluxions: from the fluxion, or some relation involving it, to find the fluent. This amounts to what we call integration, possibly of a differential equation. Newton referred to these procedures as the method of quadrature and the inverse method of tangents. The infinitely small part by which a fluent such as x increased in a small interval of time o was called the moment of the fluent and its value was shown to

be ox. Given x, Newton denoted the fluent whose fluxion was x by x' or [x]. This is the same as the integral of x (but not with respect to x). Subsequently, Newton’s methods were taught at Cambridge for more than 100 years, and the word fluxion was retained. The explanations given for the procedures were usually vague and unconvincing. It was not until almost 1820 that Leibniz’s notation of dx and the integral sign started to appear in British mathematical textbooks and papers.

Between 1689 and 1693, Newton had fallen under the influence of a little-known Swiss mathematician named Nicholas Fatio de Duillier (1664-1753), who had moved to England in 1687. Later in 1693, Newton suffered what appears to have been a nervous breakdown. After his recovery, Newton decided to abandon his career in Cambridge, devoted to research and scientific discovery, and sought more wordly fame in London as Warden of the Mint in 1696. Nonetheless, Newton was alarmed when informed by John Wallis in 1695 that many continental mathematicians considered the calculus to be the invention of Leibniz alone, perhaps on the basis of his expositions of the subject in Ada Eruditorum. In support of Newton, Duillier published a paper through the Royal Society of London in 1699 in which he implied that Leibniz had plagiarized the ideas of the calculus from Newton. In 1704, Leibniz replied in the Ada Eruditorum that he had priority of publication, and he protested to the Royal Society about the unfairness of the accusation of plagiarism. The Royal Society eventually responded by establishing a committee to investigate the dispute. In 1712, the committee published their conclusions in a report entitled Commercium epistolicum. The report affirmed that Newton had invented the calculus, a point not seriously disputed even by Leibniz. The report also noted that Leibniz had had access in the 1670’s to manuscripts and letters describing Newton’s preliminary version of the calculus (for example, De analyst), so that suspicions of plagiarism were not totally dismissed. It has become apparent that the Commercium epistolicum was essentially Newton’s own work-he dictated the conclusions that the committee reported.

The priority dispute was dominated by nationalistic concerns, British mathematicians being especially keen to defend Newton’s honour and proclaim his genius as a reflection of British superiority. As a consequence, the advantages of Leibniz’s notation, subsequently developed by the Bernoullis and Euler, were ignored in Britain, and by paying too much deference to Newton, mathematics stagnated there until the early 19th century. The dispute was led in Britain, albeit secretly, by Newton himself, and he seems to have wanted to remove all record of Leibniz’s achievements.