THE HISTORY OF THE CALCULUS

The infinitives "to calculate", "to compute" (or the Middle English equivalent couten, from which we get "to count") and "to reckon", all have similar meanings related to the carrying out of numerical processes. Of these three phrases, the last two are closely associated, etymologically and in current meaning, with mental processes. The first, on the other hand, carries from its origin a connotation of nondeliberative manipulation, for "to calculate" once meant "to reckon by means of pebbles". The word "calculus" is the diminutive of the Latin calx, meaning "stone".

It is one of the ironies of history, then, that the phrase "the calculus" should have become firmly attached to a branch of maths which calls for subtlety and sophistication of thought in the highest degree. The inappropriateness of the term is clear from the fact that mastery of the calculus would be out of the question for any one who found it necessary to fall back on pebbles for computational purposes.

Conceptions in Antiquity

In the more formal sense, the calculus was fashioned in the seventeenth cen­tury A.D. (of our era), but the questions from which it arose had been asked for more than seventeen centuries before our era began. Egyptian hieroglyphic papyri and Babylonian cuneiform tablets include problems in rectilinear and cur­vilinear measuration that are appropriate to the domain of the calculus; but the treatment of these problems fell short of fully-fledged math stature in two serious respects: (1) there was no clear-cut distinction between results that were exact and those that were approximations only, and (2) relationships through deductive logic were not explicitly brought out. Perhaps the closest approach of the Babylonians to the calculus lay in an iterative algorithm they devised for finding the square root of any (rational) number. Had the Babylonians possessed any way of knowing or showing that the process is nonterminating, they might now be hailed as the originators of the subject of infinite sequences, a basic part of modern calculus. However, Babylonian skill in algebra was not matched by concern for logic; hence, credit for adumbrations of the calculus must go to the ancient people for whom the logical approach to a subject constituted a veritable passion.

It is universally admitted that the Greeks were the first mathematicians—first, that is, in the significant sense, that they initiated the development of maths from first principles. It was a shock to the Greek math community to learn that there are such things as incommensurable line segments and that the occurrence of such situations is commonplace—that is, that concepts akin to the calculus arise in the most elementary of situations. The Greek discovery of incommensu­rability confronted mathematicians directly with an infinite process. Whenever the Euclidean algorithm for finding the greatest common divisor of two integers is applied in arithmetic, the process comes to an end in a finite number of steps, for the set of positive integers has a smallest element, the number 1.

If, on the other hand, the analogous scheme is applied in geometric garb to finding the greatest common measure of two incommensurable line segments, the process will go on forever; there is no such thing as a smallest line segment — at least not according to the orthodox Greek view, nor according to conventional modern concepts. The prospect of an infinite process disturbed ancient mathe­maticians for here they were confronted with a crisis.

This view might preclude any Greek equivalent of the calculus; Eudoxus, nevertheless, suggested an approach that seemed to mathematicians as irrefuta­ble and served essentially the same purpose as an infinite process. Euclid's cal­culus (derived, presumably, from Eudoxus' "method of exhaustion") may have been less effective than that of Newton and Leibnitz two thousand years later; but in terms of basic ideas it was not far removed from the limit concept used crudely by Newton and refined in the nineteenth century. The purpose of Euclid in the "exhaustion lemma" (Book X) was to prepare the ground for the earliest truly rigorous comparison of curvilinear and rectilinear figures that has come down to us.

Archimedes and His Anticipations of Calculus

The ancient Greeks did have an alternative approach to integration, one that served as an effective aid to invention. This device was described by Archimedes, in a letter to Eratosthenes, simply as "a certain method by which it will be possible for you to get a start to enable you to investigate some of the problems in maths, by means of mechanics". The "certain method", which Archimedes correctly saw, would enable his contemporaries and successors to make new discoveries, consisted of a scheme for balancing against each other the "elements" of geometric figures.

A line segment, for example, is to be regarded as made up of points, a plane surface area is thought of as consisting of an indefinite number of parallel line segments; and a solid figure is looked upon as a totality of parallel plane elements. Without necessarily subscribing to the validity of such a view in maths, Archimedes found this approach very fruitful. The very first theorem he discovered through a balancing of elements was the celebrated result that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and equal height. This is found by balancing lines making up a triangle against lines making up the parabolic segment. To demonstrate this result, which depends so much on brilliant intuition, Archimedes uses the method of exhaustion. He "exhausts" the parabolic segment by taking out successively inscribed triangles. The total area can be approximated by a sum of areas, which by proper grouping leads to a geometrical progression, each term of which is 1/4 the previous term. The sum of this infinite progression is 4/3 of the first term. Very carefully Archimedes shows that the area of the parabolic segment cannot exceed 4/3 of the area of the first inscribed triangle and, likewise, that it cannot fall short of it. Thus, he reaches his desired conclusion and by avoiding the pitfalls of infinitesimals and limit operations attains a level of rigour which compares favourably with anything done up to the eighteenth century. The Method of Archimedes could have been of greater significance in the development of the calculus if printing had been an invention of ancient times rather than the Renaissance. Copies of the Method evidently were never numerous, and for almost two millennia the work remained essentially unknown. In his work on areas and volumes Archimedes further developed the method of exhaustion, whereby the desired quantity is approximated by the partial sums of a series or by terms of a sequence. He obtained approximations to the area of a circle by comparing it with inscribed and circumscribed regular polygons.

No one in the ancient world rivalled Archimedes, either in discovery or in demonstration, in the handling of problems related to the calculus. Nevertheless, the most general ancient theorem in the calculus was due not to Archimedes, but to Greek mathematicians who lived probably half a dozen centuries later.

In the Mathematical Collection of Pappus (c. A.D. 320) we find a proposition awkwardly expressed but equivalent to the statement that "the volume generated by the rotation of a plane figure about an axis not cutting the figure is equal to the product of the area of the plane figure and the distance that the centre of gravity of the plane figure covers in the revolution". Pappus was fully aware of the power of this general theorem and of its analogue concerning areas of surfac­es of revolution. It includes, he saw, "any number of theorems of all sorts about curves, surfaces, and solids, all of which are proved at once by one demonstration". Unfortunately, Pappus does not tell us how to prove the theorem, and we do not know whether it was either discovered or proved by Pappus himself.

It is of importance to note that there are two aspects of the ancient beginnings in the integral calculus. One of these, stemming from Eudoxus, is represented by the rigorous method of exhaustion (illustrated in the proof of the theorem from Elements, Book XII, 2); the other arising out of atomistic views associated with Democritus, is related to the method of Archimedes. The former, not far removed from nineteenth-century concepts, was a faultless means of establishing the validity of a theorem. The latter, whether making use of indivisibles or of Archimedes' elements of lower dimensionality (more closely resembling the seventeenth-century stage of the calculus), was a device that led to the discovery of plausible conclusions. Archimedes exploited both aspects most successfully, indeed. His "mechanical" method led to theorems on areas, volumes, and centres of gravity which had eluded all of his predecessors; but he did not stop there. He went on to demonstrate these theorems in the traditionally rigorous manner through the method of exhaustion.

Up to this point we have stressed the similarity between the ancient method of exhaustion and the rigorous modern formulation of the calculus, but there are also essential differences. The ancient and the modern are in sharpest contrast with respect to motive. The method of exhaustion provided an impeccable demonstration of a theorem whose truth had been arrived at more informally. The substance of Elements, Book XII, for instance, had been assumed by earlier civilizations well over a millennium before Euclid's day. The value of the modern calculus, however, lies not so much in its power of rigorous demonstration as in its marvellous efficacy as a device for making new quantitative discoveries.

Were Eudoxus to reappear in the twentieth century, he might have difficulty in recognizing the descendants of the method of exhaustion; but in at least one respect he would feel completely at home in the maths of today. The drive for precision of thought from which the ancient integral calculus arose is matched today by a comparable insistence on rigour in analysis. Eudoxus would thus share the feeling of pride suggested in the continuing use of the phrase "the calculus", which sets the subject apart from the ordinary calculations that all too often are mistaken by the uninitiated as the preoccupation of mathematicians.

Medieval Contributions to the Calculus

Pappus was the last of the outstanding ancient mathematicians; following him, the level of maths in the Western world sank steadily for almost a thousand years. The Roman civilization was generally inhospitable to maths. Latin Europe in the twelfth and thirteenth centuries became receptive to classical learn­ing, transmitted through Greek, Arabic, Hebrew, Syriac, and other languages, but the level of medieval European maths remained far below that of the ancient Greek world.

However, a certain ingenious originality resulted in the fourteenth-century advance in a direction that had been avoided in antiquity. Archimedean maths, like Archimedean physics, had been essentially static; the study of dynamic change had been regarded as appropriate for qualitative philosophical discussion rather than for quantitative scientific formulation. But in the fourteenth century scholars began to raise such questions as the following: If an object moves with varying speed, how far will it move in a given time? If the temperature of a body varies from one part to another, how much heat is there in the entire body? One recognizes such questions as precisely those that are handled by the calculus; but me­dieval scholars had inherited from antiquity no math analysis of variables and I hey developed a primitive integral calculus of their own.

One of the leaders in this movement was Nicole Oresme (1323-1382), bishop of Lisieux. In studying, for example, the distance covered by an object moving with variable velocity, Oresme associated the instants of time within the interval with the points on a horizontal line segment (called a "line of longitudes"), and at each of these points he erected (in a plane) a vertical line segment ("latitude"), the length of which represented the speed of the object at the corresponding time. Upon connecting the extremities of these perpendiculars or latitudes, he ob­tained a representation of the functional variation in velocity with time — one of the earliest instances in the history of maths of what now would be called "the graph of a function". It was then clear to him that the area under his graph would represent the distance covered, for it is the sum of all the increments in distance corresponding to the instantaneous velocities. Oresme's results were further de­veloped by Galileo Galilei two and a half centuries later.

It is important to note, in connection with medieval studies in the latitude of forms, that there was no equivalent of the concept of differentiation. It is true that manuscript copies of the graphical representations of Oresme contain strong hints of the differential triangle. It should have been obvious in the latitude of forms that the representation of a rapidly increasing quantity called for a rapidly increasing latitude with respect to longitude; yet, no systematic terminology or method of handling such a concept was developed either in antiquity or during the medieval period.

The new role of indivisibles was adopted in maths during the Renaissance period through the work of Galileo's disciple Bonavenlura Cavalieri. In 1635 in his book, the indivisible, or fixed infinitesimal, was applied so successfully to problems in the measuration of areas and volumes that the fundamental postulate, usually bearing the name "Cavalieri theorem" has remained in elementary textbooks to this day. " If two solids (or plane regions) have equal altitudes, and if sections parallel to the bases and at equal distances from them are always in a given ratio, then the volumes (or areas) of the solids (or regions) are also in this ratio." This principle permitted Cavalieri to pass from a strict correspondence of indivisibles in a given ratio to the conclusion that the totalities of these indi­visibles (i.e., the figures of higher dimensionality) were also in this ratio. The method of indivisibles was not the property of Cavalieri, the idea behind it was not really new in 1635, for it is essentially related to the mechanical method of Archimedes and to the graphical integrations of Oresme and Galileo. Kepler had used the idea when he found the area of the ellipse to be

Cavalieri ingeniously applied the idea of indivisible to a wide variety of new problems. Cavalieri, like his contemporaries, regarded his method of indivisibles as part of geometry; but even as he was writing, an analytic revolution due to Descartes and Fermat was sweeping through Europe and inevitably changed the course of infinitesimal analysis.

Pierre de Fermat, Isaac Barrow

There is every reason to acknowledge, with P. S. Laplace, that P. Fermat was the inventor of the differential calculus, as Eudoxus has been recognized as the inventor of the integral calculus. Fermat's method of integrating x" was the most elegant of those available at the time, and it came closer to the modern Riemann integral than any other before the nineteenth century. Fermat showed that for all rational values of except n = — 1,

This work by Fermat, unfortunately unpublished at the time, brought to a climax the methods of integration initiated by Eudoxus two millennia before.

But Fermat, the greatest of all amateurs in maths, was responsible for an even more significant contribution to the development of the calculus. He was literal­ly the inventor of the process that we now call "differentiation". During the very years in which he and Descartes were inventing analytic geometry, Fermat had discovered an amazingly simple method for finding the maxima and minima of a polynomial curve. Did Fermat, who was well aware of the rules for differentiating and integrating, notice the relation between these? He apparently knew full well that in the first case one multiplied the coefficient by the exponent and lowered the exponent by one unit, whereas in the latter case one increased the exponent by one unit and divided the coefficient by the new exponent. Strangely, Fermat saw nothing significant in this striking inverse relationship, nor did his contemporaries, such as, for example, Isaac Barrow. Barrow, who was a teacher of Isaac Newton, published a rale of tangents much like Fermat's method of maxima and minima, in which the interplay of coefficients and exponents again was clear.

Fermat's method for maxima and minima and Barrow's tangent rule were by no means the only devices and formulas invented in connection here. Many other mathematicians noted the play of coefficients and exponents in finding tangents and extrema of polynomials, and the inverse nature of tangent and quadrature problems. These results were finding applications in the sciences of that time: in Fermat's principle of least time in the refraction of light and in the dynamics of Christian Huygens. By the end of the second third of the seventeenth century all the rules needed to handle such problems in areas and rates of change, in maxima and tangents, were available. The time was now ripe to build the infinitesimal analysis into the subject we know as the calculus. No specific new invention was needed; the techniques were at hand. What was wanting was a sense of the universality of the rules. This awareness was achieved first by Isaac Newton, in 1665-1666, and again, independently, by Gottfried Wilhelm von Leibnitz in 1673-1676.