3000 Years of Analysis

3000 Years of Analysis by Thomas Sonar provides a mathematical and cultural excursion through the historical development of mathematical analysis from ancient to modern times. The mathematical focus of the text is placed on models of continuous change such as the differential and integral calculus. This focus naturally includes the study of infinitely small quantities and culminates with a discussion of the formal development of nonstandard models of analysis in the twentieth century. Sonar notes that in choosing a duration of 3000 years, he is making a compromise since one cannot say with certitude exactly when notions of analysis began to germinate.

The original German language editions of 3000 Years of Analysis were published in 2011 and 2016. The focus of this review is on the English translation of the 2016 edition. Sonar states in his preface to the English translation that he was partly motivated by an intention to make “the history of analysis available to interested nonspecialists and a broader audience.” The same preface contains an elegantly stated definition of analysis: “In essence analysis is the science of the infinite; namely the infinitely large as well as the infinitely small. Its roots lie already in the fragments of the Pre-Socratic philosophers and their considerations of the ‘continuum’, as well as in the burning question of whether space and time are made ‘continuously’ or made of ‘atoms’. Thin threads of the roots of analysis reach even back to the realms of the Pharaohs and the Babylonians from which the Greek[s] received some of their knowledge.”

Despite the appeal to nonspecialists Sonar makes clear in his preface that the apprehension of analysis does not come for free: “But not later than with Archimedes (about 287–212 BC) analysis reached a maturity which asks for the active involvement of my readership. Not by any stretch of imagination can one grasp the meaning of the Archimedean analysis without studying some examples thoroughly and to comprehend the mathematics behind them with pencil and paper.” In practical terms, this means that the purely mathematical passages of 3000 Years of Analysis require at least a good undergraduate background in mathematics. In other words, the level of mathematical difficulty of 3000 Years of Analysis is comparable to that of the The Historical Development of the Calculus by C. H. Edwards, Jr Edw79. Indeed, as Sonar makes plain, there are a number of instances where he explicitly follows Edwards.

A unique feature of 3000 Years of Analysis is that it provides an exquisitely detailed treatment of the history of analysis on three distinct levels: (1) the historical and cultural sweep of the times in which key advances took place is given, (2) ample biographies of the personages behind the advances are given, and (3) the mathematical foundations of the advances are presented. Such an undertaking is both a formidable task and a delicate balancing act. The outcome is a fascinating and valuable addition to existing literature on the history of mathematics. We now pick up some of the main threads of Sonar’s monograph beginning with the fourth century BC.

At this time the ancient Greek mathematicians were well aware of the existence of incommensurable geometric magnitudes and the implied limitations of the Pythagorean theory of proportionality. These deficiencies were addressed by Eudoxus of Cnidus (408–355 BC). Eudoxus was a student at Plato’s Academy in Athens and he is universally regarded as the greatest mathematician of the fourth century BC, not least because of his theory of proportionality. Eudoxus defines two geometric ratios $a : b$ and $c : d$ to be proportional, denoted $a : b = c : d$, if and only if for any two given positive integers $m$ and $n$, it follows that either (1) $na > mb$ and $nc > md$, or (2) $na = mb$ and $nc = md$, or (3) $na < mb$ and $nc < md$.

Eudoxus’s extensive theory of proportionality appears in Book V of Euclid’s Elements. In the case of two incommensurable magnitudes $a$ and $b$, Eudoxus’s definition of proportionality partitions the rational numbers into two disjoint subsets $U$ and $O$, where $U$ consists of all rational numbers $m/n$ such that $m : n < a : b$ and $O$ consists of all rational numbers $m/n$ such that $m : n > a : b$. Of this partition Sonar notes: “At the ‘interface’ between $U$ and $O$ a new number may be defined which obviously has to be an irrational one. We had to wait well until the second half of the 19th century before Eudoxus’s theory of proportions could be utilized for the construction of the real numbers. This fundamental step was finally carried out by the mathematician Richard Dedekind (1831–1916) from Brunswick, Germany.”

Eudoxus was also the first to state what has since become known as the Archimedean axiom. Namely, given two geometric magnitudes $a$ and $b$, there exists a positive integer $n$ such that $na > b$. The germ of this axiom appears in Book V of Euclid’s Elements (Definition 4) where it is stated rather differently: Two geometric magnitudes “are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.” On the basis of this axiom, Eudoxus was able to prove that if two geometric ratios $a : c$ and $b : c$ satisfy $a : c = b : c$, then $a = b$. Eudoxus’s proof is a classical exercise in reductio ad absurdum. Sonar also notes: “With the Archimedean axiom Eudoxus also brought a method for the computation of areas to life: the method of exhaustion.”

Lurking in the work of Eudoxus and other ancient Greek mathematicians we see implicit considerations of infinite processes. By adroitly using the Archimedean axiom and the method of exhaustion, Eudoxus and other ancient Greek mathematicians were able to explicitly avoid taking limits when calculating areas and volumes. Nevertheless, notions of the infinite were hotly debated by ancient Greek scholars. Two schools of thought erupted: atomism (the existence of fundamental indivisible components in nature) and the theory of the continuum (that which is always divisible no matter how often it is divided). Leucippus (5th century BC) and his student Democritus (460–370 BC) are credited with founding atomism. They contended that matter is not infinitely divisible and that it is, indeed, composed of individual discrete particles or atoms that cannot be divided. Aristotle (384–322 BC), who drew a crucial distinction between potential and actual infinities, was a major proponent of the theory of the continuum. For Aristotle, the continuum is a potential infinity: No matter how often a continuum is divided, a continuum will remain.

Foremost in the canon of ancient Greek mathematics are the brilliant works of Archimedes of Syracuse (287–212 BC). In some ways, Archimedes was a proto-engineer whose inventions involving levers, pulleys, and screws drew great acclaim during his lifetime. However, as his extant writings make patently clear, Archimedes’s primary focus was undoubtedly on mathematics. These writings focus on area, length, and volume calculations and they hone the method of exhaustion into a tool of phenomenal precision. Scholars of Archimedes identify three codices of his writings that are generally referred to as A, B, and C. As Sonar explains: “Already with the writings contained in codices A and B Archimedes could be identified as a great mathematician and physicist, but it is codex C that catapulted Archimedes into the heaven of immortals and gave him a place of honour at the side of Newton and Leibniz.”

Codex C was lost for the best part of a millennium only to come to light as partially erased text in a medieval prayer book that resurfaced in 1899. During the summer of 1906, using nothing more than a magnifying glass, the renowned Danish philologist and historian Johan Ludvig Heiberg (1854–1928) determined that the text of the prayer book was written over several treatises of Archimedes, including two previously unknown works: The method and Stomachion. The greatest revelation in Codex C is undoubtedly The method because it presents the mechanical heuristic that Archimedes used to discover some of his most outstanding quadrature and center of gravity results. The rationale behind Archimedes’s heuristic was based on levers in equilibrium and he went on to apply the method with extraordinary success by “weighing” indivisibles.

We see in the sixth century AD a tremendous phase transition in the history of mathematics. As the Roman empires crumble a great confluence of Greek, Persian, and Indian mathematical ideas takes place in the emerging Arabic realms. The prophet Mohammed was born in Mecca in 570 AD and went on to found Islam after receiving divine inspiration in the Cave of Hira, Jabal an-Nour (mountain), in 610 AD. Within decades large swathes of the Greek-Hellenistic world, the Iberian Peninsula, and North Africa became subject to rapidly emanating influences of Arabic and Islamic culture. Manuscripts of the ancient Greek, Persian, and Indian mathematicians were translated into Arabic and disseminated throughout the Arabic realms.

The mathematician and astronomer al-Khwārizmī is thought to have lived from 780–850 AD. Not much is known about al-Khwārizmī’s life but he worked at the Grand Library of Baghdad under the patronage of the Abbasid dynasty Caliph Ma’mun. During this period Baghdad was a renowned center for the study of the works of ancient Greek, Persian, and Indian scholars. In terms of mathematics, al-Khwārizmī wrote influential textbooks on arithmetic and algebra. His textbook on arithmetic, which only survived to more modern times in Latin translation (Algoritmi de numero Indorum), dealt extensively with the Hindu art of reckoning. Manipulation of decimal numerals, positional notation, and a symbol for zero all figure prominently in Algoritmi de numero Indorum.

A central figure in the firmament of Islamic Golden Age scholarship is the physicist and mathematician al-Haytham (965–1039 AD). Considered to be a parent of modern day optics, al-Haytham underpinned his investigations into the nature of light and vision with direct experimental evidence. His multi-volume work on optics was translated into Latin as Opticae thesaurus Alhazeni in 1270 and subsequently became influential in Western Europe. Al-Haytham was a master practitioner of the method of exhaustion for calculating areas and volumes. In particular, al-Haytham obtained nontrivial generalizations of some of Archimedes’s celebrated volume results.

The Dark Ages refer to the European period of some four centuries that followed the fall of the Western Roman Empire in 476 AD. During this period of stagnation and decline the scientific posture of Western Europe was seriously corroded and, in fact, barely limped along. The masterworks of the ancient Greeks were reduced to a dim memory during the Dark Ages. Of this period Edwards Edw79 writes: “Only the Latin encyclopedists preserved any connection, however tenuous, with the intellectual treasures of the past.” This statement is beautifully unpacked by Sonar in his treatment of “the great time of the translators.” One of the earliest translators of Arabic texts was Adelard of Bath (1080–1152). Sonar notes: “One of the first Latin translations (from the Arabic) of Euclid’s Elements flew from his quill as did astronomical tables compiled by al-Khwārizmī.” And so it happens, punctuated by crusades and other calamities, that a rich mathematical tradition, nurtured and enriched by Arab scholars, slowly finds its way back into late medieval Europe.

Between 1328 and 1350 a group of logicians and natural philosophers at Merton College in Oxford developed a theory, that became known as latitudes of forms, to quantify “qualities” such as heat and speed. The Merton scholars introduced rigorous definitions of uniform motion and acceleration, and derived the mean speed theorem for uniformly accelerated bodies. Sonar describes the mean speed theorem as being the “First Law of Motion.” The theorem represents a radical departure from millennia of primarily static mathematical thought and it inaugurates kinematics as a fundamental field of scientific inquiry. Sonar deftly explains how this profound transformation arose from within Scholasticism and he provides vivid insights into some of the key historical figures, including richly detailed passages on Robert Grosseteste, Roger Bacon, Albertus Magnus, Thomas Bradwardine, and Nicole Oresme.

The ideas of the Merton College scholars spread rapidly to France and Italy in the middle of the fourteenth century. The theory of latitudes of forms was keenly studied and extended by the Parisian polymath Nicole Oresme (1320/25–1382). Oresme introduced graphical representations of intensities of qualities and provided a geometric verification of the mean speed theorem. In Oresme’s work we see glimmers of the graphical representation of functional relationships and steps being taken towards the introduction of coordinate systems.

At the beginning of the fifteenth century a clear shift away from the Middle Ages became palpable in Europe. The ensuing Renaissance lasted for roughly two centuries and provided the pathway to modernity. The period of the High Renaissance in the Italian states, which started in about 1495 and lasted for around thirty years, is of central importance to art historians, not least because of the epic artworks of Leonardo, Michelangelo, and Raphael. The vivid cultural flowering of the Renaissance encompassed a period of rapid scientific progress. As Sonar points out, a key driver of this scientific progress was the recently invented Gutenberg printing press.

One may view the Renaissance as embodying a shift away from the Church-centered Scholasticism of the Middle Ages and toward the establishment of a new era of Humanism–a return to the centrality of the individual as emphasized in classical antiquity. Renaissance Humanism included the notion that natural phenomena may be completely explained by science and mathematics. Nowhere was this notion more clearly displayed than in the radical astronomical models of Copernicus, Kepler, and Galileo. Sonar pays particular attention to Johannes Kepler (1571–1630) and this is hardly surprising because Kepler, apart from being one of the most extraordinary figures in the history of science, was a proponent of using infinitesimals as a means to simplify the calculations of areas and volumes. The tumultuous life and times of Kepler are vividly sketched by Sonar over the course of some twenty pages. These biographical musings are followed by a treatment of Kepler’s tactics for using geometric infinitesimals to calculate areas (such as Kepler’s “barrel rule” for determining the area under a parabola) and volumes of solids of revolution (such as the torus).

The first half of the seventeenth century also witnessed the majestic contributions of René Descartes (1596–1650) and Pierre de Fermat (1607–1665) to the development of analytic geometry and analysis. As Sonar implies, Descartes and Fermat were polar opposites in terms of temperament. For instance, about Descartes, Sonar writes, “After an utterly eventful life as a superb philosopher, mathematician, physicist, bon vivant, mercenary, and wrangler he died on 11th February 1650, shortly before his 54th birthday, in Stockholm.” In stark relief we have Fermat, who Sonar describes as being “the most unobtrusive of the great French mathematicians of the 17th century – no known scandals, no life as a mercenary, and no sharp turning points in his life; at least as far as we know. He was, however, one of the most profound thinkers of his age.” The historical significance of the new analytic geometry of Descartes and Fermat is succinctly summarized by Edwards Edw79: “Whereas the Greek geometers had suffered from a paucity of known curves, a new curve could now be introduced by the simple act of writing down a new equation. In this way, analytic geometry provided both a much broadened field of play for the infinitesimal techniques of the seventeenth century, and the technical machinery needed for their elucidation.”

Until the early seventeenth century the construction of tangent lines to curves were, for the most part, a rarity. In the 1630s, aided by the new analytic geometry, Fermat and Descartes introduced novel methods for constructing tangent lines to hitherto unknown classes of curves. Fermat used a poorly explained but presumably infinitesimal based “pseudo-equality” technique to construct tangent lines. In contrast, the “circle method” of Descartes is of a purely algebraic nature. Descartes was pleased to avoid the use of infinitesimal arguments in his construction of tangent lines but it came at a high cost in terms of the monotonous algebraic calculations that had to be carried out. This shortcoming of the circle method of Descartes was largely alleviated in the 1850s by algorithmic advances of Johann Hudde (1628–1704) and René de Sluse (1622–1685) that streamlined the construction of tangent lines.

Descartes’s circle method only applies to explicitly defined curves of the form $y = f(x)$, $f^{2}$ a polynomial. In the mid 1650s de Sluse took things even further and developed an algorithmic procedure for constructing tangent lines to implicitly defined curves of the form $f(x,y) = 0$, $f$ a bivariate polynomial. Sluse’s rule was published in the 1672 Philosophical Transactions of the Royal Society but without any indication of how the rule was obtained. As Edwards Edw79 points out: “Whatever may have been the means by which Sluse’s rule was first discovered, the principal significance of the rules of Sluse and Hudde lay in the fact that they provided general algorithms by which tangents to algebraic curves could be constructed in a routine manner.”

In Italy, during the time of the great works of Descartes and Fermat, Galileo’s disciple Bonaventura Cavalieri (1598–1647) put forth radical new ideas on how to apply indivisible techniques to solve previously inaccessible quadrature and cubature problems. The most well-known theorem of Cavalieri is the following simplified principle: “If two solids have equal altitudes, and if sections made by planes parallel to the bases and at equal distances from them are always in a given ratio, then the volumes of the solids are also in this ratio.” (Quoted from Edw79, p. 104.)

Cavalieri was also skilled in the manipulation of the cross sectional indivisibles of lone geometric figures. For example, by calculating “sums of powers of lines”, Cavalieri was able to determine the area under the curve $y = x^{n}$ ($n$ a positive integer) on the interval $[0,1]$, albeit not very rigorously. As Sonar remarks: “Looking at Cavalieri’s ‘summations’ today is breathtaking and hair-raising. There were lines of thickness $0$ airily ‘summed’ and put into ratios; hence it may be not surprising that opposition formed quickly against Cavalieri’s method of indivisibles.”

During the sixteenth century mounting pressure to simplify tedious arithmetic calculations led to the invention of logarithms by John Napier (1550–1617). In essence, Napier isolated his definition of a logarithm from a series of proto-logarithmic tables and a kinematic model involving points moving on a pair of line segments. In order to construct these tables Napier combined judicious numerical choices together with some subtle nonlinear interpolation schemes. Napier’s logarithmic tables were published as a slender volume in 1614, adroitly entitled Mirifici logarithmorum canonis descriptio. The impact of Napier’s wonderful logarithms was both dramatic and immediate. For example, Kepler used Napier’s tables to simplify the computations that led to his discovery of the third law of planetary motion.

At the outset of the seventh chapter of 3000 Years of Analysis Sonar presents twenty pages of thoughtfully written biographical material on Isaac Newton (1643–1727). Sonar’s treatment is concise, deft, and finely balanced. The capacity of Newton to be cantankerous is clearly stated but not overly dwelt upon and, for this, the reader can be most grateful. Sonar charts Newton’s difficult childhood days (when great ingenuity was already abundantly evident in the young man), the sublimely productive Cambridge years 1663–1687, and eventual decline as a well paid government official. Of the beginning of the sublime period, Sonar writes: “In a notebook we find Newton’s true occupation in 1663: Theorems concerning conic sections following Pappus, remarks concerning geometrical theorems by Viète, van Schooten, and Oughtred, theorems concerning arithmetic by John Wallis, methods of grinding lenses, questions of natural philosophy, theology, and alchemy.”

During the period 1663–1687, Newton became an expert manipulator of infinite series. The starting point for Newton was studying John Wallis’s (1616–1703) Arithmetica infinitorum. Newton’s reading of Wallis led him to formulate the binomial series expansion of $(1 + x)^{\alpha }$, $\alpha$ a constant. The case $\alpha = 1/2$ was previously known to Henry Briggs (1561–1630) through his work on logarithms and finite versions of the binomial theorem (when $\alpha$ is a positive integer) had been known since antiquity. Critically, Newton’s formulation of the binomial theorem allowed for the free use of negative and fractional exponents.

Newton’s interest in the binomial theorem was not idle. In developing the calculus of fluxions, Newton relied on the binomial theorem to unlock implicit differentiation for curves of the form $f(x,y) = 0$, $f$ a bivariate polynomial. As Sonar notes: “Newton thought in terms of motion and velocities when he attempted to compute tangents of curves of the form $f(x, y) = 0$. In the eyes of Newton the curve $f(x, y) = 0$ itself ‘results’ from the points of intersection of two moving lines which we can interpret as being the velocity components in $x$- and $y$-direction.”

Newton wrote up his work on fluxions in a manuscript that is dated October 1666. This so-called October tract was circulated to some English mathematicians but it was not formally published. Included in the October tract is Newton’s “inverse method of fluxions.” This was the first statement of the fundamental theorem of calculus in the history of mathematics. In the words of Sonar: “Modern analysis could only begin with the thorough knowledge that differentiation and integration are inverse operations. Barrow had this result implicitly but it was left to Newton and Leibniz to clearly acknowledge the central place of the fundamental theorem.” Newton went on to use his method of fluxions to show that problems of quadrature, constructing tangent lines, rectification of curves, extreme values, and so on, all fall under one umbrella. Newton had thus unified millennia of mathematical analysis into a single coherent whole. Gottfried Wilhelm Leibniz (1646–1716), within the space of a few short years, would independently obtain the fundamental theorem of calculus, but from a different point of view to that of Newton.

Leibniz completed a baccalaureate in philosophy and mathematics at the University of Leipzig in 1663 and was awarded a doctorate of law at the University of Altdorf in 1667. In 1672 Leibniz traveled to Paris in a diplomatic capacity for the Elector of Mainz. This placed Lebniz within the orbit of a coterie of superb European scholars, including the Dutch mathematician Christiaan Huygens (1629–1695). Huygens is credited with bringing Leibniz up to pace on the mathematical literature of the times. In this way, Leibniz was exposed to significant treatises such as Arithmetica infinitorum by John Wallis and Opus geometricum by Gregorius Saint-Vincent (1584–1667). During his four years in Paris, Leibniz began to assemble and finesse his own invention of the calculus.

In discussing Leibniz’s development of the calculus Sonar makes the following preliminary remark: “We are used to present Newton’s results in Leibniz’s notation simply because it turned out to be more feasible.” Indeed, the primacy of Leibniz’s notation in calculus is not simply a quirk or historical accident. During his life, Leibniz was intensely interested in finding a universal language or characteristica universalis that would allow complicated notions of reasoning to be distilled into simpler components. The program of characteristica universalis is a recurring theme throughout the works of Leibniz and it necessarily involves a preoccupation with symbols and notation. The outcome is summed up by Edwards (Edw79, p. 232): “His infinitesimal calculus is the supreme example, in all of science and mathematics, of a system of notation and terminology so perfectly mated with its subject as to faithfully mirror the basic logical operations and processes of that subject. It is hardly an exaggeration to say that the calculus of Leibniz brings within the range of an ordinary student problems that once required the ingenuity of an Archimedes or a Newton.”

Section 7.2.4 in 3000 Years of Analysis deals with the infamous priority dispute that erupted between Newton and Leibniz over the invention of the calculus. Newton had developed the outline and framework for his fluxion-based calculus during the years 1664–1666 but he did not formally disseminate the work until the publication of Philosophiae Naturalis Principia Mathematica in 1687. Prior to 1676, Newton’s calculus of fluxions and fluents was for the most part unknown outside of England. An exception to this was a letter that Newton had sent to de Sluse in 1672 on the matter of constructing tangents. In contrast, Leibniz developed his own differential-based version of the calculus during the years 1672–1676, almost a decade later than Newton. However, the priority of publication for works on both differential and integral calculus belongs to Leibniz. In 1684 and 1686 Leibniz published articles on differential calculus and integral calculus (respectively) in the Leipzig periodical Acta Eruditorum.

Using Henry Oldenburg as an intermediary, Newton and Leibniz corresponded directly about the origins of the calculus during the second half of 1676. Newton addressed two letters to Leibniz that have since become known as Epistola prior (13 June 1676) and Epistola posterior (24 October 1676). The first letter was open and friendly but the tone of the second letter was more circumspect and it used an insoluble anagram at a critical juncture to secrete the true scope of fluxional calculus from Leibniz. By the end of 1676 some lines had been drawn but there was, as yet, no rancorous priority dispute between Newton and Leibniz. Matters took a turn for the worse in 1684 with Leibniz’s publication of Nova methodus pro maximis et minimis in Acta Eruditorum and were overheated by 1699. As Blank Bla09 puts it: “Throw in a priority dispute, charges of plagiarism, and two men of genius, one vain, boastful, and unyielding, the other prickly, neurotic, and unyielding, one a master of intrigue, the other a human pit bull, each clamoring for bragging rights to so vital an advance as calculus, and the result is a perfect storm.”

Subsequent to giving a measured treatment of the priority dispute, Sonar turns to the actual mechanics of Leibniz’s development of the calculus. The extent to which Leibniz used infinitesimals is discussed at length, with particular attention given to the so-called characteristic triangle and its relation to Leibniz’s transmutation theorem:

$$\begin{equation*} \int _a^b y\, \mathrm{d}x= xy \big |_a^b - \int _a^b xy^{\prime }\, \mathrm{d}x. \end{equation*}$$

Leibniz was keenly aware of both the significance and versatility of the transmutation theorem. It put Leibniz in a position to re-derive virtually all previously known plane quadrature results and to provide some brilliant new applications. One such new result was Leibniz’s “arithmetical quadrature of the circle” which leads to the heavenly series:

$$\begin{equation*} \frac{\pi }{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots . \end{equation*}$$

The brightest mathematical star of the Age of Enlightenment was Leonhard Euler (1707–1783). Born and educated in Basel, Switzerland, Euler entered the University of Basel at the age of 13 or 14 and was mentored in mathematics by John Bernoulli (1667–1748). Euler’s first mathematical work, Constructio linearum isochronarum in medio quocunque resistente, was published in Acta Eruditorum in 1726. This paper marks the beginning of a prodigious and wide ranging mathematical output by Euler, the collected works of whom exceeds 70 hefty volumes.

After the immense flowering of infinitesimal analysis in the eighteenth century an unease about rigor started to become pervasive at the beginning of the nineteenth century. Or as Sonar puts it: “After the death of Euler many mathematicians believed that there would not be much left in mathematics worth[y] of study. On the other hand one felt a certain discomfort concerning the foundations of analysis which was triumphant in applications but operated still upon infinitely small quantities or even, as with Euler, upon ‘zeros’.” It is not surprising then that mathematical research during the nineteenth century focused on consolidation and a search for rigor, at least in terms of analysis. Against this backdrop a new wave of mathematicians began to apply themselves to the determination of a more rigorous basis for (infinitesimal) analysis and related notions such as continuity.

One of the earliest and most important figures in this new wave of rigor-oriented mathematical analysis was Bernhard Bolzano (1781–1848). In 1817 Bolzano published a note in Prague entitled Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. It is reasonable to assert that this note heralds the advent of nineteenth century mathematical analysis for in it we find (arguably) the first precise formulation of continuity. Bolzano defined continuity as follows: $f$ is continuous at $x$ if the “difference $f(x + \omega ) - f(x)$ can be made smaller than any given quantity, if one makes $\omega$ as small as one wishes.” Bolzano’s note goes on to give a rigorous proof of the intermediate value theorem.

As Sonar explains, Augustin Cauchy (1798–1857) provided a proof of the intermediate value theorem at around the same time as Bolzano. However, whether or not Cauchy formulated continuity in similar modern terms to Bolzano remains moot. Across a number of picturesque passages Sonar evokes the rich tapestry of the life, times and mathematics of Cauchy, writing at one point: “As a teacher Cauchy turned out to be a revolutionary. Since he considered analysis being indispensable to engineers he gave lectures on that topic. He developed a rigorous concept formation of the limit and attached much importance to utmost accuracy which discouraged his students.”

Prior to the nineteenth century there was no formal definition of the definite integral as we know it today. In the eighteenth century integration typically entailed finding anti-derivatives in the spirit of either Newton or Leibniz. Of this century, Edwards Edw79 writes: “Neither limits of sums nor areas of plane sets were sufficiently well understood to provide a solid basis for a logical treatment of the integral.” Cauchy was the first to develop a notion of the definite integral that was predicated in terms of limits of sums, rather than anti-derivatives. The “Cauchy integral” of a continuous function on a compact interval was first introduced by Cauchy in his textbook Résumé des leçons sur le calcul infinitésimal.

Despite the success of putting integration on a more secure footing, the Cauchy integral was too limited in its scope to deal with basic questions that had already arisen from the works of Joseph Fourier (1768–1830) and Lejeune Dirchlet (1805–1859) on the representation of functions by trigonometric series. This limitation led Bernhard Riemann (1826–1886), who was also profoundly interested in Fourier series, to develop a more general definite integral. Riemann gave necessary and sufficient conditions for a bounded function to be (Riemann) integrable and he pointed out that it is possible for a function with a dense set of discontinuities to be integrable.

Edwards Edw79 mentions that “Cauchy occasionally stumbled conspicuously, as in failing to distinguish between continuity and uniform continuity or between convergence and uniform convergence.” One of the first persons to grasp the importance of uniform convergence was Karl Weierstrass (1815–1897). In 1872 Weierstrass stunned the mathematical world by exhibiting a nondifferentiable continuous function. Prior to the publication of Weierstrass’s example there had been a common misapprehension that a continuous function may only have isolated points of nondifferentiability. (Bolzano was under no such illusion as he had given an example of a nondifferentiable continuous function in the 1830s. However, in a quirk of fate, Bolzano’s example was not published until the 1920s.) Part of the fallout from Weierstrass’s example was a realization that the foundations of analysis needed further attention, especially in respect of the construction of the real numbers. This led to a flurry of constructions of the real line in the early 1870s, the most enduring of which are those of Richard Dedekind (1831–1916) and Georg Cantor (1845–1918).

Chapter eleven of 3000 Years of Analysis deals with the twentieth century renaissance of infinitesimal analysis. Sonar recounts that inklings of this revival may be found in an unpublished “black book” Vom Unendlichen und der Null – Versuch einer Neubegründung der Analysis that was written by Curt Schmieden (1905–1991). In the mid 1950s Detlef Laugwitz (1932–2000) became acquainted with the “black book” and this eventually led to a joint paper with Schmieden SL58. This paper presents a largely constructive version of nonstandard analysis but it is one in which the (putative) hyperreal numbers $\mathbb{R}^{\ast }$ form only a partially ordered ring and for which the “transfer principle” from the reals to hyperreals is quite limited. A considerably more satisfactory version of nonstandard analysis was developed by Abraham Robinson (1918–1974) during the 1960s. Robinson’s nonstandard analysis is based on model theory and it is one for which the hyperreal numbers $\mathbb{R}^{\ast }$ form a non-Archimedian field equipped with a substantial transfer principle.

Sonar’s discussion of nonstandard models of analysis completes an epic walk through several millennia of mathematical discovery. It is apt that Sonar’s treatment of the historical development of analysis effectively begins and ends with the continuum. We remark that the sheer depth of historical detail included in 3000 Years of Analysis sets it apart from other sublime works such as that of Edwards Edw79. We further remark that Sonar’s monograph is richly illustrated with a plethora of geometric figures, line drawings, engravings, reproductions of historical artworks and photographs.