Both the classical enthusiasms of Paduan lawyers and notaries and the literary works of Florentines like Brunetto Latini and Dante show that about 1300 the prosperous educated laymen in the Italian cities were groping their way towards a new culture distinct from both the chivalric culture of the medieval nobility and the scholastic culture of the clergy. This was a natural response to the conditions of their life. Since the nineteenth century, historians have labelled this new culture ‘humanism’, though that abstract term was coined by a German scholar in 1808 and appears nowhere in the writings of the Renaissance itself. The term that did exist then was ‘humanistic studies’ (studia humanitatis), used to designate a cluster of academic subjects much favoured by humanists. By the first half of the fifteenth century, the term ‘humanist’ (in Latin, humanista) had come into use, originally as student slang used to designate masters who taught those particular academic subjects: grammar, rhetoric, poetry, history, and moral philosophy. ‘Humanism’, the bundle of subjects taught by ‘humanists’ in the Latin grammar schools and university faculties of liberal arts, made no claim to embrace the totality of human learning, nor even all of the traditional seven ‘liberal arts’ (embracing the trivium, or grammar, rhetoric, and dialectic, and the quadrivium, or arithmetic, geometry, astronomy, and music) that in theory were studied by all who received the bachelor and master of arts degrees from a university. The studia humanitatis did not include the subjects taught in the three higher faculties of medieval universities: law, medicine, and theology.

By the closing decades of the Quattrocento (the fifteenth century), humanism dominated the culture of the ruling groups in all important Italian cities, from Rome north to the Alps. Although Florence deservedly has received the most attention and provides the best evidence of a link between the new culture and an ethic of political action and public service, humanism fitted almost as neatly into the need of the many princely courts for a distinctive lay culture and for a cadre of classically educated administrators and advisers to the ruler. Even the papal curia at Rome found the linguistic and scholarly skills of humanists useful in the chancery, the department that drafted papal letters and many other official documents. Although the sympathy of Pope Nicholas V for humanism, expressed in his library and his programme of sponsoring translations from the Greek, quickened this process, humanists played an important role in curial administration even under Paul II (1471–84), who began his pontificate by dismissing many curial humanists and even arrested several of them in 1468 on trumped-up charges of impiety that probably masked his fear of political conspiracies. By 1500, Rome was on its way to outstripping Florence as the most brilliant centre of humanistic culture, just as it was also taking leadership in the patronage of art.

Printing and the new culture

The rise of curial and courtly humanism and the growing dominance of humanist schoolmasters in the Latin grammar schools of the peninsula did much to establish humanism as the major force in Italian culture. Another source of humanism's growing dominance was the new art of printing.

The bright hopes of a dawning golden age of peace, social justice, and religious renewal that Erasmus expressed in his famous letter of 1517 to the young humanist Capito (Ep. 541, in Collected Works of Erasmus 4: 261–68) evaporated in the fires of religious controversy. For Erasmus personally, the bitter divisions that separated Protestant from Catholic were both a social disaster and a personal tragedy, causing sharp alienation between him and many of his younger disciples. Christian humanism, Erasmian humanism, as a coherent programme of religious and cultural renewal ceased to be a unified movement. Some historians would therefore end the history of Renaissance humanism with the outbreak and rapid spread of the Reformation, taking the open break between Erasmus and Luther in 1524 as the symbolic final act. Though such a conclusion reflects the bleak despair with which Erasmus and his friends viewed the religious schism, it overlooks some deeper and more lasting consequences of the spread of humanistic learning. Despite the uncertainties and violence of the Reformation era, humanism was alive. Although divided by confessional hostilities, humanism remained a major force in religion, both Protestant and Catholic. Its penetration of schools and universities continued almost everywhere, so that even the most self-consciously conservative educational systems produced graduates equipped with humanistic knowledge and skills. The humanist penetration of the schools inevitably led also to a more shallow but unmistakable diffusion of the new Latin culture into the vernacular world, through popular and courtly literature, translations, imitations, and the fine arts.

The last thing readers want from the authors of history books is arcane historiographical debates among contending schools of interpretation. Yet in the case of Renaissance humanism, any serious reader needs some knowledge of where we have been since 1860, when the Swiss historian Jakob Burckhardt put forward what long remained the dominant concepts of the key terms, ‘Renaissance’ and ‘humanism’, in his masterpiece of cultural history, The Civilization of the Renaissance in Italy. Although few specialists today would give unqualified endorsement to his description of the age, it was profoundly appealing to his contemporaries, and it still has much to offer.

Burckhardt's book quickly captured the imagination of educated readers because it was a subtle and learned synthesis of opinions about the Renaissance that had been accumulating for centuries and had grown powerful during the Age of the Enlightenment (Ferguson 1948). Burckhardt seemed to confirm a story that had already become prevalent but had never before been given such a powerful and coherent presentation. This story, or historical myth, was the product of the secular intellectuals of the eighteenth and nineteenth centuries who were searching for the origins of their own beliefs and values. According to this story, after the collapse of ancient civilization in the fifth century AD, a thousand years of darkness and barbarism ensued, with the Christian church acting simultaneously to preserve some few shreds of ancient civilization and to suppress any intellectual or religious revivals that might weaken the stranglehold that the higher clergy and the warrior aristocracy held over the minds as well as the bodies of ordinary people.

A decade after the appearance of the first edition, the opportunity to prepare a revised second edition has permitted me to introduce a number of changes in the text. Some of these are relatively minor revisions made in the interests of clarity and readability. Obviously, the new Bibliographical essay includes a number of important publications that appeared since my earlier work. Three changes, however, are more substantial. First, new publications on Italian education have caused me to rethink the treatment on that subject, now located in Chapter 2. Unfortunately, the story of the ‘educational revolution’ of the Italian Renaissance has become more complicated, though the revised account still maintains that a significant transformation in the nature of Italian grammar-school education took place during the fifteenth century. Second, new publications on what I have called Italian pre-humanism and on the role of ‘civic humanism’ (if any) in the development of Renaissance Italy have led to some rethinking of my earlier treatment, though I remain unconvinced that the basic idea of a historically decisive connection between political and intellectual history in Quattrocento Florence should be abandoned. As a result of these two areas of revision, the original Chapter 1 has been divided, with the new Chapter 2 beginning with the emergence of humanism as the dominant culture of the elite classes not only in Florence but throughout Italy. That emergence of humanism and the mechanisms (such as education) by which it took place are the main theme of the new Chapter 2.

Petrarch raised most of the issues of humanistic culture and set many of the directions in which it moved during the two centuries following his lifetime. He also became a famous literary personality in his own time. Nevertheless, he was a deracinated intellectual with no fixed abode and no real sense of belonging to a community. His exquisite lyric poetry and his classical learning did indeed point towards a new direction for Italian culture, but in the short run his constituency was mainly the wealthy, leisured clergymen and the secular princes whose patronage assured his material existence. What saved Petrarch's work from becoming merely a literature of the court was the adoption of his ideals by the prosperous lay intellectuals of the Italian cities, particularly Florence, which in the late fourteenth century and throughout the fifteenth century was by far the most important centre of humanistic culture. Modern scholars have probed the history of this great city, trying to understand why it outstripped all the other cities where humanism was also developing. Perhaps there is no definitive answer to the mystery of Florentine cultural leadership. Florence was by no means the only city with the wealth and the literate population needed to support the new culture. One special advantage of Florence may have been a lack. Unlike Padua, Bologna, Pisa, and several rival cities, Florence did not have a university at all until 1349. Even after that date, the local university never played a dominant role in cultural life. Hence local intellectuals were relatively free from the traditionalism and professionalism of university life.

By the closing decades of the sixteenth century, Renaissance humanism had become the prevalent culture among the educated elite that dominated the political establishment, the officially recognized churches, and the schools of western and central Europe, north as well as south of the Alps. Even across the bitterly contested divide of the Reformation, and despite the limitations imposed by censorship of the press, inquisitions, and consistories, a broadly uniform higher culture prevailed. Books and ideas passed back and forth from Protestant to Catholic regions. A famous classical scholar and neo-Stoic moralist like Justus Lipsius (1547–1606) could make his academic career at different periods of his life at both Protestant and Catholic universities provided he was willing to conform to the established local form of Christianity, even though there were some bitter remarks about the slippery religious convictions of an author whose most famous book of ethics was called Of Constancy. Furthermore, though the humanistic culture that prevailed in the late Renaissance was in one sense restricted to the privileged few who could obtain an education, the work of translators and popularizers allowed a much larger proportion of the population to share the new culture. A man of the people like Shakespeare had access to themes, ideas, and values found in classical texts, even though he could never penetrate the classical heritage with the profundity of a brilliant and privileged university graduate like John Milton. The humanistic tradition remained dominant in many respects, among plain folk as well as the educated classes, into the late nineteenth century.

Although Italy in the fourteenth and fifteenth centuries was a distinct society with a distinct culture, sharply different from the rest of Europe, it was closely linked to transalpine Europe by religion, politics, and trade. Not only people but also ideas and books passed back and forth across the Alps. During the fourteenth century, many of the mercenary troops who conducted Italy's wars were foreigners, especially French and English. Large colonies of Italian merchants were established in northern commercial centres such as London, Paris, Lyons, and the cities of Flanders and Brabant. The clergy were important carriers of cultural influence, both northerners going to Rome to conduct lawsuits in the church courts, seek dispensations, or petition for appointments, and Italians going north as papal legates, collectors, and appointees. Other carriers of ideas were university teachers and students. A few of these were Italians going north to study (almost exclusively theology, at Paris) or to teach in northern schools and universities; but far more numerous were the northerners coming to Italian universities. From the thirteenth century, Italian faculties of medicine and law were regarded as the best in Christendom, and a doctorate in law or medicine from Bologna, Padua, or another Italian university provided a clear competitive advantage when students returned home. Until after 1450, few northern students came primarily to study humanistic subjects; but Italian law and medical science had some connection with humanistic studies. When transalpine students returned north, they not only carried both ideas and books with them but also in many cases gained influential positions that enabled them to become patrons who promoted humanistic learning.

By the beginning of the sixteenth century, the programme of humanistic studies and the dream of a renewal of ‘civilized’ learning and literature and an end of ‘barbarism’ had established themselves to some extent in all the major countries of western and central Europe. But the new learning, often viewed with suspicion because of its Italian origins and its reverence for pagan literature, had by no means become dominant north of the Alps. Its marginal and subordinate position was accurately reflected in the life of the universities (see Chapter 4). There were some masters in every faculty of liberal arts who criticized traditional textbooks and aspired to modify the curriculum leading to arts degrees in order to de-emphasize logic and give greater attention to humanistic studies. These men were able to offer lectures on classical authors and private lessons in Greek from time to time. A genuine and spontaneous interest in the new learning was growing up, and even in such a notoriously conservative university as Cologne, many students managed to lay the foundation of a mastery of classical Latin and ancient literature, even of Greek, that enabled them to become prominent humanists in later decades. Most humanists were perfectly content to work quietly on the margin of university life, teaching the works of classical or modern humanistic authors, and eventually working their way into more important positions as heads of colleges or attaining far more important positions in one of the three higher faculties that dominated the institutions and offered virtually the only stable, salaried or beneficed professorships.

This book aims to present a comprehensive account of the development and significance of the humanistic culture of Europe (north as well as south) in the age of the Renaissance. It is based on the researches of more than a generation of scholars active since about the end of the Second World War. At that time, critical attacks on the traditional picture of Renaissance civilization established by Jakob Burckhardt had produced so much doubt about the meaning, and even the existence, of a Renaissance that many historians abandoned use of the term itself, a situation brilliantly demonstrated in Wallace K. Ferguson's The Renaissance in Historical Thought (1948). In the aftermath of that historiographical demolition, I myself wrote a dissertation which systematically avoided use of the dread term ‘Renaissance’, and not a single member of my examining committee challenged or even mentioned the omission. Yet leaving ‘the Renaissance’ and ‘humanism’ out of the history of the Renaissance age was not a viable position, as I found when I undertook to revise my dissertation into a book that addressed the intellectual problems of the sixteenth century, and found in an even more pressing way when I faced the task of explaining the Renaissance to a class of college freshmen in a lecture of fifty minutes. Whether historians like the concepts ‘Renaissance’ and ‘humanism’ or not, the centuries to which those terms are conventionally applied really did exist and must be faced, since they contributed in important ways to the subsequent development of Western society and civilization.

We have already discussed, in chapter 4, how complicated functions may be expressed as power series. However, this is not the only way in which a function may be represented as a series, and the subject of this chapter is the expression of functions as a sum of sine and cosine terms. Such a representation is called a Fourier series. Unlike Taylor series, a Fourier series can describe functions that are not everywhere continuous and/or differentiable. There are also other advantages in using trigonometric terms. They are easy to differentiate and integrate, their moduli are easily taken and each term contains only one characteristic frequency. This last point is important because, as we shall see later, Fourier series are often used to represent the response of a system to a periodic input, and this response often depends directly on the frequency content of the input. Fourier series are used in a wide variety of such physical situations, including the vibrations of a finite string, the scattering of light by a diffraction grating and the transmission of an input signal by an electronic circuit.

The Dirichlet conditions

We have already mentioned that Fourier series may be used to represent some functions for which a Taylor series expansion is not possible.

It happens frequently that the end product of a calculation or piece of analysis is one or more algebraic or differential equations, or an integral that cannot be evaluated in closed form or in terms of tabulated or pre-programmed functions. From the point of view of the physical scientist or engineer, who needs numerical values for prediction or comparison with experiment, the calculation or analysis is thus incomplete.

With the ready availability of standard packages on powerful computers for the numerical solution of equations, both algebraic and differential, and for the evaluation of integrals, in principle there is no need for the investigator to do anything other than turn to them. However, it should be a part of every engineer's or scientist's repertoire to have some understanding of the kinds of procedure that are being put into practice within those packages. The present chapter indicates (at a simple level) some of the ways in which analytically intractable problems can be tackled using numerical methods.

In the restricted space available in a book of this nature, it is clearly not possible to give anything like a full discussion, even of the elementary points that will be made in this chapter. The limited objective adopted is that of explaining and illustrating by simple examples some of the basic principles involved. In many cases, the examples used can be solved in closed form anyway, but this ‘obviousness’ of the answers should not detract from their illustrative usefulness, and it is hoped that their transparency will help the reader to appreciate some of the inner workings of the methods described.