Chapter 28
Medieval and Early Modern Europe
Greek mathematics held on longer in the Byzantine Empire than in Western Europe. Although Theon of Alexandria had found it necessary to water down the more difficult parts of Greek geometry for the sake of his weak students, the degeneration in Latin works was even greater. The decline of cities in the West as the authority of the Roman Emperor failed was accompanied by a decline in scholarship. Only in the monasteries was learning preserved. As a result, documents from this period tend to be biased toward issues that concern the clergy.
28.1 From the Fall of Rome to the Year 1200
During the first five centuries after the fall of Rome in 476, a great deal of scholarly work was lost. While a new, and in many ways admirable, medieval civilization was being built up, only some very basic mathematics was being preserved in Western Europe. However, within the Carolingian Empire, the foundation for more advanced activity was being laid in the cathedral and monastery schools, so that when the knowledge achieved in the Islamic world was translated into Latin, scholars were prepared to appreciate and extend it. We shall mention only a handful of the scholars from this time.
28.1.1 Boethius and the Quadrivium
The philosopher Boethius (480–524) wrote Latin translations of many classical Greek works of mathematics and philosophy. His works on mathematics were translations based on Nicomachus and Euclid. Boethius' translation of Euclid has been lost. However, it is believed to be the basis of many other medieval manuscripts, some of which use his name. These are referred to as “Boethius” or pseudo-Boethius. The works of “Boethius” fit into the classical quadrivium of arithmetic, geometry, music, and astronomy. This quadrivium (fourfold path) was neatly subdivided into the categories of number (discrete quantity), magnitude (continuous quantity), statics, and kinematics. Thus number at rest is arithmetic, number in motion is music, magnitude at rest is geometry, magnitude in motion is astronomy.
The quadrivium. From left toright: Music holding an instrument, arithmetic doing a finger computation,geometry studying a set of diagrams, astrology holding a set of charts. Copyright © Foto Marburg/Art Resource.
Politically and militarily, the fifth century was full of disasters in Italy, and some of the best minds of the time turned from public affairs to theological questions. For many of these thinkers, mathematics came to be valued especially because it could inspire religious feelings. The pseudo-Boethius gives a good example of this point of view. He writes1:
The utility of geometry is threefold: for work, for health, and for the soul. For work, as in the case of a mechanic or architect; for health, as in the case of the physician; for the soul, as in the case of the philosopher. If we pursue this art with a calm mind and diligence, it is clear in advance that it will illuminate our senses with great clarity and, more than that, will show what it means to subordinate the heavens to the soul, to make accessible all the supernal mechanism that cannot be investigated by reason in any other way and through the sublimity of the mind beholding it, also to integrate and recognize the Creator of the world, who veiled so many deep secrets.
28.1.2 Arithmetic and Geometry
Besides the geometry just mentioned, Boethius also discussed the numerical part of the quadrivium, including a topic that is not in the older Greek works: the abacus. It was a ruled board, not the device we now call an abacus. The Latin word originally denoted the square stone at the top of a pillar. This computational aspect of arithmetic is not so well represented in the Greek texts. In terms of its number-theoretic content, however, Boethius' treatise is far less sophisticated than the elaborate logical system found in Books VII–IX of Euclid's Elements.
28.1.3 Music and Astronomy
The other two sections of Boethius' work on the quadrivium are also derivative and based on Greek sources. His astronomy omits all the harder parts of Ptolemy's treatise. In addition, he wrote an influential book with the title De institutione musica that is of interest in the history of mathematics, since it adopts the traditional Platonic (Pythagorean) point of view that music is a subdivision of arithmetic. Boethius divides the subject of music into three areas: Musica Mundana, which encompasses the “music of the spheres,” that is, the regular mathematical relations observed in the stars and reflected in the sounds of nature; Musica Humana, which reflects the orderliness of the human body and soul; and Musica Instrumentalis, the music produced by physical instruments, which exemplify the principles of order that the Pythagoreans allegedly ascribed to musical instruments, particularly in the simple mathematical relations between pitch and length of a string.
For over a millennium, such ideas had a firm grasp on writers such as Dante and scientists such as the seventeenth-century mathematician and astronomer Johannes Kepler. Indeed, De institutione musica was used as a textbook at Oxford until the eighteenth century, and Kepler actually wrote the music of the spheres as he conceived it.
28.1.4 The Carolingian Empire
From the sixth to the ninth centuries a considerable amount of classical learning was preserved in the monasteries in Ireland, which had been spared some of the tumult that accompanied the decline of Roman power in the rest of Europe. From this source came a few scholars to the court of Charlemagne to teach Greek and the quadrivium during the early ninth century. Charlemagne's attempt to promote the liberal arts, however, encountered great obstacles, as his empire was divided among his three sons after his death. In addition, the ninth and tenth centuries saw the last waves of invaders from the north—the Vikings, who disrupted commerce and civilization both on the continent and in Britain and Ireland until they themselves became Christians and adopted a settled way of life. Nevertheless, Charlemagne's directive to create cathedral and monastery schools had a permanent effect, leading eventually the synthesis of observation and logic known as modern science.
28.1.5 Gerbert
In the chaos that accompanied the breakup of the Carolingian Empire and the Viking invasions, the main source of stability was the Church. A career in public life for one not of noble birth was usually an ecclesiastical career, and church officials had to play both pastoral and diplomatic roles. That some of them also found time for scholarly activity is evidence of remarkable talent.
Such a talent was Gerbert of Aurillac. He was born to lower-class but free parents in south-central France some time in the 940s. He benefited from Charlemagne's decree that monasteries and cathedrals must have schools and was educated in Latin grammar at the monastery of St. Gerald in Aurillac. Throughout a vigorous career in the Church that led to his coronation as Pope Sylvester II2 in the year 999, he worked for a revival of learning, both literary and scientific. His work as secretary to the Archbishop of Reims was reported by a monk of that city named Richer, who described an abacus constructed to Gerbert's specifications. It was said to have been divided into 27 parts, and Gerbert astounded audiences with his skill in multiplying and dividing large numbers on this device (Lattin, 1961, p. 46).
While revising the curriculum in arithmetic, Gerbert wrote a tract on the use of the abacus in which the Hindu–Arabic numerals were used. This innovation required reintroduction several times, but received a strong impetus two centuries later from the Liber abaci of Leonardo of Pisa.
In some early letters written addressed to the monk Constantine of Fleury just before he became Abbot of Bobbio, Gerbert discusses some passages in Boethius' Arithmetic; and in the last letter written before he became pope, he writes to Adalbold of Liège about an inconsistency in Boethius' work (Lattin, 1961). He discusses an equilateral triangle of side 30 and height 26 (since 
), whose area is therefore 390. He says that if the triangle is measured by the arithmetical rule given by Boethius—that is, in terms of its side only—the rule is “one side is multiplied by the other and the number of one side is added to this multiplication, and from this sum one-half is taken.” In our terms this would give area s(s + 1)/2 to an equilateral triangle of side s. We recognize here the formula for a triangular number. Thus, guided by arithmetical considerations and triangular numbers, one would expect that this formula should give the correct area. However, in the case being considered, the rule leads to an area of 465, which is too large by 20%. Gerbert correctly deduces that Boethius' rule actually gives the area of a cross section of a stack of rectangles containing the triangle in question and that the excess results from the pieces of the rectangles sticking outside the triangle. He includes a figure to explain this point to Adalbold.
We can see from this discussion by one of the leading scholars of Europe regarding the extent to which scientific and mathematical knowledge had sunk to an elementary level a thousand years ago. From these humble beginnings, European knowledge of science underwent an amazing growth over the next few centuries.
Gerbert also wrote a treatise on geometry based on Boethius. His reasons for studying geometry were similar to those given by Boethius3:
Indeed the utility of this discipline to all lovers of wisdom is the greatest possible. For it leads to vigorous exercises of the soul, and the most subtle demands on the intuition, and to many certain inquiries by true reasoning, in which wonderful and unexpected and joyful things are revealed to many along with the wonderful vigor of nature, and to contemplating, admiring, and praising the power and ineffable wisdom of the Creator who apportioned all things according to number and measure and weight; it is replete with subtle speculations.
This view of geometry was to be echoed four centuries later in the last Canto of Dante's Divine Comedy, which makes use of geometric analogs to describe the poet's vision of heaven:
Like the geometer who applies all his powers
To measure the circle, but does not find
By thinking the principle he needs,
Such was I, in this new vista.
I wished to see how the image came together
With the circle and how it could be divined there.
But my own wings could not have made the flight
Had not my mind been struck
By a flash in which his will came to me.
In this lofty vision I could do nothing.
But now turning my desire and will,
Like a wheel that is uniformly moved,
Was the love that moves the sun and the other stars.
28.1.6 Early Medieval Geometry
A picture of the level of geometric knowledge in the eleventh and twelfth centuries, before there was any major influx of translations of Arabic and Greek treatises, can be gained from an early twelfth-century treatise called Practica geometriae (The Practice of Geometry, Homann, 1991), attributed to Master Hugh of the Abbey of St. Victor in Paris.
The content of the Practica geometriae is aimed at the needs of surveying and astronomy and resembles the treatise of Gerbert in its content. This geometry, although elementary, is by no means unsophisticated. It discusses similar triangles and spherical triangles, using three mutually perpendicular great circles to determine positions on the sphere. After a discussion of the virtues and uses of the astrolabe, the author takes up the subjects of “altimetry” (surveying) and “cosmimetry” (astronomical measurements).
The discussion of “altimetry” is a straightforward application of similar triangles to measure inaccessible distances. The section on “cosmimetry” is of interest for two reasons. First, it gives a glimpse of what was remembered of ancient work in this area; and second, it shows what techniques were used for astronomical measurements in the twelfth century. The author begins by giving the history of measurements of the diameter of the earth, saying that the earth seems large to us, due to our confinement to its surface, even though “Compared to the incomprehensible immensity of the celestial sphere with everything in its ambit, earth, one must admit, seems but an indivisible point.”
These views had been expressed by Ptolemy as justification for idealizing the earth as a point in his astronomy, and, of course, they are completely in accord with modern knowledge of the size of the cosmos. The author then goes on to discuss in detail the history of measurements of the circumference of the earth. He tells the famous story of Eratosthenes' measurement of a degree of latitude. (See Section 1 of Chapter 17.)
The author of the Practica geometriae continues by calculating the height of the sun by use of similar triangles. To do this, one must know the distance from the point of measurement to the point where the sun is directly overhead and then measure the length of the noontime shadow cast by a pole of known height. The author says that the Egyptians should be given credit as the first to compute solar altitude this way and that they were successful because their country was flat and close to the sun! The figure cited for the diameter of the sun's orbit (this is geocentric astronomy) is 
miles. Using the value 
, the author computes the length of the sun's orbit as 
miles. (This number is less than 6% of the true value.)
28.1.7 The Translators
During the twelfth and thirteenth centuries, European scholars sought out and translated works from Arabic and ancient Greek into Latin. We can list just a few of the translators and their works here. Our debt to these people is enormous, as they greatly increased the breadth and depth of knowledge of natural science and mathematics in Europe.
Various authors ascribe to each of these last three translators the responsibility for translating the Arabic word jayb, which had evolved from the Sanskrit jiva (bowstring), into Latin as sinus, thereby establishing a usage that has persisted for 900 years all over Europe. Most of these statements are vague as to precisely where the term occurs. According to Holt, Lambton, and Lewis (1970, p. 754), the word occurs first in the twelfth-century translation of al-Battani's Kitab al-Zij, and therefore must have been introduced by Plato of Tivoli.
28.2 The High Middle Ages
As the western part of the world of Islam was growing politically and militarily weaker because of invasion and conquest, Europe was entering on a period of increasing power and vigor. One expression of that new vigor, the stream of European mathematical creativity that began as a small rivulet 1000 years ago, has been steadily increasing until now; it is an enormous river and shows no sign of subsiding. By the middle of the twelfth century, European civilization had absorbed much of the learning of the Islamic world and was ready to embark on its own explorations. This was the zenith of papal power in Europe, exemplified by the ascendancy of the popes Gregory VII (1073–1085) and Innocent III (1198–1216) over the emperors and kings of the time. The Emperor Frederick I, known as Frederick Barbarossa because of his red beard, ruled the empire from 1152 to 1190 and tried to maintain the principle that his power was not dependent on the Pope, but was ultimately unsuccessful. His grandson Frederick II (1194–1250) was a cultured man who encouraged the arts and sciences. To his court in Sicily4 he invited distinguished scholars of many different religions, and he corresponded with many others. He himself wrote a treatise on the principles of falconry. He was in conflict with the Pope for much of his life and even tried to establish a new religion, based on the premise that “no man should believe aught but what may be proved by the power and reason of nature,” as the papal document excommunicating him stated.
Our list of memorable European mathematicians from the late Medieval period begins in the empire of Frederick II.
28.2.1 Leonardo of Pisa
Leonardo (1170–1250) says in the introduction to his major book, the Liber abaci, that he accompanied his father on an extended commercial mission in Algeria with a group of Pisan merchants. There, he says, his father had him instructed in the Hindu–Arabic numerals and computation, which he enjoyed so much that he continued his studies while on business trips to Egypt, Syria, Greece, Sicily, and Provence. Upon his return to Pisa he wrote a treatise to introduce this new learning to Italy. The treatise, whose author is given as “Leonardus filius Bonaccij Pisani,” that is, “Leonardo, son of Bonaccio of Pisa,” bears the date 1202. In the nineteenth century Leonardo's works were edited by the Italian nobleman Baldassare Boncompagni (1821–1894), who also compiled a catalog of locations of the manuscripts (Boncompagni, 1854). The name Fibonacci by which the author is now known seems to have become generally used only in the nineteenth century. A history of what is known of Leonardo's life and an exposition of his mathematical works has recently appeared (Devlin, 2011).
28.2.2 Jordanus Nemorarius
The works of Archimedes were translated into Latin in the thirteenth century, and his work on the principles of mechanics was extended. One of the authors involved in this work was Jordanus Nemorarius (1225–1260). Little is known about this author except certain books that he wrote on mathematics and statics for which manuscripts still exist dating to the actual time of composition. One of his works, Liber Jordani de Nemore de ratione ponderis [The book of Jordanus Nemorarius on the ratio of weight (Claggett, 1960, pp. 167–229)] contains the first correct statement of the mechanics of an inclined plane. We shall confine our discussion, however, to his algebraic work, in which he discussed various conditions from which the explicit value of a number can be deduced.
28.2.3 Nicole d'Oresme
One of the most distinguished of the medieval philosophers was Nicole d'Oresme (1323–1382), whose clerical career brought him to the office of Bishop of Lisieux in 1377. D'Oresme had a wide-ranging intellect and studied economics, physics, and mathematics as well as theology and philosophy. He considered the motion of physical bodies from various points of view, formulated the Merton rule of uniformly accelerated motion (named for Merton College, Oxford), and for the first time in history explicitly used one line to represent time, a line perpendicular to it to represent velocity, and the area under the graph (as we would call it) to represent distance.
28.2.4 Regiomontanus
The work of translating the Greek and Arabic mathematical works went on for several centuries. One of the last to work on this project was Johann Müller (1436–1476) of Königsberg, better known by his Latin name of Regiomontanus, a translation of Königsberg (King's Mountain). Although he died young, Regiomontanus made valuable contributions to astronomy, mathematics, and the construction of scientific measuring instruments. He studied in Leipzig while a teenager and then spent a decade in Vienna and the decade following in Italy and Hungary. The last five years of his life were spent in Nürnberg. He is said to have died of an epidemic while in Rome as a consultant to the Pope on the reform of the calendar.
Regiomontanus checked the data in copies of Ptolemy's Almagest and made new observations with his own instruments. He laid down a challenge to astronomy, remarking that further improvement in theoretical astronomy, especially the theory of planetary motion, would require more accurate measuring instruments. He established his own printing press in Nürnberg so that he could publish his works. These works included several treatises on pure mathematics. He established trigonometry as an independent branch of mathematics rather than a tool in astronomy. The main results we now know as plane and spherical trigonometry are in his book De triangulis omnimodis, although not exactly in the language we now use.
28.2.5 Nicolas Chuquet
The French Bibliothèque Nationale is in possession of the original manuscript of a mathematical treatise written at Lyons in 1484 by one Nicolas Chuquet (1445–1488). Little is known about the author, except that he describes himself as a Parisian and a man possessing the degree of Bachelor of Medicine. The treatise (see Flegg, 1988) consists of four parts: a treatise on arithmetic and algebra called Triparty en la science des nombres, a book of problems to illustrate and accompany the principles of the Triparty, a book on geometrical measurement, and a book of commercial arithmetic. The last two are applications of the principles in the first book.
28.2.6 Luca Pacioli
Written at almost the same time as Chuquet's Triparty was a work called the Summa de arithmetica, geometrica, proportioni et proportionalita by Luca Pacioli (or Paciuolo, 1445–1517). Since Chuquet's work was not printed until the nineteenth century, Pacioli's work is believed to be the first Western printed work on algebra. In comparison with the Triparty, however, the Summa seems less original. Pacioli has only a few abbreviations, such as co for cosa, meaning thing (the unknown), ce for censo (the square of the unknown), and æ for æquitur (equals). Despite its inferiority to the Triparty where symbolism is concerned, the Summa was much the more influential of the two books, because it was published. It is referred to by the Italian algebraists of the early sixteenth century as a basic source.
28.2.7 Leon Battista Alberti
In art, the fifteenth century was a period of innovation that marked the beginning of the period we call the Renaissance. In an effort to give the illusion of depth in two-dimensional representations, some artists looked at geometry from a new point of view, studying the projection of two-and three-dimensional shapes in two dimensions to see what properties were preserved and how others were changed. A description of such a procedure, based partly on the work of his predecessors, was given by Leon Battista Alberti (1404–1472) in a 1435 Latin treatise entitled De pictura, published posthumously in Italian as Della pittura in 1511.
28.3 The Early Modern Period
Sixteenth-century Italy produced a group of sometimes quarrelsome but always brilliant algebraists, who worked to advance mathematics in order to achieve academic success and for the pleasure of discovery. As happened in Japan a century later, each new advance brought a challenge for further progress.
28.3.1 Scipione del Ferro
A method of solving a particular cubic equation was discovered by a lector (reader, that is, a tutor) at the University of Bologna, Scipione del Ferro (1465–1525), around the year 1500.5 He communicated this discovery to another mathematician, Antonio Maria Fior (dates unknown), who then used the knowledge to win mathematical contests.
28.3.2 Niccolò Tartaglia
Fior met his match in 1535, when he challenged Niccolò Fontana of Brescia, (1500–1557) known as Tartaglia (the Stammerer) because a wound he received as a child when the French overran Brescia in 1512 left him with a speech impediment. Tartaglia had also discovered how to solve certain cubic equations and thus won the contest.
28.3.3 Girolamo Cardano
A brilliant mathematician and gambler, who became rector of the University of Padua at the age of 25, Girolamo Cardano (1501–1576) was writing a book on mathematics in 1535 when he heard of Tartaglia's victory over Fior. He wrote to Tartaglia asking permission to include this technique in his work. Tartaglia at first refused, hoping to work out all the details of all cases of the cubic and write a treatise himself. According to his own account, Tartaglia confided the secret of one kind of cubic to Cardano in 1539, after Cardano swore a solemn oath not to publish it without permission and gave Tartaglia a letter of introduction to the Marchese of Vigevano. Tartaglia revealed a rhyme by which he had memorized the procedure.
Tartaglia did not claim to have given Cardano any proof that his procedure works. It was left to Cardano himself to find the demonstration. Cardano kept his promise not to publish this result until 1545. However, as Tartaglia delayed his own publication, and in the meantime Cardano had discovered the solution of other cases of the cubic himself and had also heard that del Ferro had priority anyway, he published the result in his Ars magna (The Great Art), giving credit to Tartaglia. Tartaglia was furious and started a bitter controversy over Cardano's alleged breach of faith.
28.3.4 Ludovico Ferrari
Cardano's student Ludovico Ferrari (1522–1565) worked with him in the solution of the cubic, and between them they had soon found a way of solving certain quartic equations.
28.3.5 Rafael Bombelli
In addition to the mathematicians proper, we must also mention an engineer in the service of an Italian nobleman. Rafael Bombelli (1526–1572) is the author of a treatise on algebra that appeared in 1572. In the introduction to this treatise we find the first mention of Diophantus in the modern era. Bombelli said that, although all authorities are agreed that the Arabs invented algebra, he, having been shown the work of Diophantus, credits the invention to the latter. In making sense of what his predecessors did, he was one of the first to consider the square root of a negative number and to formulate rules for operating with such numbers. His work in this area will be discussed in more detail in Chapter 41.
28.4 Northern European Advances
The work being done in Italy did not escape the notice of French and British scholars of the time, and important mathematical works were soon being produced in those two countries.
28.4.1 François Viète
A lawyer named François Viète (1540–1603), who worked as tutor in a wealthy family and later became an advisor to Henri de Navarre (who became the first Bourbon king, Henri IV, in 1598), found time to study Diophantus and to introduce his own ideas into algebra. His book Artis analyticae praxis (The Practice of the Analytic Art) contained some of the notational innovations that make modern algebra much less difficult than the algebra of the sixteenth century.
28.4.2 John Napier
In the late sixteenth century the problem of simplifying laborious multiplications, divisions, root extractions, and the like, was attacked by the Scottish laird John Napier, (1550–1617) Baron of Murchiston. His work consisted of two parts, a theoretical part, based on a continuous geometric model, and a computational part, involving a discrete (tabular) approximation of the continuous model. The computational part was published in 1614. However, Napier hesitated to publish his explanation of the theoretical foundation. Only in 1619, two years after his death, did his son publish an English translation of Napier's theoretical work under the title Mirifici logarithmorum canonis descriptio (A Description of the Marvelous Law of Logarithms). This subject, although aimed at a practical end, turned out to have enormous value in theoretical studies as well.
Questions
Historical Questions
28.1 What mathematics was preserved in the Western part of the Roman Empire during the period from 500 to 1000?
28.2 What justifications do the early Medieval writers give for the study of geometry and arithmetic?
28.3 What Arabic and Greek works were brought into Europe in the eleventh and twelfth centuries, and who were the translators responsible for making them available in Latin?
28.4 How did the term sine (Latin sinus) come to have a geometric meaning as one of the trigonometric functions?
Questions for Reflection
28.5 Dante's final stanza (quoted above) uses the problem of squaring the circle to express the sense of an intellect overwhelmed, which was inspired by his vision of heaven. What resolution does he find for the inability of his mind to grasp the vision rationally? Would such an attitude, if widely shared, affect mathematical and scientific activity in a society?
28.6 What is the significance of ruling a board into 27 columns to make an abacus, as Gerbert is said to have done? Does it indicate that there was no symbol for zero?
28.7 One popular belief about Christopher Columbus is that he proved to a doubting public that the earth was spherical. What grounds are there for believing that “the public” doubted this fact? Which people in the Middle Ages would have been likely to believe in a flat earth? Consider also the frequently repeated story that people used to believe the stars were near the earth. Is this view of Medieval scholarship plausible in the light of the Practica geometriae?
28.8 What role can or should or does mathematics play in representational arts such as painting and sculpture? Does the presence of mathematical elements enhance or detract from the emotional content and artistic creativity involved in these arts?
Notes
1. This quotation can be read online at http://pld.chadwyck.com. This passage is from Vol. 63. It can be reached by searching under “geometria” as title.
2. He was not a successful clergyman or pope. He got involved in the politics of his day, offended the Emperor, and was suspended from his duties as Archbishop of Reims by Pope Gregory V in 998. He was installed as pope by the 18-year-old Emperor Otto III, but after only three years both he and Otto were driven from Rome by a rebellion. Otto died trying to reclaim Rome, and Sylvester II died shortly afterward.
3. This quotation can be read online at http://pld.chadwyck.com. This passage is from Vol. 139. It can be reached by searching under “geometria” as title.
4. Sicily was reconquered from the Muslims in the eleventh century by the Normans. Being in contact with all three of the great Mediterranean civilizations of the time, it was the most cosmopolitan center of culture in the world for the next two centuries.
5. Before modern notation was introduced, there was no uniform way of writing a general cubic equation. Since negative numbers were not understood, equations had to be classified according to the terms on each side of the equality. As we saw in the case of Omar Khayyam, this complication results in many different types of cubics, each requiring a special algorithm for its solution.