Teaching of Mathematics in Italy and in Germany in the fifteenth century

Teaching of Mathematics in Italy and in Germany in the fifteenth century

In is book Beyond numeracy, John Allen Paulos tells this story: “A German merchant of the fifteenth century asked an eminent professor where he should send his son for a good business education. The professor responded that German universities would be sufficient to teach the boy addition and subtraction but he would have to go to Italy to learn multiplication and division. Before you smile indulgently, try multiplying or even just adding the Roman numerals CCLXIV, MDCCCIX, DCL, and MLXXXI without first translating them.”

Paulos provides no source for this. Does anyone know whether this is real or not? And, if it is indeed real, who is this “eminent professor”?


This is not a real story but an illustrative description, probably invented in the 1930s.

The first Indo-Arabian numerals came to Europe in the 10th century. They had a hard time at first. In the 13th century the Italian Leonardo Fibonacci published the Liber abaci (1202) which popularised their use further, but mainly in Italy. In 1522 Adam Ries then published Rechenung auff der linihen und federn in Germany - and in German instead of Latin.

That is the pretty well established spread of that numerical writing and calculating system in Europe. That anecdote in question is intended as an illustration for that. It is doubtful that this exact conversation took place or was recorded in that way. After all, why did "the eminent professor" not teach that child himself, if he was familiar with Indo-Arabic numerals and convinced of their advantages?

After all, using Roman numerals is cumbersome compared with Indo-Arabic numerals, for us. But multiplications can be made with an abacus and that instrument is suited for the use of Roman numerals.

Then we have to look at the university system at the time. The artes liberales did include arithmetic and geometry in the Middle Ages. But that was fine even with Roman numerals. And neither Medici, nor Fugger, nor Welser merchants went there for study to learn the business.

They learnt by doing business (or in their own, rare, lay, abbacus schools.) One most prominent example, Jakob Fugger the Rich:

A document from the Austrian state archive has now shown that Jakob Fugger was already representing his family business in Venice in 1473 at the age of 14. Other research showed that Jakob Fugger spent the years between 1473 and 1487 mostly at the Fondaco dei Tedeschi, the house of German merchants in Venice. Venice being one of the most important centers of trade at the time proved to be an ideal environment for Jakob Fugger's education in banking and the metal trade. His long residence in Italy also helped bring the renaissance style to the German region, with his funding the construction of the first buildings of this style that originated in Italy. Legal and architectural structures of Venice also had a significant influence on the funding of the Fuggerei which was similar to the social housing of Venice.

So the term "university" seems to be the give-away. Without it, the anecdote looks plausible. Some merchants did sent their sons to Italy, they sent them there for learning the trade, as the Italian system was way more advanced in the high Middle Ages than anywhere else in Europe (banking). The universities were also old and good in Italy. Just that merchants didn't go there.

And why should they? Mathematics education was quite the step-child in European universities, being very impractical in nature:

We may perhaps wonder why the medieval university, with all its success in the domains of logic and natural philosophy, and in spite of the activity of several noteworthy mathematicians, never brought it far in the domain of mathematics education. [… ] As far as mathematics is concerned, lectures combined with discussion favour the development of metamathematics - that is, also philosophy. But in order to become creative in mathematics itself, and possibly to enjoy it, one has to do mathematics, not only to speak about it. Inside the curriculum of the learned schools and the universities, the areas where one could do mathematics were few. Computus was one such area - but its mathematics did not go beyond simple arithmetical computation. Rithmomachia was another one, and the game indeed remained popular until the sixteenth century. The third was computation with Hindu-Arabic numerals in the use of astronomical tables - perhaps not too inspiring either, but nonetheless a domain that was practised assiduously well into the Renaissance, whether for its own sake or (rather) because it was a sine qua non for simple astrological prediction. [… ]
We know very little about the education of burghers' children after the twelfth-century revival of city life. A few institutions like the Saint Victor school in Paris admitted them, but what they offered seems to have been badly adapted to a future in commercial life (future artisans were in any case taught as apprentices); Pirenne (1929, p. 20) relates that a Flemish merchant's son was put into a monastic school around 1200 in order to learn what was needed in trade - but then became a monk. Some clerks served as house teachers in wealthy families (Pirenne 1929, 21ff), and some probably held private schools.
That Italian merchants had been taught by Latin-writing clerics is illustrated by Boncompagno da Signa's description (1215) of their letters as written in a mixture of corrupt Latin and vernacular.21 Computation was presumably learned on the job, during apprenticeship - but even this is nothing but an educated guess built on what we know from later times.
Jens Høyrup: "Mathematics Education in the European Middle Ages", in: Alexander Karp & Gert Schubring (Eds): "Handbook on the History of Mathematics Education", Springer: New York, Heidelberg, 2014.

The same anecdote is re-told (p14) in Frank J. Swetz & David Eugene Smith: "Capitalism and Arithmetic: The New Math of the 15th Century, Including the Full Text of the Treviso Arithmetic of 1478, Translated by David Eugene Smith", Open Court Publishing, 1987. They weren't able to authenticate the anecdote either.

But the origin is traceable at least to Tobias Dantzig: "Number. The Language of Science", MacMillan, 1930 (archive.org, p27). There we find no source attribution either, but an important qualification:

There is a story of a German merchant of the fifteenth century, which I have not succeeded in authenticating, but it is so characteristic of the situation then existing that I cannot resist the temptation of telling it. It appears that the merchant had a son whom he desired to give an advanced commercial education. He appealed to a prominent professor of a university for advice as to where he should send his son. The reply was that if the mathematical curriculum of the young man was to be confined to adding and subtracting, he perhaps could obtain the instruction in a German university; but the art of multiplying and dividing, he continued, had been greatly developed in Italy, which in his opinion was the only country where such advanced instruction could be obtained.
As a matter of fact, multiplication and division as practiced in those days had little in common with the modern operations bearing the same names. Multiplication, for instance, was a suc- cession of duplations, which was the name given to the doubling of a number. In the same way division was reduced to mediation, i.e., “halving” a number. A clearer insight into the status of reckoning in the Middle Ages can be obtained from an example. Using modern notations:
We begin to understand why humanity so obstinately clung to such devices as the abacus or even the tally. Computations which a child can now perform required then the services of a specialist, and what is now only a matter of a few minutes meant in the twelfth century days of elaborate work.
The greatly increased facility with which the average man today manipulates number has been often taken as proof of the growth of the human intellect. The truth of the matter is that the difficulties then experienced were inherent in the numeration in use, a numeration not susceptible to simple, clearcut rules. The discovery of the modern positional numeration did away with these obstacles and made arithmetic accessible even to the dullest mind.

That would make the anecdote a tale of morality, with bits and pieces found in history cobbled together to form an instructive story of slow but triumphant progressivism,
which is unfounded in actual history, as evidenced by too many wrong details in the story (and all its unsourced variations (example, set even earlier, but getting it very wrong in the process).

Like another instructor concluded:

The state of science in medieval Europe can be characterized through an anecdote reported in Ifrah (2000):

A German merchant wanted to give his son the best possible education. He called for a respected professor and asked him to which university he should send him. The professor's advice was: "A German university will do if he only wants to learn addition and subtraction. If he wants to learn multiplication and division as well he should go to an Italian university."

Anecdotes are like caricatures; they exaggerate typical features, but they have a true core. The story of the medieval merchant demonstrates that spending an entire lecture on medieval science in Europe is an undeniable act of cultural bias. From the point of view of global history it cannot be justified. The only excuse I can offer is that I was born into the European civilization and therefore have an interest in even the darkest times of European history.

Matthias Tomczak: "The state of science in medieval Europe.", Science, Civilization and Society, CPES 2220: A course of 35 lectures, first given at the Flinders University of South Australia during the second half of 2004.

Curiously, this anecdote speaks of a German merchant. The English speaking web and books from the 1930s onward recount this tale numerous times, mostly with just miute variations.
Yet German books seem to copycat this story only in very recent years. Not that it would count for anything, but the earliest record for this in German language publications seems to be from 1999 (and that one even being an American originally)?

A German merchant of the fifteenth century asked an eminent professor where he should send his son for a good business education.

It is clear that this story is made up by the author for illustration purposes. In 15 century universities had nothing to do with business education. See, for example "Trivium" and "Quadrivium" on Wikipedia. Business mathematics (like "double entry book-keeping", and the use of abacus, for example) was taught privately, and Italy was indeed the place for study of business mathematics. Why Italy? Probably because of its closer connection to the Middle Eastern trade.

Liber Abacus which introduced decimal arithmetic in Europe was published by the Italian merchant Fibonacci in 13 century, so I suppose that decimal system was already in common use among the merchants two hundred years later. Fibonacci had no affiliation with any university. He learned business mathematics while traveling to Algeria.

For multiplying and dividing the numbers, and other long calculations, a simple computational aid (abacus) was used.

Education in Italy

Education in Italy is compulsory from 6 to 16 years of age, [2] and is divided into five stages: kindergarten (scuola dell'infanzia), primary school (scuola primaria or scuola elementare), lower secondary school (scuola secondaria di primo grado or scuola media inferiore), upper secondary school (scuola secondaria di secondo grado or scuola media superiore) and university (università). [3] Education is free in Italy and free education is available to children of all nationalities who are residents in Italy. Italy has both a private and public education system. [4]

Education in Italy
Ministero dell'Istruzione, dell'Università e della Ricerca
Minister of EducationPatrizio Bianchi
National education budget (2016)
Budget€65 billion
General details
Primary languagesItalian
System typepublic
Compulsory primary education1859
Literacy (2015)
Total99.2% [1]
Post secondary386,000

The Programme for International Student Assessment coordinated by the OECD currently ranks the overall knowledge and skills of Italian 15-year-olds as 34th in the world in reading, literacy, and mathematics, significantly below the OECD average of 493. [5]


During the centuries in which the Chinese, Indian and Islamic mathematicians had been in the ascendancy, Europe had fallen into the Dark Ages, in which science, mathematics and almost all intellectual endeavour stagnated.

Scholastic scholars only valued studies in the humanities, such as philosophy and literature, and spent much of their energies quarrelling over subtle subjects in metaphysics and theology, such as “How many angels can stand on the point of a needle?

From the 4th to 12th Centuries, European knowledge and study of arithmetic, geometry, astronomy and music was limited mainly to Boethius’ translations of some of the works of ancient Greek masters such as Nicomachus and Euclid. All trade and calculation was made using the clumsy and inefficient Roman numeral system, and with an abacus based on Greek and Roman models.

By the 12th Century, though, Europe, and particularly Italy, was beginning to trade with the East, and Eastern knowledge gradually began to spread to the West. Robert of Chester translated Al-Khwarizmi‘s important book on algebra into Latin in the 12th Century, and the complete text of Euclid‘s “Elements” was translated in various versions by Adelard of Bath, Herman of Carinthia and Gerard of Cremona. The great expansion of trade and commerce in general created a growing practical need for mathematics, and arithmetic entered much more into the lives of common people and was no longer limited to the academic realm.

The advent of the printing press in the mid-15th Century also had a huge impact. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their commercial needs and mathematics gradually began to acquire a more important position in education.

Europe’s first great medieval mathematician was the Italian Leonardo of Pisa, better known by his nickname Fibonacci. Although best known for the so-called Fibonacci Sequence of numbers, perhaps his most important contribution to European mathematics was his role in spreading the use of the Hindu-Arabic numeral system throughout Europe early in the 13th Century, which soon made the Roman numeral system obsolete, and opened the way for great advances in European mathematics.

Oresme was one of the first to use graphical analysis

An important (but largely unknown and underrated) mathematician and scholar of the 14th Century was the Frenchman Nicole Oresme. He used a system of rectangular coordinates centuries before his countryman René Descartes popularized the idea, as well as perhaps the first time-speed-distance graph. Also, leading from his research into musicology, he was the first to use fractional exponents, and also worked on infinite series, being the first to prove that the harmonic series 1 ⁄1 + 1 ⁄2 + 1 ⁄3 + 1 ⁄4 + 1 ⁄5… is a divergent infinite series (i.e. not tending to a limit, other than infinity).

The German scholar Regiomontatus was perhaps the most capable mathematician of the 15th Century, his main contribution to mathematics being in the area of trigonometry. He helped separate trigonometry from astronomy, and it was largely through his efforts that trigonometry came to be considered an independent branch of mathematics. His book “De Triangulis“, in which he described much of the basic trigonometric knowledge which is now taught in high school and college, was the first great book on trigonometry to appear in print.

Mention should also be made of Nicholas of Cusa (or Nicolaus Cusanus), a 15th Century German philosopher, mathematician and astronomer, whose prescient ideas on the infinite and the infinitesimal directly influenced later mathematicians like Gottfried Leibniz and Georg Cantor. He also held some distinctly non-standard intuitive ideas about the universe and the Earth’s position in it, and about the elliptical orbits of the planets and relative motion, which foreshadowed the later discoveries of Copernicus and Kepler.

Science and Philosophy:

The Arabs’ Contribution

Through the ‘Middle Ages’ the monks and clergymen had been familiar with much of the writings of the Greeks and Romans. But they had not made these widely known. In the fourteenth century, many scholars began to read translated works of Greek writers like Plato and Aristotle. In fact, the Arab translators had carefully preserved and translated ancient manuscripts in Arabic. In addition to that, some European scholars translated works of Arabic and Persian scholars for further transmission to Europe. These were works on natural science, mathematics, astronomy, medicine and chemistry. The curricula in universities continued to be dominated by law, medicine and theology. But humanist subjects slowly began to be introduced in schools.

Artists and Realism

Artists were inspired by the figures of ‘perfectly’ proportioned men and women sculpted many centuries ago during the Roman Empire. Italian sculptors further worked on that tradition to produce lifelike statues. Artists’ endeavor to be accurate was helped by the work of scientists. Artists went to the laboratories of medical schools so that they could study anatomy. Painters utilized the knowledge of geometry to understand perspective. They used proper combination of light shadow to create three-dimensional quality in paintings. The oil paint gave a greater richness of color to paintings than before. Influence of Chinese and Persian art can be seen in their depiction of costumes in many paintings. Thus, anatomy, geometry, physics, and a strong sense of what was beautiful have a new quality to Italian art. This art was later called ‘realism’ and the movement continued till the nineteenth century.


The city of Rome revived in a spectacular way in the fifteenth century. From 1417, the popes became politically stronger. They actively encouraged the study of Rome’s history. The ruins in Rome were carefully excavated by archaeologists. This inspired a revival of the imperial Roman style of architecture. It was now called ‘classical’. Popes, wealthy merchants and aristocrats employed architects who were familiar with classical architecture. Artists and sculptors were also employed to decorate buildings with paintings, sculptures and reliefs. Some artists were skilled equally as painters, sculptors and architects, e.g. Michelangelo Buonarroti (1475-1564), Filippo Brunelleschi (1337-1446). Another remarkable change was that from this time, artists were known individually, i.e. by name, not as members of a group or a guild.

The First Printed Books

Development of print technology was the greatest revolution of the sixteenth century. Print technology came from China. Johannes Gutenberg (1400-1458), a German, made the first printing press. By 1500, many classical texts, nearly all in Latin, had been printed in Italy. Print technology ensured that knowledge, idea, opinions and information moved rapidly and widely than ever before. Now, individuals could also read books. The humanist culture of Italy spread more rapidly from the end of the fifteenth century because of growing popularity of printed books.

Majority of mathematicians hail from just 24 scientific ‘families’

Evolution of mathematics traced using unusually comprehensive genealogy database.

Most of the world’s mathematicians fall into just 24 scientific 'families', one of which dates back to the fifteenth century. The insight comes from an analysis of the Mathematics Genealogy Project (MGP), which aims to connect all mathematicians, living and dead, into family trees on the basis of teacher–pupil lineages, in particular who an individual's doctoral adviser was.

The analysis also uses the MGP — the most complete such project — to trace trends in the history of science, including the emergence of the United States as a scientific power in the 1920s and when different mathematical subfields rose to dominance 1 .

“You can see how mathematics has evolved in time,” says Floriana Gargiulo, who studies networks dynamics at the University of Namur, Belgium and who led the analysis.

The MGP is hosted by North Dakota State University in Fargo and co-sponsored by the American Mathematical Society. Since the early 1990s, its organizers have mined information from university departments and from individuals who make submissions regarding themselves or people they know about. As of 25 August, the MGP contained 201,618 entries. As well as doctoral advisers (PhD advisers in recent times) and pupils of academic mathematicians, the organizers record details such as the university that awarded the doctorate.

Previously, researchers had used the MGP to reconstruct their own PhD-family trees, or to see how many ‘descendants’ a researcher has (readers can do their own search here). Gargiulo's team wanted to make a comprehensive analysis of the entire database and divide it into distinct families, rather than just looking at how many descendants any one person has.

After downloading the database, Gargiulo and her colleagues wrote machine-learning algorithms that cross-checked and complemented the MGP data with information from Wikipedia and from scientists' profiles in the Scopus bibliographic database.

This revealed 84 distinct family trees with two-thirds of the world’s mathematicians concentrated in just 24 of them. The high degree of clustering arises in part because the algorithms assigned each mathematician just one academic parent: when an individual had more than one adviser, they were assigned the one with the bigger network. But the phenomenon chimes with anecdotal reports from those who research their own mathematical ancestry, says MGP director Mitchel Keller, a mathematician at Washington and Lee University in Lexington, Virginia. “Most of them run into Euler, or Gauss or some other big name,” he says.

Although the MGP is still somewhat US centric, the goal is for it to become as international as possible, Keller says.

Peculiarly, the progenitor of the largest family tree is not a mathematician but a physician: Sigismondo Polcastro, who taught medicine at the University of Padua in Italy in the early fifteenth century. He has 56,387 descendants according to the analysis. The second-largest tree is one started by a Russian called Ivan Dolbnya in the late nineteenth century.

The authors also tracked mathematical activity by country, which seemed to pinpoint major historical events. Around the time of the dissolution of the Austro-Hungarian Empire in the First World War, there is a decline in mathematics PhDs awarded in the region, notes Gargiulo. Between 1920 and 1940, the United States took over from Germany as the country producing the largest number of mathematics PhDs each year. And the ascendancy of the Soviet Union is marked by a peak of PhDs in the 1960s, followed by a relative fall after the break-up of the union in 1991.

Gargiulo’s team also looked at the dominance of mathematical subfields relative to each other. The researchers found that dominance shifted from mathematical physics to pure maths during the first half of the twentieth century, and later to statistics and other applied disciplines, such as computer science.

Idiosyncrasies in the field of mathematics could explain why it has the most comprehensive genealogy database of any discipline. “Mathematicians are a bit of a world apart,” says Roberta Sinatra, a network and data scientist at Central European University in Budapest who led a 2015 study that mapped the evolution of the subdisciplines of physics by mining data from papers on the Web of Science 2 .

Mathematicians tend to publish less than other researchers, and they establish their academic reputation not so much on how much they publish or on their number of citations, but on who they have collaborated with, including their mentors, she says. “I think it’s not a coincidence that they have this genealogy project."

At least one discipline is trying to catch up. Historian of astronomy Joseph Tenn of Sonoma State University in California plans by 2017 to launch the AstroGen project to record the PhD advisers and students of astronomers. “I started it," he says, "because so many of my colleagues in astronomy admired and enjoyed perusing the Mathematics Genealogy Project."

Women English Renaissance

The Renaissance was a period from 14 th to the 17 th century in Europe which is defined to be the time of the revival of arts. The renaissance started in Italy which was the hub of this revolution between fourteenth and sixteenth century, between Europe and Eurasia. In this period different art forms, sculptors, paintings, and architecture took a new turn and defined new concepts in the field of art. The period begins in the fourteenth century from the hub of revolution, Italy, and slowly progressed to all parts of Europe till the fifteenth century. The aim of this revolution is to follow the culture that was part of ancient Greek and Roman History. These new concepts of wisdom and art were initially directed towards men and women were excluded from equal involvement in the revolution. It was the time when women were distributed in upper and lower classes of which the upper class was able to take part in the activities but the lower class was extremely suppressed and was meant to giving birth to the children and serve the men as servants. This revolution resulted in the empowerment of women who were suppressed in every field of life until then. This paper is focused on the role of women in the renaissance period and how they handled their families, jobs and daily life during this time, it will also compare the renaissance women with the women of Middle age. The role of women in the renaissance period and their service in the society became the reason for women empowerment which was not possible in middle ages.


Women were initially not an active part of the revolution and their social and economic status became a hindrance to their involvement. Until sixteenth-century women were not an active part of the revolution and their growth in new forms of art was suppressed by the strong power of male dominant society. We will further describe their roles in the period as mothers, working women and as an active part of the society.

Women in Renaissance held a high virtue about their family and their obligations. Women in renaissance were forced to look for the children and household and were suppressed by the males (Herlihy, David, 1995). They still managed to improve their way of life by presenting their daily obligations as part of their obligations. The disease outbreak of the 15 th century killed many of the people of the region and there was a need for someone to take the job roles that were necessary at that time, women started participating by performing in these jobs but they were suppressed by men. (Mitchell, Linda, 2012). In order to support families, many women took the jobs as nurses and in Florentine shops.

The noble and lower class women provided their services by taking jobs as wet nurses and in Florentine shops. Although they had their own families and own children, their service in jobs never changed their preference and they actively participated in their household works. Their high virtue in participating society motivated them to work as silk spinners, housemaids and in bakery shops when they knew these jobs should be done by men, but they had to empower themselves and their family so they resisted every movement made against them. (Ward, Jennifer, 2016).

The noble women expected their political rights from the government (Tomas, Natalie, 2017). They demanded their rights as respectable members of the society and asked for the opportunities to be provided in jobs (Chadwick, 1990).They demanded their rights in selecting their life partner which was a dilemma in the renaissance period and women were sold for dowry and to settle personal conflicts (Kirshner, Julius, 2015).

In the Renaissance period, women understood the need for education to keep up with the progressing world and achieve their status in their society. (Wyles, R., & Hall, 2016) the learning methods started developing in the 14 th century and were at their height by the end of 15 th century. (Charlton, 2013) The education that women were interested in was mainly Greek and Latin literature. By the initiatives taken by humanists, women started learning the art, architecture, and languages. the education opportunities had a difference in class. Most of the noble women received home education from the experts in the field and learned the major subjects popular in English education at that time. (Hexter, Jack H, 1950) The poor class and widowed women, however, are seen struggling for their basic educational rights and their progress was slow because they were not supported by the politics and government of the time (Wainwright, Anna, 2018)

The women in renaissance had a great interest in music and arts. the songs composed by women in the English renaissance described the hardships and struggle that they went through. the initial composition of songs was done in the in St. Clare, Florence by the members of churches. the songs were mostly in the form of chapels and sung by nuns. (Tomas, Natalie, 2017) The women artists of renaissance period developed amazing new techniques in their painting. Their paintings too described the suppression made by men powered community on them. their paintings highlighted the major problems women are facing and their struggle. very few female artists received recognition and appreciation. Sofonisba Anguissola is one of these few artists who attacked the women suppression by the power of her paintbrush. she described the major problems of that time by focusing on “women” as their main subject. (Chin, Lily, 2018)

Women role in renaissance became the reason of women empowerment. In Middle Ages women had no rights and their roles were limited to house premises to their husbands and families. There were concepts from Bible that women a reason to human mistakes and they have no rights to participate in society or the social politics. After the renaissance women stood up against male superiority in the society. They started participating in society jobs, politics and education (Tomas, Natalie, 2017). They asked for respectable positions in society and their efforts were fruitful when the government started giving them job opportunities and places in politics.


The women in renaissance made huge efforts for their equal rights. They were suppressed in all job role of the society and were kept ignorant on purpose. After the renaissance women started to understand their place in the society and started fighting for it. They started taking part in various jobs and learned the value of education. While they fought for the equal rights as men they never forgot their obligations and need in their family, they continued to serve their families and people depending on them and side by side worked in various jobs as servants, nurses and as silk workers.

Works Cited

Beilin, Elaine V. Redeeming Eve: women writers of the English Renaissance. Princeton University Press, 2014.

Chadwick, Whitney, and Whitney Chadwick. Women, art, and society. London: Thames and Hudson, 1990.

Charlton, Kenneth. Education in Renaissance England. Vol. 1. Routledge, 2013.


Herlihy, David. Women, family, and society in medieval Europe: historical essays, 1978-1991. Berghahn Books, 1995.

Hexter, Jack H. “The Education of the Aristocracy in the Renaissance.” The Journal of Modern History 22.1 (1950): 1-20.

Kirshner, Julius. Marriage, dowry, and citizenship in late medieval and Renaissance Italy. Vol. 2. University of Toronto Press, 2015.

Klapisch-Zuber, Christiane. Women, family, and Ritual in Renaissance Italy. University of Chicago Press, 1987.

Mitchell, Linda E., ed. Women in Medieval Western European Culture. Routledge, 2012.

Shuger, Debora K. Sacred rhetoric: The Christian grand style in the English Renaissance. Princeton University Press, 2014.

Tomas, Natalie R. The Medici women: gender and power in Renaissance Florence. Taylor & Francis, 2017.

Tomas, Natalie. “The grand ducal Medici and their archive (1537-1543) women artists in early modern Italy: Careers, fame, and collectors [Book Review].” Parergon 34.2 (2017): 179.

Wainwright, Anna. “Teaching Widowed Women, Community, and Devotion in Quattrocento Florence with Lucrezia Tornabuoni and Antonia Tanini Pulci.” Religions 9.3 (2018): 76.

Ward, Jennifer. Women in medieval Europe: 1200-1500. Routledge, 2016.

Wyles, R., & Hall, E. (Eds.). (2016). Women Classical Scholars: Unsealing the Fountain from the Renaissance to Jacqueline de Romilly. Oxford University Press.

The Mathematical Cultures of Medieval Europe - Introduction

Mathematics in medieval Europe was not just the purview of scholars who wrote in Latin, although certainly the most familiar of the mathematicians of that period did write in that language, including Leonardo of Pisa, Thomas Bradwardine, and Nicole Oresme. These authors &ndash and many others &ndash were part of the Latin Catholic culture that was dominant in Western Europe during the Middle Ages. Yet there were two other European cultures that produced mathematics in that time period, the Hebrew culture found mostly in Spain, southern France, and parts of Italy, and the Islamic culture that predominated in Spain through the thirteenth century and, in a smaller geographic area, until its ultimate demise at the end of the fifteenth century. These two cultures had many relationships with the dominant Latin Catholic culture, but also had numerous distinct features. In fact, in many areas of mathematics, Hebrew and Arabic speaking mathematicians outshone their Latin counterparts. In what follows, we will consider several mathematicians from each of these three mathematical cultures and consider how the culture in which each lived influenced the mathematics they studied.

We begin by clarifying the words &ldquomedieval Europe&rdquo, because the dates for the activities of these three cultures vary considerably. Catholic Europe, from the fall of the Western Roman Empire up until the mid-twelfth century, had very little mathematical activity, in large measure because most of the heritage of ancient Greece had been lost. True, there was some education in mathematics in the monasteries and associated schools &ndash as Charlemagne, first Holy Roman Emperor, had insisted &ndash but the mathematical level was very low, consisting mainly of arithmetic and very elementary geometry. Even Euclid&rsquos Elements were essentially unknown. About the only mathematics that was carried out was that necessary for the computation of the date of Easter.

Recall that Spain had been conquered by Islamic forces starting in 711, with their northward push being halted in southern France in 732. Beginning in 750, Spain (or al-Andalus) was ruled by an offshoot of the Umayyad Dynasty from Damascus. The most famous ruler of this transplanted Umayyad Dynasty, with its capital in Cordova, was &lsquoAbd al-Raḥmān III, who proclaimed himself Caliph early in the tenth century, cutting off all governmental ties with Islamic governments in North Africa. He ruled for a half century, from 912 to 961, and his reign was known as &ldquothe golden age&rdquo of al-Andalus. His son, and successor, al-Ḥakam II, who reigned from 961 to 977, was, like his father, a firm supporter of the sciences who brought to Spain the best scientific works from Baghdad, Egypt, and other eastern countries. And it is from this time that we first have mathematical works written in Spain that are still extant.

Al-Ḥakam&rsquos son, Hishām, was very young when he inherited the throne on the death of his father. He was effectively deposed by a coup led by his chamberlain, who soon instituted a reign of intellectual terror that lasted until the end of the Umayyad Caliphate in 1031. At that point, al-Andalus broke up into many small Islamic kingdoms, several of which actively encouraged the study of sciences. In fact, Sā&lsquoid al-Andalusī, writing in 1068, noted that &ldquoThe present state, thanks to Allah, the Highest, is better than what al-Andalus has experienced in the past there is freedom for acquiring and cultivating the ancient sciences and all past restrictions have been removed&rdquo [Sā&lsquoid, 1991, p. 62].

Figure 1. Maps of Spain in 910 (upper left), 1037 (upper right), 1150 (lower left), and 1212-1492 (lower right)

Meanwhile, of course, the Catholic &ldquoReconquista&rdquo was well underway, with a critical date being the reconquest of Toledo in 1085. Toledo had been one of the richest of the Islamic kingdoms, but was conquered in that year by Alfonso VI of Castile. Fortunately, Alfonso was happy to leave intact the intellectual riches that had accumulated in the city, and so in the following century, Toledo became the center of the massive transfer of intellectual property undertaken by the translators of Arabic material, including previously translated Greek material, into Latin. In fact, Archbishop Raymond of Toledo strongly encouraged this effort. It was only after this translation activity took place, that Latin Christendom began to develop its own scientific and mathematical capabilities.

But what of the Jews? There was a Jewish presence in Spain from antiquity, and certainly during the time of the Umayyad Caliphate, there was a strong Jewish community living in al-Andalus. During the eleventh century, however, with the breakup of al-Andalus and the return of Catholic rule in parts of the peninsula, Jews were often forced to make choices of where to live. Some of the small Islamic kingdoms welcomed Jews, while others were not so friendly. And once the Berber dynasties of the Almoravids (1086-1145) and the Almohads (1147-1238) from North Africa took over al-Andalus, Jews were frequently forced to leave parts of Muslim Spain. On the other hand, the Catholic monarchs at the time often welcomed them, because they provided a literate and numerate class &ndash fluent in Arabic &ndash who could help the emerging Spanish kingdoms prosper. By the middle of the twelfth century, most Jews in Spain lived under Catholic rule. However, once the Catholic kingdoms were well-established, the Jews were often persecuted, so that in the thirteenth century, Jews started to leave Spain, often moving to Provence. There, the Popes, in residence at Avignon, protected them. By the end of the fifteenth century, the Spanish Inquisition had forced all Jews to convert or leave Spain.

Figure 2. Papal territories in Provence

It was in Provence, and later in Italy, that Jews began to fully develop their interest in science and mathematics. They also began to write in Hebrew rather than in Arabic, their intellectual language back in Muslim Spain.

Victor J. Katz (University of the District of Columbia), "The Mathematical Cultures of Medieval Europe - Introduction," Convergence (December 2017)

Leonardo Pisano Fibonacci

Fibonacci ended his travels around the year 1200 and at that time he returned to Pisa. There he wrote a number of important texts which played an important role in reviving ancient mathematical skills and he made significant contributions of his own. Fibonacci lived in the days before printing, so his books were hand written and the only way to have a copy of one of his books was to have another hand-written copy made. Of his books we still have copies of Liber abaci Ⓣ (1202) , Practica geometriae Ⓣ (1220) , Flos Ⓣ (1225) , and Liber quadratorum Ⓣ . Given that relatively few hand-made copies would ever have been produced, we are fortunate to have access to his writing in these works. However, we know that he wrote some other texts which, unfortunately, are lost. His book on commercial arithmetic Di minor guisa Ⓣ is lost as is his commentary on Book X of Euclid's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view.

One might have thought that at a time when Europe was little interested in scholarship, Fibonacci would have been largely ignored. This, however, is not so and widespread interest in his work undoubtedly contributed strongly to his importance. Fibonacci was a contemporary of Jordanus but he was a far more sophisticated mathematician and his achievements were clearly recognised, although it was the practical applications rather than the abstract theorems that made him famous to his contemporaries.

The Holy Roman emperor was Frederick II. He had been crowned king of Germany in 1212 and then crowned Holy Roman emperor by the Pope in St Peter's Church in Rome in November 1220 . Frederick II supported Pisa in its conflicts with Genoa at sea and with Lucca and Florence on land, and he spent the years up to 1227 consolidating his power in Italy. State control was introduced on trade and manufacture, and civil servants to oversee this monopoly were trained at the University of Naples which Frederick founded for this purpose in 1224 .

Frederick became aware of Fibonacci's work through the scholars at his court who had corresponded with Fibonacci since his return to Pisa around 1200 . These scholars included Michael Scotus who was the court astrologer, Theodorus Physicus the court philosopher and Dominicus Hispanus who suggested to Frederick that he meet Fibonacci when Frederick's court met in Pisa around 1225 .

Johannes of Palermo, another member of Frederick II's court, presented a number of problems as challenges to the great mathematician Fibonacci. Three of these problems were solved by Fibonacci and he gives solutions in Flos Ⓣ which he sent to Frederick II. We give some details of one of these problems below.

After 1228 there is only one known document which refers to Fibonacci. This is a decree made by the Republic of Pisa in 1240 in which a salary is awarded to:-

This salary was given to Fibonacci in recognition for the services that he had given to the city, advising on matters of accounting and teaching the citizens.

Liber abaci Ⓣ , published in 1202 after Fibonacci's return to Italy, was dedicated to Scotus. The book was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe. Indeed, although mainly a book about the use of Arab numerals, which became known as algorism, simultaneous linear equations are also studied in this work. Certainly many of the problems that Fibonacci considers in Liber abaci Ⓣ were similar to those appearing in Arab sources.

The second section of Liber abaci Ⓣ contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China.

A problem in the third section of Liber abaci Ⓣ led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today:-

The resulting sequence is 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , . ( Fibonacci omitted the first term in Liber abaci Ⓣ ) . This sequence, in which each number is the sum of the two preceding numbers, has proved extremely fruitful and appears in many different areas of mathematics and science. The Fibonacci Quarterly is a modern journal devoted to studying mathematics related to this sequence.

Many other problems are given in this third section, including these types, and many many more:

There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series.

Fibonacci treats numbers such as √ 10 in the fourth section, both with rational approximations and with geometric constructions.

A second edition of Liber abaci Ⓣ was produced by Fibonacci in 1228 with a preface, typical of so many second editions of books, stating that:-

Liber quadratorum, written in 1225 , is Fibonacci's most impressive piece of work, although not the work for which he is most famous. The book's name means the book of squares and it is a number theory book which, among other things, examines methods to find Pythogorean triples. Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n 2 + ( 2 n + 1 ) = ( n + 1 ) 2 n^ <2>+ (2n+1) = (n+1)^ <2>n 2 + ( 2 n + 1 ) = ( n + 1 ) 2 . Fibonacci writes:-

Fibonacci's work in number theory was almost wholly ignored and virtually unknown during the Middle ages. Three hundred years later we find the same results appearing in the work of Maurolico.

The portrait above is from a modern engraving and is believed to not be based on authentic sources.

Hebrew Scholasticism in the Fifteenth Century

In their pursuit of a renewal of Jewish philosophy, a number of scholars active in Spain and Italy in the second half of the fifteenth century (Abraham Bibago, Baruch Ibn Ya‘ish, Abraham Shalom, Eli Habillo, Judah Messer Leon) turned to the doctrines and methods of contemporary Latin Scholasticism. These philosophers, who read Latin very well, were impressed by the theories formulated by their Latin colleagues (Albert the Great, Thomas Aquinas, William of Ockham, John Duns Scotus and their followers). They composed original works in Hebrew (mainly commentaries and questions on Aristotle), in which they faithfully reproduced the techniques and terminology of late Scholasticism, and explicitly quoted and discussed Scholastic texts and doctrines about logic, physics, metaphysics and ethics.

Thus, in fifteenth century Italy and Spain there came into being what we may call a "Hebrew Scholasticism": Jewish authors composed philosophical treatises in which they discussed the same questions and used the same methods as contemporary Christian Schoolmen. These thinkers were not simply influenced by Scholasticism: they were real Schoolmen who tried to participate (in a different language) in the philosophical debate of contemporary Europe.

A history of "Hebrew Scholasticism" in the fifteenth century is yet to be written. Most of the sources themselves remain unpublished, and their contents and relationship to Latin sources have not yet been studied in detail. What is needed is to present, edit, translate and comment on some of the most significant texts of "Hebrew Scholasticism", so that scholars can attain a more precise idea of its extent and character.

This book aims to respond to this need. After a historical introduction, where a "state of the art" about research on the relationship between Jewish philosophy and science and Latin Scholasticism in the thirtheenth-fifteenth centuries is given, the book consists of four chapters. Each of them offers a general bio-bibliographical survey of one or two key-authors of fifteenth-century "Hebrew Scholasticism", followed by English translations of some of their most significant "Scholastic" works or of some parts of them: Abraham Bibago’s "Treatise on the Plurality of Forms", Baruch Ibn Ya’ish’s commentaries on Aristotle’s "Nicomachean Ethics" and "De anima", Eli Habillo’s introduction to Antonius Andreas’s commentary on the "Metaphysics", Judah Messer Leon’s commentary on Aristotle’s "Physics" and questions on Porphyry’s "Isagoge". The Hebrew section includes critical editions of some of the translated texts, and a Latin-Hebrew glossary of technical terms of Scholasticism.

The Posttraditional Society

After World War II, the Cold War infiltrated European classrooms. In France and Italy the communists were supported by more than a fifth of the population moreover, regions of Eastern Europe from L󼯬k to Trieste had been transformed into Communist states which promoted a utilitarian, politically dogmatic educational pedagogy. Although the United States wanted to establish comprehensive education in its German occupation zone, West German politicians wanted to return to the pre-Nazi tripartite system. Spain and Portugal, however, remained as they were before the war�scist dictatorships where no reforms were expected.

As industrial production became more technological, demand grew for white-collar workers to supplement the traditional blue-collar labor force. In the 1970s, conventional wisdom referred to the service society in the 1980s, economists described the information society and in the 1990s, experts coined the term the knowledge society. These developments had a great impact on education. Furthermore, new scientific discoveries entered the classrooms, which necessitated new forms of teaching. For example, knowledge of computers and the Internet had to be integrated in all subjects.

In a rapidly changing society, it is not sufficient to maintain one's competence rather, it is necessary to engage in lifelong education. Given the extent of GLOBALIZATION it is not possible for nation-states to maintain their own individual standards. For example, international organizations such as the United Nations Educational, Scientific and Cultural Organization (UNESCO) have created channels to further global communication in the educational field. British sociologist Anthony Giddens described what he called the post-traditional period. He suggested that tradition should no longer be the guideline for education and for life in the modern world, risks dominate and individuals must continually assess the pros and cons of their decisions. In such a complex world, education must also be more complex, and the solutions to teaching problems could be to create new subjects or to combine existing subjects in new ways. Thus, interdisciplinary work has become common in all types of secondary schools and the universities.

There are at least two paths to choose when planning an educational approach. One is the Anglo-Saxon curriculum, popular in Great Britain and the Scandinavian comprehensive schools. All pupils follow the same core curriculum and progressively they are given more choices in order to follow their individual talents. The comprehensive system responds to the challenge of globalization by teaching a variety of school subjects. Each student's proficiency is tested periodically to ensure that the teaching objectives are being satisfied. Another approach is the German or continental didactical method. Instead of choosing elective courses, students decide to attend one of three types of secondary schools: Hauptschule (26 percent), Realschule (27 percent) or Gymnasium (32 percent). Only a few students choose to go to private schools the remaining 9 percent attend a comprehensive school. The pupils do not have a free choice between different institutions, however their teachers at the lower level decides for them. The pupils in the Hauptschule can continue their studies at the vocational training schools, those who attend the Realschule can go to technical schools, and the pupils in the Gymnasium can go to the sixth form and continue their studies at the university and academy. In fact, although there are relatively few choices between subjects in the German system, it ensures coherence and progression. Moreover, the teachers are free to develop a personal didactic approach to teaching, often with student participation, in order to prepare their pupils for the final state-controlled examinations.

In the 1990s, to prepare their citizens to contribute to the knowledge society, several European countries formulated an education plan. This approach expected 95 percent of young people to graduate from secondary school, with 50 percent of those students going on to university. In order to fulfill this plan, it was appropriate to stress the learning rather than teaching educators discussed terms such as the Process for Enhancing Effective Learning (PEEL, a method developed in Australia) in order to focus on the responsibility of the pupils. Because the individualization of education made it difficult to know whether all students had reached an acceptable proficiency level, it was therefore necessary to evaluate the educational process and its results. Swiss psychologist JEAN PIAGET's theory of children's maturation influenced these educators. They also incorporated the ideas of German philosopher Wolfgang Klafki, who promoted categorical learning as a synthesis of material and formal education.

The development of globalization presented a challenge to the European nation-state one of the responses has been the development of the European Union (EU), a trading bloc with a common currency. Another was the collaboration between the industrialized countries of the world in the Organisation for Economic Cooperation and Development (OECD). This organization developed a program called PISA (Programme for International Student Assessment) which in 1998 published a review called Knowledge and Skillsfor Life. This comprehensive account showed Ȯvidence on the performance in reading, mathematical and scientific literacy of students, schools and countries, provides insight into the factors that influence the development of these skills at home and at school, and examines how these factors interact and what implications are for the policy development." More than a quarter of a million students, representing almost seventeen million fifteen-year-olds enrolled in the schools of the thirty-two participating countries, were assessed in 2000. The literacy level among students in the European countries differed very much from one nation to the next. Finland was at the top, followed by Ireland, the United Kingdom, Sweden, Belgium, Austria, Iceland, Norway, France, Denmark, Switzerland, Spain, the Czech Republic, Italy, Germany, Poland, Hungary, Greece, Portugal, and Luxembourg. All sorts of explanations for the differences can be brought forward, and there probably is no single underlying factor. Economic variation is likely to be a contributing factor, but it is not sufficient. The report concludes that the socioeconomic background of the students, although important, does not solely determine performance. Religious affiliations are no longer a decisive factor, but combined with the fact that countries like Germany and Luxembourg have a comparatively large number of immigrants with a different cultural background, religion may have had some influence on reading proficiency. Other factors could be the regional differences in teacher training, the structure of the native language, or the reading traditions in the home.

Medieval Traditions

The writings of Boethius (ca. 480-524), Roman philosopher and statesman, constituted the major source from which scholars of the early Middle Ages derived their knowledge of Aristotle. Highly learned and industrious, Boethius hoped to make the works of Plato and Aristotle available to the Latin West and to interpret and reconcile their philosophical views with Christian doctrine. Charged with treason by Theodoric the Ostrogoth, he was executed without trial in 524, never completing his project. In prison he wrote his most popular work, De consolatione philosophiae. Boethius had a profound influence on medieval Scholasticism his Latin translations of Aristotle's Categoriae and De anima provided the Schoolmen with Aristotelian ideas, methods of examining faith, and classification of the divisions of knowledge.

Isidore of Seville
Venice: Peter Löslein, 1483

Isidore (ca. 562-636), archbishop of Seville, compiled numerous works which were instrumental in the transmission of the learning of classical antiquity to the Middle Ages. Among the most important productions of the "Great Schoolmaster of the Middle Ages" is the Etymologiae, also called the Origines, assembled by Isidore between 622-633. An encyclopedic work, unsystematic and largely uncritical, it covers a wide range of topics, including geography, law, foodstuffs, grammar, mineralogy, and, as illustrated here, genealogy. The title "Etymologiae" refers to the often fanciful etymological explanations of the terms introducing each article. The work became immensely popular and largely supplanted the study of classical authors themselves.

Eusebius Pamphili
Historia ecclesiastica
Italy, fifteenth century

The reputation of Eusebius Pamphili (ca. 260-340), bishop of Caesarea, as the "Father of Church History" rests mainly on his Historia ecclesiastica, issued in its final Greek form in 325. For over a millennium it has served as the major source for the history of the early Church. At the urging of Chromatius (d. 406), bishop of Aquileia, a Latin translation was produced in the late fourth century by Rufinus, presbyter and theologian. Rufinus made numerous changes in Eusebius' account which reflected his own theological stance and historical viewpoint, and introduced additions from original sources which are now lost. The present manuscript dates from the fifteenth century and once belonged to the marquis of Taccone, treasurer to the king of Naples late in the eighteenth century.

Basil the Great
De legendis gentilium libris
Bound with
Vita Sancti Antonii Eremitae
Italy? ca. 1480?

The writings of Basil (329-379) and Athanasius (293-373) exercised great influence upon the development of the ascetic life within the Church. Both men sought to regulate monasticism and to integrate it into the religious life of the cities. De legendis gentilium libris does not deal specifically with monasticism, but is instead a short treatise addressed to the young concerning the place of pagan books in education. The work displays a wealth of literary illustration, citing the virtuous examples of classical figures such as Hercules, Pythagoras, Solon, and others. Moral exhortations are also found in Athanasius' Vita Sancti Antonii Eremitae, a hagiography which awoke in Augustine the resolution to renounce the world and which served to kindle the flame of monastic aspirations in the West. This manuscript edition of the two works, probably originating from fifteenth-century Sicily, was written by Gregorius Florellius, an unidentified monk or friar.

De lapidibus
pretiosis enchiridion
Freiburg, 1531

Precious stones and minerals have long been prized for their supposedly magical and medicinal properties. During the Middle Ages these popular beliefs were gathered under the form of lapidaries, works which listed numerous gems, stones, and minerals, as well as the many powers attributed to them. Marbode (1035-1123), bishop of Rennes, composed the earliest and most influential of these medieval lapidaries, describing the attributes of sixty precious stones. For his work Marbode drew upon the scientific writings of Theophrastus and Dioscorides and the Alexandrian magical tradition. Christian elements, derived from Jewish apocalyptic sources, were not added to lapidaries until the next century. Marbode's work, which became immensely popular, was translated into French, Provençal, Italian, Irish, Danish, Hebrew, and Spanish. This third printed edition is one of five issued in the sixteenth century.

Notabilia dicta
Italy, ca. 1430 ­ 1450

Beginning in the twelfth century, much of the Aristotelian corpus became available for the first time to the Latin West through the medium of Arabic translations. Many Schoolmen were introduced to the philosophy of Aristotle through the extensive commentaries of Averroes (1126­ 1198), the renowned Spanish-Arab philosopher and physician who deeply inflluenced later Jewish and Christian thought. Followers saw implicit in his writings a doctrine of "two truths": a philosophical truth which was to be found in Aristotle, and a religious truth which is adapted to the understanding of ordinary men. This denial of the superiority of religious truth led to a major controversy in the thirteenth century and a papal condemnation of Averroism in 1277. Contained in this Latin manuscript are portions of Averroes' commentaries on Aristotle's De anima and Metaphysica, and his medical tract Al-Kulliyyat.

de medicinis
Naples, Italy, ca. 1500,
with sixteenth-century additions

Throughout the medieval period, the practice of medicine was more of an art than a science and required the preparation of complex "recipes" containing numerous animal, mineral, and vegetable substances. Materiae medicae, herbals, and antidotaries described innumerable recipes for everyday needs and proposed remedies which were believed to cure a wide range of human ailments. Many of the medieval prescriptions combine more than a hundred ingredients. This fifteenth-century materia medica contains prescriptions attributed to Galen (131 ­ 200), Mesuë (776 ­ 857), Avicenna (980-1037), Averroes (1126-1198), and others. Condiments and spices (pepper, ginger, cardamom, oregano) appear in most of the prescriptions, along with such favorites as camomile, mandrake, honey, camphor, aniseed, and gum arabic. Recipes are given for ink, soap, white sugar, hair-restorers and dyes, cosmetics, and colors ­ to name but a few. Remedies are suggested for such ubiquitous woes as dog-bite, headache, and gout.

Blasius of Parma
Questiones super libro methaurorum
Italy, fifteenth century

Blasius of Parma (ca. 1345 ­ 1416), a versatile, eminent, and sometimes controversial scholar, was instrumental in the dissemination and popularization in Italy of the new ideas then being debated by Scholastics at the University of Paris. Best known for his commentaries upon the works of Aristotle and more recent authors, he wrote on mathematics, physics, logic, psychology, theology, astrology, and astronomy. His discussion of Aristotle's Meteorologica found in this manuscript is distinctly anti-Aristotelian in tone and may be traced to the Platonist reaction fostered by the Medici. Blasius, also known as Biagio Pelacani, taught at Pavia, Bologna, and Padua and spent some time at the University of Paris. His wide range of interests anticipates the breed of scholar who would make Italy the center of the early Renaissance.

Book of Hours
(Use of Chalôns-sur-Marne)
Northeastern France, ca. 1400-1410

This Book of Hours is a noteworthy example of fifteenth-century Horae displaying a mixture of Parisian, Flemish, and provincial styles. The pages, adorned with elaborate borders and illuminations, contain ten miniatures depicting episodes in the life of the Virgin Mary. The elegant and mannered poses, the wave-form robe motifs, and the aerial perspectives based on graded blue skies are characteristic of early fifteenth-century Parisian illuminations. They contrast with the more provincial elements such as short, stocky figures and rustic faces which can be traced to Flemish influence. Prescribing daily worship periods, these texts served as concise breviaries for the laity. Including a liturgical calendar, psalms, hymns, anthems, and prayers, Horae were frequently produced in fifteenth-century France and Flanders.

Book of Devotions
Germany, fifteenth century

Books of Devotions, such as the example here, express the growth of a new religious consciousness and independence among the lower clerical orders and laity during the fourteenth and fifteenth centuries. The text, probably gathered and copied in or around Mainz between 1450-1475, is a collection of allegorical and devotional meditations, rules, stories, and exhortations. Of note is an allegory concerning Christ and the loving soul, using the metaphor of the human body as a castle, Christ as the master, and the soul as the mistress. Scattered through-out the final leaves are personal notes made by various lay owners of later periods. These include pious phrases in Latin and German lists of debts and interest paid the memoranda of one Ernst Lorentz Pauly (d. 1718) concerning his marriage, children, several baptisms, and a murder which occurred in 1669.

Strassburg: Johann Grüninger, 1507

Lay piety found new forms of expression with the rise of printing in the late fifteenth and early sixteenth centuries. Sources for this Altvaterbuch, a collection of lives of the saints, may be traced to late antique Byzantine hagiographies of the desert Fathers, such as Anthony, Gregory, and Hilary. The exemplary figures described in such traditional works provided personal and immediate sources of inspiration for devoted laity. The Latin Vitae patrum were subsequently translated into vernacular tongues, along with other popular devotional literature. The editions produced by the celebrated printer Johann Grüninger were known for their fine illustrations, usually produced from metal plates instead of the more frequent woodcuts. In order to facilitate the identification of pious readers with the holy figures, the illustrator depicted the Fathers in contemporary garb and placed them at work among the common people.

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