The teaching of mathematics in The Renaissance.


The Renaissance was a period of immense transformations within Europe, not the least of which involved a major shift in European educational ideas. During this period, a new way of thinking came to the fore proposing a different form of training, one which would provide the student with skills for life and not just those which were required by their occupation. These views were championed by Humanists who established schools and institutions which implemented these ideas.

Vittorino da Feltre was one of the founders of Renaissance education and his school was arguably the most liberal. The range of subjects available was extensive and Vittorino's aim was that each student should leave with a basic understanding of each, and they should also have received the time and support to study those subjects at which they excelled in greater depth. In particular, selected students were encouraged to concentrate their efforts in the mathematical field, a practise which was not promoted by many other educators. Guarino, whose school was established shortly after Vittorino's, preferred the more classical stance to learning, concentrating heavily on Grammar and Literature, utilising the work of prominent Roman authors as examples. Such subjects as Mathematics and Music were still taught but at a much more elementary level. Other Educationalists, like Palmein, who advised that a greater understanding of Arithmetic and Geometry was necessary, as practical arts and rational disciplines, were criticised. Most held similar opinions to mainstream Humanists; subjects which could not be directly applied to life were of secondary importance and an education which improved knowledge of Latin (allowing for further perusal of the ancient texts) and the art of Oratorio was considered sufficient.

The study of Mathematics in particular was disputed by many, because of its strong association with trade and commerce. Merchants and master craftsmen in many areas in Europe were not given an identical level of respect or deference as they commanded in Germany. This meant that sons of the merchant class were taught only in those subjects which would aid them in their efforts to become statesmen and politicians. What little mathematics was taught in the merchant schools therefore became highly theoretical and divorced from possible applications in the real world.

To cope with this gap in the educational system, another type of school was founded in Florence and its surrounding areas. The Scoula d'abaco taught those who wanted to improve their ability in commercial areas, and hence provided courses in Arithmetic, Algebra, Astronomy, book-keeping, and the more practical elements of Geometry, which were fast becoming important due to recent advances in Navigation.

Advances were being made in other sciences and technologies, the invention of the printing press having the most profound effect on education. This allowed for a rapid dissemination of knowledge with many more people able to afford to purchase books, especially when the practice of printing texts in monthly instalments reduced the price even further. In the beginning, there were insufficient printing presses dealing with mathematical documents but the demand for astronomical charts and commercial tables among society grew, and after the translation of important texts such as Euclid's Elements into German, French and Italian the demand increased.

Not solely the ancient texts, but also the works of more modern mathematicians such as Cardan, Bombelli and others began being printed. In England, Robert Recorde wrote what is thought to be the first series of textbooks in English. These were not intended for the highly educated mathematician but for the common man seeking to improve his understanding of such subjects as the Hindu-Arabic numeral system, conversions between weights and coins, computation with counters which would aid their work in trade and commerce. These subjects were covered in Recorde's first and most successful book The Grounde of Artes which was first printed in 1540, but which was reprinted over fifty times in nearly a hundred and sixty years. Recorde's three other major works The Pathwaie to Knowledge, The Castle of Knowledge and The Whetstone of Witte were not so popular. This is most likely due to their less practical and more advanced contents.

Characteristic of most of Recorde's texts is his question and response style of writing. This is very close to the teaching style used during the Mediaeval Ages, but it was obviously beginning to be recognised as inadequate by Recorde, since he counselled against using a similar style in the classroom because of its limitations. He did not, however, explicitly propose any other method. The Pathwaie is the only one of Recorde's four books not written in this dialogue style. It is considered almost as an abridged version of Euclid's Elements with many of the Latin terms replaced with English equivalents of Recorde's own devising. This caused much criticism and later versions of Pathwaie returned to the originals. A German attempt to replace the Latin terminology was successful although why one was accepted but the other was not, remains a mystery.

Around the middle of the sixteenth century, Ramus proposed that in France the Arts courses taught at universities should return to the seven classical liberal arts, but with the syllabus more based on applied topics. He developed "method" as a pedagogical concept taking theory towards that required for practical problems. He proposed to reorganise the seven liberal arts using the following three "laws of method":-

(i) only things which are true and necessary may be included;

(ii) all and only things which belong to the art in question must be included;

(iii) general things must be dealt with in a general way, particular things in a particular way.

Using this approach Ramus worked on many topics and wrote a whole series of textbooks on logic and rhetoric, grammar, mathematics, astronomy, and optics.

The existence of textbooks like those of Ramus and Recorde and others, and the reduced cost of purchasing books (thanks to the printing press) caused an increasing interest in the sciences, not only amongst the wealthy but also amongst the middle class. Knowledge of certain mathematical skills and techniques beyond what was available in most schools was becoming more important, especially in the towns and cities. It was soon possible for people to earn a living as a private Mathematics tutor for those with enough money, or as a Mathematics practitioner similar in style to the scriveners of previous centuries. Those with more money either became academics or amateur mathematicians, many of whom often made considerable advances in both the subject and its place in education.

John Dee, one of the editors for The Grounde after graduating from Cambridge with both a BA and an MA and later lecturing at the University of Paris on sections from Euclid's books, wrote of the need for improving the place of Mathematics in Education. He argued that Mathematics should be studied, not only for its practical use (of which there was still too little), but also for its ability to 'lift the heart to the heavens' which is reminiscent of Pythagorean beliefs. He proposed translating currently available mathematical texts into English, in order to aid the spread of knowledge to those who had not spent years learning Latin at school and University and who found studying the texts in the original language difficult. Dee himself helped to translate Euclid's Elements into English and this was then published in 1570, eighty-eight years after it was published in Latin.

Sir John Cheke, the first Professor Regius of Greek at Cambridge, also made attempts to study more advanced Mathematics including much of Euclid's work and then to pass on his knowledge to others at the University, the majority of whom became tutors at court. Few other professors made any significant efforts in this direction. The Edwardian Statues of 1549 at Cambridge did try to improve the situation by laying down that all freshmen were to be taught Mathematics at foundation level as part of a liberal education. It recommended textbooks including Cardan and Tunstall (whose De Arte Suppletandi of 1522 was very much based on the academic style found in Italy and hence of little use to the merchant classes) along with Euclid's Geometry and Astronomy. These Statues were removed only twenty-one years later during the reign of Elizabeth I, because the commissioners believed that Mathematics was applied to the practical life and was therefore more part of a technical education than that which should be provided at the University.

The two Universities of Oxford and Cambridge were no longer alone in Britain. The 14th and 15th centuries saw the foundation of over 50 new universities across Europe, including three in Scotland alone. A University at St Andrews was founded in the early 15th century because of problems posed by sending the most advanced students of the Grammar and Cathedral schools to the University of Paris and France's transferral of allegiance to the Roman Pope. Universities in Glasgow and Aberdeen soon followed. These three were very much in the same style as the Parisian University and followed the mediaeval syllabus, where subjects in the Quadrivium were considered to be useful only as training for what was soon to be considered the more important subjects at university: Natural, Metaphysical and Moral Philosophy.

This Mediaeval curriculum, and the structure of the Universities, were revised shortly after the Reformation. Andrew Melville, one of the supporters of the ideals presented in the Book of Discipline, was offered the principalships of both St Andrews and Glasgow following his return from Paris and Geneva. At this time Glasgow was in greater need, having few students and little money. The improvements that both Andrew and his nephew James made were considerable. They introduced the study of Greek for the first time in Scottish universities and the extended curriculum included lectures on Arithmetic, Geometry, and Mathematics which were given predominantly by James. The University structure was changed and the practice of teaching first year students all of their courses by the Regent was removed. In 1597 Andrew Melville was forcibly removed to St Andrews which too enjoyed a New Foundation and a short golden era of advancement. These reforms did not remain in place and by the middle of the 17th century the Universities had reverted to the old Mediaeval system. This remained in place until the middle of the 18th century.

The Reformation and the First Book of Discipline highlighted the failings of education in Scotland. Where some form of teaching was available it was usually provided by the Church, but in 1549 the Provincial Council accused cloistered schools of their 'crass ignorance or literature and all liberal arts'. The larger towns and cities fared better with almost all the burghs of any size containing a Grammar school (many of which were already over a century old) by 1560. However, the more rural areas of Scotland were often left with no form of education available at all, excepting small and poorly run schools where letters and reading might be taught. In what was later called the First Book of Discipline, the 'six Johns' chosen by the Privy Council included two chapters, specifically dedicated to education. Here they laid out detailed plans for the Universities, and guidelines proposing advances for school level education. Some of the most fundamental points included the requirement that every church should appoint a schoolmaster of some ability, and that the poor should also be educated, regardless of whether or not they intended to follow a career in the Church.

The overall effect of the Reformation in Scotland was a general improvement in the standards of education available, and a marked increase in the number and placement of schools. Much of this did not happen until the 17th century, however, and it did cause great disruption to the educational system in the late 16th century. The disruption caused by the Reformation to the Church, meant that many who would have progressed into further education and who would then have been sponsored at University chose not to do so. As a consequence, the enrolment at Glasgow and St Andrews dropped to almost nil in 1560, jeopardising their very existence. Schooling was also disrupted when rioting destroyed cathedrals and churches, with the buildings close to these structures. These specific buildings had previously housed the schools and now there existed no other suitable buildings in the area which could offer the same. However, these problems were only temporary, and Andrew Melville's work did much to restore the ailing Universities.

Article by: J J O'Connor and E F Robertson based on a University of St Andrews honours project by Elizabeth Watson submitted May 2000.


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JOC/EFR August 2000

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