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Medieval European mathematics (c. 300—1400)

Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato's Timaeus and the biblical passage that God had "ordered all things in measure, and number, and weight" (Wisdom 11:21).

The Early Middle Ages (c. 300—1100)

This period saw an almost total decline in mathematical activity in Europe; only within the Christian monasteries was a low flicker of sub mathematical studies maintained. This consisted of two interrelated areas on the one hand computus i.e. the calculation of the date of Easter and on the other the basics of astronomy to determine the times for prayers and to maintain the calendar. These rudiments of mathematical activity, as found in the works of the English monastic scholar Bede, were taught from the Eighth to Eleventh centuries in monastic schools.

Boethius provided a place for mathematics in the curriculum when he coined the term "quadrivium" to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica, a free translation from the Greek of Nicomachus's Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Geometry. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.

The Rebirth of Mathematics in Europe (1100—1400)

This period saw a substantial increase in the mathematical activity. First of all it saw the establishment of the universities whose courses for the first degree of Bachelor of Arts was based on Boethius's seven liberal arts, which consisted of the trivium and the quadrivium. In reality however at many of the universities for much of this period only lip service was paid to the quadrivium and where it was taught it did not proceed far beyond the absolute minimum. However all of the universities did teach computus and in addition algorithmus,which was the Hindu-Arabic place value decimal system of numbers as this had proved a useful aid to the computus. Algorithmus texts such as those of Grosseteste or Sacrobosco were based on the Latin manuscript of Al-Khwarizmi's arithmetic text, Algoritmi de numero Indorum. Robert of Chester had also translated Al-Khwarizmi's Al-Jabr wa-al-Muqabilah, he was one of a group of European translators which included Gerard of Cremona, Adelard of Bath, John of Seville and others who travelled in the 12th century to the cultural centres in Spain, Sicily etc. where Islamic, Christian and Jewish culture existed along side each other. Here they translated most of the major Greek and Islamic works on the mathematical sciences from Arabic into Latin making them available in Europe for the first time since the end of the Roman Empire. This period is know as the Mathematical Renaissance.

The 12th and 13th centuries also saw the first original mathematics produced in Europe since the 2nd century CE in the works of Jordanus Nemorarius and Fibonacci. Fibonacci produced works of a theoretical nature and also works of practical nature. The latter are very important as they led to the establishment in Italy in the 13th century of so called Abbacus Schools, small private schools that taught the basics of arithmetic, algebra and geometry to apprentices. By the 14th century such schools had spread to France and Germany along the European trade routes. The teachers in these schools usually wrote their own textbooks, known as Abbacus Books after Fibonacci's Liber Abbaci. These continued to be the standard text books for mathematics up till the late 16th centuries with examples being written by famous mathematicians such as Piero della Francesca, Luca Pacioli, Tartaglia, Cardano and Michael Stifel.

Following on from the work of Jordanus an important area that contributed to the development of mathematics concerned the analysis of local motion.

Thomas Bradwardine proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been invented, we can express his conclusion anachronistically by writing: V = log (F/R). Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.

One of the 14th-century Oxford Calculators, William Heytesbury, lacking differential calculus and the concept of limits, proposed to measure instantaneous speed "by the path that would be described by [a body] if ... it were moved uniformly at the same degree of speed with which it is moved in that given instant".

Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (which we would solve by a simple integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".

Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance travelled. In a later mathematical commentary on Euclid's Geometry, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.

Early Modern European Mathematics (c. 1400 - 1600)

This period saw a massive increase in mathematical activity in comparison to what had been going on before in mediaeval Europe. This activity can be split up into five different areas:

1. The rediscovery of linear perspective
2. The Universities
3. Astronomy and its applications
4. The Archimedean Renaissance
5. Algebra

The rediscovery of linear perspective

The first important mathematical development after 1400 was the rediscovery of linear perspective in the 1430s. The first mathematic description of how to construct picture using linear perspective was written by Alberti. Shortly after Piero della Francesca wrote a more extensive treatise which was never published but was heavily plagiarised by Luca Pacioli whose writings on perspective where illustrated by Leonardo. This was followed by works from Dürer and many others throughout the next two hundred years. The interest in perspective and its development led to interests and development of geometrical optics and Euclidian geometry especially in stereometry. A study of the regular Platonic Solids and the Archimedean semi-regular solids can be traced from della Francesca through Pacioli/da Vinci, Dürer, Jamnitzer and onto Kepler.

Universities

The mathematical sciences had been largely neglected at the universities during the previous period the Early Modern Period saw the gradual introduction of chairs for mathematics into the university and with them an increase in the status of mathematics within these institutions. Chairs for mathematics were introduced into the humanist universities of Northern Italy along with Krakow, Vienna and Ingolstadt in the 15th century. Philipp Melanchthon introduced chairs for mathematics into the Protestant schools and universities of Germany and Scandinavia in the second quarter of the 16th century. Christopher Clavius did the same for the Catholic educational institutions in the last quarter of the century. England lagging somewhat behind established the Gresham Chairs for Geometry and Astronomy in 1597 and the Savillian Chairs for the same disciplines at Oxford in the 1620s. The major motivation for the installation of chairs for mathematics was the need to teach iatromathematics to medical students. Iatromathematics or astrological medicine was the dominant direction in renaissance medicine so medical students required the mathematics and astronomy necessary to cast horoscopes and to construct astrological calendars. Many of the leading mathematicians of the period were also medical practitioners: Toscanelli, Copernicus, Rheticus, Robert Record, Cardano etc. Even Galileo was still teaching iatromathematics to medical students at Padua at the beginning of the 17th century.

Astronomy and its applications

The increased need for data to do astrological calculations drove the creation of the new astronomy but it was not only astrology that demanded new levels of accuracy from the astronomers. Due to demands created by social, political and economic developments the early modern period saw a massive growth of the interest in navigation, chronometry and cartography all of which required accurate astronomical data. Throughout this period most of the leading mathematicians were heavily involved in the development of the new astronomy and its application, these included Toscanelli, Peuerbach, Johannes Werner, Johannes Stabius, Regiomontanus, Peter Apian, Gemma Frisius, Copernicus, Rheticus, Pedro Nuñez, Oronce Fine, Maestlin, Tycho, Mercator, John Dee, Edward Wright and Thomas Harriot. All of these discipline are heavily dependent on trigonometry for their calculations and this period also saw the introduction of the Indian half-cord trigonometry by Peuerbach and Regiomontanus to replace the Greek cordal trigonometry and its development by Johannes Werner, Peter Apian, Copernicus, Rheticus, Gemma Frisius, Vieta, Finck and Pitiscus, the later being the first to introduced the word Trigonometry.

The Archimedean Renaissance

Throughout this period Humanist scholars collected Greek scientific and mathematical manuscripts. An interests for the contents of the manuscripts developed out of this collecting habit leading to a great interest in the works of Archimedes especially in Italy in the 15th and 16th centuries. According to both Paul Rose and Jens Høyrup this interest in the works of Archimedes was a leading factor in the turn towards the mathematisation of nature that forms the core of the scientific revolution.

Algebra

The works of Fibonacci had introduced Arabic algebra into Europe and the Abbacus Schools had kept it alive. In the 15th century algebra started to enter the mathematical mainstream. Contributions were made by Chuquet in France, Regiomontanus in Germany and Piero della Francesca in Italy. The publication of Pacioli's Summa in 1494 made the whole of the available knowledge in Algebra easily available. The discovery of the general solution of the cubic equation by Scipione del Ferro.