There is much fascinating material to be explored in the history of medieval and early modern mathematics, but perhaps the first thing to be aware of it is that, particularly for the earlier centuries, almost all original sources are in Latin. Mathematical Latin uses a specialised but relatively limited vocabulary, and if you know some Latin already (A-level or a good GCSE, say) you can build on it, but if you have none you will need to devote time to learning it.
Medieval mathematics (roughly 1100–1500)
Medieval mathematics was on the whole far removed from anything that we think of as mathematics today. Indeed to study this period at all you need to be prepared to enter a world whose preconceptions, political, religious, or mathematical, were very different from our own. There are texts that are recognisably devoted to arithmetic, geometry, or occasionally algebra, but most of the writings that were later described as 'mathematical' were concerned with astrology and astronomy (the distinction between the two was often blurred). Others border on philosophy, natural philosophy (or early science), and even theology.
The main centres for such studies in northern Europe were Paris and Oxford, and Oxford still holds major collections of medieval manuscripts, many with mathematical content. This has not been a major area of study in the last few years, but for useful introductory material and overviews see the following.
- Grant, Edward, (ed), A source book in medieval science, Harvard University Press, 1974.
- Molland, George, 'Mathematics', in David C Lindberg and Ronald L Numbers (eds), Cambridge history of science, II, Cambridge University Press, 1999.
- North, J D, Chaucer's universe, Clarendon Press, 1988.
- North, J D, 'Astronomy and mathematics' and 'Natural philosophy in late medieval Oxford', in J I Catto and T A Evans, The history of the University of Oxford, II, Clarendon Press, 1992.
- North, J D, 'Medieval Oxford', in Fauvel, Flood and Wilson, Oxford figures: 800 years of the mathematical sciences, Oxford University Press, 1999.
- Stedall, Jacqueline, 'How algebra was entertained and cultivated in Europe' in Stedall, A discourse concerning algebra, Oxford University Press, 2002, 19–54.
Early modern mathematics (roughly 1500–1700)
Like all other areas of intellectual activity in the sixteenth century, mathematics was revitalized by the translation of Classical texts from Greek to Latin. It was further stimulated by the absorption of ideas from Islamic sources, and by the new technical challenges posed by increased trade and navigation. During the seventeenth century in particular, mathematics in western Europe began to change rapidly and dramatically. At the beginning of that century, mathematicians looked back on ancient learning as something they could barely hope to emulate. By the end of it, they had far outstripped Classical achievements in both methods and results, and had developed their own tools and language, recognisably similar to those we use today. The most notable mathematical advances of the seventeenth century were the development of analytical geometry, the new acceptance of indivisibles, the discovery and use of infinite series, the discovery of the calculus, and the beginnings of a mathematical interpretation of nature. All of these changes were continued, consolidated, and argued about during the eighteenth century.
At the same time, mathematical learning was becoming more widespread, and the number of publications increased rapidly. In recent years the availability of electronic databases and searchable text has transformed research possibilities for this period. Every book published in England in the seventeenth century is now catalogued, and usually digitally available, on a database known as EEBO (Early English Books Online). Its eighteenth-century counterpart is ECCO (Eighteenth Century Collections Online). Access to these is available only through academic libraries, but they are an excellent way to begin to explore the literature.
Accounts of the early modern period are to be found in all general histories of mathematics, and you should read as many as you can. The following more specialised bibliography, by no means exhaustive, is intended to illustrate just some of the different ideas, approaches, and research methods that are to be found in recent literature.
- Bos, Henk, Redefining geometrical exactness: Descartes' transformation of the early modern concept of construction, Springer, 2001.
- Feingold, Mordechai, The mathematician's apprenticeship: science, universities and society in England 1560–1640, Cambridge University Press, 1984.
- Feingold Mordechai, 'Decline and fall: Arabic science in seventeenth-century England' in F Jamil Ragep and Sally P Ragep (eds), Tradition, transmission, transformation, Leiden: Brill, 1996, 441–469.
- Feingold Mordechai, 'Gresham College and London practitioners: the nature of the English mathematical community', in Francis Ames-Lewis (ed), Sir Thomas Gresham and Gresham College: studies in the intellectual history of London in the sixteenth and seventeenth centuries, Ashgate, 1999, 174–188.
- Guicciardini, Niccolo, '"Gigantic implements of war": images of Newton as a mathematician', in E Robson and J Stedall (eds), Oxford Handbook of the History of Mathematics, Oxford University Press, 2008.
- Hill, Katherine, 'Juglers or Schollers?: negotiating the role of a mathematical practitioner', British Journal for the History of Science, 31 (1998), 253–274.
- Mahoney, Michael Sean, 1973, The mathematical career of Pierre de Fermat 1601–1665, Princeton University Press, 1973, reprinted, 1994.
- Malcolm, Noel, and Stedall, Jacqueline, John Pell (1611–1685) and his correspondence with Sir Charles Cavendish: the mental world of an early modern mathematician, Oxford University Press, 2005.
- Stedall, Jacqueline, A discourse concerning algebra: English algebra to 1685, Oxford University Press, 2002.
- Stedall, Jacqueline, 'Symbolism, combinations, and visual imagery in the mathematics of Thomas Harriot', Historia mathematica, 34 (2007).
- Wardhaugh, Benjamin, 'Poor Robin and Merry Andrew: mathematical humour in Restoration England', BSHM Bulletin, 23 (2007).