The history of science can be approached in several different ways. It may be studied, as in the classification once favoured in the long-established Department of History and Philosophy of Science at University College London, by considering separately the history of individual sciences: physics, chemistry, biology, etc.—Partington's monumental History of chemistry is a good example of the cross-section of history of science obtained by considering a single discipline. This approach is understandable when history of science is the work of retired specialists in a particular science. On the other hand, many of those who have approached the history of science from a training in general history have tended to favour a study of a particular period as an alternative to an orientation by subject. This is particularly valuable before the nineteenth century, when subject boundaries were not so tightly drawn as some of the old science historians tended to assume. A third possibility is area studies, usually the history of science within a particular country. Sometimes this is done unconsciously, as when historians claim that they are dealing with a general theme, such as science and religion or scientific institutions, but do so with special reference to their own country. French historians of ‘the Enlightenment’ often study French authors exclusively. Language as much as country is a limiting factor here.

Even at the level of the most elementary arithmetical operations, procedures and practices change. During the twelfth and thirteenth centuries an unusually well documented development took place: at the beginning of the period the authors of elementary manuals of computation taught the use of the abacus, whereas at the end they described the method of calculation which came to be known as the algorism. Their ideas about number, however, were still largely drawn from Boethius's rendering of Nicomachus of Gerasa's Introduction to arithmetic in the Arithmetica and there had been little progress in attempting to reconcile Boethius's teaching on the theory of number with the rather different assumptions that underlie the methods of practical calculation. Boethius and Nicomachus, for example, emphasize that one is not a number but the source of number, and they are aware of the special problems posed by ‘two’. Nicomachus questions whether ‘two’ is anything more than an embodiment of the principle of ‘otherness’; for him, it is open to dispute whether it can be rated a number in its own right. For the teacher of the skills of calculation, ‘one’ and ‘two’ are merely digits like any other. By the fourteenth century, collections of textbook material on elementary arithmetic provided the student with instruction in both theory and practice. The abacus manuals are missing from many such collections because by then the abacus has apparently been relegated to the status of a simple practical aid, but the other elements in medieval arithmetical studies are variously covered.