The relation of mathematics to physics. The simplest mathematical notion is that of the number of a class. This is the property common to the class and to any class that can be matched with it by pairing off the members, one from each class, so that all members of each class are paired off and none left over. In terms of the definition we can give meanings to the fundamental operations of addition and multiplication. Consider two classes with numbers a, b and no common member. The sum of a and b is the number of the class consisting of all members of the two classes taken together. The product of a and b is the number of all possible pairs taken one from each class. We cannot always give meanings to subtraction and division, because, for instance, we cannot find a class whose number is 2–3 or 7/5. But it is found to be a great convenience to extend the notion of number so as to include negative numbers, ratios of numbers irrespective of whether they are positive or negative, and even irrational numbers. When this is done we can define all the four fundamental operations of arithmetic, and the result of carrying them out will always be a number within the system. We need trouble no more about whether an operation is possible with a particular set of numbers, since we know that it is, once we have given sufficient generality to what we mean by a number.

ROMAN LAW AND NATIONAL LAWS

In 1653 there appeared a book by the English civilian Arthur Duck on the use and authority of the Roman civil law in the realms of Christian princes (De usu et authoritate iuris civilis Romanorum in dominiis principum Christianorum). It is based on precise information about the extent to which the civil law had been received in different European countries and Duck was at pains to bring out the common ideas on the nature of law that those countries shared. Wherever one does not look merely at custom but seeks equity, he says, the laws of no nation are more suited than the civil law of the Romans, which contains the fullest rules concerning contracts, wills, delicts, judgments and all human actions.

The exact extent of the civil law component varied from country to country. Court practice (usus fori), as evidenced by collections of decisions, had for long reflected the particular amalgam of Roman civil law and customary law of the country or region. University teaching, on the other hand, had always remained tied to the civil law and ignored the customary element. By the middle of the seventeenth century the universities had to come to terms with the civil law as it was understood locally, and law faculties recognised national compounds of Roman and local law. In 1650 Michael Wexionius, professor in the university of Åbo (Turku) in Finland, then part of the Swedish kingdom, published an introduction to the study of Roman-Swedish civil law (iuris civilis Sveco-Romani).

ROMAN LAW AND GERMANIC LAW IN THE WEST

From the sixth century until the eleventh, a reference to Roman law in Western Europe was normally understood to be to the law of the so-called barbarian codes, in particular the Roman law of the Visigoths. These collections reflected not Roman law of the classical period but the ‘vulgar law’ of the fifth century. They served as quarries from which rules could be dug when required for smaller collections. Compared with the scope and complexities of Justinian's compilation, their contents reflected a low level of legal science, but even so they sometimes proved to be beyond the comprehension of those who consulted them in the sixth and seventh centuries.

In the early middle ages, the imperial system of courts, staffed by professional judges who represented a state machine that could enforce their decrees, disappeared. In its place were groups of freemen from the locality who sought to settle disputes in such a way that the disruption of community life would be minimised. The assemblies of freemen had to establish the customary rules relevant to the case before them. These rules were not applied rigidly but provided a background against which the dispute was to be settled, often by compromise. Instead of the sense of belonging to a world empire, the individual had more of the sense of being part of a community of people of similar ethnic origin with similar customary traditions.