Die conste vanden getale

Een studie over Nederlandstalige rekenboeken uit de vijftiende en zestiende eeuw, met een glossarium van rekenkundige termen

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Summary

In Europe during the Middle Ages some important developments were taking place in the field of arithmetic. The traditional way of calculating with counters was competing against the new calculating method that used written Hindu-Arabic numerals. In the course of the fifteenth and sixteenth centuries this new method was introduced in the Netherlands.

Arithmetic books in the Dutch language from that period dealing with the new arithmetic form the starting-point of this study. There are, as far as is known, 36 such arithmetic books in existence: 12 manuscript and 24 printed editions. Three of these books were written in the fifteenth century, the oldest dated 1445. The others date from the sixteenth century. Eleven of the 36 arithmetic books are either a reprint or a copy of an earlier work of the corpus. None of the other 25 arithmetic books have an identical predecessor, but in many cases there are similarities with one or more earlier works. A survey of the corpus is to be found in Appendix II.

The central research question in this study is twofold:

-	which kind of arithmetical instruction method is applied in the arithmetic books: what is taught, by whom, to whom and in which way?
-	what contribution did the arithmetic books make to the development of mathematics, education, didactics and the arithmetical terminology in the Dutch speaking countries?

The first chapter describes the appearance of the new arithmetical method with Hindu-Arabic numerals in Southern Europe at the end of the twelfth and the beginning of the thirteenth century and its spread through Europe during the centuries that follow.

In 1202 the Italian Leonardo of Pisa (also called Fibonacci, ca. 1170-1240) used the Hindu-Arabic numerals in his Liber Abaci, applying the new arithmetical method to a great many commercial problems. This part of his work was copied by the authors of hundreds of Italian arithmetic books. The new method became popular in many other European countries and practical applications appeared in arithmetic books written in several languages, amongst them books in the Dutch vernacular.

These abacus books, named after the Liber Abaci, were but one route by which the new arithmetical method entered Europe. In the twelfth century, several Arabic arithmetical manuscripts came into the Mediterranean area, especially Spain and Sicily, via trade routes. The best known is the ninth-century arithmetic manu-

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script of al-Khowarizmi (ca. 780-850). Several Latin translations and adaptations were made of this treatise. Inspired by these works, thirteenth-century scholars like John of Sacrobosco and Alexander of Villa Dei wrote their own arithmetic books. In contrast to abacus treatises, these algorismus treatises are written in Latin and do not contain practical applications of the arithmetic. They may have been intended for a learned audience.

The new arithmetical method did not immediately supersede the traditional method of calculating with counters. For a long time the methods were used side by side, the authors of the books not being partial to either in particular. There was no competition, as is sometimes wrongly suggested. In the Dutch arithmetic books of the fifteenth and sixteenth centuries, calculating with counters was still frequently found.

It is understandable why the traditional method survived for such a long time. Calculating with counters is possible for those who cannot write or know how to use zero. Another advantage is that one can show calculations with concrete objects. Nevertheless, written calculations were preferred in the end, because they make it easier to divide, extract roots, calculate with fractions and manipulate calculations. Also, they made possible new developments in mathematics and astronomy, and the same instrument (the pen) is used both for calculating and recording the result.

Outside arithmetic books, in official documents and cash-books, the transition from Roman to Hindu-Arabic numerals for recording amounts, dates, page numbers, etc. also took quite a while. Here the advantages of the new number system were apparent particularly in the reading and writing of big numbers.

Chapter 2 contains a description of arithmetical instruction in the Netherlands during the fifteenth and sixteenth centuries. In the parish schools, founded in great numbers during the twelfth and thirteenth centuries, arithmetic was hardly taught. Their main task was religious instruction.

In the course of the fourteenth century, more and more parish schools came under the jurisdiction of the town government. The pupils of these town schools still had to fulfill many tasks during religious services and did not learn arithmetic.

In the end, these town schools were divided into two types. An onder-school (or Dietsche school), where 8- to 10-year-old pupils were taught to read and write in the Dutch language, or a boven-school (or Latin school), where older pupils were taught exclusively in Latin. In these schools arithmetic hardly played a part.

This was different in some by-schools, i.e. schools founded by private initiative, in whose curriculum the town government did not have a say. These by-schools were competitors of the Latin schools. This goes especially for the French schools, where merchants sent their sons to study subjects like foreign languages, bookkeeping and arithmetic, which were very useful for their future business career. Some teachers teaching in these French schools wrote their own arithmetic book in Dutch about calculating with Hindu-Arabic numerals.

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In the Chapters 3 and 4, the contents of the Dutch arithmetic books are discussed. In the first part, the authors generally teach the basics of arithmetic, which means that they deal with the reading and writing of Hindu-Arabic numerals and the arithmetical operations: addition, subtraction, multiplication and division. The algorithms they teach largely correspond to those in use today. Only the division algorithm shows some differences.

First, calculating with whole numbers is taught, followed by fractions. The arithmetic is presented in the shape of many examples worked out in detail. Subjects appearing in many (but not all) books are: halving, doubling, calculation checking (check of nines), tables of exchange rates and calculations with money, weights and measures, extracting roots and calculating with counters.

In the second part of the arithmetic books elementary arithmetic is used to solve all kinds of practical problems. For that purpose the pupil is taught a lot of arithmetical rules, of which the rule of three is the most important one. This rule is used to find the fourth number in proportion to three given numbers. The other arithmetical rules are mostly variants of the rule of three. In general they owe their name to the situation in which they are applied: the buying, selling or exchanging of goods, partnership, changing money, the calculating of interest, insurance, profit, loss, etc.

Some authors deal with more difficult arithmetical rules not based on the rule of three. These are:

-	the regula falsi (or rule of false position). It is used to find the required unknown number with the help of two arbitrarily chosen numbers;
-	the regula cos. Here the pupil learns to compose and solve algebraic equations;
-	the rule of progressions. In this chapter the author deals with several arithmetical and geometrical progressions, which have to be added up;
-	the rule of proportions. This topic contains an enumeration of all sorts of categories of proportions.

Most of the time practical applications are missing from these chapters on more mathematical subjects.

In Chapter 5, the target group the authors had in mind is described. In general they seem to have addressed a young, male audience with a basic education, in many cases probably the pupils of the French school, who, by means of practical problems in the arithmetic books, were trained to become merchants or technical, administrative or financial practitioners.

Some authors also addressed adult pupils, who probably used the book for self-study or as a reference book to look for algorithms and to consult tables of coin values, weights and measures. It could also serve as a ready reckoner.

The tables and the many realistic problems can supply information for today's researcher about the sixteenth-century economy and society. Yet these data ought to be handled with great care because they come from school books, and need to be confirmed by data from other sources.

Occasionally, unrealistic problems appear in the arithmetic books. These are often very old and were possibly included for traditional reasons or for pleasure.

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Sometimes very large numbers or calculations appear in striking geometrical forms. These are perhaps meant to impress the reader and convince him (or her) of the power and beauty of the art of number.

It is possible to derive some didactic principles from the way in which the authors deal with the arithmetic. For instance, the books are graded from easy to difficult. Solution methods are presented in the shape of arithmetical recipes, which are described by means of a step-by-step example. Proofs or arguments for the method of solution are lacking. The pupil is advised to use the same method in all similar situations. After the example many similar problems follow, which differ only in the numbers used. Repetition may help the pupil to remember the method of solution. The aim of the authors is not primarily to teach their pupils mathematics, but rather to teach them recipes that can be used to solve the arithmetical problems they may encounter in their future professions.

Repetition plays an important part in memorizing. Mnemonic techniques like rhyme and dialogue are hardly used. In the course of the sixteenth century, some authors tried to shorten their solution methods. Rather than giving detailed descriptions, they prefer to use words with a symbolic function or have recourse to graphical means like lines, signs, frameworks and numbers. A schematic solution method is more readable, easier to learn by heart and reduces the risk of making mistakes. Modern arithmetical symbols are still absent in the arithmetic books, but the efforts to shorten calculations clear the way to the later symbolic mathematical notation.

In general, the pupil learns only one solution method for each problem. This is the standard solution, which is sometimes a bit cumbersome, but can be always applied more or less ‘blindly’. In some arithmetic books the author deals with the Welsche or Italiaanse praktijk (French or Italian practice). This is a collection of alternative methods with which the arithmetician (on certain occasions, and with suitable numbers) can speed up and simplify the calculation. This type of insightful calculation only plays a minor part in the arithmetic books.

Many mutual relations exist between the Dutch arithmetic books and between these Dutch and the non-Dutch works. In Chapter 6, all relations found are surveyed. It is important to realise that none of these links can be established with absolute certainty, as it is unknown which arithmetic books have been lost in the course of time. In the diagram (p. 279) many intermediate steps are inevitably missing.

Only five arithmetic books (all manuscript) are unrelated to other works. The authors of the other arithmetic books used the work of at least one predecessor. The way in which these sources are used differs widely. Some authors copy large portions, others revise or translate their sources, using only a few problems or compiling the material at their disposal and supplementing it with their own explanation, comment or examples.

In the course of the sixteenth century the arithmetic books hardly developed in form and content. The size of the books increased, however. This was caused by a growing number of examples and problems, and some authors also added more

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mathematical subjects to the elementary arithmetic in their works. In this way they hoped to reach a wider audience. The traditional way of calculating with counters disappeared from the arithmetic books at the end of the sixteenth century.

In Chapter 7, the development of the Dutch arithmetical terminology in the sixteenth century is described. Initially the authors use quite a few Latin arithmetical terms, but sometimes they adjusted these in form or sound to the Dutch language. There are also some French and to a lesser extent some German influences to be found. Gradually more foreign terms were translated into Dutch words. For the purpose of this translation the authors sometimes used words from everyday language, but they also invented or derived new Dutch arithmetical terms.

During the sixteenth century, the authors used many new Dutch arithmetical terms, which they coined themselves or copied from the work of their predecessors. However suitable many of these new words may have been, they did not immediately replace the existing arithmetical terms. In many cases old and new terms occur (literally) side by side.

In this way a lot of synonyms came into use. At first sight this seems undesirable in mathematics, but when there is no standardisation of arithmetical terms, placing the old and new names next to each other seems to be quite suitable. As long as the new terms are not sufficiently familiar, one cannot omit the old terms with impunity. Among the synonyms presented, there will always be at least one term that the reader may know or understand. In this way the arithmetic remains transferable.

So after more than four centuries, sums are still being done in a way that shows many similarities with the calculation methods of the sixteenth-century arithmetic books. The arithmetical vocabulary in use nowadays has its origin mainly in the sixteenth century. From this one may conclude that the sixteenth-century arithmetic masters produced accomplished works.

Most of the arithmetical words of the arithmetic books - whether they are foreign or Dutch terms or hybrid loanwords, do not appear in the Woordenboek der Nederlandsche Taal or in the Middelnederlandsch woordenboek (two standard dictionaries of Dutch and Medieval Dutch respectively). This is why a disk is appended to this study with a glossary of all arithmetical words and expressions that are found in the arithmetic books. The meaning of each word is given and quotations are added.

The search programme added to the glossary, enables the user to order the material in different ways, to make selections according to his or her own criteria and to find answers to new linguistic, historical and/or mathematical questions.