Antoine Thomas, SJ, and his Synopsis Mathematica: biography of a Jesuit mathematical textbook for the China mission

This article is an examination of a nearly forgotten massive two-volume octavo textbook of introductory (theoretical and practical) mathematics published in Douai in 1685, with a second issuing of it in 1729. The theme of mathematical training has been central to the understanding of the Jesuits in China in the late seventeenth and early eighteenth centuries, and this discussion gives a detailed survey of the mathematical ‘baggage’ of the author, Antoine Thomas, SJ, (1644-1709). Here we consider his teaching at the Colégio das Artes in Coimbra, Portugal, in the late 1670s, when he synthesized basic mathematical knowledge. Most importantly, Thomas’s Synopsis was explicitly written for the use of Jesuit candidates for the China mission, and describes in detail the minimum level of mathematical, and especially astronomical, knowledge and skills that were expected from them. Despite its two issues and its well-targeted didactic program, the book’s reception—which spans a period from 1685 until at least 1756, when there is evidence that it was still being recommended—was actually quite limited; this reception can mainly be gauged from the twenty-six extant copies, and some references in auction catalogues. These data reveal a restricted geographic spread, with some notable exceptions, including some copies which made it to South America. Soon after its appearance, the Synopsis found a secondary use outside the context of the Jesuit mission to China as a textbook of mathematics. It later enjoyed a reception as a ‘collector’s item’, although it had no further scholarly impact.

Vocabulaire Hindernissen bij Wiskunde

This study is part of a research project that investigates what problems pupils may have with language used in mathematical textbooks. Based on earlier research, the expectation is that minority pupils will have problems on micro and mesolevels of texts. The focus of this article is on the microlevel of texts. The mathematical textbook that is used for analysis is based on the ideas of Realistic Mathematics Education. In this view of education, mathematics must be connected to reality, so mathematical problems are presented in a practical context. First, the mathematical textbook was analysed on vocabulary. On the basis of this analysis of the textbook, it can be safely concluded that Realistic Mathematics Education makes strong demands on the vocabulary abilities of pupils. Mathematical texts feature many words that are not in the list of most elementary Dutch words and many of these words are difficult and have a low frequency. Second, the vocabulary of the textbook was compared to the vocabulary employed by the teacher during classroom discussions. From these analyses the conclusion may be drawn that every day words from the mathematical textbook that were used to describe the contexts were hardly used by the teacher in classroom interaction. However, the teacher did use many words from the category of mathematical words. This means that, while infrequent every day words used to describe the contexts may cause problems for pupils, the verbally presented realistic contexts are hardly discussed in class. Overall, this micro analysis reveals that the words used in mathematical texts may well pose problems for pupils, especially the infrequent every day words used to describe the realistic contexts, Further research will focus on whether minority pupils have problems with the way mathematical exercises are presented and, if so, what these problems are.

Algorithms and Turing Machines

What Precisely is an algorithm, or a Turing machine, or a universal Turing machine? Why should these concepts be so central to the modern view of what could constitute a ‘thinking device’? Are there any absolute limitations to what an algorithm could in principle achieve? In order to address these questions adequately, we shall need to examine the idea of an algorithm and of Turing machines in some detail. In the various discussions which follow, I shall sometimes need to refer to mathematical expressions. I appreciate that some readers may be put off by such things, or perhaps find them intimidating. If you are such a reader, I ask your indulgence, and recommend that you follow the advice I have given in my ‘Note to the reader’ on p. viii! The arguments given here do not require mathematical knowledge beyond that of elementary school, but to follow them in detail, some serious thought would be required. In fact, most of the descriptions are quite explicit, and a good understanding can be obtained by following the details. But much can also be gained even if one simply skims over the arguments in order to obtain merely their flavour. If, on the other hand, you are an expert, I again ask your indulgence. I suspect that it may still be worth your while to look through what I have to say, and there may indeed be a thing or two to catch your interest. The word ‘algorithm’ comes from the name of the ninth century Persian mathematician Abu Ja’far Mohammed ibn Mûsâ alKhowârizm who wrote an influential mathematical textbook, in about 825 AD, entitled ‘Kitab al-jabr wa’l-muqabala’. The way that the name ‘algorithm’ has now come to be spelt, rather than the earlier and more accurate ‘algorism’, seems to have been due to an association with the word ‘arithmetic’. (It is noteworthy, also, that the word ‘algebra’ comes from the Arabic ‘al-jabr’ appearing in the title of his book.) Instances of algorithms were, however, known very much earlier than al-Khowârizm’s book.