Abstract
In the early eighteenth century, techniques of computation for decimal fractions, especially non-terminating repeating decimals, were being developed amid a debate over their utility compared to common fractions for merchants and tradesmen facing complicated metrological and currency systems. That is, we are almost exclusively concerned with procedures for manipulating decimal representations of rational numbers; irrationals get only a brief mention. The most comprehensive exploration of these arithmetical techniques was undertaken by John Marsh in his Decimal Arithmetic Made Perfect of 1742. In this paper we explain Marsh’s achievement, locate his contribution in the context of earlier work, and consider his audience and its implications as evidence for the depth and spread of interest in mathematics in England
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Notes
- 1.
Marsh’s book has been digitized and various exemplars are available online, for example at the Internet Archive: https://archive.org/details/decimalarithmet00marsgoog (as of 10-Jan-2018); print on demand copies are also obtainable.
- 2.
Marsh has been credited with being the first to use this notation, on the strength of Cajori, who claims he was “perhaps the earliest writer to use a special notation for their [repeating decimals] designation.” (Cajori 1929) However, among Marsh’s sources, Malcolm, and, following him, Wright use the same notation. Malcolm presents a “Scholium”
If the Repetend be twice written down with an &c. after it, this will clearly shew that there is a repetition, and what the Repetend is: But this we may do more conveniently, by setting a Point over the first and last Figure of the Repetend once written down: Thus instead of .033 &c. write \(.0\dot {3}\); And for .4376376 &c. write \(.4\dot {3}7\dot {6}\); and so of others. (Malcolm 1730)
Wright adds a useful piece of terminology, which sadly did not catch on:
Note also, we shall call this the Repeating Point, to distinguish it from the Decimal Point, which stands always before it. (Wright 1734)
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Melville, D.J. (2018). John Marsh and the Curious World of Decimal Arithmetic. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90983-7_2
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