Every one knew how laborious the usual Method is of attaining to Arts and Sciences; whereas by his Contrivance, the most ignorant Person at a reasonable Charge, and with a little bodily Labour, may write Books in Philosophy, Poetry, Politicks, Law, Mathematicks and Theology, without the least Assistance from Genius or Study.

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Gulliver's Travels:
Voyage to Laputa





Laputan Logic

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Laputan Logic*
Fanciful. Preposterous. Absurd.

Posted on Thursday 16 November 2006

During the Second World War, in India, a young Indian girl found herself having to introduce one of her oriental friends to an Englishman who appeared at her home. The problem was that her girlfriend was Japanese and would have been immediately arrested were this to become known. So, she sought to disguise her nationality by telling her English visitor that her friend was Chinese. He was somewhat suspicious and then surprised them both by asking the oriental girl to do something very odd: ‘Count with your fingers! Count to five!' he demanded. The Indian girl was shocked. Was this man out of his mind, she wondered, or was this another manifestation of the English sense of humor? Yet the oriental girl seemed unperturbed, and raised her hand to count out on her fingers, one, two, three, four, five. The man let out a cry of triumph, ‘You see she is not Chinese; she is Japanese! Didn't you see how she did it? She began with her hand open and bent her fingers in one by one. Did you ever see a Chinese do such a thing? Never! The Chinese count like the English. Beginning with the fist closed, opening the fingers out one by one!'

— John D. Barrow, Pi in the Sky, Penguin 1993, p. 26
Frustratingly, the story breaks off at this point and we never do find out what happened to the Japanese girl (the story may also be apocryphal) but it is true that the Japanese do count on their hands differently to the Chinese. Any similarity between the way Chinese and the English count, however, are purely superficial.

For one thing, like the Japanese the Chinese count from one to ten using only one hand. Unlike the Japanese, however, who achieve this by counting back down using the same fingers, the Chinese represent the numbers 1 to 10 using ten separate signs. This allows the system to be used for more than merely keeping track whilst counting, it can also be used for communication as well. Chinese numeric hand gestures are a common sight in Asian marketplaces and are especially useful for specifying quanities and prices in noisy environments (i.e. Asian marketplaces).

The table below shows the signs for the numbers 1 to 10. The basic idea is that each sign is suggestive of the Chinese character that corresponds to that number. The characters for the first three numbers are familiar, just like Roman Numerals turned on their sides, but any similarity with the English system of counting starts to break down at the number four.











Only the number 10 in this table shows the use of two hands which, when compared with the other signs, appears rather inelegant. In reality, in my experience at least, the number 10 is normally represented with a "fingers crossed" gesture on one hand.

The English way of finger counting, on the other hand ;-), hasn't always been as simplistic as it is today. In 725 AD, the famous Anglo-Saxon Benedictine monk, the Venerable Bede, presented in his book Tractatus de computo, vel loquela per gestum digitorum a system for representing the numbers up to 9,999. Bede's system, which had precedents in ancient Rome and was similar to ones in use throughtout the Muslim world until modern times, worked by dividing each hand into two parts: on the left hand, the middle, ring and little fingers were used to represent the numbers 1 to 9 while the thumb and index fingers represented the tens. On the right hand, the thumb and index fingers represented the hundreds while the remaining three fingers represented the thousands. Bede extended his system to even be able to represent numbers up to one million by positioning the hands in different ways upon different parts of the body.

Bede is famous for helping to establish a common chronology for historical events based on Anno Domini (the year of the birth of Jesus Christ) and reconciling important dates within this framework. He was meticulous in preferring to work from primary sources rather than relying upon second-hand information and for doing his own calculations. This was unusual for scholars of his day (and even got him into some heretical bother at one point) and one can imagine that, with all those calculations, his finger reckoning system must have been put through a very thorough test.

Bede's system is presented here in this workcut from Luca Pacioli's most famous work the Summa de Arithmetica (1494). This book, incidentally, also contains the first ever exposition of the method of double-entry book-keeping and so Pacioli, apart from his reputation as a great mathematician, is also considered the patron saint of accountants. In this work, Pacioli added a modern touch by annotating Bede's system with Indian-Arabic Numerals. He also modified it slightly by reversing the arrangement on the right hand so that the thumb and index fingers now represented the thousands. This was, presumably, to make the numbers slightly clearer to read off.

Another scheme for reckoning using the fingers was invented more recently by a Korean school teacher named Sung Jin Pai. This one was designed to enable the use the hands as a sort of abacus to perform addition, substraction, multiplication and even division. It's called Chisan-bop and here's a tutorial.