Thursday, 22 March 2012

Multiplying devices

click to enlarge

Some mathematics-related oddments. I was pleased to find the above nice curio - a pair of circular slide rules, complete with manual and wallet - very affordably in the local Estuary League of Friends. Charity shops are getting increasingly savvy and commercialised these days, to the point where you rarely see anything as obscure and interesting as this.

Both slide rules are made by the Concise Corporation, Japan, and the instruction leaflet says the agent is Takeda Drawing Instrument Mfg. Co. Ltd. The little one, a bit over 8cm in diameter, is a Concise No. 28N, a basic model with D, C, CI, A, K scales on the front and a metric-imperial conversion table on the back. The larger 11cm one, the Concise No. 300, is a full-featured log-log slide rule:

  • K, A, D, C, CI, B, L on the front / LL3, LL2, D, C, S, T1, T2, ST on the back.
  • Especially designed for professionals, this circular slide rule enables you to make various calculations ranging from general multiplication and division to the square root, cube root, trigonometric function and even logarithm and descending series.

I know all this because, surprisingly, they're still manufactured - here's the Concise No. 28N and Concise No. 300 - and available by mail order from the Concise website. The No. 28N costs ¥1,000 (a bit under £8) and the No. 300 costs ¥2,600 (around £20). Concise have diversified, and companion products include various drawing and measuring tools, stationery, and personal kit for the traveller.

There was a kind of synchronicity to this, because I'd just run into a couple of other mathematics-related topics, one of them being the interesting link at Ptak Science Books, Alan Turing--Report Card Teachers' Comments, 1926-1931 (which references Turing’s school reports at the website of the author Alex Bellos). Alex comments that "It’s interesting to see how he changes from an untidy and careless mathematician to a distinguished scholar", and John notes the snarkiness of some of the comments that describe Turing in such terms as "absent-minded":

It may be easy to judge some of the remarks as intemperate, the teachers unable to clearly see the genius-in-the-making who (70 years later) we can so clearly see today. I think the remarks need more careful consideration than that, and that is where they become interesting.

True. There is a standard "teachers are unable to recognise genius" meme, and people who remember their schooldays with hostility latch readily on to this (especially given the regular data points from celebrities who went on to great careers after being written off as the "class clown"). It is, however, a lot more complex than that. I'm not remotely in the intellectual league of Turing, but I can recognise from my own experience the syndrome of erratic achievement in academically good and mathematically-inclined pupils in traditional British education.

In part, the syndrome lies in the system. Good teachers recognise bright students and give them appropriate work to keep them interested; poor ones don't. And I must have run into the latter. I also had a number of reports, at junior school, saying I was "absent-minded", and the description is completely unrecognisable. What I do remember is being intensely bored; in mathematics, I remember with especial loathing what were called "Problems", page after page of identically structured sums that read like primers in capitalism (and sexism):

1. A man buys 10 oranges for 2s/6d, and sells them at 4d each. What profit does he make?
2. A man buys 11 apples for 1s/7d, and sells them at 1½d each. What loss does he make?
39. A man buys 19 pears for 2s/9d, and sells them at 2d each. What profit does he make?
40. A man buys 13 carrots for 2s/0d, and sells them at 2½d each. What profit does he make?

I remember similar sets of what seemed interminable repetitions - continuing the exercise long after (at least for me) the point had been driven home - for long multiplication, and I recall not finishing the sets because I'd start playing with other ways of doing them (I knew from books at home about Russian Peasant Multiplication and what's now called Lattice Multiplication).

That said, there are equally faults of working that bright students can get into spontaneously, and a major one stems from over-confidence: a tendency to "wing it". That is, letting trust in one's being good enough to improvise solutions over-ride the need to consolidate basic knowledge. For example, I don't think I ever properly learned many of the useful trigonometric identities such as sin(2x) and cos(2x) because I trusted in being able to derive them as needed from De Moivre's Theorem cos(x) + i*sin(x) = e^(ix) - despite the extra time needed to do that.

With other faults - "untidiness" - it's really hard to tell where the fault lies. Traditional schooling placed what I think was an unnecessary emphasis on format over content - and it goes against the reality of mathematics to expect a perfectly laid-out solution first time. Again, good systems existed, that allowed sufficient space for rough work that didn't count against the final "fair copy". Handwriting has been always been an issue: for some, poor handwriting is involuntary (a symptom of, for instance, dyspraxia); for others, it's a repairable result of poor initial instruction. Personally, I've always found it very difficult, for no reason I can fathom. That may even have helped steer me away from English and History, both of which subjects I enjoyed and still do, because physics and mathematics required much less writing.

And just as I was thinking about that, along came, via Yahoo! Answers, a relevant online paper: Mathematical justification of some non-traditional methods of multiplication (Y. D. Deshpande, Bulletin of the Marathwada Mathematical Society, Vol. 10, No. 2, December 2009, Pages 8–15). This is an interesting paper on on three unorthodox multiplication methods, explaining why they work: the Urdhva Tiryakbhyam method, the Ganesh method and the Russian peasant multiplication method.

Though the paper calls them "non-traditional", they're perfectly traditional, just methods that pre-date the dominance of the standard long multiplication algorithm. The Ganesh method is a slight variant on the lattice multiplication known across a number of cultures, the Russian Peasant Multiplication goes back to Ancient Egypt, and the Urdhva Tiryakbhyam method - a method from Vedic mathematics - is in the same territory as the merchants' multiplication systems I mentioned in November in the post Tagliente's multiplication by columns.

Although ancient India produced some serious innovation in mathematics, Vedic mathematics is something different: a system presented by the Indian author and scholar Bharati Krishna Tirthaji in the early 20th century. I say "presented" because it seems extremely doubtful that it comes from ancient Hindu sources as the author claimed. It comprises a set of algorithms, presented as fundamental "sutras", for performing rapid calculations on, usually, a one-line basis. I can't really disagree with one main criticism: that it was more relevant to pre-calculator days; and furthermore, the algorithms have the fault of it not being 'transparent' as to how they work. That said, I find Vedic mathematics interesting for that very reason; as with the previously-mentioned Tagliente method and similar, untangling the algorithm can make a diverting puzzle.

There are a number of websites and books about the topic. They vary between the adulatory (see Vedic Mathematics Academy) - via enthusiastic but straight expositions (see the Google Books preview of Vedic Mathematics, 1992) to the distinctly hostile. For example, Vedic Mathematics - 'Vedic' or 'Mathematics': a fuzzy & neutrosophic analysis (WB Vasantha Kandasamy, Florentin Smarandache, 2006) frames a technically accurate description of the system with the assertion that its popularity in India reflects its appeal to those with a right-wing caste-driven fundamentalist Hindu agenda. It reports an impressive body of signatories to a statement to the effect that "it is largely made up of tricks to do some elementary arithmetic computations. Its value is at best recreational and its pedagogical use limited". On the other hand, "the fuzzy and neutrosophic" method the book uses to analyse opinions on Vedic mathematics looks a majorly unproven dialectical method (and from the description here - Neutrosophy - my immediate feeling is that it could even be pseudomathematics).

- Ray


  1. Though I've restricted myself to finding examples of slide rules I owned at school or as an undergraduate, and I have my different circular slide rule for that (plus another given to me) I find myself strangely drawn to the idea of buying one that is still in production ... you may have cost me twenty squids :-)