Addometer

Ptak Science Books just featured a very nice post - Beautiful Techno-Art & Invention: Circular Adding Machines, 1850-1900 - featuring technical illustrations from US patents for various rotary adding machines. I haven't looked into the background, but patents aren't proof of technical feasibility, and quite possibly many of these devices didn't make it to the stage of practical use. However the W Lang adding machine (pictured above) certainly was feasible, because I have one of exactly this type.

This British-sold "Addometer" works on a straightforward principle; you use a metal stylus to turn a dial to add and subtract, and overflow cascades to the adjacent dial. A slide lever clears everything to zero. (see YouTube demo - I can't show you the workings, as the casing is riveted shut, but the Vintage Calculators Web Museum Addometer page has pictures). These multi-spindle adders were a popular, robust and successful design; they recur in 19th / early 20th century patents, with no clear attribution trail, and continued in use right through to the end of the mechanical calculator era.

This particular one - dating I think from the 1920s-30s - is both technologically and conceptually obsolete: it adds Sterling currency - pounds, shillings, pence, and farthings.  As the front logo says, it's US-made (by the Reliable Typewriter and Adding Machine Co., Chicago, Illinois) but the brass logo on the edge has been milled out, and the device rebadged to "Taylor's 74, Chancery Lane, London": a firm that historically sold typewriters and other office equipment.

I'm mildly puzzled that I don't remember any such devices from my childhood. My mother worked in various offices as a secretary, and took me there sometimes when I was on school holidays (this would be very early 1960s). I don't remember seeing anything but typewriters and duplicating machines. Were these calculating devices so expensive, or so cumbersome, that they weren't widely used?

- Ray

Mindbender 5

Another puzzle that caught my eye from the Western Morning News "Mindbender" series.

See solution.

- Ray

Life in Life

Via MetaFilter, Life in Life:a video of a remarkable piece of coding - OTCAMP - that runs a Conway's Game of Life simulation inside Conway's Game of Life.

It strongly reminds me of the theme of Daniel F Galouye's 1964 SF novel Counterfeit World (a.k.a. Simulacron-3), which had a theme of nested simulations. In Counterfeit World, a virtual reality city has been set up for market research; but after a death and a mysterious disappearance, one of the researchers twigs that his own reality (i.e. ours) may itself be inside another simulation.

Counterfeit World was filmed in 1973 as Fassbinder's TV mini-series Welt am Draht (World on a Wire), which is on YouTube, though without subtitles (part 1 / part 2), and as the 1999 The Thirteenth Floor. The latter is a much under-rated film, mostly through unfortunate timing; appearing in the same year, it was eclipsed by The Matrix.  In 2011, Janus Films released a digitally restored and subtitled version of World on a Wire.


US trailer for World on a Wire, Janus Film, 2011

- Ray

Elegant route

I dabble in mathematics at a puzzle-solving level. Although I took it as a first-year subject at university, I never had that extra spark of aptitude that makes a mathematician. However, very very rarely I find problems that briefly touch what I imagine real mathematics to be.

Recently I encountered this problem:

Solve (x+2)(x+4)(x+6)(x+8) = 9

Now the temptation is to charge straight at it, and I did, first expanding terms of the left hand side and rearranging to a standard quartic ...

(x+2)(x+4)(x+6)(x+8) = 9
(x^2 + 6x + 8)(x + 6)(x + 8) = 9
(x^3 + 12x^2 + 44x + 48)(x + 8) = 9
x^4 + 20x^3 + 140x^2 + 400x + 384 = 9
x^4 + 20x^3 + 140x^2 + 400x + 375 = 0

This proved to be horrible. A lot of slog through possible small-integer zeros found x = -5 to be a solution. This enabled a long division by (x+5) to give this factorisation:

(x + 5)(x^3 + 15x^2 + 65x + 75) = 0

But the remaining cubic seemed impossible to tackle without using a maths package.

But then I noticed the symmetry of (x+2)(x+4)(x+6)(x+8) around x+5, and I tried this substitution:

Let p = x + 5 ... so the problem becomes:

(p - 3)(p - 1)(p + 1)(p + 3) = 9

This leads nicely to a reverse application of the standard "difference of two squares" factorisation, and a greatly simplified result.

(p - 3)(p - 1)(p + 1)(p + 3) = 9
(p - 3)(p + 3)(p - 1)(p + 1) = 9 ... rearrange terms
(p^2 - 9)(p^2 - 1) = 9 ... expand to a pair of "difference of two squares"
p^4 - 10p^2 + 9 = 9 ... expand
p^4 - 10p^2 = 0 ... simplify

This is easy to solve for p:

p^4 - 10p^2 = 0
p^2(p^2 - 10) = 0
So p^2 = 0, or p^2 = 10
So p = 0, or p = √10 or p = -√10

And finally we can reverse the substitution: x = p - 5

So x = -5 or x = √10 - 5 or x = -√10 - 5

It's only a puzzle, but the elegance and clarity of the second solution route compared to the murky dead end of the routine one is oddly satisfying.

(The quartic has only three zeros, all real, by the way, because its central maximum at x = -5 touches the x-axis).

- Ray

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

Shellular automata

We just had our bathroom renovated, and this rather nice seashell is hanging on the light pull-cord. I got it a while back at B&Q as a curiosity, but this is the use they were selling it for. I forget the country of origin - somewhere Indo-Pacific - but I recall the label said these particular B&Q ones were produced on a sustainable basis; the molluscs are used for food, and the empty shells sold as decorative objects or collector's pieces.

The particularly neat thing about this one is the pattern: a classic example of pattern-forming processes that have been successfully analysed mathematically. The mollusc, when it's alive, has a mantle whose edge contains cells that deposit pigment on the shell lip as it grows. These cells interact with their neighbours - they can switch on and switch off each others' pigment-making according to well-defined conditions - so the whole band of cells has a dynamically-changing linear pattern of pigment production; and as the shell grows, a record of that pattern is left behind.

The Algorithmic Beauty of Sea Shells

One of the classic expositions of the theory on this is Hans Meinhardt's The Algorithmic Beauty of Sea Shells (Springer, 4th ed. 2009, ISBN  978-3-540-92141-7). Meinhardt analyses the patterns in terms of hormone diffusion, successfully simulating the patterns on a range of shells. As part of Springer's Virtual Laboratory series, it comes with companion software for playing with the simulations. If the maths is not of substantial interest to you, it's still a compendium of beautifully -photographed shells - although at nearly £40 it's rather expensive as a coffee-table book. Springer have an online preview: see the intro page and reader.

Whatever 'computational engine' drives the pattern changes on the shells is effectively a close analogue equivalent of the digital algorithmic device called a one-dimensional cellular automaton (you can play with an applet here). The outputs are very similar in appearance, and one celebrated example, Cymbiola innexa, displays Sierpinski triangles, a classic fractal pattern produced by many 1D cellular automata.

The rather peculiar aspect is - what function do these patterns serve? In many species of mollusc, the patterns aren't visible during life, as they're covered by a dull-coloured protective "periostracum", so camouflage or identification (the usual reasons for animal patterns) don't seem to apply. Do they have a function we don't yet know? Are they fossils of a past function that's better served by the tough periostracum? Or are they "spandrels" (the term borrowed from architecture by Gould and Lewontin for non-adaptive by-products of other adaptively evolved functions)?

- Ray

Mindbender 4

I've been following for a few weeks the daily Mindbender puzzles in the Western Morning News, and documenting my solutions for some: not through any claim to brilliance, but because they often present interesting angles on techniques for solving mathematical problems.

This one proved a classic example for a very powerful technique that's not well-known outside the maths/sciences circuit.

See solution.

- Ray

Fitting designs (continued)

Further to Fitting designs - which looked at the history of the classic puzzle of making an object that will fit three holes of different shapes: square, circular, and triangular - Emily from the largely mathematics/nature blog Ephemeral Curios commented:

These might also be the inspiration for Douglas Hofstadter's "trip-lets"-- the blocks on the cover of Godel, Escher, Bach that cast three different letter shadows.

Thanks! Yes, the GEB "trip-let" (left) is identical in concept to the object with three orthogonal letter shapes objects spelling "PSM" in Popular Science Monthly for July 1927 (see Arthur L Smith's A Square Peg in a Round Hole).

But I'm not clear if (as is quite possible) it was independently conceived; Hofstadter wrote in the intro to GEB:

The trip-let idea came to me in a flash one evening as I was trying to think how best to symbolize the unity of Gödel, Escher, and Bach by somehow fusing their names in a striking design. The two trip-lets shown on the cover were designed and made by me, using mainly a band saw, with an end mill for the holes; they are redwood, and are just under 4 inches on a side.

Does "idea came to me" mean the whole trip-let concept, or just the idea of using it for the cover?

There are plenty of interesting spin-offs. Googling found a nice page by Humberto José Bortolossi of the Mathematics Department, Universidade Federal Fluminense - Triplets - which has rotatable models, along with an intriguing reference to the generalisation of the idea. Objects can exist with an arbitrary number of specified shadows when illuminated from different directions. This is not merely theoretical, but has actually been applied to the creation of the "digital sundial" designed by Scharstein, Scharstein and Krotz-Vogel: see U.S. patent 5,590,093 and Digital Sundials International.

Another good page is The magic of trip-lets at mariano tomatis blog, which shows the design (not trivial to achieve) of an "EMC" triplet for the Eseential Magic Conference. It also raises the possibility of trip-let type objects in the past - that is, objects designed to look different from different directions - via the example of a 17th century carved crucifix by the Franciscan monk Fra Innocenzo da Petralia, deemed miraculous for its ability to show different expressions on the face of Christ (suffering, dying and dead) according to viewing angle. This looks to me an equivalent of the "Noh Mask Effect": the ability of masked actors to convey expression change in Noh drama by tilting the head to give different viewpoints of their rigid masks. See Michael J Lyons, Noh Masks & Facial Expression Perception, which has several research papers. The concept seems to be pretty widespread: see, for example, Perceiving faces of Buddha statues, Same miraculous statue; different expressions, and How Moving Light Changes Expression on Face of Small Statue.

- Ray

Fitting designs

Ptak Science Books just had an interesting quick post - Tech-Quiz 6: A Square Peg in a Round Hole? - concerning an 1885 US patent by one Charles M Dewey of Jersey City, New Jersey, for "a new and Improved Puzzle", which comprises a solution to the puzzle of making an object that will fit three holes of different shapes: square, circular, and cross-shaped.

Here's the patent in full: Patent number: 328766, Filing date: Jan 13, 1885, Issue date: Oct 20, 1885.

The "Improved" appears to be an allusion to an older puzzle of similar format. As Dewey puts it: "I am aware that it is old to make a puzzle from a piece of card-board by forming three holes therein, one of which is round, another square, and a third triangular. I do not lay claim to a puzzle thus made".

I first ran into the square/ round / triangular peg puzzle in my childhood in a rather nice book that somehow disappeared during family movings, Robert Morrison Abraham's Tricks and amusements: with coins, cards, string, paper and matches (see page 132). The book actually dates from the 1930s, when it was originally published as Winter Nights' Entertainments: a book of pastimes for everybody, but we had one of a number of later Dover Books paperback imprints.

The idea is, as Dewey says, quite old: for example, it turns up as "The Triple Accommodation" on page 707 of the 1880 The Boy's Own Book by William Clarke (Internet Archive boysownbookbywc00clargoog), and on page 26 as puzzle 16 of the snappily titled Rational amusement for winter evenings, or, A collection of above 200 curious and interesting puzzles and paradoxes relating to arithmetic, geometry, geography, &c: with their solutions, and four plates, designed chiefly for young persons by "John Jackson (private teacher of the mathematics)".

The toy appears in the 1811 catalogue of scientific instrument makers W & S Jones:

The mathematical paradox, a piece of wood of one figure, fits exactly, and passes through a triangular, square, and a circular hole. - £0 2s 6d.
- page 7, An analysis of the principles of natural philosophy, Matthew Young (Bishop of Clonfert), 1811

And earlier still, I found this 1759 variant:

Queſtion 169, by Mr. C. Pagister of Greenland-Dock.
I have a regular Piece of Wood which will fill up a round Hole, a ſquare Hole, and an oval Hole : Query what is the Shape?
N.B. We think proper to propoſe this Question for the Amuſement of our young Philomaths, tho' it is not a new one.
- page 702, Miscellaneous Correspondence, Volume 2, 1759.

The solution to the basic version is, by now, well-known. You start with a cylinder of unit height and unit diameter, then mark a diameter AB on one end, and a second diameter CD at right angles to this on the other end. Then you cut off two pieces along the planes ABD and ABC. What's left is a plug with circular, triangular and square cross-sections from three orthogonal directions.

There are actually an infinite number of solutions. This Harvard Mathematics Department page - The cork plug -  has nice animations of two: one the planar cut just described, and the other one with a continuously curved surface (the "cork plug" is Martin Gardner's name for the object: Gardner, M. "The Cork Plug." Ch. 5 in The Second Scientific American Book of Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 52-59, 1961).

I don't know whether Dewey ever made anything from his patent for the Improved Version. The puzzle in variants has cropped up various times since. For example, Popular Science Monthly for July 1927 has an article by Arthur L Smith, A Square Peg in a Round Hole, featuring a puzzle similar to Dewey's and a more complex configuration spelling "PSM", and poses the problem of making an object to fit triangular, square, and T-shaped holes.

The solution is here in the August 1927 issue.

Following the US patent trail finds many more variants. I don't really know how the patent system works, because in 2002, Joanne Godin for Ideas That Matter got a patent for the probably centuries-old square / circle / triangle format (USD482080).  A nice later variant is the maze toy by Warner A Davis (USD562915) where the plug needs to be manipulated through holes in inaccessible internal partitions of a pyramid. This in turn cites earlier versions: an "educational toy" by Raymond J Scherf (US2542948) that has T-shaped, circular and notched-square holes; James Lloyd Taylor's version with "a space-aged theme" (US3804414), where the plug has to fit round, Saturn-shaped, and star-shaped holes partitioning a box representing space; and Hubert Andrew Johnson Jr's "ornamental design for a child's toy" (USD380240) where the three shapes give access to a hollow cube (directly relating the puzzle to the three orthogonal cross-sections of the object).

I'm sure the format will continue to be developed.

Addendum: see update - Fitting designs (continued)

- Ray

Mindbender 2

As mentioned previously, I've been getting into the Western Morning News's Mindbender daily mathematical puzzles. Here's another one, which raises interesting points regarding my earlier thoughts on solution methods. Have a go, then go to this backdated post (SPOILER: solution).

- Ray

Mindbender 1

 

I regularly do the Mindbender mathematical puzzles in the Western Morning News. Usually they involve fairly simple algebra, but sometimes they need number theory. This one proved even more difficult than usual. The problem comes down to finding a number that's one more than 17 times the sum of its digits. I did solve it eventually, but by a lot of intuitive jumps. Can anyone see a straightforward method? Have a go, and then go here (spoiler with solution) to compare.

- Ray

Tagliente's multiplication by columns

image: Tomash Library

An interesting challenge via the recent Ptak Science Books post An Episode in the History of Multiplication: reconstructing a 500-year-old algorithm for long multiplication.

John mentions the 16th century instructional Libro dabaco, a pocket-sized book on practical mathematics for merchants, co-written by Girolamo and Giovanni Tagliente (since only one was the nominal author, I'll use the singular Tagliente for convenience). The book is described in more detail in Paul F Gehl's online book Humanism For Sale ("Making and Marketing Schoolbooks in Italy, 1450-1650") in the section 6.03 Commercial skills.

The problem lies in the book's multiplication "per colonna" ("by columns") system, of which there are some nice images here as part of the The Erwin Tomash Library. Both John and Paul comment that Tagliente doesn't explain the algorithm, because his text says the diagrams are self-explanatory. They're not!

image: Tomash Library

This example evaluates 9876 x 6789 = 67048164. As John says, you can see where the digits in the individual rows come from: Tagliente multiplies pairs of digits in the two multipliers. But how do those rows produce the final product?

It turns out to be clear when the multiplications are assigned their correct column values: the result is just their sum.

This leaves the question of why the digits are multiplied in that order to produce the rows.

The full method turns out to be very elegant. Modern long multiplication involves carrying at each stage of producing the partial sums. For example 1234 x 5678 requires four multiplications (each with four sub-multiplications), all with carries when done by hand:

1234 x 8 = 9872
1234 x 70 = 86380
1234 x 600 = 740400
1234 x 5000 = 6170000
then you sum the results to get 7006652.

As with ordinary long multiplication, Tagliente's method involves multiplying all permutations of digit pairs taken from the two multipliers - but it does it in such an order that there are no carries within each row. Each single multiplication produces a result in the range 00-99 (i.e. occupying two decimal places) and each pass of the algorithm produces a row of such results, each shifted from the previous by two decimal places, so that they can just be written adjacently.

Applying this to 1234 x 5678, the order of multiplication could go:

1234 x
5678

Row 1: 1x5, 2x6 3x7, 4x8 (vertically aligned)
Row 2: 1x6, 2x7, 3x8 (offset 1 right)
Row 3: 1x7, 2x8 (offset 2 right)
Row 4: 1x8 (offset 3 right)
Row 5: 2x5, 3x6, 4x7 (offset 1 left)
Row 6: 3x5, 4x6 (offset 2 left)
Row 7: 4x5 (offset 3 left)

The results are:

(05)(12)(21)(32)
(06)(14)(24)
(07)(16)
(08)
(10)(18)(28)
(15)(24)
(20)

It looks like this when laid out in columns to show the place values:

I haven't tried it with multiplications between figures with different number of digits, but as long as the place values are kept correcly, the permutation routine would work fine. The order of permutation doesn't really matter as long as all the permutations are covered, and Tagliente's own examples show several different routines (see Addendum 3).

I'm sure Tagliente arranged his calculations according to this column layout, because the final sum doesn't work otherwise. But I can only guess that the woodcut creator for Libro dabaco was more interested in artistic design than mathematical correctness, and broke the layout, obscuring the method. This is further suggested by the errors in the host-and-chalice example for 927 x 789, which gets the correct answer of 731403 despite a number of mistakes in the sub-results (such as having 2x7=16).

Multiplication "per coppa" - image: Tomash Library

The procedure in the above example is meant to be structured as:

Row 1: 9x8, 2x9
Row 2: 9x7, 2x8, 7x9
Row 3: 2x7, 7x8
Row 4: 7x7
Row 5: 9x9

Results:
(72)(18)
(63)(16)(63)
(14)(56)
(49)
(81)

Corrected for place values:
72180 +
631663
14560
4900
8100
------------
731403

As I said, the order of permutations seems flexible. This third example goes through them in yet another order, and, unlike the other two, is rather more faithful to the column structure.

image: Tomash Library

Addendum:I just found a reference to the above methods in the 1908 bibliographic catalogue Rara Arithmetica (Internet Archive ID raraarithmeticac00smituoft). Again without explanation of the algorithm, its authors comment on the 'diamond' and 'triangle' plates above, in the 1541 Tagliente:

For two curious forms of multiplication see Fig. 64. Such arrangements of the work in multiplication were quite common, particularly in the early Spanish and Italian arithmetics of the first half of the sixteenth century. That they should have found place in a popular mercantile treatise is, however, rather surprising.
- page 116, Rara Arithmetica; a catalogue of the arithmetics written before the year MDCI, with description of those in the library of George Arthur Plimpton, of New York, Smith, David Eugene, and Plimpton, George Arthur, pub. Boston Ginn, 1908.

Addendum 2 (upgraded from comments): Leon has commented that:

This looks a lot like Lattice Multiplication.

Thanks: yes indeed. I didn't spot this, but it's algorithmically very similar. Lattice multiplication handles the permutations of the multiplier digits in the rather more foolproof way of laying them out along the axes of a rectangular grid. It creates exactly the same array of two-digit numbers as Tagliente's method, but at an angle: you sum them down a diagonal (from top right to bottom left) rather than vertically. For comparison:

Image from interactive app here

The Tagliente method could be described as a hybrid between the lattice method and regular long multiplication.

Addendum 3: And Thony Christie, at the interesting-looking blog The Renaissance Mathematicus, also solved it, pointing out that Taliente's method is, in effect, a sloppily-structured version of an algorithm called the Diamond (see Multiplying the Renaissance way). This led me to a source: Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (ed. Ivor Grattan-Guinness, 2003) of which pages 203-206, a section by the abacco arithmetic expert Warren Van Egmond, have clear descriptions of such algorithms. It seems they derived from methods originally for the abacus, but adapted to the needs of pen-and-paper calculation. Forms included multiplication in croce (in the form of a cross) which refers to the set patterns for multiplying the digits, as in the Tagliente method; and several forms laid out per campana (a bell shape), per coppa (in the shape of a cup), or as a diamond or circle (see diagram, page 205). Italian mercantile calculation of this period was a kind of 'Burgess Shale' of long multiplication, when diverse forms initially flourished and eventually resolved into the modern form. WVE notes that unlike many mathematical operations, these were not imported from Hindu or Arab mathematics, but "developed independently by the abaccists themselves, with much trial and error".

Victor J. Katz's 2000 Using History to Teach Mathematics: an International Perspective (page 15) has an example of the per coppa calculation.

- Ray (and thanks to John Ptak for raising the topic).

"You'll like this, it has buttons" #2

click to enlarge

David Mamet's play Edmond has as its central premise the aphorism that "Every fear hides a wish". Maybe this was behind my initial distaste of the pink Casio FX-83GT Plus, as I just decided to embrace my pink side and get one.

I confess to being a slight calculator geek, but beyond the curiosity value, this turns out to be an extremely nice calculator. As you see, it's lightweight and slimline, the battery augmented by a solar panel, and comes with a robust slip-on cover. It also features an interestingly different but highly intuitive logic.  I'm most used to the "immediate execution" style of calculator:

input ... display
7 ... 7
x ... 7
6 ... 6
x ... 42
5 ... 5
x ... 210
4 ... 4
= ... 840

The FX-83GT Plus, however, uses what Casio calls "Natural-V.P.A.M."(Visually Perfect Algebraic Method) which shows you the expression as you build it, and only returns the result when you press =.

input ... display
7 ... 7
x ... 7x
6 ... 7x6
x ... 7x6x
5 ... 7x6x5
x ... 7x6x5x
4 ... 7x6x5x4
= ... 7x6x5x4 ... 840

( ... (
7 ... (7
+ ... (7+
6 ... (7+6
) ... (7+6)
x ... (7+6)x
( ... (7+6)x(
5 ... (7+6)x(5
+ ... (7+6)x(5+
4 ... (7+6)x(5+4
) ... (7+6)x(5+4)
= ... (7+6)x(5+4) ... 117

I'm probably way behind the times in not having encountered this style of calculator logic before, but I like it a lot.

The FX-83GT Plus - see the Casio page - is aimed at the educational market and permitted for use in all UK and Irish school exams, and while it's not a graphing calculator, it does all you'd want as a basic scientific calculator for pure and applied maths up to O Level. It covers all the usual trig and transcendental functions plus stats, with a few nice bonus features such as prime factorization, exact storage of fractions and recurring decimals, assignable variables, polar/rectangular coordinate conversion, table generation from a function, and a verify mode for testing equalities or inequalities.

It has mathematical limitations - it won't do calculus, complex numbers or equation solving (for that you'd need the FX991ES Plus) - but altogether it's extremely good value. For my desktop use, anything everyday where I wouldn't use a PC maths package, I've now promoted it to be my regular calculator.

- Ray

The Pinker Calculator

A CALCLIST reference to the Greener Calculator (nothing to do with better colour or better environmental credentials) reminded me of this photo I took recently of this pair of Casio FX-83GT Plus calculators. Gender stereotypes are still with us, it seems. If I needed one, I think I'd get the pink out of sheer perversity; and it'd be far more findable in my office, which is full of gloomy black hardware. I already have a very nice pink geometry set, one of a number of Helix maths sets that come in pink and blue gendered versions.

Addendum: I also feel that JML Dryer Balls are gendered. Whose idea was it to have a pink ball with rounded protrusions, and a blue one with angular? But as described in QI - Series G - Girls And Boys - the pink/blue female/male convention is very recent - and in fact reversed - historically.

Addendum 2: I find that the pink Casio FX-83GT Plus isn't entirely unprecedented. Texas Instruments have done a dark pink special edition of their TI-84 Graphing Calculator, and there's also a Canon pink scientific calculator.  TI do scientific calculators in various nice colours, such as the blue-cyan TI-34 MultiView. They come in lime green too, such as this Canon one. And there's a serious side beyond aesthetics: this Sci-Plus Scientific Calculator range for visually impaired users has casing in colours including ruby red, blueberry, bumblebee yellow and lime green to improve keyboard visibility.

On balance, knowing that pink scientific calculators are part of a wider continuum of colours makes me less bothered about the stereotyping angle. Anything that makes mathematical tools more accessible - especially at school and student level - can't be bad.

- Ray

In praise of The Mathenauts

I just re-read Norman Kagan's 1964 short story The Mathenauts, which I first encountered in a secondhand Judith Merrill's 1965 10th Annual SF anthology. At 11-ish I didn't remotely understand it; only the sheer strangeness came across. But it gets better on each return visit.

The Mathenauts is set in a near-future where "Brill-Cohen flight" has been discovered: the ability to take a ship into the raw mathematical space underlying reality. Ships, which look like a radio minus the casing, are crewed by eccentric high-flyer mathematicians, but their internal reality is held together by a "psychic ecology" of students with more mundane mindsets.

The ship, the Albrecht Dold, was a twelve-googol scout that Ed Goldwasser and I'd picked up cheap from the NYU Courant Institute. She wasn't the Princeton IAS Von-Neumann, with googolplex coils and a chapter of the DAR, and she wasn't one of those new toys you've been seeing for a rich man and his grandmother. Her coils were DNA molecules, and the psychosomatics were straight from the Brill Institute at Harvard. A sweet ship. For psychic ecology we'd gotten a bunch of kids from the Bronx College of the New York City University, commonsense types - business majors, engineers, pre-meds.

Jimmy, the mathematician narrator, tells how there is a horrific in-flight accident when the "isomorphomechanism" (that keeps the internal reality stable) fails, affecting another member of the crew.

Instrument racks and chairs and books shrank and ballooned and twisted, and floor and ceiling vibrated with my breath. It was horrible. Ted Anderson was hanging in front of the immy, the isomorphomechanism, but he was in no shape to do anything. In fact, he was in no shape at all. His body was pulsing and shaking, so his hands were too big or too small to manipulate the controls, or his eyes shrank or blossomed.

Jimmy repairs the fault, but Anderson, occupying the same space as the "immy", is rejected by the "commonsense circuits" and disappears. After an unsuccessful search of a ship and mathematical discussion of where Ted might have gone, there's a further scare when the ship's psychic ecology - students in a facsimile of a New York streetcar - breaks down.

The walls were firm, the straw seats scratchy and uncomfortable. The projectors showed we were just entering the 72nd Street stop. How real, how comforting! I slid the door open to rejoin Johnny and Ed. The subway riders saw me slip into freefall, and glimpsed the emptiness of vector space. Hell broke loose! The far side of the car bulged inward, the glass smashing and the metal groaning. The CUNYs had no compensation training!

Johnny Pearl, the ship's "psychist", restores the ecology by singing a college anthem, and the crew finish their search and begin the tests the voyage is intended for. At that point, a ghostly Ted Anderson reappears and reveals an uncomfortable truth: that the raw mathematical space is the real universe. He gives Jimmy a glimpse of the creatures that inhabit it.

— I saw a set bubbling and whirling, then take purpose and structure to itself and become a group, generate a second-unity element, mount itself and become a group, generate a second unity element, mount itself and become a field, ringed by rings. Near it, a mature field, shot through with ideals, threw off a splitting field in a passion of growth, and became complex.
— I saw the life of the matrices; the young ones sporting, adding and multiplying by a constant, the mature ones mating by composition: male and female make male, female and male make female — sex through anticommutivity! I saw them grow old, meeting false identities and loosing rows and columns into nullity.
— I saw a race of vectors, losing their universe to a newer race of tensors that conquered and humbled them.

Reality as humans know it, Anderson explains, is the creation of a "Great Race" in this mathematical universe, who lost their powers but left mathematics as a "seed" by which humans might regain the ability to inhabit that reality. He then disappears permanently, leaving his notebook.

The surviving characters go on to live their lives, dealing with that revelation in different ways: one marries and has 15 children; one gets religion and writes a book about Ted's views; and Jimmy, the narrator, concentrates on the business side of marketing Ted's ideas, multidimensional products that make a paradise of Earth.

Me, I'll stick to the Earth. The "real" planet is a garden spot now, and the girls are very lovely.

Ted Anderson was recorded lost in topological space. He wasn't the first, and he was far from the last. Twiddles circuits have burned out, DaughtAmsRevs have gone mad, and no doubt there have been some believers who have sought out the Great Race.

The story, at one level, is a pastiche. Kagan wrote it when he was a mathematics student, and it abounds in punning use of mathematical terms and its characters' hardboiled mathematical exclamations ("Great Gauss!", "Holy Halmos!" etc). For that reason, I guess, it's often classed as humorous SF (as in the 1982 anthology Laughing space: funny science fiction). Its extensive mathematical content, and its excellent portrayal of the ethos of mathematicians, also accounts for its presence in Rudy Rucker's 1987 niche anthology Mathenauts: tales of mathematical wonder.

Nevertheless, I think it's an altogether better story than that, as an examples of "conceptual breakthrough" story (see PK Dick, Ubik and conceptual breakthrough and Breaking out of the game) that, despite the pastiche and generally positive outcome, leaves the reader with a continuing sense of unease that the assumptions of reality have been revealed hollow. It tackles the still-topical questions of Platonism and Neoplatonism: the idea that reality may have a mathematical structure, since mathematics describes it so well. An example of such theories is Antony Garrett Lisi's Exceptionally Simple Theory of Everything, which proposed that elementary particles correspond to the symmetries of a vast mathematical group called E₈. All that aside, though, the strange and bold vision in The Mathenauts still makes it as fresh reading as when I first encountered it. It's a pity it's not online anywhere, but secondhand anthologies containing it aren't too hard to find.

Norman Kagan wrote several other mathematical stories, notably the 1964 Four Brands of Impossible, but didn't go on to an SF writing career.  According to the blurb in Rudy Rucker's 1987 Mathenauts anthology:

What ever became of Norman Kagan anyway? He left math for film and still lives in Manhattan. He's written a number of books on cinema, and is currently involved in putting together a TV science news magazine to be called "Spacetime Continuum News." When I pressed having once written a mathematical mystery story called The Venn Data Vendetta — but he lost the only copy.

I don't know what he's doing now (or even if he's still alive), but I assume these books - on the cinema of Stanley Kubrick, Oliver Stone, Robert Zemeckis and Roibert Altman- are by the same Norman Kagan.

Addendum: a personal note, via discussion with Felix Grant. The story is particularly memorable for me as one of the first SF stories I read. I got into the genre via my great-uncle Dennis, a nice autodidact polymath among my Wiltshire relatives who I regret not fully appreciating at the time - but I was only 11, so I guess it's excusable - and who gave me heaps of secondhand SF Book Club editions (including the brilliant The Hole in the Zero) and copies of Analog magazine. On those grounds, this post is dedicated to my late and excellent Uncle Den.

- Ray

Kells surprise!

Upgraded from out-takes: Researcher uncovers secrets of Kells 'angels'. This PhysOrg piece summarises Cisne J L, 2009, "Stereoscopic comparison as the long-lost secret to microscopically detailed illumination like the Book of Kells’" Perception 38(7) 1087 – 1103, whch postulates a simple stereo-viewing trick as the means for drawing the detailed illuminations in the famous Book of Kells transcribed by Celtic monks around 800CE. The abstract:

The idea that the seventh- and eighth-century illuminators of the finest few Insular manuscripts had a working knowledge of stereoscopic images (otherwise an eighteenth- and nineteenth-century discovery) helps explain how they could create singularly intricate, microscopically detailed designs at least five centuries before the earliest known artificial lenses of even spectacle quality. An important clue to this long-standing problem is that interlace patterns drawn largely freehand in lines spaced as closely as several per millimeter repeat so exactly across whole pages that repetitions can be free-fused to form microscopically detailed stereoscopic images whose relief in some instances indicates precision unsurpassed in astronomical instruments until the Renaissance. Spacings between repetitions commonly harmonize closely enough with normal interpupillary distances that copying disparities can be magnified tens of times in the stereoscopic relief of the images. The proposed explanation: to copy a design, create a pattern, or perfect a design’s template, the finest illuminators worked by successive approximation, using their presumably unaided eyes first as a camera lucida to fill a measured grid with multiple copies from a design, and then as a stereocomparator to detect and minimize disparities between repetitions by minimizing the relief of stereoscopic images, in the manner of a Howard–Dolman stereoacuity test done in reverse.

The Book of Kells, which lives at the Old Library, Trinity College Dublin, is iconic both as Christian and Celtic art, and central to the Celtic revival; there are plenty of images online, and many nice books with facsimiles (at the bookshop, for instance, we have a copy of the 1914 The Book of Kells / described by Sir E.Sullivan, and illus. with twenty-four plates in colours). But despite the exposure, the techniques to produce it are still an enigma. As the PhysOrg intro says:

The Book of Kells and similarly illustrated manuscripts of seventh- and eighth-century England and Ireland are known for their entrancingly intricate artwork -- geometric designs so precise that in some places they contain lines less than half a millimeter apart and nearly perfectly reproduced in repeating patterns -- leading a later scholar to call them "works not of men, but of angels".

The phrase comes from the Welsh historian Giraldus Cambrensis, referring in the 12th century to the now-lost Book of Kildare:

Fine craftsmanship is all about you, but you might not notice it. Look more keenly at it and you will penetrate to the very shrine of art. You will make out intricacies, so delicate and subtle, so exact and compact, so full of knots and links, with colours so fresh and vivid that you might say that all this was the work of an angel and not of a man.

I've often suspected that strongly myopic artists must have played a part. I've moderate short sight (I need -8.00D glasses) and find this gives a very useful "hi mag"mode for looking at things about 12cm away, enough for sub-millimetre detail such as halftone rosettes. I'm sure someone worse affected could manage the resolution of the Kells artwork. But Cisne's paper also notes that the Book of Kells shows this kind of resolution between duplicates of the same design, suggesting some means of visual comparison with a template. The mechanism is essentially that used when viewing random dot stereograms.

But precision aside, even at manageable size the complexity of Book of Kells designs is quite baffling, particularly the interlacings of Celtic knotwork. One handy fact - while I don't pretend to understand knot theory - is that some of it is self-working. If you draw a continuous loop, following it and assigning a simple over-under alternation at each crossing automatically assigns consistent interlacing (a feature known across multiple cultures). Forgive the crudeness of this diagram:

While I think that's pretty cool theory, the practice to produce aesthetically good designs requires astonishing powers of draughtsmanship, and the key and classic book is George Bain's Celtic Art: The Methods of Construction, originally published in 1951 to not much acclaim, but reissued in 1971 to become central to the revival of Celtic artwork. See Celtic Interlace; An Overview (Stephen Walker, orig. in Dalriada Magazine). Bain's contribution was to document Celtic art from standing stones and other artefacts, and reinvent low-tech humanly feasible algorithms for drawing them and creating original designs.

click images to enlarge

The above images show Bain's reconstruction and the original of a roundel featuring stylised and interlaced birds, beasts and plants from The Book of Kells, Folio 188r, Incipit to the Gospel of Luke. "Quoniam quidem multi" (see full page at Wikimedia Commons). The actual roundel is just one-and-a-third inches across.

For a bit about Bain himself, see Groam House Museum and George Bain - A Highland Homecoming.I'm sorry to say that he comes across as a somewhat irritating presence in the book. Quite apart from his evident dislike of the modern art of his time, his commentary has a rather tiresome tone, not uncommon with autodidacts with off-the-wall theories, to the effect that everyone else is closed-minded, especially those who acquired knowledge through academic routes

... By minds already stored with information, whether it be acquired by instruction of others or by dint of personal application, prospective books will frequently be rejected. What has been diligently attained is too often assiduously hoarded; and pride and envy co-operate with avarice to render the process of knowledge difficult and expensive
- Anonymous 1795 1

... holders of Art College diplomas ... resenting the introduction of the study of a form of Art of which they knew nothing...

The two above examples show the gross travesties of Pictish Art in Publications by supposed authorities that have been for the past fifty years the only source of information for Students and others in the libraries of every university and centre of art education in the civilised world.

... writers who have been blinded by the "classical" education that still claims to be the basis of all European artistic achievements,

Maybe he's right, but parading the gripe in his book doesn't do him credit as a person. I can't over-emphasise, however, his skill as an artist and draughtsman, and the remarkable body of design techniques he devised. After his death in 1945, his work was continued by his son Iain Bain, a civil engineer who revised his father's techniques to simplify areas dependent on intuition, and published books including Celtic Knotwork and Celtic Key Patterns.

To Bain, by the way, we owe every Celtic knotwork tattoo you've ever seen (many are ripped off straight out of his book).

1. Any ideas on source? It's a distinctive quote, but Google Books finds no sign of it. I suspect Bain made it up.

Addendum For those who can fuse crossed-eye stereopairs, here's one I prepared earlier. Click to enlarge, as usual.

- Ray

Mathematics and Fiction

Kindly forwarded by Felix: the announcement for Mathematics and Fiction, a themed weekend conference run by the British Society for the History of Mathematics which will "will explore the various uses of mathematics in fiction, novels about mathematics or mathematicians, and novels based on mathematical structures". Here are some quick thoughts and associations arising from the topic list:

Melanie Bayley: "Free will and statistics in Middlemarch and Jude the Obscure".

This explores the relation of these novels to "statistical fatalism", a 19th century intellectual scare:

As more and more statistics were gathered during the first half of the nineteenth century, and crime figures were shown to be roughly constant from year to year, influential commentators began to announce that statistics were driving people to behave against their will. Mad as it may seem, the public seized on the idea that people would act in a particular way to fulfil a statistical quota.

See Society prepares the crimes in The taming of chance (Ian Hacking, Cambridge University Press, 1990, ISBN 0521388848) which mentions the particular influence of the Belgian social statistician Adolphe Quetelet.

David Bellos: author, Georges Perec: a life in words, on the Oulipo.

I'll refrain from a wealth of digressions here: this refers to the "Ouvroir de littérature potentielle" group of writers and mathematicians that specialised in bizarrely creative works produced under various constraints of style or content, such as Perec's novella Les revenentes, in which "e" is the only vowel allowed. See this section of Emily Apter's book The translation zone for a section of Ian Monk's translation, The Exeter Text: Jewels, Secrets, Sex.

William Goldbloom Bloch: "Navigating Labyrinths in Jorge Luis Borges' story The Library of Babel".

The Library of Babel is perhaps [Borges'] most famous story, and in its scant seven pages, he deploys simple combinatorial ideas to help create a miasmic atmosphere in the service of raising issues about the meaningfulness of our existence.

See The Library of Babel for a translation by James Irby; as the Wikipedia article explains, the Library contains books comprising all possible permutations of text, thus all possible truths, untruths and gibberish. Another Borges story, The Book of Sand, offers much the same in a single infinite book; you can read the text if you care to reconstruct it via Maximus Clarke's hypertext puzzle.

Andrew Crumey, novelist, author of Sputnik Caledonia and Mobius Dick

See the Guardian reviews It's Scotland, but not as we know it and Spatial awareness, and Meltdown moments.

Marilyn Gaull, "Romantic numeracy"

The paper covers a range of allusions from statistics to time, geometry and accounting, even the sizing of clothes and interpreting temperature, all new to fiction during this period as they were new to the culture. My thesis would be that novelists such as Jane Austen and Sir Walter Scott used contemporary mathematical concepts and applications to create an illusion of fact in their fiction, which also helped their readers learn and adapt to mathematics. In turn, mathematicians used literary examples and fictional style to explain mathematical concepts culminating in Russell's Tristram Shandy paradox.

Dorothy Ker, composer, "the 19th step: Borges, Maths and Music"

the19thstep project brought three artists together with Marcus du Sautoy to explore Borges' virtual universe. Composer Dorothy Ker introduces the project and talks about how working with a mathematician stimulated new ways of thinking about space in live performance.

Donald Knuth on his novel Surreal Numbers.

See Knuth's own page Surreal Numbers: How two ex-students turned on to pure mathematics and found total happiness. Knuth is an eminent computer scientist whose work explores the skill and art of programming. His Metafont system, which defines fonts mathematically, thus enabling continuous morphing of fonts through a passage - as in Carroll's The Mouse's Tale - has given rise to some delightful typographic explorations such as Daytar's Surrealey, "The poem/song Die Loreley by Heinrich Heine in a form which was inspired by the works of Guillaume Apollinaire".

Nikita Lalwani, novelist, author of Gifted

Nikita Lalwani's Gifted tells the story of a maths prodigy conflicted by her situation as the daughter of Asian immigrants in Cardiff. It was longlisted for the Booker, and won the Desmond Elliott prize (whose brief is to reward debut novels that combine "intelligence" and "broad appeal"). See the Guardian review, Freedom by numbers.

Ann Lingard, novelist, "The Embalmer’s Book of Recipes (with a nervous nod towards quasicrystals)".

Ann Lingard ... is founder of SciTalk, the free national resource that helps fiction-writers to find and talk to scientists, engineers and mathematicians
...
One of the characters in Ann Lingard’s fifth novel is a mathematician, who works on quasicrystals. Why? Ann will talk about the fun and challenges of making this decision.

The Embalmer's Book of Recipes page at Ann Lingard's website has an interesting tour of the cultural and scientific background to the novel, such as the strange anatomical dioramas of Frederik Ruysch and Frans Lemmens' remarkable aerial photos of Dutch tulip fields

Mark McCartney, "James Clerk Maxwell: A Poetic Life"

I'd run into one or two of Maxwell's poems, particularly his Valentine by a Telegraph Clerk ...

The tendrils of my soul are twined
With thine, though many a mile apart.
And thine in close coiled circuits wind
Around the needle of my heart.

Constant as Daniel, strong as Grove.
Ebullient throughout its depths like Smee,
My heart puts forth its tide of love,
And all its circuits close in thee.

O tell me, when along the line
From my full heart the message flows,
What currents are induced in thine?
One click from thee will end my woes.

Through many a volt the weber flew,
And clicked this answer back to me;
I am thy farad staunch and true,
Charged to a volt with love for thee.

... but I didn't know just how prolific he was. See PoemHunter for 44 of them (unfortunately minus date and context); the whole batch is available as a PDF download. This Selected Poetry of James Clerk Maxwell (1831-1879) page has a few with in-references to academia and physics explained. James Clerk Maxwell: Maker of Waves at the Victorian Web quotes Peter Guthrie Tait, the Professor of Natural Philosophy at Edinburgh University:

"Maxwell's early skill in versification developed itself in later years into real poetic talent. But it always had an object and often veiled the keenest satire under an air of charming innocence and naive admiration. No living man has shown a greater power of condensing the whole substance of a question into a few clear and compact sentences than Maxwell exhibits in his verses"

BTW, while the telegrapher poem is whimsical, romance by telegraph was as real as a phenomenon in its time as that by Internet now; see Tom Standage's excellent book The Victorian Internet (Phoenix, 1999, ISBN-10: 0753807033), which explores the telegraph/Internet analogy in depth.

Scarlett Thomas, novelist, author of PopCo and The End of Mr Y.

There are summaries of PopCo and The End of Mr Y at Alex Kasman's Mathematical Fiction site, which is a large browsable compilation of the many works relating to mathematics.
- Ray