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I've found a supplement for computer tree above. . The new one is interesting and has its differences from its predecessor, and divides its generations of computers in terms of logic technology. It is found in a 1960 NSF pamphlet called "The Family Tree of Computer Design", a Brief Summary of Computer Development, and I found its reference in a good book by I.B. Cohen, Howard Aiken, Portrait of a Computer Pioneer, published by MIT and available here.
JF Ptak Science Books Revisting/Expanding Post #76 (from 2008)
Saunderson (1682-1739) was an extraordinary mathematical talent—he was
also blind (from about the age of one), and invented, principally for
his own uses, what I think is the first mathematical calculator designed
specifically for the use of the blind.
He was supremely gifted and creative, and rose to become the fourth Lucasian professor at Cambridge, succeeding the expelled William Whiston, who had in turn succeeded Isaac Newton—Saunderson also held the post for one of the longest periods of time, 1711-1739. He was friend and associate to Newton, Whiston, Roger Cotes, Halley, De Moivre and others during a particularly rich intellectual period in the history of physics and the maths.
His calculator was smart and simple, based on a cribbage-board –like device, that was able to perform arithmetical and algebraic functions—it consisted of nine rows and was worked with two pins, the positioning of the pins on the engraved board telling the user their value. (There was another calculator for the blind constructed by Meyer (below, left) using a sort of reverse principle to the Saunderson model where it was the shape and placement (leaning or not, for example) of the pegs in the hole that annotated value rather than their placement on the board. The Saunderson computer was described in his The Elements of Algebra…, published at Cambridge in the first edition just after the author’s death, in 1740. The device was described in the book by John Colson (who succeeded Saunderson to the Lucasian chair), who commented that it was via the use of the device that Saunderson could compose his treatise on algebra. (Above is another Saunderson-based calculator allowing for the construction and study of geometrical figures).
"Palpable Arithmetic", the sub-heading for the sheet illustrating aspects of algebra for Abraham Rees' (1743-1825) great if not problematic 45-volume Cyclopedia, is a system that records and organizes and sometimes calculates using three dimensional objects.
For example the Egyptians (for one) calculated with pebbles; then there was the ABAX of the Greeks, and the abacus (and also called the mensa Pythagoras) of the Romans (and of the Japanese and earlier still of the Chinese), the scaccarium of the English (via the Norman conquest), and innumerable other systems that performed arithmetic and recording and archiving functions via the employment of reeds, notches on a tree or cloth or stick (etc.), reeds, knots, fingers, beans,shells, string, sand, and on and on. Palpable arithmetic also has a specialized meaning in places as a calculating device in which the numbers are recognized by touch and used by blind mathematicians or other parishioners. (Just for the record, there are a number of eminent blind mathematicians including, for example, Leonard Euler (1707–1783, who was blind in the last 17 years of his life), Nicholas Saunderson (who I wrote about in an earlier post), Louis Antoine (1888-1971), Lev Pontryagin (1908-1988.))
An interesting and very large philosophical issue that comes up here with the blind mathemaitican is the concept of image formation and its dependence upon sight for intuition, as with geometry or topology. Plato for one determined for himself that image formation issues were precognate and the same in sight and non sighted people. How would you manipulate a geometrical form if you’ve never actually seen one, or how would you extend you spatial imagination of compex forms without a reference?
But my main issue here is the image from the hees book. I'm by the meaning of this particular calculator or recording system--I just can't tell what it is. Can you? If so I'd love to hear from you.
Note on the Anthropology of Numbers:
From Levi Leonard Conant's The Number Concept Its Origin and Development we find these very descriptive definitions of words for numbers, all of which relate to the sort of implement that they were controlling their numbers with, or calculating:
"in Javanese, Malay, and Manadu, the words for 1, which are respectively siji, satu, and sabuah, signify 1 seed, 1 pebble, and 1 fruit respectively. Words as natural and as much to be expected at the beginning of a number scale as any finger name could possibly be. Among almost all...the derivation of number words from these sources can constitute no ground for surprise. The Marquesan word for 4 is pona, knot, from the practice of tying breadfruit in knots of 4. The Maori 10 is tekau, bunch, or parcel, from the counting of yams and fish by parcels of 10. The Javanese call 25, lawe, a thread, or string; 50, ekat, a skein of thread; 400, samas, a bit of gold; 800, domas, 2 bits of gold.The Macassar and Butong term for 100 is bilangan, 1 tale or reckoning..."
There aren't many other title pages in the history of mathematics that swing the doors of beauty and heaven and opportunity and logic and thought than the (appropriately-named) Bonaventura Francesco Cavalieri/Cavalerius. A Milanese (1598-1647) who taught at Pisa and who published eleven books and worked in wide fields in mathematics as well as astronomy and physics, published his Trigonometria... in Bologna in 1643, and sent a fabulous image to his new readers right there on his very illustrated and augmented title page.
Seldom have the doors of knwoledge been thrown open so wide and invitingly as this, with Trigonometria lightly awarding the reader with a new portal to fantastic opportunities--who wouldn't want to walk on in? The tools of trig are displayed on the ground at Trionometria's feet (arranged and not by happenstance), while five figures employ these same tools in the background hills, which are being bathed in a pure holy light of some sort, an inspiration from the sky. The doors are decorated with astronomical figures and light calculation, as well as polyons and geometrical drawings. And just to make sure that the author meant what he intended, we can see above the full/thick wind-blown hair of Trigonometria (which in itself seems very unusual to me, such a carefree adn tossled attitude) a fruit-garland, a horn of plenty in its way. Make no mistake--Cavalieri was making a king-sized offer to his reader to open their brains.
I did leave out a major part of the title--the section dealing with logarithms, which is basically a reprint of Cavalieri's early work (done in 1632), and which was the first work in Italian on that subject.
"Walker Evans climbed to the roof of the Fisk Building on Central Park South to photograph the web of steel struts and electric signs that were rapidly filling the skies over Manhattan. The electric signs in this photograph alternately flashed the company's name and the names of its two principal products. As there were approximately a hundred million wheels rolling over America's roads in 1928, the sale of tires quickly overtook that of rubber galoshes. The image expresses Evans' conviction that modern art could be timeless yet topical when perfectly wrought of vernacular materials."--Metropolitan Museum of Art
This is one in a series of posts made on found geometries--here are a few others:
These images come from a later (and English edition) of Cesare Ripa's essential keybook to Renaissance iconogrpahy, the Iconologica overa descittione d'imagini delle virtu, viti, affetti, passioni humane...first published in the early 1590's with Ripa's artwork. I don't much like the English version of his work, nor do I care that much for the artwork (inferior in most ways to the great craftsman-like feel of the Renaissance images), though it does serve the purpose of place-holding the symbolism and explaining the elements of the icons.
I've chosen a few emblems for the sciences to illustrate this important work of deduction and explanation, as it would be an aid to anyone interested in the art of the Renaissance--and that would be an interest at virtually any level.
I chose "Arithmetic" as my first example because of the lovely parallels found in the folds of Arithmetic's dress--musical notation, underneath which we see the faint "Pars Impars" (even and odd) to help settle the old scores of symmetry. The connection between music and mathematics is thousnads of years old, and though I might've used "mathematics" rather than its oldest branch (of "arithmetic") to show the ancinet connection between music and numbers, it should be remembered that not every aspect of arithmeic is standard, and that there is a good and strong relation between music and "higher arithmetic" (number theory). In general I just like the emblzoned connection here between the two, showing a temporal sequencing representation, a placing of music in relation to number right out there in time-and-space,an adept of the mental image of music. There is no doubt that when a viewer was finsihed with this image that there is a lasting property of the mathematical properties of sound and teh harmony of number. Its not exactly "music of the spheres", but it does work.
Good transliterated description are to be found on the Alchemy Website, here; the English-language version cane be found here.
[My thanks to Patti Digh for providing the idea for the Goedel part of this adventure into Playtex and Logic--she did so because (a) they fit together and (b) girdle/Goedel sounded almost identical to a woman who once lived in Munich!]
In the long history of Holding Things In, perhaps the newest of its
members was upon us only recently. In the long, deep past we have held
our breath, hidden our anger, stowed our emotions, and so on, but it was
only recently that we began to hold our bellies in. One of the masters of Holding Things In for this period turns out to be the sublime logician and re-inventor of modern mathematics (by putting one piece of the great Hilbert to sleep), Kurt Goedel, who towards the mistakenly-self-engineered end of his life, held on to everything, virtually--he organized and filed almost very piece of paper that he came into contact with at any level, became ever more reclusive, and at the end (due to his theories of people/institutions wanting to kill him) refused food and, of all things, water. Surrounded by the smartest people on the planet (including his friends Einstein and von Neumann) up there at the Institute for Advanced Study at Princeton, Goedel withered away until he had almost no shadow. It is a bad irony that he could be so inconceivably unmovable and restrained while at the same time, and in the same life, offered such incredible newness to the maths--both ends of the mountain at the same time.
1951, the year in which these girdle advertisements appeared in Life magazine, was also the year that Goedel present us with the Goedel metric, and also in which he received (with Julian Schwinger) the first Albert Einstein award (and of course delivered his famous Gibbs lecture "Some basic theorems on the foundations of mathematics and their implications").
The popular introduction of the girdle I think that
this happened at about the same time for the sexes, only these
conveniences were much more often advertised for women than they were
for men. Slender and non-existent waistlines for women were more of a cultural identifier
than a slim-hipped man, and the ads for his cheaters appeared far less
frequently than those for women.
The first widespread appearance of the girdle for the sake of vanity must have occurred during the 19th century, or perhaps a little later is my best guess--but the first time the device began to appear for the common woman must've come around the time when there was time for leisure, or shopping, or of being seen in public in short intervals. And that I believe is a Victorian-age invention.
But the binder doesn't come into fabulous presence until the distribution of mass population illustrated magazines, or I should say the advertisements that made these magazines possible: production like LIFE (from which these 1951 images come) reached far more women than the popular older periodicals like Harper's Weekly or other polite mid-19th century journals for women. The advertisements were certainly more enticing, the possibilities more rewarding, and the girdle comfort levels far higher than their predecessors, and the availability of disposable income for women far greater--and so incidentals like the girdle became more greatly commodified, and moved into the "essentials" category.
The idea of these ads seem horribly revolutionary: on the one hand, the badly-named and hyphenated Playtex product "Pink-Ice" squeezed women into new tight but malleable molds, while at the same time promised some sort of ballet-like freedom because of it. Like the creeping ("two steps forward and one step back") communism of the time, Playtex promised the possibilities of enhanced freedom through restrictive clothing (in a "peace through strength" vision). In any event, and in spite all of what I just wrote, the pictures are kind of amazing.
[I'm well aware that this may be one of the worst things ever written about Kurt Goedel--the Renault Dauphine of Goedeliana. But it doesn't matter, because in all of his powers, Goedel could absolutely prove that g_d existed, and that I don't.]
I found this in the Alex Cashman's lovely site, Mathematical Fiction (here). The except is from Charles Dicken's Hard Times (1854)1, and in it we see Dickens making another in a long series of assaults on what he thought to be a disturbed social layering of dealing with the underclasses and the working poor.--and this time in the form of an entire novel. In the instance sited below (of interest right now because of its maths connection), Dickens makes a case for the"success" for the government of the fictitious city of Coketown to see a relatively small percentage of people in that city starving to death on its streets to be not a success at all. The character making the complaint is Sissy Jupe, whose fault at school came when asked the percentage of the starving/dying responded by saying it really didn't matter, because people were starving anyway, and whether it was one or a thousand dying from something as fixable as hunger meant all the same thing. She was presented with the "statistics"2 from her teacher, but to Ms. Jupe it came to her ear as "stutterings", which is what Dickens felt the numbers were. Dickens was attempting to make those people suffering in the streets less blank, less hollow, and more than a number.
1. It should be pointed out here that this is Dickens' 10th novel, the 42-year-old already having a enormous success with his writing and an even grater one in his storytelling. To this date, Dickens had already written The Posthumous Papers of the Pickwick Club (Monthly serial, April 1836 to November 1837; The Adventures of Oliver Twist (Monthly serial in Bentley's Miscellany, February 1837 to April 1839); The Life and Adventures of Nicholas Nickleby (Monthly serial, April 1838 to October 1839); The Old Curiosity Shop (Weekly serial in Master Humphrey's Clock, 25 April 1840, to 6 February 1841); Barnaby Rudge: A Tale of the Riots of 'Eighty (Weekly serial in Master Humphrey's Clock, 13 February 1841, to 27 November 1841); A Christmas Carol (1843);The Chimes (1844); The Cricket on the Hearth (1845); The Battle of Life (1846);The Haunted Man and the Ghost's Bargain (1848); The Life and Adventures of Martin Chuzzlewit (Monthly serial, January 1843 to July 1844); Dombey and Son (Monthly serial, October 1846 to April 1848);David Copperfield (Monthly serial, May 1849 to November 1850); and Bleak House (Monthly serial, March 1852 to September 1853). Remarkable.
2. "Statistics" as a word has been in use for a long time, finding air as early as 1787, at least so far as the sense in which it is used here. There are earlier references, but they actually refer to the "state", as in "statecraft" and government and such, and not for interpreting a collection of data.
I found this interesting 400-word bit in the 18 July 1895 issue of Nature (A Weekly Illustrated Journal of Science) and of course still very much alive and still a great publishing scientific powerhouse. As Wells' treatment of time as a fourth dimension was still quite speculative, and his creation of the time machine as we now came from his hands, and as its reception by some literary journals was not very cordial, it is particularly interesting to see the positive review from this science journal. As reported by Paul J. Nahin in Time Machines, Time Travel in Physics, Metaphysics and Science Fiction (published by Springer Verlag in a second edition in 1999, a graceful book of splendid observations with an introduction by Kip Thorne, Cal Tech's Feynman Professor of Theoretical Physics), Wells' book was seen as "hocus pocus" and a "fanciful and lively dream" by The Spectator and "bizarre" by the Daily Chronicle. [This item is available for purchase via our blog bookstore, here.]
Nature's editors says of the book:
"...apart from its merits as a clever piece of imagination, the story is well worth the attention of the scientific reader, for the reason that it is based as far as possible on scientific data..."
"It is naturally in he domain of social and organic evolution that the imagination finds its greatest scope."
"From first to last the narrative never lapses into dullness."
And so Wells--who had cleaned up and rewritten an earlier version of this book from 1888--received a happy review from one of the leading scientific magazines of the time. It is not because of a wide interest of Wells in the fourth dimension per se that Wells wrote the book, but rather in rattling Victorian sniffy conceits of his day.
*Time as the fourth dimension is a concept not much older than Well's idea, one of the earliest publications of the idea appearing ten years earlier (and written anonymously by "S") in Nature. Charles Hinton, the person we most associate with the introduction of the fourth dimension (or at least so on a popular level) didn't really establish time as the fourth dimension. (Hinton led a colorful, or at least interesting life--he was a brilliant guy, and also a convicted bigamist who was married to George Boole's daughter, and creator of an automatic baseball pitching machine. His books on the fourth dimension certainly were influential, and in the estimation of Florence was a contributor to the ideas of the coming modern art movements of the 1890-1910 period.)
And there were certainly many who came before Wells on the subject of the fourth dimension (though not many on the subject of time as the fourth dimension): R.C. Archibald wrote on d'Alembert's (1754) use of time as a fourth dimension (in the Bulletin of the American Mathematical Society for May 1914); Cayley's "Analytical Geometry of n-Dimensions (Cambridge Mathematical Journal, 1843); Grassmann's Die Lineale aus Dehnungslehre (1844); Riemann's 1854 effort on curved space (translated in 1873 for Nature by W. Kingdom CLifford); Beltrami's introduction of the pseudosphere in 1868; J.J. Sylvester (again in Nature for 30 December 1869); Hermann von Helmholtz and his curvature for three-dimensional spaces; and Poincare's work in 1900, among others. There is a rich field of these efforts, beautifully investgated for the general reader by Linda Dalrmple Henderson in her classic The Fourth Dimension and Non-Euclidean Geometry in Modern Art (Princeton 1983).
Arithmetic of 1935 is an interesting book discussing the four prinipal operations of mathematics, though it is not quite Busby Berkeley's Gold Diggers of 1935. It is however a very neat and orderly and in some ways exhaustive treatment of its subject, and that it does share with the brilliant choreography of Mr Berkeley.
I've touched on Mr. Berkeley's work a couple of other times on this blog: Bio-Mathematics in Busby Berkeley Musicals(here) Mr. Mole, the Stationary Busby Berkeley of Mass Crowd Installation Photographs, 1918 (here).
Just a quick post here on patented mathematical tools using long wooden planks in a sliding frenzy to find displayable answers for 19th century schoolchildren and their addition problems. I know a good set of Napier's bone could come in very handy (appearing much earlier on than these instruments), but I'm just not sure what kids could learn from using these tools--like any other calculating device, early or late, they have a questionable quidity so far as appreciating the essence of numbers is concerned. That said, the drawing for the Meyers' machine is beautiful, and could no doubt lead to a number of short, noir-y stories.
Lewis Carroll created a lovely, simple cipher in the midst of his Alice and Snark and Logic and Sylvie publications. It really is just a simple bit of polyalphabetic substitution, bu tit gets the job done. (Many others have walked this royal road: Leon Battista Alberti, A Treatise on Ciphers, [De componendis cyfris]; Giovan Battista Belaso, La cifra del Sig. Giovan Battista Bel[l]aso, gentil’huomo bresciano, nuovamente da lui medesimo ridotta à grandissima brevità et perfettione, Venetia 1553 (and also his Novi et singolari modi di cifrare de l’eccellente dottore di legge Messer Giouan Battista Bellaso nobile bresciano, Lodovico Britannico, Brescia 1555); Giombatista Della Porta, De furtivis literarum notis vulgo de ziferis, G. M. Scoto, Neapoli 1563; Galileo Galilei, Intorno a due nuove scienze, Opere, . Vol. VIII, Firenze; Blaise de Vgenere, Traicté des chiffres ou secrètes manières d’escrire, Abel l’Angelier, Paris, 1586; and so on...its a very wide literature, even pre-18th century). Louis Carroll. Louis "Cipher" Carroll. Comes sort of goofily close to "Louis Cipher". Lucifer. Not the case, of course unless you were trying to figure out one of his tricky puzzles.
Perhaps it is the cipher's presentation and design and simplicity, its elegance, that I like so much. It reminds me in some ways of the Henry Holiday masterpiece of nothignness created for Carroll's Hunting of the Snark--and that of course would be the Bellman's map, a map of nothing, a map showing nothing at all to the sailors who must follow it and who were all happy that the map had nothing to obstruct their vision of possibility and blank expectation. (I wrote about that in The Most Beautiful Map in the World, here). It is interesting to note that none of the illustrators who followed Holiday chose to illustrate the nothing map with such nothingness as in Carroll--there would be hands on it, or the map would be oblique, or not the central image of the illustration. Holiday's map was just that--straightforward, simple, strong).
I've decided to make this a part of the History of Blank, Empty and Missing Things series simply because everything is missing unless you have the missing key--here you have all the parts of the puzzle, and then some, everything that you need to solve it, save for the integral part of ordering.
From Carrolls's text:
Each column of this table forms a dictionary of symbols representing the alphabet: thus, in the A column, the symbol is the same as the letter represented; in the B column, A is represented by B, B by C, and so on.
To use the table, some word or sentence should be agreed on by two correspondents. This may be called the 'key-word', or 'key-sentence', and should be carried in the memory only.
In sending a message, write the key-word over it, letter for letter, repeating it as often as may be necessary: the letters of the key-word will indicate which column is to be used in translating each letter of the message, the symbols for which should be written underneath: then copy out the symbols only, and destroy the first paper. It will now be impossible for any one, ignorant of the key-word, to decipher the message, even with the help of the table.
For example, let the key-word be vigilance, and the message 'meet me on Tuesday evening at seven', the first paper will read as follows—
v i g i l a n c e v i g i l a n c e v i g i l a n c e v i
m e e t m e o n t u e s d a y e v e n i n g a t s e v e n
h m k b x e b p x p m y l l y r x i i q t o l t f g z z v
The second will contain only 'h m k b x e b p x p m y l l y r x i i q t o l t f g z z v'.
The receiver of the message can, by the same process, retranslate it into English.
If this table is lost, it can easily be written out from memory, by observing that the first symbol in each column is the same as the letter naming the column, and that they are continued downwards in alphabetical order. It would only be necessary to write out the particular columns required by the key-word, but such a paper would afford an adversary the means for discovering the key-word.
Passing through a later and edited edition of Jacques Ozanam (RECREATIONS MATHEMATIQUES ET PHYSIQUES, Qui Contiennent Plusieurs Problemes d’Arithmetique, de Geometrie, de Musique, d’Oprique, de Gnomonique, de Cosmographie, de Mecanique, de Pyrotechnie, & de Physique. Avec un Traite des Horloges Elementaires. Nouvelle Edition, Revue, Corrigee & Augmentee... and published in Paris in 1749-1750) looking for possible expansions on what he wrote on the Knight's Tour--a chess/math problem where the knight starting at, say, the center position must be moved to touch every square of the board in 64 moves--I found this little diagram showing the spaces to which a knight may not move:
It seemed just a little unusual to me--not being a reader of chess literature--to see what seemed the negative of the knight's movements on a truncated board. But I guess this is what we calculate and just not entirely "see" while playing.
The original problem in the 1672 edition of Ozanam's Recreations looks like this (and titled "faire parcourir au cavalier toutes les cases de l'echiquer"):
in which the knight starts off life at the top right square (h8) and finishes at f3. In the 1803 edition of the work it is pointed out that the knight can be started from any square and moved 64 times to accomplish this same feat.
Here are four further examples of solutions to the knight problem:
And as they say, many more are available.
A more modern version of the solution, this on a 24x24 grid:
"The name Magic Square, is given to a square divided into several other small equal squares or cells, filled up with the terms of any progression of numbers, but generally ah arithmetical one, in such a manner, that those in each band, whether horizontal, or vertical, or diagonal, shall always form the same sum." --from the very busy Charles Hutton's translation of Jean Etienne Montucla's edition of Jacques OzanamRécréations mathématiques et physiques (1694, 2 volumes, revised by Montucla in 1778, 4 volumes) and the whole thing revised in an English edition of 1844 by the appropriately-names Edward Riddle, and available online at Cornell's collection of historical mathematical monographs.
That was sort of a simple introduction to magic squares, tortured by my note on the quote's parentage. Nevertheless, leafing through a copy of Ozanam's work I found a lovely little (literally speaking, as it is about 1/2 inch by an inch) 3x3 multiplication magic square for the happy sequence of 1, 2, 4 ,8, 16, 32, 64 and 256. (That means that each of the nine numbers may appear only once, and that the product (4096) must be the same for each column and row). It is a nice little problem, and I was just surprised to see it in such spare simplicity.
And since we're at it slightly, a few pages further on I found this nice series of 3x3 magic squares for numbers 1-25:
These also are a half-inch (or less) and about two inches long...they're just very attractive.
But I guess I cannot leave the subject of "pretty" magic squares without referencing a "beautiful" one, and this being one of the earliest inclusions of a magic square in Western printmaking, and surely one of the most beautifully-encumbered one in general, from Albrecht Durer's mega-popular masterwork, Melancholia (printed 1514). The magic square had been around for at least 2,000 years at this point, starting up evidently in China between 650-1000 BCE before making its way west through the Arab lands and then through India, and finally into Europe around the 13/14th century, and then into art prints with Durer in 1514.
I doubt that Abraham's Rees' "Magic Circle of Circles" (published ca. 1814) is "pretty", and I'm not so sure it is "beautiful", but I am sure that it is "elegant".
Ditto his "Magic Square of Squares" (published ca. 1814):
In any event these are just a few samples that I had close to the top of my head--no doubt there are endless others, but these are some that have attached themselves longest to me (with the exception of the Ozanam, which are new).
There is a terrific find on Alex Bellos' website exhibiting Alan Turing's “report cards” for his time at the great Sherborne School from 1926-1930 (and which were transcribed by archivist Rachel Hassall), from the time when Turing was 14 to 19 years old. Turing (1912-1954) I think needs no introduction for his importance to mathematics and computing (and code breaking during WWII), and it is very interesting—thrilling even—to see how his instructors were coming to grips with the developing genius. Even at such a school as Sherborne (a very old school with 39 headmasters overseeing the place since 1437) where the teachers were I am sure familiar with gifted pupils, The comments on the reports of Turing's progressed showed that many weren't quite sure about what Turing was all about. Obviously Turing as a boy was very gifted, but many instructors reported as many hindrances to his intellectual development as there were advances—more, even.
Perhaps people at the school didn't know exactly how to deal with him; perhaps they did, but still at the end of the day Turing had to meet the common standards of the school. Or perhaps not—I really can't tell from the transcripts presented by Bellos and I don't know the intricate history of the school. But certainly as time progressed Turing's abilities were more readily recognized, but early on it seems that his talents didn't overwhelm his many supposed shortcomings, the faults of the parts larger than the whole of what he could accomplish. In instructors' comments across all of his disciplines, Turing was “capricious”, “untidy”, “lacking in life”, “need(ed) concentration”, “depressing unless it amuses him”, “careless”, “absent minded”, “un-methodological”, “slovenly”, (made) “mistakes as a result of hastywork”, and so on. He “could do much better” though one instructor felt that “he may fail through carelessness”. All of which may well have been true—from the outside. These statements may have simply been the result of teachers not being able to reach a boy genius, and perhaps the boy couldn't be reached, at least early on in his academic career.
The statements in general—especially in the maths—I think are fascinating things. 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.
Here are some selection from reports on Alan Turing, 1926-1930, below; a more full list exists at the Bellos site, here.
1926. Works well. He is still very untidy. He must try to improve in this respect
1927. Very good. He has considerable powers of reasoning and should do well if he can quicken up a little and improve his style.
____. A very good term’s work, but his style is dreadful and his paper always dirty.
____. Not very good. He spends a good deal of time apparently in investigations in advanced mathematics to the neglect of his elementary work. A sound ground work is essential in any subject. His work is dirty.
____. Despite absence he has done a really remarkable examination (1st paper). A mathematician I think.
____ I think he has been somewhat tidier, though there is still plenty of room for improvement. A keen & able mathematician.
These lovely images weren’t intended to show people living in the Renaissance and Baroque eras how to actually record data on their hands—they were intended rather as templates to show how they could use their fingers and hands for calculating and as memory devices. Much like Frances Yates has shown us so beautifully in The Art of Memory and how info and data was stored in imagined and compartmented palaces in the mind (relying upon images), the hands were also used as a theatre of memory in addition to extended calculation.
These mnemonic devices were necessary—especially during the Renaissance—because of the general lack of and access to affordable vellums or paper and writing instruments. Having notebooks filled with memoir or history o calculation was generally not something that was happening for even the not-wealthy but not struggling class. These mental images were used widely in the areas of religion, palmistry, astrology mathematics, astronomy, astrology, alchemy, music, and other such fields. The first image (from a German manuscript) of the hand-theatre was found and deciphered by Claire Richter Sherman (Folger Shakespeare Theatre) and is religious in nature, an intentional piece of memory for the devoted and for devotions. The needs of religion were splayed out as the hand was opened and fingers flexed, and working from thumb to pinkie, from finger tip and joint—“do God’s will, examine your conscience, repent, confess”, and so on, and above all be content with your lowly penitente stature. If there were 28 of these admonitions or reminders at different points of the hand and you memorized them all, it would be a much simpler time to recall and keep them in order if you merely had to touch a part of your hand where that memory should be to invoke what it was you were supposed to do. Therefore you could theoretically cast about with your creator with your hands in your pockets—if you had pockets.
The next two images (including the enlargement of the hand section) are from a work from 1587 entitled Musique and are attributed to John Cousin the Younger (1522-1597). The basic premise for this device—it seems to me—was to be able to order the different chords of 20 different instruments. Another musical hand mnemonic was the Guidonian hand, a survivor of Medieval times, and possibly named after Guido of Arezzo (a musical theorist), and was an aid to singers learning to sight sing.
The entry for the Guidonian hand in Wiki explains it use rather well: “The idea of the Guidonian hand is that each portion of the hand represents a specific note within the hexachord system, which spans nearly three octaves from "Γ ut" (that is, "Gamma ut") (the contraction of which is "gamut", which can refer to the entire span) to "E la" (in other words, from the G at the bottom of the modern bass clef to the E at the top of the treble clef). In teaching, an instructor would indicate a series of notes by pointing to them on their hand, and the students would sing them. This is similar to the system of hand signals sometimes used in conjunction with solfege…” The final two examples come from Jakob Leupold’s (1674-1727) Theatrum Machinarum (1724)—this was a complex work involving nine sections and addressed the theoretical aspirations of engineering (load, flexure, that sort) and its applications to its daily practitioners. In one section of the book he sought to explain the connections (and correlations) of hand motion and symbolism to the origins of the number systems, carrying it out further still into body language, so that two people conversant in these symbols could talk and bargain between themselves in economic/body terms. Barbara Maria Stafford, in her Artful Science, Enlightenment,
Entertainment and the Eclipse of Visual Education (1994) points out the long history of this tradition, and that it reached far back into misty time: Leupold knew that Appian, the Venerable Bede, and Aventinus had been fascinated by manuloquio, or natural language with the hands. He thus linked counting to a global….medium of prearranged gestures…”