A Daily History of Holes, Dots, Lines, Science, History, Math, the Unintentional Absurd & Nothing |1.6 million words, 7000 images, 3.6 million hits| Press & appearances in The Times, The Paris Review, Le Figaro, MENSA, The Economist, The Guardian, Discovery News, Slate, Le Monde, Sci American Blogs, Le Point, and many other places... 3,000+ total posts
"The Fingers as an Aid in Multiplication" is another wonderful find coming from a general browse in the great Scientific American (October 22, 1898, page 265). It is an interesting article, using the fingers so that the multiplication tables didn't have to be memorized by kids--I think that they absolutely should be--but the images taken out of context can also be regarded in some sense for this blog's "Found Absurdist" series. (We are told that the system was devised by "a Polish mathematician", Procopovitch--he is repeatedly referred to as "the Polish mathematician".)
See "Digital “Computers 1450-1750: Memory and Calculating on the Fingers and Hands", a post on this blog from 2008, here.
The "Proportior" (sounding somewhat like an exclamation, like Excelsior!, and looking for all the world like it should be spelled some other more appropriate way) a fabulous desktop/tabletop calculator, as seen in the Science, October 4, 1889:
Thanks to Ray Girvan for the link to a successful download of images for this!
The idea behind this extraordinary image below is the construction of an 8x8 magic square capable of describing BOTH the Knight's tour (" a sequence of moves of a knight on a chessboard such that the knight visits every square only once...if the knight ends on a square that is one knight's move from the beginning square) WITH the resulting moves forming a series of magic squares. It is the product of a Kiwi engineer named Sturmer, and appeared in the Scientific American Supplement for 1888.
Eric Weinstein says in his article "There Are No Magic Knight's Tours on the Chessboard" on Wolfram's Mathworld site says, well, such a thing is not possible. "After 61.40 days of computation, a 150-year-old unsolved problem has finally been answered. The problem in question concerns the existence of a path that could be traversed by a knight on an empty numbered 8 x 8 chessboard."
Weinstein is concise: "Not surprisingly, a knight's tour is called a magic tour if the resulting arrangement of numbers forms a magic square, and a semimagic tour if the resulting arrangement of numbers is a semimagic square. It has long been known that magic knight's tours are not possible on n x n boards for n odd. It was also known that such tours are possible for all boards of size 4k x 4k for k > 2. However, while a number of semimagic knight's tours were known on the usual 8 x 8 chessboard, including those illustrated above, it was not known if any fully magic tours existed on the 8 x 8 board."
Reckoning time from city to city before the adoption of Standard Time (1883/4) was a difficult and confusing process. Local time was local, and usually dominated by a central timepiece located in a public area, as in a church steeple or town hall. The problem is is that the foundation for this timekeeping--the zenith of the sun during the day--would be different from area to area. That's not so bad when communication over periods of days was involved, bu tit was very noticeable to train travelers who would find their home-reckoned pocket watches to be not-quite-right when moving hundreds of miles over the course of a day.
And that's where these charts come in. They were made before the institution of a standardized system for keeping time worldwide, and so the "Dresden Time" of new was actually equal to about 10:48 in Madrid or 12:02 in Vienna. In the U.S. in the 1870's there were hundreds of time zones (whittled down somewhat to 100 towards the end of that era) which made it a bit of a nightmare for maintaining railroad schedules. Stanford Fleming fixed all of that (the end of a long line of people who attempted standardized systems) in 1884, and did away with the need for these beautiful but confusing time arrangements.
On the other hand the displayed results for dealing with the pre-standardized chrono-confusions are beautiful things...
Image Source: Johnson's New Illustrated Family Atlas of the World, (1874). Source: David Rumsey Map Collection, here
(Note: we'll deal with square root of 3 at another time...)
The author of this manuscript, Charles Fisher, took a solitary pleasure in calculating the square roots of numbers from 2 to 589, not bothering to write down the 24 perfect squares to 576. (The sqrt being r2 = x for every non-negative real numberx.)From the few bits that I have checked the man seems to have done a good job back there in the 1830’s.
I cannot determine where this book was written or who Mr. Fisher was, though it is possible that he was a (Baptist) Minister from the few short textual notes that pop up here and there.
[This 140pp manuscript is being offered for sale at our blog bookstore, here.]
His work is pretty elegant.Take for example his solution (and proof) for the square root (hereafter sqrt(x)) 309,
which the calculator living under this page says is 17.578395831246947Mr. Fisher’s answer is 17 10/17 = 799/17=89401/989=309 100/989, and after some more involved arithmetic comes t the lovely proof number of
4121989960986322995025 /13339773336525317136 or
Which is getting pretty close.
The only note that Mr. Fisher makes on his calculations is for the sqrt(193), which he notes as “the hardest number to find the approximate root of any between 1 and 200. I have found it after repeated trials and have this evening wrote it in as above. March 1st, 1833. CF.”
I found these extraordinary magic squares lurking in the September 1915 issue of Himmel und Erde--seeing them was a shock to me, especially since I was expecting articles dealing with more technical and also astronomy-related issues, though treated as a history of astrology/astronomy aspect I guess that I should've have been too surprised. The author, W. Ahrens (of Rostock) had written on different aspects of the magic square in the South Pacific ("Etwas von magischen Quadraten in Sumatra und Celebes") and more extensively on the Kabalah and magic squares (in "Hebraeische Amulette mit Magischen Zahlenquadraten" for example), and in general in his Mathematische Unterhaltungen und Spiele (1901).
(An interesting example of a "Venus" magic square, with seven cells).
(This magic square, a "Moon" or "Lunar" square, has nine cells, and also has a form of a "T" world map. The names of the magic squares were adopted by Cornelius Agrippa (1486-1535) who constructed squares of 3,4,5,6,7,8 and 9 cells, naming them for the seven "planetary" astrological symbols, Saturn, Jupiter, Mars, the Sun, Venus, Mercury and the Moon. The magic square as an idea seems to have been introduced in the west by Moschopulus of Constantinople in the earlier 15th century--the magic square itself though is ancient, alive and well in India more than 1500 years before its arrival in Europe).
A magic square amulette from East India.
Another 9-cell magic square, indentifying itself with "Lunae" and "Luna".
This thinking stick comes to us via the courtesy of the Scientific American Supplement (19 August 1876, page 542), and shows a lovely Victorian non-SteamPunk answer to on-the-run calculation. It is an almost-elegant device, and seems as though it should work just fine. Its utility as a writing instrument seems a bit limited, as seems the calculator part, but the whole of it seems to be full of possibility--and a very nice piece of dedicated thinking.
[Text describing the invention below.]
Other nice examples of adding/writing found in the U.S. Patent Office via GooglePatents are seen below:
QUESTON: If great but not-popularly-known scientists could be represented as a chess piece, and that chess piece was on a game board opposite Popularly-Known-Celebrities-Not-Known-to-Scientists (and etc.), what piece would the great and dusty Robert Hooke be? And conversely, (on that opposite side), what piece would someone like, say Paris Hilton be?
I wonder about poor old Robert Hooke. He was such a tremendous thinker, a terrific rush of ideas, with revolutionary insights in many fields; he was a leading architect, a physicist, a microscopist, a chrononaut, a mathematician, an everything. He carried the Royal Society for years, carried on hundreds if not thousands of experiments, and of course was famously on the other side of a bad series of arguments with Isaac Newton. At the end of his long life, Hooke was afraid of not being remembered, of not having enough money to see
himself through hi sold age, afraid of others taking credit for his
work. He just seemed not to matter, anymore, in the last decade of his
life (and a period in which he was still doing significant work), and I
just wonder why he managed to become so semi-invisible.
He was so dedicated. I have an image of him scurrying with a friend, removing the estimable library of a patron and donor to the Society, trundling the books in wheelbarrows across a mile and then-some of bumpy London streets finding a home for these great treasures. He was an older man at this point, marching these books across parts of the city that he helped to restore after the fire, passed buildings that he helped to build and design, bumping his way through London, a great and famous scientist, saving books by the handful. [An idealized portrait of Hooke, at right.]
He was a tireless,
relentless observer and experimenter, who lost little effort in a stranded idea
and pursued interesting and problematic questions relentlessly. More than others too he chased his won glory—minor
but long and insistent—the years of which wore thin on many people in the
scientific community. But there were
many characteristics of the man that made him not quite so lovable and
endearing—not that Newton was any of those things, as he was not, but if you
are going to be a secondary luminary to a super nova you’ve got to have
something else going for you that the other man doesn’t have—sharing, helpful,
greatly generous—to get you into the long pre-dusty pages of history. Also it would’ve
helped if Hooke chose his battles with a little more aplomb and ingenuity—the
war which began in 1672 with Newton went very badly for Hooke and followed him
to the grave (and far beyond).
He just didn't "catch on", I think--at least he not for the long term. His brain teemed with ideas, but perhaps by the last decade of his long life, his tireless brain still working on innumerable bits, he just sucked the air out of a room.
He also never had his likeness recorded during his lifetime. And that is saying a lot. And I still don't know why.
Back to chess: I figure Hooke to be a Knight. I prefer Knights. He moved like a Knight. He importance was "higher" than that of a Knight, but, well, the Knight seems a good fit (and so he seems to get downgraded, in a way, even in this game). And Paris Hilton? I think she might be a queen (with a small "q")--as someone who is ultra-well-known but not for anything in particular except for the quality of being well-known.
The title of this quick post seems both irresistibly attractive and horribly repelling in an oh-g_d-is-this-what-we're-down-to dissertation for a moderately-good university. But really all this post is is a title--I stumbled upon this table looking for emigration figures to illustrate a dot-matrix map from the United States Industrial Commission (printed in 1900), volume 3, which concentrated on the statics and sociology of prison labor. So what this table shows is the effect of prisoners' labor on the price of pork and pork fat in the Chicago market for a ten year period at the end of the 19th century, and what we see is that the "free" labor in prison in this area produced cheaper prices in the fat market. There you have it. [Source: Internet Archive, here.]
Can We See More or Less than We Used To Be Able To See?
An early study of attention and perception (or “How Many Items Can it Embrace at Once?”) popped out at me while muscling my way through another year of Nature magazine for 1871. The article was by the polymatic W. Stanley Jevons ("The Power of Numerical Discrimination," in Nature volume III, 18711) who contributes an interesting and very early experimental bit on the success of the brain to correctly formulate an accurate memory when in a flash shown a number of items. (That is to say, when shown a certain group of X-number of items instantaneously and then removed, how often will the mind be able to remember the correct number upon recall--and without committing them to memory per se or counting them?) In this fascinating study Jevons records not only right/wrong answers but how "close" the remembered fit is to the original number, and in effect is a pioneering scientific effort towards understanding our abilities and limits in information processing. And as it turns out the ability to precisely recognize and remember groups of objects with success and without counting stops at about four items for the vast number of people texted. (It is another display of a famous four, including the four faces of Brahma, directions, Gospels, minute mile, playing card suits, seasons, corners of a square, virtues, color problem and of course four- letter words, to name a few.)
[Source: University of Wisconsin Digital Collections, here.]
Its important to distinguish Jevons’ experimental work on apprehension from earlier (and much earlier) philosophical
and semi-scientific work on memory formation and retention. This of course goes back as far or as deep as you want to go—taking a stab at random we’ll use Simonides who while trying to organize poetry and other data in his head came upon the idea of using Mnemonics and using associative processes in art and poetry to establish his own history of memory. (I should point out that one of the early-modern experimenters in the formation of memory was Giordano Bruno (at right) who wound up being tortured and burnt at the stake for other offense against The God while trying to formulate a truthful approach to science; evidently the memories that would be threatened by his scientific approach proved to be too much for the righteous in power, and he was removed before he could threaten corrective memory any further.)
It is interesting that many of the crimes of science punished by the Catholic Church during this period (1450-1650) were as much crimes against memory than they were crimes against the future—changing and challenging collective memory often proved fatal.
Getting back to Jevons—who was a very smart guy and who applied himself to a number of fields, not the least of which was constructing a logic machine: his experiment proved to be a springboard for a host of others, some of which didn’t do Jevons justice, misreporting his finding, misrepresenting the stuffy, and so forth. Perhaps the greatest of these was the greatest of the efforts based on his—Raymond Cattell —who for some reason stated in his very influential textbook of 1907 that humans can remember around 7 things (without counting) when the objects are flashed before their eyes. And for a hundred years this figure stuck, even though the Jevons report issued a more complex summation, and saying, anyway, that the number was around 10.
And what bothered me a bit with the Jevons experiment is what people remembered when shown the objects (beans)—would his results have varied if subjects were asked how many were shiny or odd-colored or deformed or whatever rather than just a simple number, the results could’ve been more interesting. (I don’t doubt that these issues have been taken up in the 20th century but haven’t looked). Sometimes people are just looking in the wrong places—for example operant psychology labs tested rats via visual stimuli until it was discovered that rats were olfactory geniuses and that humans were using them wrong all along.
It would be interesting to know what the history is of human capacity for image formation. Considering the growth of distractions and the enormous amount of true and trash stimuli—visual bombardment from television, outdoor advertisements, the sheer amount of growth of human construction and interaction—have humans enhanced this spatial/memory information processing capacity? Consider the growth of (just) eye movements over the last few hundred years, with the visual sense being subjected (for all classes of people) to enormously and fractally-expanded print sources, television, digital communication and so on—has this expanded this neural capacity? Has "space invaders" aided bean counting?
I don’t know, though I do wonder (literally) what the effect exponentially-growing mass input of (mostly junk) data might be doing to our noggins. Maybe the effects lean more towards dissolving privacy and reflective time—when does a person think if interruptive stuff is coming into your head at all points of the day, with the brain trying to interpret incomplete and ambiguous strings of sensory inputs?
Seems scary to me. Maybe memory is affected, maybe it makes it go away, shriveled because recollection is being eliminated. Or demented like the wonderful Yossarian (his first name is John, btw) from Joe Heller’s beautiful Catch-22 who develops for himself a condition in which he remembers everything twice (whatever that means).
It would be interesting to see a war of societies in which the sides were a culture that remember
nothing versus a culture that remembered everything. The unspeakably lovely Jorge Borges wrote something touching on this in "Fumes the Memorist," in which the humble narrator is capable of forgetting nothing, being able to recall explicitly everything within eyesight, perfectly—the problem is though that it takes a day to remember the events of another day. What would happen in such a culture where everyone forgot nothing?
And I’ve just been dealing here with visual memory, really—and as Proust makes plainly clear (and Borges and other prove), vision isn’t everything.
I'm not sure that I've ever seen a list of the personal daily cost of antique drug use, though I did manage to stumble across one in a remarkable little pamphlet by Edward C. Jandy called Narcotic addiction as a Factor in Petty Larcency in Detroit (published November 1937). There's a lot packed into its 23 pages, not the least of which is a pretty sophisticated look at how to examine the costs of drug addiction to the sales economy of that city. One of the interesting historical bits that emerges from it is a list of the daily cost of the addiction of one of the target study groups--a selection of 43 local addicts with a combined 673 years of addiction (averaging an unholy 15.5 years of addiction/person).
[Full list in the Continued Reading section.]
There are immediate limitations to this info--for example there is no correlation to the number of years of addiction to the individually-reported daily drug costs--but since this data seems to be fairly rare it does at least give some idea of the strain of usage per person.
And what does it mean to spend $5/day on your heroin habit? CPI is useful, but it is better to look at what that figure means in terms of the average salary and costs of basic goods. If you were working back there in a bad spot of the Depression in 1937 the average salary was about $1,700/year, which means that if these addicts were working (and the great majority wasn't) then they would be spending about 1/3% of their annual income per day--or a little more than all of their daily salary--on their everyday habit. Spending $1,800 a year on drugs on a $1,700 salary leaves not-so-much-room for anything else but crime, and not having any income at all would mean that all of that money would have to be from criminal activities.
In another (potentially gross) way of thinking about this expense is by looking at the average salary in 1937 being about 1/30th of what the average American family income is in 2012, so the daily $5 heroin hit would be something like $150 today, which sounds about right. And if you applied that multiplier to some other standard 1937 prices1, the numbers are fairly constant from then to now--the big exception being postage stamps (which would be $1.50 for a first class stamp) and gasoline ($6/gallon), both of which would show a decline. Again, that's a very crude approximation, but it does pause.
The author then does some tricky and interesting semi-statistical work with the bottom line showing that drug addicts stole a total of about 3% of the total retail sales (of $545 million) in the U.S. That's a big number--in today's economy, which currently stands at about $33 billion in thefts (or 1.5%) that would 3% for just addicts would be an enormous number, twice the national general total which would spike drug losses at $100 billion for theft alone.
I'm thinking that these 1937 stats might be a little (or a lot) loose, but it the report still is decently argued and nicely presented though the data might be not-great--and the daily/habit numbers are a fine thing to find.
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..."
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.
"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).
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…”