A Daily History of Holes, Dots, Lines, the Unintentional Absurd & Nothing |1.6 million words, 7000 images, 3 million hits| History of Science, Math & Tech | Press & appearances in The Times, Le Figaro, The Economist, The Guardian, Discovery News, Slate, Le Monde, Sci American Blogs, Le Point, and many other places...
With "all" of the recent talk about prime numbers I thought to post this elegant and small summary of the different ways of writing numbers.
Above is a lovely detail of a collection of "ancient arithmetical characters", including notations for the numbers 1 through 9 by "Boethius, Plenudes, al Sephadi, Sacro Bosco, Indian, Roger Bacon, and AL Sephadi".
This was a surprise, finding M. Bollee's article (Sur une nouvelle machine a calculer) in this 1889 Comptes Rendus, pecking around in that big 10-pound volume looking for something else. It was very easy to miss if you weren't looking for it, just a few pages long in a 1000-page book. But there it was, nestled comfortably in pp 737-739. It these few pages Bollee describes his machine and with particular reference to his innovative approach to direct multipilication--a fine addition (ha!) to the long line of contributions by Babbage and Clement, Scheutz, Wiberg and Grant and Hamann.
Léon Bollée: "Sur une nouvelle machine a calculer", in Comptes Rendus de l'Academie Sciences (Paris), volume 109, 1889, pp. 737-9.
An image of the machine from The Manufacturer and Builder:
There is an interesting side note to this blog's series on the histories of holes and dots--a mathematical aspect involving decimal points, decimal notation and placeholders. This is exclusive of the number zero, however, which is an entirely different topic.
The book that this beautifully-illustrated counting board (below) is found is in Gregor Reisch's (1467-1525) Margarita Philosophica (1503) and depicts (amidst much else in the greatly humanist volume) representations of the mathematicians Boethius and Pythagoras working math problems on the given tools of their day. The tools on the right seem to be circles, but they're not--they're counting stones, and for our intents and purposes here, they shall be dots, and in the history of dots in math and business reckoning they have had a strong and long life.
We can see in his expression that Boethius, on the left, is rather enjoying himself, knowing the superiority of his system of counting, which was the the Hindu-Arabic number notation--he definitely has a sly, self-appreciating smile on his face. Pythagoras, working with the old counting table, definitely looks worried, or at least unhappy, unsettled. Never mind that Pythagoras (570-495 b.c.e., none of whose works exist in the original, another sort of entry in our Blank History category) was at a definite disadvantage in the calculating department, being dead and all that for hundreds of years before the Arabic notation was more widely introduced in the West, probably being introduced by Pisano/Fibonnaci in the 12th century. But it does fall to Boethius, the smirker, to have introduced the digits into Europe for the very first time, deep into the history of the Roman Empire, in the 6th century.
The numerical stand-ins in the Reisch book with which Pythagoras worked were blank, coin-like slugs used as placeholders, and would be used in place of rocks or pebbles or whatever other material was at hand. It is interesting to note that the Latin expression, "calculos ponere", which basically means "to calculate"or "to compute", is more literally translated into "to set counters" or "to place pebbles" (upon a counting board) or to set an argument2, which is exactly what some of the Roman daily reckoners would do at their work. And also used, in this case, by the unhappy Pythagoras.
The foundation for the .14159... that comes to the right of the integer 3 in pi is a relatively recent idea in the history of the maths--at least so far as the represrntation of the ideas in numbers and the decimal point is concerned.
Simon Stevin (1548-1620) introduced the idea of decimal numbers in his 36-page De Thiende ('The Art of Tenths"1) in 1585, an idea that replaced much more cumbersome earlier methods of representation. So, the number 3.14159 would be written in the Stevein notation as (where in this case numbers enclosed by brackets, i.e. "" would have been represented in print as a 9 within a circle) 314159. It is also seen here:
The importance of the introduction of this idea is difficult to underestimate, according to many and by example the The Princeton Companion to Mathematics by Timothy Gowers:
The Flemish mathematician and engineer Simon Stevin is remembered for his study of decimal fractions. Although he was not the first to use decimal fractions (they are found in the work of the tenth-century Islamic mathematician al-Uqlidisi),it was his tract De Thiende (“The tenth”), published in 1585 and translated into English (as Disme: The Art of Tenths, or Decimall Arithmetike Teaching ) in 1608, that led to their widespread adoption in Europe. Stevin, however, did not use the notation we use today. He drew circles around the exponents of the powers of one tenth: thus he wrote 7.3486 as 7�3�4�8�6�4. In De Thiende Stevin not only demonstrated how decimal fractions could be used but also advocated that a decimal system should be used for weights and measures and for coinage.
This idea would be further developed by Bartholomeus Pitiscus (1561-1613) who was the first to introduce the decimal point in 16123. It was a far more robust and simple was of dealing with decimal notation than anything that had come before.
2. The Reisch book is remarkable: it is basically a Renaissance encyclopedia of general knowledge, divided into twelve books: grammar, dialectics, rhetoric, arithmetic, music, geometry, astronomy, physics, natural history, physiology, psychology, and ethics.
3. Pitiscus was also the first to introduce the term "trigonometry" earlier in 1595 in a highly important and influential work he produced in 1595.
I wrote yesterday about found poetry in Simson's Elements of the Conic Sections (1804)--there was much more. Some of the owners of this book used it pretty hard over the years (mostly in the early decades of the 19th century), and there are numerous examples of worked problems in the margins and in the folding engraved plates of illustrations of proofs.
They are beautiful works. But one of the other finds in this book is written out on the rear paste-down, which contains (on one sheet of paper) the entire 1806 class for the bachelor of arts degree at Princeton College for 1806. ( A little more information on these Princetonians is available at the General Catalogue, Princeton Universtiy--the list checks out to a man.)
It is interesting to consider that an entire attending class of what is today a very major institution could be easily listed on one 8x5-inch piece of paper.
In my travels in and out of imagery and books, I've kept a small space allocated in what is left of my memory palace for images of the working poor and the laboring classes. Most of the world's population of course has been and still is composed chiefly of the daily (and not so) laborer, but if you were to measure images that were made of these folks while at work or with the tools of their trade, and compare it to the rest of the images of more-exalted people spending their time doing more-exalted things, I would guess that the images of the working classes would be vastly outweighed by the rest, at least in the ancinet to early-modern times (of say 1925)--that is why it is so very interesting to see third-quarter 19th century photographs of these people.
This image was made by William Carrick (1827-1878, born in Scotland but who spent his life chiefly in Russia) in the 1860's, and is found at the National Gallery of Scotland, here. Carrick made important contributions to the ethnography and the history of photogaphy, documenting the lifestyle and costume of Russian peasants in the country. It is probable that he made his series cartes de visite images of itinerant trades people as a sort of postcard to the tourist and taveling class--but what happened of course is that he recorded in great detail bits of everyday life that was in general invivible to the main sttream of people who were in the image-making class. And so Carrick produced images of knife sharpeners, and tool sellers, and milk haulsers, and woodcarvers and hackmen and chimney sweeps and so on, pictures of the people who made the city run. I was particularly struck by this image of a boy selling abacuses/abaci--Im not sure that I 've seen a photograph/engraving of a street vendor of this sort of instrument, but here one is--a boy in the 'tween years, wearing boots way too big, with a woven basket full of abacuses/abaci. (The varities here have two ranks of four beads--one for quarter-rubles and the other for quarter kopeks--while the other wires hold ten beads apiece.)
Maybe the boy isn't a 'tween--maybe he's younger than that. Maybe the boy is a girl. In any event, the basket looks heavy, and I'm sure the child returned home with a weary back at the end of the day.
Edwin Abbott’s slender Flatland, a Romance of Many Dimensions is perhaps one of the best books ever
written on perception and dimensions, a beautifully insightful book that
was quick and sharp, and in spite of all that was also a best-seller.
Written in 1884 when Abbott was 46 (Abbott would live another 46 years
and enjoy the book’s popular reception), it introduces the reader to a
two dimensional world with a social structure in which the more sides of
your object equals power and esteem. Thus the lowest class would be a
triangle (three sides) while the highest (priestly) class would be
mega-polygons whose shape would approach a circle. Abbott’s magistry
comes in explaining to the three-dimensional reader what it was like to
be in a two-dimensional world.
This interesting, unusual, and scarce publication was printed for the New York World's Fair, 1939, as part of the French pavilion. It is only seven pages long, but its a useful seven pages, and very seldom seen.
It seems that for such a great number as pi the naming of had little (where "little" = "no") fanfare in its introduction.
Since it is just after 1:59 on this pi day of 2013 (3.14159...) it might be nice to have small bit of history on the first use of the symbol for pi. Pi the idea is ancient (on the ratio between the diameter of a circle and its circumference) but using the Greek symbol for the word is relatively new. It shows up for about he first time here:
on page 267 of William Jones' (1675-1749) Synopsis Palmariorum..., which was in general a book for beginners in semi-advanced mathematics, and which was published in 1706. Pi is represented as a square or the Hebrew letter "mem" in John Wallis' Arithmetica infinitorum (published in Oxford in 1655), according to Florian Cajori in his A History of Mathematical Notations (volume II page 8), though William Oughtred had used the Greek letter in fractional form to designate the ratio as the periphery and the denominator as the diameter. Still, it was Jones who introduced the letter pi as the sign for the ratio, and he did it without saying very much at all about it. Actually, he says nothing about it at all--just states the case, and that was it.
Cajori writes on to say that Jones used the symbol earlier in the book, but in a different context, and that the immediate reaction to the symbol (if there was a reaction) was also a bit of nothing, with many writers continuing to use different symbols to talk about pi for several more decades, and that it didn't really come into use in general textbooks in German and French until nearly 1800.
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".
(Revisiting an earlier post from 2008 with a added details).
While sitting at the edge of a stream in a flowing dress and shading herself with a yellow parasol, my wife and the jewel in the crown, the brilliant Patti Digh, wrote a superb essay on following the paths less marked. (Maybe I'm making up the parasol bit. And the stream. And flowing dress.) I've always been a happy observer of found geometries, and her essay heightened my awareness to them, along with the conscious efforts that we may daily make to enforce these lines or follow a new, multi-dimensional geometric path.
So when I stumbled across this map of Alexander Pope's garden and noticed the odd paths winding their way through it, and how much in contrast this was to the very sniffy upper crust English garden of the 18th century, Patti's essay lit instantly to mind—and also because in addition to being a geometer of emotion and thought she was in a past life a U VA English Master.
Pope's garden was an irregular path in an irregular garden, that garden being his leased property at Twickenham, where he came to live in 1717, along with his mother and his boyhood nurse, Mary Beach, and his dog Bounce.
Pope's garden was an irregular path in an irregular garden, that garden being his leased property at Twickenham, where he came to live in 1717, along with his mother and his boyhood nurse, Mary Beach, and his dog Bounce. (Actually that would be a series of dogs as he named them all "Bounce".) Pope was a poet and satirist and critic and general taste-maker-the social influence extending even to garden-making, as his was famous even during his lifetime, and not for the reasons that most English gardens were famous.
His was not the planned regularity of the gentility (and far of course from the suffocating, critically flawed gardens at places like Versailles, "Is there anything more shocking than the regularity of a planned garden?" asks Batty Langley in his New Principles of the Gardening… 1728), and were removed from the "normal", more articulated garden. I've never really understood the formal garden, especially as it relates to the ancient analogies of gardens and paradise-the Garden of Eden, Pliny's Garden of Venus, Lucretius' Earthly Paradise, seem as though they might be some sort of paradisical thing (though we won’t address Bosch’s Garden of Earthly Delights) , but the formal gardens and its paradise relationship seems odd, what with the removal of nature's "chaos" and all, unless that was indeed the Master Plan.
But the mathematical garden was a desirous thing—still is—and none other than Christopher Wren wrote that it was “naturally more beautiful than any irregular figure” (1750). There were some critics of the math garden, like Mr. Langley, but most garden designers wrote and argued in favor of it, coming to the similar conclusion of John Dennis, in that the Universe, being “regular in all its parts” would dictate that its beauty be found as regular in other things too, “and it is that exact regularity that [the mathematical garden] owes its admirable beauty” (1704). Pope didn't seem to address (in print) the garden idea formally except for a few places-- Pope praised 'the amiable simplicity of unadorned Nature' in a famous essay, published in The Guardian in 1713, and then again in his message to Burlinton in 1731.
The plan of Pope’s garden was somewhat irregular, but to me the most appealing aspect was the irregularity of the paths that curled and snaked their way through the garden. And particularly, what about the lower part of the path along the bottom of J. Serle’s 1725 engraving? What was that path in Pope's garden leading to, or from? Was there a particularly nice tree there, a favorite spot, a cool shade, an odd bit of sunshine at sundown? Or was this wall-clinging path just the workman's route?
Perhaps these weren’t Pope's principal desire lines in the garden—in fact, the garden desire lines weren’t the garden at all— they might’ve been under it. Pope’s famous grotto at Twickenham may be better known than his garden-it certainly sounds much more interesting. Followed his own desire line, underground, in his famous grotto, which he began to build shortly after arriving at Twickenham, and which was still under construction at the time of his death.
The unseen desire lines of Alexander Pope seem more fitting to me, more poetic, what with Pope needing to dig out the earth in order to find what he was looking for, rather than a more simple wandering around his fine, rustic garden.
Have a look at the materials house in the grotto/cabinet of curiosity here, along with a commendable plan of the grotto itself.
This beautiful table appeared in Roswell W Smith's Practical and Mental Arithmetic, on a New Plan, in Which Mental Arithmetic is Combined with the Use of a Slate..., which was a soaringly successful book, this being the 53rd edition (!) and printed in Hartford in 1836. (Smith was a busy man, writing several other standard and widely distributed works in geography and grammar).
But what we find here, in one of the many tables in the book, is the surprising "3 times 3 are 9" rather than what we normally hear in the U.S. as "3 times 3 is 9". Perhaps what we are saying is "three threes are nine" or "seven threes are twenty-one", and so perhaps this really should be "are" rather than "is", except in the case of a number multiplied by one, when we would use "is" (as in "1x3 is 3", or "one three is three"). This would not be so for "3x1" where we are saying "three ones are three").
We can also disregard both of those words and simply say "equal". Or is that "equals"?
Its a small thing, I know, and I was attracted to the beauty and simplicity and assurance of the table--it is simply a finely designed thing.
It seems so unlikely that the Venn diagrams took so long to appear, that it hadn't been Aristotle's Diagrams, or Cartesian Diagrams; rather, these elegant creations--about a different way to represent propositions by diagrams-- belong to Mr. Venn's invention in 1880. Mostly. It was in that year that these famous more fully-developed diagrams first appeared in a paper entitled "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings" in the Philosophical Magazine and Journal of Science. They seem so beautiful, and so simple, and so powerful, and so elegant. So necessary. And so why-did-they-take-so-long-to-appear, so obvious. In a way, they are in that category of The Great Obvious Invention, in a similar vein to Mr. Luke Howard first classifying the names of clouds in 1803. Sometimes things are not obvious until they are.
[Source: Internet Archives, John Venn, Symbolic Logic, 1881, full text.]
Well, almost. The diagrams did mostly make an appearance earlier--though not in the Venn form--in the work of Christian Weise (d. 1708) and then in that of Leonhard Euler (1707-1783), that according to the great William Rowan Hamilton, explained in his work Metaphysics and Logic (1858-1860). But the final form seems to be that of Venn.
Hamilton's work appears so:
They also appeared in the Proceedings of the Cambridge Philosophical Society,
volume 4, 1880, in his article "On the employment of geometrical
diagrams for the sensible representation of logical propositions" (here) in which he discusses the history of the diagrams, including the Hamiton history. Venn also includes this beautiful logic template of Bolzano:
I found a wonderful book review of Venn's book Symbolic Logic that appeared the following year, appearing in the journal Nature on 14 July 1881, and reviewed by the inimitable W. Stanley Jevons (which is reproduced below).
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..."