A Daily History of Holes, Dots, Lines, Science, History, Math, the Unintentional Absurd & Nothing |1.6 million words, 7000 images, 3.5 million hits| Press & appearances in The Times, The Paris Review, Le Figaro, The Economist, The Guardian, Discovery News, Slate, Le Monde, Sci American Blogs, Le Point, and many other places... 3,000+ total posts
This bibliography for George Boole just came in very handy, so I decided to distribute it. It appears in Treatise on Differential Equations, printed in 1865 and edited by Isaac Todhunter, the full text of which is found here. A very good entry on Boole is found in the Encyclopedia Britannica 11th, here.
The question above is probably seldom asked and probably even less than so required, but it sure does look good for the title of a post. The pillowed polyhedra came into light while I was breezing through some issues of the British Nature magazine for 1893/4--the title of the work of course was a lovely thought and the author of the paper was the mega-gifted and highly significant Lord Kelvin (William Thomson), but the bait that made me swallow the hook on the paper was the illustration.
There are many (i.e., 1,496,225,352) different forms of the convex 14-faced polyhedron tetrakaidekahedron (see Wolfram Math
for a quick summary)--this is the first I can remember being displayed
on a 19th century pillow. Kelvin was particularly interested in
displaying geometrical figures in three dimensional space, and would
return to the issue numerous times, particularly in the Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light (see page 606ff).
Lord Kelvin, (William Thomson, 1824-1907), "On the Homogeneous Division of Space", in Nature, two issues, as follows: (1) 8 March 1894, pp 445-448; and (2) 15 March 1894, pp 469-471. A version of this paper appears slightly earlier (17 January) in the Proceedings of the Royal Society.
Both papers are available for purchase via our blog bookstore, here.
These are remarkably interesting papers--especially considering their brevity.
It is interesting to note the list of what Thomson "is known for" in his Wiki entry:
(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.”
There is something enormously appealing in the general nature of old numbers, numbers written or printed long ago, numbers making an appearance in the general sense of ordinary and commonplace, everyday garden variety numbers (like the example above and it continuation below), as well as in more famous numbers, numbers that present a concept for the first time, or offer a proof in thought and conjecture (as seen further below with Mr. Stevin).
The first example is from a worn copy of a common early-ish 19th century American math textbook by Rosell C. Smith, Practical and Mental Arithmetic, on a New Plan, in which Mental Arithmetic is Combined with the Use of the Slate... which was printed in Hartford beginning in 1829 (my copy being printed in 8136). It was a popular book, and it claimed to make math more useful by using calculations for problems to be figured in terns of dollars and cents, thus giving the exercises the chance of direct application to the daily grind. My copy of this book is very worn--not the worn that comes from mistreatment, but rather use-worn, the book being smooth and lustrous from repeated and deep use, handled so much over the years that the paper covers have a very definite and smooth patina.
In any event Mr. Smith's numbers have a special bit to them, something nor-quite-like-everything-else. The care and the design and placement of the numbers is very attractive, even if it makes the numbers sometimes a little illegible.
The numbers have a certain beauty to them, as does the space aloted for their answers:
Famous numbers have a distinct beauty as well, in the more refined and exalted antithesis as those numbers for a simple sum problem: from two ends of the spectrum ,sometimes, though they both meed in the middle where the numerological beauty occurs. A great example f famous numbers might belong with Simon Stevin (1548-1620), who introduced the idea of decimal numbers in his 36-page De Thiende ('The Art of Tenths")
in 1585 His was an idea that replaced much more cumbersome earlier methods of
representation. So, the number 3.14159 would be written in the Stevin
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:
One item that attracted my attention--easily so--was the following problem:
It was also the only illustration in the 284-page book. And it makes sense, I think, because squirrel hunting is just what people did at this time, and the calculation could be a useful one. Still, it is an unusual image to set to work illustrating a math problem--and interesting.
Babe Ruth x Hank Aaron = the product of the first seven primes multiplied in succession.
Babe Ruth hit 714 home runs over his long and incredible career, a record that would stand until the fantastic Henry Aaron broke it on 8 April 1974, hitting his own 715th home run. Multiplying the two numbers makes an interesting product:
510510=2x3x5x7x11x13x17, which is the first seven primes. And the first seven primes add up to 58, which is close to the record 60 single-season home run mark set by Ruth on 30 September 1927 off a .493 lifetime w/l pitcher in Tom Zachary, (but close doesn't count). The average of those primes is 8.2--Aaron wore the number 44. Again, "close" doesn't count, but it is a fun observation.
I must point out that this is not my own work--its a remembered bit from The Long Ago and I just can't remember the originator.
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".