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
“O time, thou must untangle this, not I. It is too hard a knot for me t'untie.”--William Shakespeare, Twelfth Night
This beautiful collection of knots and splices appeared on the front cover of Scientific American, March 18, 1871--a knot for every need. Knots become ever more complex at about the same time in the hands of the great mathematician and teacher, P.G. Tait. Tait was the major domo of knot classifiers so significant in the developing field of topology, and followed the work of Vandermonde, Gauss, and Kelvin. But here, in 1871, these were just beautiful and useful knots having open ends, and not having anything to do with the Tait conjecture--that would come later.
I've missed this pamphlet for years--tall, slender, no spine title--until this morning, when it came to life. It is the work of Miles C. Hartley, who deconstructed polyhedron for the purpose of reconstructing them in a classroom as "sensory experiments". This was his method of keeping students interested in math, or at least keeping students interested in something that hopefully led to an interest in math. Hartley recognizes that many textbooks will include models for the five regular polyhedron, and stop there, the students "stalemated". Hartley goes on to include 69 models, many of them fairly complicated, in the hope of creating a deeper interest in solid geometry and symmetry.
For example, this is a great rhombicuboctahedron, scooped out of the fourth image below, and which nicely fills a larger sheet of paper for cut-fold-paste:
If I could determine that the book was out of copyright and not just out of print I would just reprint the thing right here--but for now I've included the first 22 diagrams with scans tight enough to allow them to be printed, cropped, enlarged, and printed out again with each design on a 11x8 sheet of paper. Enjoy!
[Source: Hartley, Miles C., Patterns of Polyhedrons. This copy is the first edition of 1941, printed by Edwards Brothers lithography, 27x22cm, 29pp, and which seems to be fairly scarce. The reprints that are available in one form or another seem to all come from the "revised edition" of 1948 and 1951, which is the same size but with 48pp. I've not seen one of these and don't know if the diagrams are just spaced differently or if there are more.]
This is a slightly manipulated image of the original (below) with an overlay of a daguerreotype process. The math notes (square roots of numbers from 25-60,025 in 25 increments) occurs on the back of several pages of foolscap note "di geologiska formationrma" and by the footnotes it seems as though the notes are from 1893/5 or thereabouts. In any event the calculations have a certain beauty to them.
I'm just revisiting an earlier post about the beauty of a found mathematical notebook--its quiet assuredness, its composition, its care, its trying....just lovely, speaking of the efforts of a young person back in 1840.
For more, see http://longstreet.typepad.com/thesciencebookstore/2014/03/the-beauty-of-mathematical-manuscripts-1840.html
I was not going to write about this until it dawned on me that what happened here yesterday was a first in the history of computational logic--namely, that A.M. Turing was used to capture a baby bunny.
It was a cottontail, about 25 days old, and it got into our house of multiple cats, so it was imperative for us to find it before anything else did. The bunny was very scrappy and quick (as a bunny)--that is until it was confronted by the entry into the real world of computation and logic in the form of Turing's "On Computable Numbers" (1933) and "Computability and Definability" (1937), at which point the bunny (nicknamed "Baby") ran into a brick wall--actually, a thick wall, really.
After having cornered Baby in the kitchen I built a little fortress around it of dozens of volumes of The Journal of Symbolic Logic, which happily just happened to be on hand. Faced with this new Gargantua, the bunny simply froze in a last-ditch effort to confuse me or blend-in with the red flooring, a tactic which simply didn't work. The bunny was scooped up thanks to the gigantitude of Turing, and was able to continue its life without the bother of cats.
Even if one doesn't laugh, it might be tempting to do so with the addition of this footnote to the history of logic.
This fine engraving of Pythagoras was printed in 1739 and appeared in Veterum Illustrium Philosophorum, Poetarum, Rhetorim et Oratorum Imagines...and published by Jo. Petri Bellorii, in Rome. The engraving is very heavy and on a thick paper, and differs somewhat from image that below, which is sharper, with greater detail.
This second image I think was executed by an engraver copying Theodor Galle, which would make it 16th century, though for the life of me the print doesn't feel to me to be that age--it is old, for sure, and I would've guessed 17th century, but not 16th. I've not found this image anywhere though I've found similar that point to Galle in the 16th c.
I'm just sharing this here because it is unusual in my experience to see competing similarities in antiquarian images of scientists--except if you're talking about Newton, which is a different case.
This beautiful bit of data visualization was a significant development in the history of statistics and was employed in pioneer Sir Francis Galton's "Typical Laws of Heredity, III", which appeared in Nature magazine on April 19, 1877.
[Image: Francis Galton, "Typical Laws of Heredity, III", Nature. p 513, April 19, 1877]
This contribution by Galton is the "first major step in the development of correlation and regression analysis" according to Judy Klein's Statistical Visions in Time: A History of Time Series Analysis, 1662-1933, (page 131). She continues:
That is a provocative title, or chapter heading, but that's how it appears in a pamphlet I'm reading right now, a juicy thing that can lead to a memory palace of ideas...until you start reading the para below the head, when things get both more crystalline and fuzzy.
John Alexander Henderson, a lightning calculator and professor of math at Delaware College in New York, produced this sprightly pamphlet eponymously titled Henderson's United States Intellectual and Practical Lightning Calculator, the Unity and Decimal Method, which he published in St. Louis in 1879. (It is followed a few years later in another edition with a hundred new pages.)
What Mr. Henderson is getting at is a calculator for reducing a date to find out the day of the week a particular date falls on, from the 1st century to the 99th. On the back cover of this pamphlet he provides a tickler for this enumerating device, which is explained in another publication (Henderson's United States Unity and Decimal Method of Calculating).
And so in order to calculate the day of the week on which, say, November 26, 2014, occurs you would you the dial above as follows, but first an explanation of what is on the dial face:
"And it was then that all these kinds of things thus established received their shapes from the Ordering One, through the action of Ideas and Numbers."[Plato, Timaeus ]
Just a quick comment here about Matila Ghyka's The Geometry of Art and Life, published by Sheed and Ward in 1946. It is a beautifully-designed book, skinny even though it is 176pp (not long not short), and well and interestingly illustrated. It addresses math and art, or math in art, finding recurrent spatial proportions there, and in music, with the maths being at the center of the creative and imaginative process. The author goes further and dips into these themes in life/biology (that's the "Geometry of Life" part), where he trods in familiar antique and old grounds of the Golden Section of logarithmic spirals, and then finds himself in a very interesting position in chapter VII's "The Transmission of Geometrical Symbols and Plans". I'm not so sure that I like the book as a complete work, but the parts make for some very good reading.
"A QUARTETTE OF MATHEMATICAL GYMNASTS. "This is a pretty devastating review of the work of four mathematicians/engineers that appeared in the Scientific American on May 6, 1867: The article concludes: “We have a few other mathematical acrobats on our list, but as their summersaults were turned on another stage, we will not mention them at the present time, but we hope be- fore long to place them before the readers of the Scientific American. We will briefly observe, however, that one of them is not a thousand miles from the Navy Department, and he is still, we believe, accumulating figures with extraordi- nary cunning and industry.”It is pretty cutthroat beginning-to-end:"The errors which have lately been made in calculating the power of projectiles, the resistance of armor plates, and the force of steam vessels when used as rams, seem to indicate that a knowledge of first principles is more necessary for a correct appreciation of mechanical problems than any amount of abstract mathematical skill..."
I've written a number of posts on this site about W. Stanley Jevons, a very talent all-around smart-guy who worked in many different fields and at very high levels. Today's installment on the Jevons' from is his paper "The Power of Numerical Discrimination", published in the third year of Nature, and published in London in 1871. Jevons contributes a rather odd bit here on the success of the brain to correctly formulate an accurate memory when 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 odd and fascinating study Jevons records not only right/wrong answers but how 'close" the remembered fit is to the original number. Pretty cool, and an early effort towards understanding our abilities and limits in information processing.
"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.
Sometimes the rote and the routine as practice by young hands two hundred years ago can yield some surprising and beautiful results. And so we find the beauty in these columns of arithmetic problems, practiced by a young girl in Philadelphia in 1806. The work is determined and taken all together is just a lovely thing. (And yes there's a mistake here and there but it doesn't matter, not really.)
See also: Mathematical Art, the End of Simple Multiplication (1814), here.
And again in this "numeration table" where we see the
This is a fine detail from a late 17th century print of Gemma Frisius (1508-1555). He died young but with many accomplishments, not the least of which was being a mathematician of stature. And a cartographer. And surveyor. And instrument maker. And physician.
In the original print (available for sale in our blog bookstore) Frisius' eye occupy less than a half-square inch of paper--still they are powerful and direct in spite of their size.
The legend of the engraving read "Gemma Frisius, Doccomiensis, Medicus et Mathematicus.Ut simulat solem radiantis gemma pyropi, Sic Gemmam artifici picta tabella manu: Haec vultum dedit, ipse animi monumenta perennis; Ne quid in exstincto non superesse putes. Vita escessit Louany VIII. Kal., Iun. MCLV, Aet XLVII."
From the warehouse comes this lovely find: a manuscript notebook of a combination of elementary and slightly advanced mathematics, kept by a young person, written around 1840. It is a beautiful work, and given that it is only about 100 pages long, it is a surprisingly and refreshingly thorough review of the mathematical necessaries of the mid-19th century.