A Daily History of Holes, Dots, Lines, Science, History, Math, the Unintentional Absurd & Nothing |1.6 million words, 7500 images, 3.9 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... 4,000+ total posts
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.
[Associated posts: The Mother of all Renaissance Logical Graphs, The Knight's Tour, Porphyry and Boethius and Census Art and the Display of Quantitative Data, 1860.]
Well now:I don’t know what the provocation or inducement is here to hurtle this axe-swinging monk to attack Porphyry’s Tree1, though it would be interesting in a forensic sort of way to know what the tree’s section might reveal. The “tree” was a diagrammatic creation of a 3rd century Syrian mathematician/logician/philosopher named Porphyry who-- much taken with Aristotle (and with the Categories in particular)-- developed a systematic approach to the organization of thought in diagrammatic form.
What’s inside a tree of logic and memory?Is there axylem-y/phloem-y stuff besides a three-dimensional representation of the structure of organizational thinking?Or is the 2-dimensional rip a fatal blow to other dimensions, and like Eddington’s Turtles, it’s a simple slice of Flatland all the way down?
Perhaps Porphyry’s tree rings would look like this, a magic circle or spiral, which would make some sense, and would bring to bear an associated use of turtles—or tortoises, I should say.It turns out that perhaps the very first use of the magic circle, rolling back its origins through the Islamic world to India and to Persia and then to Japan, and then finally to China where, in about 2000 BCE, the magic square appears in an image with the Emperor Yu, inscribed on the back of a tortoise.
1.This image appears in the rareDestructio sive eradicatio totius arboris Porphirii : magni philosophi ac sacrae theologiae doctoris eximii Augustini Anchonitani ordinis fratrum Heremitarum Sancti Augustini, cũ quadã decretali eiusde, published in 1503.
JF Ptak Science Books Post 692 (from 2009) Expanded
This post was written five yers ago when I was excited to find an illustration of a dreamer and numbers with some mathematical content. I thought that it was pretty unusual to find as an illsutrastion, even though I had read many stories of people being influenced byt their dreams, evem in the sciences, but that there were few images of the process. In Post 692 I psoted a found image of such a thing that was just a simple advertisement in a German magazine. There was also a series of images from Francis Galton on imaging arithmetical processes from Nature back in 1880, which may also be the first scientific article on synesthesia. And there doesn't seem to be a lot more than this, even in canonical illustration.
So it came as a surprise today to find this photo to add to the small collection of math dreams and imagery of "doing" math (from ca. 1880):
Thinking big thoughts in dreams is generally not a common thing, as anyone who has read their own semi-conscious half-awake memory notes of a dream-based inspiration could attest. But it does happen:
Paul McCartney1 dreamed the song Yesterday, Gandhi dreamed the source of non-violent resistance, Elias Howe dreamed of the construction of the first sewing machine, and Mary Shelley the creation of her novel Frankenstein... For good or for ill, William Blake was evidently deeply influenced by his own dreams; on the other hand, Rene Magritte was deeply influenced by dreams but didn’t use any of his own for his paintings, or so it was said. Otto Loewi turned an old problem into not one in a dream, finding a solution to the prickish problem of whether nerve impulses were chemical or electrical (and resulting in the Nobel for medicine in 1935); the fabulous discovery of the benzene ring came to August Kekule in a dream as well. Artists have been representing people in dreams and dreamscapes for many centuries: Durer depicted a dream in a 1525 watercolor, for example, and thousands of artists have depicted famous biblical dreams (Joseph of Pharo) for long expanses of time.
What struck me, though, in this illustration found on the other side of the page of the Illustrirte Zeitung2 (for August 1932) that I used for yesterday’s post about damming Gibraltar and Shakespeare’s memories, was the depiction of someone dreaming mathematical thoughts…or at the very least, dreaming numbers. People have undoubtedly dreamed much in mathematics, but I can not recall seeing illustrations of these dreams.
I'm differentiating here from something like a Poincarean inspiration, or vision, or thunderstrike--I'm talking about drop-dead asleep sleep, dreaming sleep, REM and all that. Also I'm differentiating this from imaging mathematical thought, as in the work of Francis Galton in 1880 in which the subject of mentally seeing the process of mathematics is perhaps first addressed. I wrote a short piece on that here, way back in Post 9. )
The numerical sequence in this dream doesn’t look like anything to me: the backwards radicand doesn’t strike anything common in my head. The geometrical drawing under the portrait in the dreamer’s room though is the impossibly iconic Pythagorean theorem, and there is a nice picture of a conic section in the foreground; but the artist, who improbably signed the work “A. Christ”, doesn’t offer much of math in the dreamscape. Still, it is a rare depiction of someone dreaming about math.
Notes 1. "I woke up with a lovely tune in my head. I thought, 'That's great, I wonder what that is?' There was an upright piano next to me, to the right of the bed by the window. I got out of bed, sat at the piano, found G, found F sharp minor 7th -- and that leads you through then to B to E minor, and finally back to E. It all leads forward logically. I liked the melody a lot, but because I'd dreamed it, I couldn't believe I'd written it. I thought, 'No, I've never written anything like this before.' But I had the tune, which was the most magic thing!" from Barry Miles (1997), Paul McCartney. 2. This is really a great sheet of paper, coming from issue 4492, pp 518-519. Two pictures of dreams on one side, with three visionary images on the other (the Gibraltar dam, a sub-polar submarine, and a futuristic Indian railway/bridge.
Very little is pretty as pie, especially when there's more than one. Double that for pi. I happened to be breezing through an 1829 book of mathematical formulas by C.P. Biel and found this lovely section on an extended computation of pi.It occurs on page 38, and carries pi out to 155 places--which for the time was very significant. (Pi was computed to 9 places by Francoise Viete in 1579; 15 places by Adriaan van Roonan, 1593; 32 by Ludolph van Ceulen in 1596; 35 by Willebrord Snell in 1621; 38 by Christoph Grienberger; 75 by Abraham Sharp in 1699; 100 by John Machin in 1706; 137 by Jurj Vega in 1794; and 152 by Legendre in 1794.)
Crusing through some issues of Nature down in the studio--this is the current long-lived Nature, though mine are from 1869-1949, so not-so modern--I was working my way through a few issues before all hell breaks loose with the Roentgen publications in late 1895, and found this very interesting article by Francis Galton. Now Galton was a very interesting man with very advanced interests in many different fields, though he does have some unfortunate bits to his personality and eugenic-based beliefs, so he is an extremely accomplished if not a problematic man. Actually one of my earliest posts on this blog (more than 2,500 posts ago--yes there are only 2147 numbered posts but there are also something like 500 unnumbered "quick posts" as well) was relating another occasion of thumbing my way through another Nature volume, and finding Sir Francis again, though in this one he wrote what may be among the earliest papers on synesthesia ("Visualized Numerals", 1880, appearing here, with full text).
The present Galton (in 15 November 1894) is his review of work done by Alfred Binet, "Psychology of Mental Arithmeticians and Blindfold Chess-Players", who looked at the extraordinary abilities of "two groups of remarkable men", one of which possessed fantastic mental calculating abilities, and the other with the capacity of playing multiple games of chess while blindfolded. A closer look at the first group revealed two men who relied on quite different benefits: one having a great ability to calculate according to "imagined sounds" and another who "relied on...imagined sounds", both endowed with considerable numerical abilities as well as memories.
Galton reviews a number of calculators and then remarks on experiments he performed on himself, trying to "visualize" a calculating process via olfactory means. The rest of the story of this fascinating article can be found here. (Also see--from Galton.org--a bibliography of his journal articles works in psychology.)
This is somewhat off the mark, but it may be interesting--given Galton's olfactory/math experiment--to have a quick look at an earlier post on this blog regarding the first photograph of a smell, here.
I should say that Galton wrote more for Nature than any other journal, some 115 contributions from 1870-1910.
See also this take on the Galton from J.M. Stoddart's The New Science Review, 1894/5:
"When I imagine a triangle, even though such a figure may exist nowhere in the world except in my thought, indeed may never have existed, there is nonetheless a certain nature or form, or particular essence, of this figure that is immutable and eternal, which I did not invent, and which in no way depends on my mind.--Rene Descartes, Meditations on First Philosophy (1641) tr. John Cottingham, Descartes: Meditations on First Philosophy (1986)
This is the lovely response by the great mathematician J.J, Sylvester to Thomas Huxley's muckety comment on the lack of imnagination in the mathematical sciences. Huxley's remarks were made at a meeting of the British Association for the Advancement of Science, stating that (Mathematics) "is that study which knows nothing of observation, nothing of induction, nothing of experiment, nothing of causation"1 This quote is taken from Sylvester's quick and elegant responses in two articles in Nature, December 30, 1869 (231-3) and January 6, 1870 (pp 261-3), as "A Plea for the Mathematician" and "A Plea for the Mathematician II".
"For Sylvester, the ability to be able to imagine what the experience of space would be like in dimensions other than three is sufficient to establish the empirical basis of geometry--the three-dimensional Euclidean is not the science of space in general, but the science of the space of our experience."--Fact and Feeling: Baconian Science and the Nineteenth-Century Literary ...by Jonathan Smith, pp 181-182
Ex nihilo nihil fit/Nothing comes out of nothing.--R Descartes, Principia philosophiae, Part I, Article 49
"How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics."--mnemonic device by James Jeans on remembering pi to 15 places, where each word length assoicates a number in pi. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag, pp 44-5.
There are simpler ways of remembering something than developing an odd mnemonic for remembering pi, especially when the memory device will get you only 30 places or so...and so it hardly feels worth the effort to remember some posie and then write it down to perform a pi-matic numerical translation. This is especially so when you consider that the man recognized as being the world record holder but not so (Akira Haraguchi) committed the fist 100,000 places of pi and took nearly a day to recite it--megaefforts like this make the smaller accomplishments of remembering pi to 100 places seem fairly insignificant.
Some of the people set to remembering pi (piphilologists) use methods similar to this as memory devices, including an entire book of 10,000 words constructed in just this way--many more, I think, use a memory palace/method of loci method, locating numbers and identifying sub-patterns and placing them in connecting "departments" in the brain. (The Big Book on memory devices is by Frances Yates, The Art of Memory, University of Chicago Press, 1966; also Jonathan Spence, The Memory Palace of Matteo Ricci, Viking Penguin, 1984.)
I'm not sure what the need is for remembering the number to so many places when it seems as though the first seven digits will be sufficient for most (when "most" = "just about everything") things.
On remembering pi, from Nature, volume 72, p 558, 1905:
This is a very special effort on behalf of logic and the law, and may have been the first effort to codify the idea of the law in symbolic logic. It appeared in the journal Nature on 24 April 1913, and I think in spite of it being interesting and even pretty, it just doesn't work for me. But I'm reproducing the article (a long one for Nature) because it is really such an audacious thing, and a terrific idea, if not a good one.
I am probably not understanding this wonderful effort1 by the dead Lewis Carroll, establishing a new algorithm for doing long division. It came into view while I was looking for a paper in Nature by A.A. Michelson on his analog computer, the harmonic analyzer, a fantastic device that he used to help measure the speed of light back in 1898. Carroll (as Dodgson) appeared just a few pages away2; his obituary appears not much after the division article, thus making the paper the last of his career here on Earth.
It is a very interesting effort, and it perhaps is even brilliant and of a wonderful construct, but the bottom line is that it makes the process of division harder to do--perhaps it is mire understandable as a process, but the process itself is decidely not a preferred one. Perhaps it is appropriate for his last work to have been on mathematics, since the majority of his 20 published books relate to mathematics or logic.
I've reprinted the entire article, below, as well as a list of his mathematical efforts.
(The article may be purchased via this blog's bookstore, here.)
1. DODGSON, Charles Lutwidge ('Lewis Carroll'). "Abridged Long Division", in Nature, page 269, in the weekly issue for 20 January 1898, with the original outer wrappers removed from larger bound volume. Very good copy. It is here that Carroll announces (in a long 3pp letter to the editor) his implementation of a new algorithm for division--it makes the process longer and more difficult, unfortunately.
The article is abstracted here, a few weeks after publication:
Article abstract: Nature 57, 390-391 (24 February 1898), "Abridged Long Division" abstracted by ROBT. W. D. CHRISTI:
"HAVING been working on similar lines for some years, I was very much interested in the late Mr. Dodgson’s letter on abridged division in NATURE of January 20, and I should like to offer a few observations and to give a variation of the method which appears much simpler. It will be admitted that Mr. Dodgson’s plan is of limited application, and rather complicated for general use. There is nothing to hinder the method given below from being universally used, though it may not in all cases be the shortest. It also has the merit, I think, of directness and uniformity."
2. This is one of three publications of Carroll's mathematical works that appeared in Nature, being "Abridged long division", Nature 57, 269-271; "Brief method of dividing a given number by 9 or 11", Nature 56, 565-566; "Pillow problems. Curiosa mathematica. Part II", Nature XLVIII. 564.
The following list of Carroll's mathematical works is taken from the table of contents of The mathematical pamphlets of Charles Lutwidge Dodgson and related piece, compiled, with introductory essays, notes, and annotations, by Francine F. Abeles. (New York : Lewis Carroll Society of North America ; Charlottesville : Distributed by the University Press of Virginia, 1994.)
1. Notes on the First Two Books of Euclid (1860)
2. Enunciations of Euclid, Books I and II (1863)
3. Fifth Book of Euclid (1868)
4. Enunciations of Euclid I-VI (1873)
5. Formulae of Plane Trigonometry (1861)
6. Formulae (Group C) (undated)
7. Simple Facts about Circle-Squaring (1882)
8. Proof Sheets: Propositions I, II (undated)
9. Question 11530 (1893)
10. Condensation of Determinants (1866)
11. Algebraical Formulae for the Use of Candidates for Responsions (1868)
12. Formulae in Algebra (1868?)
13. Algebraical Formulae and Rules for the Use of Candidates for Responsions (1870)
14. Algebra (1877)
15. Formulae (1878)
16. Question 9995 (1889)
17. Note on Question 7695 [including Question 7695 and its solution (1885)]
18. Response to "Infinitesimal or Zero?" (1886)
19. Something or Nothing? (1888)
20. Question 9588 (1889)
Arithmetic Computation and Theory
21. Arithmetical Formulae and Rules for the Use of Candidates for Responsions (1870)
22. Examples in Arithmetic (1874)
23. Arithmetic I (1870-74?)
24. Arithmetic II (1870-74?)
25. Arithmetic (undated)
26. Practical Hints on Teaching. Long Multiplication Worked with a Single Line of Figures (1879)
27. Divisibility by Seven (1884)
28. To Find the Day of the Week for Any Given Date (1887)
29. Question 9636 (1888)
30. Question 12650 (1895)
31. Number-Guessing (1896)
32. Question 13614 (1897)
33. Brief Method of Dividing a Given Number by 9 or 11 (1897)
34. Variant of Item 33 (same title, date)
35. Rule for Finding Easter-Day for Any Date till A.D. 2499 (1892-97?)
36. Abridged Long Division (1898)
37. Key-Vowel Cipher (1858)
38. Matrix Cipher (1858)
39. Alphabet Cipher (1868)
40. Telegraph Cipher (1868)
41. Circular to Mathematical Friends (1862)
42. Proof Sheets Accompanying the Circular (undated)
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).
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.”