Category: mathematics, logic

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This is just a short bibliographic note on David Hilbert’s great paper of 1900–this a result of trying to figure out where a particular French translation occurred in the history of the printing of the paper.  

The paper in question is David Hilbert  “Problemes Mathematiques“ , printed in Paris  by Librairie armand Colin (28 Fevrier 1901, 12^e annee, No. 4) in the publication Revue generale des Sciences pures et appliques. This is a revised printing of the 2nd printing of Hilbert’s “great problems” speech (taken from The Archives of Mathematics), in which the he famously posed his 23 problems.  “The Problems of Mathematics” was delivered to the Second International Congress of Mathematicians in Paris and outlined major issues of math

“…which would challenge mathematicians to solve fundamental problems in the maths for years to come.   It was a speech full of optimism for mathematics in the coming century and he felt that open problems were the sign of vitality in the subject:  The great importance of definite problems for the progress of mathematical science in general … is undeniable. … [for] as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality. … every mathematician certainly shares ..the conviction that every mathematical problem is necessarily capable of strict resolution … we hear within ourselves the constant cry: There is the problem, seek the solution. You can find it through pure thought… Hilbert’s problems included the continuum hypothesis, the well ordering of the reals, Goldbach’s conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet’s principle and many more. Many of the problems were solved during this century, and each time one of the problems was solved it was a major event for mathematics”.–A Century of Advancing Mathematics by Paul Zorn, pg 321.

It seems as though this is the second of the early printings of this classic (though third if you include an earlier and truncated French translation):

• Hilbert, “Mathematische Probleme”, lecture held at the Paris ICM 1900, Nachrichten von der Koniglichen Gesellschaft der Wissenschaften zu Gottingen (1900), HEFT 3,  making this the third quarter of 1900), pp 253-297; revised version in Archiv der Mathematik und Physik 1 (1901) 44-63, 213-237, and [58:3, pp. 290-329]
• The first English translation (by Mary W. Newson) in Bulletin of the American Mathematical Society, 8 (1901) 437-479
• French translations by L. Laugel (with corrections and additions) in Compte rendu du deuxieme congres international des mathematiciens tenu `a Paris du 6 au 12 aout 1900, E. Duporcq, ed., Gauthier-Villars, Paris, 1902, pp. 58-114; (fragments).L’Enseignement Mathematique 2 (1900) 349-354. 

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I don’t encounter “love” and “Euclid” together very often outside of “I Heart Euclid” tee shirts and mugs and such, but what we have below is a satirical/comical proof of the existence of love (within limited contexts) employing a parallelogram and Euclid. The cartoon (from Punch, printed in London in 1879) has nothing really to do with Euclid, though it does lay the 13th book, proposition 1, of his Elements (“If a straight line is cut in extreme and mean ratio, then the square on the greater segment added to the half of the whole is five times the square on the half”) on top of things.  The court (as in “courting”) for the two players actually doesn’t look much like a parallelogram, though if it is a standard tennis court, it must be a rectangle and so a parallelogram, the odd shape a result of a cartoon drawn in probably two hours with a little bit of distortion in the perspective. The lady is coy-ish, and Mr. Punch–dressed in a heart-drench tunic and clearly removed from our mortal coil–is definitely feeling something for the young lady, as we can plainly see with the various lines connecting them.  And of course Mr. Punch is clearly a prize in anyone’s estimation.

I guess some of this must have to do with the contemporary tennis craze, as this issue of Punch hosts a number of cartoons regarding seemingly everyone at play with tennis. that part of the point tally is the score “love”, the cartoonist didn’t have to go far to spread Euclid on top of the geometric situation.  

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I’ve posted a couple of other bits about the wonderful book by Roswell Smith (Practical and Mental Arithmetic on a New Plan...which was published in Hartford in 1836) here and there, as this elementary text is really chock full of interesting insights into early 19th century life in the U.S. as well as some good, logical approaches to arithmetic. It is a small book, about five inches high, and mine is particularly worn from use–its paper cover with the title page imprinted on it is half gone and mostly illegible, and is smooth to the touch from (I guess) constant handling.  I particularly like this breakdown of components of a dollar, divided into nine sections instead of four–the dollar buying quite a bit in 1836, including the purchasing power of five copies of the Smith book.  

 

 And this fine chart, nineteen pages into the work, addressing about the most basic bits of arithmetic, being numbers:

 

 

 

JF Ptak Science Books   Post 2640    

I’m returning to this earlier post, expanding it some, and correcting it a little–usually that comes with a fresh eye to something old and done. The work on hand is a mathematics manuscript by a man named Charles Fisher. It is an impressive  collection of work on square roots, from √2 to √628.  It is 13.5 x 8.5 inches and  160 pages long–it is all gathered together in separate quires, the text now just floating inside the original binding, which is very thick paper wrappers that have been handled so much that it feels like leather.   Most of the manuscript is still bound, though several “signatures” are now loose within the binding. It is a good, solid copy in spite of what I just wrote–and it is a definite work of art.  

The book is paginated according to the sq rt that Mr. Fisher was working on, so the first page is page 2 for √2. Also, the work figuring the square roots of 2 through 196 takes place on the top half or portion of the first 80 leaves, at that point he turns around and starts on the first leaf again, working out his answer for √197 (and √198) on the first page (or page 2) and so continues to the end of the book again, ending up at 628. The square roots seem pretty complete through the 400s and then gets very spotty after that, for whatever reason Mr. Fisher leaving some numbers alone. So far as I can determine the work is complete in itself.  

The (seeming)  author of this manuscript, Charles Fisher, evidently took a solitary pleasure in calculating the square roots of numbers from 2  to 628  not bothering to write down the 24 perfect squares to 576.  (The sqrt being r² = x for every non-negative real number x.) From the few bits that I have checked the man seems to have done a good job back there in the 1830’s.   (Note: we’ll deal with square root of  3 at another time…)

I cannot determine where this book was written or who Mr. Fisher was.  There is a transcription in the last leaf of the book of a community meeting dedicating people to building a meeting house in “Wertham” (?) near “Cumberland’, and that something like the minister or preacher would be shared with the local Baptist Church or something like that, and signed in January 1767.  Fact is though that there is at least one contemporary date in the work and that is 1833 for figuring the √193.  

Fisher’s 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 17.578395831246947    Mr. 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

309.00000000000000000007496379247

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 1^st, 1833. CF.”  (This may actually be 1838–I think it is dependent on interpretation.)

The lovely cover of the work:

And a double-page sample of the work:

This fine engraving of Zeno of Elea (490-430 BCE) was printed in 1739  and appeared in Veterum Illustrium Philosophorum, Poetarum, Rhetorim et Oratorum Imagines…by Jo. Petri Bellorii, and published in Rome.  It is a fairly large image for a portrait like this, measuring 8.5×5″inches on a sheet 13×9″, engraved on a very heavy and thick paper.  

From the Stanford Encyclopedia of Philosophy:

“Zeno of Elea, 5th c. B.C.E. thinker, is known exclusively for propounding a number of ingenious paradoxes. The most famous of these purport to show that motion is impossible by bringing to light apparent or latent contradictions in ordinary assumptions regarding its occurrence. Zeno also argued against the commonsense assumption that there are many things by showing in various ways how it, too, leads to contradiction…” More here: http://plato.stanford.edu/entries/zeno-elea/

It is a little curious to me that in Augustus de Morgan’s wonderful A Budget of Paradoxes that there isn’t a single hit for Zeno in the 150 mentions of “paradox” in the work, though that is probably due to my “looking around” more than anything else.  

Full text for that here at Project Gutenberg: http://www.gutenberg.org/files/23100/23100-h/23100-h.htm

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Source: unknown (perhaps Thomas Simpson, Elements of Geometry…?)

 

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I don’t know how many of you have carried the collected works of physicists or mathematicians–it seems to be a special moment, having someone’s output for their working lives right there in your hands. Some formats of some collected works are just too big and cumbersome (the many small volumes for an early edition of Galileo or the massive 20+ volumes for Huygens), but the Sir William Rowan Hamilton (1805-1865) volumes seem to be perfect, and finely representative somehow of a fine man and excellent mathematician.  So I was saddened to see this about him and his not being able to afford his F.R.S. in a stumbled-upon section of August de Morgan’s  (1806-1871) A Budget of Paradoxes (David Eugene Smith’s second edition of Sophia de Morgan’s edited first of 1872):

“This distinction, this mark set by science upon successful investigation, is of necessity a class-distinction. Rowan Hamilton, one of the greatest names of our day in mathematical science, never could attach F.R.S. to his name—he could not afford it. There is a condition precedent—Four Red Sovereigns. It is four pounds a year, or—to those who have contributed to the Transactions—forty pounds down. This is as it should be: the Society must be supported. But it is not as it should be that a kind of title of honor should be forged, that a body should take upon itself to confer distinctions for science, when it is in the background—and kept there when the distinction is trumpeted—that the wearer is a man who can spare four pounds a year. I am well aware that in England a person who is not gifted either by nature or art, with this amount of money power, [30]is, with the mass, a very second-rate sort of Newton, whatever he may be in the field of investigation. Even men of science, so called, have this feeling. I know that the scientific advisers of the Admiralty, who, years ago, received 100 pounds a year each for his trouble, were sneered at by a wealthy pretender as “fellows to whom a hundred a year is an object.” Dr. Thomas Young was one of them. To a bookish man—I mean a man who can manage to collect books—there is no tax. To myself, for example, 40 pounds worth of books deducted from my shelves, and the life-use of the Society’s splendid library instead, would have been a capital exchange. But there may be, and are, men who want books, and cannot pay the Society’s price. The Council would be very liberal in allowing books to be consulted. I have no doubt that if a known investigator were to call and ask to look at certain books, the Assistant-Secretary would forthwith seat him with the books before him, absence of F.R.S. not in any wise withstanding. But this is not like having the right to consult any book on any day, and to take it away, if farther wanted.” –Page 30 (with the italics in the original), from the full text via Project Gutenberg: http://www.gutenberg.org/files/23100/23100-h/23100-h.htm

[Hamilton portrait source: http://www.hamilton2005.ie/]

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This fine engraving of Zeno of Elea (490-430 BCE) was printed in 1739  and appeared in Veterum Illustrium Philosophorum, Poetarum, Rhetorim et Oratorum Imagines…by Jo. Petri Bellorii, and published in Rome.  It is a fairly large image for a portrait like this, measuring 8.5×5″inches on a sheet 13×9″, engraved on a very heavy and thick paper.  

From the Stanford Encyclopedia of Philosophy:

“Zeno of Elea, 5th c. B.C.E. thinker, is known exclusively for propounding a number of ingenious paradoxes. The most famous of these purport to show that motion is impossible by bringing to light apparent or latent contradictions in ordinary assumptions regarding its occurrence. Zeno also argued against the commonsense assumption that there are many things by showing in various ways how it, too, leads to contradiction…” More here: http://plato.stanford.edu/entries/zeno-elea/

It is a little curious to me that in Augustus de Morgan’s wonderful A Budget of Paradoxes that there isn’t a single hit for Zeno in the 150 mentions of “paradox” in the work. Full text for that here at Project Gutenberg: http://www.gutenberg.org/files/23100/23100-h/23100-h.htm

If you’d like to own the original, check out “Zeno” in the blog’s bookstore:https://historyofideasblog.com/books/2014/05/catalog-1-2014-physics-math.html