JF Ptak Science Books

Mathematical Art: "The End of Simple Multiplication", 1814

JF Ptak Science Books  Post 982

I found this fabulous example of multiplication in a student's workbook, written out in 1814.  Undoubtedly the sub-title, "the end of Simple Multiplication" was not intended to imply the literal end of multiplication (though the thought does invoke some interesting bits of fiction), just the result. It is simply a gorgeous object.

A Short Look at Fifths and Fives: Joltin’ Joe, Legal Sausage and the Structure of Mathematics

JF Ptak Science Books LLC  Post 925 

I’m going to take a drive through five fifths—all of which don’t come close to making a whole—to get to my ultimate destination, a ribbon of numbers circling the earth.

In American law, “taking the fifth (Amendment)” allows you the prohibition of incriminating yourself, giving the supposed ability to look innocent whole not saying anything about anything for fear of legal repercussions, though this sausagey implosion from guilt seldom looks pretty. 

There’s the fifth column (as in Hemingway’s play by this name), the not-fourth-but-fifth force, the musical perfect fifth of highest consonance, the Keplerian nestling of the five Platonic solids,  Beethoven’s Fifth, a fifth of (good) scotch, the fifth cardinal direction (the center), the five books of the Torah, the five pillars of Islam, the worst categorization of hurricanes and tornadoes, living in the fifth Mayan world, the fifth Fibonacci number, and on and on.

And aside from being the number that Joe DiMaggio wore on his pinstripes, it is also the number of the  Euclidean postulate that called to all those who were interested in the possibilities of the postulate not being so—that there exists as perfect a non-Euclidean system as there is for the Euclidean.  The fact of the matter is that this calls into question the foundation, the very basis, of mathematics.  Or used to.

All of this was brought into mind by this 1957 ad for the RECOMP computer produced by Autonetics.  The image of a sheet of numbers circling the earth¹, the product of digital computation, suggested structure, mathematical imperative and solidification of numbers.  But the point of this was the bottom line: what was the point, and for that matter, what was the point?  They were hardly physical things floating in space, but ideas (except of course in the worlds of Flatland). 

Euclidean structure was the infallible, rigid, predictive, explanatory structure and the most accurate descriptor of physical space.  This had certainly been the case for fifteen hundred years or so, or at least until the struggling fifth postulate of Euclid’s Elements became really, well, unwieldy.  More to the point, the axiom, the parallel postulate axiom, wasn’t as self-evident as the first four, and wasn’t provable from the other nine of the first ten axioms.  And so came the unbelievable question, what if the fifth postulate wasn’t true?  What if there were other geometries—non-Euclidean geometries—that were as provable as that of the Master?  The possibility of relegating the understanding of nature and the physical surroundings to something other than the existing geometry was simply, positively, extraordinary. 

The origins for this movement away from Euclid started with Girolamo Saccheri (his book published in 1733 but not “discovered” for its non-Euclidean importance for another 150 years) Georg Kluegel (1763), Johann Lambert (Theorie der Parallellinien, 1766) and Adrien-Marie Legendre (1752-1833, who was sort of in a similar boat with Saccheri in that his work wasn’t actually published until after that of the next two mathematicians); and coming to a point of real invention in the work of Nikolai Lobachevsky (1792-1856) and Janos Bolyai³ (1802-1860). (It would be incorrect to credit these last two men [plus Gauss] with the creation of non-Euclidean geometry given the longish and complex history of its development.)

This work brought into question the way in which the world was seen, and the very foundation of recording visualized space.  This is all much messier than had been planned (so to speak) by almost all previous mathematicians, questioning the very foundations of mathematics.  The orderliness of the earth-circling numbers doesn’t seem to be quite as they were, but as Henri Poincare stated, the “new” system isn’t anything better or worse, just different,  a new and more convenient way of looking at things⁴.

Notes

1. To be honest about it, the foundations of math bit didn’t suggest itself right away.  The first thing I thought when looking at what this paper trail was supposed to represent—a string of numbers 41 billion zeroes long—didn’t look right.  Given the size of the numbers coming out of the “printer” there would need to be five times as many rings around the earth.  But when you read the text the numbers are supposed to be “hand written”.  Therefore if you assume the numbers to be less than an inch high and with no spaces in between, then this 2.5 times around the earth works out to be accurate to the number of inches in the circumference. 
2. Still earlier work is found in the work of  Omar Khayyam's Discussion of Difficulties in Euclid, John Wallis and Nasir al-Din al-Tusi... among others.
3. Bolyai’s work—published as an appendix to his father’s work on the foundations of geometry—was privately reviewed by the great and impossibly smart Carl Gauss, who gave it a backhanded series of compliments while at the same time saying “I thought so…”  Which was true, evidently, as it looks as though he was working on the problem beginning in 1799.
4. "If geometry were an experimental science, it would not be an exact science. it would be subject to continual revision ... the geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. In other words the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise. What then are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates are false. One geometry cannot be more true than another; it can only be more convenient.” --from:  M J Greenberg, Euclidean and non-Euclidean geometries: Development and history. (1980).

The History of the Internet: Remembering Vannevar Bush and the Memex (1945)

JF Ptak Science Books LLC  Post 812  Blog Bookstore

On this day, which may or not be the 40^th anniversary of the first use of the internet (and the 80^th anniversary of the collapse of the stock market of ’29, and Lucy A’s anniversary of her birth) it might be good to remember an earlier version of the internet and the semi-invention of the concept of hypertext: the MEMEX machine of the under-remembered Vannevar Bush. 

Vannevar (pronounced “van ee var”) was a flinty no-nonsense New Englander who was an organizational and mechanical genius who as a professor at MIT developed a remarkable analog computer that greatly advance computation capacities for solving differential equations.  This was in the 1930’s, and even after creating an improved electromechanical version of the machine still chose the wrong way to go in the soon-to-materialize digital computer revolution.  During WWII Bush was one of the most important Americans in the war effort, overseeing the entire scientific effort of the U.S.—an enormous effort dispatched beautifully (and successfully).  His importance in this regard is difficult to overstate.

But right there at the end of the war, when he was organizing a still-secret effort of controlling the spread of atomic weapons (before Hiroshima), Bush was also thinking about the future of the organization of human knowledge and memory.  And published it, too, in the popular Atlantic magazine in July 1945. (It was republished in a shortened version a few months later in September—it remains to this day, according to an editor at Atlantic, one of the most often-requested reprints) where it reached a reading audience of millions.  The article, “As We May Think”, outlined his idea for a device he called the MEMEX¹ (“memory” and  “index”), which compressed and organized all that its user could remember and whatever information would be obtained in the future via electromechanical apparatuses, and available with associative tracking between the microtext frames.  So all manner of book and papers and reports and newspapers and images would be microphotographed and stored in a beautifully-indexed mechanical retrieval system where it would live with the information of others.  There was also the possibility of real-time textual additions to whatever it was that was retrieved, the genetic precursor of hypertext.  Bush’s thinking has been deeply recognized (and also by the creators of the internet) as the intellectual grandparent of the modern internet. 

I just thought on this day that the man who saw more clearly into the future on advanced global communication and societal memory should be remembered on a day when his grandchild came into being.

Notes:

1. According to Bush, “the MEMEX is a sort of mechanized private file and library…. It uses methods such as microfilm storage, dry photography, and analog computing to give postwar scholars access to a huge, indexed repository of knowledge-any section of which can be called up with a few keystrokes."  Wardrip-Fruin, Noah and Nick Montfort, ed (2003). The New Media Reader. p. 35. The MIT Press

The physical plant of the MEMEX:  “A memex consists of a desk, where on top are slating translucent screens on which material can be projected for convenient reading. Within the desk were mechanisms that stored information through microphotography. Most of the memex contents are purchased on microfilm ready for insertion. On the top of the memex is a transparent platen. When a longhand note, photograph, memoranda, and other things are in place, the depression of a lever causes it to be photographed onto the next blank space in a section of the memex film, dry photography being employed.”  Manovich, Lev. "As We May Think", The New Media Reader and wiki on The Memex. 

Big Mathematics 1816: Manuscript Notes and Found Math Art

JF Ptak Science Books LLC  Post 757  Blog Bookstore

In my science bookstore business one of my principle interests is antique manuscript notebooks in the sciences.  It is a pleasure to see someone working through a problem, or finding what the writer thought to be the most interesting point in a lecture on a particular subject, and to read the occasional doodles and see the timely gloss every now and then.  I’ve had notebooks from an advanced and very careful student for a course given by the great Robert Bunsen that was filled with hundreds of pages of what the master was trying to communicate to his pupils.

There’s also a  series of notebooks from the spectacular period of Cornell physics history (1948-1949), including 6 notebooks of classes with Richard Feynman, including several hundred pages from his 1948/9 course in advanced quantum mechanics, which included (a very early) use of the Feynman diagrams.  The thing that makes these observant, advanced notes taken during Feynman’s classes have more than a pedagogical interest (which is substantial and enough on its own)—they have a capacity to show the logic behind the man’s thinking and how he presented his ideas over a period of some months.  Also, and curiously, notebooks like these seem to have not survived. 

Then there’s the notebooks taken by the team of Charlotte and Fritz John when they were pursing doctoral degrees at Goettingen (1929-1933), until forced to leave with the enforcement of the Nazi laws against non-Aryans.  Goettingen happened (and happens) to be the seat of a long, great mathematical heritage, home of Felix Klein, Emmy Noether, David Hilbert and Bernhard Riemann (among many others).  The notebooks—some two linear feet of them—represented four years of advanced interactions with some of the finest mathematical minds in Germany.  I was told by Mrs. John that they were required to take such full notes—but let me tell you, these notebooks were typed transcriptions of the daily handwritten notes, compiled over the course of a class, and then bound.  They are beautiful, and go far beyond my concept of “required”.  Mrs. John told me the story of the class notes that she and Dr. John took while taking a course with the great Hermann Weyl (who had taken his doctorate under the iconic David Hilbert, and who was also forced to flee Goettingen because his wife was Jewish):  everyone of course was required to take extensive notes during the day and work on them during the night, turning them in the following day for appraisal.  She told me that she still vividly remember placing their notes on a long wooden table at the front of the class, and then picking that set of note up the next day.  After a month of this, she, said, they became convinced that Weyl was not looking at the notes at all, as everything seemed always to be in the spot that they left it in.  One week they decided to see if the notes were being moved, and placed a hair carefully on the interior page, reasoning that if the notes were reviewed, the hair would be gone.  For a week, the hairs stayed in place.  “And so, what did that mean to you?” 

Mrs. John replied that it really didn’t mean anything, and that they continued to do the notes as they would, and leave them on the long wooden table.  They never slacked off (which I can fully imagine).  She said that they did it for themselves, of course, but also they did it for Dr. Weyl, fully expecting that the day that they missed handing in their notes would be the day that they were looked for by Dr. Weyl.  Plus they were studying with a famous and fabulous thinker, and did not want to disappoint themselves or their instructor. 

I do enjoy the historic or significant material like this, but I also quite the naïve, homegrown, elementary manuscript attempts by children at doing math or understanding physics.  Cipher books have always been a little hard to find in my field, wit many of them (I think) winding up in the genealogy area), and they just don’t seem to be very much on the findable side of things anymore. 

The example I’ve reproduced here are from an anonymous (but dated 1818) work on different elementary areas of mathematics.  What makes this special to me is the precision of the penmanship and the extreme effort that is being used for a simple display of division.  It would certainly convey the message if the author had used two five-figure numbers to display division rather than these enormous, never-to-be-encountered numbers. 

Surely there were pages and pages of scrap used to get to this point as the operator machined his numbers down, showing us the result rather than the real process.  But what is left to us in this effort is a thing of considerable beauty—the found art of practicing elementary math. 

Ludwig Wittgenstein's Adventure in Aeronautics, 1911

JF Ptak Science Books LLC  Post 751  Blog Bookstore

The Austrian Ludwig Wittgenstein (1889-1951) was one of the 20^th century’s most gifted and influential philosopher-logicians, though he started his advanced educational career in an aeronautics program at Manchester, in England.  And by “though” I mean a very soft “though”, as the engineering students educated in Germany were commanded to have a wide intellectual calling, including heavy backgrounds in mathematics and physics.  He came to Manchester to pursue a career as aeronaut and engineer, which is  a little odd because following its initial great exploits in flight in the mind-19^th century, England certainly was not a hotbed of activity for flight in the first (“flight”) decade of the 20^th century.  Be that as it may, he went to Manchester, pushed by the great influences in the work of Ludwig Boltzmann and Heinrich Hertz. He also received nourishment for his interest in mathematics, leading to a deeper pursuit in the foundations of mathematics, all of which fueled his developing study away from aeronautics and engineering.

And it was here at Manchester that Wittgenstein patented his innovative approach to propulsion—a sort of powered series of ejectors that were attached to the ends of the blades of a propeller.  The image that accompanies the patent application is here, above, and entitled “Improvements in Propellers Applicable for Aerial Machines”, dated 9 September 1911.  This was admittedly an early period in the development of aviation—especially in England—and although an interesting product it was not very practicable, though a design similar did appear on helicopter rotors in the 1950's.    

It was not so practicable that Wittgenstein pursue a career as an aeronaut—within  a year of the publication of his patent, he was on his way to Cambridge to join Bertrand Russell (who had just published the20th century’s most difficult-to-read work in logic in the Principia Mathematica) and begin his work in philosophy. 

It is fitting in a way that Wittgenstein had this background, as he described his philosophy of language in terms of projective geometry...

Notes:

1) Legend for the patent image: Fig. 1 is the gaseous or vaporized fuel inlet; a1 is the air inlet. Fig. 2 and Fig. 4 are alternative methods of fuel flow to the combustion chamber C. In Fig. 2 fuel is mixed with the air just after both emerge from the hub.

2) I've read that Wittgenstein was very deeply influenced by the following passage from Hertz' Principles of Mechanics:  "When these painful contradictions are removed, the question as to the nature of force will not have been answered; but our minds, no longer vexed, will cease to ask illegitimate questions" If I squint a little I can see the seeds of the Tractatus in there. 

3) For a real treatment of this invention of Wittgenstein's see the article at the Royal Society.

4) Also see Ian Lemco's Wittgenstein's Aeronautical Experiments 

5) "My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) He must surmount these propositions; then he sees the world rightly."--Wittgenstein's Tractatus, 6.54

6) See Klagge's online biography and philosophical history of Wittgenstein

The Color Blind Geometer and the Color-Coded Euclid

JF Ptak Science Books Post  315

Here's a bit of a reach, a very-long-distanced connection, between to unrelated items sharing a similar design and operating at the zenith and nadir of color theory.  The first work is the great and nearly unusable work by Oliver Byrne on the first six books of Euclid (The First Six Books of Euclid in which coloured diagrams and symbols are used instead of letters for the greater ease of readers..., printed at the Chiswick Press by C. Whittingham for WIlliam Pickering in 1847).  The book is, well, unusual and probably not at all useful.  Bryne replaced all of the algebraic notation, identifying letters and almost all of the descriptive text with color and color codes, leaving Euclid mysterious, hidden, awkward, impervious, and, yes, beautiful.  as a matter of fact this work presages the cubists (and especially Piet Mondrian) by 60 years, and all by accident. 

Augustus De Morgan, mathematician and logician, wrote a very highly critical book A Budget of Paradoxes in which he describes hubbub, fakirs, frauds, perpetual motion machines, squaring the circle efforts, millstones and other useless books in the maths, and in here he sniffily dismisses Byrne's work. At best, to De Morgan, the Byrne book is "curious". But useless and curious, as I have seen many thousands of times, does not mean that it can't be attractive and beautiful, and the Byrne book is probably the leading candidate in the Bad & Beautiful category..In Victorian Book Design McLean calls the Byrne book "...one of the oddest and most beautiful books of the whole century...a decided complication of Euclid, but a triumph for Charles Whittingham [the printer]".  (I should point out that the Chiswick press returned to Euclid again in 1893, publishing Gilbert Redgrave's address on Erhard Ratdolt, who was the first scientific printer in history and also the first publisher of Euclid in 1482. See HERE for the digital record of Redgrave.)

Curiously the Pythagorian theorem illustrated on the title page of Byrne's book seems at a fast glance to appear on the front cover of Phillip S. Newton's Color Blindness, Suggested Aids for Correcting 1946.  It really isn't all that close except by fleeting recognition, but close enough to stop me in my tracks, to make me think of where my copy of Byrne was and put them side by side.  Curious doubly because in Byrne's case color is used exclusively in place of numbers and words and is totally and completely dependent upon it. In Newton (who also has the same name as Sir Isaac, author not only of the Principia but also the second-best book of the 18th century, Opticks) we find the confusion of color and plans to correct the color vision, and in a twisty way, on the opposite end of the interests of Byrne.