JF Ptak Science Books

A Beautiful Idea: the Knight's Magical Magic Square Tour (1888)

JF Ptak Science Books  Post 2243  

The idea behind this extraordinary image below is the construction of an 8x8 magic square capable of describing BOTH the Knight's tour (" a sequence of moves of a knight on a chessboard such that the knight visits every square only once...if the knight ends on a square that is one knight's move from the beginning square) WITH the resulting moves forming a series of magic squares. It is the product of a Kiwi engineer named Sturmer, and appeared in the Scientific American Supplement for 1888.

Eric Weinstein says in his article "There Are No Magic Knight's Tours on the Chessboard" on Wolfram's Mathworld site says, well, such a thing is not possible. "After 61.40 days of computation, a 150-year-old unsolved problem has finally been answered. The problem in question concerns the existence of a path that could be traversed by a knight on an empty numbered 8 x 8 chessboard."

Weinstein is concise: "Not surprisingly, a knight's tour is called a magic tour if the resulting arrangement of numbers forms a magic square, and a semimagic tour if the resulting arrangement of numbers is a semimagic square. It has long been known that magic knight's tours are not possible on n x n boards for n odd. It was also known that such tours are possible for all boards of size 4k x 4k for k > 2. However, while a number of semimagic knight's tours were known on the usual 8 x 8 chessboard, including those illustrated above, it was not known if any fully magic tours existed on the 8 x 8 board."

Source:  http://mathworld.wolfram.com/news/2003-08-06/magictours/

 

And this, from the British Chess Maagzine, 1888, page 278: 

Related articles

Magical Magic Squares of Some Advanced Age

Math Magic: Magic Squares, Magic Circle of Circles, Magic Spirals

Being in Regret--Verdun, 1916-1918

JF Ptak Science Books  Post 2215

I was looking through a very heavy volume from 1924 of Military Engineer and came upon a very large folding map¹.  It was vanilla on the outside and stayed so, opening only the back of the map through one, two, three, four (!)  unfoldings,  not yet revealing itself, until it was at it full width, and then unfolded once down, which opened to the middle of the map, which was a mass of lines and shading of brown and gray.  Another unfold up, and then another down, and more of the same, so much detail that the context still was hidden. I unfolded the bottom half of the map three more times and at the bottom was the town "Regret".  My right hand obscured the name of the much larger and antique-fortified town, which turned out to be the military-sacred city of Verdun.  When I unfolded the top of the map--making it about four feet long--I saw that it was for development of the battle and lines of communication and placement of troops and so on for part of the Autumn of 1918. And trenches.  

(This was actually "Verdun B", the mate of "Verdun A", which together form a huge and wildly complex 4x4' map. )

This was a reprint six years later of the 34th edition of this particular map--that is a lot of editions.  But this was a lot of place, Verdun.  A fluid place of ordered killing chaos that was as dynamic as it was occasionally static, starting in a very contained space of massive fighting that took place from 21 February to 18 December, 1916. The map is of a place that was about the longest, costliest, and deadliest battles that humans have come to, so far.   Casualties were about the same on each side (370,000 French and 340,000 German) and totaled about 710,000 people, though scholars argue the point, some coming to a figure much higher, approaching a million.  That makes 70,000-100,000 per month for the battle, which was like a war in itself. 

There were probably two million soldiers in motion here, at Verdun, in a relatively tiny area, with front lines extending about five miles or so, the battlefield being fairly narrow from point to point, perhaps totaling 20 square miles of heavily bombarded/shelled ground.  It was a terrifying place to be, and I'm certain that it must have scarred forever hundreds of thousands who survived the ordeal.  

Unfortunately Verdun 1916 was only about the half-way point of the war, with these maps generated to show artillery targets for the American entry in the war for action that would take place just before the end of the war on the 11th hour of the 11th day of the 11th month of 1918. This was the French Fourth Army and the American First Army which attacked on a front from Moronvillers to the Meuse on 26 September 1918 through the end of September, and from which the German army began its gradual withdrawal from the area, continuing right up to Armistice Day.  

Ultimately whenever I think of this war things usually boil down to Dalton Trumbo's Johnny Got his Gun, which was required reading in my freshman year of high school in 1970.

 

 

Notes:

1.  Journal of the Society of Military Engineers, volume 16, 1924; the map titles are "Tranchee Francaise schematique a la date de 1^er Sept 1918" and "Carte Generale des Objectifs d'Artillery".  

The Endless Immensity and Finite View of the Infinite of Thomas Wright, 1754

JF Ptak Science Books   Post 2064

Thomas Wright (1711-1786) saw about as deeply into the deep as just about anyone else--he looked into the night sky and pretty much saw all of it. In his book, An Original Theory or New Hypothesis of the Universe, Founded upon the Laws of Nature¹, he described a version of the universe that was influential in the thinking of Kant and Herschel, finding a rectangular/squashed "finite infinity" of stars, "a vast infinite Gulph, or Medium, every Way extended like a Plane, and inclosed between two Surfaces".

 

Our Milky Way, which at the time was thought to be the entire universe rather than a galaxy as it was later discovered to be--one galaxy in a seemingly endless sea of galaxies--was presciently seen by Wilkins as being but one assembly of stars in an "endless immensity" of stars:

"And farther since without any impiety; since
as the creation is, so is the Creator also magni-
fied, we may conclude in consequence of an in-
finity, and an infinite all-active power; that is
the visible creation is supposed to be full of si-
derial systems and planetay worlds, so on, in
like similar manner, the endless immensity is an
unlimited plenum of creations not unlike the
known Universe."--page 143. (Again, the "Universe" eferred to here is the Milky Way galaxy.)

Wright's vision of this plethora of Universes, in which each creation is one like the Milky Way--a radical thought in 1750:

[Part of me wants to include the first Wright engraving in this blog's series on the History of Lines, seeing as how they represent the great Something that seem to be infinitely binding the infinity of universes...]

Wright also writes on the minuteness of the human condition, of the perfect sense of nothingness that is the Earth in a sea of infinite possibilities of other earths and earthy creations, which was definitely an outpost of thinking in 1750:

"In this great celestial creation, the catastro-
phe of a world, such as ours, or even the to-
tal dissolution of a system of Worlds, may pos-
sibly be no more to the great author of nature,
than the most common accident in life with us,
and in all probability such final and general
doom-days may be as frequent there, as even
birth-days, or mortality with us upon the Earth.
This idea has something so cheerful in it, that
I own I can never look upon the Stars without
wondering why the whole world does not be-
come Astronomers; and that men endowed with
sense and reason, should neglect a science they
are naturally so much interested in, and so ca-
pable of enlarging the understanding, as next to
a demonstration, must convince them of their
immortality, and reconcile them to all those lit-
tle difficulties incident to human nature, with-
out the least anxiety."--page 132

 

Notes

1.  The full title:  An Original Theory or New Hypothesis of the Universe, Founded upon the Laws of Nature, and solving by Mathematical Principles the General Phenomena of the Visible Creation ; and particularly the Via Lactea. Comprised in Nine Familiar Letters from the Author to his Friend. And illustrated with upwards of thirty graven and mezzo-tinted Plates by the best Masters. London, MDCCL." Full test, here.

 2. An odd note about Thomas Wright's personal history, from Science, 1902: "A word, in passing, about Wright. Like many another, so unfortunate as to live ere the times were ripe, he has been consigned to unmerited oblivion. Even the writer of the entry upon him in the ' Dictionary of National Biography '—a work so uniformly accurate — is unaware of the sources from which information could have been obtained, and so has nothing to tell, — does not even know the dates of his birth and death, or why he was called 'of Durham."--[Science, N. S. Vol. XIII. No. 321. 2-22-1902

An interesting poem by Rafinesque to start of his edition of Wilkins:

"Where ends the range and limits have been set
To mortal eyes, there mental sight begins
To fathom space, and worlds invisible
Surveys, admires
The mind must feel that space can have no bound*,
Whatever number be of things or thoughts
Others may be beyond—and thus behind
The Nebulas and Belts, our Galaxies
Of stormy clouds and oceans
There stands the central land and throne
Of our wide Universe, the home of Angels,
The seat of Love Divine"
Rafinesque, Poem on Instability, found at the beginning of Rafinesque's 1837 American edition of Wright's 1750 work. 

History of Lines--Big Lines, June 1944

JF Ptak Science Books   Quick Post 

[Source:  the Library of Congress, here.]

The idea of being on the receiving end of these lines on 6 June 1944 is terrifying.  General Rommel pretty much figured out what was going to happen, and sort of when it was going to happen, but he was kept out of the strategy loop even though he was in charge of the German defences here, unable to convince Hitler to move men and machines southward to meet the invasion where he thought it was going to come rather than strengthen the position of defence in a place where he knew the invasion wasn't coming, which was Pas de Calais. The pull of war by this time had destroyed the Luftwaffe, and German high command had been destroyed by Hitler--or at least communications and straegy within the command system of the German army was very highly compromised.  In any event, once the invasion had begun, there was not much hope for the Germans--it had been a complete surprise, with the huge efforts of misdirection playing themselves out beautifully. So beautifully, as a matter of fact, that once the invasion was well underway it was still a matter of no small debate as to where the "real" invasion would take place.  Even after the airborne divisions began landing some hours before the assault began, it was only the elderly and problematic General von Rundstedt who reacted appropriately, believing that the airborne assault was far too large to be a feint, and ordered two reserve panzer divisions to Normandy.  the amount of men and materiel moving onto Normandy was gigantic, impossible, overwelming, as some part of this map makes clear. 

The Mathematical Garden--Alexander Pope and Found Geometries

JF Ptak Science Books (Revisiting an earlier post from 2008 with a added details).

While sitting at the edge of a stream in a flowing dress and shading herself with a yellow parasol, my wife and  the jewel in the crown, the brilliant Patti Digh, wrote a superb essay on following the paths less marked. (Maybe I'm making up the parasol bit. And the stream. And flowing dress.) I've always been a happy observer of found geometries, and her essay heightened my awareness to them, along with the conscious efforts that we may daily make to enforce these lines or follow a new, multi-dimensional geometric path. 

So when I stumbled across this map of Alexander Pope's garden and noticed the odd paths  winding their way through it, and  how much in contrast this was to the very sniffy upper  crust English garden of the 18th century, Patti's essay  lit instantly to mind—and also because in addition to being a geometer of emotion and thought she was in a past life a U VA English Master. 

Pope's garden was an irregular path in an irregular garden, that garden being his leased property at Twickenham, where he came to live in 1717, along with his mother and his boyhood nurse, Mary Beach, and his dog Bounce. 

Pope's garden was an irregular path in an irregular garden, that garden being his leased property at Twickenham, where he came to live in 1717, along with his mother and his boyhood nurse, Mary Beach, and his dog Bounce.  (Actually that would be a series of dogs as he named them all "Bounce".)  Pope was a poet and satirist and critic and general taste-maker-the social influence extending even to garden-making, as his was famous even during his lifetime, and not for the reasons that most English gardens were famous.

His was not the planned regularity of the gentility (and far of course  from the suffocating, critically flawed gardens at places like Versailles,  "Is there anything more shocking than the regularity of a planned  garden?" asks Batty Langley in his New Principles of the Gardening… 1728), and were removed from the "normal", more articulated garden.  I've never really understood the formal garden, especially as it relates  to the ancient analogies of gardens and paradise-the Garden of Eden, Pliny's Garden of Venus, Lucretius' Earthly Paradise, seem as though  they might be some sort of paradisical thing (though we won’t address  Bosch’s Garden of Earthly Delights) , but the formal gardens and its  paradise relationship seems odd, what with the removal of nature's "chaos" and all, unless that was indeed the Master Plan.

But the mathematical garden was a desirous thing—still is—and none  other than Christopher Wren wrote that it was “naturally more beautiful than any irregular figure” (1750).  There were some critics of the math garden, like Mr. Langley, but most garden  designers wrote and argued in favor of it, coming to the similar  conclusion of John Dennis, in that the Universe, being “regular in all its parts”  would dictate that its beauty be found as regular  in other things too,  “and it  is that exact regularity that [the mathematical garden] owes its admirable beauty” (1704). 
Pope didn't seem to address (in print) the garden idea formally except for a few places--  Pope praised 'the amiable simplicity of unadorned Nature' in a famous essay, published  in The Guardian in 1713, and then again in his message to Burlinton in 1731.

The plan of Pope’s garden was somewhat irregular, but to me the  most appealing aspect was  the irregularity of the paths that curled and snaked their way through the garden. 
And particularly,  what about the lower part  of the path along the bottom of J. Serle’s 1725 engraving?  What was that path in Pope's garden  leading to, or from?  Was there a particularly nice tree there, a  favorite spot, a cool shade, an odd bit of sunshine at sundown?  Or was this  wall-clinging path just the workman's route? 

Perhaps these weren’t Pope's principal desire lines in the garden—in fact, the garden desire lines  weren’t the garden at all— they might’ve been under it.  Pope’s famous grotto at Twickenham  may be better known than his garden-it certainly sounds  much more interesting. Followed his  own desire line, underground, in his famous grotto, which he began to build shortly after arriving at  Twickenham, and which was still under construction at the time of his death. 

The unseen desire lines of  Alexander Pope seem more fitting to me, more poetic, what with  Pope needing to dig out the earth in order to find what  he was looking for, rather than a more  simple wandering around his fine, rustic garden. 

Have a look at the materials house in the grotto/cabinet of curiosity here, along with a  commendable plan of the grotto itself.

Exercise in Irony: the Usually-Invisible Chinese Rail Worker and the Highly Visible Photographer, 1869

JF Ptak Science Books   Post 1903   Part of the series on the History of Blank, Empty and Missing Things:  Chinese Rail Workers, 1869. 

One thing that you seldom see so juxtaposed in the photographic history of the American West are the people who were so important to it placed in proximity to those who were no well disposed to recording their existence. 

For example:  this image by J.B. Silvis, "China Section Gang Promontory", a photograph published as a stereocard between  1869 and 1870, shows a small group of Chinese workers setting two rails in Utah--they are the ubiquitously missing elements in historical images of transcontinental railroad construction. There were many thousands of these workers who were set to notoriously difficult tasks in bad situations and under time constraints with little liberty or pay (and not well-afforded much protection under the law), and in the history of documenting the expansion of the railroad across the West these workers are very seldom seen.

They are set incongruously in front of a photography car that so often saw through or around them.   The  car was labeled  "U.P.R.R. Photograph J.B. Silvis Stereoscopic & Landscape Views of Notable Points on line of Pacific R.R. Always on Hand", and I found it tempting to try to see in the photograph that it was made solely for the photographer's advertisement, but the central placement of the workers makes a good argument against this.  On the other hand, when you look at some of the other work of Silvis documenting his railway car (see here), there are a number of group images where he gathers cowboys or Indians  or makes a photo with his car in an unusual place, he is definitely using those people and locales to establish the idea of remoteness and wildness of the places in which he was working. It seems to me that in the end, he was using the Chinese laborers as a piece of his advertising ensemble more than documenting them. Again, this is very speculative.

It is a very unusual image which I think is knee-deep in irony. 

I should also like to point out that most of the workers have taken care to sew  large leather patches on their breeches--that to help their thighs when pulling that heavy metal lever across their bodies while setting and moving track. 

[All images via Denver Public Library, here.]

Vernacular Found-Polygonism 2: Walker Evans on the Roof

JF Ptak Science Books  Quick Post

How many polygons can you find? 

"Walker Evans climbed to the roof of the Fisk Building on Central Park South to  photograph the web of steel struts and electric signs that were rapidly filling the skies over Manhattan. The electric signs in this photograph alternately  flashed the company's name and the names of its two principal products. As there  were approximately a hundred million wheels rolling over America's roads in  1928, the sale of tires quickly overtook that of rubber galoshes. The image  expresses Evans' conviction that modern art could be timeless yet topical when  perfectly wrought of vernacular materials."--Metropolitan Museum of Art

This is one in a series of posts made on found geometries--here are a few others:

Found Alphabets and Geometry in Stone.(here)

Magritte-Like Disorienting Geometry in Found-Art Urban Design, 1925 (here)

Sierpinski Triangles & the Empire State Building (here)

 

Images of Shadows in Storytelling

JF Ptak Science Books   Post 1836

'There is seldom in shadow a mystery less hidden"--Ombra Nichts Schatten, The Non-Mystery of the Shadow, London, 1842

Shadows have long played havoc in the imagination, for good and for evil, for elucidation and of mystery--they have hidden nothing and everything, and displayed as little as possible and as much as can be imagined. A simple shadow in a hole had proved the circumference of the Earth twenty centuries ago, and before that the Earth's shadow on the face of the Moon showed us to be a sphere; the shadow allowed us to measure the heights of mountains on the Moon before we could do so (accurately) on Earth; we measure shadows in X-rays, and follow the trails of subatomic particles, and on and on.  Shadows are particularly interesting in story telling, whether they be suggested  aurally or in text, or of course in illustration: 

Chalres H. Bennet, Shadow and Substance, London 1860.

A superior example of shadow-art is seen in Indonesian shadow puppet theatre, "Wayang Kulit", where the shadows of Javanese-Indonesian characters are projected onto a cloth screen by a strong light from behind.

Audiences were bathed in direct and brilliant light while being swamped in shadow in the early 19th century as part of an early form of non-cinema cinema, as we see here in the frontispiece from Robertson’s Mémoires récréatifs, which depicts a phantasmagoria of lantern projections:

 

And more forthright illusions as seen in this engraving showing the spectre “Dr. Pepper’s Ghost,” from Theodor Eckardt, Die Physik in Bildern Eßlingen, 1881:

 

The shadow may have been invented in cinema by the German Expressionists, with  Friedrich Wilhelm Murnau's Nosferatu (1919) as a shining example:

And of course the shadow is used with great effect in the  very black-and-white and no-gray-tones Film Noir world, particularly in the hands of a master like Mr. Welles in films like The Third Man:

And in Fritz Lang's diabolically-bad M, earlier, in 1931 (featuring an impossible shadow in the movie still below):

 

Shadow has been used to given questionable dimensionality in addition to giving perspective to dimensions that we are familiar with:

 

 

And the artificial shadow made for the geographical clock dial to tell the story of passing time:

And also for shadow puzzle toys:

 

And on and on, deep into the night...

 

An Episode in the History of Anticipation: Death at Archimedes' Door

JF Ptak Science Books   Post 1665

Earlier in this blog there appeared an entry on the death of Archimedes--actually, it was a piece on an image depicting the killing of the great mathematician.  It is a pretty gruesome affair, even by the very early 18th century standards under which the engraving was made--gruesome still by today's standards, what with the man's head being cleaved laterally in two by the summoned Roman Centurion. I'm not an historian of these images of Archimedes, but I can say that for whatever I am, I have not seen anything quite so graphic dealing with the man's death.

The usual images of Archimedes coming to his end show him working a problem in his dust/sand/earth "chalkboard", absorbed to such an extent in his thinking that he doesn't hear the chaos outside his rooms, and doesn't sense the approach of the soldier who finds Archimedes unresponsive to his calls and therefore a threat, and so kills him.

[Image source:  Titus Livius, Romanae historiae principia, libri omnes.  Printed in Frankfurt by Feyerabend, 1578.]

Looking at this image (above) brought to mind another in the history of anticipation--not knock knock knocking on Heaven's door, but the opposite, and the opposite of that, the opposite of  knocking on Death's door, or at Death's door:  Death knocking at our door. 

The title of this emblem--Cunctos mors una manet/death is all that remains--is meant to remind us

of our common foe.  It is the work of  one of the great engravers of his time, Otto van Veen (teacher of Rubens),  and appears (along with next image and 100 others) in his Q. Horati Flacci emblemata.  Imaginibus in acs incisis notisque illustrata, printed in Amsterdam in 1607 by H. Verduss.

--All images and most translations courtesy of the splendid work at The Emblem Project/Utrecht, here.--

Part of the legend (which appeared in four languages) of the image leaves us with the following reminders, choice bits on the folly of forgetting our common end:

La terre embrasse tout comme mere commune --Mother Earth embraces us all

Moritur sutor eodem modo ac Rex.--The King died the same as the shoemaker

Death is the thing that sucks the rivers of lives, "Del rico al pobre, del soldado al Papa", From rich to poor, soldier of the Pope.

and so in image and word telling us that death will seek us out, whether king or shoemaker, rich or poor--and that in the coming end the Earth will take us all.  Archimedes didn't hear that one coming; nor did he care, I think. 

THis last image brings to mind Archimedes' work in the dust, though the intent of teh emblem ("Estote prudentes" //Be ye therefore wise // Be Cautious) wouold have been completely lost on him.

The legend translates (Thanks to the Emblem Project for this):

When the serpent sees that it is going to be attacked with, Lethal wound, it protects its head with artful care. Here lie the dwellings of the soul, the sanctuary of the truth. Hence comes life that is to be hoped for by every body.

Surely Archimedes could relate to his lines and the expressions of his thought as the dwellings of his soul (if he actually believed in a soul), though I guess you could say that the artful deception of the serpent's head finally wound up killing him via the Roman soldier.

Letali serpens cùm se videt esse petendum
Vulnere, sollicita contegit arte caput.
Hîc animæ sedes positæ, verique recessus: (here placed the seat of the soul, in artful covering...)
Hinc spiranda omni corpore vita venit

In any case I was drawn to the similarities of these two events, and the bit of anticipation that seems common between them. 

History of Lines: Clouds

 JF Ptak Science Books  Post 1629  (On Clouds and Not-Clouds, Part I)

"...and they drew all manner if things--everything that begins with an M---'

    'Why with an M?' said Alice

   'Why not?', said the March Hare." --Lewis Carroll,  from "A Mad Tea Party", in Alice's Adventures in   Wonderland.

Why not, indeed?  I've long had an interest in clouds, and since this blog is ostensibly on the surfacing or definition of Cloud Bones, I though to spend a little time on the origins of the depictions of printed clouds in art. Not being an expert by any means on the history of art, it just seems to me that after long exposure clouds are not well-represented in woodcuts, wood engravings and other early engravings, or at least not so well depicted as their landsmen in paintings.  There seems to be no shortage of effort to reveal gorgeous clouds  through the early Renaissance--but the doesn't seem to apply very often to their representation in prints.

And that of course is when the detail of the sky is depicted at all. It seems that more often than not, in woodcuts printed from say 1460 through 1550, that the sky is left blank, like this image from Ovid (Accipe Studiose Lector P. Ouidij Metamorphosin...printed in Venice in 1509:

"You could not see a cloud, because
No cloud was in the sky:
No birds were flying overhead --
There were no birds to fly".--Lewis Carroll, Through the Looking Glass. 

I could go on with many examples of the blank sky--the white, blank, settled nothingness--but exhibiting variations of nothing at this point isn't necessary.  But illustrating clouds is.

And for the most part, clouds don't look very much like themselves in prints, not really, not for more than a hundred years.  For example, Martin Schoengauer's spectacular  The Temptation of St. Anthony, which was printed in 1470/5, has a beautiful, fluid circularity to it, full of an earthy roundness even as the saint is pursued by demons.  But the sky in the background is populated with nothing but dashes to suggest clouds. 

Perhaps clouds would have taken away from the moment; but if that were the case, why bother with the dashes?

There are considerable exceptions to this practice, with Durer and Altdorfer coming quickly to mind--but in my experience, the oddly-formed cloud seems to be the majority rule in the woodcut.  

At this time there were of course many depictions of skies with no clouds in paintings, bare suggestions of themselves, even in some of the most famous works--Botticelli's Birth of Venus and Michelangelo's Fall of Man in the Sistine Chapel are fair examples.  But once you move away from the golden Sienese skies of the14th century¹, clouds are given full and beautiful shapes from early periods. (Giotto's Christ's Entering Jerusalem, 1305/6; Simone Martin The Road to Calvary, 1340; Duccio's Christ Entering Jerusalem, 1308-1311; Fra Angelico's The Dposition, 1436; Masaccio's The Tribute Money, 1424; Lippi's Annunciation, a perfect cloud treatise, 1442, and on and on. Other examples are found in continued reading below.)

Clouds in prints are a different matter--I'm not sure if it was just too technically difficult to create clouds in wood, or if there was some sort of languishing Byzantine sensibility inherent in woodblocks that was not so in oil. I'm really not so sure about this, and as much as anything else I'm thinking out loud about this lack of cloud definition.  I don't have an answer for it.  But I do want to share the beginning of a timeline for undefined clouds as they appeared (or not) over the course of about one hundred years,  from 1485 to about the end of the 16th century.   As I mentioned clouds appeared beautifully in paintings from earlier periods, and there doesn't really seem to be a substandard-cloud complement in the painting world as there was in the print world.  Of course, in western philosophy and science, clouds were pretty much left alone from inquiry, even by a series of the great natural history classifiers--they wouldn't really be given a fully taxonomic appreciation until 1803 by Luke Howard (which I wrote about in this blog earlier, here).

A Timeline of Clouds in Prints, 1485-1596

Franciscus de Retza, De Generatione Christi, sive Defensorium Castiatis Beatae Virginis Mariae, a beautiful work in Gothic type--a picture book, really, with two illustrations of magnets.  Printed in 1485, this image depicts Isidorus in a magnetic coffin which is floating in the air:

"The Adoration of the Magi", from Legenda Sanctorum Trium Regum, printed in Modena in 1490:

The Triumph of Love, 1492/3:

A Forty-Year Precursor to Modernism--Found Cubism in 1870

JF Ptak Science Books  Quick Post     [History of Lines #18]

I was very surprised to come across this illustration while browsing the (3 April)  1870  volume of Punch, or the London Charivari .

I cannot think of any particular peculiar or unusual edition of Euclid that was published at about this time to demand such an unusual illustration. (The only newish edition that comes into any possible play here is Oliver Byrne's semi-insane First Six Books of Euclid, published by the Chiswick Press in 1849 and one of the most unusual and beautiful books of any sort printed in the 19th century--but then again that was 21 years before this volume of Punch.) And by "unusual" what I really mean to say is "highly unexpected" , as this all-angular depiction of a woman appears for all the world to be a Cubist- or Expressionist-like image, predating those movements by four decades. 

But the Angular Woman just seems to be happenstance, a happy precursor to a revolution-in-the-making, an accident forty+ years too early.  She is extraordinary--and for some reason she is smoking a cigarette. 

[Below--an illustration from Oliver Byrne, illustrating the propositions of Euclid via the use of color rather than words, a beautiful but non-functioning work.]

History of Lines, #17: the Artist Painting the Negative, Camille Corot, 1874

JF Ptak Science Books   Post 1584   [Part of the History of Lines series]

An interesting appearance of lines in art that seemingly crosses several disciplines and chronological development is Jean-Baptiste Camille Corot's The Dreamer and the Large Trees (1874).  It is vivid artwork full of motion, expressing itself in an impressionistic sense that seems almost more abstract expressionist than mid-early Impressionist. (As an Impressionistic work it would be well into the second decade of the movement; if Abstract Expressionist, it is three decades early.)  It is a brusque, almost savage portrayal of a somber person in a deep wood, though the lines that is is composed of say anything but that. 

The print was made via the cliche verre process¹--Corot used a semi-photographic medium in which there was no camera, as it was the artist who drew the negative, who created the image from nature without a lens and without a camera body (thus eliminating lensless photography like pinhole), drawing directly on the medium (which in this case is glass, insinuated by the cliche) and then exposed/printed onto a photo-sensitive paper (the verre).  The artist is in a sense making a photograph directly of the image in their mind, recording it on the medium, and then printing it from there. 

Of course there are other important lines in the history of art: Vincent Van Gogh Starry Night (1889), Giacomo Balla, Street Light (1909), Umberto Boccioni States of Mind (1911), Michael Larionov's Rayonist Painting (1913),   Robert Delaunay Windows, (1912), Roberto Crippa (1921-1972), Jackson Pollock (1912-1956, particularly Number 1, 1948), and any numbers of works by Barnet Newman.  And of course the Cubists: Braque, Gris, Picasso, and the rest; the Constructivists like Malevich, and on and on into the sunset.  But the reason I've chosen Corot is because of the unexpected, wildish impressionist qualities that seem so far out of place, even with the movement. 

Notes:

1.  Cliche Verre was quite popular with French artists of this period, used by Theodore Rousseau, Charles Daubigny, Charles Jacques, Francois Millet and others.  The process itself appeared very early on in the history of photography, being described in Robert Hunt's technical manual on artistic photography as early as 1841, two years after photography's beginning.

The images:

History of Lines: Looking Straight Down from Straight Up, 1786

JF Ptak Science Books   Post 1581

I was struck by the mosaic possibilities of the image (below) found in Thomas Baldwin's Airopaidia (printed in 1786) and so made it into one.  But what the engraving is,m really, is a view from a balloon, looking straight down.

We're looking straight down, through the clouds, seeing two towns (at bottom-left and top-center)--why the rivers are red, I don't know.

There are other posts on this blog on looking straight down, which is an attractive subject--to me, at least. First-hand images of looking straight down fro a balloon are rare things, even through the early 19th century, and offer a prospective seldom seen in human history.

Another view from this work attempts to show some depth:

Here's another view of that same image from the Baldwin experience, reprinted around 1810, and left in black-and-white:"

 Halton Tuner makes these observations on Baldwin's flight in his iconic Astra Castra: Experiments and adventures in the atmosphere ¹(published in 1864 in London): 

History of Lines: the Desire Lines of William Blake

JF Ptak Science Books   Post 1578    [Part of this blog's History of Lines series}

 The post dedicated to Patti Digh, brilliant writer and birthday girl, who wrote a lovely, image-filled  essay on Desire Lines back in December 2005 on her award-winning blog, 37days. I think the idea of Desire Lines is one of her favorites, and so I tried to imagine my own in her honor, a small Festschrift for the birthday girl.  Happy Birthday, P.D.!

 William Blake had a hard time of things, what with being thought of in unflattering terms of crazy vs. insane by many people rather than, say, being though of as "creative".  He led a good and relatively long life (1757-1827), and by his artistic and poetic creations inspired generations of other creatives, though he was very little recognized during his lifetime.  He was a seeker, and was somehow able to do his art and printmaking and poetry without a substantial income, living mostly on adequate income and patrons, slowly advancing to semi-poverty as he aged , but dying without a debt to his name.

The "line" that I site above to his unusual work, The Gates of Paradise--not the Baptistery doors by Ghiberti--, was actually a ladder and a collection of lines, but with the application of perspective becomes one combined line quicker and longer than it could ever be a ladder.   Blake's character, as with Blake himself, wants whatever it is that he can have, conceivable or not; and som, if he wants the Moon, he just goes to get it.  This version of The Gates of Paradise, which is largely a story of pictures and a few words as legend, was intended for children, perhaps--or at least that's what the title says.  In any event, its an interesting message to send to children whether it was actually intended for children of the chronological age, or not.  The "I Want, I Want" legend really wasn't necessary, as the image stands on its own.

Blake liked to tell a story with lines like this, as we see in this famous image of Newton:

Blake was an impossible anti-Newtonian--at least in my limited reading of the man--making a career of being a spectacular and beautiful anti-rationalist,  a weaver of smoke, playing with and distorting images like a theramin player of words, using language to produce dichotomies in even the most standard of sensate ideas.  Blake was the poster child for thing anti-tech, an anti-scientist out to save the world from its scientific self,  the rationalist world set to destroy imagination, the “Antichrist science” hell bent on destroying the soul of art and religion. 

“Art is the tree of life; Science is the tree of death” wrote Blake, and so he chose the “Athesistic”  Sir Isaac to stand for the beast, depicting him in art as a naked geometer intent on subjugating the world with a compass and a keen brain, reducing glory to quantification, the work of Satan..  For most of my life I thought that this image by Blake was celebrating Newton—I knew little of the poet/artist/poet, and thought the painting a reverence.  When I understood Blake a little,  I saw that the image was intended as a mockery of Newton and the idea of science—this was an unusual sensation, because absent the unspoken intention of the artist,  you could still take the subject matter of the artwork two ways. 

And of course the Creator:

 

God is presented as the Great Geometer, the creative power of the universe issuing forth with compass/dividers, was perhaps a kind of Greek conception of the rule of the universe and its interaction with all things on Earth. 

But it is interesting that in the end, Blake displays the sublime reasoning powers of Heaven and Earth, in the forms of God and Newton, both with desire lines.  I think he enjoyed doing that. 

The Series "For Children" of 1793:

History of Lines--a Note on the First Appearance of the Addition ("+") Sign

JF Ptak Science Books   Post 1575  [Part of the History of Lines series]

The history of lines is full and rich with symbols, some of which have absolutely nothing to do with the extension of trust and belief and hope and fear--unless you're a young student studying math or chemistry or physics.  One symbol in the maths that seems to have no exact age except to say that it is semi-ancient is the addition sign.  (It will be interesting in this series on the history of lines to address the origins of  other symbols like the sign for pi, division. multiplication, percent, inequality,  equality and so on.)

I say "semi-ancient" because the symbol for addition is a hieratic form of the Egyptian hieroglyphic, looking like a capital "A" in a way, without the horizontal crossbar.  The earliest appearance of a symbol for addition in Europe was in the 13th century, with the letter "p", appearing in various forms, as well as the word "plus", "used in connection with the Rule of False Position"¹.  The concept also appears as the word itself, as "et", or "and", which just so happens to closely resemble the symbol that the concept would become a few centuries later.  The earliest appearance of the symbol we would recognize instantly today as an addition sign came in the 1489 book by Wideman, the Behenunde und kuepsche Rechnung (seen above in detail and in full, from the 1532 edition of the work), while the first time the "+" appears in an algebraic equation for the first time 25 years later, in a work by Vander Hoecke.  

It is interesting to think of the life of these symbols before they became themselves. 

I am not certain of this, but it seems that the first "+" printed in the United States did not occur until the 20th edition of James Hodder's² Arithmetic³ was printed in Boston in 1719, and this at the press of James Franklin (1697-1735), the fourth child of ten in the Franklin family that included Benjamin (1706-1790).  So it is entirely possible that 13-year-old Ben helped his older brother print the first addition sign in the British Colonies. 

Notes:

1. D.E. Smith, The History of Mathematics, volume II, p. 397

2. Hodder in 1672 wrote "note that a + (plus) sign doth signifie Addition, and two lines thus = Equality, or Equation, but a X thus, Multiplication," no other symbols being used."  (Smith p395)

3.  In his introduction, Hodder says: