The Rainbow Connection

Check this out, then come back.

Theodoric (or Thierry, or Dietrich) of Freiberg (c.1250-c.1310) was a Dominican, a philosopher, and a physician. His name is often written with the title magister (master), so we know he had an advanced university education, almost certainly at Paris. In 1293 he was named Provincial of the Dominican Order, Albertus Magnus' old post.

Freiberg's description of the geometry of the rainbow.

We have 21 works written by him, although a list of works by Dominican authors compiled in 1330 lists 31 under his name. Somewhere between 1304 and 1310, Theodoric produced De iride et radialibus impressionibus (Concerning the rainbow and impressions of radiance). In it, he presents the correct explanation for the rainbow. He explains the primary rainbow, the secondary rainbow and why the colors are reversed, and the path light takes to make the rainbow.

That last is important, especially if you've read the link I gave you above and are aware of the competing theories for refraction and reflection, and the place of water droplets versus clouds. Freiberg accurately describes how the path of sunlight is refracted when it enters the droplet, reflected off the other side of the droplet, and refracted again when it leaves the droplet and becomes visible to the observer. Freiberg determined much of this by experimenting with glass spheres filled with water, an extraordinary act in itself in the history of scientific experimentation.

Perhaps, however, the mechanics of the rainbow was an idea whose time had come. In one of those examples of synchronicity that crop up in history from time to time, there was another scientist who came to some of the same conclusions as Frieberg. His name was Kamal al-Din al-Farisi, and he and Freiberg had no contact—although they did have one thing in common: they both knew the 11th century seven-volume work called The Book of Optics by Ibn al-Haytham. But that's for another day.

Chasing Rainbows

The formation of a rainbow is a complex matter, inspiring both wonder and curiosity. How they come about took a great deal of time, speculation, and ultimately experimentation.

Aristotle was sure that water droplets were involved, and he knew there was a relationship between rainbow, sun and observer. In his model, however, each water droplet in the air is a tiny mirror that reflects toward the observer a piece of color.

Since each of the mirrors is so small as to be invisible and what we see is the continuous magnitude made up of them all, the reflection necessarily gives us a continuous magnitude made up of one color; each of the mirrors contributing the same color to the whole. We may deduce that since these conditions are realizable there will be an appearance due to reflection whenever the sun and the cloud are related in the way described and we are between them. ... So it is clear that the rainbow is a reflection of sight to the sun. [Meteorologica, Book III, Part 4]

Among his other theories, Robert Grosseteste (c.1175-1235) rejected Aristotle's view that the rainbow was created by reflection; instead, he believed that light passing through clouds, rather than bouncing off them, produced the spectrum. Since every schoolchild knows that refraction breaks white light up into the spectrum, this seems to us like Grosseteste knew what he was talking about.

Then came Roger Bacon a generation later. Some believe he studied under Grosseteste. What is certain is that Bacon knew of Grosseteste's works, because he sometimes quotes them verbatim in his own writing. When it comes to the rainbow, however, Bacon does something that seems baffling on the surface. He rejects the refraction theory and returns to Aristotle's reflection theory. Modern historians shake their heads over this apparent retrograde thinking.

Christ on a rainbow, the Macclesfield Psalter

Bacon had his reasons, however, which make more sense once you know the details of Grossesteste's theory. Grosseteste required three separate refractions to take place, using the borders of the clouds in a complicated lensing effect. Bacon pointed out that a rainbow could appear in a simple spray of water, as in a fountain, and the clouds and interfaces needed for the complex refractions described by Grosseteste were clearly not involved. Bacon also pointed out that the view of the rainbow changed as the observer moved, which meant the rainbow was being reflected toward the observer while keeping its proportions and color. It did not stay "painted on the clouds" as if it were just projected there by light refracted through a cloud lens. (At this point, it is obvious that they did not yet understand "seeing" as light reflecting off objects and into the eye.)

Bacon didn't have all the answers, of course. He struggled to explain the curve in the rainbow, and the fact that it was not a solid half-sphere: why wasn't there color in the center? And he ignored refraction completely when discussing the rainbow, even though he used refraction to explain the occasional halo around the moon.

Did Bacon hold back scientific progress? Hard to say. Grosseteste's theory was valuable in that refraction is crucial in the formation of a rainbow, but he made several assumptions that could not be supported. He ignored the part played in the process by water droplets, even though Aristotle and—more recent to Grosseteste, Albertus Magnus (c.1200-1280)—had insisted on the part they played. Grosseteste thought the entire cloud was the refracting lens. Rainbows were still not properly understood, but the efforts made to comprehend something that could not be touched and experimented on were impressive.*

...and what of the accurate explanation of the rainbow?  A few years after Bacon's death, a disciple of Albertus Magnus would work it out. But that's for another day.

*More on Grosseteste, Bacon and their theories can be found in an article by David C. Lindberg in Isis, Vol.57, #2 (1966).

Phantom Time

In June 2005, at a conference in Toronto on "Anomalous Eras - Best Evidence: Best Theory," Heribert Illig presented a paper he called "The Invented Middle Ages." It was not the first time this theory of history had been presented to the public—it had been known in Germany since 1996—but the first time it had been presented outside of Europe. In it, he explained his path to finding an anomaly in the historical record: that 300 years of our history did not exist! This theory is called the "Phantom Time Hypothesis."

Illig was born in 1947 in Germany. He studied economics, mathematics, physics, some art history and Egyptology, and describes himself as "not a historian in the narrow sense of the word." While reading the theories of Immanuel Velikovsky (that Earth has barely survived closes passes by Venus and Mars, before they settled into their present orbits, and that these fly-bys took place within the memory of ancient man and were recorded as myths), he began to question the historical record in Egypt, which led him to co-author a book, Wann lebten die Pharaonen? (When Did the Pharaohs Live?).

Diagram of missing and "recalibrated" years.

Once he was comfortable with questioning the accepted history of the human race, he started looking at the Middle Ages. He asked himself questions. Why did certain documents with earlier dates only get discovered later? How far off might the calendar have been by the time Pope Gregory insisted it be fixed? Could the engineering of Charlemagne's time really produce a building like the Chapel of Aachen, which looks to be part of Romanesque architecture style, which only existed two centuries after Charlemagne? As for Charlemagne himself: did he really create a re-birth from 768-814, when everything on either side of him is still "dark," and could one man possibly have done all that scholars say he did? How much can we trust those periods in western Europe that we now call "Dark Ages"?

His conclusion: there is a gap of years, from 614 to 911, for which any dates and events ascribed did not in fact take place. Essentially, a 300-year span has been "presumed" by historians who have tried to make sense of the unclear and inaccurate data we have; methods of radiometric and dendrochronological dating are unreliable, et cetera. Others have picked up on this and added to it; of course, he also has his opponents.

Illig has to assume enormous errors on the part of archaeologists and historians, as well as an elaborate conspiracy taking place in the centuries after 911 to "record" history that took place in the three centuries previous. Some of his arguments result from his misunderstanding of Gregorian calendar reform and dating methods. Some are just assumptions that contemporary witnesses are untrustworthy.

Is there a chance he's right? Is it possible that we are living in the year 1715 CE? Fortunately, astronomy helps. The Persian Wars between Greece and Persia lasted from 499-449 BCE.* The Greek historians of the wars tell of two solar eclipses taking place not far apart. The only times for two solar eclipses near each other in that part of the world were 2492 years ago and 2490 years ago, on 2 October 480 BCE and 14 February 478 BCE.

So there it is. No missing time. Thanks, science!

*One of these battles, Marathon, is remembered in the present day in footraces across the world. Another battle, Thermopylae, gave us the plot for the movie "300."

Geometry in Nature

The Fibonacci series, mentioned here, has been found to correlate to patterns found in nature. There is no evidence that Fibonacci himself ever made the connection between his arithmetical progression and the world around him, but the link between math and nature was recognized in other ways in the Middle Ages.

There is a unique manuscript called "The Sketchbook of Villard de Honnecourt." We know nothing about Villard except what can be gleaned from the internal evidence of this manuscript, which seems to be a notebook of his travels and interests. MS Fr 19093 now is in the Bibliotheque Nationale in Paris; it has been extensively copied and studied since it became widely known in 1849.

From the language and the 250 drawings found in his 33-page manuscript, we assume he was an architect from Picardy in northern France (there is a village called Honnecourt-sur-l'Escault in Picardy), traveling in the early-to-mid 1200s. Although architectural features make up the majority of his sketches (and presumably his interest) by far, there is no documentary evidence that he ever was connected with the design or construction of a building; nor did he sign his name with a title such as "master," to which he would have been entitled had he trained as an architect.

Still, he seems particularly introduced in buildings, drawing detailed façades of Laon and Reims Cathedral, a clocktower, the geometry of buildings; but he also depicts animals and insects, sculptures and mechanical devices such as a trebuchet, a machine for lifting heavy objects, a self-operating saw, a crossbow that never misses, and a perpetual motion machine. Alas, he draws them without explaining how to make them work.

Some of his notes, such as the one attached to the Lion shown above, are suspect. It is a very good drawing, but the caption tells us "Here is a lion seen from the front. Please remember that he was drawn from life." It seems unlikely that he saw a lion himself. The porcupine on the same page has the note "a little beast that shoots its quills when aroused," which tells us that he was getting his information from traditional bestiaries, not from real-life observation. On the lion's face you can still see the faint symmetrical guidelines he drew to start the artistic process.

Sketching guidelines was not an unusual start for medieval artists. Villard, however, offers us several examples of the correlation between geometry and representations of organic figures. Shown to the right is Page XXXVI of his manuscript, in which he shows how geometry fits into faces and figures.

His note in the bottom of the page depicted here says "Here begins the method of representation as taught by the art of geometry, to facilitate work. Elsewhere you will find the method of masonry."

A brief video showing some of his pages is here:

Medieval Bluetooth

The symbol shown here is used for the modern wireless communications protocol called "Bluetooth," created by the telcommunications company Ericsson. If you are at all familiar with the Runic system, it might look familiar in a different way because of its straight lines and angles. That is because it is a combination of the Runes Hagall ("h") and Bjarkan ("b"). And the reason for using the letters B and H is because they are the initials of the 10th century King of Denmark and Norway, Harald Bluetooth.

Harald "Bluetooth" Gormsson (c.935-986) was the son of the first historically recognized King of Denmark. Harald created the largest of the Jelling Stones (his father had set up the first). These commemorative stones carry inscriptions that include the earliest reference to Denmark as a nation. He also conducted important building projects, including a half-dozen massive stone ring forts and the oldest known bridge in southern Scandinavia, the half-mile long stone Ravninge Bridge (no longer extant).

He seemed to prefer negotiating over fighting, and managed to join and keep Danish tribes together, and briefly ruled Norway (okay, that was by force, after their king was assassinated). Perhaps it was his less-aggressive nature that made him amenable to Christianity, although the stories of his conversion are varied. One says he converted on a dare, when a monk named Poppa "proved" the power of God by carrying a heavy brand from the fire without being harmed. One story says he was converted against his will when he had been defeated by Otto I (founder of the Holy Roman Empire). Another account (written centuries after Harald's death) says it was Otto II who forced him to convert. Whatever the case, Harald converted in the 960s, and took it seriously: he transferred his father's body from a pagan-style grave in an ancient mound to a church.

This commemorates Harald's conversion.

But how did this all turn into a modern wireless protocol being named after Harald's nickname? And where did the nickname come from? The commonly repeated legend is that Harald loved and ate blueberries so much that his teeth were stained blue. A different (and not as attractive) story is that at least one of his teeth was diseased and took on a dark tinge, looking "blue" to some. This ties into one of the many legends of his Christian conversion: that he suffered from toothache and converted because Christian prayer was the only thing that took the pain away.

Whatever! What we can document is that one of the developers of the Bluetooth technology was reading a historical novel about Harald on the side. He felt that his protocol would unite different devices in a way analogous to Harald uniting different tribes in Denmark, instead of having them conflict with each other. He proposed calling the protocol Blåtand, Harald's nickname in Scandinavian. Although early Ericsson documents use this name, it formally became the English word "Bluetooth™"; I have read that folk in Scandinavian countries frequently use Blåtand instead of the official English name.

Frederick II

Frederick II of Sicily (1194-1250) has crossed the path of this blog more than once, but has not yet been featured.

He declared the Edict of Salerno, separating physicians and pharmacists.
Frederick was interested in math and science, and was friendly to and supportive of Fibonacci.
He promised to go on the Fifth Crusade, mentioned here, but never participated; he was blamed for its failures by Christians all over Europe as well as Pope Honorius III (who had been Frederick's tutor while young).

From the time he was declared Holy Roman Emperor in 1220 until his death 30 years later, he was a tremendous influence on science and culture, but a difficulty for popes and religion—odd, considering he willingly took the title Holy Roman Emperor. Although Pope Innocent III was his guardian growing up, Frederick often said blasphemous things, supposedly mocking Moses and Jesus and Mohammed for being frauds. His public attitude toward religion was unusual for his era and position, and Dante's Inferno places him in the circle of hell reserved for heretics.

He was, however, also possessed of a rationalism that was unusual for his era. He hired Arabs/Muslims as soldiers and personal guards; he hired Jewish scholars to be at his court. He pointed out the unfairness of trials by ordeal, because the stronger man would always win regardless of guilt or innocence. He hired the mathematician and scholar Michael Scot (of whom Honorius III thought very highly) to, among other things, make new translations of Aristotle and Arabian works into Latin. Michael Scot's translation of Aristotle was done with the help of Hermannus Alemannus ("Herman the German").

He had three wives and several mistresses. His third wife was Isabella of England, the daughter of King John Lackland. It was a political marriage, taken on because marrying an English princess would make his political opponents lose support from England. Once Isabella arrived in Sicily, she was sent to live in seclusion in Padua with only two of her English retainers.

Although Frederick had a profound and positive impact on laws and science, his personal manner made him many enemies and detractors. The Hohenstaufen lineage, which had included Frederick Barbarossa, lost power after Frederick II's death.

The Abacus

After mentioning Fibonacci's work, the Liber Abacus, it occurred to me that the place of the abacus in history deserved a little attention.

The Salamis Tablet, 300 BCE

Like the etymology for book, the word "abacus" does not start out to "mean" a frame with wires and beads. The word "abacus" first enters print in the English language in 1387. The Latin word from which it is lifted refers to a sandboard, a counting board covered in sand that allows you to draw with your finger. Latin took the word from the Greek abax, abakos, a board covered with sand for the purpose of drawing figures and calculating. At some point, the sand was replaced with counters of wood or stone that were moved from column to column for calculations, and the board itself was designed to facilitate calculations

In 1846, on the island of Salamis, a white marble counting board was discovered. The Salamis Tablet has been studied extensively, and one scholar has made a video of its proper use.

But when did abacus come to refer to the wooden frame with beads on wires? A reconstruction of a 1st century Roman abacus shows a board with grooves to keep the round beads in line. Visually, it resembles the abacus with which we are familiar. Gerbert of Aurillac (c.946-1003), one of the most influential scientific minds of his era, pushed the use of the abacus as a method of calculating much more swiftly than when using Roman numerals. He was able to promote its use even more when he became Pope Sylvester II.

The abacus in the form we think of it seems to come from China in the 2nd century BCE. Called a suanpán ("counting tray"), it was built with rods that held beads, 2 on an upper deck and 5 on a lower. Now called the "2/5 abacus," the two decks allowed the user to use larger numbers without adding 1+1+1+1, etc. Other versions had different numbers of rods, and different numbers of beads on them.

Abacus showing 87,654,321

Visually, it is very much like the Roman abacus mentioned above. Commerce between Rome and China was not unknown, but a direct influence cannot be proven. Still, the wooden-framed Chinese suanpán was so much like the Roman abacus that it was natural that the West would use the same name for the new device. In fact, no one type of the many objects used for calculating universally replaced the others. Counting boards of clay or wax were used well past the Middle Ages. In fact, until just after 2000, some accounting schools in China required proficiency in using the bead abacus.

Fibonacci

While the foundation of the Tower of Pisa was being being laid, a man was born nearby who developed math skills that might have helped the ill-fated architectural wonder.
Leonardo Pisano ("of Pisa") (c.1175-c.1250) was the son of Guglielmo Bonacci. Although known as "Leonardo Pisano" during his lietime, he signed his name "Bonacci" on his writings; an 1838 writer referred to him as "Fibonacci"—short for filius Bonacci ("son of Bonacci")—and the name stuck with his modern fans.

His father was a customs officer in Algeria, and between living there and traveling around the coast of the Mediterranean, Leonardo grew up exposed to education outside of the Greco-Roman/Western European tradition. He recognized the advantages of the Hindu-Arabic system of numbers over using Roman numerals, and worked to popularize it in Europe, starting with his 1202 work Liber Abaci ("Book of the Abacus" or "Book of Calculating"). In it, he presented to Europe the decimal system by which we all learn the four basic mathematical functions in school.

Calculating with the four functions in a decimal system.

The decimal system, with its "places" for ones and tens and hundreds, etc.,  was much "neater" than the system of Roman numerals and included a digit for "zero." Roman numerals had no "zero," and the words null or nihil were used to express a lack of something. The Roman tradition had great difficulty with the concept of "nothing" in math, because it seemed inappropriate to have a "something" that would indicate a "nothing."

If people have heard of Fibonacci in the present day, it is usually because of a particular sequence of numbers associated with him. In Chapter 12 of the Liber Abaci, he presents a math problem: how many rabbits are created in one year starting with one pair? After describing the progression in words, he shows the number progression as 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Examples of this series existed prior to Fibonnaci, and it is likely that he was simply repeating something he had learned, but 19th century mathematician Edouard Lucas called this sequence the Fibonacci numbers. They have been found to relate to many phenomena found in nature. (A thorough discussion is impossible here, but look.)

A webpage with some simple representations of the Fibonacci sequence is here.

The Groma

[We had so much fun with the Jacob's Staff (well, I did), that I hope you don't mind talking so soon about another tool for more measurement.]

Roman roads, the grid system for Roman camps, sections of Hadrian's Wall—how did they get their lines so straight? They used a groma. The groma had several parts:

• The Upright: a long pole with a metal point on the bottom for driving it into the ground.
• The Rostro: an arm that extended horizontally from the top, then angled upward, designed to swivel where it attached to the Upright.
• The Groma: a cross of four equal-length arms, with its center at the top-point of the Rostro. From the end of each arm, an equal length plumb line descended.
• The Marker Peg: a plumb line that descended from the center of the Groma's cross-piece all the way to the ground.

The Upright would be stationed so that the Marker Peg fell to what you wished to be the center of your surveying. From there, you could swivel the Rostro so that each arm extended to the cardinal points of the compass. Once the four plumb lines ceased swaying, you could send your assistant toward the horizon (standard distance was 125 paces). By getting behind the groma and sighting so that two opposite plumb lines were aligned, you could, with a wave of your hand, guide your pole-carrying assistant laterally until his vertical pole aligned with the plumb lines. You now had a straight line of hundreds of feet between two points, exactly along the direction you wanted.

Who used the groma? Land surveyors, or as they were called in Latin, Agrimensores (from ager, "field" and mensura, "measure). Which brings me to the Corpus Agrimensorum Romanorum ("The Work of Roman Surveyors"), considered the earliest scientific or technical manuscript that exists, from a time (the 5th-6th centuries) when most documents were about religion or literature (or politics). The page seen here is from the section on laying out a house as opposed to a settlement. One picture is of the house, and one is of the property lines around it.

A 1554 edition of the Corpus added illustrations to the text with grid-lines to show the accuracy of the process. A few examples of its pages and illustrations can be seen here. And if you are interested in robots, you might get a chuckle from this after today's post.

Tide Goes In, Tide Goes Out

The Classical World and the Middle Ages wrestled with the cause of the tides for centuries. Although one early scholar (Alpetragius, who flourished in the late 1100s) felt it was caused by some general motion of the world/celestial spheres that ran from east to west, most others (such as Bede and Gerald of Wales) felt it had a stronger connection to the movement of the Moon.
Alpetragius died in 1204, and his theory on the motion that caused the tides was translated into Latin by Michael Scot. This brought it to the attention of Robert Grosseteste (c.1175-1235), who had an explanation for the tides that relied on his theories of light. (The following is from the Questio de fluxu et refluxu maris, attributed to Grosseteste, although that attribution is disputed.)

Remember that there was no working theory of gravity yet; just a feeling that substances could be heavier or lighter depending upon their composition and gravitate (see? in this enlightened age, the concept of gravity pervades even our language) toward like substances: solids fall to earth; liquid (containing more of the element of water) flows to a lower spot to find its kind; fire yearns upward through air, because fire is even "lighter" than air.

For Grosseteste, light imparted force. Rays of light could carry with them the power to generate heat, for instance (see his theory on the sun). He postulated that, when the Moon rose above the horizon, its rays impressed against the waters and pushed them ahead of it, toward the west. This was not as simple and direct as a physical object pushing against water, and so water didn't rush to the shore as soon as the Moon rose. The rays of the Moon started pushing against the sea closest to it, pushing that water toward the observer. When the Moon was overhead, its rays had pushed as much water as it could at that time. Once the Moon passed the zenith and was over land, then the waters started to recede. The Moon then passes west and under the earth, at that point causing (somehow) the tides again.

Grosseteste admits that we don't know everything about this process, and my summary is a radical simplification of his detailed analysis. He notes the changes in tides as the Moon changes its declination, and theorizes that the Sun also "helps" the Moon in some manner.

For more detail, find the Question on the flow and re-flow of the sea (available in Isis, Vol. 57, No. 4 (Winter, 1966), pp. 455-474 in an article by Richard C. Dales) and enjoy.

How Does the Sun Work?

Robert Grosseteste (c.1175-1235) is considered by some to be the founder of modern English intellectualism. Among other topics, he focused (pun intended) on light. One of his works seeks to explain how the sun produces heat.

He first explains the three methods of heat generation:

1. An object that is hot
2. Motion/Friction
3. The scattering of rays

He determines that Method 1 cannot apply here. For heat to transfer from a hot object, there must be a medium through which it travels, and that medium will heat up during the transfer of heat. Clearly everything between the sun and us does not heat up.

He decided that Method 2 is also insufficient to explain the heat, because the motion that creates heat is caused by two substances moving in opposite directions—for instance, rubbing your hands together to warm them up—and the sun's circular motion does not act upon a second substance moving in an opposite direction: everything up there moves from east to west.

Method 3, he decides, must pertain. He reminds his reader that Euclid explains how a concave mirror can focus the sun's rays to cause a fire. He states that the sun's rays falling upon the earth are scattered, but reflection by a mirror or refraction by a (clear) spherical body can change the direction of the rays, focusing them via the medium of the dense air and generating heat. For him, this has much to do with the denseness of the medium: he tells us that the same amount of light falls on a mountaintop and scattering can be observed there, but the thinness of the medium of air disallows the generation of heat.