Saturday, August 9, 2014

The Discovery Of Middle Earth, Graham Robb

c. 2013

Every once in awhile one comes across a book that is simply the "cat's meow." For me, The Discovery of Middle Earth appears to be one such book.

I happened to see it in an obscure location on the very top shelf -- almost out of my reach -- in the corner of the museum bookstore at the Getty Museum, Santa Monica, California, yesterday. I was simply looking for a new book to read; there were several possibilities but then I saw this one, and that ended the search. I was only going to buy one book and this would be the book.

I am only a few pages into the book but I can already tell I am really, really, going to enjoy this book for so many reasons.

The premise is this: while planning a cross-European bicycle riding route, the author claims to have discovered the Via Heraklean to be the Celtic backbone of Europe. I do not know if this is an original thought with this author, but early on he met with his editor/publisher and swore them to secrecy, when asking their opinion whether it was worth pursuing as a publishable book.

If so, I put this book among the handful of books that opens up an entirely new world for me:
  • Mrs Dalloway, Virginia Woolf
  • The Truth Will Out: Unmasking the Real Shakespeare, Brenda James
  • Thh Discovery of Middle Earth, Graham Robb

I have only begun reading; I have only begun taking notes.

while writing the book, author living in Oxford, England
Matthew Arnold's Scholar Gypsy
a spindle tree; it is interesting -- the author happening to mention this tree (see link)
Iron Age
Uffington White Horse -- Iron Age hill figure
Bablock Hythe, a ferry crossing; would have been much like the one described in The Lord of The Rings, no doubt; described in Matthew Arnold's Scholar Gypsy

Cumnor Village
flood plain of the Thames at Farmoor
Via Heraklea -- fabled route of Hercules
"Sacred Promontory" to the Alps
Roman Via Domitia, modern A9 autoroute
Alpine pass of Montgenevre (Matrona, Celtic)
Matrona: spring of the mother goddesses

ancient cultures -- Celts, Etruscans, occasionally the Romans -- angled temples, tombs, streets -- NE -- rising sun solstice
21 June
21 December
several days the sun appears to stand still (solstice) -- rising, setting exactly same spot
ancient cultures afraid sun would disappear forever if not worshipped
author noted two coincidences, and, later a third:
1) the diagonal of Via Heraklea
2) orientation of Celtic, Etruscan, Roman structures (NE)
3) Mediolanum -- an enigmatic name; Celts called about 60 locations between Britain and Black Sea: Mediolanum

our sacred sites on Middle Earth correlate to places in the upper and lower worlds
Midgard: the word for Middle Earth /Mediolanum in Norse and Germanic mythology
1974: Yves Vade -- scholar who noted a Celtic network of these "middle places" -- Mediolanum -- each site equidistant from two other sites; on the Via Heraklea there are no less than 6 places found to named Mediolanum
author started to connect the dots; finds ancient birth of modern Europe
the geography of the western world had been organized into a grid of "solstice lines"
the network was based on the Via Heraklea
much of this was lost; Romans destroyed so much

 a solar-lunar calendar discovered near a lake in the Jura, the Coligny Calendar; 2 AD; considered the earliest map of the world; it is very, very possible, per this author, that the Via Heraklea is an even earlier map (it had its genesis around 700 - 600 BC) -- seven centuries before the Roman calendar/map of the world
author claims the Heraclean Way and all that follows from that is the earliest map of the world

Monday, July 28, 2014

Sorting Out Virginia Woolf's "The Waves"

Virginia Woolf remains my favorite author, or perhaps better said, the author that is most important for me.

I have transcribed -- completely, word for word -- two of her novels: Mrs Dalloway and The Waves. In transcribing Mrs Dalloway I discovered on my own that it was a "prose poem," something I did not know existed until then, and then discovered I had "re-invented the wheel," as they say. Having said that, it was one of the best "literature" things I have ever done, transcribing Mrs Dalloway in free verse.

I transcribed The Waves for a number of reasons. It is perhaps the most difficult to follow, and yet it is considered by many to be her best novel. In addition, closer to home, a close family friend, Ellen, considers it her favorite novel.

Mostly because I could not understand it, I transcribed The Waves.

Today I added the following to that transcription:

Perhaps somewhere else I tried to correlate the Greek party-goers and the characters in The Waves with Virginia Woolf’s circle, but if I did not, a couple of thoughts:
Jinny: serial lover of men, can only be Nessa, (Vanessa, Virginia’s sister; who had at least three lovers)
Percival: can only be Thoby; a he-man who died falling off a horse; Virginia worshipped her brother Thoby
Neville: homosexual; could only be Clive Bell, Nessa’s husband
Susan: possibly Virginia – Jinny’s life partner through extension of Greek counterparts
Socrates: could Virginia’s husband Leonard Woolf be Socrates?
Bernard is the storyteller in The Waves which is most likely Lytton Strachey. From wiki: he is best known for establishing a new form of biography in which psychological insight and sympathy are combined with irreverence and wit. His biography Queen Victoria (1921) was awarded the James Tait Black Memorial Prize. He was perhaps best known for his Eminent Victorians.
Rhoda is the youngest; I can’t think of a third woman in Virginia Woolf's circle; it was just Virginia and Vanessa, and many men: Leslie, Thoby, Adrian, Clive, Duncan Grant, Lytton Strachey. Roger Fry was also one of Vanessa’s lovers – she had at least three lovers: Clive, whom she married; Duncan Grant, whom she probably loved most, if I remember correctly; and, Roger Fry. There were several women in the group, but less well-known: Dora Carrington, Angelica Garnett, Julia Strachey, Molly (Mary) MacCarthy, Lydia Lopokova. Based on the linked essay below, Mary MacCarthy.  MacCarthy would have been, by far, one of the youngest. Virginia was born in 1882 and MacCarthy was born in 1912.
A superb essay, by the way on The Waves and the Bloomsbury Group: Utopian Wholes: Virginia Woolf's The Waves and the Bloomsbury Group

Friday, June 27, 2014

The 40s: The New Yorker, The Editors, c. 2014

I think if I could bring only one “new” book to the beach this year, it would be: The 40s: The New Yorker.

Almost 700 pages of short essays writing by the best (or at least some of our most famous authors) about one of the most exciting decades of our parents and/or grandparents. Below is a sampling of the short essays or articles that were originally printed in The New Yorker. If it’s a best-seller, perhaps The New Yorker would consider duplicating the effort with one on the 60s, perhaps the most important decade of my generation.

So, here goes, a sampling of selections from The 40s: The New Yorker.

 “Survival,” John Hersey, June 17, 1944 (On Lieutenant John F. Kennedy). Worth the price of the book. The quintessential story of "PT-109." I think as a teen-ager I wanted to build a model of the boat.

 “The Great Foreigner,” Niccolo Tucci, November 23, 1947 (On Albert Einstein). The author takes his mother-in-law, who is visiting from Italy, to visit Albert Einstein in Princeton. It turns out that Einstein’s sister came later to join her brother and now lives with Einstein. Einstein’s sister had been the “surrogate mother” for the author’s own mother. Along with his mother, the author brought his 6-y/o daughter. Very, very enjoyable on so many levels.

 “Come In, Lassie,” Lillian Ross, Febraury 21, 1948, On the Red Scare in Hollywood. Besides being written by a writer I wanted to know more about, a great subject.

“D-Day, Iwo Jima,” John Lardner, March 17, 1945. What can one say, to read this in “real-time” by a great writer?

“The Birch Leaves Falling,” Rebecca West, October 26, 1946 (On the Nuremberg Trials) This helps one understand events following the US toppling of Saddam Hussein. From wiki: Time called her "indisputably the world's number one woman writer" in 1947.

 “Letter from London,” Mollie Panter-Downes, September 14, 1940 (On the Blitz). An excellent first-person account. Reminds me of the biography of Graham Greene when he was in London during the Blitz.

 “Cross-Channel Trip,” A. J. Liebling, July 8, 144 (On D-Day). Superb.

 “La France Et Le Vieux,” Janet Flanner, February 12, 1944 (On Marshall Petain). In addition everything else great about this article, it helps explain a bit of trivia in Casablanca, the greatest movie ever.

 “The Suspended Drawing Room,” S. N. Behrman, January 27, 1945 (On Post-Blitz London). Superb.

“Greek Diary: Communists, Socialists, and Royalists,” Edmund Wilson, October 20, 1945.

“The Beautiful Spoils: Monuments Men,” Janet Flanner, March 8, 1947 (On Nazi Art Theft). I might have skipped this article, saving it for later, had it not been for current interest in the subject and the movie on the subject.

Sunday, June 8, 2014

Radioactivity, Marjorie C. Malley

c. 2011

Chapter 1
The Beginnings

Antoine-Henri Becquerel, director of Paris' Museum of Natural History; responsible for large collection of luminescent minerals that his father had assembled. When these minerals absorbed light, they would emit light of wavelengths (colors) different from the original source.

Luminescence: while the incident light is present
Fluorescence: continues to emit the light even when the incident light (e.g., sunlight) is removed
Phosphorescence: if the luminescence continues

Becquerel started testing various minerals.

Uranium: named in 1789 after the newly discovered planet Uranus; it was a heavy metal found mainly in European mines; it was used to color ceramics and glass; no evidence there was anything special about it

Becquerel discovered that uranium phosphoresced even without an antecedent light; laying a piece of uranium on a photograhic plate in a darkened drawer; some time later, the developed play would reveal the outline of the mineral shape that had been placed there.

Concept of ionizing rays.

Chapter Two
The Curies

Madame Curie tested uranium compounds for emitting ionizing rays (which she called "activity"): the ability of an element to emit ionizing rays depended directly upon the amount of uranium it contained, rather than on its physical or chemical state. - p. 24

Madame Curie decided to test many minerals looking for ionizing rays. Only two metals she tested gave off invisible ionizing rays: uranium and thorium.

Thorium, a mineral first identified in Norway, was named in 1829 after the Norse god Thor.

Curie named this "power": "radio-activity," in 1898. These rays became generally known as "Becquerel rays," a term first used by the Curies in the same year.

"If radioactivity was a property of certain elements regardless of their physical or chemical state, radioactivity must be a property of the atoms of these elements, an atomic property. At that time it was considered very important to distinguish between atomic properties and molecular properties. Atomic properties were presumed to be unchanging characteristics of individual atoms, while molecular properties characterized combinations of atoms, such as chemical compounds.."

"As an atomic property, radioactivity would take its place among the established atomic properties of weight, spectra, and valence."

"Looking back, it is tempting to read more into the term atomic property than it meant at the time, and Marie Curie herself encouraged this extrapolation. Becquerel had already concluded that radioactivity was a property of a specific element. Curie went one step further by stating that radioactivity was an atomic property. This insight was significant. However, the term atomic in 1898 did not have the associations it acquired after the discovery of atomic transmutation, especially after atomic reactors and bombs entered the picture." -- p. 26

New Elements! -- p. 26

DNA: The Secret Of Life, James D. Watson

c. 2003

The Double Helix: This is Life

Watson got "hooked on the gene" in his third year at the University of Chicago. His change of heart was inspired by a little book that appeared in 1944, What Is Life:, by the Austrian-born father of wave mechanics, Erwin Shrodinger. The book grew out of several lectures he had given at the Institute for Advanced Study in Dublin, 1943.

DNA located exclusively on chromosomes, but at the time it was too big a molecule for chemists to study. In addition, at the time, most biologists felt that genetic information would be carried by proteins, not by DNA.

DNA had been known for 75 years by then. In 1869, Friedrich Miescher, a Swiss biochemist working in Germany, had isolated DNA, which he called "nuclein."

1930s: DNA shown to be a long molecule containing four different chemical bases: T, A, G, and C. In 1944, it was unknown how these subunits, called deoxynucleotides) of the molecule were chemically linked.

DNA did not move into the genetic limelight until 1944: Oswald Avery's lab, Rockefeller Institute, NYC, reported that the composition of the surface coats of pneumonia bacteria could be changed. For decades, scientists had known there were two strains of Pneumococcus: "smooth" (S) and "rough" (R).

The Pneumococcus coat could transform.

In part because of its bombshell implications, that DNA was the holy grail (genetic material), the resulting February 1944 paper by Avery, MacLeod, and McCarty met with a mixed response.

Avery did not get the Nobel prize. The Nobel committee makes it records public 50 years following each award; it turns out that Swedish physical chemist Einar Hammarsten blocked the nomination of Avery. Hammarsten had produced high quality DNA but still believed that genes to be an undiscovered class of proteins. Even have the double helix was found, Hammersten still said Avery not eligible for the Nobel prize if it couldn't be explained how DNA transmitted information. Avery died in 1955; had he lived a few years long he certainly would have won a Nobel prize for identifying DNA as the genetic material.

When Watson arrived at Indiana University in 1947, Avery's paper came up in discussions all the time.

Cambridge, England: the canny Scottish chemist Alexander Todd identified the chemical bonds that linked together the nucleotides; they were all the same, and thus regular.

At Columbia University, Erwin Chargaff, developed process for measuring relative amounts of A, T, C, and G in each DNA molecule. None were the same.

The Phage Group at Indiana University began after Watson arrived; formed under Watson's Ph.D. supervisor, Salvador Luria (Italian) and Max Delbruck (German) and Alfrey Hershey (American).

Delbruck and Luria (fled Europe; banned from war effort in US) collaborated on phage experiments during successive summers at Cold Spring Harbor. Their theory: phages were "naked genes."

[I had an "aha" moment in medical school when I saw the same thing, that phages/viruses were "naked genes." That was in 1973 - 1974.]

The concept of "naked genes" had been first proposed in 1922 by the imaginative American geneticist Herman J. Muller, who three years later demonstrated that X-rays cause mutations. His belated Nobel Prize came in 1946, just after he joined the faculty at Indiana University. It was his presence, in fact, that led Watson to Indiana.

Watson felt that research on Muller's fruit flies was "the past." The future was Luria's phages, and that's where Watson headed.

Because he was weak in chemistry, Watson would not have survived in a chemistry lab. He therefore took a postdoctoral fellowship in the Copenhagen lab of the biochemist Herman Kalckar in the fall of 1950, studying the small molecules that make up DNA. Watson knew that this would also be a dead end, but his year in Copenhagen turned out to be productive.

To escape the cold Danish spring, Watson went to the Zoological Station at Naples during April and May. During his week there, he attended a small conference on X-ray diffraction methods for determining the 3-D structure of molecules. Initially, he was disillusioned by the conference. And then the last-minute talk on DNA by a 34-year-old Englishman named Maurice Wilkins from the Biophysics Lab of King's College, London.

And then the rest is history as they say.

Wilkins was a physicist; he had worked on the Manhattan Project.

He, too, had read Schrodinger's book and was tackling DNA with X-ray diffraction.

But Wilkins not much interested in talking to Watson at the time.

Watson returns to Copenhagen. Back in America, Caltech's Linus Pauling announced a major triumph: he had found the exact arrangement in which chains of amino acids fold up -- the alpha helix. Short bio of Linus Pauling, p. 43; fascinating.

Then the short history of how Watson ended up at Cavendish, starting at the bottom of page 43.

Chapter Three
Reading The Code: Bringing DNA To Life

The RNA Tie Club -- absolutely fascinating. A small group --
  • G. Gamow -- inducted Edward Teller
  • A. Rich
  • P. Doty
  • R. Ledley
  • M. Yoas
  • R. Williams
  • A. Dounce
  • R. Feynman
  • M Calvin
  • N. Simons
  • E. Teller
  • E. Chargaff
  • N. Metropolis
  • G. Stent
  • J. Watson -- inducted Richard Feynman
  • H. Gordon
  • L. Orgel
  • M. Delbruck
  • F. Crick
  • S. Breener
Notice whose name is missing: Maurice Wilkins.

DNA: molecule model, 1953. Awarded the Nobel Prize in Physiology or Medicine (Wilkins, Crick, and Watson, in 1962). Had Rosalind Franklin lived, the problem would have arisen whether to bestow the award upon her or Maurice Wilkins. The Swedes might have resolved the dilemma by awarding them both the Nobel Prize in Chemistry that year. Instead, it went to Max Perutz and John Kendrew, who had elucidated the three-dimensional structures of hemoglobin and myoglobin respectively.

Chapter Four
Playing God: Customized DNA Molecules

Friday, May 30, 2014

God's Equation: Einstein, Relativity, And The Expanding Universe, Amir D. Aczel

c. 1999

Nova (temporary brightness of a white dwarf star):
A nova (plural novae or novas) is a cataclysmic nuclear explosion in a white dwarf, which causes a sudden brightening of the star. Novae are not to be confused with other brightening phenomena such as supernovae or luminous red novae. A nova is caused by the accretion of hydrogen onto the surface of the star, which ignites and starts nuclear fusion in a runaway manner. Novae are thought to occur on the surface of a white dwarf in a binary system. If the two stars are close enough, material can be pulled from the companion star's surface onto the white dwarf.
Supernova (death of a star):
A supernova is a stellar explosion that briefly outshines an entire galaxy, radiating as much energy as the Sun is expected to emit over its entire life span, before fading from view over several weeks or months.
The extremely luminous burst of radiation expels much or all of a star's material at a velocity of up to 30,000 km/s (10% of the speed of light), driving a shock wave into the surrounding interstellar medium. This shock wave sweeps up an expanding shell of gas and dust called a supernova remnant. A great proportion of primary cosmic rays comes from supernovae.
Supernovae are more energetic than a nova. Nova means "new" in Latin, referring to what appears to be a very bright new star shining in the celestial sphere; the prefix "super-" distinguishes supernovae from ordinary novae which are far less luminous. The word supernova was coined by Walter Baade and Fritz Zwicky in 1931.
Supernovae can be triggered in one of two ways: by the sudden re-ignition of nuclear fusion in a degenerate star; or by the gravitational collapse of the core of a massive star.
A degenerate white dwarf may accumulate sufficient material from a companion, either through accretion or via a merger, to raise its core temperature, ignite carbon fusion, and trigger runaway nuclear fusion, completely disrupting the star. The core of a massive star may undergo sudden gravitational collapse, releasing gravitational potential energy that can create a supernova explosion.
Although no supernova has been observed in the Milky Way since Kepler's Star of 1604 (SN 1604), supernova remnants indicate that on average the event occurs about three times every century in the Milky Way. They play a significant role in enriching the interstellar medium with higher mass elements. Furthermore, the expanding shock waves from supernova explosions can trigger the formation of new stars.
Chapter 1: Exploding Stars

Supernova: a massive star, much more massive than itself, runaway fusion, hydrogen --> helium --> carbon; collapses in on itself; becomes a neutron star. Inside the interior of the dense, collapsed star, ordinary protons and electrons can no longer co-exist; they fuse and become neutrons. TYPE IIA.

Super-supernova: TYPE 1A. Six times as bright as an ordinary supernova. Again, a white dwarf attracts surrounding matter, becomes 1.4 times as massive as our Sun, and then explodes.

Perlmutter, late 1990s, noted that the universe's expansion was accelerating; unheard of at that time, and not expected. This meant something frightening: our universe is infinite. Announced to the world in 1998. The mass of the known universe was too small to stop the universe from expanding forever.

"Sunsets are red, the sky is blue" -- Rayleigh's Blue Sky Law that every beginning physics student recites.

Chapter 2: Early Einstein

Chapter 3: Prague, 1911

Chapter 4: Euclid's Riddle
p. 59 -- "the happiest thought"

Famous thought experiment: imagined a circle spinning in space -- the center of the circle did not move but its circumference was moving quickly in a circular directions....concluded that the boundary of the disk contracted as it spun...a force acting on the circle at the boundary -- the centrifugal force -- and its action as analogous to that of a gravitational force. But the same contraction that affected the outer circle left the diameter unchanged. Thus Einstein concluded, in a way that surprised even him, the ratio of the circle to the diameter was no longer pi. He deduced that in the presence of a gravitational force (or field), the geometry of space in non-Euclidian.

Chapter 5: Grossman's Notebooks
  • concept of a tensor
  • increasingly complex mathematics needed to solve the problems of relativity; Grossman, a contemporary student, helped him
  • 1913: the visit to Zurich that would change his life forever; Max Planck (1858 - 1947) and Hermann Nernst (1864 - 1941) visited Zurich where Einstein was (ETH); at their insistence he moved to Berlin, center of anti-Semitism at the time
Chapter 6: The Crimean Expedition
  • the Crimean, August 1, 1914
  • collaboration with Freundlich
  • August 2, 1913: excited; "I am quite convinced that the rays of light get bent."
  • also very interested in Freundlich's interest in double stars
  • December 7, 1913: the eclipse viewed from the Crimean would prove/disprove whether rays of light bend
  • Einstein spends considerable time countering Freundlich's assertion that detection of starlight shifts near the Sun could be done in daytime without a total solar eclipse. Einstein said "no" in no uncertain terms. Author's footnote: this feat is still not possible today. Even during an eclipse, detecting the bending of light requires a complicated procedure.
  • July 19, 1914: reach the town of Feodosiya in the Crimea
  • [June 28, 1914: Archduke Ferdinand assassinated.]
  • a rift develops between Freundlich and Einstein; author suggests it was Einstein's fault
  • Einstein reading astronomer Arthur Eddington's work; coincidentally, and unknown at that time, March 1, 1919, Eddington was planning to embark on a trip to to an island off the coast of equatorial Africa to watch an eclipse of the Sun to try to detect the bending of star light to prove Einstein's theory of general relativity
Chapter 7: Riemann's Metric
  • Riemann attracted to Gottingen where Gauss was teaching
  • Gauss assigned Riemann his third choice for a presentation to start Riemann on his road to a position at Gottingen: a whole new theory in geometry requiring the fields of complex numbers and number theory
  • Riemann's presentation was a theory that would change the face of both geometry and the physical sciences forever. What was Riemann's groundbreaking idea?
  • Riemann decided that the property of a surface that he needed to understand and capture was the notion of a distance (also called a metric). Riemann used a non-flat surface, developed a new formula, and 60 years later, Einstein would use that formula to finally derive the equations of general relativity; a crucial element in Einstein's tensor equation, signifying the metric tensor. The metric tensor would allow Einstein to account for the curvature that the gravitational field imposed upon the space of the universe
  • Riemann's idea led to a new theory: differential geometry
  • the general approach had implications in the field of topology
  • topology is the study of spaces and continuous functions; it deals with questions such as whether a surface is connected or made up of several disconnected components; whether sequences of points converge to a point in the space itself or outside it; and whether it is possible to cover an infinite space with a finite collection of subsets
  • equivalences: a cup with a single handle and a doughnut for example
  • Reimann died of consumption, at a villa on Lago Maggiore in northern Italy, 1866; age 39
Chapter 8: Berlin: The Field Equation
  • Einstein: voted into the Prussian Academy in Berlin, 1913, war
  • "The race to finish the general theory of relativity is a story of mathematical trial-and-error in solving the great puzzle of matter and gravitation, which Einstein performed at an amazing speed during one incredible month: November, 1915."
  • all of his problems would be solved in a whirlwind of mathematical research using Riemannian geometry during November
  • a great chapter explaining, almost day-by-day, how Einstein solved the problem
Chapter 9: Principe Island, 1919
  • the story of smuggling Einstein's paper on general relativity to Arthur Eddington in England
  • Eddington, born 1882
  • fascinated by large numbers; as a youngster memorized the entire 24 x 24 multiplication table
  • Dyson and Eddington, near the end of WWI -- set out to prove general relativity; another eclipse, Brazil
  • their trip was successful; verified Einstein's equations
Chapter 10: The Joint Meeting
  • 1919: results were verified by Eddington, but Einstein was not informed
  • September 1919: Einstein informed
  • it is now clear, that near a massive object, space is non-Euclidean -- it is curved
  • the question may then arise: what is the shape of the entire universe, not just hte local neighborhood of a massive object such as a star? But here again, Einstein was way ahead of the crowd
  • Einstein had already begun to consider the shape and evolution of the entire universe two years before the big event of the 1919 eclipse
  • his work would lead him to the most controversial hypothesis of his life
  • in 1917, while manipulating his field equation, Einstein unwittingly opened a Pandora's box
Chapter 11: Cosmological Considerations


Tuesday, May 13, 2014

Three Books On "Math" To Consider Reading

From Zero to Infinity, Constance Reid, c. 1992. A small, short. soft cover. This little classic, first published in 1955, was Constance Reid's first book, and it has earned a special place in popular mathematical literature -- from the back cover.

Conquering Calculus: The Easy Road to Understanding Mathematics, Jefferson Hane Weaver, c. 1998. Almost all prose, very few formulas or even numbers in the book. Deceptive title: not on "calculus" as I thought, but on "calculus" as in calculating. The author is a lawyer which explains a lot.

Trigonometric Delights, Eli Maor, c. 1998.
The first nine chapters require only basic algebra and trigonometry; the remaining chapters rely on some knowledge of calculus (no higher than Calculus II). Much of the material should thus be accessible to high school and college students.

Rhind Papyrus:
  • bought in 1858, by a Scottish lawyer and antiquarian, A. Henry Rhind
  • found a few years earlier in the ruins of a small building in Thebes (near present-day Luxor), Upper Egypt
  • 84 mathematical problems
  • a scroll 18 feet long; 13 inches wide
  • Rhind died at age 30; British Museum gets the papyrus; a bit of it missing; miraculously, the missing portion possessed by the New-York Historical Society; complete text is now available
  • originally copied around 1650 BC (reign of King A-user-Re, Hyksos dynasty)
  • probably written during reign of King Ne-ma'et-Re, Amenem-het III, 1849 to 1801 BC
  • the word degree originated with the Greeks
  • Greek word "moira"
  • Arabs translated "moira" into daraja (akin to the Hebrew dar'ggah, a step on a ladder or scale); this in turn:
  • Latin: de gradus, --> degree
  • Greeks: the sixtieth part of a degree the "first part," the sixtieth part of that the "second part," and so on
  • Latin: the former was calls pars minuta prima ("first small part") and the latter pars minuta secunda ("second small part") from which came our minute and second
  • why we use radians instead of degrees
  • eliminates the unwanted factor, π /180
  • also, a small angle and its sine are nearly equal numerically (p. 17)
  • sine of one degree (sin 1°) = 0.0174524
  • one degree = 0.0174533 radian, so the angle and its sine agree to within one hundred thousandth; for an angle of 0.5° (again expressed in radians), the agreement is within one millionth, and so on -- the smaller than angle the closer the sine (of its measure in radians) and its measure in degrees (again: the sine of one 1°  = 0.0174524 and 1° = 0.0174533 radians;
  • again, 1° = 0.0174524 radians
  • sine (1) = 0.0174533
  • radian: modern vintage; coined by Lord Kelvin, 1871
  • Greek trigonon = traingle
  • Greek metron = measure
  • ghomon: an analog device for computing cotangent function; ghomon = "shadow reckoning"
  • Hipparchus: trigonometry in the modern sense began with Hipparchus of Nicaea (ca. 190 - 120 BC); stars
  • Ptolemy: first major work on trigonometry to have come to us intact from Ptolemy (ca. 85 - ca. 165 AD); Alexandria, the intellectual center of the Hellenistic world (unrelated to the Ptolemy dynasty that ruled Egypt after the death of Alexander the Great in 323 BC); star catalog based on Hipparchus' work; names 48 constellations (still in use today); standard map used well into Middle Ages; greatest work, Almagest, a summary of mathematical astronomy, 13 books, reminiscent of the 13 books of Euclid's Elements (forms core of classical geometry); similarities go even farther; evolution of Almagest:
  • Ptolemy's title translated to "mathematical syntaxis"
  • later generations added the superlative megiste ("greatest")
  • Arabs translated the work into their own language, kept the word megiste but added the conjunction al ("the"); in due time it became known as the Almagest
  • Almagest became cornerstone of geocentric world picture well into 16th century; became the canon of the Roman Church
  • Of particular interest in this chapter: Ptolemy's table of chords; subject of chapters 10 and 11 in the first book of the Almagest; essentially a table of sines. It was the Hindus that shortened the process to come up with a table of sines.

Plimpton 322:
  • item #322 in the G. a. Plimpton Collection at Columbia University in New York
  • earliest trigonometric table?
  • regardless: it proves that the Babylonians were not only familiar with the Pythagorean Theorem a thousand years before Pythagoras, but that they knew the rudiments of number theory and had the computational skills to put the theory into practice
Chapter 3: Six Functions Come Of Age

Aryabhatiya of Aryabhata (ca 510) is considered the earliest Hindu treatise on pure mathematics
it is also the first work to refer explicitly to the sine as a function of an angle
  • ardha-jya: the half-chord, sometimes turned around to jya-ardha (chord-half)
  • jya or jiva evolved as shortened version
  • Arabs translated the Aryabhatiya; retained the word jiva without translating its meaning
  • in Arabic -- as also in Hebrew -- words consist mostly of consonants
  • jiva could be pronounced as jiba or jaib, and jaib in Arabic means bosom, fold, or bay
  • Arabic translated into Latin, jaib was translated into sinus, which means bosom, bay, or curve
  • sine became the English version of the Latin sinus
  • abbreviated notation sin was first used by Edmund Gunter (1581 - 1626), an English minister who later became professor of astronomy at Gresham College in London; invented the forerunner to the familiar slide rule; and the notation sin (as well as tan) first appeared in a drawing describing his investion, the "Gunter scale"

Note: sin^2φ = square of sinφ [not, sin(sinφ)]. So sin^2φ = sinφ x sinφ.

The chapter goes on to say how the other trig names originated.

Tangents not really needed until navigational tables were computed in the 15th century.

Tangents and cotangents originated with the gnomon and shadow reckoning. [Coincidentally, and completely unexpected, I came across the "gnomon" again a few days later when reading John North's Stonehenge, c. 1996, p. 401: "Of the Heel Stone he writes that it 'was a gnomon for the purpose of observing the rising of the Sun on the auspicious morn of the summer solstice.'"]

Chapter 4: Trigonometry Becomes Analytic (17th and 18th centuries)
  • Wallis introduced the infinity symbol we use today
  • there are two "types" of trigonometry: computation and analytic
  • computational: associated with the triangle; Napier's table
  • analytic: relationships among the trig functions
  • trig numbers: numbers in their own right; don't have to be associated with the triangle
  • three big areas of study at this time: a) range of cannon projectile; b) navigation on open sea; c) music
  • range of canon projectile; known for a long time, but now had analytic basis
  • navigation on open sea: major area of study in 17th and 18th century oscillations; navigation depended upon clocks of ever greater precision; led scientists to study the oscillations of pendulums and springs of various kinds
  • increased skill and sophistication in building musical instruments; scientists motivated to study vibrations
  • all of this underscored the role of trigonometry in describing periodic phenomena and resulted in a shift of emphasis from computational trig (the compilation of tables) to the relations among trig functions
  • that's why one study "computational trig" in geometry, but then there is a whole new subject called analytic trigonometry that is taught a year or so later
  • developments from trig even farther from its original connection with a triangle; now, trig functions were defined as pure numbers rather than as ratios; for example cos x is now defined as an independent variable itself as a real number rather than an angle
  • Fourier's theorem marks one of the greatest achievements of 19th century analysis: he showed that the sine and cosine functions are essential to the study of all periodic phenomena, simple or complex, in much the same way prime numbers are the building blocks of all integers
  • Fourier's theorem was later generalized to non-periodic functions (in which case the infinite series becomes an integral), as well as to series involving nontrigonometric functions -- crucial in all branches of science
Chapter 5: Measuring Heaven and Earth

Chapter 6: Two Theorems From Geometry

Chapter 7: Epicycloids and Hypocycloids
  • the study of the toy that was put on the market in the 1970s -- the spirograph
Chapter 8: Variations on a Theme by Gauss
  • the story of Gauss summing the numbers 1 to 100
Chapter 9: Had Zeno Only Known This!

Chapter 10: (sin x)/x

Chapter 11: A Remarkable Formula

Chapter 12: tan x
  • of the numerous functions we encounter in elementary mathematics, perhaps the most remarkable is the tangent function, for a couple of reasons, but particularly this reason: tan x has a period π (a function f(x) is said to have a period P if P is the smallest number such that f(x+P) = f(x) for all x in the domain of the function). This fact is quite remarkable: the functions sin x and cos x have the common period 2π, yet the ratio, tan x, reduces the period to π. When it comes to periodicity, the ordinary rules of the algebra of functions may not be valid: the fact that two functions f and g have a common period P does not imply that f + g or fg have the same period.
  • as we saw in chapter 2, the tangent function has its origin in the "shadow reckoning" of antiquity; during the Renaissance it was resurrected -- though, it was not called "tangent" -- in connection with the fledging art of perspective. 
Chapter 13: A Mapmaker's Paradise
  • a cylinder projection of the earth appears to be identical to Mercator's projection, but they resemble each other only superficially; they are based on entirely different principles 
  • Mercator's story
  • Mercator: one of the first to bind in one volume a collection of separate maps: called it an "atlas," in honor of the legendary globe-holding mythological figure that decorated the title page; this work was published in three parts, the last appearing in 1595, one year after his death
  • how he implemented his plan, the spacing between successive parallels had first to be determined. Exactly how Mercator did this is not know (and is still being debated by historians of cartography); he left no written record of his method except for a brief explanation (in the book, p. 173). 
Chapter 14: sin = 2: Imaginary Trigonometry

Chapter 15: Fourier's Theorem
three developments transformed trigonometry
Ptolemy's table of chords
de Moivre's theorem and Euler's formula (e^ix = cos x + i sin x)
Fourier's theorem

Hemingway: The Final Years, Michael Reynolds

c. 1999 (first set of notes completed)

Part One: The Fortunes Of War, July 1940 to November 1944

Chapter 1: Ringing the Changes, July to Early Winter, 1940
WWII -- Germans were in Paris
First wife Hadley and son Jack were in Chicago
Second wife Pauline Pfeiffer with two sons Patrick and Gregory waiting for divorce papers (on grounds of desertion)
for past 1.5 years, Hemingway had been living in Cuba with Martha Gellhorn while still married to Pauline

Chapter 2: To Mandalay and Back, January to September 1941

Chapter 3: Voyagers, September 1941 to Christmas 1942
Back in Sun Valley with Hollywood movers and shakers
Submarine threat in Gulf of Mexico intense
Ernest Hemingway's private war against German submarines

Chapter 4: American Patrol, January to July 1941
The Pilar; hunting submarines

Chapter 5: Intermezzo, August 1943 to May 1944
Martha and Ernest had been a couple for seven years; four clandestine; three married
Apart: sweet, tender letters
Together: huge arguments; Martha wanted to travel, report, WWII; constrained with H. in Cuba
H. moves in on Martha's Collier's; becomes the magazine's lead journalist; to NYC, on way to Europe as war correspondent
Hemingway's and Martha's marriage over; only on paper

Chapter 6: Putting on the Ritz, June, July, August 1944
June 6, 1944 -- D-Day, Hemingway on the Empire Anvil, one of a hundred transports headed to France
Saw Sword, Juno, Gold, Omaha, and Utah on his briefing maps
Arrived at Omaha; with binoculars used Coleville church steeple as his guide but brought back to England per military policy; few correspondents allowed to go ashore that first day
Writing from Dorchester Hotel, London; talked to RAF pilots
Hemingway actually flew bombing missions with RAF pilots on Normandy (other correspondents did also)
Spends nights with Mary Welsh in the London hotel, 1944; in peacetime, scandalous; in wartime, no one thought about it; she was married to a correspondent who was in France
Welsh: twice married; once divorced at the time she was with Hemingway; she had multiple liaisons, as did he
Hemingway assigned to Patton's Third Army grouping at Nehou
Then assigned to First Army just as Omar Bradley broke out
Joins up with 22nd Infantry Regiment Colonel Charles "Buck" Lanham
Ernest: part-time journalist; irregular soldier; gatherer of intelligence

Chapter 7: Down Among the Dead Men, September to November 1944

Part Two: A Fall From Grace, 1945 to 1952

Chapter 8: Starting Over, March to December 1945
Eager to return to Cuba; Mary Welsh agrees with trial union in Cuba
End of 1945: divorce final for Martha and Hemingway; Mary now felt like a fiancee
Mary was getting used to Cuba; Hemingway did not care for the nightlife; was cutting down on his drinking and getting back in shape
Working on For Whom The Bell Tolls

Chapter 9: Rules of the Game, 1946
Remembering all the monumental authors he had ever read -- Homer, Proust, James, Joyce -- Hemingway loosely envisioned a book that woudl bring together everything he had learned about structure, landscape, and character
The book went through a huge metamorphosis over 15 years -- the sea war became what is now called Islands in the Stream, and The Old Man and the Sea; the air was eventually abandoned; and the ground war would become the memories of a bitter Army colonel dying in Venice, Across the River and into the Trees.
For the next 15 years, often fishing with his sons off Bimini, Hemingway's memory and his fiction would return again and again to the apartment above the sawmill on rue Notre Dame des Champs where he first found his voice.
Finally, of things past would produce his Paris memoir -- A Moveable Feast.
Mary depressed; not "up" to his previous three wives; -- p. 158; had not "laid any bricks" as one of my girls friends once said
Hemingway himself depressed; many arguments with Mary; often sulked; if all else failed, he threatened suicide
A week after Hemingway's 47th birthday, he learned that gertrude Stein his lieterary mother and godmother to his first son, had died in Paris, leaving tiny Alice to nurture her memory
Hemingway saves Mary's life; burst fallopian tube; hemorrhaging; on way to Idaho; Mary never forgot that her life was saved by Hemingway
Sun Valley Lodge for Mary (see p. 138): with Hadley, he left Chicago for the Latin Quarter of Paris; with Pauline he moved into the St Germaine area and then Key West; with Martha he moved to Finca in Cuba. With each marriage; he gave up some of his favorite spots and few friends at each; now moving to Idaho for Mary.
Sun Valley Lodge for hunting and fishing with his three boys
Returned to the novel he now titled Islands in the Stream
Ketchum, Idaho

Chapter 10: Year of the Dog, 1947
47 years old; aging quickly; hairline receding; liver disease
had not published a book in six years; would not publish another for three more years
1940: a lion among writers
1947: a literary relic
evolving: Allen Ginsberg, William Burroughs, Jack Kerouac
a young veteran Norman Mailer was finishing The Naked and the Dead
a 1000-page manuscript; apparently ignoring his own advice to F. Scott Fitzgerald to write short novels
Max Perkins dies, unexpectedly, from pneumonia; for 20 years was Hemingway's editor
Pauline nurses Mary back to health
Failed revolution, Dominican Republic

Chapter 11: Enter Biographers, Stage Left, December 1947 to September 1948
end of chapter, in September, Mary and Ernest to Italy
talk of suicide continues
young assistant editor from Cosmopolitan magazine visits Hemingway in Cuba; Aaron Edward Hotchner wants to do series on Future of Literature; 

Chapter 12: Sentimental Journey, September 1948 to May 1949
meets 18-y/o Adriana Ivancich; falls in love with her

Chapter 13: Venice Preserved, May to December 1949
erratic mood swings continue; talks of death continue
by end of chapter, back in France

Chapter 14: The Middle Parts of Fortune, January to October 1950
Back to NYC and then back to Cuba

Chapter 15: Roadstead of  the Heart, November 1950 to February 1952
listless, depressed; lovesick for Adriana
Adriana and her mother show up in Cuba; live with Hemingways
The Old Man and the Sea
Hemingway's mother in nursing home in Memphis at age 79; hallucinations; senile;

Part Three: End Game, 1952 to 1961

Chapter 16: The Artist's Rewards, March 1952 to June 1953
everyone agreed: The Old Man and the Sea -- stunning, told as simply as a fable, and as tenderly as a love letter
Cuban coup; Batista in power; business as usual

Chapter 17: The Phoenix, June 1953 to March 1954
to Spain, Kenya, and the plane wreck that almost them their lives

Chapter 18: Fortune and Men's Eyes, March 1954 to January 1956
back to Venice; a changed man after the plane crash: his beard whiter, his eyes frequently vacant, his moods mercurial
thirteen months after leaving Cuba, now returned
Nobel Prize winner
now writing his story of his African adventure; Mary was a character in the book and would have a larger role than Pauline had in Green Hills of Africa
father's death February, 1954; mother's move to new nursing home in Minnesota
end of chapter: receives a letter from Sylvia Beach with re: to her memoirs

Chapter 19: Intimations of Mortality, January 1956 to March 1957
movie-making, fishing off coast of Peru
sailed to Spain, but unable to get to Africa due to conflicts on the continent

Chapter 20: Cuba Libre, April 1957 to December 1958
thinking of his memories as a series of short stories

Chapter 21: Exiles from Eden, January 1959 to January 1960
from Ketchum, following the Cuban revolution
Hemingway really starting to lose his sanity; becoming paranoid

Chapter 22: The Body Electric, January 17, 1960, to July 2, 1961
board the Union Pacific's "City of Portland" on first leg back to Cuba
arrive in Cuba
Hotchner his editor
clearly mentally ill by now; refusing treatment
anonymously treated as patient George Saviers
discharged from NYC hospital; back to Ketchum, Idaho
Bay of Pigs
Mayo Clinic, Rochester, MN
chapter ends as clock ticks to 7:30 am July 2, 1961
born in July, blown up in Italy in July, Pamplona in July, for Hemingway, July was a memorable month
61 years old

Coda, October 26, 1998
a short one-page chapter

Chronology at the end of the book