The Duodecimal System

by William H. Benson

October 3, 2019

     For centuries, the ancient Romans calculated sums with their clunky numerals: I, V, X, L, C, D, and M; or one, five, ten, fifty, one hundred, five hundred, and one thousand. They knew nothing better.

     Then, Western Europe’s Crusades to the Holy Land introduced Arabic-Hindu numbers to the Europeans. “The medieval world required nearly five hundred years to make the change from the archaic Roman numerals to the Arabic notation, with its miraculous 0.”

     Still, the Romans, like other civilizations, relied upon the Base 10 system for counting because of a biological accident. People stared at their hands and counted ten fingers. It made sense to attach a name to each of the ten digits: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

     But for measuring things, people tend to gravitate to a Base 12 system.

     For example, a foot has twelve inches, bakers sell donuts in lots of twelve’s, and a day has two twelve-hour segments, one for a day, another for a night.

     “A carpenter uses a ruler divided into twelve subdivisions, a grocer deals in a dozen eggs and a gross of cans, and apothecaries and jewelers divide a pound into twelve ounces.”

     The ancient Babylonians preferred a Base 60 system, or five twelve’s, a system that we still retain for time-keeping. An hour has sixty minutes, a minute sixty seconds. Mariners and geographers prefer the number 360, or thirty twelve’s, or six sixty’s, for their compass and for pinpointing locations.

     The number ten has only two factors, 2 and 5, but the number twelve has four factors: 2, 3, 4, and 6. Twelve is the smallest number with four factors. Twelve divides into exact thirds, but ten divided into thirds results in an unwieldy number, 0.33333…

     In the twentieth century, certain mathematicians, who argued for a Base 12 system, came together to form an organization that they named the Duodecimal Society of America, or the Dozenal Society. In the 1930’s, one of the society’s first presidents, F. Emerson Andrews, wrote a book, New Numbers: How Acceptance of a Duodecimal Base would Simplify Mathematics.

      In 1962, another mathematician, A. C. Aitken, wrote, “The duodecimal tables are easy to master, easier than the decimal ones. Anyone having these tables will do these calculations more than one-and-a-half times faster than in the decimal system.”

     How does Base 12 work? The society’s members assign a symbol to the numbers ten and eleven, such as X and E, for a total of twelve digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, and E. They pronounce the numbers ten, eleven, and twelve in Base 12 as “dek,” “el,” and “doh.”

     Instead of placing a decimal after the units, such as in “13.25” in the decimal system, society members distinguish their numbers with a semicolon, such as “13;25.”

     A member writes twelve as 10, because the second number to the left of the semicolon represents the number of twelve’s. For example, 20 represents twenty-four, 30 is thirty-six, and 40 is forty-eight.  

     The third number to the left of the semicolon represents 12², the number of grosses, or the number of 144’s. Thus, 723 in duodecimal = (7 x 144) + (2 x 12) + 3 units, or 1035. in the decimal system.

     The fourth number to the left of the semicolon represents 12³, or the number of 1728’s, the fifth digit represents 12 to the fourth power, or the number of 20,736’s, and so on.

     Because you and I are accustomed to thinking in terms of the decimal system, we feel a real need to convert duodecimal numbers back to decimal numbers. The good news is that mathematicians have created a conversion calculator that we can find on the internet.

     But how do you count on your fingers in Base 12? A very ingenious person noticed twelve segments on the four fingers: three on the index finger, three on the middle finger, three on the ring finger, and three on the pinkie. He or she then used the thumb to point to each segment when counting.

     One, two, three on the index finger; four, five, six on the middle finger; seven, eight, and nine on the ring finger; and ten, eleven, and twelve on the pinkie. A person can count on either hand up to twelve.

     Or a person can count on the left hand, and then use the thumb and finger segments on the right hand to indicate the number of twelve’s counted on the left hand. Thus, a person can count up to 144.

     Are people ready to add, subtract, multiply, and divide numbers in the duodecimal system? No. It is too complicated to convert. We are too ingrained in the way things are now. “Old habits die hard.”

      On November 23, 1793, during the French Revolution, clocks converted to ten hour days, one hundred minute hours, and one hundred second minutes. “The system though proved unpopular.”   

     The modern world will stick with the decimal system for counting and for mathematics, but retain certain portions of the duodecimal system for measurement and time-keeping.

Studs Terkel’s “Working”

by William H. Benson

September 19, 2019

     I consider 1974 a great year, perhaps my best. In May, I graduated from Northeastern Junior College in Sterling, Colorado, spent the summer driving a tractor and a combine, and in September, I moved out of my parent’s house and into a basement apartment, owned by a retired North Dakota Methodist minister and his wife, Ray and Ella McClure, in Greeley, Colorado.

     On September 25, classes at the University of Northern Colorado commenced, and everyday that school year, I attended a sequence of History Department classes. There was Dr. Rowe’s Colonial History, Dr. Knott’s Diplomatic History, Dr. Boeck’s Early American History, among others.

     I sat at the feet of, what I believed then, were the best professors. Although poor, hungry, bookish, and overstudied, I was fascinated. The lectures stimulated my thinking. I felt as if the encyclopedias that I had read so often when a child had come alive and were teaching me.

     With a little reflection, I began to contrast my work on the farm with that of a university professor. Life seemed easier at the university, more mentally enlightening, more pleasant working conditions, more gratifying, whereas life in the field was monotonous, dirty, dusty, dangerous, smelled constant of oil and diesel fuel. It was more physical than mental, more blue than white collar work.

     Yet, with a little further reflection, I realized that the university professor owns no tractors or equipment, no wheat ground, no pastures, no sheds, no cattle. Save for boxes of books and a pension, she or he has little opportunity to accumulate capital, as does a farmer. 

     In 1974, the radio host Studs Terkel published his book, Working: People Talk About What They Do All Day, and How They Feel About What They Do, a bestseller. Studs had traveled around Illinois, and interviewed over a hundred people, recording their stories about their work on reel-to-reel tapes.

     In the book’s “Forward,” Terkel said that he found that “work was a search, sometimes successful, sometimes not, for daily meaning as well as daily bread.”

     Once Terkel published his book, he packed his tapes into boxes, and there they remained until he passed away in 2008. People who enjoy archival work discovered his boxes and began to listen to the tapes, still in good condition. On a recent podcast on “Planet Earth,” I listened to him interview people. 

     For example, Terkel interviewed Sharon Griggins, a teenager who worked for the telephone company, as an operator, in Waukegan, Illinois. She agreed that it was “a lot of talk, but no human communication.” Her talk to callers included “seven to eight stock phrases” that she stated over and over, for eight hours, with nothing personal permitted. Sharon did not like her work, and soon quit.

     Terkel interviewed Gary Briner, an automotive worker at a plant in Illinois, who described his job, as “boring and monotonous.” He said, “we show up for work, but we are not robots. We sweat, we have emotions, we are hungover, but we want the company to treat us with some dignity and respect.”

      Briner said he did not set out to join the autoworkers union, but he soon learned “that every gain we made for better working conditions was because the labor union voted to strike, and because of those gains, the labor union forged a middle class.”

     Terkel interviewed Renault Robinson, an African-American police officer in Chicago, who said that the officer in charge would pair a different white officer with each black officer, and the two would ride around in a car for eight hours together, and never speak, with “very little or no communication.”

     Robinson became quite disenchanted by the way the police expected him to stop cars for almost no cause in the African-American communities. Out of hundreds of stops, the policemen would find some violations, but the process upset the people stopped. They felt harassed and persecuted.

     Robinson was one of the founders of the African-American Patrolmen’s League, a labor union for black police officers. One of the things his league pushed for, and was successful, was the removal of sawed-off shotguns loaded with double 0 buckshot from the squad cars. He said, “Those guns could take a kid’s head off,” an unnecessary resort to violent force.

     The early 1970’s was a different era for America’s labor force. Unions were stronger then, and certain jobs, like telephone operator, have disappeared. Today, “computers have transformed the workplace. Productivity has soared, but job satisfaction has plummeted.”

     Also, the absence of strong labor unions means that the majority of workers cannot strike for increased wages and better working conditions, and this absence may have caused the gap between rich and poor to widen even further.

     What has stayed the same over the past forty-five years is that each worker has a predilection for certain work and a disgust for other types of work. America today has low unemployment, and if a worker does not like his or her job, he or she can quit and find other work.

The 1619 Project

by William H. Benson

August 22, 2019

     A staff writer for The New York Times named Nikole Hannah-Jones came up with an idea for a series of essays that appeared in last Sunday’s edition of The New York Times Magazine. She called it the 1619 Project. The year 1619 is not a date well known among Americans, and yet slavery in North America began 400 years ago this month, on an unknown day in August of 1619.

     In that month, between 20 and 30 African slaves stepped off an English pirate’s ship in chains and shuffled into Jamestown, in Virginia, where English land owners purchased them to work their tobacco fields. African slaves would toil for free on North American soil for the next 250 years.

     Jamestown was then only 12 years old. The Pilgrims would not land at Plymouth Rock for another year, and the first Puritans would not land at what became Boston for another 11 years, in 1630. At that moment, in 1619, America incorporated slavery into its history, into its very DNA. 

     A Portuguese slave trader had stolen these 20+ slaves from their homes in what is today Angola, in southwestern Africa, weeks before, but English pirates had attacked the Portuguese slave ship on the high seas, confiscated the Africans, and then brought them to Point Comfort, near Jamestown.

     The Spanish and Portuguese were the first Europeans to initiate the capture and transport of African slaves to the Americas. Because landowners needed laborers to work their sugar fields in South and Central America, in the Caribbean Islands, and in Florida, they kidnapped millions of Africans.

     Now the stage was set for the English to carry forward and expand the global slave trade, because Virginia tobacco farmers now needed laborers, hundreds and then thousands of laborers.

     In the first essay, The Idea of America, Nikole Hannah-Jones provides a sobering series of statistics, first that Europeans captured some 12.5 million Africans and transported them to America, “the largest forced migration in human history until the Second World War,” that millions failed to survive the gruesome journey, and that of the millions who did survive, some 400,000 arrived in North America.

     She mentions that slavery, as it adapted in North America, was commercial, but in addition, it was “racial, heritable, and permanent, not temporary.” Children born to slaves did not belong to their parents, but to their parents’ masters, who could sell them to other masters, splitting up families.

     “Enslaved people were not recognized as human beings but as property that could be mortgaged, traded, bought, sold, used as collateral, given as a gift, and disposed of violently.”   

     She points out that the British in England noted the hypocrisy among the British in the American colonies, at the beginning of the American Revolutionary War. Samuel Johnson in London, asked a question, “How is it that we hear the loudest yelps for liberty among the drivers of Negroes?”

     Thomas Paine arrived in Philadelphia in early December of 1775 from England and gaped in astonishment at the slave market going on just outside his door. He asked, “With what consistency, or decency, they complain so loudly of attempts to enslave them, while they hold so many hundred thousands in slavery; and annually enslave many thousands more?”  

     Nikole Hannah-Jones then jumps to the Civil War, to the year 1862, when Union forces were struggling to subdue the Southern Confederacy. On August 14, 1862, “President Abraham Lincoln called a group of five esteemed free black men to the White House for a meeting.”

     The men walked in with some hope, but Lincoln dashed that little hope when he informed them that “he had convinced Congress to appropriate funds to ship black people, once freed, to another country.”

     The President’s next words chilled them. “You and we are different races. Your race suffers very greatly, many of them, by living among us, while ours suffer from your presence. In a word, we suffer on each side. It is better for us both, therefore, to be separated.”

     Nikole Hannah-Jones writes, “You can imagine the heavy silence in that room, as the weight of what the president said stole the breath of these five black men.” Edward Thomas, the delegation’s leaders, promised the president that “they would consult on his proposition.”

     Lincoln said, “Take your full time. No hurry at all.”

     After the Civil War ended, when “four million black Americans were suddenly free,” a group of black leaders submitted a resolution against black colonization to Africa or to Haiti or to Panama. They said, “This is our home, and this is our country. Beneath its sod lie the bones of our fathers. Here we were born, and here we will die.” They rejected Lincoln’s colonization proposal.

     Next time in these pages, we will consider the second essay in the 1619 Project, Matthew Desmond’s thoughts on Capitalism, and how King Cotton solidified slavery in the Southern states

Charles Manson and Sharon Tate

by William H. Benson

August 8, 2019

     On occasion, a brief sentence captures a facet of human wisdom better than does a lengthy essay or a philosophical tome of hundreds of pages.

     For example, “Pride goes before a fall,” and “Disaster follows achievement,” are two aphorisms that describe how human life can move, how it ebbs and flows.

     The ancient Chinese of the third century B.C. noted this phenomenon and arrived at the philosophy of “yin and yang,” that “all things exist as inseparable and contradictory opposites, including male-female, dark-light, and old-young.”

     Fifty years ago this summer, “flower power” captured the nation’s attention. Hippies and flower girls in California and elsewhere were dropping out of middle-class America, with its pursuit of jobs, homes, and children, and were choosing to live in communes.

    There, the drug of choice was LSD. Guys wore jeans, sandals, and psychedelic-colored t-shirts, refused to cut their hair, shave, bathe, or work. Instead, they strummed their guitars, sang of free love, and accepted every idea that jumped out.

     Girls wore granny-styled dresses, and rimless glasses. The “Mamas and the Papas” sang, “If you’re going to San Francisco, be sure to wear some flowers in your hair.”

     Yet, flower power, for all its ideals, in the hands of the wrong person, can go awry.

     An example of that is Charles Manson, who in the late 1960’s gather about him in Los Angeles a gaggle of young and impressionable girls and guys, who listened to his crazed lectures, and followed his instructions to murder people.

     In July of 1969, all of America was enthralled with its new heroes, astronauts who dared to walk on the moon. That same month, Charles Manson instructed Mary Brunner, Susan Atkins, and Bobby Beausoleil to murder a musician named Gary Hinman.

     Then, on the night of August 8-9, 1969, Manson ordered Susan Atkins, Charles “Tex” Watson, Patricia Krenwinkel, and Linda Kasabian to drive to 10050 Cielo Drive in Beverly Hills, California and commit murder again. Five people lost their lives.

     They were: the actress Susan Tate, then 26 years old, and 8 ½ months pregnant; Steven Parent, then 18; Jay Sebring, a Hollywood hairdresser, then 35; Abigail Folger, then 25, and heir to the Folger’s coffee fortune, and Voytek Frykowski, then 32, and friend of Roman Polanski, Sharon Tate’s husband.

     The next night Leno LaBianca, 44, a supermarket executive, and his wife Rosemary, 38, lost their lives, but this time Manson himself participated. Also, in August of 1969, the Manson Family ended Donald Shea’s life.

     When the truth came out, all of America was appalled. Nine people were dead. “Flower power” had gone to seed. Freedom to live was transformed into freedom to kill. Instead of heroes whom we could admire, we now had anti-heroes who disgusted us, shocked us.

     The essayist Joan Didion said, “Many people I know in Los Angeles believe that the Sixties ended abruptly on August 9, 1969, ended exactly the moment when word of the murders on Cielo Drive traveled like brushfire through the community.”

     The movie director Quentin taratino just released his newest movie, “Once upon a Time . . . in Hollywood,” a spoof that uses the Tate murders as backdrop to Taratino’s own fictional rewriting of the factual history.

     Yet, it is true that Charles Manson died in prison on November 19, 2017, but Leslie Van Houten, Patricia Krenwinkel, and Charles “Tex” Watson are still there. Krenwinkel is now California’s longest-serving woman prisoner.

Apollo 11

by William H. Benson

July 25, 2019

     Apollo 11’s Saturn V rocket lifted off from Kennedy Space Center’s Launch Complex 39A, on Merritt Island, Florida, on Wednesday, July 16, 1969, at 9:32 a.m. EDT. At that moment a clock began running. Hence, NASA officials set liftoff at zero hours Mission Time.

     The Saturn V rocket contained almost a million gallons of kerosene, liquid oxygen, and liquid hydrogen. Michael Neufeld, a Smithsonian curator, said, “If the Saturn V had blown up on or near the launch pad, it would have had the force of a small nuclear weapon.”

     After one complete orbit around Planet Earth, the three astronauts—Neil Armstrong, Edwin Aldrin, and Michael Collins—set a trajectory for a rendezvous with the Moon, 240,000 miles distant. Three days later, on Saturday, July 19, at 75:49 hours, the three entered into the first of numerous lunar orbits.

     On Sunday, July 20, at 100:0:12 hours, the Command Space Module, CSM, and the Lunar Module, LEM, undocked from each other. Armstrong and Aldrin commanded the LEM, code-named “Eagle,” while Collins remained alone to pilot the CSM, code-named “Columbia.”

     At 102:33:05 hours, when 50,000 +/- feet above the lunar surface, Armstrong and Aldrin flipped the switches that ignited the “powered descent engine.” Less than thirteen minutes later, at 102:45:39 hours, Armstrong landed the LEM on the Moon’s surface.

     He reported, “Houston. Tranquility Base here. The Eagle has landed.”

     NASA officials, historians, and journalists have dissected and rehashed those thirteen minutes often. The BBC even built a podcast around those tense, nail-biting, thirteen minutes.

     First, the computer called out a series of “1202” computer malfunctions that alarmed Armstrong and Aldrin, plus Collins and Houston.

     Later, NASA officials determined that they were asking the computer to complete too many tasks, that it felt overloaded, forced to drop certain tasks and stick with others. NASA officials chose to set aside the “1202” warnings, deal with them later, but let Armstrong land the Eagle, rather than abort.

     Then, when Armstrong neared the Moon’s surface, he noticed that the computer wanted to land the Eagle in a crater filled with boulders, some as big as cars. Armstrong took over the landing, keeping the Eagle airborne long enough to pass over that crater until he saw a level plain, and there he landed.

     One NASA official estimated that the Eagle had seventeen seconds of fuel remaining.

     Six and a half hours later, at 109:24 hours, or 9:56 p.m. EDT, Neil Armstrong climbed down the ladder and planted a footprint onto the Moon’s surface. Buzz Aldrin followed at 109:43.

     Together they planted the Stars and Stripes, set up a video camera, unveiled a plaque with their  signatures engraved, collected dirt samples, snapped photos, heard President Nixon offer his congratulations from the Oval Office, and ambled about the Moon.

     Buzz Aldrin looked about him, and called the scene, “Magnificent Desolation.”

     After two hours outside, at 111:37, the two were back inside the Eagle. There they spent the night. Altogether, the two astronauts spent 21 hours and 36 minutes on the Moon’s surface.

     The next day, July 21, at 12:54 p.m. EDT, or at 124:22 hours, the two flipped switches to ignite the Lunar module’s liftoff from the Moon’s surface. The rocket scientists had not built redundant machinery for a liftoff. If Armstrong and Aldrin were to leave the Moon, and return to Earth, that single liftoff system had to work, and it did. One newspaper though had a headline prepared if it failed, “Marooned.”

     At 128:03 hours, three hours and forty-one minutes after liftoff, Armstrong and Aldrin docked their  module with Columbia. They then joined Michael Collins inside the Command Module.

     Three days later, on July 24, at 195:07 hours, the Command Module entered the Earth’s orbit, and at 195:18, it splashed down in the Pacific Ocean, some 800 miles southwest of Hawaii, some 12 miles from the USS Hornet.

     After 8 days, 3 hours, 18 minutes, and 35 seconds, the three called their mission completed.

     A helicopter carried the three to a quarantine trailer, where they remained until August 10. NASA officials feared that the astronauts would bring back a germ that would unleash an epidemic on Earth.

     Soon, they were reunited with their families, and a New York City ticker tape parade awaited them.

     Fifty years have passed since those moments in 1969, but the three astronauts’ story and adventure, their unimaginable achievement, continues to astonish and surprise everyone. They expected their mission to succeed, and it did.