School begins this month, perhaps this week. Teachers again will introduce students to questions on math, science, English, social studies, and foreign languages. I wish them all well. No subject is easy, but some say math is the hardest.

I remember geometry as my hardest. Theorems spooked me. The following year I found algebra more enlightening. Graph coordinates with x’s and y’s, as well as logarithms seemed to click in place.

Jordan Ellenberg, a math whiz from Wisconsin, just published a new book, Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else.

In an early chapter he describes Abraham Lincoln’s admiration for the Greek mathematician Euclid, who wrote Elements of Geometry, “the most successful textbook of all time,” around 300 BCE, in Alexandria, in Egypt, in northern Africa.

Lincoln’s law partner, Billy Herndon, remembers watching Lincoln struggle for two days to square a circle, “almost to the point of exhaustion.” He “was trying to construct a square with the same area as a given circle, using two tools, a straightedge and a compass.” He gave up. He found the job too difficult.

For centuries, “squaring the circle” has been synonymous with “completing an impossible task,” or like “passing an act of Congress.” Yet, Lincoln tried.

The job involves first finding the area of a circle, using the formula πr². Then, finding the square root of that product will produce the length of each of the four sides of a square. Both the circle and the square will have the same area.

It sounds easy, and it is with computers and hand calculators, but only to an approximation, and the reason is because π is an irrational number. It is 3.141 . . . , without any recurring pattern.

Lincoln though took to heart Euclid’s talent for making sense out of the bewildering. Euclid started with 35 definitions, a handful of postulates, a dozen or more axioms, a series of postulates, and argued for a set of theorems, all about triangles, lines, angles, squares, and circles.

In so doing, he created a complete body of mathematical work, 2300 years ago.

As a young man, Lincoln was reading a stack of law books, hoping someday to practice law. He said, “I constantly came upon the word demonstrate.” He looked the word up in the dictionary. He read, “Certain proof,” and “proof beyond the possibility of doubt.” But then, “what is proof?”

Finally, he gave up, and said to himself, “Lincoln, you can never make a lawyer if you do not understand what demonstrate means. I went home to my father’s house, and stayed there till I could give any propositions in the six books of Euclid at sight.

“I then found out what ‘demonstrate’ means, and went back to my law studies.”

Lincoln read a lot of Thomas Paine, the American revolutionary writer, who in his The Age of Reason, declared his unwavering admiration for Euclid.

“I know, however, but of one ancient book that authoritatively challenges universal consent and belief, and that is Euclid’s ‘Elements of Geometry,’ and the reason is, because it is a book of self-evident demonstration, entirely independent of its author, and of everything relating to time, place, and circumstance.”

In Shape, Jordan Ellenberg points out that Euclid understood the “extreme and mean ratio.” Form a line, A to B. At a certain point on that line, between A and B, but closer to B, place a point, called C. If you divide the length of AB by the length of AC, you will get a number about 1.6.

If you divide the length of AC by the length of CB, you also will get 1.6. If the point C was placed correctly, the actual ratio will be 1.6180339887 . . . , another irrational number, but less popular than π.

Mathematicians since Euclid now call this the Golden Mean, identified by the Greek letter φ, that that pronounce as “fee.” They find this number when they count leaf arrangements in botany, when they measure the swirl of galaxies, and when they look into number theory.

For example, consider the Fibonacci numbers, a series in which the next number is the sum of the previous two numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The ratio of 3 to 2 is 1.5, 5 to 3 is 1.666, 8 to 5 is 1.6, 13 to 8 is 1.625, 21 to 13 is 1.61538, and 144 to 89 is 1.61797.

The ratio keeps getting closer to 1.618 . . . .

Ellenberg writes, “In Lincoln, we find a more appealing character: enough ambition to try, enough humility to accept that he hadn’t succeeded.” He further says, “The ultimate reason for teaching kids to write a proof is not that the world is full of proofs, but that it is full of non-proofs.”

Assertion is not evidence. To assert is not to demonstrate a proof.

I wish all this year’s crop of math students the best. You might ask your teacher about Euclid.

Rarely do men and women seem free, even for a moment, from the evils that have plagued human beings for millennium: war, poverty, famine, slavery, racism, diseases, pestilence, and natural disasters.

The U.S. armed forces first attacked the Taliban in Afghanistan on October 7, 2001, and now, the twin wars in Afghanistan and Iraq are winding down after two decades, wars that have cut short many human lives and caused painful, debilitating injuries. On the horizon is a menacing conflict with China.

Poverty and famine often arrive together. During a famine, the poorest starve. In the past hundred years, the worst famines have struck hard the poorest peoples of India, China, and Russia.

Racism begets slavery. When one race, class, or culture views another as weak and less-than-human, the superior will try to enslave the weaker. “Enslave them, work them, refuse to pay them.”

On occasion, some have resorted to mass murder to eradicate another race, class, or culture. The Nazi’s sought to destroy the Jewish population across Europe in the 20th century, and in a prior century, greedy European settlers looked through Native Americans, hoping to steal their land.

Certain diseases—cancer, heart attacks, diabetes, and arthritis—unleash an ocean of misery, pain-filled days, and shortened lives. No one signs up for these very human diseases.

Modern human beings almost forgot about pestilence. Then, eighteen months ago, from out of China, came a coronavirus carried by bats that has cut a deadly swath through the world’s human population. Now, men, women, boys, and girls, must prepare for another wave, the delta variant.

Vets in Weld County, Colorado have determined that a horse there has tested positive for West Nile virus, and local health officials in La Plata County, Colorado, near Durango, have reported that a sample of fleas has tested positive for Yersinia pestis, the bacteria that causes bubonic plague.

We dare not forget the other form of pestilence, insects. Sawflies can wreak havoc upon a wheat field, and swarms of Africanized honey bees called killer bees can attack human beings and livestock.

Then, because we are required to live on Planet Earth, we have little say-so in what Mother Nature decides. She outfoxes all of our pitiful attempts to predict her variable moods.

In a matter of minutes, a hail storm can destroy fields of wheat, sugar beets, corn, or beans. A heavy rain miles away can cause a flood here, where it is dry. Then, men and women continue to gape in astonishment at the devastation that tsunami’s, earthquakes, volcanoes, and forest fires can produce.

For example, on June 24, 1976, my dad’s wheat fields, located fifteen miles southeast of Sterling, Colorado, stood more than waist high and was difficult to walk through, a better crop than usual. He was most anxious to cut it in three or four weeks and put it into the bins.

That night, a five or six-inch rain fell on my dad’s farm. Hail and high winds pulverized the wheat crop. The next morning my dad and I stared at what was once a wheat field, but now looked more like summer-fallowed ground. The wheat crop was gone, shredded beyond recognition.

Then, the soil atop the hills of the summer-fallowed ground washed down into the draws, leaving a hardpan. I had to bounce on an International 806 tractor for days, pulling sweeps and a rod back and forth across that pitted hardpan, in order to work up enough soil to plant next year’s wheat crop.

The night of the storm we also lost the use of one of our two tractors for the rest of that season, because I had parked it the night before in a gravel pit, that filled with water and covered the tractor.

Those were hard days. I thought, “this has to be the worst,” but Mother Nature had a different idea.

Six weeks later, on Saturday evening, July 31, 1976, an estimated twelve inches of rain fell in just four hours in the Big Thompson Canyon, east of Estes Park.

A wall of water twenty feet high rushed down the canyon, toward Loveland, at a speed of 14 miles per hour. It picked up gigantic boulders and pitched them aside. In all, “the flood destroyed 418 homes, 52 businesses, and 400 cars,” along with campers, and dozens of propane bottles that floated away.

Of the 4,000 people in the canyon that Saturday night, 144 people drowned, and of those, five bodies were never found. One who died was Colorado State Trooper Willis H. Purdy, who stayed long enough in the canyon to turn back all travelers coming up from Loveland.

His last transmission came at 9:00 p.m. “I’m not going to make it out of here. Warn everybody below. The water is coming, and they need to get out.” His superior officers said, “That transmission, that order, that directive, is credited with saving hundreds of lives.”

If I remember correctly, I met Officer Purdy once. In the mid-1960s, he was a patrolman based in Logan County, in northeastern Colorado, and he came to a 4-H club meeting that I attended. He spoke to our club’s members, about driving safely, and he showed us pictures of numerous accidents.

Last weekend marked the 45th anniversary of the Big Thompson flood.

He begins with the Vikings, and then steps forward, chapter by chapter, until he finishes in the 21st century. Along the way, he brings in plenty of fascinating details about the island’s towns, people, weather, government, and the Northern Lights, an enjoyable and readable geography primer.

In his introduction, he tells of the day he received a special bonus as a cub reporter at his town’s newspaper, “a twenty-seven-gear Mongoose bicycle, a touring bike with fat tires and a rear rack.”

He rode the bike out of Selfoss, a town 50 kilometers east of Reykjavik, intending to ride the full Route 1, the Ring Road, “an 821-mile loop that connects most towns and villages in the country.”

Egill discovers that the terrain is “famously uneven,” and that “along the coast the wind blows hard,” and “directly against you while bicycling. Always, I tell you. Always.”

He made it half-way, from Iceland’s southwest corner to Húsavík, a town on Iceland’s north shore, that lies just under the Arctic Circle. At that latitude in the north Atlantic, the sun never rises or sets between June 11and June 30 each year, truly a “land of the midnight sun.”

There, Bjarnason took a job on a cruise vessel that carried paying passengers out to sea to observe whales, or to the west to see Greenland’s massive fjords. He discovers how the ocean’s cold and its chronic wind can make him feel most miserable. “Water. Water. Water. Land!” he cries.

In his first chapter, he recites the stories of the most famous Vikings: of Erik the Red, his son Leif Erickson, their settlement in Greenland, and their discovery of Vinland in North America. Leif named it Vinland because one of their party found and ate fermented grapes. “Wineland or Vinland.”

Throughout the book, Bjarnason drops in certain interesting facts about Iceland.

For example, Iceland has the smallest army in Europe, not a single soldier. It has never participated in a foreign invasion. It has no railway system. Most of Iceland’s towns have a pool that is open all year with geothermal-heated water, and many of its residents enjoy a daily swim all year.

The only mammal native to Iceland is the Arctic fox. There are few, if any, reptiles or amphibians in Iceland, and no forests. Centuries ago, the earliest settlers cut down the trees. Instead, it has “boundless Icelandic deserts, shaped by volcanic eruptions and covered in different shades of lava.”

In and around the bareness, a certain purple-flowered plant has now emerged, “the Alaskan lupine that arrived in Iceland in 1945 in a suitcase,” in a misguided attempt to provide “an efficient cover for the eroded land.” Today, “the lupine is considered an invasive plant.”

Bjarnason points out that “Iceland is basically Hawaii upside down.” The first is located in the northern Atlantic, and the other is in the southern Pacific. But whereas Hawaii has 11,000 square miles divided among a number of islands, Iceland’s single island covers 40,000 square miles.

Iceland’s population approaches 357,000, and of that total, 123,000 live in Reykjavik, the world’s most northern capital.

Bjarnason tells of the two most famous visitors to Iceland: Neil Armstrong and Bobby Fischer.

In July of 1967, the American astronaut Neil Armstrong trained in Iceland for the scheduled moon landing two years later. NASA chose Iceland because of its lunar landscape, described as “volcanic geology with no vegetation cover.” During his days off, Neil found an Icelander who took him fishing.

In the summer of 1972, the American Bobby Fischer challenged the Russian grand master chess champion, Boris Spassky, to a 21-game match that would convene in Reykjavik. Bobby won 12 ½ games to Spassky’s 8 ½, and became the eleventh world chess champion.

In April of 2005, Japanese officials released Bobby into the custody of Iceland’s officials, after the Japanese government had held him in detention there for nine months. U.S. officials had wanted to extradite him back to the U.S. for income tax evasion, and because he had played chess in Yugoslavia.

Bobby lived in Iceland for the next 27 months, although Bjarnason says, “The truth is that Bobby Fischer hated living in Iceland. He could not travel abroad with the U.S. still pursuing a case against him.” Also, the cold and the constant wind frayed his “madness, paranoia, and aimless years.”

He died in Iceland on Jan. 17, 2008. Bjarnason says, “In the end he defeated himself.”

A most honored American, Neil Armstrong, vs. a dishonored American, Bobby Fischer. Each spent time in Iceland, and each helped Iceland go about in a quiet and Nordic way to “change the world.”

As the year 1776 unfolded, American colonists were confronted with the question of independence. Some favored it, others rejected it, and a third group remained uncommitted.

This political question caused hard feelings between colonial Americans. More and more colonists were forced to take sides in this bitter conflict. Some chose. Some refused. Battle lines were drawn. The question divided families, communities, churches, schools, and local governments.

Those who spoke out in favor of separating from England’s political system—some 30 to 40% of the population—took the name of Patriot, even though they were rebels, committing treason against King George III and Parliament, and if caught, tried by jury, and found guilty, were subject to execution.

Ben Franklin said, “We must all hang together, or, most assuredly, we shall all hang separately.” He understood fully the possible tragic consequences when he signed the Declaration of Independence.

Then, those who remained faithful to the British government—some 20% of the colonies’ existing population—took the name of Loyalist, or Royalist, or Tory.

Thus, some 40 to 50% of the remaining population considered themselves neutral, uncommitted. “Some neutrals did not much care who governed them, so long as the government left them alone. Others did want to be on a losing side. It was a great risk to stake out a position.”

A recent arrival from England, known only as “an Englishman,” published a pamphlet that he entitled Common Sense on January 10, 1776, in Philadelphia, Pennsylvania. Readers quickly learned that the author’s name was Thomas Paine. His pamphlet sold 150,000 copies, a run-away best-seller.

In it, he argued for independence, and based his argument upon two ideas.

First, that “the nature of the British monarchy and constitution was so corrupt that the only solution was immediate and complete separation.” And, second, that “the resources of the colonies were sufficient to defeat any military force that the British might dispatch to the colonies.”

Many colonists read his words and agreed. They dared to hope for independence, and in their ability to govern themselves. Others were appalled. Whereas Paine saw a quick and easy military victory, others saw “rivers of blood,” and where he saw democracy, others saw a mob’s chaotic rule.

For two months no one dared to step forth and argue for loyalty to the British government.

Then, on March 13, 1776, a short pamphlet entitled Plain Truth appeared in bookstores. Its author identified himself as “Candidus.” In recent years, historians have determined that the author was James Chalmers, a wealthy landowner then in Kent County, Maryland, on the eastern shore.

In Plain Truth, Chalmers said of Paine, “His first indecent attack is against the English constitution, which, with all of its imperfections is, and ever will be, the pride and envy of mankind.”

Second, Chalmers did not believe that “the colonists could ever defeat Great Britain’s might army and navy. The English outgunned the colonists’ rag-tag militia. Alone, they could not win. They must join forces with a great European power, like France or Spain, in order to gain even a slim chance.”

Then, on April 17, 1776, “Candidus” published a second pamphlet, Additions to Plain Truth, and in it, Chalmers warned the colonists of the high price of an ugly war.

“Should this war prove unsuccessful on the part of Great Britain, we cannot imagine that it will terminate, e’er many bloody fields are lost and won; I say, it will not end in less than ten years.”

Actually, the war for independence spread across six and half years, from Lexington and Concord, in April of 1775, to Cornwallis’s surrender at Yorktown, Virginia, in October of 1781.

The colonists now had three pamphlets in front of them to read. They could agree with Thomas Paine and fight for independence, or side with “Candidus” and hope that the British will defeat the colonists with little loss of life and property, and that the British will reconcile with the colonists.

Historians since 1776 have searched in vain for reasons to explain why one colonist would chose the Patriots’ cause and another the Loyalists’ cause. The choice was not due to “a person’s educational level, occupation, social class, or economic status.” It was individual, made at a personal level.

Some Loyalists, who could see that Paine, Jefferson, and other Patriots would win the propaganda war, gave up and fled to Canada or to England, but only between 80,000 and 90,000 did so.

One Patriot, Benjamin Franklin, remained forever bitter about one Loyalist’s choice, that of his son William Franklin, then the Governor of New Jersey. In his will, dated July 17, 1788, Ben left to William, “all the lands I hold in the province of Nova Scotia,” 2,000 acres of forest.

Ben then says, “The part he acted against me in the late war will account for my leaving him no more of an estate [than] he endeavored to deprive me of.” Much to William’s regret, he chose King and Parliament, rather than his dad’s experiment in self-government and independence.

You and I, and all others who claim American citizenship, now have reason to celebrate a new Federal holiday, Juneteenth, our 12th legal public holiday.
Last week, on Tuesday, June 15, the Senate unanimously passed legislation to make June 19, or Juneteenth, a national holiday. On Wednesday, June 16, the House passed it with only 14 “no” votes.

On Thursday, June 17, President Joe Biden signed into law the Juneteenth National Independence Day Act in the White House’s East Room. In his remarks there, Biden said, “Great nations don’t ignore their most painful moments. They don’t ignore those moments of the past. They embrace them. Great nations don’t walk away. We come to terms with the mistakes we made. And in remembering those moments, we begin to heal and grow stronger.”

On Friday, June 18, federal workers enjoyed a day off, to reflect upon slavery’s extinction.

The last time Congress and a president brought into existence a new federal holiday was in 1983, almost four decades ago, when Ronald Reagan signed Martin Luther King Jr. Day into law.

Today is Saturday, June 19, and I am writing this column now. You and I now recognize Juneteenth as, “A holiday celebrating the emancipation of those who had been enslaved in the United States.”

On June 19, 1865, a Union Army major general named Gordon Granger arrived in Galveston, Texas, to inform all Texans, both free and slave, that slavery was over, finished, an ugly and painful memory, and that he will enforce Abraham Lincoln’s Emancipation Proclamation of 1862.

Two months before, on April 9, 1865, the Civil War had ended, when Robert E. Lee surrendered to Ulysses S. Grant. Five days later, on April 14, John Wilkes Booth shot Lincoln in the back of his head, and he died the next day. Lincoln was dead, but so too was slavery. He had set free four million slaves.
Granger issued General Order Number 3, a document now housed in the National Archives Building in Washington, D.C. The major general delivered just four sentences, split into two paragraphs.

First sentence, “The people of Texas are informed that, in accordance with a proclamation from the Executive of the United States, all slaves are free.” One can only imagine the wild joy that tore across the Texas prairie, once the slaves heard the welcome news that President Lincoln had freed them.
Second sentence, “This involves an absolute equality of personal rights and rights of property between former masters and slaves, and the connection heretofore existing between them becomes that between employer and hired labor.”

In terse words, Granger states that no man or woman can henceforth buy, sell, or claim to own another person, that each citizen is “absolute” equal to another, and that each retains “personal rights.” Granger then replaces the idea of “masters and slaves,” with “employer and hired labor.”
That distinction in the changed relationship between management and its work force is immense. A slave would receive little compensation for her or his work, had no opportunity to leave an employer and find work elsewhere, and suffered the most brutal beatings if tempted to run away.

Now he or she is free to find work and opportunity wherever. Gone are the chains, the leg irons, the whips, the ropes, and the merciless whipping post that kept a slave in bondage forever.

Third sentence, “The freedmen are advised to remain quietly at their present homes and work for wages.” Granger suggests that the former slaves should stay, for the moment, where they live now, refrain from seeking revenge, and that if they work, their former masters must pay them money.

Fourth sentence, “They are informed that they will not be allowed to collect at military posts and that they will not be supported in idleness either there or elsewhere.”

Granger fears that the former slaves will seek protection from their former masters’ cruelty by congregating “at military posts,” and that they will then expect the U.S. Army to support them “in idleness either there or elsewhere.”

Granger dashes that notion. Although he will set the Texas slaves free and will grant them a set of astonishing political rights, he expects the former slaves to work. Granger knows that an economy will not function without a work force, that management needs to hire workers, and that workers need jobs.

Last April, the historian Annette Gordon-Reed, a native Texan and a descendant of slaves, published a slim volume she entitled “Juneteenth.” She writes that, “she remembers Juneteenth celebrations from her childhood, drinking red soda and setting off firecrackers that her grandfather bought for her.”

As of last week, Juneteenth is no longer just a Texas holiday, but a national holiday, an attempt to “embrace” our “most painful moments, “to come to terms with the mistakes we made,” early in our history, and “to heal and grow stronger.”

How does one recognize great writing in a novel, a work of history, or a scientific treatise? The typical answers include: if it sells 5,000 copies, if it makes the “New York Times Best Seller” list, if it wins a literary prize, if a literary critic gives his or her stamp of approval, or if it is printed for decades.

Each generation of young people discover for themselves the wealth of ideas that they can dig out of Plato’s Dialogues, Ralph Waldo Emerson’s Essays, or Shakespeare’s plays. Those works have lasted.
Another question. How does one recognize a great teacher?

Years ago, I attended a University of Northern Colorado alumni meeting and met UNC’s president then, Hank Brown. He asked a series of questions: how does a university train a teacher to teach? What variables make a teacher great? How can a school district evaluate their teachers?

We brainstormed for a moment, and came up with some quick answers: effective presentation of content, wise selection of a well-written text, maintain class order, challenge the lazy, direct the smart, and include a fun factor, that is a teacher must present neat and intriguing ideas everyday.

How does one recognize great art, or a great preacher, or great politics, or a great religious faith? Harold Bloom, Yale’s long-time English scholar, wrote a book thirty years ago, The American Religion, and in it he stepped away from literary criticism to write as “a self-appointed religion critic.”
Here is an appropriate question for today. How does one recognize great parents? By a casual look at the calendar, we see we are midway between Mothers’ Day and Fathers’ Day.

Great parents endure many hours seated in a gym / stadium / theater, watching their daughters and sons perform, either a musical or theatrical performance; or a football, volleyball, or basketball game; or a wrestling match; or a cheerleading contest; or a gymnastics meet.
A comment I heard years ago, “The best gift that a dad can give his son or daughter is that he loves his child’s mother.” And, to a child, “TIME = LOVE.” A child measures the depth of a parent’s love by the amount of time that that parent devotes to that child, or at the least, pays attention.

It might make our jobs easier if a scientist could reduce any number of human activities to an equation. If we add x, y, and z, the good things in life, and then subtract out the sum of a, b, and c, the bad parts, we might get q, a successful writer, teacher, preacher, parent.
Is an equation for these items even possible? I wonder, “is there a mathematical law that predicts these kinds of human achievement?”

Three hundred years ago, people did not know that mathematics directed the universe’s motion of planets around the sun, until Sir Isaac Newton, an English scholar and scientists, published Principia Mathematica, System of the World.
His law of gravity states that if a mathematician multiplies the mass of one body times the mass of a second body, and then divides that product by the distance between the two bodies, squared, she or he will get F, the attractive force called gravity. He gave the world an equation that measures gravity.
Newton’s Law of Gravity “tells us that the motion of bodies within the solar system is determined by a mathematical law.” What other mathematical laws exist that we know nothing about?

In recent days, I have worked my way through a book I found, In Pursuit of the Unknown: 17 Equations That Changed the World. Ian Stewart, mathematician at University of Warwick in Coventry, England, divides his book into 17 chapters, a chapter for each equation.
The first chapter / equation is Pythagoras’s theorem, the second is the logarithm, the third is Isaac Newton’s calculus, and the fourth is Newton’s law of gravity.
I find the seventh chapter most interesting, “Patterns of chance, normal distribution.” Men and women are individuals. No one knows what each will do or say, but Stewart points out, “people en masse behave more predictably than individuals.”

Scientists in the last two hundred years learned of a bell curve when it came to variables of a given group of men and women. Some of those variables include height, weight, crime, births, deaths, test scores, IQ. “No one expected these variables to conform to any mathematical law,” and yet they did.

The equation states, “the probability of observing a particular data value is greatest near the mean value, the average, and then dies away rapidly as the difference from the mean increases. That difference from the mean is called standard deviation.”
For example, out of a given classroom, one or two students get an A, one or two get an F, because they are distant from the classroom average. They are outliers. The majority get a B, a C, or a D.

We all can look forward to a modern-day Isaac Newton who will devise an equation that will guide each of us to perhaps, write a novel that will win a Pulitzer Prize, to win accolades as a great teacher, or to parent exceptional boys and girls. Alas, we are too average.