Math ...History?

A Course on How Humans Have Used Math Through the Ages

At The Lukeion Project, we offer a unique course which covers the history of math. Students considering taking a math history course sometimes wonder if the course is a mathematics course or a history course. As with most good questions, the answer is complicated. 

It’s a Math Course

In Counting to Computers (C2C) we discuss many of the topics mathematics students encounter from the beginning of their math career to the end of calculus and beyond. Despite covering so much content, because these concepts developed in a natural way over time we cover them in a format accessible to students in algebra or are advanced prealgebra.

At the “dawn of numbers,” the only tools prospective mathematicians had were their brains and their fingers. It took thousands of years to build up the tools to do calculus and computing. In C2C, we begin where all children begin, with counting. Even counting turns out to be not so simple! 

Over the centuries people have come up with many different number systems which have different characteristics and strengths. The Egyptians developed fractions, but they only used 1 in the numerator (with a few exceptions). Instead of writing ⅔ you would have to write ½+⅙. The two forms are equivalent, but the second seems much more complicated. On the one hand that seems like a lot of work for a simple concept! On the other hand, for solving certain types of division problems, Egyptian fractions were much superior to the our simpler fractions.

In another example of crazy counting Babylonians developed a base 60 system for doing arithmetic. At first glance, base 60 seems much more complicated than our good old familiar decimal system. Why would they make things more complicated? As it turns out, when we use minutes in an hour or degrees in an angle, we actually pay tribute to the Babylonian system. For all its faults, the Babylonian system was miles ahead of Roman numerals.   

Some of the problems we do in the course come directly from problem sets developed to train ancient scribes and mathematicians. We can do problems from the Ahmes papyrus (1650 BC) and decipher cuneiform tablets found in the trash pile from a Babylonian school. Fibonacci’s Liber Abaci, originally written to demonstrate Hindu-Arabic numerals to merchants used to cumbersome Roman numerals, becomes a source for puzzles for homework.

Gradually the math that we discuss becomes more powerful. Even students who are a long way from trigonometry can look at the ratios created by Hindu mathematicians to talk about astronomy, or practice interpolation to find difficult values for sine and cosine using prealgebra techniques. We don’t have to do calculus problems, but we can practice the methods of Archimedes which were precursors of those developed in the 17th century by Newton and Leibniz. The key to thinking about these more advanced concepts is that they developed gradually and not all at once. By following along with the history, we can do the math along side the ancient mathematicians.

It’s a History Course

C2C is also a history course. At its most basic, any history course is a discussion of past events. We discuss events in the history of math such as the development of zero and the conflict between Leibniz and Newton over the development of the Calculus. We show how the development of mathematical symbols over hundreds of years makes the calculations we can do easier and more powerful.

However, there is more to math history than just the events in the subject of mathematics. We discuss how world events led to the export of the modern Hindu-Arabic number system to Europe. The development of the printing press helped spread trigonometry across Europe and made it useful for the scientists discovering how orbits worked and how the stars seemed to move.  In addition, understanding cultures and their motivations makes clear why the mathematics developed by one culture differs from others. The differences in cultures michte explain why the Greeks valued rigorous proof, while Indian mathematicians were able to see  the concept of zero, and Islamic mathematicians developed the idea of algebra. When we study the origins and history of mathematics, we see that mathematics didn’t arrive full blown in an Algebra 1 textbook, but was a gift from many places and times.

In addition to the events in history, we discuss the character and biographies of mathematicians. The people who do math turn out to be as interesting as the ideas they discover. We discuss what is known about the biographies of great mathematicians and why they became involved in mathematics, from Hypatia to al-Kwarizmi to Leonhard Euler. The personalities and culture of these mathematicians influenced what they thought of as important, and therefore what mathematics they developed.

It is a Mix of Both

The power of studying history and math together is that, no matter your background or preferences, there is something for every student.  If you would rather read an article on ancient history than solve a single linear equation, this course is for you.  If you love math and can’t get enough of it but really hate thinking about people, places and things, this course is for you.

Here are two comments from past students, first from a more advanced student (post geometry) and then from a student who is closer to the beginning of her math journey.   

"The history was the most useful part, because while I knew quite a bit of the mathematics, I knew very little of the history of those who discovered it. The brief overviews of the mathematics in the form of the homework was also incredibly valuable, because it helped refresh and cement mathematical principles and even teach me a few new things."

"I enjoyed the connections between math and history and the new math concepts like calculus and logarithms."

Mathematics informs history, and history informs mathematics.  There is no reason to be satisfied with just one or the other when you can get both in one course!
 

What the Romans Did for Us

Thinking about Rome?

By Amy Barr with The Lukeion Project

“Rome,” is now apparently a common answer to the question, “what are you thinking about?”
I agree that anyone hoping to better understand the modern world should look at Rome. The parallels are clear. Comparisons will lend insight to students of history. The best takeaways still serve us well today. Her worst ideas continue to echo through current events. Take five minutes to think about Rome. I’ll keep things brief.
The Romans were famous for adapting and adopting good ideas from other nations and then turning them into great ideas. Architecture, for example, was given a tremendous boost by the ancient Greeks. The Romans enjoyed their artistry and then tweaked the Greek post-and-lintel system by adding the arch. Adapting arches into barrel vaults and groin vaults allowed architects to craft massive structures like the Colosseum and Pantheon. If they tucked those vaults underground, they crafted effective sewer systems that allowed towns to become massive cities. Further adaptations meant Romans had fresh water year-round. This lead to all sorts of luxurious spa experiences that would fetch high prices today though they were usually free in ancient Rome.
The list of Roman contributions is long, and this blog is brief. Highlights include roads (many still used after 2000 years), Latin (still used today in all academic fields worldwide), cement, plumbing, reliable water pressure, good sanitation, heated floors, and hot tubs the size of swimming pools, and unforgettable sports venues. Rome got close to a global currency that worked just about everywhere (we have found Roman coins in India, China, and throughout Europe. Their legal system was sensible, reliable, and included basic human rights applicable to citizen and foreigner alike.
Neutral contributions include the introduction of fast-food restaurants with take-out, massive free public entertainment events that would last for days of blissful vacation time, and satire as a national pastime. The Romans had a relaxed lifestyle with delicious and readily available food but still loved to crack jokes about all of it. Such is human nature.
The Roman Republic was first established in response to the emergent need to topple a violent and dysfunctional monarchy. As is often the case in historical matters, the Romans had to quickly find a good way to prevent tyranny’s return through a decent political and legal system with plenty of checks and balances on her politicians (they considered term limits a great idea, for example).  Early American founders had much in common with early Romans. They too sought a better future for our nation when it became clear that none of us needed kings nor queens.
The Roman Republic worked very well for a little more than 400 years. It served the people and senate of Rome in practical ways while Rome grew from a swampy village to city-state, to nation, to world power, to mega power. Things went well until her politicians discovered loopholes in Rome’s systems of checks and balances. The last century of the Republic left the city in tatters and too much of her population dead. A lengthy period of civil war marked the end of the system. Her politicians had learned how to do whatever they wanted whenever they wished. Once laws are gone, tyranny is the result.  
Chaos diminished under the very strong leadership of Rome’s first emperor. In a series of astounding turns of events, the 19-year-old successor to Julius Caesar would overthrow his competitors and, after having united Rome for the first time in a century, he would rewrite the Roman constitution and then quit. The senate begged their new commander-in-chief to remain and power while they gave him a new name, Augustus. In a little over a decade he would change everything, all while heralded as a patriotic and traditional man-of-the-people. He did love Rome and her empire. He was quick to rebuild, reunite, and expand. Augustus had tremendous organizational abilities, but he also had two big problems. First, he couldn’t live forever. Second, though he was long lived, by the time he did die nobody could remember living under any other system.
Wanting a proper heir, Augustus promoted a disappointing stepson who lacked interest in Rome or her people. Under Tiberius, the whole system could have reverted to a Republic, but the Romans bypassed many opportunities because they were too busy attending sports events. They pressed on and installed a long series of truly dreadful emperors who were easily pressured and conned by those that worked for them. The Roman equivalent of the secret service/FBI/CIA (the emperor’s Praetorian guard) learned that they could easily manipulate any emperor who served. The Roman empire was barely beyond infancy when the Praetorian learned how to obtain whatever commands they placed on Rome’s weak rulers.
The most successful emperors learned how to appease the Praetorian guard with lavish perks and extra power, but it was difficult to maintain positive momentum during transitions to a new ruler. Very rarely was an emperor’s son a worthy candidate so success was limited to emperors chosen for practical skills and intellect rather than legacy.
The Roman empire became massive while her enemies became clever and powerful. They would learn to pick and peck at her boundaries while Rome slowly died of a thousand cuts.   
What caused the fall of Rome? That’s the subject of many books and theories. Rome fell when she outlived her own success by a couple of centuries. Once there were no more major money-making-wars to fight, she turned her attention to micro-managing the empire. Too much governmental oversight was disastrous. Towards the end, the powers governing Rome split her into quarters and made her rulers and sub-rulers bustle around in mobile command units as needed (or even not as needed). Rome as a city lost her identity as capital while new city-centers grew in importance for the sake of convenience.
Perceiving that the leviathan formerly known as Rome was aging and weakening, her enemies kicked at all her doors and windows until they were eventually invited in. Under the pressures of disease, overextended military forces, squandered taxes, under-performing leadership, and a disinterested population, Rome invited her enemies to join their rat race. They were invited to serve in her military and guard her boundaries. Such soldiers had no interest in a Roman way of life. Rome fizzled, sputtered, and died.   

History Was not Boring

Your Teacher Was

Regan Barr with The Lukeion Project

Recently I was speaking to a friend who admitted she is still trying to overcome a lifelong hatred of history. She related a story that I have heard dozens of times: all her "formal" history education was nothing more than memorizing faceless lists of names and dates. She is still bitter. I had some awful history class experiences, as well.
For example, I remember taking state history in junior high...I do not remember any state history, I just remember TAKING state history. And I remember the room clearly. The powers-that-be called these things “portable classrooms,” but the students called them “trailers.” They were rectangular boxes made of corrugated metal that had been strategically placed at the farthest corner of a black-topped parking lot. The Oklahoma summer sun blazed down on that parking lot in late August heating it up to at least four hundred degrees. Crossing from the main building to the trailer for history was dangerous business. The furnace blast would hit you and you could feel your skin melting. Your only hope was to make it to the trailer before your blood began to boil. When we finally stumbled up the rickety wooden steps and into the air-conditioned coolness of that trailer, we collapsed into a dehydrated heap and gasped for air. As we paused to admire our own survival skills, it would hit us. “Oh, man!! They have done it to us again. They tricked us into coming to history.”
The state history teacher was young, just out of college, and he had flaming red hair. He was a wrestling coach who had been forced to teach state history. He must have done something awful in college because he was being punished in this assignment. History was as torturous for him as it was for us. He was easily flustered and when he became angry or frustrated, he turned bright red in the face. Some classmates decided that evoking this reaction from our uninspired and unmotivated instructor was a lot more interesting than any old boring state history. Those students smelled fear, and when their pack instincts kicked in, the poor man had no chance. He was like a wildebeest separated from the herd and the lions were intent on bringing him down. He would get louder, redder, and more animated the angrier they made him. To his credit, he never acted upon those thoughts of violence that surely filled his head. That class took years off his life.
Now some might be under the naïve impression that teachers should spend time learning the subject matter they will be teaching, but I think in teachers’ colleges today you get right to the crowd control training. It is a rare junior high that has an excited and knowledgeable state history teacher, so any poor fool who gets stuck in that classroom better have some crowd control training – some riot gear might help, too. No experienced teachers ever made that blistering journey across the blacktop to see how things were going out in the “state history trailer.” He was on his own!
That is the sum of my memories of history prior to college. I cannot even recall what gender my high school history teacher was, though I am certain I took some history in high school. Is it any surprise that I hated history? I remember my civics teacher because he had some funny mannerisms that were more interesting than civics. I remember my kindergarten teacher because she mispronounced my name for half the school year. I remember…or rather feared…my fourth-grade teacher because she was a large, gray-haired gal who thought her students’ idea of a fun time was skipping the lunchroom pizza crowd to eat vegetable soup (!) with her in the classroom. The image of her brandishing a yardstick as she attempted to chase down some miscreant on the playground is emblazoned in my memory.
I remember those teachers, but not the history I was supposed to learn. Why? Because of the interesting details. My teachers, classmates, and classrooms were swirling vortexes of interesting details, but history itself was not. Today I understand that it is the details that make history fascinating – the sort of details I remember about angry teachers or the walk through the inferno to the “state history trailer” – not the sorts of details you find on a timeline.
By the time it got to me, history had been scoured clean of all the interesting details and only drudgery was left. History does not occur in expansive decades, centuries, and millennia. It does not even happen in timeline tick marks. It happens the same way that life does: one moment at a time, one blistering step at a time. History is not the story of enormous hordes of faceless automatons. It is the story of unique people like you and me. They sometimes cooperate, sometimes get into fights, and sometimes act with dignity and reason, but quite often act in silly, dangerous, or unpredictable ways. It was not until I got to college that I had instructors who loved what they were teaching and filled their history lessons with little-known facts and engaging anecdotes. Guess what?? History is about real people! History cannot be boring unless life itself is boring. But sometimes we stand so far away that we cannot see any of the interesting bits.

Check out Lukeion Classical History for high school level and above

The History of Zero

By Dr. Kim Johnson, The Lukeion Project (Check out Dr. Johnson's class Counting to Computers: Math from the Dawn of Time to the Digital Age)

It turns out that one of the greatest accomplishments in the history of mathematics is---nothing.

From where we sit today, the tools of mathematics seem so obvious that they hardly bear thinking about. It might surprise you, then, to learn that one of these everyday tools took thousands of years, deep philosophy and even controversy to become part of our everyday tool kit. What is this development that we use every day without even thinking about, and yet was so difficult for the most brilliant and advanced mathematicians in ancient times to grasp? The number zero.

Before Zero

In the mists of time, mathematics seems to have been used primarily for counting or measuring things. Whether you are counting the number of animals in your flock or the distance to the next town, the concept of assigning a number to “no animals” or “no distance” seemed unnecessary and overly complicated. After all, “zero cows” is the same thing as “zero apples,” while “one cow” and “one apple” are decidedly different from each other.

This is not to say that ancient civilizations never used any concept related to zero. The Egyptians had massive building projects which required sophisticated engineering: they used a number to refer to the ground level. They also had a concept of a “zero balance” when no money was owed. The symbol for these was called nef, which also meant “perfect” or “complete.” 

The Greeks had an even harder time with the concept of zero than the Egyptians. Pythagoras declared, “All is number,” but what he really meant was “All is ratio.” Numbers to him represented the ratio between two quantities or lengths. This is useful in geometry (recall the concept of similar triangles) and in music (the ratio between lengths of string that produce an octave is 1:2) but terribly difficult for the concept of zero--a ratio of zero is almost nonsensical. Aristotle, whose philosophy reigned over the western world through the Middle Ages, stated that, “Nature abhors a vacuum.” Not only was zero not a number, but philosophically speaking, it was impossible. The paradoxes of Zeno of Elia (490-430 BC) point out the difficulties that the existence of zero would imply.

The absence of the number zero became a problem for Babylonian computation. Unlike the Egyptians, Romans, and Greeks, the Babylonians had a positional place value system where each number meant different things depending on where it occurred. In our system, where the positions represent powers of 10, the digit 2 in 2,528 means either 2,000 (the first 2) or 20 (the second 2). In the Babylonian system different positions represent different powers of 60. The leftmost 2 would mean 2x(60x60x60) while the rightmost means 2x60. This is fine unless you need a number which contains a zero. For the Babylonians, 11 could mean 101, 1001, 110, and so on, depending on context. They eventually developed a place holder to mean zero, but it only worked in the middle of a number, not at the end. They could distinguish 202 from 22, but not 220 or 2200.

The Discovery of Zero

Fortunately for mathematics, something changed in about 300 AD. Indian mathematicians began developing the symbol sunya which referred to the number zero. Rather than just a place holder, sunya (first a dot, then a ring) was used as a number. It makes sense that zero should be developed in the context of eastern rather than western philosophy. The concept of nothingness or emptiness, anathema to the Greeks, plays a much larger part in Indian religious philosophy.

 

In the 7th century, the great mathematician Brahmagupta (598-668 AD) created a whole system explaining how zero worked with other numbers. Some of his rules included the fact that a fortune, take away zero, is a fortune; or a fortune multiplied by zero is zero. He did commit the mathematical sin of dividing by zero, but that concept remained difficult for mathematicians well into the modern era. Brahmagupta also talked about negative numbers with more comfort than western mathematicians would into the 18th century but that’s a blog post for another day. 

 

After Zero

It seems obvious to us now but zero took a while to catch on in the rest of the world. The first people to adopt zero—and see how useful it was—were the Arabs. Most notably, Al Khwarizmi (c. 780-850 AD) used Hindu-Arabic numbers as he was developing algebra at the House of Wisdom in Baghdad. He described zero, which he called sifr, in his book Algoritmi de Numero Indorum as “The tenth figure in the shape of a circle.”

Several individuals were instrumental in introducing zero to Europeans. One of the earliest, Pope Sylvester II (c 946-1003) had studied the Hindu-Arabic numerals when he went to Muslim controlled Spain to study mathematics and science. Because of his facility with these numbers (and probably for other reasons) he was accused of consorting with the devil.   

The next great advance in using this Hindu-Arabic numeral came when Leonardo di Pisa—better known as Fibonacci (c.1170-1240) wrote his great work Liber Abici. In addition to problems about rabbits, he introduced the numerals he had learned about when he studied in North Africa among Muslim mathematicians. He also introduced algorithms for doing basic mathematics including 28 pages about how to do long-division. We still use his method today.

 

 

“These are the nine figures of the Indians 9, 8, 7, 6, 5, 4, 3, 2, 1. And so, with these nine figures, and with the symbol 0, which is called zephyr in Arabic, whatever number you please can be written, as is demonstrated below.”

Even after zero came to be used in the west, it wasn’t immediately accepted by the public. Financial institutions in Florence banned the use of zero in 1299. They felt that there were too many opportunities for fraud because someone could use place value to change the values on the checks that were being written. Even today we write “seven hundred fifty and no/100 dollars” on our checks---perhaps as a deterrent to adding zeros and increasing the value of your check.

The medieval philosophy textbook Margarita Philosophica (Gregor Reisch, 1503) shows a picture of a contest between someone computing with the old counting board on the right, and the new-fangled Hindu-Arabic numerals on the left. You can see who is happier. The Hindu-Arabic number system eventually won out, including its use of zero.

Zero has proven useful beyond allowing a more efficient number system. Zero eventually became the star of the number line and of the Cartesian plane, invented by Pierre de Fermat and Rene Descartes, and used to visualize algebraic functions and equations. Zero is essential in the invention of calculus. Newton and Leibnitz use infinitesimals to find a tangent line to a curve, quantities that are almost zero but not quite. Don't forget that zeros make up about half of every binary computing system.

The history of zero spans thousands of years, multiple civilizations and continents---and yet it is something so simple that we use it every day.

If this story of the connection between math and culture piques your interest, you might want to sign up for the course Counting to Computers: the history of math from the dawn of time to the computer age. This course is appropriate for strong pre-algebra students through high school students taking higher math.

100

The Lukeion Project in Retrospect

By Amy Barr of The Lukeion Project

This is our 100^th Sassy Peripatetic Blog! Sassy Peri began September 21, 2018, after a full decade (likely more) of people pestering us about starting a blog. Of course, we knew we should start a blog, but for the first five years of The Lukeion Project, we hadn’t worked out how to get a full night’s sleep much less how to add anything new to our workload. It took us another 5 years to catch up. Many of you may not know our story, so gather around. This is the abbreviated version.

The idea of The Lukeion Project was hatched toward the end of 2005 by two Classical archaeologists, namely me and Regan Barr, as we painted our daughter’s bedroom (pink and purple, if you must know). We had settled into very mundane lives after over a decade in the archaeological field, digging at places like ancient Troy. A financial crisis forced us to get creative about our future. As crises almost always do, we were given the chance to return to our roots, namely our passion for dead languages, Classical history, archaeology, and Classical literature. We wanted to teach they way we wish we had been taught. We wanted to take our students seriously.

When we began our small venture, we had to pick a name. We had no intention of becoming a full school or academy offering a comprehensive curriculum to all ages. We were well skilled in Latin, Greek, and related fields but had no intention of adding things like art, sports, and science nor did we have the patience of saints necessary to teach all ages.

As everyone should, we turned to Aristotle for answers. His philosophical hangout in Athens was known in Latin as The Lyceum. In Greek – as Aristotle would have used—it was called Λύκειον (Lukeion). Aristotle established his school in Athens in part of the temple to Apollo in 335 BC (he worked with what was available). He lectured there, led his students on walk-abouts (thus their name Peripatetics), and established the first European library in history. He had always been a book collector and then his student, Alexander the Great, helped by sending him crates of scrolls and copies of important works as gifts to his former tutor.

The word Project in our name was chosen because we didn’t want people thinking we were a full school nor an academy since our courses were meant to supplement the curriculum of our primary audience, home educated students working at the high school level.

When we first began classes in the spring of 2006 there was only one other online program out there. Most of their classes were still conducted by correspondence but a few had gotten high tech like our classes. This meant we were the first of our kind. Every big flashy online program that we see at conferences today first talked to us way back then about what we were doing and said, “hey, that’s a good idea.”

When we first went to homeschool conventions to spread the word about our classes, we spent half our time explaining what an online class was. Terms like webinar were not commonly understood. Words like blog, vlog, and podcast were still in the formative stages. The wizardry of having students synchronously join us online, hearing, and seeing material as we taught was also virtually unknown at the start. Our first online learning platform was for business meetings. It often had as much as a 7-second delay making online discussions a bit tricky. Such technology has now become so mundane that even 10-year-olds get irritable if our internet gets crackly or because we don’t let them play with the video filters in class.

In 2012 we successfully cajoled our archaeological colleague Dr. Sue Fisher to join us as she expanded our Latin program and then expanded our Classical literature offerings with her Muse series. Since then, we’ve added Prof. Baty (literature including AP/writing/rhetoric), and then Dr. Johnson (logic and more) who stepped in right after the sudden tragic passing of Dr. Haggard (logic/philosophy). Prof. Powell came on board last year to expand our Latin program further. Regan Barr (beloved teacher of Witty Wordsmith/Barbarian Diagrammarian) also directs our 8 years of Classical Greek plus Greek history, and Philosophy. I (Amy Barr) direct the large Latin program (also 8 years and expanding) plus Roman History.

The Lukeion Project has offered semester classes since 2006. As we now begin our 17^th year, we have had over 3000 (close to 4000) students, 100 blogs, hundreds of semester courses, and at least a thousand workshops. We’ll never win awards for fame nor size, neither breadth of influence nor endowments. We’ve labored to keep standards high (despite direst pressure to drop them low) and to keep tuitions low (despite direst need to increase them). We work without the safety net of government support nor impressive private capital. Our few faculty and staff are dedicated, humble, and talented beyond measure. there will never be enough thanks nor money to truly pay them.

Our rewards are less earthly and more eternal. Like Aristotle getting gifts from Alexander the Great, our former Lukeion students send back crates of greatness. Their lives have now spread into the stars at NASA, into the lights of Broadway or Carnegie Hall, into the classrooms of Ivy League schools, into excavations, laboratories, and libraries as doctors, lawyers, writers, scholars, linguists, dancers, artists, and (perhaps best) as moms and dads who delight in educating their own children whom we hope to teach—in turn—in the years to come.

The poet Horace (Odes 3.30) expressed our hope most fittingly in 23 BC:

exegi monumentum aere perennius

regalique situ pyramidum altius,

quod non imber edax, non Aquilo inpotens

possit diruere…

I have built a monument more lasting than bronze,

higher than the Pyramids’ regal structures,

that no consuming rain, nor wild north wind

 can destroy…