It’s not the beginning of a joke. It happened a couple of years ago, when two friends and I began meeting to share our experience teaching Euclid’s Elements. Ben teaches it to sixth graders, Dillon to tenth, and I teach eighth. We thought about publishing a Companion to Euclid, teacher notes on how to teach it well. Then we realized a companion has to be a companion to some particular edition. Dillon made the case for Sir Thomas Heath’s translation, which is masterful and the standard for English speakers. Ben argued for Oliver Byrne’s edition, for the diagrams and the small drawings set into the text. No edition combined them, so we built it.

Proclus, a Neoplatonist, recalls an anecdote about Euclid and King Ptolemy I of Egypt in his commentary on Book I of the Elements. The king asked Euclid if there was an shortcut to learning geometry that avoided using the Elements. Euclid replied,

The edition we built does not offer a royal road or a shortcut. It just makes starting the long, difficult journey easier. That’s why we stopped at Book I. Heath’s Euclid in Color is a new edition of Book I of Euclid’s Elements that uses Sir Thomas Heath’s complete 1908 translation, with color added to the diagrams. Most teachers already color diagrams at the board to help a student follow “the red line” or “the green angle.” Oliver Byrne tried something like this in 1847. He shortened Euclid’s text to make room for colored figures, and he removed the letters that name the points in the diagrams. Andrew Seeley’s review in Principia makes the case that those letters matter, because the points are the principles of geometry. Heath’s translation appears in the new edition in full, with the lettering intact. The added color just helps you find things in the diagram.

In May, paid subscribers got a digital preview of this project.

That project is now a real book. I have a copy on my desk. Hardcover and paperback, both on Amazon, co-edited with Benjamin George and Dillon Raum.

Most educational publishers design books to get you to buy the next version, whether that’s a workbook series, an app with a new paid level, or a curriculum that only works if you buy the whole set. Euclid in Color could have gone that route, but we decided against it. It guides a reader through Book I, and then leaves the rest of the Elements to other standard editions.

Tackling the Elements is a difficult project. And Book I is where a first-time reader needs the most help. The propositions are unfamiliar and the diagrams are confusing. But by Book II, a student who has actually worked through Book I can read a Euclidean diagram without help.

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I don't plan to extend it, and it would take someone who knows Euclid better than I do to convince me otherwise. In preface of this edition, I argue that a student should graduate off the color. If you’re using this to study Euclid for the first time, use it the way it’s designed. After Book I, switch to a standard Heath edition for Book II and beyond.

Paid subscribers still have the free digital edition of Book I. That’s linked below, along with the Amazon listings if you’d rather have it in hand.

I'd like to know if it works.

If you teach Euclid, or you're working through Book I yourself, tell me what works and what doesn't. Whether the color clarifies or distracts. Whether Book II in color is a project worth pursuing. Whether you'd give this version to a student. Anything at all is useful to me right now.

Purchase the hardcover of Heath’s Euclid in Color on Amazon.

Purchase the paperback of Heath’s Euclid in Color on Amazon.

Never.

The idea that we study mathematics in order to function in the real world is absurd.

People learn the most useful skills on the job through training or experience. So if the content is useless, the value of mathematics must lie somewhere beyond utility.

I teach my students geometry, specifically Euclid's Elements, for three reasons.

1. Geometry exposes them to mathematical beauty.

2. It trains them to reason deductively.

3. It gives them the experience of knowing things for certain.

The Beauty of Math

The image below is of proposition 47 from Book I of Euclid’s Elements, the same diagram as the one at the top of the page, but with color added for clarity. It is a proof of the Pythagorean theorem: the areas of the squares built on the small sides of a right triangle add up to be equal to the area of the square on the longest side. A student studying this proposition begins confused. That’s fine.

As the proposition unfolds in the text of the Elements, he begins to understand. The line AL is key, which divides the big square into two rectangles.

The green square is double the triangle FBC. And the triangle FBC is equal to the triangle ABD. But the blue rectangle is double the triangle ABD. So the green square is equal to the blue rectangle. Similarly, the red square is equal to the yellow rectangle.

The two small squares are squeezed into two rectangular shapes, which exactly fill the square on the longest side.

With effort, our confused student discovers an immaterial beauty in the proof.

Bertrand Russell, in his essay The Study of Mathematics, says,

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."

To teach mathematics is to train the palate of students so that they can appreciate that beauty. In elementary school, this training forms a child’s attitude towards math, getting him excited about what more it has in store for him. In middle school, this looks like getting students to say “wow” (they will never say, “that’s beautiful”).

Aquinas says in the Summa Theologiae, Prima Pars, Q. 5 A. 4:

"… beauty consists in due proportion; for the senses delight in things duly proportioned…"

What is geometry but the art of continuous quantity, finding proportionality among magnitudes?

Training in Logical Deduction

The physical sciences train students in induction. Mathematics trains them in deduction. In the science lab, students look for patterns in the way a ball drops to try to guess at the underlying principles. In geometry, students learn the underlying principles and definitions, then they use their reason to determine particular properties of triangles and squares.

Look back at the proof you just read. The green square is equal to the blue rectangle because triangle FBC is equal to triangle ABD. But why are those two triangles equal? Because of proposition I.4, the ancient version of Side-Angle-Side, proved forty-three propositions earlier. And I.4 rests on the definitions, postulates, and common notions at the front of Book I.

Euclid doesn’t ask us to trust him that the Pythagorean theorem is true. There is a continuous chain of deductions throughout the book. Each proposition is a consequence of things a student has already agreed to, one at a time.

In class, this has to be worked on intentionally. A student looks at two triangles and says they’re equal.

“Why?”

“Because they look equal.” That’s not an answer, and my students eventually learn this.

“What’s your justification?” He has to cite a proposition, either by number or by describing what it says.

A student who has been made to justify every step of a proof has been trained to ask what follows logically from what, and to notice when a conclusion has been asserted rather than argued.

Knowing Things For Certain

A friend of mine who teaches Euclid to sixth graders tells his students partway through the year, “I have something to admit. It was all a lie. Everything we’ve talked about so far in Euclid is actually not true.” The students revolt, and he loves it. “Just kidding,” he replies, “isn’t that incredible, though, that you know it so strongly that you are willing to disagree with your teacher about it?”

A student can be certain of the Pythagorean theorem in a way he cannot be certain of almost anything else he learns in school. This is why Brian T. Kelly, Dean of Thomas Aquinas College, argues that mathematics “frees the student from the grips of skepticism.”

Some students think knowing a proof just means memorizing it. Others think it means reading it and understanding it once through. Neither of these is true knowledge. To know a proposition means understanding it and being able to produce a demonstration for it. That is scientia, true knowledge. A student who has done this has experienced absolute truth. He knows it exists and cannot be convinced otherwise.

If you want to know whether a student really knows a proposition, give him a blank sheet of paper. Ask him to draw the diagram and talk through the proof without the book. Two days ago he was confused, but now he can confidently prove a timeless truth.

Our work as math teachers is to foster habits of thinking, and to train the minds of our students not just to know the truth, but also to love and desire it.

What Schools Forget is reader-supported. Paid subscribers get the Timeline of American Education in Primary Sources, an annotated reading list running from 1647 to 1987, with a note on each source explaining what it is and why it matters. I revise and expand it as I find more. They also get Euclid in Color.

That was my problem too. I started out trying to find out why every math class in the United States looks the same, and this led me to study the history of education generally. So I began collecting the primary sources and putting them in order. It runs from 1647 to 1987 and it is still growing.

In 1913, the General Education Board, funded by Rockefeller, published this:

"In our dream, we have limitless resources and the people yield themselves with perfect docility to our molding hand. The present educational conventions fade from their minds; and, unhampered by tradition, we work our own good will upon a grateful and responsive rural folk. We shall not try to make these people or any of their children into philosophers or men of learning, or men of science…The task we set before ourselves is very simple as well as a very beautiful one, to train these people as we find them to a perfectly ideal life just where they are."

That document is hard to find. Many of them are. Finding out which ones are important is even harder, and that’s what this timeline is for.

The timeline is a paid post.

So is Euclid in Color, Book I.

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K-6 grade: arithmetic. 7th grade: pre-algebra. 8th grade: algebra I. 9th grade: geometry. 10th grade: algebra II. 11th grade: pre-calculus. 12th grade: calculus.

Every school I know of has some version of this, and so does every school my students transfer in from, public and private, classical and progressive. Ask a teacher why geometry sits between the two algebras and you will get an answer about scheduling or about what algebra II assumes. Ask who chose it, and no one can name a person or a committee.

The same is true of almost everything else on the schedule.

Parents Are the Primary Educators

From a natural perspective, who cares more about a child's education than their parents? Some argue that government officials or teachers have a greater stake in children's educational outcomes than parents. Tell that to any mother, if you will. I wouldn’t dare. While there are exceptions, of course, these exceptions actually reinforce the general rule that parents are the most invested in their children's education.

The Catechism of the Catholic Church teaches that parents are the primary educators of their children (CCC 2222-2224). In fact, the very purpose of marriage, in the eyes of the Catholic Church, is the procreation and education of children.

In the early years of life, parents form their children in manners, language, culture, religion, and character. As children grow older, parents may partner with schools and teachers to assist in this task. But parents can never hand over the responsibility entirely.

Schools should seek to work with parents, and parents should stay engaged with the education of their children. Otherwise, education and even parenting begin to break down.

Teachers Are a School’s Most Valuable Resource

Schools depend on teachers to be the heart of education. A good teacher cannot be replaced by a curriculum, a textbook, or a standardized test.

Yet for nearly two centuries, American education has been built on the assumption that most teachers cannot be trusted to teach well on their own.

In the 1800s, reformers like Horace Mann doubted the competence of local teachers. To solve this problem, they created teacher training schools. These institutions were designed to standardize what teachers taught and how they taught it because poor teaching should not be tolerated. The problem is that this should be taken care of locally, school by school. Teachers were no longer free educators making professional judgments.

This shift from trust to control accelerated through the 20th century. As states took authority over education, they built elaborate standards, curricula, and assessment systems, absorbing what little autonomy teachers had. Federal involvement only tightened the grip. Performance metrics, standardized benchmarks, and scripted lessons became the norm nationwide. Teachers who wanted to teach well, responding to their students, following genuine curiosity, adapting the lesson to the class, were given a checklist to follow instead.

Horace Mann wanted better teachers, but the system he set in motion selects for compliant ones. Schools that micromanage teachers end up with teachers who execute orders rather than think deeply about education. The teachers most likely to think for themselves and to teach their students to do likewise are the ones most likely to leave.

Real education doesn’t come from following checklists or prepping for tests. It comes from teachers free to teach what their students need, when they need it.

How Parents and Teachers Were Sidelined

Parents and teachers were sidelined gradually over the span of nearly two centuries, and it is unlikely that those who took over intended any harm.

1. Mann and his contemporaries built normal schools to train teachers to a state standard. Distrust of the local teacher became a permanent institution rather than one man’s opinion.

2. Compulsory attendance laws made the state’s judgment about a child’s education override the parents’. Massachusetts passed the first in 1852. By 1918 all the states had followed suit.

3. Dewey and Thorndike recast schooling as a science that requires experts. Neither mothers nor teachers qualified.

4. Federal money arrived attached to nationwide standards in reading, math, and behavior, and to performance on the benchmarks that measured them.

5. Private schools adopted the same methods anyway, on the grounds that this was how it was done.

We are left with a system where the methods are nearly universal and questioning them is a risk few are willing to take. The reformers who built this system ruled out parents and teachers and put their confidence in school boards and standards committees instead. Nearly every structure built since has assumed they were right.

The remedy for all this is not going to come from a movement, but rather from the two people closest to the child making their own decisions.

As a parent at home, the freedom you already have is broad and deep. Don’t worry if your daughter is nine and still counts on her fingers. She is nine. If your son finished the arithmetic book in February, the next book is not the only place to go. Euclid is sitting right there and has been for two thousand years. View the scope and sequence as a recommendation, not a hard line you have to follow.

For a teacher at a school, there is less latitude. It’s unlikely that you can rewrite the scope and sequence, but you can decide that this class gets three more days on fractions because they need it. You can teach the geometrical origins of trigonometry so that your students actually understand the mathematical definitions. You can answer a student’s question instead of dismissing it because of the textbook’s pacing guide.

Parents and teachers may be told that these are decisions for people with more training, but you know your child better than them.

What Schools Forget is reader-supported. Paid subscribers get the Timeline of American Education in Primary Sources, an annotated reading list running from 1720 to 1987, with a note on each source explaining what it is and why it matters. I revise and expand it as I find more. They also get Euclid in Color.

My teachers told me that cotangent is the reciprocal of tangent. So I decided “co” might mean “reciprocal”.

Then we learned cosecant, the reciprocal of sine. My hypothesis was promising: “'co' must have something to do with reciprocal,” I thought.

Then we learned secant is the reciprocal of cosine. That’s weird. No “co” in sight, and we had another reciprocal. So “co” doesn’t mean reciprocal.

Most people hate trigonometry. There are six functions with two of them named as if “co” means something like reciprocal.

Fortunately, I didn’t end up hating trigonometry, but that’s only because I tricked myself with a fake explanation. Looking back at cosine, I realized you could think of it as the reciprocal of secant. Maybe that was a key… I concluded that sine, tangent, and secant are the fundamental trigonometric functions and cosine, cotangent, and cosecant are their reciprocals.

It turns out that “co” does not mean anything like “reciprocal.” I just got lucky.

That’s not a naming inconsistency. It’s because these six “functions” originated as six lengths in one diagram all related to each other. Once you can see the diagram, the “co” stops being a memory trick you have to reverse-engineer. It just tells you which line you’re looking at.

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Sine and the Archer’s Bow

A couple of weeks ago I wrote about where “sine” actually came from. It wasn’t born as a ratio. Aryabhata measured it as a length: half a chord, like a bowstring in part of a circle. If sine is a length, the other five functions should be too. They are. All six are lengths hiding in the same diagram, the same bow and arrow, just measured along different lines.

Once you can see where each one lives, “co” stops being a memory trick and starts being a direction: go find the complementary angle.

Let’s start with cosine.

Cosine

“Co” is short for “complementary.” Cosine is just the sine of the complementary angle. That’s it. That’s the whole definition.

A cosine is really just a sine, measured for a different angle, one that's already in our original diagram. And the “cosine” length is there, too.

Go back to Aryabhata’s bow.

Draw a circle with radius AB = 1. Draw AE perpendicular to the diameter, DC. AE is the sine (or “half-chord”) of angle ABE. The angle BAE is the complement of ABE because the other angle in triangle ABC is a right angle.

Draw another radius, FB, so that angle FBE is equal to angle ABE. Both triangles have a right angle, an equal angle, and an equal radius. By Angle-Angle-Side (or Euclid’s Elements I.26), FG = BE and BG = AE. But FG is the sine (half-chord) of angle FBE. Since FG also equals BE, then BE is equal to the sine of angle FBE, which is the complement of ABE. In other words, BE is equal to the “cosine” of ABE.

Edmund Gunter invented the term “cosine” in 1620 in a book of trigonometric tables called Canon Triangulorum. He structured his tables so that the same row of his table also gives the sine of the complement, which for the sake of brevity we call the cosine (“sinus complementi, quem brevitatis causa co.sinum dicimus”).

Gunter was being efficient. He needed a shorter word for a phrase he was tired of writing out in full, and “co” was born as an abbreviation, not a mystery.

There’s a third length hiding in that same small triangle, one that never made it into the modern curriculum at all: the versine, EC, the gap between the chord and the arc itself. Indian and Arabic astronomers called it the arrow, or sagitta in Latin, because in the bow-and-arrow picture, it’s the part that actually gets pulled back and released. We kept the bow. We kept the string. We dropped the arrow.

One clean rule: “co” means “complementary angle.” Keep that in mind. It will also explain cosecant and cotangent.

Tangent

Next, let’s take a look at tangent. Go back to the bow. At point A on the circle, the top endpoint of the half-chord, draw a line perpendicular to the radius.

It touches the circle at exactly one point.

That’s called a “tangent” line, from the Latin root tangens, “touching.” It’s not “opposite over adjacent.” It’s a line, touching a circle, at one point.

Here’s the problem. As drawn, that line touches the circle, but it doesn’t measure anything yet. It has no defined length. Perhaps we could say the starting point is point A, but it has no ending point in either direction.

We need something to cut it.

Secant

Take the cosine line, the base of our triangle, and keep going. Don’t stop where the triangle ends. Extend it straight out until it runs into the tangent line you just drew.

Now the tangent line has a length. It runs from point A where it touches the circle down to point H where the extended cosine line crosses it. And that extended cosine line has a length too, from B to H.

BH is called a “secant” line. From the Latin secans, cutting. And that’s exactly the job it’s doing: it cuts across the circle, and in this case it also cuts the tangent line and gives it length.

AH is the tangent, touching the circle and intersecting with the secant.

The tangent touches the circle. The secant cuts it.

You can trace a new right triangle, ABH, with the new lines we’ve drawn. The tangent AH is one leg, the radius AB is the other leg, and the secant BH is the hypotenuse.

And this large triangle is split up into the triangle we started with and a similar right triangle, triangle AEH.

Four lengths in, and every one of them has been a real, measurable line. No ratios involved. Just a bow, an arrow, a line that touches, and a line that cuts.

Cosecant and Cotangent

That tangent line doesn’t stop at the point where it touches the circle. It runs in both directions from that single point of tangency.

We already used one direction, extending the cosine line until it cuts the tangent to get tangent and secant. This time, instead of extending a line we already have, we draw a new one.

Draw a BJ perpendicular to the diameter, DC, at point B. J is the point of intersection with the tangent line from earlier.

BJ drawn from the center out to where it cuts the tangent line is the cosecant. Same word, same secans, cutting, just cutting the tangent line from the opposite side this time.

The piece of the tangent line trapped between the point of tangency and that same cut is the cotangent. Tangent’s other half, running the other way, measuring the other angle.

One tangent line. Two cuts. Tangent and secant on one side, built by extending cosine. Cotangent and cosecant on the other side, built by extending sine.

That’s not a coincidence.

It’s not a different rule for the “co” functions. Cotangent really is the tangent of the complementary angle. Cosecant really is the secant of the complementary angle. Not by definition someone made up. Because it’s the same line, cut from the other side, measuring the other acute angle in the triangle.

Six lengths. One bow and arrow. One tangent line, cut twice.

Wrap-up & a Quick Proof

Here are all six lengths at once, changing together as the angle changes.

Watch the plots of sine and cosine. As one grows, the other shrinks, and they meet in the middle at 45 degrees, exactly where you’d expect the complementary angle to give you the same length twice.

Now watch tangent and secant race off the chart as the angle approaches 90 degrees. That’s not a graphing glitch. They race off the circle diagram, too. The tangent line and the cosine line are nearly parallel by then, so the cut happens further and further away. Cotangent and cosecant do the same thing in reverse, blowing up as the angle approaches 0.

None of this needed a single ratio to explain. It’s five similar triangles, AEH, ABE, ABH, ABJ, and JBH, all sharing angles, all growing and shrinking together as one triangle stretches.

That similarity is also where the modern ratio definitions come from:

1. SOHCAHTOA tells us that sin𝛉 = AE/AB.

2. But AB = 1

3. So sin𝛉 = AE.

4. Similarly, cos𝛉 = BE.

What about tangent?

1. Well, tan𝛉 = AE/BE in triangle ABE.

2. But triangle BHA is similar to triangle ABE. And similar triangles have corresponding sides in the same proportion.

3. So AE/BE = AH/AB = AH.

4. So, tan𝛉 = AH.

We can also show the following:

• BH = 1/BE (secant = 1/cosine)

• BJ = 1/AE (cosecant = 1/sine)

• AJ = 1/AH (cotangent = 1/tangent)

Don’t take my word for it: prove it to yourself that these last three statements are true.

This is also where my lucky pattern from the beginning of this article stops being pure luck. Cosine really is 1/secant. Cotangent really is 1/tangent. Cosecant really is 1/sine. Not because someone decided sine, tangent, and secant were the “real” functions and bolted reciprocals onto them.

Because in a circle of radius 1, dividing any of these lengths by the radius doesn’t change the number, and dividing by a length is the same operation, in reverse, as the length itself. The reciprocal relationships were always going to fall out of the similar triangles. You just weren’t shown the triangles.

Gunter knew exactly what he was doing when he wrote his tables. In the same book where he coined “cosine,” he stopped to make sure nobody confused his tables for the real thing:

Now, our Sines are not half-chords, and our Tangents are not perpendiculars raised from the end of the diameter: rather, proper numbers have been substituted in their place, which for that reason we call…“artificial.”

He knew his numbers were an artificial (artificiales) substitute. He said so. Four hundred years later, we teach the substitute and have forgotten there was ever an original.

Before you introduce SOHCAHTOA, hand your students a compass, a straightedge, and a circle. Let them draw the bow, the arrow, the tangent, the two cuts. Let them find all six lengths with their own hands before you ever hand them a ratio.

Six lengths. One diagram. The bow and arrow were never gone. We just stopped giving students the thread that ties it all together.

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Your grandparents already relearned subtraction once, during the onslaught of the New Math movement in the 1960s. It turns out they relearned the wrong method.

Some Disturbing Comedy

Tom Lehrer was a musical comedian from the 1950s and 1960s who wrote, sang, and played satirical songs. In the video below, he runs through the old shorthand method of subtraction, then sings a song about the “New Math” method. The audience of our grandparents and great grandparents finds it absolutely hilarious. To a modern viewer it should be unsettling.

If you watched that video and were not shocked, you misunderstood it.

Watch it again.

Lehrer is mocking the subtraction algorithm that we teach in schools nationwide.

Everyone in that room easily followed along with his first shorthand version. They knew the older method, and they found the new one absurd.

The Method Nearly Everyone Learns

In the early 20th century, there was a “great debate” over which subtraction algorithm was the best to teach in schools. The “simple borrowing” or “decomposition” algorithm gradually won out and the other algorithms were abandoned.

We use the decomposition algorithm in schools nationwide today.

Consider the following subtraction problem:

Nearly every person in the United States today would solve this using the same algorithm. We solve it column by column, moving from right to left as follows:

1. 2 minus 3 cannot be done, so borrow 1 from 4, which becomes 3, and do: 12 - 3 = 9

2. 3 minus 7 cannot be done, so borrow 1 from 3, which becomes 2 and do: 13 - 7 = 6

3. Finally, do: 2 - 1 = 1

4. So the final answer is 169

Where this algorithm actually comes from

In The Teaching of Arithmetic (1909), David Eugene Smith tells us that the decomposition algorithm in common use today appears in the oldest known English arithmetic manuscript, in 13th-century Spain, in Italy in the Middle Ages, and in India even earlier.

New Math did not invent the algorithm. It just made it universal.

The designers of New Math promoted the algorithm because they wanted to foster understanding, a noble ideal. But the way the algorithm was taught in the classroom buried the explanation.

The Expanded Version: Why the Trick is Allowed

The goal of the New Math movement was understanding. Seeking understanding in mathematics is absolutely the right approach, but New Math failed because it was implemented from the top-down, without first getting the buy-in and understanding of the teachers.

Here is a way to present the decomposition method with an eye towards understanding.

Start with the example from the Lehrer video. But first, we rewrite the numerals in expanded form so that place value is apparent.

Now we subtract 10 from 40 and add it to the 2 in the ones place. We perform the far-right subtraction: 12 - 3 = 9:

Then we subtract 100 from 300 and add it to the 30 in the tens place. We perform the next subtraction: 130 - 70 = 60.

We do the final subtraction in the hundreds place: 200 - 100 = 100.

And so, our final answer is 100 + 60 + 9 = 169.

What Your Grandparents Actually Did

The “Austrian” or “equal additions” method, which Smith described as his preferred method is what many of our grandparents probably learned. He notes that it is not as old as other methods of subtraction, although it appears in some manuscripts as early as the sixteenth century. Smith describes two alternatives, one of which is very similar to the decomposition method, the other of which is what we now call the “equal additions” method. He says the second is preferred. Both methods use addition to find the answer.

We’ll work through the same problem as before.

Going from right to left, we do the following:

1. 3 + 9 = 12

2. Add the 1 from 12 to 7 and get 8: 8 + 6 = 14

3. Add the 1 to 1 and get 2: 2 + 1 = 3

Written out in expanded form, the reasoning looks like this:

Note that for the shorthand version, we don’t even talk of borrowing from the 4 in the tens place or from the 3 in the hundreds place. We simply keep in mind that we borrowed and change the numbers accordingly. Lehrer's audience proves Smith’s preference. They were able to easily follow along with his shorthand example.

Smith’s Verdict, 1909

David E. Smith describes no fewer than five different algorithms in use in the year 1909 when he wrote The Teaching of Arithmetic. Although he states that some algorithms were more common than others, there was, as yet, no dominant subtraction algorithm.

He argues that the best subtraction algorithm a priori is the “Austrian” method because it is more natural, requires less memorization, and is faster:

1. It is more natural: it is the common method for making change:

   If I owe $7.65 and pay $10, the merchant finds the change and I verify his work by saying, “65¢ and 5¢ are 70¢ and 30¢ more makes $1, and $2 more makes $10.

   This reason is less applicable now when the majority of people rarely use bills and coins, but it is the way our minds naturally tend to think. Most adults do mental subtraction by using addition.

2. It requires less memorization: it removes the need to memorize subtraction tables. You only need to know addition tables.

3. It is faster: addition facts get used, and practiced, far more often than subtraction facts, so they’re faster and more reliable to call up.

That said, he was not criticizing the other methods. He wrote that switching to another algorithm after a child has already learned one well is not worth it. But for his audience in 1909, when there had as yet been no standardization, he says the better method should be taught.

The “great subtraction debate” of the early 1900s, alluded to earlier, arose as proponents of each method tried to get their own approach adopted as the standard. Early on, the Austrian method was mandated by the boards of education in New York City and San Francisco. Multiple studies found that students learning the Austrian method performed better than students using the decomposition method.

The debate continued until the School Mathematics Study Group (AKA “New Math”) wrote a series of textbooks calling for the use of the decomposition method.

Unfortunately, we did not take Smith’s advice in 1909. That said, none of this is an argument that Austrian is a silver bullet.

Understanding comes first, always, no matter which method you teach.

A child who genuinely understands decomposition will outperform a child who has merely memorized Austrian as an unexplained ritual. Smith's case for Austrian is real, but it's modest: a small, honest edge in speed and reliability once understanding is already in place.

It was never a substitute for it.

Two Kinds of Readers

For teachers or administrators in larger schools with a standard curriculum, this is a case study. When your institution opts for a curriculum change, the teachers in the classroom need to be allowed to give their input. Ease of institutional rollout should not be the primary criterion, nor should theory be prioritized over practical experience.

For homeschoolers and smaller schools, you don’t have Smith’s problem. You don’t need to follow state standards, use a specific textbook, or train hundreds of teachers. You can teach the method you choose from day one. If your goal is that your children or your students learn subtraction, opt for the best method, which may not be the one you were brought up on.

The best algorithm is the understood algorithm. Try it with your kids this week.

He was a sharp kid. He already understood that it equaled opposite over hypotenuse. He wanted the etymology, but I didn’t know.

I told him so, and I promised an answer the next day.

After doing a little bit of research, what I gave him was accurate but lifeless: sine comes from the Latin sinus, meaning fold, translated from an Arabic word that was itself a mistranslation of a Sanskrit word.

I had the facts, but I didn’t have the archer, the bow, the half-drawn string. I didn’t know that story, so I couldn’t tell it.

I had a chance to bring that class to life, and instead I gave a curious kid a dictionary entry.

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Sine, Cosine, and Tangent Were Not Always Ratios

I rarely look forward to introducing sine, cosine, and tangent to my students.

I do my best to make it coherent. I explain the labels on the sides of a right triangle: opposite, adjacent, hypotenuse. My students have already met these terms in earlier theorems, so they aren’t foreign. They understand “opposite” and “adjacent” easily enough, and the etymology of hypotenuse is interesting and makes sense once you break it down.

But the real trouble starts when I try to explain what sine, cosine, and tangent actually are.

I’ve always taught them as ratios. Just a few weeks ago I learned that when sine, cosine, and tangent were first discovered, they weren't ratios at all.

And yet, in nearly every geometry class, sine and cosine are introduced as functions that take an angle and output a ratio of the sides of a right triangle.

The standard tool for teaching this is the abbreviation SOHCAHTOA:

SOH: sine = opposite/hypotenuse

CAH: cosine = adjacent/hypotenuse

TOA: tangent = opposite/adjacent

This is mathematically correct. But it is also historically backward, and I think that backwardness is exactly why so many students memorize the acronym without ever understanding what the words mean.

Aryabhata’s Bow and Arrow

Trigonometry gets its name from the Greek for “measure of triangles,” but the discipline itself descends from Indian astronomy, not Greek geometry. The mathematician Aryabhata, writing around 500 AD, modeled the arc of a circle as an archer’s bow, the chord connecting the two ends of the arc as the bowstring, and half of that chord as the arrow drawn back from the string’s center.

Aryabhata called that half-chord ardha-jya, “half-chord,” and shortened it over time to jya. From there the word took a strange trip. When Arabic scholars translated Aryabhata’s work, they didn’t translate jya, they transliterated its sound as jiba, a string of letters with no actual meaning in Arabic. Arabic is written without vowels, so jiba was recorded as just its consonants. Later readers, no longer aware jiba was a foreign import, assumed those consonants stood for a real Arabic word, jaib, meaning the fold or pocket of a garment. When European scholars translated Arabic astronomy into Latin in the 12th century, they translated jaib literally, using the Latin word for a fold or a bay: sinus. Sinus became sine.

The word “sine” is a linguistic accident, the translation of a mistranslation of a faulty transliteration of a word that had meaning: “half a bowstring.”

This story isn't a trivial footnote. It gives the trigonometric function names meaning. It puts the invention of trigonometry in its historical context. Finally, it gives the concrete staging to an generally abstract concept.

A Length, Not a Ratio

Aryabhata’s half-chord was not a ratio, but a length measured in a circle of a specific, fixed size.

Aryabhata built his sine table using a circle with a radius of 3438 units, a number he chose because it made the circle’s circumference come out to a tidy 21,600 minutes of arc. For any given angle, his “sine” was simply the number of those units in the actual half-chord. Nothing was divided by anything. You could go measure it with a ruler on a circle of the right size.

Here's how the two connect. Draw a 30-degree angle in a circle of radius 2. The half-chord measures 1 unit, physically, with a ruler. Divide by the radius and you get 0.5, which is sin(30°) in the modern sense. Aryabhata would have stopped at "1 unit." We keep going and divide by the radius. That extra step is the whole difference between his sine and ours.

Now try it again with a circle of radius 1. The half-chord still comes out to whatever it measures, but this time dividing by the radius does nothing, because you’re dividing by 1. The length and the ratio become the same number. That’s the whole trick: SOHCAHTOA is built on what mathematicians typically call normalization, but that we don’t shows students. We teach the ratio and skip the length it came from.

That circle of radius 1 has a name students will meet again: the unit circle. Except, by the time most students see it, it shows up as a brand new topic in precalculus, disconnected from the ratio they learned years earlier in geometry. Historically, the circle came first and the ratio was what fell out of it once the radius was fixed at 1. In the classroom, we’ve reversed the order entirely: ratio first, with no circle in sight, and the circle bolted on years later as if it were a new idea rather than the thing sine was measured on all along.

This is the same mistake as teaching decimals before fractions, and fractions before ratios. We hand out the portable shortcut before students have held the concrete thing it replaced.

David E. Smith says that the first time that sine, cosine, and tangent were described as ratios instead of lengths was in Georg Rheticus’ Canon doctrinæ triangulorum (1551). François Viète took the ideas of Rheticus and published them later in the more famous Canon Mathematicus (1579). Once the ratio took hold in print, the length quietly disappeared from how the subject was taught, and it never came back.

What To Do Instead

Before SOHCAHTOA, before any right triangle at all, give students a circle, a fixed radius, and an arc. Have them draw the chord. Have them measure the half-chord with a ruler, in the units of that circle, the way Aryabhata’s astronomers did. Do this for a few different angles in a few different circles. Let them notice, on their own, that the half-chord grows with the angle, but not in a straight line.

Only then introduce the shortcut: pick a circle of radius 1, and the half-chord length becomes a portable number you can use in any circle, any triangle, once you scale by the radius. That’s sine. It’s a measurement that became universal once someone standardized the yardstick.

Then tell the story of sine. The story of a term that traveled between civilizations over millennia, whose name and even meaning was and is misunderstood by most of the world. But not by them. They know the truth.

Now tell them about SOHCAHTOA.

Right triangle, adjacent, opposite, hypotenuse: all of that is real geometry and none of it needs to disappear. It comes after a lot of ground work. SOHCAHTOA is a fine shortcut once a student has held the length in their hand and watched it become a ratio. Taught first, it’s an unmotivated rule attached to a word that used to mean “bowstring” and now, through two thousand years of translation error, means nothing at all.

That boy who asked me about the meaning of the word "sine" has probably graduated from high school by now.

This is the answer I owe him.

Housekeeping:

Dear readers,

A while back I made Euclid in Color, Book I free for all subscribers. I liked the idea of removing friction for people encountering Euclid for the first time.

The feedback made it worth it.

I want this newsletter to keep some real value reserved for paid subscribers, and this e-book is the clearest thing I have to offer for that. So it’s moving behind the paywall.

If you’re a free subscriber and want your own copy, download it in the next week. After that, it’s for paid subscribers only.

I put real time into this project and into this newsletter more broadly, time I could spend on work that pays. Paid subscriptions are what justify choosing this instead.

If this newsletter has been worth your time, consider becoming a paid subscriber. It’s the best way to tell me to keep going.

P.S.

Check out my recent article on Liberty Sentinel commemorating the 250th.

It is a warning to take a closer look at the philosophical foundations of our public school system.

It started with ten names. In a few months, my list had grown to fifty. I began the project trying to understand why all math classes in the United States look the same. Soon I realized I had to expand my vision to include the history of education more generally.

This is the reading list that grew out of that rabbit hole, organized by date, and a key to understanding the history of education in the United States.

“Do Your Own Research” with regard to history of education means read primary sources, and then draw your own conclusions. Reading primary sources trains your own powers of reading and discernment. It also gives you a sense for the style and rhetoric of the original authors. Knowing the dates of the writings gives you historical context, how they connect to the greater world around them and how their own ideas evolve. An author’s ideas are not static. They change and develop over the course of his life in response to people and experiences.

Pestalozzi, for example, writes with emotion, drama, and Christian conviction. Robert Owen writes with vigor and emotion while insisting that his own arguments are purely logical. Owen’s earlier writings are against any form of organized religion, and his later writings appeal to spiritualism and mediumism. Horace Mann is factual, with Christian conviction, but sometimes shoots stinging critiques at his opponents. In his early life, he was very political and measured, but as he neared the end, he explicitly expressed his views against sectarian Christianity. Dewey’s writing is challenging and must be deeply studied. He uses common terms in uncommon ways. To read Dewey, you really need to immerse yourself in his writings. Dewey began his professional life deeply Christian, but he ended it as a secular humanist.

Googling these questions does not work. The algorithm is built to give you what resonates with what you already think. Asking an AI does not work either, at least not for answers or quotations, though it is useful for finding out what exists.

Use them to locate sources. Then read the sources.

The hard part is knowing which ones matter. That is what this is for: A Timeline of American Education History in Primary Sources.

Read more

Our education system is based on his ideas, transmitted to us through the Prussian education system by Horace Mann.

On the 50th anniversary of the adoption of the Declaration of Independence, he gave a speech calling for a new kind of “Mental Independence.”

He was a socialist opposed to ownership of private property

He was a spiritualist opposed to organized religion.

And he was opposed to marriage.

Check out my article on Liberty Sentinel for more.

I never expected to find it again in a document about a king.

Read the Declaration of Independence slowly, the way you’d read Proposition I.1, and the parts show through. A general principle, stated in the abstract. A grounding in the specific case. A restatement of what’s about to be shown. A long list of justifications, cited one after another like margin notes. An act, not just an argument. A conclusion.

This isn’t an accident, and it isn’t just a nice metaphor for a summer newsletter. Jefferson’s own draft shows him reaching for the word. He first wrote that these truths were “sacred and undeniable.” He crossed it out. He wrote “self-evident” instead, the word Euclid used for an axiom, a thing you don’t prove but grant, and build everything else on top of.

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Jefferson studied Euclid at William and Mary under a tutor, William Small, who historians believe drilled him in exactly this kind of reasoning: definitions, postulates, axioms, and the long chain of justified steps that follows once you accept them. Lincoln did the same thing decades later with a saddlebag copy of the Elements, teaching himself what Jefferson had been taught.

Today, everyone else is writing about 1776. I want to write about geometry, because Jefferson did too.

Euclid begins with his definitions, postulates, and common notions. On these, he builds mathematical arguments that reach certain conclusions, because mathematical objects are simple and abstract. A geometric proof can grow enormously complex and its conclusion will still be certain, so long as the logic holds.

Jefferson is not proving a geometric fact. He’s making a political argument, using the closest thing to Euclid’s certainty that he can build: a king’s actions standing in for a circle’s radii. But a king’s actions aren’t radii. They have to be weighed and judged, not measured. Jefferson compensates for that harder, messier subject matter the only way he can: he keeps the logical structure as tight and simple as possible, so that once a reader grants the axioms, the conclusion is difficult to escape.

Euclidean Structure

The entire Declaration of Independence can be read through the parts of a Euclidean proposition.

The Introduction Explains the Necessity

Euclid never explains why he’s writing. He simply begins. Jefferson can’t afford that luxury. The opening sentences of the preamble to the Declaration of Independence are not part of the argument. They explain why the document is necessary.

When in the Course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature’s God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.

The Axioms: Self-Evident Assumptions

Before Euclid proves anything, he asks you to grant his definitions and postulates. Jefferson does the same thing, and does it in one sentence, before he ever mentions the King.

We hold these truths to be self-evident,

1. That all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.

2. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed,

3. That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.

Strip away the king, the grievances, even Euclid, and a single conditional is doing all the work: if a government becomes destructive of these rights, the people may abolish it. Everything from here to the end of the document exists to establish one thing, and one thing only: that the antecedent is true in this case. Prove that, and the conclusion follows whether anyone likes it or not.

Here the analogy briefly strains. Euclid's postulates are genuinely generic: "draw a line between any two points" has nothing to do with triangles yet. Jefferson's third axiom already names the exact principle his whole document will run on, that a destructive government may be abolished. His Enunciation, one sentence later, restates that same principle almost word for word. Where Euclid moves from the fully general to the fully specific in two clean steps, Jefferson circles the same claim twice before he ever names a king. Perhaps this is not a flaw so much as emphasis.

He wanted this axiom, above the other two, to be unmistakable before the grievances began.

The Enunciation Declares the General Principle

Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed. But when a long train of abuses and usurpations, pursuing invariably the same Object evinces a design to reduce them under absolute Despotism, it is their right, it is their duty, to throw off such Government, and to provide new Guards for their future security.

Only now does a name enter the argument.

The Exposition and Specification Name the Parties and State the Charge

Euclid often folds these two moves into a single sentence: Let AB be the given line... I say that an equilateral triangle can be constructed upon it. Jefferson does something similar.

Such has been the patient sufferance of these Colonies; and such is now the necessity which constrains them to alter their former Systems of Government. The history of the present King of Great Britain is a history of repeated injuries and usurpations, all having in direct object the establishment of an absolute Tyranny over these States.

The general principle now has its two parties: a King, and a people who have suffered patiently long enough. What remains is to show it.

The Construction Builds the Argument, Indicting King George III

Later editors added an extra part, the Justifications, which are marginal notes. Jefferson doesn’t include them. But you can supply the citation yourself for any grievance on the list, just trace it back to one of the three self-evident truths above.

To prove this, let Facts be submitted to a candid world.

1. He has refused his Assent to Laws, the most wholesome and necessary for the public good.

2. He has forbidden his Governors to pass Laws of immediate and pressing importance, unless suspended in their operation till his Assent should be obtained; and when so suspended, he has utterly neglected to attend to them.

3. He has refused to pass other Laws for the accommodation of large districts of people, unless those people would relinquish the right of Representation in the Legislature, a right inestimable to them and formidable to tyrants only.

4. He has called together legislative bodies at places unusual, uncomfortable, and distant from the depository of their public Records, for the sole purpose of fatiguing them into compliance with his measures.

5. He has dissolved Representative Houses repeatedly, for opposing with manly firmness his invasions on the rights of the people.

6. He has refused for a long time, after such dissolutions, to cause others to be elected; whereby the Legislative powers, incapable of Annihilation, have returned to the People at large for their exercise; the State remaining in the mean time exposed to all the dangers of invasion from without, and convulsions within.

7. He has endeavoured to prevent the population of these States; for that purpose obstructing the Laws for Naturalization of Foreigners; refusing to pass others to encourage their migrations hither, and raising the conditions of new Appropriations of Lands.

8. He has obstructed the Administration of Justice, by refusing his Assent to Laws for establishing Judiciary powers.

9. He has made Judges dependent on his Will alone, for the tenure of their offices, and the amount and payment of their salaries.

10. He has erected a multitude of New Offices, and sent hither swarms of Officers to harrass our people, and eat out their substance.

11. He has kept among us, in times of peace, Standing Armies without the Consent of our legislatures.

12. He has affected to render the Military independent of and superior to the Civil power.

13. He has combined with others to subject us to a jurisdiction foreign to our constitution, and unacknowledged by our laws; giving his Assent to their Acts of pretended Legislation:

14. For Quartering large bodies of armed troops among us:

15. For protecting them, by a mock Trial, from punishment for any Murders which they should commit on the Inhabitants of these States:

16. For cutting off our Trade with all parts of the world:

17. For imposing Taxes on us without our Consent:

18. For depriving us in many cases, of the benefits of Trial by Jury:

19. For transporting us beyond Seas to be tried for pretended offences:

20. For abolishing the free System of English Laws in a neighbouring Province, establishing therein an Arbitrary government, and enlarging its Boundaries so as to render it at once an example and fit instrument for introducing the same absolute rule into these Colonies:

21. For taking away our Charters, abolishing our most valuable Laws, and altering fundamentally the Forms of our Governments:

22. For suspending our own Legislatures, and declaring themselves invested with power to legislate for us in all cases whatsoever.

23. He has abdicated Government here, by declaring us out of his Protection and waging War against us.

24. He has plundered our seas, ravaged our Coasts, burnt our towns, and destroyed the lives of our people.

25. He is at this time transporting large Armies of foreign Mercenaries to compleat the works of death, desolation and tyranny, already begun with circumstances of Cruelty & perfidy scarcely paralleled in the most barbarous ages, and totally unworthy the Head of a civilized nation.

26. He has constrained our fellow Citizens taken Captive on the high Seas to bear Arms against their Country, to become the executioners of their friends and Brethren, or to fall themselves by their Hands.

27. He has excited domestic insurrections amongst us, and has endeavoured to bring on the inhabitants of our frontiers, the merciless Indian Savages, whose known rule of warfare, is an undistinguished destruction of all ages, sexes and conditions.

Twenty-seven grievances, and not one of them is the argument itself. They are evidence, the raw material a proof still has to reason over.

The Proof Demonstrates Legitimacy of Independence

In every stage of these Oppressions We have Petitioned for Redress in the most humble terms: Our repeated Petitions have been answered only by repeated injury.

A Prince, whose character is thus marked by every act which may define a Tyrant, is unfit to be the ruler of a free people.

Nor have We been wanting in attentions to our British brethren. We have warned them from time to time of attempts by their legislature to extend an unwarrantable jurisdiction over us. We have reminded them of the circumstances of our emigration and settlement here. We have appealed to their native justice and magnanimity, and we have conjured them by the ties of our common kindred to disavow these usurpations, which, would inevitably interrupt our connections and correspondence. They too have been deaf to the voice of justice and of consanguinity. We must, therefore, acquiesce in the necessity, which denounces our Separation, and hold them, as we hold the rest of mankind, Enemies in War, in Peace Friends.

The reasoning is done. What’s left is to say, formally, what has now been shown.

The Conclusion Restates the General Principle in the Specific Case as Emphatically Proven: The Colonies Demand Independence from Great Britain

We, therefore, the Representatives of the united States of America, in General Congress, Assembled, appealing to the Supreme Judge of the world for the rectitude of our intentions, do, in the Name, and by Authority of the good People of these Colonies, solemnly publish and declare, That these United Colonies are, and of Right ought to be Free and Independent States; that they are Absolved from all Allegiance to the British Crown, and that all political connection between them and the State of Great Britain, is and ought to be totally dissolved; and that as Free and Independent States, they have full Power to levy War, conclude Peace, contract Alliances, establish Commerce, and to do all other Acts and Things which Independent States may of right do.

A Euclidean geometric proposition ends here with a Q.E.D. or Q.E.F., at the moment of proof, because the thing is now certain.

Jefferson’s cannot.

Jefferson ends the Declaration with a pledge, because the thing is not certain yet. Fifty-six men signed their names, risking everything on a future they could only promise, not prove.

And for the support of this Declaration, with a firm reliance on the protection of divine Providence, we mutually pledge to each other our Lives, our Fortunes and our sacred Honor.

Euclid’s proposition ends this way because a mathematical proposition, once proved, is certain forever. Jefferson’s document was never a theorem, and never a construction either. It was something Euclid never had to write: a pledge on a future that hadn’t happened yet.

We are that future.

Two hundred and fifty years of a nation still standing, still free, still governed by consent, is not a geometric proof. But it is a kind of confirmation Euclid would have recognized.

The intended thing has been done.

Quod erat faciendum.

Today, we get to say it.

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Being able to understand the puzzle and solve it depends on whether you understand prime factorization.

Prime factorization is my favorite part of the elementary math curriculum.

With prime factorization, students begin to understand the properties of numbers. Instead of just being a word from a math fact or a symbol on a page, the inner life of each distinct number is revealed.

Prime factorization is the one place in the elementary curriculum where students can still touch arithmetike, the study of pure number.

Most teachers treat it as a computation unit. It isn't.

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Prime Factors are the DNA of Numbers

So, what’s a factor? A factor of a number is a number that exactly divides the number with no remainder. A prime factor is a factor that is prime. The proper factors of a number are all the factors of that number, excluding the number itself.

Just as the DNA of living organisms determines their traits, the prime factors of numbers determine properties of numbers. All evens have 2 as a prime factor. All odds have only odd prime factors. All known perfect numbers are composed only of the prime factors 2 and some prime number. (It is still an open question if odd perfect numbers exist). All prime numbers have only one prime factor (itself).

Each prime factorization of a number is unique to that number, like a genetic code. The number 12 must be factored as 2 • 2 • 3. This is not true when we decompose a number into its addends. 12 = 1 + 11, 2 + 10, 3 + 9, 1 + 1 + 10, 3 + 4 + 5…

Square numbers have an odd number of factors. Perfect numbers are equal to the sum of their proper factors.

It should come as no surprise that a common method used to determine prime factors is called the “factor tree,” descriptive of the tree-like structure that takes form when the primes are factored from proper factors of a number.

Divisibility Rules: The Practical Payoff

Knowing a number’s factors tells you things about it before you’ve done any calculation.

This is the practical payoff of thinking in primes. Some of these follow cleanly from prime structure. Others are accidents of base ten. Either way, they reward students who already think about numbers in terms of what divides them.

The clean ones first. The prime factors of 10 are 2 and 5. That’s why multiples of 2 end in an even digit, multiples of 5 end in 0 or 5, and multiples of 10 end in 0. The rules for 4 and 8 extend the same logic: 4 = 2 × 2, so check the last two digits; 8 = 2 × 2 × 2, so check the last three. The rule for 6 is just the rules for 2 and 3 combined, because 6 = 2 × 3. These aren’t tricks. They’re direct consequences of prime structure.

The rules for 3, 9, and 11 are different. They work because of how our base-ten system handles remainders, not because of anything intrinsic to those primes. In base seven, the digit-sum rule for 3 would vanish entirely. These rules are useful. But they’re accidents of our number system, not revelations about the numbers themselves.

Most books will tell you there is no divisibility rule for 7. There are several, all of them awkward. That awkwardness is itself worth noting: 7 shares no factors with 10, so our base gives us no foothold. The number resists the system.

The deeper point is this. A student who thinks in factors doesn’t need to memorize most of these rules. He derives them. He knows 6 = 2 × 3, so the test for 6 is obvious. He knows 4 = 2², so he applies the rule for 2 twice. The rules become consequences of something he already understands, not a list to be recalled under pressure.

Making Factors and Addends Do the Work

Students should habitually think of numbers as the product of factors and the sum of their addends. Thinking about numbers in terms of sums of their addends aids in the development and understanding of addition and subtraction facts. Thinking about numbers in terms of products of factors aids in the development and understanding of multiplication and division facts. It’s the understanding, not the rote memorization, of number facts that makes them stick.

Mental math must be a core part of every math class. The simplest trick when doing mental math is to break apart the number into its addends or its factors.

Take, for example, 284 + 343. 284 = 300 - 16, so we can get the answer by adding 300 to 343, then subtracting 16:

284 + 343 = (300 -16) + 343 = 643 - 16 = 627.

Or if it’s multiplication:

36×22 = (2 × 2 × 3 × 3)×(2 × 11) = (2 × 2 × 2 × 2)×(3 × 3 × 11) = 16×99 = 16×(100 - 1) = 1584.

Division is particularly interesting. We can just divide out the common factors from each number:

648 ÷ 24 = 648 ÷ (2 × 2 × 2 × 3).

We can simply divide 648 by 2 three times, then once by 3: 648 ÷ 2 = 324, 324 ÷ 2 = 162, 162 ÷ 2 = 81, 81 ÷ 3 = 27.

Alternatively, if we do the prime factorization of both numbers, its just a matter of cancelling the common factors:

648 ÷ 24 = (2 × 2 × 2 × 3 × 3 × 3 × 3) ÷ (2 × 2 × 2 × 3) = (3 × 3 × 3) = 27.

Instead of teaching a mindless division algorithm to students, give them mental math division problems they can solve in their heads.

The Ancient Greeks certainly thought of numbers this way. They saw the “perfect number” as a number which was equal to the sum of its proper factors. 6 is perfect because the sum of its factors is 1 + 2 + 3 = 6. 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28. This is truly the “liberal arts” or “classical” approach to thinking about numbers.

Nicomachus’ Algorithm for Calculating Perfect Numbers

The Ancient Greeks developed deep insights into the nature of numbers based on their grasp of numbers as sums of addends and products of factors. In his Introduction to Arithmetic, Nicomachus explains a beautiful method for finding perfect numbers.

1. Set out the sequence of powers of 2: 1, 2, 4, 8, 16, 32, 64…

2. Sequentially add each number together: 1, 3(=1+2), 7(=3+4), 15(=7+8), 31(=15+16), 63 (=31+32)…

   OR Take each number and subtract 1: 1, 3, 7, 15, 31, 63… (this is the crux of the proof)

3. If a number in the second list is prime, multiply it by the power of 2 in the previous list that appears once space behind it.

4. 1 is not prime. 3 is prime. Multiply it by 2, and we get 6, a perfect number.

   7 is prime. Multiply it by 4 and we get 28, another perfect number. 15 is not prime. 31 is prime. Multiply it by 16 and we get 496, another perfect number.

This can be written algebraically as

Fibonacci’s Beautiful Algorithm

Since prime factorization is my favorite part of the elementary curriculum, I was ecstatic when I discovered a beautiful algorithm that Leonardo of Pisa (Fibonacci) described in Liber Abaci. With an understanding of prime factors, the algorithm is beautiful and simple. Without that understanding, it is just another boring algorithm.

This algorithm does demand a high level of number sense to understand how it works. But if students have been trained to see numbers as their prime factorizations, it will come easily.

Fibonacci presents it as a way to divide by large composite numbers. Consider the problem 654 ÷ 45. The first step is to set up the problem, writing 45 as its prime factorization: 5 × 3 × 3.

The steps are self-explanatory thanks to these great pictures from an article by Anna Long. She points out that “this algorithm uses prime numbers instead of base-10 place values as a way to reduce the size of the computations.”

The process is remarkably simple. Divide sequentially by the prime factors.

654 ÷ 5 = 130 + 4/5

(130 + 4/5) ÷ 3 = (43 + 1/3) + 4/15

(43 + 1/3 + 4/15) ÷ 3 = 14 + 1/3 + 1/9 + 4/45 = 14 + 15/45 + 5/45 + 4/45 = 14 + 24/45 = 14 + 8/15

We can write these steps in an abbreviated form. We simply sequentially divide by each of the prime factors of 45 and we write the remainders in the space above the number they correspond to. Because we divided by 5, the remainder of 4 is really 4/5.

We continue the process, dividing each quotient by the next prime factor.

Finally, we arrive at the quotient, 14, with a remainder of 1.

If we wanted an approximate answer to 654 ÷ 45, 14 works. If we want to be more precise, we have to include the remainders from each step. The remainder of 4 from the first step is divided by 5, then by 3 two times, so it becomes 4/45. Similarly, the remainder of 1 from the second step was divided by 3 twice, so it is 1/9. The final remainder of 1 is just 1/3.

Adding up these fractions, we get 4/45 + 1/9 + 1/3 = (4 + 5 + 15)/45 = 24/45 = 8/15

So our final answer is 654 ÷ 45 = 14 + 8/15.

It is beautiful, and understanding the algorithm tells you something about what actually happens when you divide two numbers.

The Answer to the Puzzle

Which light bulbs are left on?

Assign the numbers 1-100 to each of the hundred lightbulbs.

Each time you do a pass, you flip the switch for a number

1st pass: if it has 1 as a factor

2nd pass: if it has 2 as a factor

3rd pass: if it has 3 as a factor…etc

All switches started in the OFF position, so the condition for a switch being ON is that you flipped it an odd number of times.

What numbers have an odd number of factors? Factors come in pairs, but for square numbers, the square root is paired with itself. Only the lightbulbs whose number is a square will be left on: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Prime factorization isn't a computation unit. It's a key. The lightbulb puzzle is just one door it opens.

Numbers stacked. Operation symbol on the left.

Addition does it. Subtraction does it. Multiplication does it.

Then we get to division, and suddenly there's a half-box shape with no symbol in sight, and we expect children to just accept that.

They don't ask why. We don't either. But we should.

Is it possible to do division using an algorithm that is more similar to the other algorithms we teach? It seems like we should be able to. And we can.

I didn’t invent it, I only discovered it by reading Leonardo of Pisa’s Liber Abaci (1202).

Leonardo of Pisa (AKA Fibonacci) describes an algorithm for solving a division problem like this:

Starting with this simpler case, I will then extend it to a more complex one.

Fibonacci’s “Stack ‘em In Order” Algorithm

Surprisingly, Fibonacci demonstrates the algorithm moving from left to right.

Step 1: 3 ÷ 2 = 1, with a remainder of 1 (1 R1).

Write the 1 under the line, in the space beneath the 3, and write the remainder 1 above the 3, like so:

Step 2: 16 ÷ 2 = 8, with no remainder.

Write the 8 under the line, in the space lined up with the 6, like so:

Step 3: 5 ÷ 2 = 2 R1.

Write the 2 under the line, in the space lined up with the 5, followed immediately by the remainder of 1, like so:

That’s it. Really not that complicated. It’s formally similar to other algorithms, and it’s not hard to understand: we approximated the quotient one place value at a time, then corrected for the approximation.

I love it. But what happens when the numbers get bigger?

When The Numbers Get Serious

Let’s look at a more complicated example, the one addressed by David E. Smith in The Teaching of Arithmetic. I have added some extensions to Fibonacci’s method in order to clarify place value.

Step 1: Moving from left to right, we don’t reach a number greater than a multiple of 72 until 102. Then, we approximate 102,528 as 102,000 and correct for the approximation in subsequent steps.

102,000 ÷ 72 = 1000, with a remainder of 30,000. We write 1000 below the line. Then we add “528” to the remainder of 30,000 to make 30,528. This is our new dividend, so we cross out the 102,528.

Step 2: We approximate 30,528 as 30,500. 30,500 ÷ 72 = 400, with a remainder of 1700. We write 400 below the 1000 and add “28” to our remainder of 1700 to make 1728, and we cross out 30,528.

Step 3: We approximate 1728 as 1720. 1720 ÷ 72 = 20, with a remainder of 280. We write 20 below the 400, 288 above the 1728, and we cross out 1728.

Step 4: We have 288 left as a dividend. 288 ÷ 72 is 4 with no remainder. Adding up all our partial quotients, we get 1424 as our final answer. If a remainder is left over, we include that in our quotient in the normal way.

Place value is preserved at each step. Division is treated as repeated subtraction. Students can understand each step of the algorithm and thereby remember it years later.

Why Don’t We Teach It In Schools?

Perhaps we want an algorithm that students can do without thinking. Servile mathematics, what the Greeks called logistike, has been the focus of math curricula for many years. Before calculators, mindless rote calculation had its own justification. We no longer have that excuse. It is both useless and demeaning to teach students to perform an algorithm without thinking or understanding.

Most people are simply unaware of this algorithm. Most teachers and parents did not learn division this way, and teaching a “new” algorithm will be met with strong resistance. The slow path is ultimately the fast path and is well worth the difficulty.

This algorithm fosters understanding of place value, shows division’s true nature as repeated subtraction, and is formally similar to other algorithms.

It should be taught in schools.

The Galley Method

The “Galley Method” of doing division can be just as mindless as the standard long division algorithm, but it is far more beautiful. It requires close attention, and the result is so elegant it's easy to mistake for decoration rather than mathematics.

Anna Taylor explains the galley method solution to 65284 ÷ 594 in her article on The History of Long Division. Here is the completed problem:

Just look at the beautiful galley ship generated by solving this monster of a division problem:

315,843,843,309,862,431,584,384 ÷ 189,810,001,989,100,018,981

We’ll save the full mechanics for a future article.

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David E. Smith argued in The Teaching of Arithmetic that the best algorithm should be taught, and that no algorithm should be standardized. He also argued that algorithms should lead to understanding, not mindless computation. When the standard division algorithm was standardized with the centralization of education, we ignored both pieces of advice.

Fibonacci’s “stack em” method is easy to understand, preserves place value at each step, treats division as repeated subtraction, and is formally similar to other algorithms.

It would be a mistake to standardize any one method. But teachers who aim to teach mathematics as a liberal art should choose a division algorithm that fosters understanding not servile obedience to arbitrary rules.

Next time a student finishes the worksheet early, hand them this instead.

I didn’t see that coming.

Here’s how it started.

Several years ago, the faculty at my school were reading and discussing Plato’s Republic. In Book VII, Socrates gives the famous “Allegory of the Cave”, but what was more striking was what Socrates discusses afterwards.

Socrates argues that mathematics is the study that most compels the soul toward true philosophy.

Then it would be fitting, Glaucon, to set this study [mathematics] down in law and to persuade those who are going to participate in the greatest things in the city to go to calculation and to take it up, not after the fashion of private men, but to stay with it until they come to the contemplation of the nature of numbers with intellection itself, not practicing it for the sake of buying and selling like merchants or tradesmen, but for war and for ease of turning the soul itself around from becoming to truth and being.

…Do you see, then, my friend," I said, "that it's likely that this study is really compulsory for us, since it evidently compels the soul to use the intellect itself on the truth itself?

This also caught the attention of the administration at my school.

Since then, I started meeting weekly with the headmaster and the heads of lower and middle school to discuss math. Every week, for an entire hour, we sought to get to the bottom of what math really is and how to teach it well.

While it is impossible to answer either of those questions completely and with certainty, we started to make sense of them. One thing we realized is that we were not teaching what the ancients considered to be true mathematics.

This drove me to start exploring the history of math curricula. I wanted to know where our curriculum came from, who created it, and why.

The answers to these questions led me deeper and deeper into a rabbit hole of the history of education in general. I realized that I could not merely focus on mathematics, but I had to look into history of our educational system as a whole.

The Rabbit Hole

As I began exploring the history of education in the United States, I realized that the historical narrative was still very much up for debate.

The more I read, the more confused I became. Conflicting histories, competing narratives, unreliable secondary sources. So I went to the primary sources. I started taking copious notes on the writings of what seemed like the 10 most important historical figures. Quickly the list of important people grew to 20, then 50…

Eventually, I decided to write something. People needed to hear about what I was learning.

I hadn’t written an essay since college, over 15 years prior. The last long-form article I wrote was a paper in graduate school on a tool for studying the brain. That was ten years ago.

What if nobody cared?

I tried submitting articles to various online journals and finally one was published in Public Discourse.

I am forever grateful to Public Discourse for giving me the momentum and confidence to continue writing. Since then I have published in the CiRCE Institute, the Washington Examiner, The Imaginative Conservative, The Liberty Sentinel, The Civitas Institute at UT Austin, and the Australian Classical Education Society

But I had a lot to say, and not enough time to get those ideas published in media outlets. That’s why I started What Schools Forget.

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Publishing What Schools Forget

A friend who writes for the Washington Examiner recommended that I start a Substack to gain notoriety and readership.

When I started writing What Schools Forget a little over a year ago, I had no idea what Substack was and how to use it. Nervous? Yes. Excited? Yes. Confident? No.

In April of 2025, I started building up my Substack profile and making the website look halfway decent. For two months, I had three subscribers: myself, my work email, and my friend from the Washington Examiner.

Getting an article published in Public Discourse gave the confidence I needed to really commit to Substack. In June, I “officially published” my Substack by sending out an announcement on LinkedIn and emailing friends and family to ask for their support.

Within a few weeks, I had over 30 subscribers, with more trickling in each day.

Then in July I started writing a series called “Schools and Schooling.” Initially, there was not a whole lot of traction, but then on July 23 I struck some sort of chord with an article on the one-room schoolhouse.

One hundred of you joined me in a single week. I still don’t fully understand why that article struck such a chord, but your response told me I was onto something real.

In early August, I started to intersperse my writing on history of education with articles on math. Many people really enjoy thinking about mathematics, almost as much as I do. These articles sparked a lot of interest, and people kept on subscribing and giving (mostly) positive feedback on my work.

As the school year began, I had to dedicate more time to teaching and less to writing.

I published a couple of articles in media outlets, but did less writing on Substack.

In early 2026, my commitments at school increased. I began coaching basketball, which was an incredible experience, but that meant I had far less time to write.

In March, when the basketball season ended, several hundred of you had subscribed. I started writing a series of weekly articles on mathematics, announcing the series with a passionate post.

This series arose from years of experience and several conversations with parents and teachers about how to teach math better. It crystallized when my brother asked me for advice because his daughter, my goddaughter, was struggling with algebra.

One article on the number line was particularly popular.

My readership was growing. My growth rate increased to around 5-10 new subscribers per day.

That brings us to today. One thousand of you reading about what schools forget. I still find that remarkable.

Coming Next

The word “classical” gets thrown around a lot. But classical education as most schools practice it today is not actually ancient, or medieval, or even early modern. It's a 20th century movement and it largely traces back to a single 1947 essay by Dorothy Sayers.

Many classical schools say they are committed to reading the “Great Books.” I do not doubt that they are. The “Great Books” movement, however, also has an interesting history, arising from a group of professors at University of Chicago in the early 20th century.

The liberal arts: that’s where the treasure lies. With roots in ancient Greece, the liberal arts were not fully systematized as such until Late Antiquity and the early Middle Ages by people like Martianus Capellus and Boethius. Modern education, if it truly seeks to be classical and to teach the liberal arts, must return to these great writers.

That's what I want to explore next.

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I was stumped. I was proctoring a study hall in my first year of teaching, and a student came to me for help with this question.

I had just graduated from Rice University with a Master’s in Applied Physics, and I was stumped by a 4th grade math problem.

Complex Analysis? No problem.

Linear Algebra? Piece of cake.

Partial Differential Equations? Got it.

Long Division? Nothing.

But I could figure it out, right? Wrong.

I had no foundation of understanding to build on. I had no tools to figure out how to do long division again. I was never taught why it works.

If someone had asked me to do Proposition 1 from Book III of Euclid’s Elements, I could have easily demonstrated it. In that class, I really learned why math worked.

I understood it. So I remembered it.

Nobody ever taught me to understand long division. So I couldn’t remember it.

How Kids Learn Long Division

The algorithm that most of us learned in school goes by the name “long division.” It is actually the middle of three similar algorithms. There is a longer one that shows why it works, and a shorter one that is more efficient. Most teachers only teach the middle one.

For the 99% of readers who, like me, forgot how to do the standard long division algorithm, here’s a recap.

Teachers typically start by explaining that we are going to learn a new method for dividing big numbers.

Let’s use this as our example:

2756 ÷ 13 = ?

We first set it up in the half-box diagram.

Which is unexpected…

The addition algorithm: stack ‘em in order, addition symbol to the left.

The subtraction algorithm: stack ‘em in order, subtraction symbol to the left.

The multiplication algorithm: stack ‘em in order, multiplication symbol to the left.

The division algorithm: half-box shape with no division symbol in sight.

We should at least acknowledge these odd features, but most teachers don’t.

It’s hard enough for kids to learn all the steps to setting up the problem, let alone learning the steps to solving one.

“First, draw this funny shaped half box. Then, write the big number inside the box and the little number outside the box.”

Now the teacher asks the students a very silly question: “How many times does 13 go into 2?”

The 2 in 2,756 is not really a 2, but a 2,000. At the age when students are struggling to understand place value, we completely ignore it for this algorithm.

Right off the bat, we’re confusing students.

Here is the way most students learn the standard algorithm:

1. Divide: How many times does the divisor (13) go into the “active” digit(s) of the dividend without going over? Write the quotient on top of the box, above the smallest “active” digit.

2. Multiply the number outside the box by the number we wrote on top of the box, and write this product underneath our dividend.

3. Subtract the product from step 2 from the active digit(s) of the dividend.

4. Bring down the next digit from the dividend and stick it on the end of the difference from step 3.

5. Repeat, now using the Frankenstein of a number we created in step 4 as our new “active” digits.

When kids can do these steps without assistance, they have “learned long division.” What they have actually learned is a ritual: Divide, Multiply, Subtract, Bring down…

Why does it work? Nobody knows…

The Extended Algorithm: The Real Long Division

In the early 20th century, there was a “great debate” over which algorithms (subtraction was especially contentious) should be taught in schools nationwide. There are in fact many other division algorithms that were used at various points in history, from the ancient “Galley Method” used by Muhammed Ben Musa in Kitab al-Jabr wa’l-Muqabala to Fibonacci’s prime factorization method in Liber Abaci.

David Eugene Smith, historian of mathematics at Columbia University, argued in The Teaching of Arithmetic that it does not really matter what algorithm was taught. It was a means to an end. What matters is how the algorithm is taught. Students need to understand.

In the case of division, Smith points out that it is better to first teach an extended algorithm to facilitate a child’s understanding. The extended method is explained, and only after his students understand does a teacher go on to teach an abbreviated form, like the standard algorithm.

He argues the following:

Most of us are accustomed to learning only the second of these two algorithms, but it is the first one that really shows us what’s going on. Just as multiplication is repeated addition of multiples, division is repeated subtraction of multiples. In the example above, it is clear from the extended algorithm (left) that we subtract 1000 × 72, then we subtract 400 × 72, then we subtract 20 × 72, then finally we subtract 4 × 72. Nothing remains of the initial 102,528, and so there is no “remainder.”

These steps are not made clear by the standard long division algorithm on the right side, as the picture below exemplifies.

But if we are going to teach an abbreviated form of the extended long division algorithm, we might as well teach the real short division algorithm.

The Short Division Algorithm

There is a shorthand to the standard long division algorithm that is actually worth teaching to students after they understand the extended algorithm outlined above.

Step 1: Set up the problem as is typically done. Divide, multiply and subtract mentally. Write the remainder (30,528) in the line below.

Step 2: Repeat, this time dividing into the remainder from step 1 (30,528).

Step 3: Repeat again for the new remainder (1,728). If the final number can be divided evenly by the divisor, complete the quotient. If the remainder cannot be evenly divided, it is the “remainder”. There is no need to write the final step of 288-288 = 0.

The Real Reason to Learn Long Division

We have calculators in our pocket that tell us the answer to a division problem if we ask, “Hey Siri, what is 2756 divided by 13?” …“212”

So why teach algorithms? Long division should be taught because it is one of the foundation stones that students will need to build on to understand higher level math.

In Algebra 2, students typically learn about polynomial division. Polynomial division is a more generalized version of long division.

2756 ÷ 13 is equivalent to the polynomial division problem:

In both cases, you are subtracting multiples of the divisor repeatedly until nothing remains. The variable x just stands in for the base. If x =10, the answer simplifies to 212.

With true understanding of the division algorithm, children have the strong foundation they need to easily understand polynomial division in algebra. In my experience as a teacher, most algebra students struggle to understand polynomial division and eventually just memorize the steps. Few truly understand why it works.

If we are going to teach algorithms without understanding, we’d be better off handing them a calculator.

The slow path is the best path. In fact, it is the only path if we want to be teachers worthy of the name.

When I dug my heels in as a teacher and forced myself to understand the division algorithm, it genuinely opened a new world of understanding and computational ability. If only someone had taught me that way thirty years ago, I might have had something to offer the student in study hall.

The art teacher at my school said this while substituting for my geometry class when we were doing proposition 5 of Book I. In retrospect, this was a bad idea. This prop is traditionally referred to as the “pons asinorum” or “bridge of donkeys.” The name arose both from the bridge-like shape of the diagram and the difficulty of the proposition. It proposes to prove that the base angles of an isosceles triangle are equal. The teacher substituting for the class was confounded by it.

He was unable to separate in his mind the definition of an isosceles triangle from its property of having the base angles equal.

It is important to distinguish between the definition of a mathematical object and a description of its properties.

An isosceles triangle by definition has two sides equal. A necessary property that logically follows from this definition is that it will have base angles equal, but this is not part of the definition.

The reason definitions are so important is that they say what a word means. In the phrase “isosceles triangle,” the word “isosceles” means that two of the sides are equal.

Aristotle says, “Well begun is half done” (Aristotle, Politics, Book 5, 1303b).

To arrive at the level of absolute certainty that mathematical proof can provide, a deep study of the definitions, postulates, and common notions is necessary. They aren't mere preamble, they're the rules of the game.

This article will focus on beginning well. Your task is to finish the second half of the enterprise.

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Failure to Launch

It was my third year of teaching Euclid’s Elements, and things were not going well.

About a month or so into the school year, all my students were struggling to learn and understand the propositions. Only the “memorizers” were able to reproduce a proposition on the board, but they did so without understanding.

I took stock of the situation and soon realized that my students did not know or understand the definitions, postulates, and common notions. In previous years, I had taken these parts of the Elements slowly, in part because the material was so new to me, too.

But in my third year I rushed it, and I did a disservice to my students.

We revisited the definitions, postulates, and common notions, essentially starting all over again. After deep discussions and a couple of quizzes, we built the foundation needed.

Most of the students that year ended up loving the Elements.

A Traveller’s Pack

A few weeks ago I spent a week in Moab, Utah with a group of 8th grade boys. We learned how to pack a hiking backpack with the bare necessities for a week of camping: tent, sleeping bag, food, cookware, and minimal clothing.

After an hour on the trail, hiking through rivers and over hills and rocks, each hiker was either grateful that he had not taken more clothing or regretting how much he had taken.

Studying the Elements is like traveling on a journey, encountering new challenges with each new proposition.

Before beginning the journey, a prudent traveller packs a bag with the bare necessities and no more. No need to carry extra weight if you don’t have to.

These bare necessities are the definitions, postulates, and common notions.

But a good traveller also fully understands what he has packed. It’s pointless to pack a tent if you don’t know how to set it up.

Likewise, the student of the Elements needs to understand the definitions, postulates, and common notions before beginning his studies.

This is what it means to begin studying the Elements well.

Definitions: What Is It?

Euclid begins the Elements with definitions.

They tell you what Euclid is talking about. They are the precisely defined geometrical terms that he will use.

Contemplate them deeply, and if possible discuss them with peers. The definitions of Euclid’s Elements are carefully worded statements describing mathematical objects, the properties of which are explored in the propositions.

Diving into Some Definitions

“A point is that which has no part.”

This first definition seems to say what a point is not. It doesn’t say exactly what a point is.

This is an opportunity for thought and discussion. What can be said about a point in a positive way? It has location. Anything else?

Great Books like Euclid’s Elements are best read with others so that you can discuss them aloud and share insights, of course. Short of this, however, have a discussion with Euclid about it. Does each definition make sense in the context of the other definitions?

The next definition, “A line is a breadthless length,” at least tells us something about a line, that it is a length. But it is also negative, similar to the first definition. A line has no breadth.

What can this possibly mean?

On a first pass through Euclid, it is better to reflect on the definitions without the aid of a commentary, but if you would like to dig deeper, Sir Thomas Heath’s extended commentary on the Elements is the place to start.

Some definitions are more important than others, either because they are more frequently referenced in the text or they are more fundamental. At the start of the year, I require my students to memorize definitions 1-16 and 23. As we move through Book I, I add definitions to the list once we encounter them in the text. I do not require students to use Euclid’s exact words. But it is difficult to get as precise as he does with different words.

When studying the definitions, don’t skip over a single one. Some will crop up more often than others in the text. The definitions are the foundation of an ancient text that summarized the geometrical wisdom of its time, and so each definition deserves attention.

Postulates: Euclid’s Rules

The postulates are axioms specific to the art of geometry that Euclid demands we assent to in order to do his math. They read as follows.

Let the following be postulated:

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any center and distance.

4. That all right angles are equal to one another.

5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Memorize the postulates.

Most of them will be used quite often, so at least know what they say. Even Postulate five, that surprisingly long one, should be well enough understood to be able to see when it might be used.

Postulate five is a famous one. Its length and complexity led many geometers to suspect that perhaps it need not be a postulate. It seems like something that might be provable from the other four postulates. Mathematicians did their utmost, but were unable to. Any attempts to do geometry without this postulate led to strange outcomes: the angles of a triangle did not add up to 180 degrees, or there were no parallel lines or there were infinite parallel lines through a given point. Leaving out postulate five led to other “non-Euclidean” geometries like spherical or hyperbolic ones.

Perhaps Euclid saw that Postulate 5 was different from the others, but he still included it because it was needed. Twenty centuries later, mathematicians proved that postulate 5 wasn’t only necessary, but it holds the flatness of Euclidean space together.

Postulates Practice

Practice doing the postulates.

If you never had practice drawing a straight line carefully with a straight edge, do so. It takes care because a pencil has a definite size and width and a ruler has height. Trying to represent geometrical reality in physical reality is no easy task.

Drawing circles with a compass likewise takes some practice. Take care not to tear the paper with the anchor point of the compass. Learn how to handle the compass to get one complete rotation without lifting the attached pencil.

There is a reason Euclid must postulate these things. There is no such thing as a physical line or circle. What we call lines and circles have attributes similar to geometrical lines and circles, but they are imperfect. We need to do our best to imagine that the lines and circles we draw on paper are intended to depict geometrical realities.

Postulates four and five are not about things to be done, but about truths of geometry to accept. These should not therefore simply be passed over. Do you agree with them? Do they make sense? Draw different examples of postulate five. Review Euclid’s definition of “right angle.” Can you see why it implies that all right angles are equal?

Common Notions: Isn’t It Obvious?

Euclid takes nothing for granted. That art teacher took something for granted, that equal sides mean equal angles, and that’s where he went wrong.

The common notions are even more obvious than the definitions and postulates. They are common sense.

Aristotle describes them as the general axioms not specific to the subject matter under study (geometry). They are not specific to geometry, but rather are general principles that Euclid uses to justify claims in the proofs.

1. Things which are equal to the same thing are equal to each other.

2. If equals be added to equals, the wholes are equal.

3. If equals be subtracted from equals, the wholes are equal.

4. Things which coincide with one another are equal to one another.

5. The whole is greater than the part.

These should be carefully read and understood.

I give a quiz on the definitions, postulates, and common notions after we have thoroughly read, discussed, and (hopefully) understood them. Only a selection of the definitions are required, but students must memorize all the postulates and common notions.

During class I call on students randomly to provide definitions, postulates, and common notions. After a week or so, I repeat the quiz, and most students ace it.

Finishing

Many readers, I hope, will take this series to heart and study Euclid on their own. This is a demanding task. If possible, get some friends to join you. There are few things more pleasurable than enjoying the certainty of mathematical truth in good company, especially over some beer or wine.

But arriving at certainty is no easy task.

“To begin is for everyone. To persevere is for saints.” (St. Josemaria, The Way, 290)

Persevere through the definitions, postulates, and common notions. You are laying the foundation, and a solid foundation is always difficult. Students who put in the work, however, will achieve something that no other discipline can provide: logical certainty.

And the eternal truths in this age-old text that stand on these foundations are beautiful beyond compare.

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That’s impossible! You can’t divide by a fraction! That’s crazy!

The class was buzzing. They wanted to get to the bottom of this mysterious problem.

Now, place yourself in the shoes of one of these fifth graders who had only earlier that year become accustomed to adding and subtracting fractions. And only the week before, we had completed the lesson on multiplying fractions. Fractions were still very mysterious, unlike anything they had seen in math.

Instead of responding to their wonder and anticipation with understanding, I gave them a trick. The same trick I learned at their age.

These problems are really easy. You just “invert and multiply.”

Then I wrote the following on the board:

The cries were silenced. Instead of enchantment and fascination I saw confusion and blank stares.

That’s it? That’s all we have to do?

They complied. They followed the rule I gave them.

But they didn’t understand, and so many of them probably began to hate math that day…

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The Slow Path is the Best Path

When I taught fifth grade math for a year, I took the easy way out. And I still regret it.

Teaching students to understand dividing fractions is difficult. It is slow, but it is the best path.

There are better ways to teach it so that students understand what is going on and why the “invert and multiply” technique works.

In a previous post, I encouraged learning math facts through understanding, not just rote memorization. The approach to teaching dividing fractions is similar.

Develop the techniques with students, instead of simply providing an algorithm.

Start With What Division Actually Means

To begin, students should get familiar with seeing fractions in a division problem.

If students understand what division means and what fractions are, they should be able to figure out the answer. The question is asking “How many thirds are there in one?” If a student does not already see the answer intuitively, we should draw a picture of 1, split it into thirds, and then count by thirds until we reach 1: ⅓, ⅔, ³⁄₃.

There are 3 thirds of a circle in 1 whole circle, so the answer is 3.

Similarly, there are 12 fourths of a circle in 3 circles, so the answer to the second problem is 12.

Students are often surprised (or confused) to find that the answer to a division problem can be greater than the dividend: 3 ÷ ¼ = 12. This confusion can be greatly reduced when they understand division, especially in the case of fractions, as asking “How many of this (divisor) are inside of that (dividend)?”

A seemingly harder problem stops most students cold:

We should first check student understanding of what this problem is asking. It is more straightforward than it looks.

The problem is asking “How many sixths are inside of five sixths?” The answer is obviously 5. We should have students draw a picture of this.

The answer is found in exactly the same way as the previous problems. We can count the sixths: ⅙, ²⁄₆, ³⁄₆, ⁴⁄₆, ⅚. So the answer is 5.

Now let’s look at why that worked and how to make it work for every problem.

The Common Denominator Unlocks Everything

Looking at the last example problem, above, we can actually find the solution very simply by basic division. Simply dividing the numerators by each other and the denominators by each other, we get the following:

And this is the same answer we get by counting by sixths. Wow!

But it doesn’t seem so straightforward with our earlier examples, like 3 ÷ ¼ or 6 ÷ ⅖. Why?

The reason why the example above was so simple is that the two fractions have a common denominator.

For the earlier examples, when we divided circles into thirds or fourths, we did the visual equivalent of finding a common denominator.

Converting these other problems formally so that there is a common denominator, we can apply our new method:

It helps to provide problems with bigger numbers, too:

Counting by multiples of ³⁄₁₀ to reach 120 or drawing 120 circles is a very difficult task, so at this point, the method sells itself.

When the Picture Starts to Get Messy

The pictures start to get messy when the divisor doesn’t divide evenly into the dividend, but the common denominator method holds.

If students have understood the previous problems, they should be able to work out, with proper guidance, the next one:

Clarify what the problem is asking: “How many groups of four fifths are there inside three wholes?”

We are counting in groups of four fifths. So we divided the circles into fifths, making a total of 15 sectors. Now we’re counting groups of four fifths at a time. The first group is in red, the second in blue, the third in gray, but the final group in green is only partial. We don’t have all four! What do we do? It’s very simple. Three out of four is…¾. So our final answer is 3 ¾.

Problems with unlike denominators follow the same logic. Convert to a common denominator first, then divide across.

Multiply by the Denominator, Then Divide by the Numerator

At this point, many students may have figured out a shortcut. For any fraction division problem, all you need to do is multiply the dividend by the denominator and divide the product by the numerator.

Why does this work?

We multiply by the denominator because we need to split the dividend into those kinds of parts. Three whole circles yields a total of 15 fifths, so we do 3 × 5 = 15.

We divide by the numerator because that is how many of those parts we are counting by. Of the 15 fifths, we can make 3 ¾ groups of 4, so we do 15 ÷ 4 = 3 ¾.

Finale

At this point, after practicing multiple problems like the ones above until they master the methods, students are ready to be taught “invert and multiply.” Some may have already figured it out on their own.

But don’t just have them memorize it!

They have to earn the algorithm.

When a student can tell you why we multiply by the denominator and divide by the numerator, they own the algorithm. “Invert and multiply” becomes a shortcut they discovered, not a spell they memorized.

That’s the difference between teaching mathematics and teaching calculation.

The mathematics is everything that earns the trick. The trick is just a shortcut.

Euclid spent four books on ratios. He never wrote a fraction. That's not an oversight but a roadmap we've ignored.

Euclid’s Ratios

The ratio was developed as part of accepted mathematical knowledge earlier than the fraction. In the Elements, Euclid dedicates four books, Books V, VI, VII, and VIII, to the exploration of ratios. Books V and VI are dedicated to ratios of continuous magnitudes, whereas Books VII and VIII focus on ratios of discrete numbers.

Fractions as such do not appear in Euclid’s Elements.

This makes sense because for the Greeks, it was absurd to split the number one. One was the fundamental unit, multiples of which make other numbers.

For a young child, it is also absurd to split the number one. They don’t know it by the definition of unit: “that by virtue of which each of the things that exist is called one.” But they still have a deeply seated conviction that it cannot be divided.

Even the ancient Egyptians, the first people to use fractions, intuited this sense of the unit. They represented every fraction as a sum of unit fractions. The fraction 5/7, for example can be expressed as 1/2 + 1/7 + 1/14. Teaching “Egyptian fractions” can be a great early lesson for children because it is easier to understand unit fractions than non-unit fractions.

The natural development of intellectual ability from childhood to adulthood tends to parallel the history of mathematics. A child develops an understanding of counting and natural numbers first, and only later do we introduce fractions. But part of understanding and mastering the natural numbers is understanding ratio.

First, the Ratio

Ratios are much easier to grasp for children than fractions. For one who is experienced with fractions, it is difficult to see the difference. The fraction, 1/2 can be understood as a ratio. For example, 1 dollar for 2 candies means that each candy is half of a dollar.

But there is a big difference between ratios and fractions.

Fractions express parts of unity or one (1). Ratios express a relationship between two natural numbers. Fractions require introducing a new kind of number (rational numbers) into the child’s world. Ratios understood as the relationship between discrete counting numbers does no such thing.

Before we start splitting apart natural numbers to introduce fractions, we need to teach children to work with ratios at a deep level. In this way, they learn to scale ratios up and down, reduce them to the simplest form, etc. Equivalent ratios behave exactly like equivalent fractions. Reduced ratios are the same as reduced fractions.

Combining ratios and adding fractions are quite different. This is one area where students may get into trouble.

Adding Ratios

Here is an example problem where combining ratios is very unlike adding fractions.

Ben and Paul are adding marbles to a box. Ben adds 2 blue marbles for every 3 red marbles. Paul adds 5 blue marble for every 6 red marbles. What is the ratio of blue marbles to red marbles in the box?

The answer should be evident, though if you think about the problem as a fraction addition problem, you’ll make a mistake: 2/3 + 5/6 = 9/6 = 3/2. The final ratio is not 3:2, but 7:9. Let’s say Ben adds 10 blue and 15 red marbles, and Paul adds 25 blue and 30 red marbles. Combining these numbers gives us 35 blue marbles and 45 red marbles, or a ratio of 35:45 = 7:9. A much simpler way to solve the problem is just to add the terms of the ratios: 2+5 : 3+6 = 7:9.

When students are confused about adding fractions and write down things like 1/3 + 1/2 = 2/5, they are doing one of two things wrong:

1. They are simply not thinking about what they are working with, and they are just adding the numbers they see on the page.

2. They are thinking about fractions as the same thing as ratios.

The first is the more likely mistake, but it is important to not simply dismiss a students mistake, but to ask questions to identify if he is actually thinking carefully or not. The second mistake is an insight that a sharp student may need to be clearly explained, or you risk losing his buy-in.

When are adding ratios and fractions similar?

Ben makes lemonade at a rate of 2 gallons every 1 hour. Paul makes lemonade at a rate of 3 gallons every 2 hours. How many gallons will they make if they work together?

The answer to this problem is not 2:1 + 3:2 = 5:3. Instead, we must find a common denominator to solve the problem. 2/1 = 4/2. So Ben can make 4 gallons in 2 hours. Working side by side they will make a total of 4 gallons + 3 gallons every 2 hours or 7 gallons every 2 hours. In mathematical terms, 4/2 + 3/2 = 7/2.

The Denominator as Unit

What makes a number countable is the multitude of units that compose it. A fraction is not countable in the same sense that counting numbers are because they contain parts of one, which is what we most typically mean by “units.”

A helpful way to understand fractions is to realize that we are in essence redefining what the unit is. The denominator indicates the size of the unit for a given fraction. Thus, 3/4 (three fourths) has a unit size of fourth and we have three of these units.

Mathnasium instructors talk about the denominator as the “naming part” of the fraction and the numerator as the “numbering part.” Denominator has its root in the Latin word nomen, which means “name.” Numerator has its Latin root in the word numer, which means “number.”

This property of fractions makes them difficult to work with. We cannot add two fractions with different denominators because that would be like trying to add apples and bananas. 3/4 and 5/6 are not 8/10. In the same way, three apples and four bananas are not seven “banan-a-pples.” We need to find a common denominator, 12ths, or a common name, like “fruits.”

It is important that fractions be understood in this way: numbers of units, with the unit size determined by the denominator.

Concrete-Pictorial-Abstract

With nearly every math concept in elementary grades, we ought to first model the concept concretely, then pictorially, then abstractly.

Starting with the ratio before teaching fractions aids in the presentation of concepts in a very concrete way. Food is a great tool for teaching. We could use fruits or baked goods or candy: “I have 8 chocolate bars and 12 lollipops. What is the ratio of chocolate bars to lollipops? 8 to 12 or 4 to 3.” We should actually have these objects in order to really be teaching concretely.

We move on to the pictorial stage if we ask them to, for example, “Draw a picture of at least 6 balloons, where the ratio of red to blue balloons is 1 to 2.” This open-ended task is great, and students love the opportunity to be creative (especially in the lower grades).

It is important that during the concrete and pictorial stages, we do not yet introduce formal abstract representations of ratios, like “8:12” or “4:3” because this will only get in the way of the understanding. The abstract representations are just a shorthand for what is happening concretely or pictorially.

The concrete-pictorial-abstract approach is exactly why the ordering below matters: ratios first, then parts of a whole, then decimals, then rational numbers. Each step earns the next.

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The Different Meanings of a Fraction

It is important to differentiate between the multiple ways that one can explain and use fractions. Without clearly delineating the benefits and shortcomings of each, we risk losing a lot of students.

I present below what I think is a comprehensive list of different representations of fractions at an elementary level, and also the order in which I think we ought to introduce them.

Fractions as Ratios

Only after students understand ratios should we teach fractions. And the best place to start is to explain fractions in the context of ratios.

Students should have already seen problems like the following word problem.

There are 10 boys and 15 girls in a class. Write the ratio of boys in the class to total number of students. Reduce it to simplest form.

If they can do this, solving the next problem is pretty much just a matter of replacing the “:” in the ratio with a “/” to make it a fraction.

There are 10 boys and 15 girls in a class. Write the fraction of the class that is boys. Reduce it to simplest form.

The ratio of students that are boys to all students in the class is 10:25. The number of boys as a fraction of the students in the class is 10/25 or 2/5 in simplest terms.

But what is the meaning of saying 2/5 of the students are boys? A teacher should be careful not to say, “2 out of every 5 students are boys.” It is not quite so simple, and some students may be confused. We can easily select 5 students that are all girls, in which case 2 of them are not boys.

Saying that 2/5 of the students are boys is saying that if we split up the class into equal groups of 5 students (“equal groups” here means the same number of boys and girls in each group), there will be 2 boys in each group of 5 students.

What would it mean in this case to add fractions or multiply the fraction 2/5 by a whole number? These questions are not easy to bring meaning to in the given case, and the same goes for many cases in which a ratio presented as the same as a fraction. This is one defect of thinking of fractions as the same thing as ratios, and thus it is a defect with only using this representation.

Fractions as Parts of a Whole

Fractions are typically first introduced to students as parts of a whole, but I think this should be done only after students have completely mastered ratios. Really, the “parts of a whole” representation of fractions is a ratio of some of the parts of a whole to all of the parts of a whole.

Shapes help students to see what 4/7, 3/4, 1/2, or 1/3 “look like.” Students are given shapes like the figures below, and they become very skilled at identifying the fractions that these shapes represent.

Shapes, however, are not numbers. These images are not fractions. They are useful representations of parts of the whole, and help students to learn to compare sizes of fractions, but they are not entirely adequate.

The first image is not the fraction 4/7. We often tell students that “4/7 of the circle is shaded.” We should begin by saying, “the ratio of shaded parts to total parts is 4 to 7.” Then we should move on to say “4 parts out of 7 total parts are shaded.” Only after this, should we introduce the abbreviated shorthand, “4/7 of the circle is shaded.”

4/7 has more parts than 3/4, but 3/4 of the circle is more than 4/7 of the circle. For a student who only has experience of counting numbers, this is bizarre. It is one of the stereotypical confusions of students when they first learn about fractions. To avoid it, we need to be precise with our language.

Saying that these images are 4/7, 3/4, 1/2, and 1/3 is a complete misuse of language and only causes confusion. Any mathematician will tell you that precision of language is essential to truly understand mathematics.

Fractions as Decimals

At some point, students do need to be able to convert fractions to decimals and vice versa. But, if introduced too soon, this can hide important details about fractions.

When we convert fractions to decimals, we are essentially converting them to have common denominators. One example of a task often given to students is to compare fractions:

Order the fractions 1/2, 2/5, and 7/10 from least to greatest.

Converted to decimals, these fractions become easy to compare: 1/2 = 0.5, 2/5 = 0.4, 7/10 = 0.7. We can say exactly how much bigger or smaller they are.

Many textbooks, however, introduce decimals before fractions.

This is a mistake: decimals are fractions!

What we typically refer to as “decimal” is shorthand for “decimal fraction.” This term, I think is preferable, though it may not be possible to salvage it. What may be possible, however, is to convince teachers to teach fractions first.

If students do not learn to work with fractions first, we lose an opportunity to help them develop their number sense. Ordering the fractions 1/2, 2/5, and 7/10 from least to greatest is not hard at all. Compare each to one half. 1/2 is equal to one half, 2/5 is less than one half, and 7/10 is greater than one half. Voila! The order is 2/5, 1/2, and 7/10.

Only after introducing techniques like this should we reveal to students the beauty of using decimals as a shorthand, by which we can also know with greater precision exactly how much greater or less the numbers are.

Fractions as Rational Numbers

The goal in teaching fractions is for students to eventually accept them as numbers. Functionally, this means that we want students to be able to add, subtract, multiply, and divide fractions by each other and by other kinds of numbers. We want them to be able to compare the sizes of fractions with each other and with other kinds of numbers.

But fractions do not have the same status as counting numbers. Only if fractions share a common denominator do they represent multitudes of units that are countable in the same way as counting numbers.

In many classrooms, students draw fractions on a number line way too early. Done correctly, they will arrive at a placement like the one shown here. This is a perfectly legitimate thing to introduce, but only after students have mastered the other previously mentioned representations of fractions.

The number line is useful but introduces its own confusions. I've written about these confusions separately and won't rehearse them here.

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Concluding Thoughts

A general principle in teaching mathematics is to start with more concrete representations, move to pictorial representations, and only then, after students have fully understood what they are doing, move to abstract representations.

Abstract representations of fractions like 1/2 or 3/4 are confusing unless a student has understood ratio first, then gone through the steps of moving to parts of a whole, decimals, and finally rational numbers.

A student who truly understands fractions isn't performing acrobatics. He's standing on solid ground that Euclid built, and that we keep forgetting to lay.

I ask him: “Do you have a proof?”

Silence.

Not because he is unprepared, but because he doesn’t understand what a proof actually is and he doesn’t know that the text he’s been staring at follows a hidden architecture that makes the proof almost inevitable, if you know how to look.

That’s what this article is about.

Euclid’s Elements is one of the most carefully ordered books ever written. Nearly 500 theorems and constructions. Each one built on the last, like bricks in a wall. The argument of Book I is one long, unbroken logical tapestry. The diagram below beautifully maps the entirety of Book I with all its interdependencies.

Don’t skip propositions. Don’t move on until you understand the previous proposition. Once you truly understand one, it’s very hard to forget, at least in outline.

What some students and teachers don’t realize is that every single proposition in the Elements has the same underlying structure. If anything changes, only one or two of the steps in the structure are missing. Once you see this, the whole book opens up.

The ancient geometer Proclus described it best. Sir Thomas Heath quotes him at length in his translation of the Elements:

“Every problem,” says Proclus, “and every theorem which is complete with all its parts perfect purports to contain in itself all of the following elements: enunciation (πρότασις), setting-out (ἔκθεσις), definition or specification (διορισμός), construction or machinery (κατασκευή), proof (ἀπόδειξις), conclusion (συμπέρασμα).”

Six parts. Always the same six parts. Here’s what they mean in practice.

The Six Parts of Every Proposition (Actually Seven)

Proclus counts six. Modern editions sneak in a seventh. More on that in a moment.

Let’s use Proposition I.1 as our example throughout. The goal: construct an equilateral triangle on a given line.

1. Enunciation The opening sentence states what is given and what is sought, in the most general terms possible. In most editions, it’s italicized.

“On a given finite straight line to construct an equilateral triangle.”

Philosophically, this comes first. Practically, it’s often too abstract to be a useful starting point for students. Start with the exposition instead.

2. Exposition (Setting-out) Begins with “Let...be the given...” This is where the general enunciation gets grounded in a specific diagram.

“Let AB be the given finite straight line.”

First step when studying a proposition: find this phrase, locate the given in your diagram, and draw it accurately with a straightedge and compass.

3. Specification Begins with “Thus it is required...” for problems, or “I say that...” for theorems. Restates what is sought, now in terms of the specific diagram.

“Thus it is required to construct an equilateral triangle on the straight line AB.”

This one phrase tells you everything. Problems and theorems are fundamentally different: problems ask you to construct something; theorems ask you to prove something. Proposition I.4, the famous Side-Angle-Side theorem, is the first theorem in Book I.

4. Justifications This is the bonus part. Proclus didn't count it because Euclid's original manuscripts didn't include it. But for a student facing that wall of text, it may be the most important of all.

Most modern editions include margin references, “Post. 3,” “Def. 15,” “C.N. 1,” that cite the definition, postulate, common notion, or prior proposition being used at each step.

Before reading the construction and proof, scan the justifications first. They function as an outline. In Proposition I.1, the justifications are:

• Post. 3: Draw a circle

• Post. 3: Draw a circle

• Post. 1: Draw a line between points

• Def. 15: All radii of a circle are equal

• Def. 15: All radii of a circle are equal

• C.N. 1: Things equal to the same thing are equal to each other

Six steps. Now the construction and proof are no longer a mystery. There is a clear path to follow.

5. Construction Everything beyond the given gets drawn here. Euclid instructs you to draw circle BCD with center A and radius AB, then circle ACE with center B and radius BA.

Draw it yourself, following the construction step by step. Don’t copy the figure from the book. The constructed diagram isn’t decoration. It is central to the argument.

6. Proof The rigorous argument. In Proposition I.1: AC and AB are radii of the same circle, so AC = AB. BC and BA are radii of the same circle, so BC = BA. By Common Notion 1, things equal to the same thing are equal to each other, therefore AC = BC. All three sides are equal.

Euclid’s proofs are essentially chains of syllogisms. You remember the form from logic class:

• Major premise: All men are mortal.

• Minor premise: Socrates is a man.

• Conclusion: Socrates is mortal.

In Euclid, it looks like this:

• Major premise (C.N. 1): Things equal to the same thing are equal to each other.

• Minor premise: CA and CB are each equal to AB.

• Conclusion: Therefore CA = CB.

The older manuscripts omit the major premises entirely. The justifications restore them in abbreviated form.

7. Conclusion Euclid restates the enunciation, typically applied to the specific diagram, confirming that we did what we said we’d do.

“Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB.”

Each problem ends: Quod erat faciendum, Q.E.F., “We did what we said we would do.”

Each theorem ends: Quod erat demonstrandum, Q.E.D., “We proved what we said we would prove.”

I tell my students this is the final mic drop. They get a kick out of writing it on the board with some flair after a successful demonstration. Getting Q.E.F. and Q.E.D. mixed up is like stumbling across the finish line.

Back to Johnny at the Board

After the silence, I don’t give Johnny the proof. I start asking questions.

Not leading ones. Not “Isn’t AB equal to BC?” That’s just giving the answer in disguise. More like: “What is our goal with this proposition?” or “Do you remember what justifications Euclid uses in the proof?” And then I wait.

I learned this from a teacher/friend who also teaches Euclid’s Elements. Students must memorize each proposition at least through the construction. Once they’ve drawn it at the board, he asks: “Do you have a proof?” If not, he asks questions until they find it themselves, the way Socrates drew geometry out of an uneducated slave boy in the Meno.

As the year goes on, the other students start asking the questions.

That’s when you know something real is happening.

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It Finally Clicks

I teach this structure to every student at the start of the year. I walk them through it carefully, show them how it maps onto the text, explain why it matters. For most students, it helps immensely.

For some, it goes in one ear and out the other.

Johnny came to me outside of class. He was drowning. He couldn’t understand the propositions, couldn’t memorize them, couldn’t see any shape in the wall of text in front of him. He was bright. He worked hard. But Euclid was a locked door.

I sat down with him one on one. We went through two propositions together, slowly, keeping the structure in front of us the whole time. Enunciation. Exposition. Specification. Justifications. Construction. Proof. Conclusion.

Something shifted. I could see it on his face.

Once he had the key, the door opened. He started acing every test. More than that, he became one of the best presenters in the class at the board, not because he had memorized the propositions, but because he actually understood them. There is a difference, and it shows.

A student who has memorized a proof recites it. A student who understands it can answer questions about it, defend it, and explain any step on demand.

That’s what the structure gives you. Not a shortcut, but a path to deep understanding. It's the thing that turns a wall of text into an argument you can hold in your head.

There’s a reason historians believe Euclid’s Elements was originally communicated orally. The verbosity, the repetitive chorus-like structure, the same form repeated nearly 500 times with all the propositions. It’s built for recitation, not just reading.

When your students can stand at the board and give a proof from memory, they’re not just doing geometry. They’re participating in a 2,300-year-old tradition of human beings reasoning together about what is certainly true.

Q.E.D.

This is the third time this article (with minor changes) has been published by sites other than my own Substack. First published on the Australian Classical Education Society, then by the CiRCE Institute in their Forma Journal, now most recently it was picked up by the Imaginative Conservative.

Schools have forgotten how to truly teach arithmetic. What exists in schools nowadays by that name is a shadow of what it was in its former glory.

Check out the article here

The most important math textbook ever written hasn't been in most classrooms for a century.

Based on my own experience and that of others, I believe studying Euclid’s Elements is essential to a real mathematics education.

I will dedicate a few articles to the Elements, drawing on nearly ten years of teaching Euclid and the experience of numerous wise guides, more equipped to teach this masterpiece than myself.

In this first article, I will focus on some background and an introduction to the structure of the Elements. The next post in the series will have some practical ideas about how to study the book itself.

I will be referencing my new e-book in these posts, so I encourage you to grab a copy and follow along! I’ve removed the paywall from the book, so that now all subscribers have access.

Euclid’s Elements: Textbook of Textbooks

For centuries, Euclid’s Elements was the most read math textbook in the world. It still remains the most influential mathematical work ever composed.

His genius was not his originality. Many of the proofs included in the Elements were likely taken from other mathematicians.

His genius was in his ordering.

There are numerous theories about why Euclid ordered the Elements the way he did. One thing is for certain: logical flow was uppermost in his mind.

Each proof is based on previous axioms, theorems, and constructions. Over the course of the book, Euclid builds a complex tapestry of interwoven threads that on the whole compose a beautiful study of basic geometry and arithmetic.

Basic? Yes. Easy? No.

It is not a textbook in the modern sense of the word. It contains no exercises. The margins are left for notes, not for cutesy, unrelated, and unnecessary pictures of kids on skateboards or smiling dogs.

Praise for Euclid’s Elements

It is more than a textbook. It is a Great Book. It is widely recognized as such by highly intelligent individuals and by the entirety of civilization. It is the second most printed book in the Western world, only after the Bible.

Abraham Lincoln famously decided to study Euclid when he was practicing law:

“In the course of my law-reading, I constantly came upon the word demonstrate. I thought, at first, that I understood its meaning, but soon became satisfied that I did not.” Lincoln said to himself, “You can never make a lawyer if you do not understand what demonstrate means.” Lincoln later said, “I left my situation in Springfield, went home to my father’s house, and staid 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.”

- Abraham Lincoln, as cited by David B Hayes in his article on LinkedIn

Albert Einstein said the following of Euclid’s Elements:

We reverence ancient Greece as the cradle of western science. Here for the first time the world witnessed the miracle of a logical system which proceeded from step to step with such precision that every single one of its propositions was absolutely indubitable - I refer to Euclid’s geometry. This admirable triumph of reasoning gave the human intellect the necessary confidence in itself for its subsequent achievements. If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker.

-Albert Einstein, On the Method of Theoretical Physics (source), emphasis added

Einstein goes on to explain that Euclid’s Elements provides an excellent analogy for how theoretical physics should progress.

The Structure of Euclid’s Elements

Euclid’s Elements is actually a collection of 13 Books total, each with about 50-150 proofs, called “propositions,” which demonstrate truths about geometry and arithmetic.

Each book begins with definitions, postulates, and common notions.

Definitions

The definitions are about what certain words mean. He defines a point as “that which has no part,” and a line as “a breadthless length.”

These are descriptions of mathematical objects which cannot physically exist. Once they are clearly defined, however, it is possible to make logical deductions about their properties and relations.

Postulates

Next up are postulates. These are statements about mathematical reality which are assumed either because they are unprovable or almost unprovable. For example, he postulates that it is possible to draw perfectly straight lines between points or that it is possible to extend a line indefinitely. These are statements about what is possible or true in the context of the Euclidean world of mathematics. They are not statements about what is possible or true in physical reality. A perfect line or a perfect point are physical impossibilities. But in the world of Euclid’s Elements, we must accept them as real.

The postulates can almost be thought of as mathematical rules. I often explain to my students that Euclid’s postulates are him telling us, “If you want to enter into my mathematical world, these are the rules. Accept the rules or do not enter.”

Common Notions

The common notions immediately follow upon the postulates.

If the postulates are assumed mathematical axioms, the common notions are general axioms.

They almost do not need to be assumed because they are part of common sense. Common sense tells us that if A = B and B = C, then A = C. And this is one of the five common notions. Common sense tells us that “the whole is greater than the part.” This is also one of the common notions.

Common notions are axioms of logic that every sane person knows to be true.

Mathematicians and Euclid’s 5th Postulate

It is important to keep in mind that Euclid’s axioms are his rules for his math. His math is often referred to as Euclidean geometry and arithmetic. Mathematicians like breaking the rules and playing games with new rules, however. So over the centuries, mathematicians have developed various other geometries and algebras based on different mathematical axioms.

One of the axioms of Euclidean 2-dimensional geometry (geometry on a flat plane) is that through any point not on a straight line there is exactly one straight line parallel to the first line. This is an equivalent statement of Euclid’s 5th postulate.

In hyperbolic geometry (geometry on a saddle-like curved surface), there is not just one but infinite straight lines parallel to a given straight line through a given point. In spherical geometry (geometry on a sphere, like a globe, for example), there are none.

If the surface itself is curved, how can we talk about “straight lines”?

If a straight line is redefined as the path which traces the shortest distance between two points, we can talk about “straight lines” in spherical and hyperbolic geometries.

And what about “parallel” straight lines?

Parallel straight lines, as constructed in Euclid’s Elements, run perpendicular to a common straight line. Straight lines so constructed in hyperbolic geometry diverge from one another. In spherical (or more generally, elliptic) geometry, such straight lines converge. In Euclidean geometry, they never meet.

Euclid chose his axioms because they bear the most obvious relevance to the natural world. They are directly abstracted from experience with physical reality. More advanced geometries like hyperbolic and elliptic are fascinating, but Euclid’s world is the place to begin.

Final Thoughts

Little is known about Euclid the man. Most historians agree that he worked in the 4th century BC, a little after Plato and before Archimedes. His work was used in the Pythagorean school to teach mathematics, and it eventually made its way to the Western world.

For over 2,000 years after Euclid lived, the masterpiece survived as the textbook for geometry and arithmetic. You had not completed your liberal arts education until you had studied Euclid.

With the scientific revolutions of the 15th-19th centuries, the liberal arts in general (and Euclid’s Elements in particular) fell out of favor. Only with the classical education movement of the 20th century has Euclid’s Elements begun to make a comeback.

With this series, I hope to make the case that Euclid's Elements still belongs in any school that seeks to provide a serious education for its students.