Chapter 2
Greece: a method that could check, but could not find
The crisis that started it
Greek mathematics had a theory of ratio built on counting: two lengths have a ratio because some common unit fits a whole number of times into each. Then the Pythagoreans found that the diagonal of a square and its side have no common unit. The entire theory of proportion broke.
Eudoxus of Cnidus (c. 408 to c. 355 BCE) repaired it, and the repair is one of the great pieces of mathematical engineering. Instead of asking what unit fits both, he compared all the multiples: two ratios are equal when, for every pair of whole numbers, the multiples fall on the same side of each other. That definition survives as Euclid, Elements V, Definition 5, and it is why Greek geometry could continue at all (Source 8, O'Connor).
Out of the same repair came the method Eudoxus is remembered for. Squeeze the unknown quantity between figures you can measure, from inside and outside, and show that any proposed answer other than the right one leads to a contradiction.
The name "method of exhaustion" is a seventeenth-century coinage. No Greek called it that, and it does not exhaust anything: it brackets (Source 8, O'Connor). The name has been misleading students for three hundred years.
Archimedes, and the difference between finding and proving
Archimedes of Syracuse (c. 287 to 212 BCE) is where this chapter earns its place, and not for the bathtub.
Guess before you read on
He proves that a parabolic segment is four thirds of the triangle drawn inside it, and the proof runs on the sum 1 + 1/4 + 1/16 + 1/64 and so on forever. He gets the right answer, and he gets it without one move you would make. Name the move he refuses.
I have a guess
Taking the limit. He writes the finite sum and the leftover term, and stops there. He never writes that the infinite series equals 4/3 (Source 1, Heath). He does not need to: the leftover can be made as small as he likes, and that is enough to rule out every wrong answer by contradiction.
If you expected the limit, you were reading a proof from 250 BCE with a habit bought for you in the 1820s. He would not put in print a step he could not defend.
In Quadrature of the Parabola he proves that a parabolic segment has exactly four thirds the area of the triangle inscribed in it. The proof rests on summing a geometric series, and he does not take a limit. He proves a finite identity and then argues by contradiction. In modern notation, the identity is:
Read this equation in words
The sum from k equals zero to n of one quarter to the k equals one, plus one quarter, plus one sixteenth, and so on, up to one over four to the n. That sum equals four thirds minus one third times one quarter to the n. As n grows without bound, the sum approaches four thirds.
I checked both the finite identity and the limit symbolically; they agree (Appendix E.1). Archimedes has the first line and the second. He does not write the third, because he will not take a limit in print. He does not have to: the second line lets him make the leftover as small as he likes, which is enough for a proof by contradiction.
Then in 1906 we found out how he was really working.
The palimpsest
Johan Ludvig Heiberg found it because of a footnote. A catalog notice from 1899 mentioned a mathematical palimpsest in Constantinople; Heiberg read a few printed lines of the Greek, concluded it "must contain something by Archimedes", traveled there, and was right (Source 2, Heath). The lesson for a classroom: read the boring catalog.
What he found was The Method of Mechanical Theorems, a letter from Archimedes to Eratosthenes, the man who measured the Earth with a stick and a shadow. In it Archimedes explains how he finds his results in practice: he slices figures into infinitely thin sections and balances them on an imaginary lever, using mechanics to discover the answer, and only afterwards constructs the rigorous geometric proof. He tells Eratosthenes plainly that publishing only the polished proof wastes everyone's time (Source 2, Heath).
This is the oldest surviving argument for showing your working, and it is from around 250 BCE.
Two more things in it are worth the whole chapter:
He credits Democritus. The preface says the credit for the cone being one third of its cylinder should go "to Democritus who was the first to make the assertion... though he did not prove it" (Source 2, Heath). The preface to On the Sphere and Cylinder credits only Eudoxus. So the recovered text changed the story: Democritus, the atom man, was probably also the first person to think of a solid as a stack of infinitely thin discs, and was honestly troubled by the paradox that creates (Source 2, Heath).
And in Proposition 14 he compares two infinite collections. Reviel Netz, Ken Saito, and Natalie Tchernetska, reading the palimpsest with modern imaging, found Archimedes calling two infinite sets "equal in multitude" and setting up a proportion between infinite aggregates (Source 3, Netz). This is contested territory. The specialists who found it hedge hard: the key step is implicit, the text may carry an interpolation, and the argument, in their own word, "does remain strange". Popular accounts routinely overstate it into "Archimedes had actual infinity". The honest classroom framing: the people who found the exciting thing are the ones telling you to calm down about it. That is what expertise sounds like (Source 3, Netz).
How the book survived, which is a better story than the mathematics
In April 1229 a scribe named Ioannes Myronas scraped the ink off a tenth-century copy of Archimedes and wrote a prayer book on the recycled parchment (the colophon date is read as either 14 or 29 April 1229) (Source 4; Source 6, Miller). Parchment was expensive and Archimedes was not. He rotated the pages ninety degrees and folded them in half, so a single Archimedes page is now split across two prayer-book leaves, sideways and upside down relative to the prayers (Source 4).
The book survived because it was destroyed. As a prayer book it was worth keeping in a monastery for seven hundred years.
Then, at some point after 1938, a forger painted gold-leaf Byzantine evangelist portraits over four of the pages to raise the sale price (Source 6, Miller). It sold at Christie's in 1998 for two million dollars (sources give 28 or 29 October) to an anonymous buyer, who immediately handed it to the Walters Art Museum, paid for a decade of imaging, and put the results online for free (Source 6, Miller).
To read through the forged gold, researchers used X-ray fluorescence at the Stanford Synchrotron Radiation Lightsource to detect the iron in the original ink. Vandalism defeated by a particle accelerator.
And the detail that belongs in front of students: the first imaging attempt failed. The technology was impressive and the output was useless until the classicists and the physicists argued with each other about what they were looking for (Source 5). Interdisciplinary work is not a slogan in this story. It is the reason the text was read.
Three manuscripts of Archimedes existed in the Middle Ages. One vanished in 1311, one in the 1550s, and the third was scraped clean and prayed over (Source 5). That is the entire margin by which we still have him.
Zeno, and a question that outlived civilizations
Zeno of Elea's paradoxes are usually taught as a puzzle about infinite series, which handles Achilles and the tortoise and misses the interesting one. The Arrow paradox is not about series at all. It asks whether an instant can contain motion. You cannot answer that without the idea that velocity is defined at a point by a limit (Source 7, Huggett).
Zeno posed it around 450 BCE. The mathematics that answers it arrives in the 1820s. A question can outlive dozens of civilizations, and Zeno was not even trying to do mathematics: per Plato he was defending Parmenides, and the paradoxes were a debating weapon (Source 7, Huggett).
Did Greek mathematics "stop short"?
The standard line is that the Greeks had the pieces and never assembled them. That framing is contested, and the contest is worth showing students, because it is the moment history of mathematics stops being a list of dates.
In 1975 Sabetai Unguru argued that reading Greek geometry as disguised algebra is an anachronism that destroys what the Greeks were doing. Three of the most eminent mathematicians of the century, B. L. van der Waerden, Hans Freudenthal, and André Weil, wrote furious replies in the same journal. Weil titled his "Who betrayed Euclid?" (Source 491, Katz). Fifty years later the argument is still running.
Victor Katz's 2024 survey lays out the whole bibliography and the argument about whether the question is even well posed (Source 491, Katz).
Before you read on: van der Waerden looked at Greek geometry and saw algebra in disguise. Unguru looked at the same pages and said that reading destroys what is there. They are not disagreeing about the mathematics. What are they disagreeing about?
What the fight is really about
Whether the notation is the thinking, or just clothing on it.
Van der Waerden's position: a Greek proposition about rectangles on a line and the identity (a+b)² = a² + 2ab + b² are the same result, written two ways. Translate one into the other and nothing is lost.
Unguru's position: translating it loses the thing you were trying to study. The Greeks reasoned about magnitudes you can construct, not about symbols you can manipulate, and a history that rewrites them into symbols has described the historian instead of the Greeks.
You cannot settle this by rechecking the mathematics, which is why three of the century's most eminent mathematicians could write furious replies and change nobody's mind (Source 491, Katz).
So the short version: asking why the Greeks did not invent calculus may be like asking why Shakespeare did not write a screenplay.
One question before you go
Archimedes told Eratosthenes how he found his results, as opposed to how he published them. What was the method?
Show the answer
He balanced the slices on an imaginary lever. He cut a figure into infinitely thin sections and weighed them mechanically to find the answer, then built the rigorous geometric proof afterwards, and he told Eratosthenes that publishing only the polished proof wastes everyone's time (Source 2, Heath).
You meet that lever again in Integrals 3.5, where mass, density and moments look like an application of the integral. They are not an application. They are where it came from.