Reviel Netz’s New History of Greek Mathematics contains a number of factual errors, both mathematical and historical. Netz is dismissive of traditional scholarship in the field, but in some ways represents a step backwards with respect to that tradition. I argue against Netz’s dismissal of many anecdotal historical testimonies as fabrications, and his “ludic proof” theory.

Transcript

A new book just appeared: A New History of Greek Mathematics, by Stanford Professor Reviel Netz, Cambridge University Press. Let’s do a book review.

It will be a critical review. The main theme will be the sciences versus the humanities. Note the title of the book: “a New History.” Netz’s “New History” represents the new humanities-centred dominance in the field. As opposed to the “old” histories written by more mathematically oriented people. In my opinion, “new” does not mean better in this case. And I will tell you why.

Let’s start by attacking a city. The enemy are hunkering down behind their city walls. We are going to have to scale the walls with ladders. How long should we make the ladders? The ancient historian Polybius has the answer:

“The method of discovering right length for ladders is as follows. … If the height of the wall be, let us say, ten of a given measure, the length of the ladders must be a good twelve. The distance from the wall at which the ladder is planted must, in order to suit the convenience of those mounting, be half the length of the ladder, for if they are placed farther off they are apt to break when crowded and if set up nearer to the perpendicular are very insecure for the scalers. … So here again it is evident that those who aim at success in military plans and surprises of towns must have studied geometry.”

Great stuff. But Netz gets it wrong, in my opinion. Here is how he concludes:

“And then, of course, we are supposed to apply – Polybius leaves this implicit – Pythagoras’s theorem.” (223)

I don’t think so. I don’t think that’s what Polybius intended.

Sure enough, you can solve for the length of the ladder using the Pythagorean Theorem, but that is a clumsy and inefficient way to do it. If you did this the modern way you would need to do some algebra followed by some calculation involving a square root. They didn’t have calculators on their phones back then, you know. Do you expect carpenters in the military to be able to calculate square roots by hand?

In fact, Polybius has already told you everything you need to know with his numerical example. If the wall is 10, the ladder should be 12, he says. But it scales! So what Polybius is really saying is that, whatever the height of the wall is, the ladder is always 20% longer than that. That’s all you need to know. No Pythagorean Theorem needed.

Those numbers are a rule of thumb. You can also do it more exactly if you want, according to Polybius’s more theoretical characterisation of the optimal length. But you don’t need the Pythagorean Theorem for that either. There’s a much better way, that you can easily teach to an illiterate carpenter in five minutes.

Draw an equilateral triangle, just as Euclid does in Proposition 1 of the Elements. Cut it down the middle. Now you have a right-angled triangle, where the base is exactly half of the hypothenuse. This corresponds precisely to Polybius’s rule: the distance along the ground is half the length of the ladder.

So now we have a scale model of what we want. The height down the middle of the equilateral triangle represents the city wall; the side of the equilateral triangle represents the ladder, and it is precisely half its own length from the foot of the wall, exactly as Polybius says it should be for optimal stability.

So if we are given that the height of the wall is for example 10 meters, then we divide the height of the triangle into ten equal parts. We take a blank ruler and mark those ten marks on it. Then we take this ruler, with this length unit, and measure the hypothenuse of the triangle. However many marks long it is, that’s how many meters our ladder needs to be.

Piece of cake. Easy to improvise in the field without any specialised knowledge or tools. While Netz is busy trying to teach his carpenters the algebra of quadratic expressions and how to extract square roots, I have already scaled his walls using my much quicker methods. That is what you get when you put humanities people in charge of mathematics.

So I wouldn’t trust Netz when it comes to mathematics, even when he says “of course,” as he does here.

Here is another example: Did you know that parabolas are pointier than hyperbolas? At least if we are to believe Professor Netz. This claim occurs in a discussion of Archimedes. Archimedes studied solids of revolution obtained by rotating a conic section around its axis. Here are Netz’s words:

“In the case of a parabola, this will be of a more pointed shape; in the case of the hyperbola, this may be more bowl-like.” (140)

This is BS. Parabolas are not “more pointed” than hyperbolas.

This is clear for example from the following fact: you can draw a hyperbola having any two given lines as asymptotes and passing through any given point. So in other words, you can draw a V, an arbitrarily pointy letter V, and then pick an arbitrary point inside that V, for instance a point super close to the vertex of the V. Then there is always a hyperbola that fits inside the V and that passes through the designated point. You can hardly get any pointier than that, now can you? Yet parabolas are nevertheless “more pointed”, somehow, Netz apparently believes.

By the way, this fact I just mentioned, about constructing a hyperbola within a given V (that is to say, with given asymptotes), that is Proposition 4 of Book II of the Conics of Apollonius.

Or is it? Here we have another interesting point. It seems that this proposition was actually not in the original version of the Conics. Because Eutocius, in late antiquity, needs this theorem at a certain point and he says he better prove it since it’s not in the Conics of Apollonius. But then in the text we have of the Conics, what we call Apollonius’s Conics today, this proposition clearly is there, with the exact same proof.

And in fact the standard text that we call Apollonius’s Conics today comes to us only through that very same author, Eutocius, who wrote a commentary on the Conics and also preserved the text at the same time. So it seems that Eutocius inserted this proposition into Apollonius’s original text, because he had noticed in other works that it was a useful thing to prove.

Netz describes this correctly, which is all the more reason why he should know that a hyperbola can be as pointy as you’d like, since this follows immediately from this proposition that he discusses at length.

But anyway, there is another kind of error here in Netz’s discussion of this. The point that this proposition of the Conics is an insertion by Eutocius — that insight, says Netz, is due to Wilbur Knorr, Netz’s predecessor as a classics professor at Stanford.

“No one noticed that prior to Knorr” (431-432), says Netz.

But that is not true. Wilbur Knorr was not the first to discover this. In fact, Knorr clearly says so in his own article, the very article cited by Netz, which Netz has evidently not read very carefully. Already in the 16th century, Commandino, in his Latin edition of the Conics, very clearly and explicitly made the exact same point as Knorr, using the exact same evidence and arguments. And this in turn was cited in a 19th-century German edition of the Conics, as Knorr himself says. So Knorr didn’t discovery anything except what people had already known for hundreds of years.

This is not such an innocent mistake. How are we supped to trust anything Netz says if he makes blatantly false statements that are clearly and unequivocally seen to be factually incorrect by simply glancing at the very article that Netz himself cites in support of his own claims?

But it’s even more problematic than that. Because it’s clearly not just a random mistake. It is an ideologically driven error. By saying that Stanford humanities professor Wilbur Knorr was the first to make this important scholarly discovery, Netz is obviously indirectly boosting the impression that his own claims are important and novel, since he too is a Stanford humanities professor.

Netz is not only saying that Wilbur Knorr was the first to discover this particular thing. He is implicitly saying that earlier generations of scholars missed important insights, and that only people like him — Stanford humanities professors — are true experts.

That is of course the point of the title of the book: A *New* History of Greek Mathematics. In the past everybody did it wrong, and we need people like Netz to finally do it right. There is indeed a lot of explicit posturing to this effect throughout the book.

Let’s look at another example of this. Let me read a passage where Netz is attacking Thomas Kuhn’s account of the history of astronomy. Thomas Kuhn wrote in the mid-20th century and he worked on the history of science even though his PhD was in physics. So that is exactly the kind of people Netz wants to denigrate. He wants to say that only specialised humanities professors, with their “new” histories, are actual experts in the field.

Here is what Netz says about Kuhn: “Like most nonspecialists, Kuhn supposed …” See? I told you. It’s not just that Kuhn was wrong. It is that Kuhn epitomises the kind of people (people with a PhD in physics, for example) who need to be eliminated from the field because they make so many hopelessly naive assumptions without even realising it. Anyway, let’s continue with the quote:

“Like most nonspecialists,