{\huge Did Euclid Really Mean `Random'?}

Euclid writes randomness into his Elements \vspace{.25in}

Euclid is, of course, the Greek mathematician, who is often referred to as the "Father of Geometry."

Today I want to talk about an ``error'' that appears in the famous volumes written by Euclid a few years ago---about 2300 years ago.

The ``error'' is his use of the word ``random'' when by modern standards he should be saying ``arbitrary.'' I find this surprising, since I think of random as a modern concept; and I find it also surprising, since the two notions are not in general equivalent.

It seems clear that Euclid said `random.' The root-word he used, tuchaios, endures as the principal word for ``random'' in modern Greek and is different from words meaning ``arbitrary'' or ``generic'' or ``haphazard'' or even ``stochastic.'' The only meaning of tuchaios we've found that would be strictly correct in Euclid's statement is, ``it is unimportant which.'' However, the way Euclid's exact phrase was put in the voice of Pope Clement I, seems not to square with that meaning either.

Euclid and Random

I never studied the Elements, Euclid's famous collection of thirteen books on geometry. Not that long ago, every school used the Elements as the textbook for the introduction of mathematics. Abraham Lincoln is said to have studied the Elements until he could recite it perfectly. I never looked at any part of it until I came across Book II while doing some research. And I was quite surprised to see the use of the notion of ``random'' there in the text.

Book II is focused on a geometric approach to identities, which are much easier to understand as algebraic identities today. The proposition that caught my eye is Proposition II.4.

Proposition 1 If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.

Perhaps this is easy to understand as a geometric statement. Today we would write algebraically that it stands for the identity

$latex \displaystyle (a+b)^{2} = a^{2}+b^{2}+2ab. &fg=000000$

Why did Euclid state it geometrically? Perhaps the main advantage was that it allowed him to reason directly about geometric objects. After all he was writing about geometry, so a square with sides of length $latex {a}&fg=000000$ stood in nicely for the term $latex {a^{2}}&fg=000000$. An even better reason might have been his lack of modern algebraic notation---the equals symbol was invented by Robert Recorde a few years after Euclid, in 1557.

Here is the proof as it appears in the Elements. The length compared to simply expanding $latex {(a+b)^2}&fg=000000$ shows the power of modern notation:

\includegraphics[width=5.5in]{proofprop4.png}

The Issue

What surprised me was the exact statement of Proposition II.4. Note that it starts If a straight line be cut at random $latex {\dots}&fg=000000$

In online Greek editions such as this the phrase hos etuchen meaning ``at random'' is set off with commas. Euclid reiterates this phrase in the first line of his proof. However, the result is actually true for any cut of the line, which is more than saying ``at random.'' So why does Euclid say ``random''?

Euclid seems nowhere to define in any precise way what ``random'' means in this context. Recall that one of the great achievements of the Elements was its claim to be a precise and axiomatic approach to geometry. But using an undefined term like ``random'' seems to run overtly counter to that goal.

Looking for other usages doesn't clearly let Euclid---or his main ancient editor, Theon of Alexandria---off the hook. Pope Clement I was St. Peter's first, second, or third successor. A novelized creation of his acts and homilies has him using the same Greek phrase at the beginning of Book 1, chapter IV:

Our Peter has strictly and becomingly charged us concerning the establishing of the truth, that we should not communicate the books of his preachings, which have been sent to us, to any one at random, but to one who is good and religious, and who wishes to teach...

This aligns with the modern meaning: the writer was saying that most people would be unqualified to preach Peter's sermons. It does not mean that all people would be bad or that it is unimportant who receives the books. There is support in other examples for the reading, ``to anyone you happen to meet,'' but even then the inference again is one of ``mostness'' not ``all.'' In any event, Euclid's proposition is correct with ``all.''

The difference is not a quibble. It is easy to make statements in Euclidean geometry that are true for ``random'' but not for ``all'': A random triple of points in the plane makes a triangle. A random line through a point outside a circle is not tangent to the circle.

However, there are also cases where holding for ``random'' is sufficient for holding for ``all.'' Equalities like Euclid's $latex {(a+b)^{2} = a^{2}+b^{2}+2ab}&fg=000000$ have that property. So does the Schwartz-Zippel lemma: if $latex {p}&fg=000000$ is a polynomial and $latex {p(\vec{x}) = 0}&fg=000000$ for a random $latex {\vec{x}}&fg=000000$ in a large enough field then $latex {p}&fg=000000$ is the zero polynomial. In fact Euclid's identity is a case of this---could we add Euclid as sharing credit for the lemma?

This led Ken and me to think about a problem: Can we make something out of Euclid's use of randomness?

Random vs Generic vs Arbitrary

There is a third concept lurking here: generic. Generic usually implies random but is more special and does not require probability or (Lebesgue) measure. In fact it basically means ``not special.'' Three collinear points are special; a line tangent to a sphere is special.

The exact notion of ``generic'' is context-dependent, but at the interface of geometry and algebra we can pin it down: a set of elementary objects (points, lines, etc.) is special if it satisfies some finite set of simple arithmetical equations and is generic otherwise. Collinear points and tangent lines are clearly special in this sense. More formally, the special sets are those closed in the Zariski topology, apart from the whole (Euclidean) space. So now we ask:

Could Euclid have been in any sense aware of the idea of genericity?

If so, then Euclid could have been led into deep waters. Consider just a line segment going from 0 to 2. The midpoint $latex {x = 1}&fg=000000$ is special because it satisfies the equation $latex {x+x = 2}&fg=000000$. Similarly so are the points $latex {x = \frac{1}{2}}&fg=000000$ and $latex {x = \frac{3}{2}}&fg=000000$. It quickly follows that all rational numbers are special. Now so is $latex {x = \sqrt{2}}&fg=000000$ since it satisfies $latex {x*x = 2}&fg=000000$. And likewise $latex {\sqrt[3]{2}}&fg=000000$, so all points for algebraic numbers are special too. Euclid would certainly have suspected that those points might not be constructible. So although he might have suspected that a ``random'' point was not constructible, he would have had a hard time realizing that the non-constructible points have a special subset too.

Of course, it took until the 1600s to articulate modern meanings of ``random'' and until the 1800s for Georg Cantor and the topological notions underlying genericity to arise. It still interests us what ``hints'' might have been perceived in the intervening centuries.

A `Random' Road to Geometry?

Alfred Tarski, the famous 20th-century logician, created formal axioms for geometry. His axioms modeled that part of geometry that is called "elementary." This includes statements of plane geometry that can be stated in first-order logic and only refer to individual points and lines: arbitrary sets are not allowed. The above reference has the details of his axioms. They were built on two primitive notions:

Tarski proved that this theory is decidable. And actually it has a remarkable property: any statement in the theory is equivalent to a sentence $latex {S}&fg=000000$ that is in universal-existential form, a special case of prenex normal form. In this form all universal quantifiers precede any existential quantifiers:

$latex \displaystyle S = \forall u \forall v \ldots\exists a \exists b\ldots (\cdots)&fg=000000$

This form is close to just having equations, so it is tantalizing to ask, given any formula $latex {\phi}&fg=000000$, does either $latex {\phi(\vec{x})}&fg=000000$ or $latex {\neg\phi(\vec{x})}&fg=000000$ hold for generic $latex {\vec{x}}&fg=000000$? Or for some notion of ``random'' $latex {\vec{x}}&fg=000000$? The basic $latex {B}&fg=000000$ and $latex {C}&fg=000000$ formulas have this property: their negations hold generically---even though they do not hold for all $latex {\vec{x}}&fg=000000$.

However, the following seems to be a valid counterexample to any kind of ``zero-one law'' holding here: Tarski's system can define $latex {Axyz}&fg=000000$ to mean that the points $latex {x,y,z}&fg=000000$ are the vertices of an acute triangle. Now fix $latex {x = (0,0)}&fg=000000$ and $latex {z = (2,0)}&fg=000000$ in the plane. Then any $latex {y = (a,b)}&fg=000000$ satisfies $latex {Axyz}&fg=000000$ if and only if $latex {0 < a < 2}&fg=000000$ and $latex {b \neq 0}&fg=000000$. Neither the set of such $latex {y}&fg=000000$ nor its complement is a nullset.

This is curious because if one regards Euclid-type diagrams as finite structures like graphs, then the first-order zero-one law by Ronald Fagin comes into play: as $latex {n\rightarrow\infty}&fg=000000$ the proportion of size-$latex {n}&fg=000000$ structures that satisfy a given first-order sentence (pure: no parity or counting) goes either to 0 or 1. Still we can ask two questions:

So what we are asking is, exactly when does Euclid's use of ``random'' for ``arbitrary'' remain correct? Which geometric statements are guaranteed either to hold or to fail for ``random'' arguments?

Open Problems

Do our questions have nice and simple answers? Are we the first to wonder how Euclid's words fare when given a modern mathematical interpretation?