The Greeks were wary of infinities of any kind, and nothing like an exact definition of 'ratio' was possible for them...
Irrationality Tamed
Bill Casselman
University of British Columbia
Not so long ago, the internet magazine Quanta posted an article titled "How the square root of two became a number". It's an interesting subject. The article covered it mostly with a discussion of Dedekind's construction of the positive real numbers by means of 'Dedekind cuts', but it's a topic that deserves more.
By his own admission, Dedekind was inspired by Euclid's treatment of irrational ratios in the Elements of Geometry. This treatment has caused much trouble over many years, and even now is not well understood. This is only natural, since the Greeks' views on magnitudes and ratios were so different from ours.
Origins
The beginning of the story is the beginning of mathematics, in about 1800 B.C.E. in the valley of the Tigris and Euphrates rivers. The following cuneiform tablet is YBC 7289 in the Yale Babylonian Collection.
The Babylonians expressed numbers in base 60, but had a floating and unspecified 'decimal' point. (They did not have a zero. Where we would use one, they just added space. Needless to say, this caused problems.) In the diagram, each side of the square is of length 30, presumably chosen somewhat arbitrarily. The length of the diagonal is expressed as 42, 25, 35, which is to be taken as $42 + 25/60 + 35/60^{2}$. It is obtained as the product of $30$ with $1 + 24/60 + 51/60^{2} + 10/60^{3}$, which is written along the diagonal. This is $1.41421296$ to 9 significant decimal figures, whereas the true 9-figure decimal approximation to the square root of 2 is $1.41421356$. (Very close!) The 30 is a length, expressed in unknown units, but the factor $1 \, 24\, 51\, 10$ is the ratio of lengths, and is dimensionless.
The point is that the Babylonians knew that the ratio of diagonal to side in a square was exactly the square root of $2$. In other words, they knew at least some form of Pythagoras' theorem, long before Pythagoras. Even more remarkably, they knew how to compute a good approximation to that ratio. You certainly can't get such accuracy with physical measurements.
Incommensurability
The next step in the story took place in Greece. First of all, Pythagoras' theorem was discovered, although we have only legend to link it to Pythagoras. It was even proved very early, not just stated, although we do not know exactly how the first reasoning went. It was also realized very early that an estimate, such as that produced by the Babylonians, could only be an approximation—as we would say, the square root of two is not a rational number.
How was the irrationality of the diagonal : side ratio formulated? Two line segments in the plane were said to be commensurable if each of them was an integral multiple of some common segment. We would say that in this case their ratio was a rational number, but that is not how Euclid would have put it.
Proposition.
The side and diagonal of a square are not commensurable.
I offer two proofs.
Even and odd
There is a simple argument, involving only even and odd integers, which goes by contradiction. Suppose the length of the side of a square is $q$ multiples of some smaller unit segment and that of the diagonal $p$ such multiples, with both $p$, $q$ positive integers. Choosing unit length suitably, we may assume at least one of these to be odd. But $p^{2} = 2q^{2}$, by the Pythagorean Theorem, so $p$ is certainly even, say $p=2n$. Then $2 q^{2} = 4n^{2}$, $q^{2} = 2 n^{2}$. So $q$ must also be even, a contradiction.
This reasoning made its way into the mainstream of classical Greek culture! The following excerpt is an illustration of proof by contradiction, taken from the end of part 23 in Aristotle's logic manual Prior analytics:
For all who effect an argument per impossibile infer syllogistically what is false, and prove the original conclusion hypothetically when something impossible results from the assumption of its contradictory; e.g. that the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate. One infers syllogistically that odd numbers come out equal to evens, and one proves hypothetically the incommensurability of the diagonal, since a falsehood results through contradicting this.
A geometric argument
I prefer a geometric proof. The basis of the argument is that if each of two line segments is an integral multiple of some one smaller segment, then so is their difference.
Let $s$ be the side of a square and $d$ its diagonal. Assume them to be commensurable—both being multiples of some smaller segment $\Delta$.
Euclid's I.20 (the triangle inequality) applied twice implies that $d/2 < s < d$, and in particular $d - s > 0$.
Lay out along its diagonal a segment of the same size as a side. Because of the previous remark, the segment's endpoint lies between the center and a corner of the square. The segment $s' = d-s$ will also be a multiple of $\Delta$.
Construct a line segment perpendicular to the diagonal at $P$. Let $Q$ be its intersection with the top of the square. On grounds of symmetry, the segment $PQ$ has length $s'$.
Lay off a segment at the upper left hand corner of the square, with endpoint $Q$. Again by symmetry, it also has length $s'$.
Construct a new square of side $s'$ on $PQ$. But because of I.20 again, as I have already remarked, $s' < s$. The diagonal of the new square will be $d'= s - s'$, commensurable with $s'$. If $s =m\cdot \Delta$, we'll have $s' = m'\cdot \Delta$ with $m' < m$. Continuing, we get an infinite decreasing sequence of positive integers $m > m' > m'' > \cdots$. Such a sequence is impossible. So $d$ and $s$ cannot be commensurable.
Consequences
What was the significance of incommensurability? What problems did it raise? There are two principal ones that come to mind:
- it made it very difficult to say exactly what a ratio was;
- it made the theory of similar figures much more complicated.
What is a ratio?
In common usage, the ratio of two things is a measure of their relative size. Thus, one thing is twice as large as another if their ratio is $2\colon 1$, and one thing is $50{\%}$ bigger than another if their ratio is $3\colon 2$. In modern terms, a ratio can be any positive real number, but the definition of real numbers didn't appear until a couple of millennia after Euclid's time, after decimal fractions (which first turned up in Europe during the Renaissance) had become a familiar tool. This meant that specifying a ratio involved specifying a potentially infinite decimal expansion.
The Greeks were wary of infinities of any kind, and nothing like an exact definition of 'ratio' was possible for them. This is a severe restriction. It meant that they could not refer to the length of an arbitrary line segment, or the area of an arbitrary planar region—because, after all, these are ratios to unit measures! It's much worse for planar regions than for line segments. A unit measure in that case is a unit square, and no proper parallelogram is the union of any finite number of squares. So Euclid does not say exactly what he means by 'ratio'—he doesn't know, except intuitively. All he says is the somewhat vague
Definition V.3.
A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
But now we ask, what is a magnitude? Again, it is not so clear. Something measurable, at least, something that has a size. Line segments and polygons qualify. But when are they are of the same kind?
Definition V.4.
Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
That is to say, magnitudes $a$ and $b$ have a ratio if there exist positive integers $m$, $n$ such that $ma > b$, $nb > a$—when they are mutually comparable. Two line segments have a ratio, but a line segment and a square do not. Implicit in the Elements is that we can add magnitudes, multiply them by a positive integer, and tell whether one is greater than another. In the case of lengths, congruent copies of a pair of lengths can be aligned, and one is less than another if its copy is strictly contained in the other. Addition or integer multiplication of lengths is done by laying segments end to end.
If we don't know what a ratio is, how can we say anything about them? Euclid's predecessor Eudoxus is credited with the brilliant observation that even if we don't know exactly what a ratio is, we can compare two of them—we can tell whether two are equal or not..
In fact, Eudoxus' observation is consistent with the fact that throughout all of the Elements, Euclid's statements about magnitudes and ratios are comparisons. For example, he never states that the area of a parallelogram is the product of its base and height, but just that two parallelograms lying between two parallel lines and on the same base have the same area in a very particular sense—they may be dissected, possibly with a negative sign, into congruent pieces. For us modern mathematicians, who define area as a limit, it is a theorem that congruent shapes have the same area and that area is additive, but for Euclid it is self-evident.
Anyway, here is Eudoxus' criterion for comparison:
Definition V.5.
Magnitudes are in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
Hard to digest. We can at least put into something a bit clearer. Suppose we are given magnitudes $a$, $b$ of the same kind as well as $c$, $d$ also of the same kind. Then to see that $a\colon b = c\colon d$ means that we must verify that for any pair of positive integers $m$, $n$ we have
$$ \eqalign { \hbox { if } &ma > nb \hbox { then } mc > nd \cr \hbox { if } &ma = nb \hbox { then } mc = nd \cr \hbox { if } &ma < nb \hbox { then } mc < nd \, . \cr} $$
On the face of it, it seems that there are an infinite number of things to check, but in practice that is not so, as we shall see soon.
We can at least translate this into something more comprehensible to us. Given any real number $x$ (equal to $a/b$ here), we can partition the positive rationals $n/m$ into three classes: those $< x$, $= x$, $> x$. Two real numbers $x$ and $y$ are equal if the corresponding partitions are the same.
We'll see shortly how to use this criterion.
Similarity
Similarity is a basic feature of plane geometry, strongly related to the notion of ratio. It should not be a surprise that incommensurability causes trouble here, too. The following, which is the first half of Euclid's Proposition VI.2, is one of the basic results in Euclid concerning similarity.
Proposition VI.2.a.
If a straight line be drawn parallel to one of the sides of a triangle, it will cut the side of the triangle proportionally.
That is to say, in the following diagram the ratio of $a$ to $b$ is the same as that of $c$ to $d$.
If $a$ and $b$ are commensurable, this is almost trivial, since results from Book I of the Elements tell us that all the small triangles in the following figure are congruent. So if the left side is divided into $N$ equal segments, so is the right side.
But if $a$ and $b$ are not commensurable, this doesn't work. In modern times, we could take a limit, but that is not an option for Euclid. Limits as we know them did not appear before the introduction of decimal fractions, and did not become rigorous mathematics until the mid nineteenth century, in a development initiated by Cauchy. Instead, Euclid bases his proof on Proposition VI.1, which in turn applies Eudoxus' criterion.
Proposition VI.1.
Triangles and parallelograms which are under the same height are to one another as are their bases.
What do "under the some height" and "are to one another" mean? In modern terminology, we would say that the ratio of the areas of two triangles of the same height is the same as the ratio of the lengths of their bases. And for us, the assertion is trivial. The area of a parallelogram is equal to the product of base and height: $A = bh$. Under the hypotheses of the Proposition, we have $A_{1} = b_{1}h$, $A_{2} = b_{2}h$. Hence
$$ A_{1}/A_{2} = b_{1}/b_{2}. $$
But this line of reasoning relies on the technology that evolved in 2,000 years of mathematics since Euclid.
How can Proposition VI.1 can be applied to prove VI.2?
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Apply VI.1 to each of the triangles above. It says, for example, that the ratio of $a$ to $b$ is equal to the ratio of the area of $A$ to that of $\Xi$. A similar relationship holds for the ratios of $c$, $d$ and $C$, $\Xi$.
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But as the diagrams above illustrate, the triangles $A$ and $C$ have the same bases and equal heights.Hence, by Euclid's Proposition I.41 and in Euclid's terminology, they are equal. Thus the ratio of $a$ to $b$ is equal to that of $c$ to $d$. In brief:
$$ a \, \colon b = A \, \colon \Xi = C \, \colon \Xi = c \,\colon d \, . $$
Ratios
Now for the proof of VI.1, which directly applies Eudoxus' criterion. I'll begin with a simple observation:
If $A$ and $B$ are triangles with the same height, then $A$ is equal to (greater than, less than) $B$—i.e., their areas are the same (greater than, etc.)—if and only if the base of $A$ is equal to (greater than, etc.) that of $B$.
The proof: suppose we are given triangles $A$, $B$. We want to show that the ratio of$A$ to $B$ is equal to that of their bases $a$, $b$. Following the definition V.5,we must show that if (for example) $mA > nB$ then $ma > nb$. But that's obvious, since $ma$ is the base of $mA$,$mb$ that of $mB$.
Dedekind
In the course of the 2,000 years following Euclid, the situation changed drastically. At some point before the current era, Babylonian astronomers had developed remarkable tools for computing astronomical phenomena (such as eclipses), and (of course) in base $60$. Greek astronomers adopted these. Sometime around 500 C.E., Indian astronomers also adopted Babylonian tools, but introduced decimal place value notation for integers, and the arithmetic of rational fractions (including a true '0'). These techniques were transmitted to Islamic scientists, who added decimal fractions to the tool kit. European scientists adopted all this in turn, so that by about 1600 notation for calculation was pretty much what it is now. With the new tools came new applications, eventually leading to methods for dealing casually, and sometimes incorrectly, with convergence and limits. With Cauchy, the paradox became most evident: he formulated an invaluable criterion for the convergence of a sequence, but relied on geometrical intuition—the apparent continuity of a straight line—to justify it.
Richard Dedekind took up a position teaching calculus in Zürich in the fall of 1858, and in preparing his lectures found himself unhappy with the justification of the simple assertion that a bounded increasing sequence of numbers had a limit. What he did was to define the positive real numbers in terms of what are now called 'cuts' in English: a partition of the set of all positive fractions into two non-empty sets, say $A$ and $B$, with the property that every element of $A$ is less than every element of $B$. According to this, for example, to $\sqrt{2}$ is assigned the set $A$ of all fractions $p/q$ with $p^{2}/q^{2} \le 2$, and $B$ its complement. Similarly, an increasing sequence $(x_{n})$ bounded from above converges to the real number whose set $A$ is that of all rational $x$ such that $x$ is less than some $x_{n}$. Dedekind claimed—with justification!—that he was the very first person to prove that $\sqrt{2}$ times $\sqrt{3}$ is equal to $\sqrt{6}$.
Curiously, at just about the same time Georg Cantor defined the real numbers to be the set of all Cauchy sequences, with the proviso that two sequences $(a_{n})$ and $(b_{n})$ represent the same real number if $(a_{n}-b_{n})$ converges to $0$. Defining things as sets was involved in both constructions, and initiated a completely new role for logic in mathematics.
Reading further
- How the square root of two became a number
- Aristotle's Prior Analytics
- Sir Thomas Heath's English edition of Euclid's Elements of Geometry
- Cauchy's Cours d'analyse
- Judith Grabiner, The origins of Cauchy's rigorous calculus, MIT Press, 1981.
- Dedekind on the nature of numbers