Welcome back to Numberholic! I’m your host, Sejari, and it’s always a pleasure to have you here. Today, we’re diving into the eighth episode of Proven Yet Hard to Believe, a series where we uncover the fascinating stories and directly explore the core ideas behind mathematical discoveries that have stood the test of time. This time, we’re once again delving into the brilliance of ancient Greek mathematics—specifically, a method that laid the foundation for integral calculus long before its formal development.
Are you ready to explore the genius of the past? Let’s get started!
Today’s topic is the method of exhaustion, which we have discussed several times before. Many may find this term unfamiliar, as it is not commonly heard. However, understanding this method is a key to comprehending the core of ancient Greek mathematics. Moreover, when interpreted in a modern sense, it provides insight into how calculus, the foundation of modern mathematics and science, is understood and why establishing its foundations was not an easy task. From the perspective of mathematical history, this makes the method of exhaustion highly significant.
Let us briefly introduce the method of exhaustion. The term itself means “a method of depletion,” which suggests that when attempting to measure a certain quantity precisely, if there is always a remaining amount no matter how finely it is measured, the method entails progressively reducing this remainder to an infinitely small value.
Since this is a complex topic, it is challenging to present it clearly. However, there is another important concept that must be discussed, which will be introduced alongside it. This concept is “proportion.”
The most concise condition for the proportion a:b = c:d is, as well known, ad = bc. In other words, the product of the outer terms is equal to the product of the inner terms. While this may seem simple, this relationship could only be naturally expressed during the time of the Persian mathematician Omar Khayyam in the 11th century.
Did the Greeks think of it this way? Surprisingly, in Proposition 19 of Book 7 of Euclid’s Elements, it is stated that if four numbers are in proportion, the product of the first and fourth is equal to the product of the second and third. Conversely, if the product of the first and fourth equals the product of the second and third, then the four numbers are in proportion. This means that the principle “the product of the outer terms equals the product of the inner terms” was already recognized.
However, the historical reality is more complicated due to the existence of irrational numbers. According to Greek mathematical expressions, some quantities could not be measured by a common unit.
The Pythagorean school had a specific view of proportion. For example, is the ratio 4:6 equal to 6:9?
In the first ratio, using the common unit of 2, the terms can be measured as 2 and 3.
In the second ratio, using the common unit of 3, the terms can be measured as 2 and 3.
Since both ratios reduce to the same values, they are considered equal.
The Pythagoreans held proportion in high regard. They believed that “All is Number.” By measuring numbers with a single fundamental unit, they established the cosmic order and the harmony of all things. However, their theory of proportion had a weakness—the existence of irrational numbers, or quantities that cannot be measured by a common unit. The discovery of irrational numbers caused confusion in the concept of number and proportion.
To modern people, the concept of real numbers is natural, making it difficult to understand the struggles of ancient Greek mathematicians. Imagine their perspective: different types of magnitudes existed, but could they be unified under the higher concept of “number”? This was not an easy question.
A satisfactory definition of proportion came from Eudoxus, who was regarded as the greatest mathematical genius in Greece before Archimedes. Eudoxus boldly shifted Greek mathematics from a number-centric approach to a geometric one. His decision, along with his brilliance, propelled Greek mathematics toward axiomatic geometry. However, the development of number theory, including algebra, had to wait another 2,000 years.
So what is the complete definition of equality of ratios? It is given in the fifth definition of Book 5 of Euclid’s Elements. It is believed that nearly all of Book 5 consists of Eudoxus’ ideas.
Given four magnitudes, the statement that the ratio of the first to the second is equal to the ratio of the third to the fourth means that:
If we multiply the first and third by the same number and take their multiples, and multiply the second and fourth by another number and take their multiples,
Then, the multiples of the first and third will be greater than, equal to, or less than the multiples of the second and fourth simultaneously.
In modern terms, a:b = c:d means that for any natural numbers m and n:
If ma > nb, then mc > nd.
If ma = nb, then mc = nd.
If ma < nb, then mc < nd.
This is quite complex. But why was this necessary? Suppose a and b can be measured by a common unit, and c and d can also be measured by a different common unit, but a, b, c, and d together cannot be measured by the same unit. How can we compare the two ratios? By simply stating that the product of the outer terms equals the product of the inner terms? Such a claim is impossible. Eudoxus’ unique definition of proportion solved this difficulty in proportion theory.
Now, let’s examine the theory of proportion combined with the method of exhaustion. This is the proof of Proposition 2 in Book 12 of The Elements, which states that the area of a circle is proportional to the square of its diameter.
“The ratio of the areas of similar polygons is equal to the square of their similarity ratio.”
This statement is easy to prove. The real question, however, is whether this claim can also be applied to circles. In other words, does the same principle hold for the areas of circles?
The key question is how the logic used for polygons can be applied to circles. The reason this is a challenge is that, unlike polygons, the area of a circle cannot be measured using a common unit, as it does not have a shared measure with polygonal areas.
How did Euclid address this?
Let’s follow Euclid’s explanation.
Consider two circles. First, assume that the ratio of their areas is not equal to the ratio of the squares of their diameters. Now, using one of the circles (Circle 1) as a reference, construct a third circle (Circle 3) so that its area follows the ratio of the squares of the diameters.
This newly constructed Circle 3 will either be larger or smaller than Circle 2. Let’s consider the case where it is smaller, since the reasoning remains the same even if it is larger.
Now, think about the difference in area between Circle 2 and Circle 3. Let’s call this difference D.
At this point, the key theorem of the method of exhaustion comes into play:
“If more than half of a given magnitude is removed, and then more than half of the remainder is removed, and this process continues, eventually, the remaining amount will be smaller than any predetermined small quantity.” (The Elements, Book 10, Proposition 1)
Now, inscribe a square inside Circle 2. Let the difference in area between this square and Circle 2 be D1. If D1 is greater than D, inscribe an octagon inside the circle.
What happens to the difference in area D2 between the octagon and Circle 2? Since the octagon removes more than half of the remaining difference, D2 is even smaller than D1.
Looking at the next figure, this becomes clear.
We then compare D2 with D.
By continuing this process and increasing the number of sides, we eventually reach a point where Dn becomes smaller than D.
For convenience, let’s assume that D2 is already smaller than D. Since the difference in area between the octagon and Circle 2 is smaller than the difference between Circle 2 and Circle 3, it follows that the area of the octagon must be greater than the area of Circle 3.
But now we have a problem.
Next, we inscribe a polygon inside Circle 1 that is similar to the octagon inside Circle 2. Since these are both polygons, the ratio of their areas must exactly match the ratio of the squares of their diameters. That is, the area ratio of Circle 1 to Circle 3.
However, since the inscribed octagon in Circle 1 is smaller than Circle 1 itself, the octagon inside Circle 2 must also be smaller than Circle 3.
This creates a contradiction.
Thus, by proof by contradiction, we establish that the ratio of the areas of circles is indeed equal to the ratio of the squares of their diameters.
By combining the method of exhaustion with the rigorous definition of proportion, Greek mathematicians developed an incredibly precise method of proving the properties of geometric figures.
The method of exhaustion showcases how ancient thinkers sought to overcome limitations in mathematical understanding with remarkable ingenuity. Their work continues to inspire mathematicians today, reminding us that the pursuit of knowledge is a journey spanning centuries.
That wraps up today’s episode! Thank you for joining me on this deep dive into mathematical history. If you enjoyed this discussion, don’t forget to like, subscribe, and share with fellow math enthusiasts. Until next time, keep questioning, keep exploring, and stay curious!
See you in the next episode of Proven Yet Hard to Believe!