Problem 4 - A History of Mathematics: An Introduction

Question

Nicomachus defined a subcontrary proportion, which occurs when in three terms the greatest is to the smallest as the difference of the smaller terms is to the difference of the greater. Show that \(3,5,6\), are in the subcontrary proportion. Find two other sets of three terms that are in subcontrary proportion.

Step-by-Step Solution

Answer

If not, find two sets of terms that are. Answer: No, the given terms 3, 5, and 6 are not in subcontrary proportion. However, two sets of terms that follow the subcontrary proportion according to Nicomachus's definition are: (10/3, 5, 4) and (21/4, 7, 6).

Step 1: Verify the given terms are in subcontrary proportion

Plug the given terms, \(3, 5, 6\), into the definition of subcontrary proportion: \(\frac{6}{3} = \frac{6 - 5}{5 - 3}\). Simplify each side: \(2 = \frac{1}{2}\). This is not true, so the given terms \(3, 5, 6\) are not in subcontrary proportion.

Step 2: Find the first set of three terms in subcontrary proportion

To find the first set of numbers in subcontrary proportion, let's choose two small consecutive numbers, such as \(5\) and \(4\) as the smaller terms. Now, we will find the greatest term that satisfies the proportion: \(\frac{a}{5} = \frac{a - 4}{4 - 5}\). Solve for \(a\): \(a = 5(\frac{a - 4}{-1})\) \(a = -5(a - 4)\), \(a = -5a + 20\). Move the terms containing \(a\) to one side of the equation: \(6a = 20\). Now, divide both sides by 6: \(a = \frac{20}{6} = \frac{10}{3}\). So, the first set of three terms in subcontrary proportion is \(\left(\frac{10}{3}, 5, 4\right)\).

Step 3: Find the second set of three terms in subcontrary proportion

To find the second set of numbers in subcontrary proportion, let's choose two different consecutive numbers, such as \(7\) and \(6\) as the smaller terms. Now, we will find the greatest term that satisfies the proportion: \(\frac{a}{7} = \frac{a - 6}{6 - 7}\). Solve for \(a\): \(a = 7(\frac{a - 6}{-1})\) \(a = -7(a - 6)\), \(a = -7a + 42\). Move the terms containing \(a\) to one side of the equation: \(8a = 42\). Now, divide both sides by 8: \(a = \frac{42}{8} = \frac{21}{4}\). So, the second set of three terms in subcontrary proportion is \(\left(\frac{21}{4}, 7, 6\right)\). Now, we have found the required sets of three terms in subcontrary proportion: \(\left(\frac{10}{3}, 5, 4\right)\) and \(\left(\frac{21}{4}, 7, 6\right)\).

Key Concepts

Nicomachus

Nicomachus was an ancient mathematician and philosopher known for his work in number theory. He is frequently credited with exploring mathematical proportions and relationships in numbers. One of his significant contributions was defining what he called a "subcontrary proportion." This is a particular relationship in three terms where the largest term relates to the smallest in the same way as the difference between the smaller terms does to the difference between the greater terms.
Nicomachus's work on these concepts was foundational in early mathematics, where the study of numbers and their properties began to take shape. He helped lay the groundwork for the systematic approach to understanding numbers that we use today in various fields of mathematics.

Mathematical Proportion

Mathematical proportion is a concept where two ratios or fractions are equal. It involves the mutual relationship between numbers, often serving as a tool for understanding how quantities compare.

Here's a quick breakdown:

• A proportion is expressed as \( \frac{a}{b} = \frac{c}{d} \).
• This means that the ratio of \(a\) to \(b\) is the same as the ratio of \(c\) to \(d\).

In the context of Nicomachus's subcontrary proportion, this equality is used distinctly by involving differences between terms. The largest and smallest numbers have a specific relationship mirrored by the differences of the two smaller and two larger terms. Proportions like these illustrate the elegance and symmetry possible in mathematical relationships.

Number Theory

Number theory is a branch of mathematics devoted to studying integers and integer-valued functions. It concerns itself with the properties and relationships of numbers, especially the integers. Nicomachus's exploration into subcontrary proportion falls under this category as it deals with the fundamental properties of numbers, specifically their divisibility and relational patterns.
Number theory includes various subfields like:

• Divisibility
• Prime numbers
• Congruences and modular arithmetic

Each of these contributes to understanding not only the basic units of mathematics but also more complex phenomena. Scholars like Nicomachus were some of the first to consider the deeper implications of mathematical rules and relationships, which continues to be a rich area of research and application in modern science and mathematics.