Problem 3 - A History of Mathematics: An Introduction

Question

An alternative method by which Archimedes could have solved the crown problem is given by Vitruvius in \(\mathrm{On}\) Architecture. Assume as in the text that the crown is of weight \(W\), composed of weights \(w_{1}\) and \(w_{2}\) of gold and silver, respectively. Assume that the crown displaces a certain quantity of fluid, \(V\). Furthermore, suppose that a weight \(W\) of gold displaces a volume \(V_{1}\) of fluid, while a weight \(W\) of silver displaces a volume \(V_{2}\) of fluid. Show that \(V=\frac{w_{1}}{W} V_{1}+\frac{w_{2}}{W} V_{2}\) and therefore that \(\frac{w_{1}}{w_{2}}=\frac{V_{2}-V_{1}}{V-V_{1}}\). Fig. 4.7)

Step-by-Step Solution

Answer

Question: Given a crown of weight \(W\) made of gold and silver with weights \(w_{1}\) and \(w_{2}\) respectively, show that the volume \(V\) can be represented as \(V=\frac{w_{1}}{W} V_{1}+\frac{w_{2}}{W} V_{2}\), where \(V_{1}\) and \(V_{2}\) are the volumes of gold and silver when they have the same weight \(W\). Also, derive the relationship between \(w_{1}\) and \(w_{2}\) as \(\frac{w_{1}}{w_{2}}=\frac{V_{2}-V_{1}}{V-V_{1}}\). Answer: We showed that the volume of the crown, \(V\), can be represented as \(V=\frac{w_{1}}{W} V_{1}+\frac{w_{2}}{W} V_{2}\) by adding the volumes of gold and silver components. We also derived the relationship between \(w_{1}\) and \(w_{2}\) as \(\frac{w_{1}}{w_{2}}=\frac{V_{2}-V_{1}}{V-V_{1}}\) by solving for the ratio of gold and silver weights and equating the expressions for \(\frac{V_{gold}}{V_{silver}}\).

Step 1: Represent gold and silver in crown by their volumes

As we know that the crown is made up of gold and silver with weights \(w_{1}\) and \(w_{2}\), respectively, we can represent their volumes as: $$ V_{gold}=\frac{w_{1}}{W} V_{1} $$ and $$ V_{silver}=\frac{w_{2}}{W} V_{2} $$

Step 2: Add the volumes of gold and silver components

To find the total volume of fluid displaced by the crown, we need to add the volumes of gold and silver components. $$ V = V_{gold} + V_{silver} $$ Plugging in the expressions for \(V_{gold}\) and \(V_{silver}\) found in step 1, we get: $$ V = \frac{w_{1}}{W} V_{1}+\frac{w_{2}}{W} V_{2} $$

Step 3: Find the ratio of gold and silver weights

After finding the relationship between \(V\), \(w_1\), and \(w_2\), we can now derive a ratio \(\frac{w_{1}}{w_{2}}\). We will begin by finding \(\frac{V_{gold}}{V_{silver}}\): $$ \frac{V_{gold}}{V_{silver}} = \frac{\frac{w_{1}}{W} V_{1}}{\frac{w_{2}}{W} V_{2}} $$ Simplifying the above expression, we get: $$ \frac{V_{gold}}{V_{silver}} = \frac{w_{1} V_{2}}{w_{2} V_{1}} $$ Now, we can rewrite the ratio \(\frac{V_{gold}}{V_{silver}}\) as: $$ \frac{V_{gold}}{V_{silver}}=\frac{V_{2}-V_{1}}{V-V_{1}} $$

Step 4: Solve for \(\frac{w_{1}}{w_{2}}\)

Now we can equate the expressions for \(\frac{V_{gold}}{V_{silver}}\) from Step 3: $$ \frac{w_{1} V_{2}}{w_{2} V_{1}}=\frac{V_{2}-V_{1}}{V-V_{1}} $$ Cross multiplying, we have: $$ w_{1}(V_{2}-V_{1})=w_{2}(V-V_{1}) $$ Dividing both sides by \(w_{2}(V_{2}-V_{1})\), we get the desired relationship: $$ \frac{w_{1}}{w_{2}}=\frac{V_{2}-V_{1}}{V-V_{1}} $$ Thus, we have shown the required relationship between \(w_{1}\) and \(w_{2}\).

Key Concepts

Fluid Displacement

Imagine submerging an object into water; it pushes the water aside, occupying space that was filled with water. This phenomenon is known as fluid displacement. Archimedes' Principle states that the volume of fluid displaced by an immersed object is equal to the volume of the object itself.

In the context of the Archimedes crown problem, fluid displacement helps to determine if the crown is made of pure gold or if it has been mixed with silver. The logic here is straightforward: gold and silver have different densities and thus displace different volumes of water, despite having the same weight.

Analysing Fluid Displacement

When various materials like gold and silver are formed into a crown and submerged, the total volume of water displaced, denoted by V, must equal the sum of the volumes displaced by the individual materials making up the crown. Mathematically, if a weight W of gold displaces a volume V1, and the same weight of silver displaces a larger volume V2 (because silver is less dense than gold), the total volume displaced by the crown V can be represented as a sum of the parts proportional to their weights in the crown, w1 and w2. The equation V = (w1/W) * V1 + (w2/W) * V2 thus reflects the concept of fluid displacement applied to a composite object.

Density and Buoyancy

The concepts of density and buoyancy are central to understanding how objects behave when submerged in a fluid. Density is a measure of how much mass is contained in a given volume. An object that is denser than the fluid will sink, while an object that is less dense will float.

Buoyancy Explained

Buoyancy is a force exerted by the fluid, which resists the weight of an immersed object. According to Archimedes' Principle, the buoyant force on an object is equal to the weight of the fluid it displaces. If the weight of the displaced fluid is greater than the weight of the object, the object will float. If not, the object will sink.

When applying this principle to the crown problem, Archimedes could analyze the crown's buoyancy by comparing the weight of the crown to the weight of the fluid displaced. Using the proportionality relation between volumes displaced by gold and silver, Archimedes could unveil any discrepancies that suggest the presence of a less dense metal like silver instead of pure gold, affecting both the density and buoyancy of the crown.

Historical Mathematics

The exploration of Archimedes' Principle is not merely a lesson in physics but also a glimpse into historical mathematics. Archimedes of Syracuse (c. 287 – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer who influenced the world with his discoveries.

His principles laid the foundation for hydrostatics, the branch of physics that deals with fluids at rest. These principles are not only important because of their applications in fluid mechanics but also because they showcase the application of mathematical reasoning to solve real-life problems.

Archimedes' Legacy

Archimedes' contribution to mathematics extends beyond principles; his work includes techniques for calculating areas and volumes, anticipating integral calculus. The story of his role in determining the composition of Hiero's crown by using water displacement represents one of the earliest known examples of applying scientific principles to investigate a practical concern. The blend of theory and application found in Archimedes' work symbolizes the essence of mathematical investigation and its potential to solve complex problems.