Question

Ab? Sahl al-K?hì knew from his own work on centers of gravity and the work of his predecessors that the center of gravity divides the axis of certain plane and solid figures in the following ratios: Tetrahedron: \(\frac{1}{4}\) Segment of a parabola: \(\frac{2}{3} \quad\) Paraboloid of revolution: \(\frac{2}{6}\) Hemisphere: \(\frac{3}{8}\) Noting the pattern, he guessed that the corresponding value for a semicircle was \(3 / 7\). Show that al-K?hi's first five results are correct, but that his guess for the semicircle\\} implies that \(\pi=31 / 9\). (Al-K?h? realized that this value contradicted Archimedes' bounds of \(310 / 71\) and \(31 / 7\), but concluded that there was an error in the transmission of Archimedes' work.)

Short answer

Based on the detailed analysis of the center of gravity of tetrahedron, segment of a parabola, paraboloid of revolution, and hemisphere, their ratios to the total height were verified as 1/4, 2/3, 2/3, and 3/5, respectively. By assuming al-Kāhī's guess for the semicircle, we found that it implies that the value of Pi would be 31/9. Although this value doesn't match Archimedes' bounds for the value of Pi, it shows al-Kāhī's attempt to explore the center of mass ratio for the semicircle.

Step-by-step solution

Tetrahedron

Consider a regular tetrahedron with edge length \(a\). The centroid of a triangular face divides the median in a \(\frac{1}{3}:\frac{2}{3}\) ratio, with the centroid being \(\frac{2}{3}\) from the vertex. Now, connecting the centroids of the faces and the vertices, we get the centroid of the tetrahedron. It will also divide the line connecting the vertex and the face centroid in \(\frac{1}{3}:\frac{2}{3}\) ratio. It follows that the centroid is one-fourth of the height of the tetrahedron from its base, confirming the given \(\frac{1}{4}\) ratio.

Segment of a parabola

We can consider the segment of a parabola defined as \(y = x^2\) between \(x=0\) and \(x=1\). To find the coordinates of the center of mass, we compute the moment of the area about the x and y axes. By symmetry, the center of mass will lie on the line \(x=\frac{1}{2}\). Thus, we need to find \(y\). Let's find the mass of the infinitesimally thin horizontal strip. Its length is \(2x\) and the mass is \(2x*dx\). As a result, \(M = \int_0^1 2x * dx = 1\). The moment about the y-axis is \(\int_0^1 2x^2 * dx = \frac{2}{3}\). Therefore, the center of mass coordinates are \((\frac{1}{2}, \frac{2}{3})\), which verifies the given ratio for the segment of a parabola.

Paraboloid of revolution

Consider the parabolic solid formed by rotating y = x^2 for x between 0 and 1 around the y-axis. The volume element is a disk with radius x and thickness dy, centered at height y. The volume is \(\pi x^2 dy\) or \(\pi y dy\) and the total volume is given by \(V=\int_0^1 \pi y dy=\frac{\pi}{2}\). The moment about the x=0 line is given by \(M_x = \int_0^1 y\pi y dy=\frac{\pi}{3}\). The ratio of the center of mass to the total height is: $$\frac{M_x}{V} = \frac{\frac{\pi}{3}}{\frac{\pi}{2}}=\frac{2}{3}$$. This confirms the given ratio for the paraboloid of revolution.

Hemisphere

Consider a hemisphere with radius R. The ratio of the distance of the center of mass to the total height is: $$\frac{M_x}{R} = \frac{\frac{3}{5}MR}{R}=\frac{3}{5}$$. This confirms the given ratio for the hemisphere.

Semicircle and Pi Calculation

Assume al-K?hi's guess for the semicircle is correct, and the center of mass lies at a distance of \(\frac{3}{7}\) from the diameter. Let's consider a semicircle of radius 1 defined by \(y=\sqrt{1-x^2}\). We want to find the moment about the x-axis. $$M_y=\int_{-1}^1 \sqrt{1-x^2}*x dx = \pi$$. Now, we compute the mass of the semicircle. $$M = \int_{-1}^1 \sqrt{1-x^2} dx = \frac{\pi}{2}$$. The ratio of the center of mass to the total height should be: $$\frac{M_y}{M} = \frac{\pi}{\frac{\pi}{2}}=\frac{31}{9}$$, as it was guessed by al-K?hi, implying that \(\pi=31 / 9\). While this result did not match Archimedes' bounds for the value of Pi, it demonstrates al-K?hi's attempt to discover the ratio for the semicircle's center of mass.

Key concepts

Centers of Gravity

The concept of center of gravity or centroid is a fundamental principle in geometry and physics, serving as the point at which an object's mass is considered to be concentrated. In the historical context, mathematicians like Abū Sahl al-Kūhī made significant strides in understanding these centers for various geometrical shapes.
For a tetrahedron, the center of gravity divides the height from a base vertex to the opposing face in a ratio of 1:4. This means the centroid is a quarter of the total height from the base.
Abū Sahl al-Kūhī, building on the work of his predecessors, identified how different shapes' centers of gravity align in elegant ratios, such as a segment of a parabola having a 2:3 ratio. This exploration extends to involve complex shapes like paraboloids and hemispheres which share distinct center of gravity properties. These insights not only showcase the beauty of mathematics but also laid the groundwork for future advancements in mechanics and physics.

Algebraic Computation

The practice of algebraic computation plays a vital role in determining the centers of gravity for various shapes. By using coordinate systems and integral calculus, mathematicians can compute centroids with precision. For example, when dealing with the segment of a parabola defined by the curve \(y = x^2\), algebra is employed to find the area and the corresponding centroid location.
This involves setting up and solving integrals that measure mass distribution and moments around axes. Similarly, for a paraboloid rotated around an axis, calculus allows for determining volume and the precise location of the center of gravity. Through these algebraic processes, patterns and rules emerge, leading to a better understanding and ability to predict behaviors of physical shapes.
The ability to use algebra to describe geometrical phenomena demonstrates the interlinking of arithmetic with spatial reasoning, a skill that has been used for centuries to solve complex physical and theoretical problems.

Mathematical Error Analysis

Mathematical error analysis involves evaluating the precision of calculations and the assumptions made. In historical contexts, such as with Abū Sahl al-Kūhī's conjectures about centers of gravity, error analysis becomes a crucial skill.
For instance, al-Kūhī's estimate for the semicircle resulted in a calculation that implied an incorrect value for \(\pi\). This type of error showcases the importance of using accurate data and the need to verify assumptions against established benchmarks, such as Archimedes' \(\pi\) bounds between \(\frac{310}{71}\) and \(\frac{31}{7}\).
By comparing results with known values and scrutinizing calculation processes, mathematicians can identify where approximations may have misled conclusions. This method fosters a more robust mathematical foundation and helps prevent such discrepancies. Error analysis not only improves the accuracy of mathematical models but also highlights the evolution of mathematical understanding through history.