Welcome back to Numberholic! I’m your host, Sejari, and today we’re diving into a story that beautifully blends ancient genius with modern admiration. This is the sixth episode of Proven Yet Hard to Believe, where we explore the brilliance behind mathematical proofs that have stood the test of time. Today, we’re uncovering a truly mind-blowing achievement: Archimedes’ discovery of the volume of a sphere. How did he manage to do it? Let’s step into the mind of one of history’s greatest mathematical geniuses and find out!
Who was the first person to calculate the volume of a sphere? It was none other than the legendary Archimedes. But how exactly did he figure it out?
In 1899, a 10th-century parchment was discovered in the Monastery of the Holy Sepulchre in Constantinople (modern-day Istanbul). This parchment, partially erased and overwritten with a prayer dated April 13, 1229, contained lost works of Archimedes. Amazingly, it included his Method of Mechanical Theorems, a text thought to have been lost forever.
The Method of Mechanical Theorems
Addressed to Eratosthenes, this remarkable work reveals how Archimedes used his understanding of levers and centers of mass to analyze and balance different geometric shapes.
Let’s walk through Archimedes’ method step by step to understand how he calculated the volume of a sphere.
Archimedes relied on two principles of the lever, which we’ll call the Axiom of Balance and the Axiom of Centers of Mass.
The Axiom of Balance states that for two objects to balance on a lever, the product of their distance from the fulcrum and their mass must be equal. This concept aligns with the modern idea of torque, or as we’ll call it here, the “distance-mass product.”
The Axiom of Centers of Mass states that if two objects on one side of a lever balance with a single object on the other, the lever will remain balanced if the two objects are replaced by a single one located at their combined center of mass. For two objects of equal mass, their center of mass lies at the midpoint of their positions.
Archimedes used three solid figures in his calculations: a sphere, a cone, and a cylinder. He considered a sphere with a diameter of 2r and configured the cone and cylinder so the cylinder perfectly enclosed the sphere.
To begin, Archimedes imagined suspending the lever in space. On one side of the lever, he placed the sphere and the cone at a distance equal to the sphere’s diameter.
Why?
Because determining the cone’s center of mass directly would be difficult.
He then sliced the cone and the sphere into infinitesimally thin disks along their heights, with each disk’s thickness denoted as Delta x.
- At height x, the cone’s cross-section is a circle with a radius of x.
- The sphere’s cross-section, also a circle, can be determined using the properties of circles and similarity ratios.
Assuming uniform density, he calculated the combined mass of the thin disks from the cone and sphere and determined their distance-mass product by multiplying the mass by a distance of 2r.
This setup was equivalent to placing four disks with radius r on the opposite side of the lever at a distance of x. Or, to simplify, he treated it as a single disk with radius r and four times the density at the same position.
Archimedes then moved along the lever from x=0 to x=2r, ensuring that the total distance-mass product of the cone and sphere matched that of the dense disk on the other side.
Once this process was complete, the right-hand side of the lever formed a cylinder. And as if by magic, perfect balance was achieved.
By the Axiom of Centers of Mass, the cylinder’s center of mass was known to lie at x=r. Comparing the distance-mass products on both sides of the lever revealed the volume of the sphere.
Isn’t it incredible?
But that’s not where the story ends.
This brief explanation only scratches the surface of Archimedes’ genius. As Archimedes himself wrote:
“… Some things were first made evident by the mechanical method, though they had to be demonstrated by geometry later, since their investigation by the mechanical method did not furnish an actual proof. But certainly, the proof is easier to supply after a preliminary knowledge of the questions, than it is to find it without any prior knowledge.” (Archimedes, The Method)
Archimedes also addressed the potential for errors in his method by introducing what is now called the Method of Exhaustion.
This rigorous technique involved inscribing and circumscribing a sphere with frustums of cones to confine the true volume within increasingly smaller bounds. By increasing the number of frustums, he demonstrated that the margin of error could be reduced to virtually nothing.
The process of comparing the volumes of spheres and frustums was incredibly intricate because the shape of the frustums varied based on their position. While we can’t replicate all of Archimedes’ calculations here, I hope this glimpse into his thought process inspires you to appreciate his genius even more.
And with that, we conclude our journey into the mind of Archimedes and his groundbreaking method for calculating the volume of a sphere. His brilliance continues to inspire mathematicians and thinkers to this day. If you enjoyed this episode, don’t forget to like, comment, and subscribe for more incredible stories from the world of mathematics. This is Sejari, signing off. Until next time, stay curious and keep exploring!