Archimedes and the volume of the ball

The greatest scientist of the Ancient World - Archimedes (circa 287-212 BC) is widely recognized as one of the greatest geniuses in the history of mankind. His contribution to mathematics is huge, and the name is covered with legends. It was Archimedes who came up with a formula for determining the area of a triangle along its sides and came very close to the notion of a definite integral, ahead of humanity for almost two millennia. Archimedes has exact formulations of the laws of nature, preserved intact for all time. Archimedes first dared to calculate the dimensions of the world around us. He defined the limits for the number π, proving that: 3 10/71 < π < 3 1/7. But most of all, Archimedes was proud of the formula he found, with which you can find the volume of the ball, and in memory of this, the descendants depicted a ball and a cylinder on his gravestone.

Following the ideas of Archimedes, you can prove the result, which gave him the highest creative joy. For example, we prove the theorem: the volume of a ball of radius 1 is equal to 4/3 π.

Proof of the test. We will rely on the following two stereometry formulas: the volume of the cylinder with the radius of the base R and the height H is equal to πR2H and the volume of the cone with the radius of the base R and the height H is equal to 1/3 πR2H. The last formula was also found by Archimedes. Let's move on to the proof. For this you need to remember children's toys, which are called pyramids. Let's recall their device: there is a stand with a vertical stick and a set of rings of different sizes, but made of the same material. You need to string these rings on the stick so that the size of the rings increases as you approach the stand. Then you get a shape that looks like a cone.

According to Archimedes, the proof of the theorem is very easy to understand with the help of such toys. Only it is necessary to make not one - conical, but three different - cylindrical, when the thin rings will have a radius of 1, and if they are assembled together, they form a cylinder of height 1, the conic - of the same thin rings, but different radii from which you can to collect a cone of the radius of the base 1, and a half-ball, collecting from the rings a hemisphere of radius 1.

And now we take the pharmacy scales with flat bowls and, like Archimedes, we put on the one bowl the toy-cylinder collected from the rings, and on the other - the cone and the hemisphere, and put the cone on the bottom of the scales, and the half-ball on the head, the base of the hemisphere was on top and is located horizontally.

Let the heights of the rings be the same and equal δ, where δ is a very small number. Let us calculate the volume of rings that are at the same height h. For a cylindrical ring this volume is πδ for a conic π(1 - h)2δ, and for a hemispherical ring π(1 - (1 - h)2 (for the radius of the ring at the cone is equal to 1 - h, and in the hemisphere, according to theorem of Pythagoras, it is equal to (1 - (1 - h)2)1/2.

The total volume on each of the scales was the same. But if δ is very small, then the conical toy will be almost indistinguishable from the cone, the hemisphere is from the hemisphere, and the cylindrical toy is always the cylinder.

In the limit, we obtain that the volume of a half-ball of radius 1 is equal to the volume of the cylinder with the radius of the base and height 1, minus the volume of the cone with the radius of the base and height 1. From this follows the proof of Archimedes' theorem: the volume of a ball of radius 1 is 4/3 π.