Archimedes Makes his Greatest Discovery

archimedes small sphere

Archimedes was fascinated by curves. His powerful mind had mastered straight line shapes in both 2D and 3D.

He needed something more intellectually challenging to test him. This came in the form of circles, ellipses, parabolas, hyperbolas, spheres and cones.

Calculation of the Volume of a Sphere

He rose to the challenge masterfully, becoming the first person to calculate and prove the formulas for the volume and the surface area of a sphere.

The way he found his formulas is both amazingly clever and shows him to be a mathematician of the first rank, far ahead of others of his time, doing mathematics within touching distance of integral calculus 1800 years before it was invented.

Taming Curves in 3-Dimensions – Not for the Timid

The surface of a sphere is incredibly hard to get to grips with compared with a shape like a cube. Cubes only change at the corners and edges. The surface of a sphere changes its direction at every point. How could you work with this?

sphere cut into hemispheres

Sphere cut into hemispheres.
Image by Jhbdel

First, Archimedes imagined cutting a sphere into two halves – hemispheres.

Taking one hemisphere gave him a shape with a flat surface to work with – easier than a sphere, and if he could find the volume of a hemisphere, doubling it would give him the volume of a sphere.

He then imagined placing the hemisphere face down on a flat surface.

Next, in his mind’s eye, he fitted a cylinder around his hemisphere.

The circle at each end of the cylinder was the same size as the circle at the bottom of the hemisphere, and the cylinder’s height was equal to the hemisphere’s height, as shown in the image below:

hemisphere within cylinder

Archimedes imagined a hemisphere within a cylinder

Salami Tactics

Archimedes then did something incredibly clever. To anyone who has studied university mathematics, you’ll recognize something very similar to integral calculus.

Archimedes imagined cutting horizontal slices through the cylinder.

He took his first slice of mathematical salami at the very top of the cylinder. Here the hemisphere is at its smallest. Looking at this first slice from above, the radius of the circle from the very top of the hemisphere is infinitesimally small.

Then, in his mind’s eye, he moved his attention a tiny bit lower down the cylinder and took another salami slice through the cylinder and hemisphere. In this slice, the hemisphere circle had grown a little larger.

Then he moved his attention a little lower again, cutting another salami slice. The cylinder circle stayed the same size, while the hemisphere circle was again a little larger than the previous slice.

He then moved down the cylinder, taking slices all the way to the bottom. In each slice, the size of the inner circle got larger, while the size of the outside circle stayed the same, as shown in these images.

Archimedes takes salami slices

The cross sections Archimedes imagined of the hemisphere and the cylinder.

Putting Everything Together

Archimedes considered each salami slice. In particular, he was interested in the gap between the two circles in each slice – shown in blue in the images above.

He took all of these blue areas – there were as many of them as he liked to imagine, with the depth of each slice as close to infinitesimally thin as he liked. He then multiplied the areas of the blue rings by their depths to find the volume represented by all of the blue salami rings stacked up on one another. (He didn’t consider an infinite number of infinitely thin slices, because if he had, he would have invented integral calculus over 1800 years before Isaac Newton did it.)

Archimedes found that the volumes of the blue rings added up to the volume of a cone whose base radius and height were the same as the cylinder’s.

This meant the volume of the hemisphere must be equal to the volume of the cylinder minus the volume of the cone.

The formula for the volume of the cylinder was known to be πr2h and the formula for the volume of a cone was known to be 13πr2h. In this example, r and h are identical, so the volumes are πr3 and 13π r3.

Subtracting one from the other meant that the volume of a hemisphere must be 23πr3, and since a sphere’s volume is twice the volume of a hemisphere, the volume of a sphere is:

V = 43πr3

Archimedes also proved that the surface area of a sphere is 4πr2.

Archimedes saw this proof as his greatest mathematical achievement, and gave instructions that it should be remembered on his gravestone as a sphere within a cylinder.

More about Archimedes

Archimedes - the sphere within the cylinder

The sphere within the cylinder. Image by André Karwath.

 

How Archimedes Invented the Beast Number

Archimedes Beast Number

The Beast Number – a number so big, that to write it out in full would take more space than there is in the observable universe

Archimedes had become fed up of people saying you couldn’t calculate the number of grains of sand on a beach.

In response to this nonsense (as he saw it) he invented new, enormous numbers. Then he calculated not just how many grains of sand there were on the beach, but how many there were in the universe.

The trouble Archimedes faced was the Greek number system. It was a primitive system in which letters became numbers: A = 1, B = 2, C = 3, etc.

Very large numbers were a problem, because there weren’t enough letters in the alphabet! The Greeks’ biggest number was a myriad, which in the modern, Hindu-Arabic number system, we write as 10,000.

Off with the old…

In The Sand Reckoner, Archimedes demolished the commonly held idea that the number of grains of sand on the shores around his home city of Syracuse could not be calculated.

In fact, he showed that he could produce numbers so large that they were bigger than the number of sand grains in the whole universe.

the sand reckoner

Can you count the sand grains?

His calculation relied on his invention of what we now call exponents (often called powers, or index numbers). For example 104 is ten to the power of four. We usually call this ten thousand. The Greeks would have called it a myriad.

Archimedes’ New Number System

Archimedes introduced a new classification of numbers.

He said that ‘first order’ numbers went up to a myriad myriads, meaning 10,000 x 10,000.

We would write this as 100 million, or 100,000,000, or 108.

Numbers of the ‘second order’ went up to 100 million multiplied by 100 million – i.e. 108 x 108 or (108)2.

Numbers of the third order were those up to 108 x 108 x 108 – i.e. (108)3,

and so on.

Ultimately, Archimedes calculated that to count the number of grains of sand in the universe he needed numbers up to the eighth order, i.e. (108)8 = 1064, which is equal to:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000.

This was the biggest number anyone would need in the universe Archimedes imagined. (By the way, electronic calculators will happily work with numbers as big as this.)

But Archimedes was not content with discovering this huge number. He went on to write numbers that dwarf it.

He moved from ‘orders’ of numbers, to what he called ‘periods.’

Mind Bogglingly Big Numbers

Archimedes said that numbers of the ‘first period’ will be those numbers up to the mind-bogglingly large:

archimedes-beast-1

This number is much too big for everyday electronic calculators to work with.

mind boggled ape

It could be written as 1 followed by 800 million zeros. If you were to publish this number as a book, the zeros would take up about 380,000 pages. That’s a long book!

Also, given that the number of atoms in the sun is 1 followed by just 57 zeros, you will have to work hard to find anything big enough to need 1 followed by 800 million zeros to describe it.

Archimedes’ Beast Number

However, Archimedes was still not ready to let things rest. He wanted to write even bigger numbers. He continued logically until he reached:

archimedes-beast-2

archimedes beast number

Archimedes called this number a myriad-myriad units of the myriad-myriadth order of the myriad-myriadth period. We’ll just call it Archimedes’ Beast Number. It’s one followed by 80 quadrillion zeros.

What could we use this number for practically?

Well, how about writing down the volume of the observable universe in cubic centimeters? Surely that would need a number close to the Beast Number? If we take it that the observable universe has a diameter of 93 billion light years, then…..

nope, the universe’s volume in cubic centimeters ‘only’ needs 35 followed by 85 zeros.

Hmmmmm. We’ll need to try harder.

Most of the universe is pretty much a vacuum. How about filling the universe with air. How many molecules of air would we need to fill the universe to Earth’s air pressure? The number of molecules needed must at least approach the Beast Number, mustn’t it?

Actually, no, we can fill the universe with just 1 followed by 106 zeros molecules of air.

How about the number of bacteria that has ever lived on Earth? Well, again… no, that’s only about 1 followed by 40 or so zeros.

DNA

Okay, one last try. Life is based on that well-known molecule, DNA. Given that each different human has different DNA, how many different human beings are possible genetically before we start creating humans with identical genotypes?

As far as we can calculate, the upper limit is believed to be somewhat less than 10106.

That’s a much bigger number than our earlier efforts: it’s 1 followed by a million zeros.

But again, it’s no match for the Beast Number.

Archimedes’ Number utterly dwarfs these huge, but more practical numbers. In fact, the Beast Number is nothing more than a measure of Archimedes’ towering mathematical ambition.

More about Archimedes

p.s. Bigger than the Beast

Before clicking the ‘publish’ button, it came to me that I ought to mention that the Beast Number does not even begin to approach infinity. In fact, it’s no closer to infinity than the number 1 is, because it’s still infinitely far away. No number we can name can get close to infinity.

And remember, the Beast Number is a rational number. The infinity of irrational numbers is even bigger than the infinity of rationals!