- Dalia Ventura
- BBC World News
Ambition is sometimes frowned upon, and Archimedes, the 3rd century BC Greek mathematician, was definitely guilty of that sin.
But his was greed to know, the most commendable.
Tired of being said that things difficult to count – like sand – were infinite, he set himself the task of doing so.
As you hear it: he set out to count all the grains of sand that could fit in the known Universe.
Born in Syracuse, Sicily, Magna Graecia in 287 BC, Archimedes was a genius obsessed with mathematics.
Among many things, it was he who came up with a value for π, one of the building blocks of science, and its estimation is still used today.
And yes it was he who ran naked through the streets of Syracuse screaming “Eureka” after solving a doubt that King Hieron II (c. 306-215 BC) had while he was in his tub, inventing hydrostatics in the process.
It was to the successor of that king, Gelón II, to whom he directed his essay “The sand counter“, a work considered as a jewel, not only for being one of the first scientific publications in history, but because…
- included the only existing reference to his own father, the astronomer Phidias
- showed that it is possible to express very large numbers in some kind of notation
- presented a way to extend the Greek numbering system to name large numbers
- estimated the size of the Universe as it was known at that time
- contains an account of an ingenious procedure that Archimedes used to determine the apparent diameter of the Sun by observation with an instrument
- Crucially, it gives the most detailed description of the heliocentric system of Aristarchus of Samos (c. 310-230 BC), showing that the latter was defending the Copernican system two millennia before Copernicus
Uncountable is not infinite
“There are some, Rey Gelón, who believe that the number of grains of sand is infinite in a multitude“, Archimedes begins.
What’s more, he writes, “there are some who, without considering it as infinity, believe that no number has been named that is large enough to exceed such magnitude“.
By this they mean, he explains, that they are convinced that any number that could express that magnitude, would be outweighed by the amount of sand there would be.
“But I will try to show, by means of geometric proofs that you will be able to follow that, from the numbers named by me, (…), some exceed not only the number of the mass of sand of equal magnitude that the Earth (…), but also that of the mass of equal magnitude that of the Universe“.
And that’s what he did in about 8 pages.
Let’s specify: Archimedes did not calculate the number of grains of sand in the Universe, but the number of grains of sand that would fill the entire space of the Universe if it were filled with sand.
In a finite world, there could not be an infinite number of grains of sand. There was a cap … but what was it?
Myriads of myriads
At that time, the highest number the Greeks had a name for: 10⁴ = 10,000, which they called μυριος (murious), which meant uncountable and was also a word for ‘infinity’ in Ancient Greece.
The Romans made that word myriad and that is how we know it now.
In order to do that immense calculation, he had to invent what we now call exponents or powers.
He started from the myriad and introduced a new classification of numbers.
He said that the ‘first order’ numbers were the ones that came to a myriad of myriads.
That is 10,000 x 10,000 = 100 million or 100,000,000 or 10⁸.
Those of ‘second order’ went from there to 100 million x 100 million = 10⁸ x 10⁸, that is to say (10⁸)².
The ‘third order’ were those up to 10⁸ x 10⁸ x 10⁸, that is, (10⁸)³, and so.
So, what order of numbers was needed then to calculate the number of grains of sand that could fit in the Universe?
According to Archimedean calculations, numbers of the eighth order were needed, that is (10⁸) ⁸ = 10⁶⁴, that is…
And who would argue with him!
The other thing that was indisputable was that he had created such a large number that it was highly unlikely that a larger number would be needed to tell anything in the Universe that he imagined.
Mmm no. Archimedes didn’t seem to like limits.
Contemporary historians say that he got euphoric when he discovered increasingly complex mathematical forms, known as Archimedean solids, ranging from the truncated tetrahedron -of 8 faces- to the blunt dodecahedron -of 92 faces-.
And in the case of numbers, it didn’t have to be any different. After all, their field of action – unlike grains of sand – was infinite.
So could not resist the temptation to continue discovering enormities.
To achieve this, he went from “orders” of numbers to what he called “periods”.
The first of these periods was (10⁸) raised to the (10⁸) power. That is, 1 followed by 800 million zeros.
In this case, I can’t show you: Doug Stewart of the Famous Scientists Organization estimates that if written down, the number would occupy 380,000 pages of a book.
Not content with this, Archimedes went on to (10⁸) raised to (10⁸) raised to (10⁸), a number he called “a myriad-myriad units of the myriad-myriad order of the myriad-myriad period.”
If its number to express the top of the grains of sand that could exist in the known universe at its time -10⁶⁴- was already too large to count what was counted at that time, there is still nothing we can count in our known universe today that is close to the enormity of that number that left us.
But, to leave you with an idea of its magnitude, perhaps it is clearer if I tell you that it is 1 followed by 80 quadrillion zeros … A measure of the genius of the mind of its creator?
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