Archimedes! Oh, how I love what he's done to mathematics. He's the one that inspired me to think so much. It's so cool that
he was given so many problems that no one knew the answers to. How do you measure the volume of such an irregular shape as a
crown? He's the one that inspires me to know that math isn't just about getting an answer, it's about figuring out a way to
solve the problem. Let's go watch him
and see what happens. I think in your younger years, a lot of people had trouble thinking about a problem like Archimedes. It seems a
lot of people didn't want to persue the process in solving a problem and would just look up the answer. They made it sound like thinking
for an hour or so was tortue and awful. Archimedes would have preferred if the king didn't always ask him to solve such realistic problems.
He kept complaining that he'd rather do the things that didn't have immediate application. Nevertheless, a king is a king, and you can't very
well tell them no. When the Romans started attacking, the king had him invent ways to beat them. He invented things like catapults,
enourmous claws that could grab ships from the ocean, and many other strange things. It got to the point that when Romans came too close to the
Greek walls in their ships, the Greeks would just throw a rope over the wall and the Romans would sail away thinking it was another invention
that could beat them.
Sometimes I find, just like Archimedes, the best way to solve a problem is to sit and think about it for a while. The king wasn't
very happy with him when he took a lot of time to think about a problem, but when no one else has done anything like it, you
can't just look it up in your math book, and there wasn't even anything close to Google in his time. When the king gave Archimedes a
problem to solve, Archimedes had to do a lot of thinking.
Ooo, before we go, let's fast forward a few years to when he is discovering the equation for a sphere! Oh, this is awesome! Here is an applet
that shows how Archimedes thought through this problem.
It takes some serious thinking out of the cylinder to come up with reasoning like that. Archimedes was so proud of that proof that he
asked for it to be put on his headstone.
He also worked a lot with very large numbers. In his day, they only had names for numbers up to a myriad (10,000) but Archimedes wanted to count how
many grains of sand it would take to fill the entire universe! ...Even though what they thought of as the universe back then was really about half of
our solar system, that would still be a lot of sand particles. Much more than a myriad. The method he used was brilliant, though simple. Instead of
filling the entire universe with sand, then counting it, he counted how many particles of sand would fit in a poppy seed. After that, he calculated how
many poppy seeds would fit in a man's finger, then how many fingers in a stadium, and so on until he deduced how many sand particles would fit in the
universe. He worked on this problem, not because the king asked him to, or because it would help in a war, there was no immediate application to his work,
he just figured it out because people said that the amount of sand in the universe was infinite. He named a whole slew of numbers just so he could
count high enough because something someone said made him think.
Of course, with as many problems as Archimedes was thinking about, it makes sense that he would eventually find some that he may or may not
have actually found the answer to. Let's go see him work on that very problem. It has to do with cows and bulls. The problem is as follows.
If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of
Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls,
mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the
whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe
further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the
proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the
fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts were
equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If
thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females
according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.
But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm,
equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and the dappled
bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular
figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and
gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this
species of wisdom.
It looks pretty daunting.
Here is the equation set up and
here is the final answer of how many cattle there are.
Mmm, now I just want to go eat
an astro-burger. If we got the beef from the Sun God, we would have enough to feed the entire universe! And not just what Archimedes thought was the universe.
Well, we better move on. This time it will be quite the ride, because to get more accurate than Archimedes did in estimating pi,
we're going to go to ancient China to pay a visit to our friend ...