I am reading this book on Archimedes, famous mathematician of his days and now, and it was found that he caculated the amount of sand in the known Universe at his days, which is from Mercury to Jupiter I believe. Solving how much for a teaspoon and a finger was complicated enough, but the task for the universe seems impossible. Usually he should have some equation or some way to caculate that large number. In the process, he made exponents, I know, but how did Archimedes complete his challange? What mathematical process or equation(s) did he use?
I recommend that you read Archimedes himself. This is easily available, translated to most languages, and you will not regret. This is really a very entertaining reading.
Of course this book of Archimedes was a kind of joke (like the Cattle problem attributed to him). He did not "calculate" the number of pieces of sand. He only estimated it from above.
His purpose was only to explain how one can represent arbitrarily large numbers. Which is not a big deal. But the details he gives are indeed very interesting and amazing. For example from this book we learn what people thought of the size of the Universe, and various related parameters involved. Especially interesting are some experimental procedures he describes. For example how he found the size of one piece of sand. And visible diameter of the Sun, etc.
Archimedes simply uses the best available at that time estimates of the size of the Earth, distance to the Moon, distance to the Sun, and how much the distance to the "fixed stars" is greater than the distance to the Sun. He obtains the volume of the Universe. Then he measures the size of the sand piece, and does the arithmetic. What is the volume of the ball of given radius he knew:-) But the main point is introducing a simple notation which can represent arbitrarily large numbers.
Remark. The Cattle problem (which I believe is also due to Archimedes) shows the same kind of the sense of humor). The point here is that every Pell's equation has a solution. The smallest solution of the Cattle problem is much greater than the number of sand pieces in the Universe:-)