Assumption 1: The Earth is no bigger than 529,200 km round. (I converted it from his 3,000,000 stadia.) This means that the radius of this "no bigger than" Earth is 84,225 km.

Again, Archimedes is saying, "You can't say my numbers are too small. I will take the biggest estimate and then make the one I use even bigger."

We know very little about this Phidias. Some think he was Archimedes' actual father and others think he was only a teacher that Archimedes thought of as a father-figure.

King Gelon was the 5th century BCE ruler of Gela) and Syracuse.

Google Earth of Sicily, now a part of Italy. But, in Archimedes' time considered part of "Magna Graecia"

Archiomedes' main reason for writing the Sand Reckoner is not so that he can get an accurate estimate of the number of sands in the Universe. It is so that he can put forth his arguments about the number system the Greeks used at this time and how that system should deal with very large numbers. The Greeks knew about the concepts of zero and infinity and even negative numbers. But, they rejected them as things that should be used in math. Archimedes is making his case for very large numbers.

Zeuxippus was the immediate successor to Euclid as a mathematics professor at Alexandria and a correspondent with Archimedes during his life. From Wikisource.org

"A term for "universe" among the ancient Greek philosophers from Pythagoras onwards was τὸ πᾶν tò pân ("the all"), defined as all matter and all space, and τὸ ὅλον tò hólon ("all things"), which did not necessarily include the void." from Wikipedia.org Image from UCR.edu

Aristarchus of Samos Source: Wikimedia.org

Google Earth link to the Island of Samos.

The book was called On the Sizes and Distances.

Image is fron an excellent article on Aristarchus' theories at AncientGreeceReloaded.com. (Use google translate to translate text to English.)

For my GHF class "Aristotle Leads the Way" notice the use of our friend for the year: proportionality. Image is from a great article on proportionality at ExamRefresh.com

It's really hard to understand what he is saying here. So, most scholars use the description of the math below to figure out what he meant to say here.

Archimedes is saying that he will prove that he has thought up numbers that are bigger than the number of sands in the Universe. Unfortunately, we have lost all copies of his work called "The Principles" that he is referring to. So, we use this work to talk about Archimedes' very large numbers until we find a copy of that original work (if one still exists).

Archimedes' good friend Eratosthenes is the person who has estimated the perimeter of the Earth to be about 300,000 stadia. Eratosthenes actually said that the circumference of the Earth was 252,000 stadia which converts to about 44,452.8 km. The actual value of the polar (north-south) circumference of the Earth is approximately 39,940.6 km. While this yields a 10% error it was a vastly superior estimate than any other we have a record of from a 1,000 years before or after Eratosthenes.

Is Archimedes saying his good friend, Eratosthenes was wrong? Not really. He is saying, "Look, I don't want you to criticize my number as too small. So, let's put the Earth at 10 times the current estimate just to prove my point..."

Eudoxus said that the Universe was made of a system of spheres. Image Credit: Prof. Richard Pogge, Ohio State