Archimedes was born about 287 BC, so he was a little younger than Euclid. Archimedes’ father was an astronomer. Because Archimedes was related to the tyrant who ruled Syracuse, on the island of Sicily, he lived his whole life in Sicily, except when he went to study at the University of Alexandria in Egypt. In Egypt, he probably met and worked with the other great scientists of his time: Euclid, Aristarchus, and Eratosthenes. Archimedes worked mainly in a losing battle to defend his city-state from the Romans, who were attacking Syracuse in the course of the First Punic War.
Archimedes invented, or people said he had invented, a bunch of different kinds of machines. The most important of these machines was the pulley. Another invention was the screw pump, which uses a screw to lift water from one place to another. (But the screw pump may well be earlier, too).
But Archimedes was also interested in why things worked, and whether they would work the same way every time. He explained why levers worked, and he worked on getting a more accurate number for pi, and wrote a proof for calculating the circumference of a circle. The accomplishment Archimedes himself was most proud of was that he proved that if you fit a sphere inside a cylinder, the sphere will have two-thirds the volume and two-thirds of the surface area of the cylinder.
Archimedes also worked out a way to calculate the volume of an irregular object. With a regular cube, you can just multiply the lengths of all the sides. But what about the volume of a complicated necklace or a Greek vase? According to one story, Archimedes noticed that the water level of the bathtub rose when he got in, and he realized that you could measure volume by putting the necklace in a cup full of water and measuring how much the water level rose. (Supposedly he got so excited that he ran through the streets naked yelling “Eureka!”, which is Greek for “I figured it out!”)
Archimedes went on from his Eureka moment to figure out the Archimedes Principle, which explains why some things (like wooden logs and metal bowls) float and other things (like metal spoons or gold rings) don’t float. Any object will displace some water, and that water will weigh a certain amount. If the object weighs more than the water it displaces, it will sink. If it weighs less than the water it displaces, it will float.
Some of Archimedes’ work may be related to the work on infinity and combinations that Indian mathematicians were doing at the University of Nalanda around the same time. Archimedes did some work on infinity, getting close to developing calculus, that may be building on Indian work. Also like Indian mathematicians, Archimedes worked on the different ways numbers can be combined into patterns.
This may be related to Archimedes’ effort to work with very large numbers, trying to figure out how many grains of sand it would take to fill up the universe. To do this, Archimedes worked with exponents, and proved that Xa x Xb = Xa+b. To get an idea how big the universe was, Archimedes followed his friend Aristarchus in thinking that the earth went around the sun. He figured that the universe was about two light-years across, and would hold about 1063 grains of sand.
Archimedes died when the Romans were conquering Syracuse; a Roman soldier killed him (even though the commanders had ordered the soldiers to capture the great man alive).
Learn by doing: put things in water and see if they float, and discuss why
More about Roman science
Greek and Roman Science, by Don Nardo (1998). Nardo has written a lot of good easy reading books about the ancient world; this one is no exception.
Ancient Science: 40 Time-Traveling, World-Exploring, History-Making Activities , by Jim Wiese (2003). Activities, as the title says – how to make your own sundial, and so on. The author is a science teacher.
Early Greek Science: Thales to Aristotle, by Geoffrey Lloyd (1974).
History of Greek Mathematics: From Aristarchus to Diophantus, by Thomas L. Heath (1921, reprinted 1981). A lot of Euclid, but also describes who the other major Greek mathematicians were and what they did.
Episodes from the Early History of Mathematics, by Asger Aaboe (1997).