different from a, b, and c, contradicting the assumption that these are the only primes.
This proof shows that there are infinitely many primes but it does not provide a formula for listing them. That formula was still to be discovered, over 2,000 years in the future.
See: Euclid’s Elements, page 35; The Fundamental Theorem of Arithmetic, page 39
3rd century BC Greece
Measurement of a Sphere
Archimedes (287–212 BC)
This concerns formulae for the volume and surface area of a sphere.
_______________
Archimedes is often described as the archetype of the absent-minded professor. Yet, he achieved more in mathematics, science, and technology than anyone else in the ancient world.
The diagram shows a sphere inside a cylinder. From this diagram Archimedes found two things:
1. The surface area of the sphere is the same as that of the cylinder (without the ends).
2. The volume of the sphere is two-thirds that of the cylinder.
The formulae 4πr2 for the surface area, and 4/3 πr3 for the volume followed. Archimedes was so proud of these results that the diagram was engraved on his tombstone.
Archimedes is famous for mathematics, scientific discoveries concerning centers of gravity and floating bodies, and such inventions as the Archimedes screw. He is also famous for leaping out of his bath and running around shouting “Eureka!”.
While the Roman army besieged Syracuse, where Archimedes was living, he invented terrifying war machines to drive them back. Despite this, they finally took the city. While he was busy tracing a geometrical figure on the sand, a soldier summoned him to attend the Roman governor. “Don’t disturb my circles!” he protested. These were his last words.
Sphere inside cylinder.
3rd century BC Greece
Quadrature of the Parabola
Archimedes (287–212 BC)
This is about finding the area between a chord and a curve.
_______________
To evaluate the area between a chord and a curve, Archimedes found a way to add infinitely many numbers.
A parabola is an example of a conic which occurs in many places in science as well as mathematics. The reflecting surface of a space-telescope, such as that at Jodrell Bank in Manchester, England, is formed from a parabola.
The diagram (page 43) shows part of a parabola curve and a straight line crossing it. Archimedes found the area of the shaded region, between the line and the curve.
This area is 4/3 of the area of the triangle with the same base and height. Although this was not a terribly interesting result, it was the method of proof that was ground breaking. Put one triangle in the shaded region, which leaves two gaps. Put two triangles in the gaps, leaving four gaps. Repeat indefinitely; the area of all these triangles approaches the area required. The first two stages are shown.
The method of proof is known as the method of exhaustion. The infinite succession of triangles “exhausts” the area between the curve and the line.
Greek mathematicians distrusted any infinite process. Zeno’s paradoxes are about the adding of infinitely many terms and obtaining something finite. Archimedes showed that it was possible.
To find the area, Archimedes had to sum infinitely many smaller areas. He showed that this sum of infinitely many fractions:
1/1 + 1/4 + 1/16 +1/64 + 1/256 + ... has a finite value, 4/3.
This is the first recorded example of the summation of an infinite series, a key part of mathematics, and moreover it was found rigorously. The formal rigor of Greek mathematics was not achieved again until the 19th century.
Area between line and curve
“Exhausting” the area
It was rumored that Archimedes had a secret method to find these results. In 1906 a palimpsest (a document hidden under another, when the parchment was recycled) was found in a monastery in Constantinople (modern-day Istanbul). It contained The Method, a lost work by Archimedes, which showed how he had first obtained his results via informal reasoning. Essentially, it was the same as the integral calculus of Newton and Leibnitz. In 1998, Christie’s auction house in New York sold the manuscript for 2 million dollars to an unidentified collector in the United States.
See: Zeno’s Paradoxes, pages 30–31; Conic Sections, pages 33–34; Integration, pages 86–87
3rd century BC Greece
The Sand Reckoner
Archimedes (287–212 BC)
Archimedes asked the question:
How many grains of sand would fill the universe?
_______________
To answer this, Archimedes had to invent a way of writing numbers much larger than any used before.
At the time, the universe was thought to be finite in radius, being bounded by the sphere of the stars. In the “Sand Reckoner,” Archimedes made estimates for both the size of the universe and of a grain of sand, and had to find how many of the latter would fit into the former.
The problem was that at the time no notation existed to express such a huge number. The largest number word the Greeks had was “the myriad,” which means 10,000. They also used “the myriad myriad,” in other words, 10,000 x 10,000, or a hundred million, or 108 in modern notation.
In ordinary notation, we go up in steps of 10, then 100, then 1,000, and so on. Single-digit numbers are less than 10, two-digit numbers are less than 100, and three-digit numbers
