C = 2rπ = πD where C is the circumference, r is the radius and D is the diameter of a circle. Using
geometric construction we can calculate the perimeter for the inscribed hexagon (Pi) and the perimeter
for the circumscribed hexagon (Pc) in units of the radius. Pi
= 6.00r and Pc
= 6.93r. Therefore, 6.00r <
C < 6.93r and since C = 2rπ, 3.00 < π < 3.46. By using a 96-gon, Archimedes determined: 3.1409 < π <
3.14292, thereby establishing 3.14 for the first three significant figures of π. (Archimedes numbers are
estimates because he had to estimate the square roots involved in determining the perimeters. The local
book store was not selling pocket calculators in Syracuse in 250 BCE!)
Archimedes could also determine the area within irregular shapes by drawing ever smaller triangles in
the shape and adding up the areas of the triangles. This approach, much like the estimation of π, borders
on calculus and solves problems like Zeno’s paradoxes. Had the ancient Greeks discovered algebra, it is
possible that Archimedes would have invented calculus almost 2000 years before Newton!
When the Romans conquered Syracuse a soldier killed Archimedes. The soldier did not realize who he
had captured. The Roman General, Marcellus, had wanted to use Archimedes’s knowledge and, finding
he had been killed, had the tomb built for Archimedes with the marble monument of the sphere in the
cylinder that he had requested.
Archimedes lived about 100 years after Aristotle and, of course, had the advantage of the scientific and
mathematical knowledge of his time. Archimedes had the cumulative knowledge of the Pythagoreans,
Euclid, and others. It is a great misfortune that the cosmology and physics of Aristotle became dominant.
Clearly Archimedes’s physics was much more modern and would have been a much better foundation
for science. The advancement of science might have been more rapid if Archimedes, instead of Aristotle,
had become the standard.
The Greek astronomer, geographer, and mathematician, Claudius Ptolemy lived in Roman Egypt from
about 85 to 165 CE. Ptolemy set out to correct the problems of Aristotle’s astronomy but wanted to
maintain the principle of circular movement in the heavens. As we mentioned before, the planets were
often observed to reverse their directions, something that was not possible if they were moving in circular
orbits around the Earth. However, Ptolemy found that he could correct these motions by using epicycles
that were themselves combinations of circles upon circles.