Some of Archimedes' mathematical proofs involve the use of infinitesimals in a way that is similar to modern integral calculus.
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics.
Plutarch wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.”
By assuming a proposition to be true and showing that this would lead to a contradiction, Archimedes was able to give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay.
This technique is known as the method of exhaustion, and he employed it to approximate the value of Pi.
He did this by drawing a larger polygon outside a circle, and a smaller polygon inside the circle.
As the number of sides of the polygon increases, it becomes a more accurate approximation of a circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value of ¼ lay between 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408).
This was a remarkable achievement, since the ancient Greek number system was awkward and used letters rather than the positional notation system used today.
He also proved that the area of a circle was equal to Pi multiplied by the square of the radius of the circle.
He used the method of exhaustion to show that the value of the square root of 3 lay between 265/153 (approximately 1.732) and 1351/780 (approximately 1.7320512).
The modern value is around 1.7320508076, making this a very accurate estimate.
Another noted mathematical work by Archimedes is The Sand Reckoner where he set out to calculate the number of grains of sand that the universe could contain.
Archimedes suggested that there are some who think that the number of grains of sand on earth is infinite, not only that which exists about Syracuse and the rest of Sicily, but also throughout the world.
He, therefore, challenged the notion that the number of grains of sand was too large to be counted.
To solve the problem, Archimedes devised a system of counting based around the myriad.
This was a word used to mean infinity, based on the Greek word for uncountable, murious.
The word myriad was also used to denote the number 10,000.
He proposed a number system using powers of myriad myriads
Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus .
In his other works, Archimedes often proves the equality of two areas or volumes with his method of double contradiction assuming that the first is bigger than the second leads to a contradiction, as does the assumption that the first be smaller than the second; so the two must be equal.
These proofs, still considered to be rigorous and correct, used what we might now consider secondary-school geometry with rare brilliance though later writers often criticised Archimedes for not explaining how he arrived at his results in the first place.