is more or less the same as the modern formula given above, where a = 1.
1850 BC Eygpt
The Greatest Pyramid
A frustum of a pyramid is a pyramid with its top cut off. An ancient Egyptian manuscript gives a method for calculating the volume of this.
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Ancient Egypt is particularly famous for the construction of the Pyramids. The engineering skills that went into their construction have, unfortunately, been lost to us. Likewise we can now only guess at the mathematical skills the Egyptians possessed.
Take a solid like a cone or a pyramid, which slopes uniformly from its base to a point at the top. If we cut a slice off the top the result is a frustum. A yoghurt pot is an example of a frustum of a cone.
The Moscow papyrus, dating from about 1850 BC, contains a set of rules for finding the volume of a frustum of a pyramid. It goes:
Given a truncated pyramid of height six and square bases of side four on the base and two at the top. Square the four, result 16.
Multiply four and two, result eight. Square the two, result four.
Add the 16, the eight and the four, result 28.
Take a third of six, result two.
Multiply two and 28, result 56.
You will find it right.
Following these rules, this method gives a formula for the volume as:
1/3 x 6 (42 + 2 x 4 + 22) = 56.
This does give the correct volume.
Generalizing, if the frustum has height h, a square top of side r and a square base of side R, the method gives the following formula for its volume:
1/3h (R2 + Rr + r2)
The illustration shows truncated pyramids.
which is correct. No indication is given for how this method was reached. Was it by experiment, or from theory?
This mathematical result was described (by a mathematician, mind you) as the “Greatest Egyptian Pyramid.”
c. 3rd century BC Global
π
The ratio of the circumference of a circle to its diameter.
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The value of π has been found to higher and higher accuracy. It occurs in many places in mathematics besides the measurement of circles.
Circles come in different sizes of course. As the diameter (the length across) increases, so also does the circumference (the length around). The ratio between these two is the same for all circles and it is given the name π (Greek letter p, pronounced “pie”).
All civilizations have needed to find an approximation for π. An early Egyptian value was 4 x (8/9)2, which is 3.16, close to 3.14. In the Bible, I Kings 7, verse 23, the more approximate value of three is given.
The first-known reasoned estimation of π is due to Archimedes in the 3rd century BC. By drawing polygons inside and outside a circle, with more and more sides, he was able to close in on the value of π. With polygons of 96 sides, he found that π lies between 223/71 and 22/7. The latter value is still used. In the fifth century, a Chinese mathematician, Zu Chongzhi (429–501), found the more accurate fraction 355/113.
Further progress was made possible by the development of trigonometry. In the 14th century the Indian mathematician, Madhava, used trigonometry to discover the following series (which continues forever:
π/4 = 1 – 1/3 + 1/5 – 1/7 + …
This can be used to find π, but it is a very inefficient method. Using a variant of the series Madhava was able to calculate π to 11 decimal places.
Until the 20th century all the calculations were done by hand but with the invention of computers, much greater accuracy is possible. In 1949, the ENIAC calculated π to 2,037 decimal places, taking 70 hours to do so. Modern computers have calculated π to well over a million places.
The number π occurs throughout both pure and applied mathematics. Often these applications have nothing to do with the measurement of circles. For example, the equation of the normal or bell curve, which is central to statistics, is:
See: ENIAC, page 181; The Normal Distribution, pages 94–95
6th century BC Greece & Italy
The Pythagoreans
The Pythagorean slogan was: All Things Are Numbers.
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The Pythagoreans were a religious, mystical, and scientific sect mainly based in Southern Italy in the 6th century BC.
Their leader, Pythagoras himself, may or may not have existed. Many incredibly important discoveries are credited to the Pythagoreans, of which some will appear in this book.
The Pythagoreans are credited with discovering the following:
• That the Earth is a sphere.
• That the Earth is not the center of the universe.
• That musical harmony depends on the ratio of whole numbers.
No one knows what the Pythagoreans’ slogan, All Things Are Numbers, means.
Does it just mean that all things can be described in terms of numbers?
Or is it something stronger, that the solid world is an illusion and that the reality behind it consists of numbers?
More important, however, than any single discovery is the Pythagoreans’ contribution to mathematics, extending it from a practical subject concerned with areas of land or weights of corn to the study of abstract ideas.
6th century BC Greece
