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# Pappus

Pappus was a geometer in the tradition of Euclid, Archimedes and Apollonius, and lived at the end of the third century and the beginning of the fourth century AD. If for no other reason, Pappus would be notable as the last of the great mathematicians of the Greek era living in Alexandria. His exact dates are not known, but his extensive work known as the *Mathematical* *Collection* dates from around 320 AD.

Pappus also wrote commentaries on Euclid’s *Elements* and *Data* and on Ptolemy’s *Almagest* , but we know nothing about these works of Pappus except from references made to them by later commentators. Another tract on the multiplication and division of sexagesimal numbers (numbers base sixty – as used for measuring time and angles) may also have been written by Pappus.

But the *Mathematical Collection* is the only work of Pappus that is extant. This collection consisted of eight books, but the first and part of the second of these have also been lost. The first book was probably on arithmetic. The second considers a method for writing and using large numbers first proposed by Apollonius. The next three deal mainly with geometry, but not including conic sections. They provide largely a commentary on earlier works on geometry, together with some new propositions, and improvements or extensions on the proofs previously given by other geometers. The sixth book is on astronomy, optics and trigonometry, the seventh on analysis and conic sections, and the eighth deals with mechanics. The last two books in particular contain a lot of original work by Pappus.

One interesting result occurs in Book V where Pappus showed that if two regular polygons have the same perimeter, then the polygon with the greater number of sides also has the greater area. A consequence of this result is the suggestion that bees must have some mathematical knowledge since they construct their cells from hexagonal prisms rather than square or triangular prisms.

**A contribution of Pappus to mathematics**

In mathematics we define various types of **mean** to be used in certain circumstances.

If we take two numbers, x and y, then three of the most commonly used means are defined as follows:

In Book III of his *Mathematical Collection*, Pappus showed a way to represent these means geometrically.

The diagram shows a semicircle with centre O and diameter AB. C is another point on the circumference. DC is drawn perpendicular to AB and DE is drawn perpendicular to CO.

Pappus showed that the means of AD and DB are given respectively by:

a) Arithmetic mean of AD and DB = OC

b) Geometric mean of AD and DB = CD

c) Harmonic mean of AD and DB = CE

**Proof**

a) is very easy to prove. Let AD = x and DB = y

b) can be proved using similar triangles:

c) can also be proved using similar triangles:

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