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

Pappus also wrote commentaries on Euclid’s

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

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.

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

1.

A.M. = (x+y)/2

The arithmetic mean is used as the best kind of average in statistics.

2,

G.M. = √xy

The geometric mean is used by economists. It is used, for example, to find an average rate of return on investments earning different rates each year.

3.

H.M. = 2/(1/x+1/y) or 2xy/(x+y)

The harmonic mean is related to some important formulae used in Physics.

As an example, if x = 2 and y = 8, then

A.M. = (x+y)/2 = (2+8)/2=10/2 = 5

G.M. = √(2×8)=√16=4

H.M. = (2×2×8)/(2+8)=32/10=3.2

In Book III of his

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

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

b) can be proved using similar triangles:

c) can also be proved using similar triangles: