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:
1.
The Arithmetic mean (A.M.), is the most commonly used mean, often simply called ‘the average’, and is defined by:
        A.M. = (x+y)/2
 The arithmetic mean is used as the best kind of average in statistics.
2,
The Geometric mean (G.M.) is defined by:
        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.
The Harmonic mean (H.M.) is defined by:
       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
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: