These texts contain detailed accounts of much of Greek
mathematics, some of which would have otherwise been
lost to us.
Extraordinarily little is known of Pappus’s life.
Many historians suspect that he lived in Alexandria
all his life and that he may have headed his own
school there.
Pappus’s famous piece, Synagoge, was written as a
general guide to all of Greek geometry, more specifi-
cally, a companion guide to read alongside the Greek
original works. As such, books I and II cover basic
arithmetic and methods of expressing large numbers.
Book III examines the
ARITHMETIC MEAN
,
GEOMETRIC
MEAN
, and
HARMONIC MEAN
of proportions, properties
of basic geometry, and the representation of these
means in geometry, as well as some consideration of
the P
LATONIC SOLIDS
. Book IV examines special curves,
including the
SPIRAL OF
A
RCHIMEDES
, and the problem
of
TRISECTING AN ANGLE
.
In book V Pappus presents a mathematical analysis
of the structure of honeycombs, showing that the
hexagonal structure uses the least amount of perimeter
to partition the plane. He also examines
POLYHEDRA
.
Book VI explores the topic of astronomy, chiefly as a
review of the works of Theodosius, Autolycus, Aris-
tarchus, and Euclid on the topic as a means to correct
common misinterpretations.
In book VII the following problem, today known
as Pappus’s problem, appears:
Given four lines in the plane, find the locus of
points Psuch that the proportion of the prod-
uct of the distances of Pfrom any two lines to
the product of distances to the remaining two
lines is constant.
Pappus shows that the resulting curve is always a
CONIC SECTION
. He also generalizes the problem to
consider the case of greater than four lines in the plane.
Much of the geometric work of the French mathemati-
cian R
ENÉ
D
ESCARTES
(1596–1650) was inspired by
this problem.
Book VIII of Synagoge deals with mechanics. It
begins with a written definition of the
CENTER OF
GRAVITY
of a figure (Pappus is the only Greek scholar
to have ever provided a definition of this concept) and
discusses a number of geometric results that employ the
notion. He also describes the principles of levers, pul-
leys, wedges, axels, and of the screw.
Pappus wrote Synagoge partly as an attempt to
revive fervent interest in classical Greek mathematics
and the style of mathematical research they conducted.
Unfortunately, this did not occur. Pappus’s work,
nonetheless, greatly influenced the course of thinking in
the development of projective geometry one millennium
later. Many of his geometric results can be best stated
and understood within this setting.
Pappus’s theorems Greek mathematician P
APPUS OF
A
LEXANDRIA
(ca. 320
C
.
E
.) established two fundamen-
tal results about the surface area and the volume of a
SOLID OF REVOLUTION
.
Consider the solid of revolution formed by
revolving a curve in the plane about a line that
does not intersect the curve. Then the surface
area of that solid is equal to the product of the
length of the curve and the distance the cen-
troid (
CENTER OF GRAVITY
) of the curve trav-
eled about the line of revolution. The volume
of the solid is equal to the product of the area
bounded by the curve (or the area between the
curve and the line of revolution) and the dis-
tance traveled by the centroid of the curve.
These results can be proved by approximating the
curve as a series of straight line segments, establishing
the claim for this polygonal path, and then verifying
that the result remains valid as one works with finer
and finer approximations of the curve. This is a process
that involves a
LIMIT
. It is remarkable that Pappus was
able to establish these results before the advent of
CAL
-
CULUS
some 1,350 years later.
In the study of
PROJECTIVE GEOMETRY
, the follow-
ing result is also attributed to Pappus:
Consider a hexagon ABCDEF in the plane.
This hexagon may be of any shape and may
even self-intersect. Suppose the points A, C,
and Elie on one straight line, and the points B,
D, and Flie on another. For each of the three
pairs of opposite sides of the hexagon, find the
location at which these sides (if extended)
intersect. Then these three points of intersec-
tion themselves lie on a straight line.
B
LAISE
P
ASCAL
(1623–62) later extended this
result to prove that if a hexagon has vertices lying on
372 Pappus’s theorems