MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

152 Biographical Vignettes
Apollonius). Property 16 of Chapter 2 defines Apollonian circles and points.
Ptolemy informs us that he was also one of the founders of Greek mathematical
astronomy, where he used geometrical models to explain planetary motions.
Source material for Apollonius is available in [167].
Vignette 6 (Pappus of Alexandria: Circa 290-350).
Pappus was the last of the great Greek geometers and we know little of
his life except that he was born and taught in Alexandria [166]. The 4th
Century A.D. was a period of general stagnation in mathematical development
(“the Silver Age of Greek Mathematics”). This state of affairs makes Pappus’
accomplishments all the more remarkable. His great work in geometry was
called The Synagoge or The Collection and it is a handbook to be read with
the original works intended to revive the classical Greek geometry. It consists
of eight Books each of which is preceded by a systematic introduction. Book I,
which is lost, was concerned with arithmetic while Book II, which is partially
lost, deals with Apollonius’ method for handling large numbers. Book III
treats problems in plane and solid geometry including how to inscribe each
of the five regular polyhedra in a sphere. Book IV contains properties of
curves such as the spiral of Archimedes and the quadratix of Hippias. Book V
compares the areas of different plane figures all having the same perimeter and
the volumes of different solids all with the same surface area. This Book also
compares the five regular Platonic solids and reveals Archimedes’ lost work on
the thirteen semi-regular polyhedra. Book VI is a synopsis and correction of
some earlier astronomical works. The preface to Book VII contains Pappus’
Problem (a locus problem involving ratios of oblique distances of a point from
a given collection of lines) which later occupied both Descartes and Newton,
as well as Pappus’ Centroid Theorem (a pair of related results concerning the
surface area and volume of surfaces and solids of revolution). Book VII itself
contains Pappus’ Hexagon Theorem (basic to modern projective geometry)
which states that three points formed by intersecting six lines connecting two
sets of three collinear points are also collinear. It also discusses the lost works
of Apollonius previously noted. Book VIII deals primarily with mechanics
but intersperses some questions of pure geometry such as how to draw an
ellipse through five given points. Overall, The Collection is a work of very
great historical importance in the study of Greek geometry. Pappus also wrote
commentaries on the works of Euclid and Ptolemy. Source material for Pappus
is available in [221].
Vignette 7 (Leonardo of Pisa (Fibonacci): 1170-1250).
Leonardo of Pisa, a.k.a. Fibonacci, has been justifiably described as the
most talented Western Mathematician of the Middle Ages [143]. Fibonacci
was born in Pisa, Italy but was educated in North Africa where his father held