Between 800 BCE and 800 CE, the world witnessed a shift in the
intellectual centres of the time from Egypt and Babylon to the
cities of the Mediterranean - Greece and Rome. The first portion of
this time period, now known as the Thalassic Age, belonged to the
Greeks. Initially, Greece kept itself isolated from its neighbours.
Ships were used only for war, not for trade. Only when they opened
their ports - and their minds - to other civilisations like the
Egyptians and Babylonians did they begin to develop mathematics as a
science. Unfortunately, their literary skills showed much less
development and accounts of the advancements in science and
technology were passed down orally for generations before being
recorded. The two pioneers of Greek mathematics, Thales and
Pythagoras, therefore have no surviving texts to their credit. Much
of what we know from them is obtained from a summary by Proclus
(fifth century CE) of the lost History of Geometry by Eudemus
(fourth century BCE), who is regarded as the first mathematical
historian. The reader must be warned that there is little physical
evidence of these men and their deeds; in fact, they could be
regarded as folklore.
Early Greek Mathematics (650-400 BCE)
Thales Of Miletus (ca. 624-548 BCE)
Thales was, by legend, a clever man, who was said to have learned
much from the Egyptians and Babylonians. He is reputed to have
demonstrated that the angle inscribed in a semicircle is a right
angle (Theorem of Thales) and put down a series of rules regarding
the angles of triangles. He was reported to have measured the height
of the Pyramids by comparing the length of their shadows to that of
a vertical stick. At the moment the length of the stick’s shadow
was equal to its height, the length of the Pyramids’ shadow would
indicate their height.
Though much of the knowledge attributed to Thales originated from
Egypt and Babylon, he is credited with organising them in a rational
manner. Thales also broke away from the rigidity of using geometry
solely for measurement and tried to apply it in practical methods.
This logical structure he provided to geometry set forth a great
idea followed by later mathematicians such as Pythagoras and Plato.
Pythagoras of Samos (ca. 580-500 BCE)
Pythagoras was one of the most influential persons of his period,
having a say not only in mathematics but also in astronomy and
religion. The school he founded in 530 BCE in Croton (in present-day
Italy) gave rise to a whole new school of thought, the members of
which, the Pythagoreans, would later be responsible for a barrage of
new discoveries and ideas in mathematics, astronomy and philosophy.
Pythagoras probably proved that the plane space around a point
could be divided into six equilateral triangles, four squares or
three regular hexagons; and that the sum of angles of a triangle was
half the central angle of a circle.
However, the theorem that is most often credited to Pythagoras
(the sum of the squares of the shorter sides of a right triangle is
equal to the square of the third side) is in fact not his own. The
Babylonians had knowledge of this fact as far back as 1900 BCE.
Perhaps Pythagoras was able to provide a proof for it.
Though few mathematical discoveries can be directly attributed to
Pythagoras, it was the Pythagoreans who linked mathematics with
everything else in the universe. They saw numbers in life, nature
and religion and help make mathematics a liberal science.
Early Greek Mathematics (After 400 BCE)
This period saw a group of Greek mathematicians, many of the
Pythagoreans, who laid the foundation for modern geometry.
Hippocrates of Chios (ca. 530 BCE) described a method for
constructing a quadrilateral equal in area to a lune - a figure
bound by two circular arcs of unequal radii. This was the first ever
representation of a curvilinear area as a quadrilateral of equal
area. Around 430 BCE, Hippias of Elis discovered the quadratrix, the
first curve that could be defined but could not be constructed with
a straightedge and compass. Hippasus, a Pythagorean, defined the
dodecahedron, a regular polyhedron with twelve faces. Some also
attribute to him the proof that certain line segments (the diagonal
of a unit square, for example) are incommensurable. Democritus (born
ca. 460 BCE) was the first to show the relation between the volume
of a cone and that of a cylinder of equal base and height. He also
computed the volume of a pyramid for the Egyptians. During this time
also appeared a crude method of integral calculus called the method
of reduction, which involved increasing the number of sides of a
rectilinear figure (such as a polygon) to approximate it to the area
of a curvilinear one (such as a circle).
Plato and Aristotle
Plato (born ca. 420 BCE), Socrates’ famed pupil, was himself
not involved in technical mathematics, but his interest in the
subject as a military and philosophical tool led him to establish
his Academy at Athens, which was a training ground for many
pioneering mathematicians. His powerful influence helped Greece
survive its defeat at the hands of the Spartans (404 BCE) and revive
itself to its Old Glory.
Plato’s teacher, Theodorus of Cyrene, proved around 400 BCE
that the square roots of non-square integers from 3 to 17 are
irrational. Eudoxus of Cnidus wrote a work on ratio and proportion.
Menaechmus, a pupil of Eudoxus, was the first to discover and state
the properties of conics - curves formed by the intersection of
planes with cones. Dinostratus, his brother, succeeded in using
Hippias’ quadratrix to “square” the circle; that is, to
construct a square equal in area to a circle.
Aristotle, Plato’s brightest student, was a prolific writer.
Two of his books are mathematical in nature, dealing with
indivisible lines and mechanics. Aristotle’s discussions on what
was truly infinite in magnitude and on the importance of hypotheses
in mathematics influenced many later writers.
The University of Alexandria
After the death of Alexander the Great (323 BCE), Egypt passed
under the rule of Ptolemy I who made Alexandria a commercial and a
scientific centre. Here were established the greatest library of the
ancient world and its first international university. To this
university is linked the names of the mathematicians who would make
the few centuries from 300 BCE to the beginning of the common era
the Golden Age of Greek mathematics.
Undoubtedly the most prominent of these mathematicians is Euclid
(born ca.300 BCE) who succeeded in encompassing the essential parts
of all known mathematical knowledge of his time in one book, The
Elements. The book also contained many original propositions, but it
became famous for its simple and logical explanation. The book had
the effect of implying to mathematicians that the field of basic
geometry was all but exhausted; it was time to move on to higher
geometry and mensuration.
Archimedes (born 287 BCE) is famous for having set besieging
ships on fire with the aid of mirrors and the sun’s rays, and for
discovering the principle of buoyancy in his bathtub. Heralded as a
“genius more divine than human” by an Italian historian, his
contributions to mathematics are as far-reaching as those in the
field of mechanics.
Among his mathematical achievements are the summation of squares
of consecutive numbers, the solution of cubic equations and the
computation of the area of a parabolic segment. He put down
twenty-eight propositions about spiral curves in his work On
Spirals. Archimedes also proved that the volume of a sphere is
one-third the volume of a cylinder with the same radius and height,
and that the surface area of a sphere is four times the area of its
As the Roman invasion of Greece began during the first century of
the Common Era, Greek mathematics suffered, but it did not perish.
Apollonius of Pegra studies conics in detail and wrote on plane
loci. Heron (ca. 50 CE) developed the famous Heron’ fomula for the
area of a triangle (sq.root.(s(s-a)(s-b)(s-c))). During this time,
the Greeks mastered the solution of quadratic equations.
Diophantus (ca. 250 CE) was one of the last great Greek
mathematicians. His principal work, the Arithmetica, covers much
that is known in modern algebra. He described various methods of
solving determinate and indeterminate equations.
When the whole of Western Europe came under the Roman Empire, the
centres of mathematical intellect shifted towards the east - China,
India and the Arabs. But the contribution of the Greeks to
mathematics cannot be overlooked. They set down the basics of
algebra, geometry and mensuration which form a large part of
mathmatics even today.