Running head: EARLY GREEK MATHEMATIC CONTRIBUTIONS 1
Early Greek Mathematic Contributions
Student's Name
Institution
EARLY GREEK MATHEMATIC CONTRIBUTIONS 2
Early Greek Mathematic Contributions
Thales of Miletus was a Greek mathematician and one of the seven sages of Greece who
came from Miletus Ionia. Thales is regarded as the first philosopher in Greek tradition and hence
referred to as the father of science since he was the first person to engage in scientific
philosophy. Thales was a scientist, philosopher, and mathematician known as one of ancient
Greece's seven sages (López-Astorga, 2017). Thales is known to have traveled to Egypt at a
young age, where he went to nurture his career as a scientist and a philosopher. He then returned
to the Greek city of Miletus after completing his academics and opening up a school. He was the
first individual to get involved in scientific philosophy by performing experiments along with his
school teaching. Thales is known to be the first person to start explaining natural objects and
phenomena using naturalistic hypotheses and theories rather than describing them using
mythology. Thales used mathematical geometry to calculate the heights of pyramids and the
distance of ships from the shore, hence being the first known individual to be attributed with
mathematical discovery (O'Grady, 2017). However, it is difficult to talk about all of his
achievements because no writings are available. Therefore, Thales contributed to mathematics
through his Theorem, which involves a triangle inscribed in a circle and having the circle's
diameter as one leg.
Pythagoras was an ancient Greek philosopher and the founder of Pythagoreanism born in
Samos and visited Babylon and Egypt as a young man, according to contemporary accounts.
Pythagoras' religious and political teachings were widely known in Magna Graecia, and the
teachings also influenced the philosophies of Plato and Aristotle, which led to Western
philosophy. Pythagoras traveled to Croton, a city and commune in Calabria, at around 530,
where he founded a school that swore initiates to secrecy and lived a communal, ascetic lifestyle.
Pythagoras was a mathematician, a philosopher, and the founder of the Pythagorean
Brotherhood, which contributed massively to developing Western rational philosophy and
mathematics (Maor, 2019). Pythagoras had disciples, and it is challenging to distinguish his
teachings from those of his disciples in the Pythagorean School. However, Pythagoras has
accredited the Pythagorean Theorem for right angles, which is likely to have been used in
practice in Greece. In addition, much intellectual tradition originating with Pythagoras belongs to
mystical wisdom, not scientific scholarship. Therefore, Pythagoras's contribution to mathematics
is through the Pythagorean Theorem of a right-angled triangle, which states that the square of the
hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.
Hippias of Elis is regarded as a sophist philosopher who contributed to mathematics by
discovering quadratics, a particular curve he used in trisecting angles. Hippias of Elis was a
philosopher and a statesman who in his life used to travel from place to place, lecturing and
collecting wealth. Elis was a small state whose inhabitants controlled the Olympic festival and
was located northwest of the Peloponnesus. Plato refers to Hippias of Elis as arrogant but blessed
with tremendous superficial knowledge. Hippias of Elis is regarded to have been a bright fellow
who had an excellent system of mnemonics, which enabled him to carry immense knowledge
with his mind (Taylor, 2016). Hippias also bragged about claiming some practical skill in the
ordinary arts of life since he used to boast of wearing his own made artefacts, including cloak,
shoes, and seal-ring. Through his continuous lectures, he won himself a considerable reputation
and mass followers, including youths of higher classes. Hippias of Elis is also regarded as a
teacher who lectured in diverse sectors, including grammar, poetry, politics, history, astronomy,
archaeology, and mathematics. He is credited with his significant contribution and work on
EARLY GREEK MATHEMATIC CONTRIBUTIONS 3
collections of Greek, Homer, and foreign literature and archaeological treaties, though nothing
remains for reference other than a few fragments. Hippias of Elis is depicted in two of Plato's
minor dialogues named after him and Plato's Protagoras. Hippias is accredited to have discovered
the quadratics of the circle, which is his significant contribution to mathematics (Taylor, 2016).
The fundamental property of the curve is that it can be used to divide an angle in any given ratio,
hence trisecting the circle. Therefore, Hippias discovered that a turn could be used to trisect an
angle and divide an angle in any given ratio, which was a significant problem in Greek
mathematics when he flourished.
Thales, regarding mathematics, was motivated to solve the problems regarding height and
distance through geometry. For example, Thales used geometry to calculate the height of the
Egyptian pyramids and measure the distance of the ship from offshore. While Thales was in
Egypt, he determined a pyramid's height by measuring its shadow's length while his own shadow
was equal to his height. He used the same scientific methods that we're alluding to in mythology
since his scientific methods were deductive and based on reasoning (O'Grady, 2017). Thus,
Thales was excellent at inventing or discovering new scientific ways in mathematics, hence
being the first true mathematician. He also derived a theorem, commonly referred to as Thales,
based on deductive reasoning. Thales' perception of geometry can be realized from his firmly
held notion that space is the greatest thing as it contains all things. However, Thales used a more
scientific and theoretical approach to discover and develop geometry. In most instances, Thales
used his knowledge in practical ways. For example, Thales used his knowledge of right and
similar triangles to find the height of Egyptian pyramids. In addition, he invented his own
Theorem related to the inscription of triangles in a circle and the Theorem regarding the circle's
diameter to be equivalent to one leg (López-Astorga, 2017). Thales went further in inventing and
developing new mathematical theorems and came up with the intercept theorem. Thales is
known to have discovered a circle that states that if a circle is cut in half by its diameter, the base
angles and vertical angles of a triangle are equal. Thales is credited with the five theorems of
geometry, which include:
1) Angles at the base of an isosceles triangle are equal
2) A circle is bisected by its diameter
3) If two straight lines intersect, the opposite angles formed are equal
4) Any angle inscribed in a semicircle is a right angle, referred to as the Thales theorem.
5) If one triangle has two angles and one side is equal to another triangle, the two triangles
are similar.
Thales theorem
DE
BC =AE
AC =AD
AB
Thales used two of his earlier discoveries, which include the total sum of the angles in a
triangle equaling two right angles and the fact that the base angles of an isosceles triangle are
equal, to come up with the Thales theorem. Thales believed all phenomena could be explained in
natural terms, contradicting his popular belief, arguing that supernatural forces determined
everything. Thales's contributions elevated measurements from practical to philosophical logic.
Thales also bridged the worlds of reason and myth, believing that one must know the world's
nature to understand the world. Modern application of the Thales Theorem states that if three
points A, B, and C lie on a circle's circumference, whereby the line AC is the circle's diameter,
then the angle <ABC is a right angle.
Thales theorem;
Let point M be the midpoint point of line AC.
EARLY GREEK MATHEMATIC CONTRIBUTIONS 4
Also let ∠MBA = ∠BAM = β and ∠MBC =∠BCM =α
Line AM = MB = MC = the radius of the circle.
ΔAMB and ΔMCB are isosceles triangles.
Therefore, using the Thales Theorem, we can accurately draw a tangent to a circle and
find the centre. Applying Thales theorem requires mathematical tools, including a set square and
a sheet of paper.
Pythagoras discovered that a complete system of mathematics could be constructed
where geometric elements corresponded with numbers. Pythagoras discovered that integers and
ratios were necessary to establish an entire system of truth and logic. Pythagoras is commonly
known for his discovery of the Pythagoras' Theorem, which states that for any right-angled
triangle, the length of the hypotenuse (the longest side opposite the right angle) squared is equal
to the sum of the square of the other two sides referred to as legs (opposite and adjacent) (Maor,
2019). However, Pythagoras and his followers did not notice that the Theorem also applied to
any shape. The simplest and most commonly quoted example of a Pythagorean triangle is one
with sides of 3, 4, and 5 units. Pythagorean Theorem and the properties of right-angled triangles
seem to be the most ancient and widespread mathematical development after basic arithmetic
and geometry (Maor, 2019). The Theorem of Pythagoras was also present in some of the most
ancient mathematical texts from Babylon and Egypt, dating long before Pythagoras was born.
However, it was Pythagoras who gave the Theorem its definitive form. The Theorem has ever
since had more than 400 different proofs in modern mathematics, including some algebraic and
some geometrical, while some involve advanced differential equations.
Pythagoras' Theorem
a2+b2=c2
.
The height of the Egyptian pyramids can be determined using modern techniques but
applying the same approach of the Pythagorean Theorem. Modern techniques include advanced
measuring pieces of equipment to ensure the accuracy of the lengths and the measurement units
used. Therefore, a pyramid's height is determined by measuring the length of the slant height
(edges) and half the length of the base to develop a right-angled triangle. Applying the
Pythagorean Theorem,
a2+b2=c2
The edges' length becomes the c, while the half base length
becomes b. Therefore, the height of the pyramid is equal to the square root of
c2−b2=a2
.
Therefore, the pyramid's height is determined by applying Pythagorean theory using the known
measurements of the base length and the length of the edge.
Hippias discovered quadratrix because he used curves to trisect and square the circle.
Hippias' discovery of quadratics states that a curve can be used to divide an angle in any given
ratio and trisect the circle. However, using it for squaring the circle is a more sophisticated
matter and might not be obvious to the original discovery, which can be seen from how the curve
is generated and described by Pappus. The Theorem can be extracted as follows: let ABCD be a
square and BED a quadrant of a circle with centre A. If the circle's radius moves uniformly from
AB to AD while the line BC moves parallel to its original position from BC to AC, then it means
that at any given time, the intersection of the moving radius and the moving straight line will
determine a point F (Sparavigna, 2021). The path traced by F becomes the curve. If one wishes
to trisect the angle EAD, let H be taken on the perpendicular FK to AD such that FK=3. Let a
straight line be drawn through H parallel to AD, and let it connect the curve at P. Let AP be
produced to meet the circle at Q. Then, based on the curve's definition, <QAD is one-third of
<EAD. Therefore, it is obvious that the curve can be used to trisect an angle and divide an angle
in any given ratio. Trisection was a great problem in Greek mathematics when Hippias came into
the limelight.
EARLY GREEK MATHEMATIC CONTRIBUTIONS 5
The quadratrix of Hippias is applicable in modern mathematics in squaring circles since
squaring a circle using a ruler and a compass is impossible. The quadratrix of Hippias enables
turning a quarter circle into a square of the same area. Therefore, a square twice the length has
the same area as the full circle.
The mathematical discoveries by the Greek mathematicians had several impacts on Greek
culture, including architecture. Greek mathematics was influenced by the time's magical,
mythological and philosophical thinking. Influenced by mathematics developed by previous
civilizations, including Egypt or the Phoenicians, Greeks perceived the discipline of mathematics
as the only key to understanding the world and reaching absolute truth. Mathematics was the
supreme form of beauty and truth to the Greeks since it was above obvious usefulness. The
people's philosophical conceptions of mathematics made the Greeks deny their intuition. For
example, an ancient mathematician named Iamblichus developed the number zero, which is still
known as zero today, but it had less impact on the Greeks since it contradicted their conception
of reality at the time. Other philosophers concluded that there is no ratio in which the void is
exceeded by the body since there is no ratio of zero to a number; hence, the void could not bear a
ratio to the full. In addition, due to ancient mathematical advancements in Greece, the Greeks
addressed the notion of infinity differently from the modern definition. Greeks perceived infinity
as an enumerative vision or a quantity that grows indefinitely. Mathematical ideas also had a
significant impact on the Greeks' magical meaning. They used numbers to represent different
archetypes, including masculinity, feminism, and family. They considered ten a magic number
since it equals the sum of its positive divisors, and also, 10 had transcendental quality in its
recurring appearance in the physical world. The Greeks also considered the straight line and the
circle to be the purest forms. Greeks also personified mathematics in their myths. An example is
the myth of "the wedding of mercury and philosophy," where geometry is a character who talks
about her sister's principles, arithmetic, and her principles, affirming they are both incorporeal.
Mathematics influenced Greek architecture, which remains an illustration of ancient
Greeks' beliefs regarding the cosmos and chaos and their curiosity to achieve the highest degree
of perfection. There are several structures in Greece which are regarded as a marvel created by
human civilization, including the Parthenon and many other temples. The main pillar of
architecture art has been mathematics, and its development in ancient Greece enabled Ancient
Greek architects to build excellent buildings (Chiotis, 2021). Several branches of mathematics
provided Greek architects with the necessary tools for proper design and construction, like
geometry. In developing ancient Greek architecture, concepts like proportion, golden ratio, and
grid were very instrumental.
In addition, numerical symbolism developed within mathematics also significantly
impacted the development of ancient Greek architecture. Ancient Greek architecture employed
the concept of proportion, which Pythagoras developed after being influenced by Egyptian and
Persian mathematical advances (Chiotis, 2021). Symmetry also enabled ancient Greek
architecture to build numerous outstanding structures and buildings. For example, the concept of
symmetry developed by Pythagoras was utilized in building the Parthenon since the temple
construction was based on the use of ratios. The idea provided architects with a fundamental
strategy of moving from a minor part to more significant amounts, leading to the construction of
many more buildings characterized by the mathematical symmetry concept. In addition, Greek
architects also benefited significantly from the idea of the grid, which was developed by a Greek
mathematician named Euclid. The concept helped Greek architects create an abstract
representation of space and generalizations from particular to whole. An example of an
EARLY GREEK MATHEMATIC CONTRIBUTIONS 6
illustration of a grid concept in design is the city of Olynthus since the idea made the city layout
neat and easy to navigate. The Greek architects created the effect of order within cities by
building standard rectangular rooms in temples and other buildings. The ancient Greek towns
also had streets that were characterized by the utilization of the grid mathematical concept since
the roads were at right angles to each other.
Most of the mathematical advancements contributed by the ancient Greek
mathematicians gave birth to modern mathematics, hence referred to as the co-founders of
mathematics. Modern mathematics is a result of developments of the ancient mathematical
concepts that ancient Greek mathematicians discovered. Several formulae and equations used in
modern mathematics are even named after ancient Greek mathematicians, including Pythagoras
and Thales, among other mathematicians. Therefore, the ancient Greek mathematicians gave the
basis for modern mathematics and references to rely on while advancing and developing more
accurate formulae. However, some ancient Greek mathematicians' discoveries remained
unchanged and applied as the ancient mathematicians developed them. For example, Pythagoras'
theory of right angles, which Pythagoras developed, has remained unchanged over the years and
is still applicable in modern mathematics as it was in ancient mathematics. However, the
approach has been used to solve more mathematical problems since it is not limited to only right-
angled triangles. Therefore, the ancient Greek mathematicians gave birth to modern
mathematics, which is crucial in understanding the world through encouraging logical reasoning,
creative thinking, critical thinking, and problem-solving abilities.
EARLY GREEK MATHEMATIC CONTRIBUTIONS 7
References
Chiotis, E. D. (2021). PYTHAGORAS'MATHEMATICS IN ARCHITECTURE AND HIS
INFLUENCE ON GREAT CULTURAL WORKS.GScientific Culture,G7(1), 57-77.
https://www.academia.edu/download/64934524/Pythagoras_Mathematics_in_Architectur
e_and_his_Influence_on_Great_Cultural_Works.pdf
López-Astorga, M. (2017). Thales of Miletus and the semantic possibilities of his view of the
soul.GAisthema, International Journal,G4(1), 101-112.
http://www.aisthema.eu/ojs/index.php/aisthema/article/view/42
Maor, E. (2019).GThe Pythagorean Theorem: a 4,000-year history. Princeton University Press.
https://books.google.co.ke/books?
hl=en&lr=&id=XuWZDwAAQBAJ&oi=fnd&pg=PP9&dq=Pythagoras+contribution+to
+mathematics&ots=5kTra5Fcl5&sig=vbULMOKM16ZqC3uXOgK4OHK_Pq4&redir_e
sc=y#v=onepage&q=Pythagoras%20contribution%20to%20mathematics&f=false
O'Grady, P. F. (2017).GThales of Miletus: the beginnings of western science and philosophy.
Routledge. https://www.taylorfrancis.com/books/mono/10.4324/9781315241548/thales-
miletus-patricia-grady
Sparavigna, A. C. (2021). Some Curves and the Lengths of their Arcs. https://hal.archives-
ouvertes.fr/hal-03236909/
Taylor, C. C. W. (2016). Hippias (2), of Elis, sophist. InGOxford Research Encyclopedia of
Classics.
https://oxfordre.com/classics/view/10.1093/acrefore/9780199381135.001.0001/acrefore-
9780199381135-e-3111
