History of Mathematics: Hellenic Traditions

History of Mathematics: 

Hellenic Traditions 

Chaogui Zhang 

Department of Mathematics 

Marywood University 

Chaogui Zhang History of Mathematics: Hellenic Traditions

The Era and the Sources 

The civilizations along the Nile, Tigris and Euphrates rivers 

started losing their leading postions gradually well before 

the Christian era, as new cultures sprung up along the 

shores of the Mediterranean Sea. 







The Thalassic Age (the ”sea” age, from approximately 800 

BCE to 800 CE) indicates this shift in the centers of 

civilization. 









civilization. 

The first portion of the Thalassic Age is labeled the 

Hellenic era and older cultures are known as pre-Helenic. 









civilization. 

The first portion of the Thalassic Age is labeled the 

Hellenic era and older cultures are known as pre-Helenic. 

The ancient Greek history, starting from the second 

millenium BCE, is a history of the Greeks rapidly learning, 

absorbing, improving the knowledge they got in contact. 



The early Greek alphabet seems to have originated 

between the Babylonian and Egyptian worlds, through a 

process of drastic reduction in the number of cuneiform or 

hieratic symbols. 







The alphabet soon found its way to the new colonies – 

Greek, Roman, and Carthaginia – through the activities of 

traders, and the Greeks made their way to the center of 

learning in Egypt and Babylonia. 







The alphabet soon found its way to the new colonies – 

Greek, Roman, and Carthaginia – through the activities of 

traders, and the Greeks made their way to the center of 

learning in Egypt and Babylonia. 

The first Olympic Games were held in 776 BCE, and the 

Greek literature had developed by then, whereas there was 

no record of any mathematical development (directly or 

indirectly). 



Another two centuries would pass before accounts of the 

Greek mathematics would appear. The main figures, in the 

6th century BCE, were Thales and Pythagoras. 






A number of definite mathematical discoveries were 

ascribed to Thales and Pythagoras by tradition, although 

no work of theirs survived and it is not even certain that 

theycomposed any such masterpieces. 






A number of definite mathematical discoveries were 

ascribed to Thales and Pythagoras by tradition, although 

no work of theirs survived and it is not even certain that 

theycomposed any such masterpieces. 

There are no extant mathematical or scientific documents 

until those from the days of Plato in the 4th century BCE. 



During the ”Heroic Age of Mathematics” (the second half of 

the 5th century BCE), there circulated persistent and 

consistent reports concerning a handful of mathematicians. 






Through the work of Archytas, Hippasus, Dmocritus, 

Hippias, Hippocrates, Anaxagoras, and Zeno that survived, 

we can see some of the fundamental changes that took 

place in the years just before 400 BCE. 






Through the work of Archytas, Hippasus, Dmocritus, 

Hippias, Hippocrates, Anaxagoras, and Zeno that survived, 

we can see some of the fundamental changes that took 

place in the years just before 400 BCE. 

Mathematical sources from the 4th century BCE are 

almost as scarce, but the works by philosophers such as 

Plato and Aristotle give us a far more dependable account 

of what happened in their days than we could about the 

Herioc Age. 


Thales and Pythagoras 

The two major figures, Thales and Pythagoras, in the 6th 

century BCE were the first ones in Greek history with 

significant mathematical work attributed to. The accounts 

of the contributions of these men, although widely 

accepted by tradition, are not by direct historical 

documents. 


Thales and Pythagoras 

The two major figures, Thales and Pythagoras, in the 6th 

century BCE were the first ones in Greek history with 

significant mathematical work attributed to. The accounts 

of the contributions of these men, although widely 

accepted by tradition, are not by direct historical 

documents. 

Both Thales and Pythagoras were said to have traveled to 

Egypt andBabylon, and learned about the Egyptian and 

Mesopotamian knowledge on astronomy and mathematics. 


Thales of Miletus (ca. 624 - 548 BCE) 

Little is known for certain regarding the life and work of Thales. 




His birth and death are gross estimations only and are 

based on stories that are not completely trustworthy. For 

example, reportedly he predicted an eclipse of the Sun on 

May 28, 585 BCE, but the authenticity of such legend is 

seriously doubted by many. 









He is unanimously regarded in ancient opinion as an 

unusually clever man and the first philosopher (the first of 

the Seven Wise Men). 









He is unanimously regarded in ancient opinion as an 

unusually clever man and the first philosopher (the first of 

the Seven Wise Men). 

He was believed to have learned from the Egyptians and 

the Chaldeans (the 11th dynasty of Babylon). 



Legend has it that Thales proved the following results in 

geometry and he is often regarded as the first true 

mathematician. 






The Theorem of Thales (most likely learned from the 

Babylonians). 







Babylonians). 

A circle is bisected by a diameter. 







Babylonians). 


The base angles of an isosceles triangle are equal. 







Babylonians). 



The pairs of vertical angles formed by two intersecting 

lines are equal. 







Babylonians). 



The pairs of vertical angles formed by two intersecting 

lines are equal. 

If two triangles are such that two angles and a side of one 

are equal respectively to two angles and a side of the 

other, then the triangles are congruent. 



There is no direct evidence supporting these achievements of 

Thales. Indirectly, we have the following: 





Eudemus of Rhodes (fl. ca. 320 BCE) wrote a history of 

mathematics (now lost). 







Before its disappearance, someone summarized part of 

the history by Eudemus, the original of which is again lost. 









The Neoplatonic philosopher Proclus (410 – 485) included 

a summary of the summary in his Commentary on the First 

Book of Euclid’s Elements. 









The Neoplatonic philosopher Proclus (410 – 485) included 

a summary of the summary in his Commentary on the First 

Book of Euclid’s Elements. 

It is widely accepted today that the Greeks added logical 

structure to the study of geometry. What’s uncertain is 

whether Thales deserves all the credit he has been given. 


Pythagoras of Samos (ca. 580 - 500 BCE) 

Pythagoras, born at Samos, is himself another controversial 

figure due to the many legends surrounding his life and work. 





Some believe Pythagoras was a student of Thales, but this 

is unlikely given their age difference. 







It is certain though that he traveled to Egypt and Babylon, 

just like Thales did. He may have traveled to as far as 

India. 









India. 

Biographies of Pythagoras were written, including one by 

Aristotle, but they did not survive. 









India. 

Biographies of Pythagoras were written, including one by 

Aristotle, but they did not survive. 

Pythagoras was, incidentally, a virtual contemporary of 

Buddha, Confucius, and Laozi (Lao-tzu) – prominent 

figures in the East. 





His obviously secret society (established after he returned 

to the Greek world) did not attribute discoveries to 

individual members. So it often is best to speak of the work 

of the Pythagoreans, rather than that of Pythagoras 

himself. 







himself. 

The Pythagorean school was politically conservative and 

with a strict code of conduct. 







himself. 



They regarded the pursuit of philosophical and 

mathematical studies as a moral basis for the conduct of 

life. 







himself. 



They regarded the pursuit of philosophical and 

mathematical studies as a moral basis for the conduct of 

life. 

The new emphasis in mathematics on philosophical 

discussion of principles, although started by Thales, was 

primarily due to the Pythagoreans according to Eudemus 

and Proclus. 





The motto of the Pythagorean school ”All is number” is 

often seen as a strong link to the Mesopotamians, who 

also attached numerical measures to things around them. 






The Pythagorean Theorem was likely derived from the 

Babylonians, while some suggested that the Pythagoreans 

provided the first demonstration or proof, which cannot be 

verified by any historical accounts. 










It is not certain whether the Pythagoreans can be credited 

with the construction of the ”cosmic figures” (i.e., the 

regular solids), as the Eudemus-Proclus summary did. 










It is not certain whether the Pythagoreans can be credited 

with the construction of the ”cosmic figures” (i.e., the 

regular solids), as the Eudemus-Proclus summary did. 

A scholium in Euclid’s ”Elements XIII” reports that the 

Pythagoreans knew only three regular polyhedra: the 

tetrahedron, the cube, and the dodecahedron. 


The Pythagorean Pentagram 

The pentagram (a five-pointed star) is said to be the special 

symbol of the Pythagorean school. 









Earlier Babylonian art 

contained the star 

pentagon and it could be 

that Pythagoras first got in 

contact with such a shape 

during his travel to 

Babylon. 





Earlier Babylonian art 

contained the star 

pentagon and it could be 

that Pythagoras first got in 

contact with such a shape 

during his travel to 

Babylon. 

One tantalizing question 

in Pythagorean geometry 

is the construction of a 

pentagram. 



The ancient Greeks were aparently very familiar with what we 

call ”golden section” today. The construction of a pentagram is 

closely related to ”the section”. 






B 

A 

D ′ 

E ′ 

C 

C ′ 

A ′ 

B ′ 

E 

D 






B 

A 

D ′ 

E ′ 

C 

C ′ 

A ′ 

B ′ 

E 

D 

Note that, for example, A ′ 

divides the diagonal BD 

such that 

BD : BA ′ = BA ′ : A ′ D 






B 

A 

D ′ 

E ′ 

C 

C ′ 

A ′ 

B ′ 

E 

D 

Note that, for example, A ′ 

divides the diagonal BD 

such that 

BD : BA ′ = BA ′ : A ′ D 

It is likely that the 

Pythagoreans used a 

geometric construction 

process rather than an 

algebraic approach to the 

”golden section” problem. 


Number Mysticism 

The Pythagoreans, like many others in early civilizations, 

attached special meanings to numbers. 





Pythagoreans were not the only people who associated 

odd numbers with male attributes and even numbers 

female. 







female. 

They believed everything is about numbers (their motto: All 

is number). 







female. 


is number). 

Each number had its peculiar attributes in the Pythagorean 

school of thought. 







female. 


is number). 

Each number had its peculiar attributes in the Pythagorean 

school of thought. 

The number ten was viewed as the number of the 

universe, representing all dimensions(1+2+3+4). 


Arithmetic and Cosmology 



The Pythagoreans reserved the word ”number” for whole 

numbers and viewed fractions as a ratio or relationship 

between two whole numbers. 






A fraction is not simply a value any more. A deeper, more 

theoretical understanding of the number concept started to 

emerge. 








emerge. 

Their admiration of the number ten seems to have provided 

the inspiration for the earliest nongeocentric astronomical 

system. 








emerge. 

Their admiration of the number ten seems to have provided 

the inspiration for the earliest nongeocentric astronomical 

system. 

Philolaus (ca. 470 - ca. 385 BCE), who is a member of the 

Pythagorean school proposed that there was a central fire 

in the center of the universe about which the earth and the 

seven planets (including the sun and the moon) revolved 

uniformly. He further assumed there must be a tenth 

”counterearth”. 


Figurate Numbers 



The Pythagoreans studied the triangular, square, 

pentagonal, and in general polygonal numbers. 





Triangular numbers: 1 + 2 + 3 + · · · + n = 

n(n + 1) 

2 






n(n + 1) 

2 

Square numbers: 1 + 3 + 5 + · · · + (2n − 1) = n 2 






n(n + 1) 

2 


Pentagonal numbers: 1 + 4 + 7 + · · · + (3n − 2) = 

Chaogui Zhang History of Mathematics: Hellenic Traditions 

n(3n − 1) 

2





n(n + 1) 

2 



n(3n − 1) 

2 

Hexagonal numbers: 1 + 5 + 9 + · · · + (4n − 3) = 2n 2 − n 






n(n + 1) 

2 



n(3n − 1) 

2 

Hexagonal numbers: 1 + 5 + 9 + · · · + (4n − 3) = 2n 2 − n 

Simple laws of music were said to be discovered by 

Pythagoras. These could be the earliest quantitative laws 

of acoustics, and the oldest of all quantitative physical laws. 


Proportions 

The theory of proportionals is one of the two SPECIFIC 

mathematical discoveries that Proclus ascribed to Pythagoras. 

Even if this may not be literally true, it is certainly plausible and 

consistent with the Pythagorean school of thought. 


Proportions 





Reportedly Pythagoras learned in Mesopotamia of the 

three means: the arithmetic mean, the geometric mean 

and the subcontrary (harmonic) mean. He also learned of 

the ”golden proportion”, which relates the arithmetic mean 

to the harmonic mean. 


Proportions 










These ideas were generalized by the Pythogareans to 

make a total of ten different means. (Remember how much 

they loved the number ten?) 


Proportions 










These ideas were generalized by the Pythogareans to 

make a total of ten different means. (Remember how much 

they loved the number ten?) 

All ten of these means are expressed with unifying 

equations of proportions. 


Proportions: Related Number Theory Results 

It is hard to know when the Pythagoreans studied the 

proportions and equality of ratios. It is presumed that such 

studies were part of their pursuit in the theory of numbers, 

which later became a study of geometric magnitudes. 







The Pythagoreans classfied numbers to different 

categories, depending on the attributes being studied. 

Examples are: odd and even numbers, polygonal numbers, 

and the so-called odd-odd and even-odd numbers. 











We do know that by the time of Philolaus, the concepts of 

prime and composite numbers were well defined. 













Traditionally, the Pythagorean triads have been ascribed to, 

as the name suggests, the Pythagoreans. It could have 

been learned from the Babylonians. 













Traditionally, the Pythagorean triads have been ascribed to, 

as the name suggests, the Pythagoreans. It could have 

been learned from the Babylonians. 

If a = (m 2 − 1)/2, b = m, and c = (m 2 + 1)/2 for an odd 

integer m, then a 2 + b 2 = c 2 . 



The classifications of perfect, abundant and deficient numbers 

are also attributed to the Pythagoteans, but it is unknown at 

what time period they studied these numbers. It is conjectured 

that these are probably later developments, as is the concept of 

amicable numbers. 








A perfect number is one such that the sum of its proper 

divisors is equal to the number itself. 









divisors is equal to the number itself. For example, 

1 + 2 + 3 = 6, so 6 is a perfect number (the smallest). 











By changing ”equal to” to ”greater than” or ”less than”, we 

get abundant and deficient numbers. 











By changing ”equal to” to ”greater than” or ”less than”, we 

get abundant and deficient numbers. For example, 

1 + 3 < 9, so 9 is a deficient number. 

1 + 2 + 3 + 4 + 6 > 12, so 12 is an abundant number. 


Numeration 

There have been two chief systems of numeration in Greece: 

the Attic (Herodianic) notation and Ionian (alphabetic) system. 


Numeration 



The Attic notation (the earlier one), is a ten-scaled, iterative 

scheme, similar to the Egyptian hieroglyphic numerations 

that came earlier and the Roman numerals that came later. 


Numeration 






Repeated vertical strokes were used for numbers one to 

four and the letter Π , first letter of the word five (pente), 

was used for five. The powers of ten also use the initial 

letters of the corresponding number words: Δ for ten 

(deka), Η for hundred (hekaton), Χ for thousand (khilioi), 

and Μ for ten thousand (myrioi). 


Numeration 






Repeated vertical strokes were used for numbers one to 

four and the letter Π , first letter of the word five (pente), 

was used for five. The powers of ten also use the initial 

letters of the corresponding number words: Δ for ten 

(deka), Η for hundred (hekaton), Χ for thousand (khilioi), 

and Μ for ten thousand (myrioi). 

One difference between the Attic and the Roman 

numerals: The Greeks combined (multiplicatively) the 

symbol for five with the powers of ten to represent numbers 

like 50, 500 etc. For example, the following represents 

45,678: ΜΜΜΜ

Numeration 

The later one of the two chief numeration systems is the 

Ionian or alphabetic numerals. 


Numeration 



The Attic numerals were found in inscriptions at various 

dates from 454 BCE to 95 BCE. The Ionian system started 

to take over by the early Alexandrian Age. 


Numeration 



The Attic numerals were found in inscriptions at various 

dates from 454 BCE to 95 BCE. The Ionian system started 

to take over by the early Alexandrian Age. 

The following is the association between letters and 

numbers: 

α β γ δ ε ϛ ζ η θ 

1 2 3 4 5 6 7 8 9 

ι κ λ μ ν ξ ο π ϟ 

10 20 30 40 50 60 70 80 90 

ρ ς τ υ φ χ ψ ω ϡ 

100 200 300 400 500 600 700 800 900 


Arithmetic and Logistic 

Due to a lack of documents from the 600 to 450 BCE, the 

early development of mathematics in Greece from that 

period of time is in many respect unknown to us, compared 

to the Babylonian algebra or Egyptian geometry. 







Some form of counting board or abacus was used for 

calculations, but not much more is known regarding these 

devices. 









devices. 

The little known descriptions of these devices show the 

tendancy of ancient civilizations to avoid excessive use of 

fractions, simply by subdivide units into ever smaller ones 

so that most calculations can be done as integral multiples 

of such smaller units. 









devices. 

The little known descriptions of these devices show the 

tendancy of ancient civilizations to avoid excessive use of 

fractions, simply by subdivide units into ever smaller ones 

so that most calculations can be done as integral multiples 

of such smaller units. 

The Pythagoreans regarded the technical details in 

computation as a separate discipline, called logistic. Their 

study of the essence and properties of numbers then was 

at a much more abstract and philosophical level. 


Fifth-Century (BCE) Athens 

The period in the fifth century BCE between the defeat of 

the Persian invaders and the surrender of Athens to Sparta 

is the great Age of Pericles. Scholars from all parts of the 

Greek world were attracted to the prosperity and 

intellectual atmosphere of Athens. 








A systhesis of diverse aspects was achieved with the 

convergence of scholars to Athens. Anaxagoras from Ionia 

had a practical turn of mind, while Zeno from southern Italy 

came with stronger metaphysical inclinations. 








A systhesis of diverse aspects was achieved with the 

convergence of scholars to Athens. Anaxagoras from Ionia 

had a practical turn of mind, while Zeno from southern Italy 

came with stronger metaphysical inclinations. 

Anaxagoras represented the spirit of rational inquiry. He 

was imprisoned in Athens for his assertion that the sun 

was nothing but a huge red-hot stone, and that the moon 

was an inhabited earth that borrowed its light from the sun. 


Three Classical Problems 

Anaxagoras attempted to square the circle while he was in 

prison, according to Plutarch. No details about the origin or 

rules were mentioned. 






This problem is one of the three classical problems that 

fascinated mathematicians for over 2,000 years, all of 

which require that only a compass and a straightedge are 

allowed in the construction. 










Squaring the circle: construct a square that has the same 

area of a given circle. 












Duplication of the cube (a.k.a. Delian problem): construct a 

cube that has twice the volume of a given cube. 












Duplication of the cube (a.k.a. Delian problem): construct a 

cube that has twice the volume of a given cube. 

Trisecting an angle: construct an angle one-third as large 

as the given angle. 


Quadrature of Lunes 

Hippocrates of Chios lost his money on his way to Athens 

in about 430 BCE, as a merchant. He turned to the study 

of geometry after that and achieved remarkable success. 






According to Proclus, Hippocrates composed a textbook 

titled Elements of Geometry, more than a century before 

the more famous Elements by Euclid. It has been lost, like 

another by Leon (a later associate of the Platonic school), 

although it was known to Aristotle. 











No mathematical treatise from the fifth century BCE 

survived, but we have a fragment concerning Hippocrates 

that is claimed to be a literal copy by Simplicius (fl. ca. 520 

CE) from the History of Mathematics (now lost) by 

Eudemus. 











No mathematical treatise from the fifth century BCE 

survived, but we have a fragment concerning Hippocrates 

that is claimed to be a literal copy by Simplicius (fl. ca. 520 

CE) from the History of Mathematics (now lost) by 

Eudemus. 

The brief statement describes a portion of the work of 

Hippocrates on the quadrature of lunes. 



A lune is a figure bounded by two circular arcs of unequal 

radii. The problem of quadrature of lunes is closely related 

to the the problem of squaring the circle. 






Eudemus believed that Hippocrates gave a proof of the 

following theorem: Similar segments of circles are in the 

same ratio as the square on their bases. 









From the above theorem, Hippocrates found the first 

rigorous quadrature of a curvilinear area in the history of 

mathematics. 









From the above theorem, Hippocrates found the first 

rigorous quadrature of a curvilinear area in the history of 

mathematics. 

Hippocrates also recognized that the continued proportion 

a : x = x : y = y : b problem is equivalent to duplication of 

the cube when b = 2a. 


Hippias of Elis 

Hippias of Elis was active in Athens in the second half of 

the fifth century BCE. He was a representative of the 

Sophists, described as vain, boastful and acquisitive, 

typical characteristics of the Sophists. 







He is one of the earlist mathematicians that we have 

firsthand information from Plato’s dialogues, although none 

of his work has survived. 










Hippias introduced the first beyond the circle and the 

straight line, known as the trisectrix or quadratrix of 

Hippias. 










Hippias introduced the first beyond the circle and the 

straight line, known as the trisectrix or quadratrix of 

Hippias. 

The curve allows one to trisect an angle easily as well as 

to square a circle. 


Philolaus and Archytas of Tarentum 

Philolaus of Tarentum was among those that received 

instructions from the scholars escaped the massacre at the 

Pythagorean center at Croton. 






He is said to have written the first account of 

Pythagoreanism, from which Plato derived his knowledge 

of the Pythagorean order. 









Archytas was a student of Philolaus’ at Tarentum, who 

believed firmly in the efficacy of number, but his 

enthusiasm for number had less of the religious and 

mystical admixture found earlier in Philolaus. 









Archytas was a student of Philolaus’ at Tarentum, who 

believed firmly in the efficacy of number, but his 

enthusiasm for number had less of the religious and 

mystical admixture found earlier in Philolaus. 

Archytas gave a three-dimensional solution of the Delian 

problem, but his most important contribution to 

mathematics may have been his intervention with the 

tyrant Dionysius to save the life of Plato. 


Incommensurability 

It’s one of the fundamental beliefs of the Pythagoreans that 

the essence of all things is explainable in terms of 

arithmos, or intrinsic properties of whole numbers or their 

ratios. 






ratios. 

Consequently, the Greek mathematical community was 

stunned by the discovery that even in geometry itself, 

whole numbers and their ratios are inadequate to account 

for even simple fundamental properties. 






ratios. 





The diagonal of a square or a cube or a pentagon is 

incommensurable with its side, no matter how small a unit 

of measure is chosen. 






ratios. 





The diagonal of a square or a cube or a pentagon is 

incommensurable with its side, no matter how small a unit 

of measure is chosen. 

It is uncertain when or how the earliest incommensurable 

line segments were recognized, but it is assumed that it 

happened with the application of the Pythagorean theorem 

to the isosceles right triangle. 



One proof of of incommensurability of the diagonal of a square 

with its side: 





Let d and s be the diagonal and the side of a square, and 

assume they are commensurable. In other words, 

d/s = p/q for some integers p and q with no common 

factors. 








factors. 

Then by Pythagorean theorem, d 2 = s 2 + s 2 = 2s 2 , and 

(d/s) 2 = p 2 /q 2 = 2. 








factors. 


(d/s) 2 = p 2 /q 2 = 2. 

This means p 2 = 2q 2 must be even, and p itself must also 

be even. Therefore q must be odd. 








factors. 


(d/s) 2 = p 2 /q 2 = 2. 



If we let p = 2r, then (2r) 2 = 2q 2 , i.e., 4r 2 = 2q 2 or 

2r 2 = q 2 . 








factors. 


(d/s) 2 = p 2 /q 2 = 2. 



If we let p = 2r, then (2r) 2 = 2q 2 , i.e., 4r 2 = 2q 2 or 

2r 2 = q 2 . 

This shows q 2 and q must be even, but it was shown to be 

odd earlier. This contradiction means that the assumption 

that d and s are commensurable must be false. 



There are other ways in which the discovery of 

incommensurable segments could have come about. 





One observes easily that the diagonals of a regular 

pentagon form a smaller regular pentagon inside. 







Most importantly, this process can be continued 

indefinitely, and consequently one can construct 

pentagons as small as desired in this way. 










That implies the ratio of a diagonal to a side in a regular 

pentagon is not rational, i.e., they are incommensurable. 










That implies the ratio of a diagonal to a side in a regular 

pentagon is not rational, i.e., they are incommensurable. 

In this case, it would be √ 5 instead of √ 2 that 

demonstrates the incommensurability. 


Paradoxes of Zeno 

The Pythagorean doctrine faced a very serious challenge 

from the discovery of the incommensurability, and there 

were more. 





were more. 

The Eleatics, a rival phlosophical movement, started by 

Parmenides of Elea (fl. ca. 450 BCE). 





were more. 



Zeno of Elea was the most best known student of 

Parmenides’, whose paradoxes caused a great deal of 

trouble, particularly the following on motion: 





were more. 






The Dichotomy, the Achilles, the Arrow, the Stade 





were more. 






The Dichotomy, the Achilles, the Arrow, the Stade 

These paradoxes are at the heart of the continuity concept 

for space and time, and they had a profound influence on 

the development of Greek mathematics. 


Deductive Reasoning 

There is great uncentainty about when deductive 

reasoning, or incommensurability entered into the Greek 

mathematics, but by the time of Plato, it had undergone 

drastic changes for sure. 


Deductive Reasoning 

There is great uncentainty about when deductive 

reasoning, or incommensurability entered into the Greek 

mathematics, but by the time of Plato, it had undergone 

drastic changes for sure. 

A ”geometric algebra” took the place of the older 

”arithmetic algebra”. 


Democritus of Abdera 

Democritus of Abdera is better known today as a chemical 

philosopher, but he also had a reputation of a geometer. 





He traveled more widely than anyone of his day, learning 

all he could whereever he went. 







In his physical doctrine of atomism, he argued that all 

phenomena were to be explained in terms of indefinitely 

small and infinitely varied (in size and shape), impenetrably 

hard atoms moving about ceaselessly in empty space. 











Not surprisingly, the mathematical problems with which he 

was chiefly concerned were those that demand some sort 

of infinitesimal approach. 











Not surprisingly, the mathematical problems with which he 

was chiefly concerned were those that demand some sort 

of infinitesimal approach. 

Archimedes later wrote that the volume formula of a 

pyramid was due to Democritus but he did not prove it 

rigorously. 


The Main Problems of the Heroic Age 

The chief mathematical legacy of the Heroic Age can be 

summed up in the following six problems: 





The squaring of the circle 






The duplication of the cube 







The trisection of the angle 








The ratio of incommensurable magnitudes 









The paradoxes on motion 










The validity of infinitesimal methods 










The validity of infinitesimal methods 

To some extent, one may associate these problems, 

although not exclusively, with the men considered in this 

Chapter: Hippocrates, Archytas, Hippias, Hippasus, Zeno, 

and Democritus. 


The Academy 

Although Archytas was included among the 

mathematicians of the Heroic Age, he really was a 

transitional figure during Plato’s time, as one of the last 

Pythagoreans, literally and figuratively. 


The Academy 





The fourth century BCE had opened with the death of of 

Socrates, a scholar who repudiated the Pythagoreanism of 

Archytas. Deep metaphysical doubts precluded a Socratic 

concern with either mathematics or natural sciences. 


The Academy 









Surprisingly, Plato, a student and admirer of Socrates, 

became the mathematical inspiration of the fourth century 

BCE. It was undoutedly Archytas, a friend, who converted 

Plato to a mathematical outlook. 


The Academy 









Surprisingly, Plato, a student and admirer of Socrates, 

became the mathematical inspiration of the fourth century 

BCE. It was undoutedly Archytas, a friend, who converted 

Plato to a mathematical outlook. 

There were also six mathematicians who lived between the 

death of Socrates in 399 BCE and the death of Aristotle in 

322 BCE that are described here. They are all associated 

with the Academy, more or less. 


The Academy 

Plato wrote about the regular solids in a dialogue titled 

Timaeus, and he applied them to the explanation of 

scientific phenomena by associating the four elements with 

the regular solids. 


The Academy 





Although Proclus attributes the construction of the ”cosmic 

figures” (regular solids) to Pythagoras, A scholium to Book 

XIII of Euclid’s Elements reports that only three of the 

regular solids were due to the Pythagoreans. 


The Academy 









The scholium reports that Theaetetus, a friend of Plato’s, 

that discovered the octahedron and the icosahedron 


The Academy 









The scholium reports that Theaetetus, a friend of Plato’s, 

that discovered the octahedron and the icosahedron 

Theaetetus made one of the most extensive studies of the 

five regular solids, and it was probably due to him the 

theorem that there are five and only five regular polyhedra. 


The Academy 

In the Platonic dialogue named after him, Theaetetus was 

also discussing the nature of incommensurable 

magnitudes with Socrates and Theodorus. 


The Academy 




Theodorus was another mathematician whom Plato 

admired and who contributed to the early development of 

the theory of incommensurable magnitudes. He was the 

teacher of both Plato and Theaetetus. 


The Academy 








Theodorus was said to have proven the irrationality of the 

square roots of the nonsquare integers from 3 to 17 

inclusive. 


The Academy 








Theodorus was said to have proven the irrationality of the 

square roots of the nonsquare integers from 3 to 17 

inclusive. 

There’s evidence that Theodorus made discoveries in 

elementary geometry that later were incorporated into 

Euclid’s Elements, but the works of Theodorus are lost. 


Eudoxus 

The Platonic Academy in Athens became the 

mathematical center of the world, and it was from this 

school that leading teachers and researchers came during 

the middle of the fourth century BCE. Eudoxus was the 

greatest among them. 


Eudoxus 

The Platonic Academy in Athens became the 

mathematical center of the world, and it was from this 

school that leading teachers and researchers came during 

the middle of the fourth century BCE. Eudoxus was the 

greatest among them. 

One of the main achievements of Eudoxus was the 

formulation of a rigorous theory of proportion, which was 

later used in Book V of Euclid’s emphElements: 

Magnitudes are said to be in the same ratio, the first to the 

second and the third to the fourth, when, if any 

equimultiples whatever be taken of the first and the third, 

and any equimultiples whatever of the second and fourth, 

the former equimultiples alike exceed, are alike equal to, or 

are alike less than, the latter equimultiples taken in 

corresponding order. 


Eudoxus 

The other main achievement of Eudoxus was on the 

method of exhaustion. 


Eudoxus 



Archimedes credited Eudoxus with the lemma now called 

the axiom of Archimedes, or axiom of continuity. 


Eudoxus 





This axiom served as the basis for the method of 

exhaustion, the Greek equivalent of the integral calculus. 


Eudoxus 







The axiom states that given two magnitudes having a ratio 

(i.e. neither being zero), one can find a multiple of either 

one that will exceed the other. 


Eudoxus 







The axiom states that given two magnitudes having a ratio 

(i.e. neither being zero), one can find a multiple of either 

one that will exceed the other. 

From this, it is easy to prove, by a reductio ad absurdum, 

the exhaustion property, which is equivalent to 

lim 

n→∞ M(1 − r)n = 0 for M > 0 and 1/2 ≤ r < 1. 


Eudoxus: Mathematical Astronomy 

Eudoxus was not only a great mathematician. He is known 

in the history of science as the father of scientific 

astronomy. 





astronomy. 

Eudoxus gave a geometric representation of the 

movements of the sun, the moon, and the five known 

planets, through a composite of concentric spheres with 

centers at the earth and with varying radii, each sphere 

revolving uniformly about an axis fixed with respect to the 

surface of the next larger sphere. 





astronomy. 







Eudoxus was undoubtedly the most capable 

mathematician of the Hellenic Age, but all of his works 

have been lost. 





astronomy. 







Eudoxus was undoubtedly the most capable 

mathematician of the Hellenic Age, but all of his works 

have been lost. 

He also saw that he could describe the motions of the 

planets in looped orbits along a curve known as the 

hippopede, which can be obtained as the intersection of a 

sphere and a cylinder tangent internally to the sphere. 


Menaechmus 

A strong thread of continuity of tradition existed in Greece, 

from teachers to students, as Plato learned from Archytas, 

Theodorus, and Theaetetus; Eudoxus learned from Plato, 

and the brothers Menaechmus and Dinostratus received 

their Platonic influence from Eudoxus. 


Menaechmus 






Menaechmus is known for his discovery of the conic 

section curves: the ellipse, the parabola, and the 

hyperbola. 


Menaechmus 






Menaechmus is known for his discovery of the conic 

section curves: the ellipse, the parabola, and the 

hyperbola. 

Beginning with a right circular cone having a right angle at 

the vertex, Menaechmus found that when the coneis cut by 

a plane perpendicular to an element, the curve of 

intersection has the property of, in modern analytic 

notations, y 2 = lx, where l is a constant. 


Menaechmus: Duplication of the Cube 

Proclus reported that Menaechmus was one of those who 

”made the whole of geometry perfect”, but we know little 

about his actual work. 






It is probable that Menaechmus knew that duplication of 

the cube could be achieved by the use of a rectangular 

hyperbola and a parabola. 









We do know he taught Alexander the Great, and legend 

has it that he told the king there is no shortcut to geometry, 

”but in geometry there is one road for all.” 









We do know he taught Alexander the Great, and legend 

has it that he told the king there is no shortcut to geometry, 

”but in geometry there is one road for all.” 

The main evidence that attributes the discovery of conic 

sections to Menaechmus is a letter from Eratosthennes to 

King Ptolemy Euergetes. 


Dinostratus and the Square of the Circle 

Dinostratus, a brother of Menaechmus, was also an 

eminant mathematician. 





Dinostratus noted a striking property of the end point Q of 

the trisectrix of Hippias, which made the quadrature 

problem a simple matter. 








If we write the trisectrix equation as πr sin θ = 2aθ, where a 

is the side of the square ABCD associated with the curve, 

then the limiting value of r as θ tends to zero is 2a/π. 











This is obvious if one has had calculus and use the fact 

lim sin θ/θ = 1. 

θ→0 











This is obvious if one has had calculus and use the fact 

lim sin θ/θ = 1. 

θ→0 

However, the proof given by Pappus and likely due to 

Dinostratus is based only on considerations from 

elementary geometry. 


Autolycus of Pitane 

Autolycus of Pitane wrote the oldest surviving Greek 

mathematical treatise, On the Moving Sphere, part of a 

collection known as the ”Little Astronomy”, which was 

widely used by ancient astronomers. 







Although not a profound or very original work, it indicates 

that Greek geometry at the time had reached the form that 

we regard as typical of the classical age. 










Theorems are clearly enunciated and proved, and the 

author uses other theorems without proof, when he 

regards them as well known. 










Theorems are clearly enunciated and proved, and the 

author uses other theorems without proof, when he 

regards them as well known. 

The conclusion is then that a thoroughly established 

textbook tradition in geometry existed. 


Aristotle 

Aristotle, most widely learned scholar, was a student of 

Plato’s like Eudoxus, and a tutor of Alexander the Great 

like Menaechmus. 


Aristotle 




Primarily a philosopher and a biologist, Aristotle was 

thoroughly au courant with the activities of the 

mathematicians. 


Aristotle 







Through his foundation of logic and his frequent allusion to 

mathematical concepts and theorems in his voluminous 

works, Aristotle can be regarded as having contributed to 

the development of mathematics. 


Aristotle 











However, his statement that mathematicians ”do not need 

the infinite or use it” may have had a negative influence on 

others. 


Aristotle 











However, his statement that mathematicians ”do not need 

the infinite or use it” may have had a negative influence on 

others. 

His more positive contribution is the analysis of the roles of 

definitions and hypotheses in mathematics.

History of Mathematics: Hellenic Traditions

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