By the 3rd Century
BC, in the wake of the conquests of Alexander the Great, mathematical
breakthroughs were also beginning to be made on the edges of the Greek
Hellenistic empire.
In particular, Alexandria in Egypt became a great centre of learning under the beneficent rule of the Ptolemies, and its famous Library soon gained a reputation to rival that of the Athenian Academy. The patrons of the Library were arguably the first professional scientists, paid for their devotion to research. Among the best known and most influential mathematicians who studied and taught at Alexandria were Euclid, Archimedes, Eratosthenes, Heron, Menelaus and Diophantus.
During the late 4th and early 3rd Century BC, Euclid was the great chronicler of the mathematics of the time, and one of the most influential teachers in history. He virtually invented classical (Euclidean) geometry as we know it. Archimedes spent most of his life in Syracuse, Sicily, but also studied for a while in Alexandria. He is perhaps best known as an engineer and inventor but, in the light of recent discoveries, he is now considered of one of the greatest pure mathematicians of all time. Eratosthenes of Alexandria was a near contemporary of Archimedes in the 3rd Century BC. A mathematician, astronomer and geographer, he devised the first system of latitude and longitude, and calculated the circumference of the earth to a remarkable degree of accuracy. As a mathematician, his greatest legacy is the “Sieve of Eratosthenes” algorithm for identifying prime numbers.
It is not known exactly when the great Library of Alexandria burned
down, but Alexandria remained an important intellectual centre for some
centuries. In the 1st century BC, Heron (or Hero) was another great
Alexandrian inventor, best known in mathematical circles for Heronian
triangles (triangles with integer sides and integer area), Heron’s
Formula for finding the area of a triangle from its side lengths, and
Heron’s Method for iteratively computing a square root. He was also the
first mathematician to confront at least the idea of √-1 (although he
had no idea how to treat it, something which had to wait for Tartaglia and Cardano in the 16th Century).
Menelaus of Alexandria, who lived in the 1st - 2nd Century AD, was the first to recognize geodesics on a curved surface as the natural analogues of straight lines on a flat plane. His book “Sphaerica” dealt with the geometry of the sphere and its application in astronomical measurements and calculations, and introduced the concept of spherical triangle (a figure formed of three great circle arcs, which he named "trilaterals").
In the 3rd Century AD, Diophantus of Alexandria was the first to recognize fractions as numbers, and is considered an early innovator in the field of what would later become known as algebra. He applied himself to some quite complex algebraic problems, including what is now known as Diophantine Analysis, which deals
with finding integer solutions to kinds of problems that lead to equations in several unknowns (Diophantine equations). Diophantus’
“Arithmetica”, a collection of problems giving numerical solutions of
both determinate and indeterminate equations, was the most prominent
work on algebra in all Greek mathematics, and his problems exercised the
minds of many of the world's best mathematicians for much of the next
two millennia.
But Alexandria was not the only centre of learning in the Hellenistic
Greek empire. Mention should also be made of Apollonius of Perga (a
city in modern-day southern Turkey) whose late 3rd Century BC work on
geometry (and, in particular, on conics and conic sections) was very
influential on later European mathematicians. It was Apollonius who gave
the ellipse, the parabola, and the hyperbola the names by which we know
them, and showed how they could be derived from different sections
through a cone.
Hipparchus, who was also from Hellenistic Anatolia and who live in the 2nd Century BC, was perhaps the greatest of all ancient astronomers. He revived the use of arithmetic techniques first developed by the Chaldeans and Babylonians, and is usually credited with the beginnings of trigonometry. He calculated (with remarkable accuracy for the time) the distance of the moon from the earth by measuring the different parts of the moon visible at different locations and calculating the distance using the properties of triangles. He went on to create the first table of chords (side lengths corresponding to different angles of a triangle). By the time of the great Alexandrian astronomer Ptolemy in the 2nd Century AD, however, Greek mastery of numerical procedures had progressed to the point where Ptolemy was able to include in his “Almagest” a table of trigonometric chords in a circle for steps of ¼° which (although expressed sexagesimally in the Babylonian style) is accurate to about five decimal places.
By the middle of the 1st Century BC and thereafter, however, the Romans had tightened their grip on the old Greek empire. The Romans had no use for pure mathematics, only for its practical applications, and the Christian regime that followed it even less so. The final blow to the Hellenistic mathematical heritage at Alexandria might be seen in the figure of Hypatia, the first recorded female mathematician, and a renowned teacher who had written some respected commentaries on Diophantus and Apollonius. She was dragged to her death by a Christian mob in 415 AD
EUCLID
The Greek mathematician Euclid lived and flourished in Alexandria in
Egypt around 300 BC, during the reign of Ptolemy I. Almost nothing is
known of his life, and no likeness or first-hand description of his
physical appearance has survived antiquity, and so depictions of him
(with a long flowing beard and cloth cap) in works of art are
necessarily the products of the artist's imagination.
He probably studied for a time at Plato's Academy in Athens but, by Euclid's time, Alexandria, under the patronage of the Ptolemies and with its prestigious and comprehensive Library, had already become a worthy rival to the great Academy.
Euclid is often referred to as the “Father of Geometry”, and he wrote perhaps the most important and successful mathematical textbook of all time, the “Stoicheion” or “Elements”, which represents the culmination of the mathematical revolution which had taken place in Greece up to that time. He also wrote works on the division of geometrical figures into into parts in given ratios, on catoptrics (the mathematical theory of mirrors and reflection), and on spherical astronomy (the determination of the location of objects on the "celestial sphere"), as well as important texts on optics and music.
The "Elements” was a lucid and comprehensive compilation and
explanation of all the known mathematics of his time, including the work
of Pythagoras,
Hippocrates, Theudius, Theaetetus and Eudoxus. In all, it contains 465
theorems and proofs, described in a clear, logical and elegant style,
and using only a compass and a straight edge. Euclid reworked the
mathematical concepts of his predecessors into a consistent whole, later
to become known as Euclidean geometry, which is still as valid today as
it was 2,300 years ago, even in higher mathematics dealing with higher dimensional spaces. It was only with the work of Bolyai, Lobachevski and Riemann in the first half of the 19th Century that any kind of non-Euclidean geometry was even considered.
The "Elements” remained the definitive textbook on geometry and mathematics for well over two millennia, surviving the eclipse in classical learning in Europe during the Dark Ages through Arabic translations. It set, for all time, the model for mathematical argument, following logical deductions from inital assumptions (which Euclid called “axioms” and "postulates") in order to establish proven theorems.
Euclid’s five general axioms were:
His five geometrical postulates were:
Among many other mathematical gems, the thirteen volumes of the
“Elements” contain formulas for calculating the volumes of solids such
as cones, pyramids and cylinders; proofs about geometric series, perfect
numbers and primes; algorithms for finding the greatest common divisor
and least common multiple of two numbers; a proof and generalization of
Pythagoras’ Theorem, and proof that there are an infinite number of
Pythagorean Triples; and a final definitive proof that there can be only
five possible regular Platonic Solids.
However, the “Elements” also includes a series of theorems on the properties of numbers and integers, marking the first real beginnings of number theory. For example, Euclid proved what has become known as the Fundamental Theorem of Arithmethic (or the Unique Factorization Theorem), that every positive integer greater than 1 can be written as a product of prime numbers (or is itself a prime number). Thus, for example: 21 = 3 x 7; 113 = 1 x 113; 1,200 = 2 x 2 x 2 x 2 x 3 x 5 x 5; 6,936 = 2 x 2 x 2 x 3 x 17 x 17; etc. His proof was the first known example of a proof by contradiction (where any counter-example, which would otherwise prove an idea false, is shown to makes no logical sense itself).
He was the first to realize - and prove - that there are infinitely many prime numbers. The basis of his proof, often known as Euclid’s Theorem, is that, for any given (finite) set of primes, if you multiply all of them together and then add one, then a new prime has been added to the set (for example, 2 x 3 x 5 = 30, and 30 + 1 = 31, a prime number) a process which can be repeated indefinitely.
Euclid also identified the first four “perfect numbers”, numbers that are the sum of all their divisors (excluding the number itself):
6 = 1 + 2 + 3;
28 = 1 + 2 + 4 + 7 + 14;
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248; and
8,128 = 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064.
He noted that these numbers also have many other interesting properties. For example:
They are triangular numbers, and therefore the sum of all the
consecutive numbers up to their largest prime factor: 6 = 1 + 2 + 3; 28 =
1 + 2 + 3 + 4 + 5 + 6 + 7; 496 = 1 + 2 + 3 + 4 + 5 + .... + 30 + 31;
8,128 = 1 + 2 + 3 + 4 + 5 + ... + 126 + 127.
Their largest prime factor is a power of 2 less one, and the number
is always a product of this number and the previous power of two: 6 = 21(22 - 1); 28 = 22(23 - 1); 496 = 24(25 - 1); 8,128 = 26(27 - 1).
Although the Pythagoreans may have been aware of the Golden Ratio (φ, approximately equal to 1.618), Euclid was the first to define it in terms of ratios (AB:AC = AC:CB), and demonstrated its appearance within many geometric shapes.
ARCHIMEDES
Another Greek mathematician who studied at Alexandria in the 3rd
Century BC was Archimedes, although he was born, died and lived most of
his life in Syracuse, Sicily (a Hellenic Greek colony in Magna Graecia).
Little is known for sure of his life, and many of the stories and
anecdotes about him were written long after his death by the historians
of ancient Rome.
Also an engineer, inventor and astronomer, Archimedes was best known throughout most of history for his military innovations like his siege engines and mirrors to harness and focus the power of the sun, as well as levers, pulleys and pumps (including the famous screw pump known as Archimedes’ Screw, which is still used today in some parts of the world for irrigation).
But his true love was pure mathematics, and the discovery in 1906 of previously unknown works, referred to as the "Archimedes Palimpsest", has provided new insights into how he obtained his mathematical results. Today, Archimedes is widely considered to have been one of the greatest mathematicians of antiquity, if not of all time, in the august company of mathematicians such as Newton and Gauss.
Archimedes produced formulas to calculate the areas of regular
shapes, using a revolutionary method of capturing new shapes by using
shapes he already understood. For example, to estimate the area of a
circle, he constructed a larger polygon outside the circle and a smaller
one inside it. He first enclosed the circle in a triangle, then in a
square, pentagon, hexagon, etc, etc, each time approximating the area of
the circle more closely. By this so-called “method of exhaustion” (or
simply “Archimedes’ Method”), he effectively homed in on a value for one
of the most important numbers in all of mathematics, π. His estimate was between 31⁄7 (approximately 3.1429) and 310⁄71 (approximately 3.1408), which compares well with its actual value of approximately 3.1416.
Interestingly, Archimedes seemed quite aware that a range was all that could be established and that the actual value might never be known. His method for estimating π was taken to the extreme by Ludoph van Ceulen in the 16th Century, who used a polygon with an extraordinary 4,611,686,018,427,387,904 sides to arrive at a value of π correct to 35 digits. We now know that π is in fact an irrational number, whose value can never be known with complete accuracy.
Similarly, he calculated the approximate volume of a solid like a sphere by slicing it up into a series of cylinders, and adding up the volumes of the constituent cylinders. He saw that by making the slices ever thinner, his approximation became more and more exact, so that, in the limit, his approximation became an exact calculation. This use of infinitesimals, in a way similar to modern integral calculus, allowed him to give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay.
Archimedes’ most sophisticated use of the method of exhaustion, which
remained unsurpassed until the development of integral calculus in the
17th Century, was his proof - known as the Quadrature of the Parabola -
that the area of a parabolic segment is 4⁄3 that
of a certain inscribed triangle. He dissected the area of a parabolic
segment (the region enclosed by a parabola and a line) into infinitely
many triangles whose areas form a geometric progression. He then
computed the sum of the resulting geometric series, and proved that this
is the area of the parabolic segment.
In fact, Archimedes had perhaps the most prescient view of the concept of infinity of all the Greek mathematicians. Generally speaking, the Greeks’ preference for precise, rigorous proofs and their distrust of paradoxes meant that they completely avoided the concept of actual infinity. Even Euclid, in his proof of the infinitude of prime numbers, was careful to conclude that there are “more primes than any given finite number” i.e. a kind of “potential infinity” rather than the “actual infinity” of, for example, the number of points on a line. Archimedes, however, in the "Archimedes Palimpsest", went further than any other Greek mathematician when, on compared two infinitely large sets, he noted that they had an equal number of members, thus for the first time considering actual infinity, a concept not seriously considered again until Georg Cantor in the 19th Century.
Another example of the meticulousness and precision of Archimedes’ work is his calculation of the value of the square root of 3 as lying between 265⁄153 (approximately 1.7320261) and 1351⁄780 (approximately 1.7320512) - the actual value is approximately 1.7320508. He even calculated the number of grains of sand required to fill the universe, using a system of counting based on the myriad (10,000) and myriad of myriads (100 million). His estimate was 8 vigintillion, or 8 x 1063.
The discovery of which Archimedes claimed to be most proud was that
of the relationship between a sphere and a circumscribing cylinder of
the same height and diameter. He calculated the volume of a sphere as 4⁄3πr3, and that of a cylinder of the same height and diameter as 2πr3. The surface area was 4πr2 for the sphere, and 6πr2
for the cylinder (including its two bases). Therefore, it turns out
that the sphere has a volume equal to two-thirds that of the cylinder,
and a surface area also equal to two-thirds that of the cylinder.
Archimedes was so pleased with this result that a sculpted sphere and
cylinder were supposed to have been placed on his tomb of at his
request.
Despite his important contributions to pure mathematics, though, Archimedes is probably best remembered for the anecdotal story of his discovery of a method for determining the volume of an object with an irregular shape. King Hieron of Syracuse had asked Archimedes to find out if the royal goldsmith had cheated him by putting silver in his new gold crown, but Archimedes clearly could not melt it down in order to measure it and establish its density, so he was forced to search for an alternative solution.
While taking his bath on day, he noticed that that the level of the
water in the tub rose as he got in, and he had the sudden inspiration
that he could use this effect to determine the volume (and therefore the
density) of the crown. In his excitement, he apparently rushed out of
the bath and ran naked through the streets shouting, "Eureka! Eureka!"
(“I found it! I found it!”). This gave rise to what has become known as
Archimedes’ Principle: an object is immersed in a fluid is buoyed up by a
force equal to the weight of the fluid displaced by the object.
Another well-known quotation attributed to Archimedes is: “Give me a place to stand on and I will move the Earth”, meaning that, if he had a fulcrum and a lever long enough, he could move the Earth by his own effort, and his work on centres of gravity was very important for future developments in mechanics.
According to legend, Archimedes was killed by a Roman soldier after the capture of the city of Syracuse. He was contemplating a mathematical diagram in the sand and enraged the soldier by refusing to go to meet the Roman general until he had finished working on the problem. His last words are supposed to have been “Do not disturb my circles!”
DIOPHANTUS
Diophantus was a Hellenistic Greek (or possibly Egyptian, Jewish or
even Chaldean) mathematician who lived in Alexandria during the 3rd
Century AD. He is sometimes called “the father of algebra”, and wrote an
influential series of books called the “Arithmetica”, a collection of
algebraic problems which greatly influenced the subsequent development
of number theory.
He also made important advances in mathematical notation, and was one of the first mathematicians to introduce symbolism into algebra, using an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. He was perhaps the first to recognize fractions as numbers in their own right, allowing positive rational numbers for the coefficients and solutions of his equations.
Diophantus applied himself to some quite complex algebraic problems, particularly what has since become known as Diophantine Analysis, which deals
with finding integer solutions to kinds of problems that lead to
equations in several unknowns. Diophantine equations can be defined as
polynomial equations with integer coefficients to which only integer
solutions are sought.
For example, he would explore problems such as: two integers such that the sum of their squares is a square (x2 + y2 = z2, examples being x = 3 and y = 4 giving z = 5, or x = 5 and y =12 giving z = 13); or two integers such that the sum of their cubes is a square (x3 + y3 = z2, a trivial example being x = 1 and y = 2, giving z = 3); or three integers such that their squares are in arithmetic progression (x2 + z2 = 2y2, an example being x = 1, z = 7 and y = 5). His general approach was to determine if a problem has infinitely many, or a finite number of solutions, or none at all.
Diophantus’ major work (and the most prominent work on algebra in all Greek mathematics) was his “Arithmetica”, a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of the “Arithmetica”, only six have survived, although some Diophantine problems from “Arithmetica” have also been found in later Arabic sources. His problems exercised the minds of many of the world's best mathematicians for much of the next two millennia, with some particularly celebrated solutions provided by Brahmagupta, Pierre de Fermat, Joseph Louis Lagrange and Leonhard Euler, among others. In recognition of their depth, David Hilbert proposed the solvability of all Diophantine problems as the tenth of his celebrated problems in 1900, a definitive solution to which only emerged with the work of Robinson and Matiyasevich in the mid-20th Century.
One of the problems in a later 5th Century Greek anthology of number games is sometimes considered to be Diophantus’ epitaph:
In particular, Alexandria in Egypt became a great centre of learning under the beneficent rule of the Ptolemies, and its famous Library soon gained a reputation to rival that of the Athenian Academy. The patrons of the Library were arguably the first professional scientists, paid for their devotion to research. Among the best known and most influential mathematicians who studied and taught at Alexandria were Euclid, Archimedes, Eratosthenes, Heron, Menelaus and Diophantus.
During the late 4th and early 3rd Century BC, Euclid was the great chronicler of the mathematics of the time, and one of the most influential teachers in history. He virtually invented classical (Euclidean) geometry as we know it. Archimedes spent most of his life in Syracuse, Sicily, but also studied for a while in Alexandria. He is perhaps best known as an engineer and inventor but, in the light of recent discoveries, he is now considered of one of the greatest pure mathematicians of all time. Eratosthenes of Alexandria was a near contemporary of Archimedes in the 3rd Century BC. A mathematician, astronomer and geographer, he devised the first system of latitude and longitude, and calculated the circumference of the earth to a remarkable degree of accuracy. As a mathematician, his greatest legacy is the “Sieve of Eratosthenes” algorithm for identifying prime numbers.
 |
 Menelaus of Alexandria introduced the concept of spherical triangle |
Menelaus of Alexandria, who lived in the 1st - 2nd Century AD, was the first to recognize geodesics on a curved surface as the natural analogues of straight lines on a flat plane. His book “Sphaerica” dealt with the geometry of the sphere and its application in astronomical measurements and calculations, and introduced the concept of spherical triangle (a figure formed of three great circle arcs, which he named "trilaterals").
In the 3rd Century AD, Diophantus of Alexandria was the first to recognize fractions as numbers, and is considered an early innovator in the field of what would later become known as algebra. He applied himself to some quite complex algebraic problems, including what is now known as Diophantine Analysis, which deals
with finding integer solutions to kinds of problems that lead to equations in several unknowns (Diophantine equations). Diophantus’
“Arithmetica”, a collection of problems giving numerical solutions of
both determinate and indeterminate equations, was the most prominent
work on algebra in all Greek mathematics, and his problems exercised the
minds of many of the world's best mathematicians for much of the next
two millennia.
 |
|
Conic sections of Apollonius |
Hipparchus, who was also from Hellenistic Anatolia and who live in the 2nd Century BC, was perhaps the greatest of all ancient astronomers. He revived the use of arithmetic techniques first developed by the Chaldeans and Babylonians, and is usually credited with the beginnings of trigonometry. He calculated (with remarkable accuracy for the time) the distance of the moon from the earth by measuring the different parts of the moon visible at different locations and calculating the distance using the properties of triangles. He went on to create the first table of chords (side lengths corresponding to different angles of a triangle). By the time of the great Alexandrian astronomer Ptolemy in the 2nd Century AD, however, Greek mastery of numerical procedures had progressed to the point where Ptolemy was able to include in his “Almagest” a table of trigonometric chords in a circle for steps of ¼° which (although expressed sexagesimally in the Babylonian style) is accurate to about five decimal places.
By the middle of the 1st Century BC and thereafter, however, the Romans had tightened their grip on the old Greek empire. The Romans had no use for pure mathematics, only for its practical applications, and the Christian regime that followed it even less so. The final blow to the Hellenistic mathematical heritage at Alexandria might be seen in the figure of Hypatia, the first recorded female mathematician, and a renowned teacher who had written some respected commentaries on Diophantus and Apollonius. She was dragged to her death by a Christian mob in 415 AD
EUCLID
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Euclid (c.330-275 BC, fl. c.300 BC) |
He probably studied for a time at Plato's Academy in Athens but, by Euclid's time, Alexandria, under the patronage of the Ptolemies and with its prestigious and comprehensive Library, had already become a worthy rival to the great Academy.
Euclid is often referred to as the “Father of Geometry”, and he wrote perhaps the most important and successful mathematical textbook of all time, the “Stoicheion” or “Elements”, which represents the culmination of the mathematical revolution which had taken place in Greece up to that time. He also wrote works on the division of geometrical figures into into parts in given ratios, on catoptrics (the mathematical theory of mirrors and reflection), and on spherical astronomy (the determination of the location of objects on the "celestial sphere"), as well as important texts on optics and music.
 |
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Euclid’s method for constructing of an equilateral triangle from a given straight line segment AB using only a compass and straight edge was Proposition 1 in Book 1 of the "Elements" |
with higher dimensional spaces. It was only with the work of Bolyai, Lobachevski and Riemann in the first half of the 19th Century that any kind of non-Euclidean geometry was even considered.The "Elements” remained the definitive textbook on geometry and mathematics for well over two millennia, surviving the eclipse in classical learning in Europe during the Dark Ages through Arabic translations. It set, for all time, the model for mathematical argument, following logical deductions from inital assumptions (which Euclid called “axioms” and "postulates") in order to establish proven theorems.
Euclid’s five general axioms were:
- Things which are equal to the same thing are equal to each other.
- If equals are added to equals, the wholes (sums) are equal.
- If equals are subtracted from equals, the remainders (differences) are equal.
- Things that coincide with one another are equal to one another.
- The whole is greater than the part.
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Euclid’s Postulates (1 - 5) |
- It is possible to draw a straight line from any point to any point.
- It is possible to extend a finite straight line continuously in a straight line (i.e. a line segment can be extended past either of its endpoints to form an arbitrarily large line segment).
- It is possible to create a circle with any center and distance (radius).
- All right angles are equal to one another (i.e. "half" of a straight angle).
- If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.
 |
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Part of Euclid’s proof of Pythagoras’ Theorem |
However, the “Elements” also includes a series of theorems on the properties of numbers and integers, marking the first real beginnings of number theory. For example, Euclid proved what has become known as the Fundamental Theorem of Arithmethic (or the Unique Factorization Theorem), that every positive integer greater than 1 can be written as a product of prime numbers (or is itself a prime number). Thus, for example: 21 = 3 x 7; 113 = 1 x 113; 1,200 = 2 x 2 x 2 x 2 x 3 x 5 x 5; 6,936 = 2 x 2 x 2 x 3 x 17 x 17; etc. His proof was the first known example of a proof by contradiction (where any counter-example, which would otherwise prove an idea false, is shown to makes no logical sense itself).
He was the first to realize - and prove - that there are infinitely many prime numbers. The basis of his proof, often known as Euclid’s Theorem, is that, for any given (finite) set of primes, if you multiply all of them together and then add one, then a new prime has been added to the set (for example, 2 x 3 x 5 = 30, and 30 + 1 = 31, a prime number) a process which can be repeated indefinitely.
Euclid also identified the first four “perfect numbers”, numbers that are the sum of all their divisors (excluding the number itself):
6 = 1 + 2 + 3;
28 = 1 + 2 + 4 + 7 + 14;
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248; and
8,128 = 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064.
He noted that these numbers also have many other interesting properties. For example:
Although the Pythagoreans may have been aware of the Golden Ratio (φ, approximately equal to 1.618), Euclid was the first to define it in terms of ratios (AB:AC = AC:CB), and demonstrated its appearance within many geometric shapes.
ARCHIMEDES
 |
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Archimedes (c.287-212 BC) |
Also an engineer, inventor and astronomer, Archimedes was best known throughout most of history for his military innovations like his siege engines and mirrors to harness and focus the power of the sun, as well as levers, pulleys and pumps (including the famous screw pump known as Archimedes’ Screw, which is still used today in some parts of the world for irrigation).
But his true love was pure mathematics, and the discovery in 1906 of previously unknown works, referred to as the "Archimedes Palimpsest", has provided new insights into how he obtained his mathematical results. Today, Archimedes is widely considered to have been one of the greatest mathematicians of antiquity, if not of all time, in the august company of mathematicians such as Newton and Gauss.
 |
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Approximation of the area of circle by Archimedes’ method of exhaustion |
Interestingly, Archimedes seemed quite aware that a range was all that could be established and that the actual value might never be known. His method for estimating π was taken to the extreme by Ludoph van Ceulen in the 16th Century, who used a polygon with an extraordinary 4,611,686,018,427,387,904 sides to arrive at a value of π correct to 35 digits. We now know that π is in fact an irrational number, whose value can never be known with complete accuracy.
Similarly, he calculated the approximate volume of a solid like a sphere by slicing it up into a series of cylinders, and adding up the volumes of the constituent cylinders. He saw that by making the slices ever thinner, his approximation became more and more exact, so that, in the limit, his approximation became an exact calculation. This use of infinitesimals, in a way similar to modern integral calculus, allowed him to give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay.
 |
|
Archimedes’ quadrature of the parabola using his method of exhaustion |
In fact, Archimedes had perhaps the most prescient view of the concept of infinity of all the Greek mathematicians. Generally speaking, the Greeks’ preference for precise, rigorous proofs and their distrust of paradoxes meant that they completely avoided the concept of actual infinity. Even Euclid, in his proof of the infinitude of prime numbers, was careful to conclude that there are “more primes than any given finite number” i.e. a kind of “potential infinity” rather than the “actual infinity” of, for example, the number of points on a line. Archimedes, however, in the "Archimedes Palimpsest", went further than any other Greek mathematician when, on compared two infinitely large sets, he noted that they had an equal number of members, thus for the first time considering actual infinity, a concept not seriously considered again until Georg Cantor in the 19th Century.
Another example of the meticulousness and precision of Archimedes’ work is his calculation of the value of the square root of 3 as lying between 265⁄153 (approximately 1.7320261) and 1351⁄780 (approximately 1.7320512) - the actual value is approximately 1.7320508. He even calculated the number of grains of sand required to fill the universe, using a system of counting based on the myriad (10,000) and myriad of myriads (100 million). His estimate was 8 vigintillion, or 8 x 1063.
 |
|
Archimedes showed that the volume and surface area of a sphere are two-thirds that of its circumscribing cylinder |
Despite his important contributions to pure mathematics, though, Archimedes is probably best remembered for the anecdotal story of his discovery of a method for determining the volume of an object with an irregular shape. King Hieron of Syracuse had asked Archimedes to find out if the royal goldsmith had cheated him by putting silver in his new gold crown, but Archimedes clearly could not melt it down in order to measure it and establish its density, so he was forced to search for an alternative solution.
 |
|
An experiment to demonstrate Archimedes’ Principle |
Another well-known quotation attributed to Archimedes is: “Give me a place to stand on and I will move the Earth”, meaning that, if he had a fulcrum and a lever long enough, he could move the Earth by his own effort, and his work on centres of gravity was very important for future developments in mechanics.
According to legend, Archimedes was killed by a Roman soldier after the capture of the city of Syracuse. He was contemplating a mathematical diagram in the sand and enraged the soldier by refusing to go to meet the Roman general until he had finished working on the problem. His last words are supposed to have been “Do not disturb my circles!”
DIOPHANTUS
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Diophantus of Alexandria (c.200-284 AD) |
He also made important advances in mathematical notation, and was one of the first mathematicians to introduce symbolism into algebra, using an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. He was perhaps the first to recognize fractions as numbers in their own right, allowing positive rational numbers for the coefficients and solutions of his equations.
Diophantus applied himself to some quite complex algebraic problems, particularly what has since become known as Diophantine Analysis, which deals
with finding integer solutions to kinds of problems that lead to
equations in several unknowns. Diophantine equations can be defined as
polynomial equations with integer coefficients to which only integer
solutions are sought.
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Diophantine equations |
Diophantus’ major work (and the most prominent work on algebra in all Greek mathematics) was his “Arithmetica”, a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of the “Arithmetica”, only six have survived, although some Diophantine problems from “Arithmetica” have also been found in later Arabic sources. His problems exercised the minds of many of the world's best mathematicians for much of the next two millennia, with some particularly celebrated solutions provided by Brahmagupta, Pierre de Fermat, Joseph Louis Lagrange and Leonhard Euler, among others. In recognition of their depth, David Hilbert proposed the solvability of all Diophantine problems as the tenth of his celebrated problems in 1900, a definitive solution to which only emerged with the work of Robinson and Matiyasevich in the mid-20th Century.
One of the problems in a later 5th Century Greek anthology of number games is sometimes considered to be Diophantus’ epitaph:
“Here lies Diophantus.The puzzle implies that Diophantus lived to be about 84 years old (although its biographical accuracy is uncertain).
God gave him his boyhood one-sixth of his life;
One twelfth more as youth while whiskers grew rife;
And then yet one-seventh ‘ere marriage begun.
In five years there came a bouncing new son;
Alas, the dear child of master and sage,
After attaining half the measure of his father's life, chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.”
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