HISTORY OF MATH CONC

EPTS
Essentials, Images and Memes

SergeyFinashin
 Modern
Mathematics having
roots in ancient
Egypt and
Babylonia, really
flourished in ancient
Greece. It is
remarkable in
Arithmetic (Number
theory) and
Deductive
Geometry.
Mathematics written
in ancient Greek
was translated into
Arabic, together
with some mathematics of India. Mathematicians of Islamic Middle East significantly developed
Algebra. Later some of this mathematics was translated into Latin and became the mathematics of
Western Europe. Over a period of several hundred years, it became the mathematics of the world.

Some significant mathematics was also developed in other regions, such as China, southern India,
and other places, but it had no such a great influence on the international mathematics.

The most significant for development in mathematics was giving it firm logical foundations in
ancient Greece which was culminated in Euclid’s Elements, a masterpiece establishing standards of
rigorous presentation of proofs that influenced mathematics for many centuries till 19th.

Content
1. Prehistory: from primitive counting to numeral systems
2. Archaic mathematics in Mesopotamia (Babylonia) and Egypt
3. Birth of Mathematics as a deductive science in Greece: Thales and Pythagoras
4. Important developments of ideas in the classical period, paradoxes of Zeno
5. Academy of Plato and his circle, development of Logic by Aristotle
6. Hellenistic Golden Age period, Euclid of Alexandria
7. Euclid’s Elements and its role in the history of Mathematics
8. Archimedes, Eratosthenes
9. Curves in the Greek Geometry, Apollonius the Great Geometer
10. Trigonometry and astronomy: Hipparchus and Ptolemy
11. Mathematics in the late Hellenistic period
12. Mathematics in China and India
13. Mathematics of Islamic Middle East

Lecture 1. Prehistory: from primitive counting to Numeral systems

Some of primitive cultures included just words for “one”, “two”, and “many”.

In addition to finger, the most usual tools of counting were sticks and pebbles.

The earliest (20-35 000BC) archeological artefacts used for counting are bones with a number of cuts.
Numeral Systems

The origin of the earliest civilizations such as Sumer (in Mesopotamia), Egypt and Minoan (in Crete) goes
back to 3500-4000BC. Needs of trade, city management, measurement of size, weight and time required a
unified system to make calculations and represent the results. The earliest Sumerian Systems of Measures
and Calendars are dated by 4000BC. Special clay tokens were invented to count sheep, days and other
objects (different ones were counted with different tokens and often in a different way).

In 3000BC in the city Uruk there were more than a dozen of different counting systems in use. About this
time, Abacus as a tool of calculation was invented. Later, as a writing system was developed (pressing
cuneiform signs on clay tablets with a reed stylus), the Sumer sexagesimal numeral system based on
powers of 60 was elaborated (do not confuse with hexadecimal system based on 16). Nowadays Sumerian
system is used for time (hour, minutes, seconds) and angle measurements (360 o).

Babylonian numerals

Initially asign-value system,

was gradually transformed into
a place-value system. In the
place-value (aka positional)
systems, the same symbols are used with a different magnitude depending on their place in the number.

Egyptian numerals(2000BC)
To compare, the Egyptian numeral
system (that also appeared about
2500-3000BC) is decimal: based on

powers of 10. But it is a sign-value system, and so, for 10, 100, 1000, etc., different symbols are used.

Maya numerals (650BC)

Maya developed a vigesimal (based on 20) place-value numeral system. They were the first ones who used

a sign for zero

(before Indians).
 Chinese rod
numbers
(decimal
place-value
system
1300BC)
In addition to
a hieroglyphic
sign-value
numeral system
in ancient China,
Rod numbers
were invented: they existed in vertical and horizontal forms. In writing they were alternating: vertical form
was used for units, hundreds, tens of thousands, etc., while horizontal
rods were used for tens, thousands, etc. Chinese developed (100BC)
negative numbers and distinguished them from positive ones by color.
Lecture 2. Archaic Mathematics in Mesopotamia (Babylonia) and Egypt
Babylonian Mathematics: not much of geometry, but amazing arithmetic and algebra
Babylonian mathematics used pre-calculated clay tablets in cuneiform script to assist with arithmetic. For
example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the
squares of numbers up to 59 and the cubes of numbers up to 32. Together with the formulae

the tables of squares

were used for
multiplication. For
division a table of
reciprocals was used together with the formula .

Numbers whose only prime factors

are 2, 3 or 5 (known as 5-smooth or regular numbers) have finite reciprocals in sexagesimal notation, and
tables with lists of these reciprocals have been found. To compute 1/13 or to divide a number by 13 the
Babylonians would use an approximation such as

To solve a quadratic equation the standard quadratic

formula was used with the tables of squares in reverse to find square
roots.

Here c was always positive and only the positive root was considered “meaningful”. Problems of this type
included finding the dimensions of a rectangle given its area and the amount by which the length exceeds
the width.

The tables for finding square and cubic root were up to 3 sexagesimals (5 decimals). To improve an
approximation x1  a the formula x2=1/2(x1+a/x1) was used. For example, to find the square root of 2
2

one can take x1=1.5 as the first approximation, x1  2. Then x2=1/2(x1+2/x1) 1/2(1.5+1.3)=1.4 is a
2

better approximation.
2
Other tables did exist to solve a system x+y=p, xy=q that is equivalent to x -px+q=0.
3
Tables of values of n + n2 were used to solve certain cubic equations, like

Multiplying the equation by a2 and dividing by b3 and letting y = ax/b we obtains

where y can be found now from the table.

Babylonian algebra was not symbolic, but it was rhetoric: instead of symbols for unknown and signs just
words were used, for example, an equation x+1=2 was expressed as “a thing plus one equals two”.
For finding the length of a circle and the area of a disc an approximate value
was known, although an approximation was also often used.
The Plimpton 322 tablet (1800 BC) in Plimpton collection at Columbia University. It contains a
list of Pythagorean triples, i.e., integers (a,b,c) such that a2+b2=c2. It seems that a general formula
for such triples was known, although no direct evidence of this was ever found.

Problems related to growth of loans were well-developed.

Astronomical calculations allowing to predict motion of planets were developed at a high level.

During the Archaic period of Greece (800-500BC) Babylon was famous as the place of studies.

Egyptian mathematics: unit fractions, more geometry, but less algebra

Rhind (or Ahmes) mathematical papyrus (1650 BC) in British

Museum, 6m length. It was found during illegal excavation and sold in Egypt
to Scottish antiquarian Rhind in 1858. It is a problem book that was copied
by scribe Ahmes from an older papyrus dated by 1800-2000BC.

There are 87 problems with solutions in arithmetic, algebra and geometry. The most of arithmetical
problems are related to the unit Egyptian fractions and involve in particular finding least common multiples
of denominators and decomposition of 2/n into unit fractions. A dozen of problems are related to linear
equations, like x + x/3 + x/4 = 2 (in modern notation) and a few more are devoted to arithmetic and
geometric progressions.

The geometric problem include finding areas of rectangles, triangles and trapezoids, volumes of cylindrical
and rectangular based granaries, and the slopes of pyramids.
The volume V of a cylindrical granary of a diameter d and height h was calculated by formula

or in modern notation where d = 2r,

and the quotient 256/81 approximates the value π  3.1605.
Another famous Moscow Mathematical Papyrus (1800BC) contains 25 problems, and some of them are of a
different kind: on finding the area of surfaces such as a hemisphere and a truncated pyramid.

From these and a few more papyri one may conclude that Egyptians knew arithmetic, geometric and
harmonic means. They had a concept of perfect and prime numbers, and used sieve of Eratosthenes.

Questions to Lectures 1-2:

• What did China, India, Egypt and Babylon have in common?
• What were the earliest causes for the creation of mathematics?
• Why were so many different bases (i.e. 2, 3, 5, 10, 20, 60) used?
• Was early mathematics recreational, theoretical, applied, or what?
• Was the idea of proof or justification used or needed?
• Why conic sections were never considered?
• How are nonlinear equations considered, solved? What do the Egyptians do? What do the
Babylonians do?
• What was the relation between the exact and the approximate? Was the distinction clearly
understood?

Lecture 3. Birth of mathematics as a deductive science in Greece: Thales

and
Pythagoras
• Archaic Period 776 BC (The first Olympic games)/500BC (Beginning
of Persian Wars)
• Classical Period 500 /323BC (death of Alexander)
• Early Hellenistic Period 323BC/146AD
• Late Hellenistic Period 146/500AD
Words:- knowledge, ς- number,-
geometry. 

Thales of Miletus (ς) 624-546 BC the first philosopher and mathematician in Greek
tradition, one of seven Sages of Greece,
founder of Milesian natural philosophy school
Recognized as an initiator of the scientific revolution: rejected mythological explanation and searched for a
scientific one. He was interested in physical world and for application of knowledge to it.

Thales introduced a concept of proof as a necessary part of mathematical knowledge (proofs did not look
that important in previous mathematics
commonly viewed rather as a collection of facts and practices in calculation). So, he distinguished
mathematics as a science from application of it to engineering and other purposes. He separated in
particular arithmetic as a science about numbers from the art of computation that he called logistic.

He considered separately two kinds of numbers: “arithmetical” natural numbers and “geometric” numbers
that are results of measurements (say, length) with a scale.

Thales introduced the idea of “construction” problems in

geometry, in which only a compass and straightedge can be
used. Giving a
solution to the
problem of bisection
of angle, he stated
the problem of
trisection.

Some theorems
usually attributed to
Thales:

1) on isosceles triangles: two sides are equal if and only if the angles are equal
2) the sum of angles of a triangle is 180 o 3) opposite angles between two lines are equal
4) similar triangles (with the same angles)
have proportional sides
5) if AC is a diameter, then the angle at B is a right angle.

Some famous applications of his knowledge

to practical needs:
How to measure the height of a pyramid?
 Pythagoras from Samos
( ς580-500 BC
After leaving Samos,where Pythagoras
had a conflict with its tyrant, he settled
in Croton and established a school, a kind of esoteric society
and brotherhood with somewhat strict rules of life, called
“Mathematikoi”. The following achievements are attributed
to Pythagoras or his followers:

1) “Principle of the world harmony”; Pythagorean tuning,

“music of spheres”
2) Theory of primes, polygonal numbers, squares and ratios of integers and other

How to find the distance from a ship to a shore? 3)

How to measure the width of a river? magnitudes
3) Irrationality of square root of 2, etc. (some attribute to his students, e.g., Hippasus)
4) Studying of the Golden Ratio and Pentagram (symbol of
Pythagoreans) a sign of math perfection
5) The problem of construction regular polygons (pentagon and some others were constructed)
2
6) Geometric algebra: solving equations like a(a-x)=x geometrically
7) Regular solids (Pythagoras himself knew possibly only three of them)
8) Doctrine of quadrature: “to understand the area means to construct a square by means of compass
and straightedge”; the problem of Quadrature of circle
9) “Pythagoras theorem” with numerous proofs, “Pythagoras triples” (although known in Babylon)
 ones)
12) Medicine: brain is a locus of the soul

“The Pythagoreans, who were the first to take up mathematics, not only advanced this subject,

but saturated with it, they fancied that the principles of mathematics were the principles of all things.”
ARISTOTLE, 384 – 322 BC Metaphysica

Great Construction problems of Ancient Greece

:
1. Trisecting the angle (stated possibly by Thales)
2. Squaring the circle (stated possibly by
Pythagoras)
3. Doubling the cube
(attributed to Plato)
----------------------------
4. Construction of a
regular n-gon
(attributed to
Pythagoras)

Lecure 4. Important developments

10)
of ideas in the classical period
Anaximenes
585- 528BC
10) Four Pythagorean Means, their geometric presentation and comparison
Anaximander
11) Astronomy: spherical shape of the Earth, Sun as the center of the world, 610-546 BC
Venus as a morming and evening star (it was considered as two different
Milesian school (of Miletus): founded by Thales. His student Anaximander:
claimed apeiron as the primary element, introduced gnomon, created a map of
the world. For Anaximenes air was primary.

Heraclites of Ephesus 535-475BC known as “weeping philosopher”

“Panta rei” (everything flows), “No man ever steps in the same river twice”; “The path up and down are one
and the same” (on the unity of the opposites); “All entities come to be in accordance with Logos” (here
Heraclites Logos is a word, reason, plan, or formula).
 Eleatic school

(of Elea): founded by Parmenides (540-?BC)

Parmenides Disputed with

Heraclites and claimed that “anything that changes cannot
be real” and that “truth cannot be
known through perception, only Zeno
Logos shows truth of the world”; "You say there is a void; therefore the void
is not
nothing; therefore there is not the void."

Zeno of Elea (490-430BC student

of Parmenides, stated aporias
(paradoxes) such as “Achill and tortoise”, “Arrow”, etc.

Democritus 460-370BC “laughing philosopher” born in Abdera also

some links him with the Milesian school. With his teacher Leucippus
proposed an atomic theory as an answer to
the aporias of Zeno.

Sophists (Protagoras, Gorgias, Prodicus,

Hippias, etc.) were a category of teachers
(mostly in 500-
3500 BC) who specialized in using the
techniques of philosophy, rhetoric (skill of public speaking) and dialectic
(skill to argue in a dialogue by showing contradictions in opponent’s
viewpoint) for the purpose of teaching arête
(excellence, or virtue) predominantly to
young statesmen and nobility.

Protagoras 490-420 BC: Taught to care about

proper meaning of words
(orthoepeia). “Man is the measure of all
things”;
Gorgias “Concerning the gods, I have no means of
knowing whether they exist or not, or what sort they
may be, because of obscurity of the subject, and the brevity of human
life.” Athenians expelled him from the city, and his books were collected
and burned on the market place.

Gorgias 485-380: performed oratory, like “Encomium of Helen”; ironic parody “On the nature
of non-existent”: 1) Nothing exists. 2) Even if something exists, nothing can be known about
it. 3) Even if something can be known about it, this knowledge cannot be communicated to
the others. 4) Even if it can be communicated, it cannot be understood. True objectivity is
impossible. “How can anyone communicate an idea of color by means of words, since ear
does not hear colors but only sounds?” Love to paradoxologia.

Socrates 470-399BC credited as

one of the founders of
Western philosophy.
"Socratic Method”
 of teaching (possibly invented by Protagoras) through
a dialogue is demonstrated in the book
of his student, Plato. Proposed to switch attention to a human and his thinking
from nature of the physical world. "I know that I know nothing."

Hyppocrates of Chios 470-410BC (do not

confuse with Hippocrates of Kos, father of Western Medicine) was Pythagorean, but then quitted.
He has written the first “Elements” (Euclid 3-4) and discovered
quadrature of Lunes as a partial quadrature of circle. He stated the
principle to avoid neusis constructions (otherwise, trisection of an
angle would be possible).

Hippias of Elis 460-400BC a sophist lecturing on poetry,

grammar, history, politics, and math Invented Trisectrix (known also
as Quadratrix after Dinostratus 390-320BC used it for squaring circle) .

Theodorus of Cyrene 465-398BC student of Protagoras

and tutor of Plato. Spiral made of right triangles whose hypotenuses are square roots from 2 to 17

Greek Mathematicians with their Home-Cities

 Abdera: Democritus  Chios: Hippocrates,  Nicaea: Hipparchus,
 Alexandria: Oenopides Sporus, Theodosius
Apollonius,  Clazomenae:  Paros: Thymaridas
Aristarchus, Anaxagoras  Perga: Apollonius
Diophantus,  Cnidus: Eudoxus  Pergamum:
Eratosthenes, Euclid,  Croton: Philolaus, Apollonius
Hypatia, Hypsicles, Pythagoras  Rhodes: Eudemus,
Heron, Menelaus,  Cyrene: Geminus,
Pappus, Eratosthenes, Posidonius
Ptolemy, Theon Nicoteles, Synesius,  Rome: Boethius
 Amisus: Theodorus  Samos: Aristarchus,
Dionysodorus  Cyzicus: Callippus Conon, Pythagoras
 Antinopolis: Serenus  Elea: Parmenides,  Smyrna: Theon
 Apameia: Posidonius Zeno  Stagira: Aristotle
 Athens: Aristotle,  Elis: Hippias  Syene: Eratosthenes
Plato, Ptolemy,  Gerasa: Nichmachus Syracuse:

Socrates, Theaetetus  Larissa: Dominus Archimedes
 Byzantium  Miletus: Tarentum: Archytas,

(Constantinople): Anaximander, Pythagoras
Philon, Proclus Anaximenes, Thasos: Leodamas

 Chalcedon: Proclus, Isidorus, Thales Tyre: Marinus,

Xenocrates Porphyrius
 Chalcis: Iamblichus
Lecture 5. Academy of
Plato and his circle. Aristotle and his
Logic
Plato 428-348 philosopher and
mathematician, the author of
Dialogues (the first original philosophical text that came to us almost untouched)
Established the Academy in a park of Athens, the first higher education
the Western center in World. The object of interest are pure forms or ideas
called (of a human)
archetypes
“Nobody
without can be considered
arithmetic, educated learning five
disciplines of math:
plane geometry, solid
geometry,

astronomy and harmony”.

Associated an element with each regular solid: fire for tetrahedron, Earth for
cube, air for octahedron, water for icosahedron and ether or prana of the whole
universe for dodecahedron.

Legend about Delian Problem (Doubling of a cube)

Eudoxus of
Cnidus (now
Datcha) 410-
355 one of the
greatest ancient
mathematicians,
astronomer,
studied at
Academy of Plato
for 2 months, but
had no money to
continue He.
Studied
irrationals, developed a theory of
proportions which was taken by Euclid into
Elements 5, “two magnitudes are
comparable is a multiple of one is greater than the
other”.
He invented Method
of

Exhaustion (a form of integration) that was later

advanced by Archimedes, created a School and
criticized Plato, who was his rival. Eudoxus
constructed an observatory, proposed a
planetary model, the first astronomer to map
stars.
Theaetetus 417-369BC studied in Academy, a friend of Plato and a character in
“Dialogues”Theory of irrational (incommensurable) magnitudes (taken to Euclid’s Elements
10) Developped construction of regular solids.

Menaechmus 380-320BC brother of Dinostratus, student of Eudoxus, friend of

Plato, tutor of Alexander the Great. The first person who studied the Conic
Sections and used them for solution of the doubling cube problem.
 Aristotle (ς384-

322BC philosopher and scientist,
student of Plato, tutor of Alexander the
Great After death of Plato quitted from

Academy
and founded
Lyceum in
Athens, in 335BC. Aristotle was the first who analyzed the Formal
Logic and developed its “grammar”, notion of syllogism. “Reason
rather than observation at the center of scientific effort”.
 Euclid (ςof
Alexandria
BC
"father of
geometry”, the
author of Elements,
one of the most
influential works in the history of mathematics,
serving as the main textbook for teaching
1. mathematics (especially geometry) from the
time of
its Other books of Euclid: Data, On division of Figures,
Catoprics, Optics, Phaenomena, Conics, Porisms , etc.

5.
publication until the late 19th or early 20th century. In
the Elements, Euclid deduced the principles of what is
now called Euclidean geometry from a small set of
axioms. 13 books of Elements the
whole math knowledge of that time was summarized. The first 6 books of
Elements are devoted to Plane Geometry, next 3 to arithmetic, and last 3 to
spatial geometry.
Basis plane Geometry: angles, areas (up to Pythagoras theorem)

2. Geometric Algebra (Pythagoras)

3. Circles, inscribed angles, tangents
(Thales, Hippocrates)
4. Incircle, circumcircle, construction of regular polygons
(with 4,5,6,15 sides).
Lectures 6 . Hellenistic Golden Age, Euclid of Alexandria
Mouseion (or Musaeum) at Alexandria, included the Library of Alexandria,
was a research institution similar to modern universities founded in the end of middle of
3d century BC. In addition to the library, it included rooms for the study of astronomy,
anatomy, and even a zoo of exotic animals. The classical thinkers who studied, wrote, and
experimented at the Musaeum worked in mathematics, astronomy, physics, geometry, engineering,
geography, physiology and medicine. The library included about half million of papyri. Hellenistic Golden
Age includes primarily Euclid, Archimedes and Apollonius.
Proportions of magnitudes (Eudoxus), arithmetical
and geometric Mean
6. Proportions in Geometry: similar figures (Theon, Pythagoras)
7. Arithmetic: divisibility, primes, Euclid’s algorithm for g.c.d. and l.c.m., prime decomposition
8. Proportions in arithmetic: geometric sequences
9. Infiniteness of number of primes, sum of geometric series, a formula for even perfect numbers
10. Theory of irrationals and method of exhaustion (based on Eudoxus)
11. Extension of the results of Books 1-6 to space: angles, perpendicularity, volumes.
12. Volumes of cones, pyramids, cylinders and spheres (Theaetetus)
13. Five Platonic solids, their size, proof that there is no other regular solids (Theaetetus).
 Lecture 7. Elements and their role in the
history of Mathematics

The structure of Propositions:

1. Enunciation (statement of the proposition).
2. Setting-out (gives a figure and denote its
elements by letters).
3. Specification (restates the general
statement in terms of this figure).
4. Construction (extends the figure with new
elements needed for the proof).
5. Proof (using previous Propositions).
6. Conclusion (connects the proof to the
initial claim in the enunciation).

Aristarchus of Samos
310-230BC astronomer and mathematician 
Estimated the size of Moon, found that the Earth revolves around the Sun and the Moon around the Earth.
4. Applying mathematics to physical phenomena, founding (Archimedes’
principle concerning the buoyant force) and (the principle of th .
5. Designing innovative such as his
and defensive war machines (Claw of Archimedes, Heat Ray,
etc.) to protect his native from
invasion.
6. Investigating the Archimedean
7. 13 Archimedean (semiregular) solids
8. Archimedes’ twin-circles in arbelos

Eratosthenes of Cyrene 276-194BC a Greek

and
He was a man of learning and
became the chief librarian at the
He invented the discipline of
including the terminology used today. He is the
founder of scientific and revised the
dates of the main political
events from the conquest
of In Math his most
famous invention is Sieve of
Eratosthenes for prime
numbers.
 Lecture 8 . Archimedes, Eratosthenes
Archimedes     ς287-212BC
inventor

regarded as one of the leading (in fact, the greatest)
in

1. Concepts of
infinitesimals and

the method of
exhaustion were
developed to
derive and rigorously prove a range of geometrical theorems, including
the area of a circle, the area of a parabolic sector, its centroid, , the
surface area and volume of a sphere and other rotational solids,
parabolic and hyperbolic conoids
2. Proved an approximation 310⁄71 < 31⁄7 using inscribed and circumscribed polygons
3. Creating a system for expressing very large numbers (Sand Reckoner)
 Focal
property
of
parabola

Lecture 9. Curves in the Greek Geometry,

Apollonius, Great Geometer
Nicomedes 280-210BC “On conchoid lines” Conchoid of Cissoid of Diocles
Nicomedes

Diocles 240-180BC “On burning mirrors” studied

the focal property of parabola, cissoids of Diocles

Apollonius of Perga
(Ἀπολλώνιος) 262-190 BC “The Great
Geometer” and astronomer. Famous work (7
books) on conic sections where the ellipse, the
parabola, and the hyperbola received their
modern names. The hypothesis of eccentric
orbits (deferent and epicycles) to explain the
apparent motion of the planets and the
varying speed of the Moon, is also attributed to him. 7 books of Conics (Κωνικά)
Books 1-4: elementary introduction (essential part of the results in Book 3 and all in Book 4 are original)

Book 5-7 (highly original): studies of normals,

determines centers of curvature and defines evolute.

A method similar to analytic geometry is developed;

difference: no negative numbers and the axis are
chosen after coordinates are chosen depending on a
given curve

5. Found the focal property of parabola

6. Studied cylindrical helix

1. Apollonian definition of a circle

2. Division a line in a given ratio, harmonic section

3. Apollonian problem: construct a circle touching three things (point, lines, or circles)

4. Apollonius Theorem

Lecture 10. Trigonometry and astronomy:

Hipparchus and Ptolemy
Hipparchus of Nicaea (now Isnik) 190-120BC the greatest
astronomer of antiquity, also geographer and mathematician
1) A founder of trigonometry (at least he used it
systematically for calculation of orbits), tabulated
the values of the Chord Function (length of chard
for each angle).
2) Accepted a sexagecimal full circle as 360o.
3) Transformed astronomy from purely
theoretical to a practical predicative science.
4) Proved that stereographical projection is
conformal
(preserves angles,
send circles to
circles).
 Claudius Ptolemy
90-168AD Roman mathematician, astronomer, geographer,
worked in Alexandria, the author Almagest
of (The Great Treatise)
that is
theonly surviving treatise in astronomy, which is based generally on the works
of Hipparchus. It contains a star catalogue
with 48 star constellations and
handy tables convenient for calculation the apparent orbits of Sun, Moon, and
planets. He tried to adopt horoscopic Astrology to Aristotelean Natural
Philosophy.

Lecture 11. Mathematics of the Late Hellenistic Period

Hero (Heron) of Alexandria 10-70 AD mathematician and engineer-inventor

Heron’s formula for the area of a triangle

Heronian Mean (related to the volume of a truncated cone)

Heronian triangle is a triangle that has side lengths and area that are all integers (like Pytagorian
ones).

In Optics: formulated the principle of the shortest path of light (stated by P.Fermat in 1662)

minutes

for coins
 Inventions:

1) Aeolipile (steam turbine) known also as “Hero’s Ball”

2) Syringe
3) Automatic temple Door opener
4) Dioptra (for geodesic measurements)
5) The first programmable robot to entertain audience at the theatre: could move in a
preprogrammed way, drop metal balls, etc., for 10

6) The first vending machine to dispense holy water

7) Fountain using sophisticated pneumatic and hydraulic principles

8) Wind powered organ (the first example of wind powered machine)

Books: Pneumatica, Automata, Mechanica, Metrica, On the Dioptra, Belopoeica,

Catoptrica, Geometria, Stereometrica, Mensurae, Cheiroballistra, Definitiones,
Geodesia, Geoponica

Nicomachus of Gerasa 60-120AD the author of Introduction to Arithmetic where for the
first time Arithmetic was separated from Geometry.
As a Neo-Pythagorean, he was interested more in some mystical and divine properties of numbers, than in
conceptual and deep mathematical questions. One of his “divine” examples is an observation about cubes:
1=13, 3+5=23, 7+9+11=33, 13+15+17+19=43, etc.
Introduction to Arithmetic was a popular and influential textbook for non-mathematicians for about 1000
years, and Nicomachus was put in one row with Euclid, Pythagoras and Aristotle,
although serious scholars did not respect him. Another popular book of Nicomachus was Manual of
Harmonics based on Pythagoras and Aristotle.
 Menelaus (Μενέλαος) of
Alexandria 70 – 140 AD
mathematician and astronomer,
the first to recognize geodesics on a
curved surface as natural analogs of
straight lines. The book Sphaerica
introduces the concept of spherical
triangle and proves Menelaus'
theorem on collinearity of points on the
edges of a triangle (which may have been previously known) and its analog for spherical triangles.

algebra"

are sought.
notation and

The first one

rational

Diophantus of Alexandria (Διόφαντος) 210 -295 called "the father of

Among 13 published books just 6 survived. A treatise called Arithmetica deals with
solving algebraic equations. Diophantine equations (including the ones known as Pell’s
equation and Ferma’s equation) are usually algebraic equations with integer coefficients,
for which integer solutions Diophantus also made advances in mathematical passed to
syncopated algebra from rhetorical algebra.

who recognized rationals as numbers and studied solutions of equations.

Pappus of Alexandria (Πάππος) 290–350 the last great mathematicians of Antiquity

 polyhedra. Famous for:

1) Pappus's Theorem in projective geometry,

2) Pappus Hexagon Theorem
3) Pappus chain of circles (centers are on an
ellipse)
4) Pappus centroid (two) theorems
5) The polar line, its construction
6) Directorial property of conics
7) Perspective, the cross-ratio and harmonic ratio, complete quadrangle
8) Analyzed curves (Spirals, concoid, Quadratrix, Helix on a sphere
9) Inscribing regular polyhedra in a sphere, 13 semiregular
polyhedra
10) Regular polygons, isoperimetric property
11) Explained terms “analysis” and “synthesis”, “theorem”,
“problem”, “porism”
12) Problem: how to draw an ellipse through 5 points, and similar
mate calculation of the square root of p/q.

Burning of
His work the Library of
Alexandria
Collection
(Synagoge) in 8 volumes
mainly survived: it including
geometry, recreational
mathematics, doubling the
cube, polygons and
Serapis Emperor Caracalla suppressed Column of
derived from the Diocletian
Osiris+Apis
Greco-Egyptian
Musaeum in 216, and on the orders of Emperor Aurelian it was then destroyed by fire in 272.
god, patron of Remains of
Ptolemaic kings the Library of Alexandria were moved to Serapeum (temple of the god Serapis) where
scholars moved the center of their studies and lectures. Roman Emperor Theodosius (after
Constantine) by a decree in 380 AD forbade non-Christian worship and destructed the Temple of Apollo
in Delphi (in 390 AD) and the Serapeum in Alexandria (in 391 AD). Later Diocletian erected a column at
the place of destruction of Serapeum.
 Theon of Alexandria 335- 405 Professor of math and
astronomy in Alexandria. A competent but unoriginal mathematician
famed for his commentaries on many works such as Ptolemy's
Almagest and the works of Euclid. These commentaries were written
for his students and some are even thought to be lecture notes taken
by students at his lectures. He corrected mistakes which he spotted,
tried to standardise Euclid’s writing, and amplified Euclid's text to
make it easier for beginners. Till recently, Theon's version of
Euclid's Elements (written with the assistance of his daughter
Hypatia) was the only Greek text of the Elements. Commented
also Almagest and other classical books.

Hypatia ( of Alexandria 370-415 math, astronomy,

philosophy, inventions
Theon’s daughter, the first in history woman recognized as a mathematician and
scientist. She helped her father in commentaries to Euclid and Ptolemy, then commented herself to
Apollonius, Diophantus, became the head of Neoplatonic School in Alexandria and taught mathematics and
philosophy there.

She was kidnapped and later murdered by a Christian mob (500 monks from mountains) in a conflict
between two prominent figures in Alexandria: the governor Orestes and the Bishop of Alexandria. Later (in
7th century) accused in “demonic charm” and witchery.
Her murder marks downfall of Alexandrian intellectual life and the end of Classical Antiquity.

Proclus 411-485 one of the major last classical philosophers

(Neoplatonist) born in Constantinople, lived and worked at Alexandria, then in Athens Became the head
of Neoplatonic Academy in Athens. His commentaries on Euclid and stories about other mathematician is
our principal source of knowledge about history of Greek Geometry.

Express your idea about Greek Mathematics

 The idea of proof.  The concept of modeling natural phenomena on a unified basis.
The introduction of axioms. Can you identify problems that could not be solved by ancient
 
Prime numbers and number methods, but which are very near to ones that can?
 theory. Numbers: the division of valuations to length and number - a
 necessary consequence of incommensurables.
Greek counting and
 calculation. What were the stimuli for the particular methods and algorithms
Greek astronomy and the  developed?
 impact of geometry on it. Can you identify sufficient mathematics to handle the needs of
The earliest concepts of limit.  commerce?


Mathematicians of Ancient Greece

 Thales of Miletus (c. 630-c 550)  Euclid (c. 295)
 Anaximander of Meletus (c. 610-c. 547)  Aristarchus of Samos (c. 310-230)
 Pythagoras of Samos (c. 570-c. 490)  Archimedes of Syracuse (287-212)
 Anaximenes of Miletus (fl. c. 546))  Aristaeus the Elder (fl. c. 350-330)
 Cleostratus of Tenedos (c. 520)  Philo of Byzantium (fl. c. 250)
 Anaxagoras of Clazomenae (c. 500-c. 428)  Nicoteles of Cyrene (c. 250)
 Zeno of Elea (c. 490-c. 430)  Strato (c. 250)
 Antiphon of Rhamnos (the Sophist) (c. 480-411)  Persius (c. 250?)
 Oenopides of Chios (c. 450?)  Eratosthenes of Cyrene (c. 276-c. 195)
 Leucippus (c. 450)  Chrysippus (280-206)
 Hippocrates of Chios (c. 450)  Conon of Samos (c. 245)
 Meton (c. 430) *SB  Apollonius of Perga (c. 260-c. 185)
 Hippias of Elis (c. 425)  Nicomedes (c. 240?)
 Theodorus of Cyrene (c. 425)  Dositheus of Alexandria (fl. c. 230)
 Socrates (469-399)  Perseus (fl. 300-70 B.C.E.?)
 Philolaus of Croton (d. c. 390)  Dionysodorus of Amisus (c. 200?)
 Democritus of Abdera (c. 460-370)  Diocles of Carystus (c. 180)
 Hippasus of Metapontum (or of Sybaris or Croton)  Hypsicles of Alexandria (c. 150?)
(c. 400?)  Hipparchus of Nicaea (c. 180-c. 125)

Archytas of Tarentum (of Taras) (c. 428-c. 347) Zenodorus (c. 100?? BCE?)

Plato (427-347) Posidonius (c. 135-c. 51)
 
Theaetetus of Athens (c. 415-c. 369) Zeno of Sidon (c. 79 BCE)
 
Leodamas of Thasos (c. 380) Geminus of Rhodes (c. 77 BCE)
 
Leon (fl. c. 375) Cleomedes (c. 40? BCE)
 
Eudoxus of Cnidos (c. 400-c. 347) Heron of Alexandria (fl. c. 62 CE) (Hero)
 
Callipus of Cyzicus (fl. c. 370) Theodosius of Tripoli (c. 50? CE?)
 
Xenocrates of Chalcedon (c. 396-314) Menelaus of Alexandria (c. 100 CE)
 
Heraclides of Pontus (c. 390-c. 322) Nicomachus of Gerasa (c. 100)
 
Bryson of Heraclea (c 350?) Ptolemy (Claudius Ptolemaeus) (100-178)
 
Menaechmus (c. 350) Diogenes Laertius (c. 200)
 
Theudius of Magnesia (c. 350?) Diophantus of Alexandria (c. 250?)
 
Thymaridas (c. 350) Iamblichus (c. 250-c. 350)
 
Dinostratus (c. 350) Pappus of Alexandria (c. 320)
 
Speusippus (d. 339) Theon of Alexandria (c. 390)
 
Aristotle (384-322) Hypatia of Alexandria (c. 370-415)
 Eudemus of Rhodes (the Peripatetic) (c. 335) 
Proclus Diadochus (410-485)
 Autolycus of Pitane (c. 300) 

 Lecture 12.
Mathematics of
Ancient China and
India

9 Chapters of the Mathematical Art fundamental work dominating

the history of Chinese mathematics and playing in China the role of Euclid’s Elements. It looks
like a practical handbook consisting of 246 problems for practical needs: in engineering,
surveying, trade and taxation. Unlike Elements 9 Chapters are not concerned about rigorous
proofs.

Some Chinese (for example, Liu Hui) believe that the basis of 9 Chapters was written about 1000 BC, and
later some mathematicians contributed to it. But in Qin-dynasty time (213 BC) all the copies were
burned which led to the destruction of the classical knowledge. A new version is basically due to Zhang
Cang in 170 BC. But mostly historians believe that 9 chapters were originally written after 200BC.

Content of the Chapters:

1) Land Surveying: 38 problems about area (triangles. Rectangles, trapeziums, circles) including addition,
subtraction, multiplication and division. The Euclidean algorithm for the

g.c.d. is given. Approximation of is presented.

2) Millet and Rice: 46 problems about exchange of goods, with the rates of 20 different types of grains,
beans and seeds. Solving proportions and percentage problems.
3) Distribution by proportion: 20 problems with direct, inverse and compound proportions. Arithmetic and
geometric progression is used.
 4) Short Width: some extremal problems, then square and cubic roots. Notion of limit and infinitesimal.
5) Civil Engineering: 28 problems on construction canals etc. Volumes of prisms, pyramids, wedges,
cylinders. Liu Hui discussed a “method of exhaustion”.
6) Fair distribution of Goods: 28 problems about ratio and proportions (travelling, taxation, sharing).
Filling a cistern through 5 canals. A pursuit problem.
7) Excess and Deficit: 20 problems with the rule of double false position. Linear equations are solved by
making two guesses at the solution, then finding answer from two errors.
8) Calculation by Square Tables: 18 problems with linear systems solved by Gaussian elimination (just
equations are placed in columns, that is why column operations are used). Negative numbers are used.
9) Right angled triangles: 24 problems. Pythagoras theorem is known as “Gougu rule”. Pythagorean
triples, similar triangles. Quadratic equations are solved using a geometric square-root algorithm. To
summarize, the main goals are

1) Systematic treatment of fractions 5) Introducing concepts of

2) Dealing with various kinds of positive and negative numbers
proportions 6) Finding a formula for
3) Devising methods for extracting Pythagorean triples
square and cubic roots 7) Calculating areas and volumes
4) Solution of linear systems of of different shapes and figures
equations

Liu Hui 263 AD one of the greatest mathematicians of

ancient China

1) Used decimal fractions and obtained , using a

3072-gon.
2) Prove a formula for Pythagoras triples.
3) Devised a method for solving linear systems (Gaussian elimination).
4) Solving many practical questions (finding the height, width of a river, depth, size of a
city, etc.).

Indian Mathematics: originates in Vedic mathematics in Sanskrit sutras with

multiplication rules and formulas (like areas of geometric figures, may be Pythagoras Th)
hidden between the Vedic hymns.

Aryabhata 476-550 AD book Aryabhatiya (on Astronomy) in 121 verses

1) The place-value system; he did not use
2) zero, but
irrational number
3)

some argue that its knowledge was implicit Approximation of = 3.1416;
he had possibly a guess that is an
 The oldest use of alphabet numerals in place of oldstyle word numerals 4)
Used arithmetic and
Geometric progressions 5) Used trigonometry for the computation of eclipses

-Solution of quadratic equation -Positive and negative numbers:

-Solution of systems by elimination properties of their addition and
method multiplication
-Syncopated algebra -Found Pythagorean triples
-Sum of squares and cubes of the 2
-Pell’s equation Nx +1=y
2

first n integers -Brahmagupta theorem on inscribed

-Zero as a number is mentioned for quadrilateral with perpendicular
the first time in his book; diagonals
Brahmagupta
operations with zero are described -
(a problem with division) Formula for the area A=

598- 668

Bhāskara I 600–680 was the first to write numbers in the Hindu decimal system, with a circle for the
zero. He gave a rational approximation of the sine function in commentary on Aryabhata's work.
Bhāskara II 1114–1185 worked in astronomy, calculus, algebra, and spherical trigonometry.

Lecture 13. Mathematics of Islamic Middle East

House of Wisdom founded by the Abbasid Caliph Abu Ja'far Al

Mansour in the 8th century, for translating foreign books. Many foreign works were translated into Arabic
from Greek, Chinese, Sanskrit, Persian and Syriac. Scholars produced important original research. New
discoveries motivated revised translations that commented, corrected
or added to the work of ancient authors. Ptolemy's Almagest, is an Arabic modification of the original name
of the work: Megale Syntaxis.
 Muḥammad ibn Mūsā al-Khwārizmī 780 –  850
(Algoritmi) “father of algebra” a Persian
mathematician, astronomer and geographer during
the Abbasid Caliphate, a scholar in the House of
Wisdom in Baghdad
His systematic approach in solving linear and quadratic
equation, leaded to algebra, a word derived from the title of
his 830 book on the subject, The Compendious Book on
Calculation by Completion and Balancing (ḥisāb al-jabr wal-
muqābala). Algebra was a unifying theory
which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as
"algebraic objects". It gave mathematics a whole new development path.

Another book, On the Calculation with Hindu Numerals written about 825, was
principally responsible for spreading the Hindu–Arabic numeral system throughout the
Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum.
AlKhwārizmī, rendered as (Latin) Algoritmi, led to the term algorithm. He assisted a
project to determine the circumference of the Earth and in making a world map.

When, in the twelfth century, his works spread to Europe through Latin translations, it
had a profound impact on the advance of mathematics in Europe.

Al-Ṣābiʾ Thābit ibn Qurra al-Ḥarrānī 826–901 Arabic mathematician, physician, astronomer,
and translator of the Islamic Golden Age who lived in Baghdad. In mathematics, Thabit discovered
an equation for determining amicable numbers. He is known for having calculated the solution to a
chessboard problem involving an exponential series. He also described a Pythagoras theorem.

ʿAbd al-Hamīd ibn Turk (830), Turkic mathematician, the author of Logical Necessities in Mixed
Equations, which is similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or
even possibly earlier than, Al-Jabr. The
manuscript
gives
exactly the
Al Hajjaj ibn Yusuf ibn Mattar ?-833
same the first translator of Eucid’s Elements
geometric
Al-Abbas ibs Saed ibn Jawhari Commentary Fateh al Harrani al Hasib ?-825
on Euclid’s Elements, 50 new propositions, Addition and Separation, The cubes,
attempts to prove 5th postulate Fundamentals of Arithmetic

demonstration as is found in Al-Jabr, and in one case the same

example as found in Al-Jabr, and even goes beyond AlJabr by
giving a geometric proof that if the discriminant is negative
then the quadratic equation has no solution.
Omar Khayyám
(1038-1123) Persian mathematician, astronomer,
philosopher, and poet, who is widely considered to be one
of the most influential scientists of all time. He wrote
numerous treatises on
mechanics, geography, mineralogy and astrology. His Treatise on Demonstration of Problems of Algebra
containing a systematic solution of third-degree equations, going beyond the Algebra of Khwārazmī,
Khayyám obtained the solutions of these equations by finding the intersection points of two conic
sections. This method had been used by the Greeks, but they did not generalize the method to cover all
equations with positive roots. A book Explanations of the difficulties in the postulates in Euclid's Elements
where he analyzed the parallel postulate contributed to the development of nonEuclidean geometry.
In considering Pascal's triangle, known in Persia as "Khayyam's triangle".

Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī (or al-

Karkhī) 953 –1029 mathematician and engineer working at
Baghdad.
1) His work on algebra and polynomials gave the rules for arithmetic operations for adding,
subtracting and multiplying polynomials; though he was restricted to dividing polynomials by
monomials.
2) He systematically studied the algebra of exponents, and was the first to realize that the sequence x,
x2, x3,... and the reciprocals 1/x, 1/x2, 1/x3,... could be extended indefinitely.
3) He wrote on the binomial theorem and Pascal's triangle.
4) In a now lost work, he introduced the idea of argument by mathematical induction.
 Sharaf al-Dīn al-Ṭūsī 1135-1213 Persian
mathematician and astronomer.
In treatise on cubic equation analyzed the maximum of
y=bx-x3 by taking the derivative. Then Al-Tusi deduces
that the equation has a positive root if D = b3/27 - a2/4 ≥
0, where D is the discriminant of the equation. Al-Tusi
then went on to give what we would essentially call the
Abu Yousuf Ali bun Ahmet bin
Abu Al Vafa Al Buzajani
?-1000 Abu Ali Al Hasan ibn Al Yaqub Al Razi Omran Al Mousili ?-955
sperical trigonometry, book using Haytham (Al Hazen) ?-1039
Ahmed bin Muhannad al negative numbers, introduced sec, elliptic and hyperbolic geometry
Hasib ?-927 bookAddition cosec
and Separation

Ruffini-Horner method for approximating the root of the cubic equation. A treatise on the two lines that
approach each other, but never intersect.
Nasir al-Din al-Tusi 1201-1274 Persian polymath and
prolific writer: an architect, astronomer, biologist, chemist, mathematician, philosopher, physician,
physicist, scientist, theologian. Perhaps, the greatest of the later Persian scholars.

Main works in Mathematics:

Book on the complete quadrilateral: A five volume summary of trigonometry including spherical
trigonometry. This is the first treatise on trigonometry independent of astronomy.

On the Sector Figure, appears the famous law of sines for plane
triangles (stated also for spherical triangles), the law of tangents for
spherical triangles, including the proofs.

Al-Kāshī 1380–1429 Persian astronomer and mathematician

In French, the law of cosines is named Theorem of Al-Kashi, as al-Kashi was

the first to provide its explicit statement.

In The Treatise on the Chord and Sine, al-Kashi computed sin 1° to nearly as
much accuracy as his value for π, which was the most accurate approximation
of sin 1° in his time. In algebra and numerical analysis, he developed an
iterative method for solving cubic equations, which was not discovered in
Europe until centuries later.

A method algebraically equivalent to Newton's method was known to his predecessor Sharaf al-Dīn
al-Tūsī. Al-Kāshī
improved on this by using a form of Newton's method to solve to find roots of N.
 In order to determine sin 1°, al-Kashi discovered the
following formula often attributed to François Viète in the
16th century:

He correctly computed 2π to 9 sexagesimal digits in 1424, and he converted this approximation of

2π to 17 decimal places of accuracy. Al-Kashi's goal was to compute the circle constant so precisely
that the circumference of the largest possible circle (ecliptics) could be computed with highest
desirable precision (the diameter of a hair).
Kamāl al-Dīn al-Fārisī 1267-1319,
impossibility of solution x4+y4=z4,
amicable numbers

Al Samaw’al ibn al-Maghribi

1130-1180, book at age 19
approximation of n-th root

Abu Bakr al-Hassar 12th century invented modern notation for fractions
Al Murrakishi
1258-1321

Timeline of (some) Math Events till 13th Century

Date Name Nationality Major Achievements

35000 BC African First notched tally bones

3100 BC Sumerian Earliest documented counting and measuring system
2700 BC Egyptian Earliest fully-developed base 10 number system in use
2600 BC Sumerian Multiplication tables, geometrical exercises and division problems
2000-1800 BC Egyptian Earliest papyri showing numeration system and basic arithmetic
1800-1600 BC Babylonian Clay tablets dealing with fractions, algebra and equations
1650 BC Egyptian Rhind Papyrus (instruction manual in arithmetic, geometry, unit fractions,
etc)
1200 BC Chinese First decimal numeration system with place value concept
1200-900 BC Indian Early Vedic mantras invoke powers of ten from a hundred all the way up
to a trillion
800-400 BC Indian “Sulba Sutra” lists several Pythagorean triples and simplified

Pythagorean theorem for the sides of a square and a rectangle, quite

650 BC Chinese Lo Shu order three (3 x 3) “magic square” in which each row, column
and diagonal sums to 15
624-546 BC Thales Greek Early developments in geometry, including work on similar and right
triangles
570-495 BC Pythagoras Greek Expansion of geometry, rigorous approach building from first principles,
square and triangular numbers, Pythagoras’ theorem
500 BC Hippasus Greek Discovered potential existence of irrational numbers while trying to
calculate the value of √2
490-430 BC Zeno of Elea Greek Describes a series of paradoxes concerning infinity and infinitesimals
470-410 BC Hippocrates Greek First systematic compilation of geometrical knowledge, Lune of
of Chios Hippocrates
460-370 BC Democritus Greek Developments in geometry and fractions, volume of a cone
428-348 BC Plato Greek Platonic solids, statement of the Three Classical Problems, influential
teacher and popularizer of mathematics, insistence on rigorous proof
and logical methods
410-355 BC Eudoxus of Greek Method for rigorously proving statements about areas and volumes by
Cnidus successive approximations
384-322 BC Aristotle Greek Development and standardization of logic (although not then considered
part of mathematics) and deductive reasoning
300 BC Euclid Greek Definitive statement of classical (Euclidean) geometry, use of axioms
and postulates, many formulas, proofs and theorems including Euclid’s
Theorem on infinitude of primes
287-212 BC Archimedes Greek Formulas for areas of regular shapes, “method of exhaustion” for
approximating areas and value of π, comparison of infinities
276-195 BC Eratosthenes Greek “Sieve of Eratosthenes” method for identifying prime numbers
262-190 BC Apollonius of Greek Work on geometry, especially on cones and conic sections (ellipse,
Perga parabola, hyperbola)
200 BC Chinese “Nine Chapters on the Mathematical Art”, including guide to how to solve
equations using sophisticated matrix-based methods
190-120 BC Hipparchus Greek Develop first detailed trigonometry tables
36 BC Mayan Pre-classic Mayans developed the concept of zero by at least this time
10-70 AD Heron (or Greek Heron’s Formula for finding the area of a triangle from its side lengths,
Hero) of Heron’s Method for iteratively computing a square root
Alexandria
90-168 AD Ptolemy Greek/Egyp Develop even more detailed trigonometry tables
tian
200 AD Sun Tzu Chinese First definitive statement of Chinese Remainder Theorem
200 AD Indian Refined and perfected decimal place value number system
200-284 AD Diophantus Greek Diophantine Analysis of complex algebraic problems, to find rational
solutions to equations with several unknowns
220-280 AD Liu Hui Chinese Solved linear equations using a matrices (similar to Gaussian
elimination), leaving roots unevaluated, calculated value of π correct to
five decimal places, early forms of integral and differential calculus
400 AD Indian “Surya Siddhanta” contains roots of modern trigonometry, including first
real use of sines, cosines, inverse sines, tangents and secants
476-550 AD Aryabhata Indian Definitions of trigonometric functions, complete and accurate sine and
versine tables, solutions to simultaneous quadratic equations, accurate
approximation for π (and recognition that π is an irrational number)
598-668 AD Brahmagupta Indian Basic mathematical rules for dealing with zero (+, - and x), negative
numbers, negative roots of quadratic equations, solution of quadratic
equations with two unknowns
600-680 AD Bhaskara I Indian First to write numbers in Hindu-Arabic decimal system with a circle for
zero, remarkably accurate approximation of the sine function
780-850 AD Muhammad Persian Advocacy of the Hindu numerals 1 - 9 and 0 in Islamic world,
Al-Khwarizmi foundations of modern algebra, including algebraic methods of
“reduction” and “balancing”, solution of polynomial equations up to
second degree
908-946 AD Ibrahim ibn Arabic Continued Archimedes' investigations of areas and volumes, tangents to
Sinan a circle
953-1029 AD Muhammad Persian First use of proof by mathematical induction, including to prove the
 Al-Karaji binomial theorem
966-1059 AD Ibn al- Persian/Ara Derived a formula for the sum of fourth powers using a readily
Haytham bic generalizable method, “Alhazen's problem”, established beginnings of
(Alhazen) link between algebra and geometry
1048-1131 Omar Persian Generalized Indian methods for extracting square and cube roots to
Khayyam include fourth, fifth and higher roots, noted existence of different sorts of
cubic equations
1114-1185 Bhaskara II Indian Established that dividing by zero yields infinity, found solutions to
quadratic, cubic and quartic equations (including negative and irrational
solutions) and to second order Diophantine equations, introduced some
preliminary concepts of calculus
1170-1250 Leonardo of Italian Fibonacci Sequence of numbers, advocacy of the use of the HinduArabic
Pisa numeral system in Europe, Fibonacci's identity (product of two sums of
(Fibonacci) two squares is itself a sum of two squares)
1201-1274 Nasir al-Din Persian Developed field of spherical trigonometry, formulated law of sines for
al-Tusi plane triangles
1202-1261 Qin Jiushao Chinese Solutions to quadratic, cubic and higher power equations using a method
of repeated approximations
1238-1298 Yang Hui Chinese Culmination of Chinese “magic” squares, circles and triangles, Yang
Hui’s Triangle (earlier version of Pascal’s Triangle of binomial
coefficients)