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History of Mathematics – OLLI 2018

I. Mathematics starts with counting. Write the same numbers with each system: 13, 22, 66, 143

A. Babylonian mathematics

Developed from 2000 BC.

Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC.

The Babylonian system of mathematics was sexagesimal (base 60) numeral system.

13<

22<<

66

143 <<<<

From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360

degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons.

Firstly, the number 60 is a superior highly composite number, having factors of 1, 2, 3, 4, 5, 6, 10, 12,

15, 20, 30, 60 (including those that are themselves composite), facilitating calculations with fractions.

Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where

digits written in the left column represented larger values (much as, in our base ten system, 734 = 7×100

+ 3×10 + 4×1

The Babylonians used pre-calculated tables to assist with arithmetic. Dating from 2000 BC, there are lists of squares of numbers up to 59 and the cubes of numbers up to 32. They used the squares to

simplify multiplication: 2 2 2 2 2( ) ( ) ( )

2 4

a b a b a b a bab ab

n n² n n²

1 1 11 121

2 4 12 144

3 9 13 169

4 16 14 196

5 25 15 225

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The Babylonians didn’t have an algorithm for long-division. Instead they used a table of reciprocals and

the formula: 1a

ab b . When the reciprocal didn’t have a finite decimal representation, they used

approximations. For example: 1 7 1 1 40 280 4 40

7 7 713 91 91 90 3600 3600 60 3600

B. Roman mathematics (Etruscans – part of Italy; Hellenistic)

13 XIII 22 XXII 66 LXVI 143 CXLIII

The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are

represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, are

based on seven symbols:

Symbol I V X L C D M

Value 1 5 10 50 100 500 1,000

The Roman numerals are NOT positional or place-value based.

The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Hindu-

Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some

minor applications to this day.

Although Roman numerals came to be written with letters of the Roman alphabet, they were originally

independent symbols. The Etruscans, for example, used I, , , ⋔, 8 , and ⊕ for I, V, X, L, C, and M,

of which only I and X happened to be letters in their alphabet.

Tally Marks: One hypothesis is that the Etrusco-Roman numerals actually derive from notches on tally

sticks, which continued to be used by Italian and Dalmatian shepherds into the 19th century.

Thus, ⟨I⟩ descends not from the letter ⟨I⟩ but from a notch scored across the stick. Every fifth

notch was double cut i.e. ⋀, ⋁, ⋋, ⋌, etc.), and every tenth was cross cut

(X), IIIIΛIIIIXIIIIΛIIIIXII...), much like European tally marks today. This produced a positional

system: Eight on a counting stick was eight tallies, IIIIΛIII, or the eighth of a longer series of

tallies; either way, it could be abbreviated ΛIII (or VIII), as the existence of a Λ implies four

prior notches. By extension, eighteen was the eighth tally after the first ten, which could be

abbreviated X, and so was XΛIII. Likewise, number four on the stick was the I-notch that could

be felt just before the cut of the Λ (V), so it could be written as either IIII or IΛ (IV).

Thus the system was neither additive nor subtractive in its conception, but ordinal. When the tallies were transferred to writing, the marks were easily identified with the existing Roman

letters I, V and X. The tenth V or X along the stick received an extra stroke. Thus 50 was written

variously as N, И, K, Ψ, ⋔, etc., but perhaps most often as a chicken-track shape like a

superimposed V and I: ᗐ. This had flattened to ⊥ (an inverted T) by the time of Augustus, and

soon thereafter became identified with the graphically similar letter L. Likewise, 100 was

variously Ж, ⋉, ⋈, H, or as any of the symbols for 50 above plus an extra stroke. The form Ж (that is, a superimposed X and I like: �) came to predominate. It was written variously

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as >I< or ƆIC, was then abbreviated to Ɔ or C, with C variant finally winning out because, as a

letter, it stood for centum, Latin for "hundred". The hundredth V or X was marked with a box or

circle. Thus 500 was like a Ɔ superimposed on a ⋌ or ⊢, becoming D or Ð by the time of

Augustus, under the graphic influence of the letter ⟨D⟩. It was later identified as the letter D; an

alternative symbol for "thousand" was (I) (or CIƆ or CꟾƆ), and half of a thousand or "five

hundred" is the right half of the symbol, I) (or IƆ or ꟾƆ), and this may have been converted into

⟨D⟩.[14] This at least was the etymology given to it later on. Meanwhile, 1000 was a circled or

boxed X: Ⓧ, ⊗, ⊕, and by Augustinian times was partially identified with the Greek

letter Φ phi. Over time, the symbol changed to Ψ and ↀ. The latter symbol further evolved

into ∞, then ⋈, and eventually changed to M under the influence of the Latin

word mille "thousand".

The advent of the printing press in the mid-15th Century also had a huge impact. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their

commercial needs and mathematics gradually began to acquire a more important position in education.

Europe’s first great medieval mathematician was the Italian Leonardo of Pisa, better known by his nickname Fibonacci. Although best known for the so-called Fibonacci Sequence of numbers, perhaps

his most important contribution to European mathematics was his role in spreading the use of the Hindu-

Arabic numeral system throughout Europe early in the 13th Century, which soon made

the Roman numeral system obsolete, and opened the way for great advances in European mathematics.

Medieval abacus, based on the Roman/Greek model

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By the 3rd Century BCE, 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 center 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 BCE, 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 BCE. 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.

C. Hindu-Arabic

By the 11th century, Hindu–Arabic numerals had been introduced into Europe from al-Andalus, by way

of Arab traders and arithmetic treatises. Roman numerals, however, proved very persistent, remaining in

common use in the West well into the 14th and 15th centuries, even in accounting and other business

records (where the actual calculations would have been made using an abacus). Replacement by their

more convenient "Arabic" equivalents was quite gradual, and Roman numerals are still used today in

certain contexts.

the 10th Century Arab mathematician Abul Hasan al-Uqlidisi, who wrote the earliest surviving text

showing the positional use of Arabic numerals, and particularly the use of decimals instead of fractions

(e.g. 7.375 instead of 73⁄8)

The Sieve of Eratosthenes

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D. Egyptian

13 22 66 143

Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt c. 3000 to c. 300 BC, from the Old Kingdom of Egypt until roughly the beginning of Hellenistic

Egypt. The ancient Egyptians utilized a numeral system for counting and solving written mathematical

problems, often involving multiplication and fractions. Evidence for Egyptian mathematics is limited to

a scarce amount of surviving sources written on papyri. From these texts it is known that ancient

Egyptians understood concepts of geometry, such as determining the surface area and volume of three-

dimensional shapes useful for architectural engineering, and algebra, such as the false position

method and quadratic equations.

Ancient Egyptian texts could be written in either hieroglyphs or in hieratic. In either representation the

number system was always given in base 10. The number 1 was depicted by a simple stroke, the number

2 was represented by two strokes, etc. The numbers 10, 100, 1000, 10,000 and 1,000,000 had their own

hieroglyphs. Number 10 is a hobble for cattle, number 100 is represented by a coiled rope, the number

1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000

is represented by a frog, and a million was represented by a god with his hands raised in adoration.

Hieroglyphics for Egyptian numerals [2]

1 10 100 1000 10,000 100,000 1,000,000

The Egyptians almost exclusively used fractions of the form 1/n. One notable exception is the fraction 2/3, which is frequently found in the mathematical texts. Very rarely a special glyph was used to denote

3/4. The fraction 1/2 was represented by a glyph that may have depicted a piece of linen folded in two.

The fraction 2/3 was represented by the glyph for a mouth with 2 (different sized) strokes. The rest of

the fractions were always represented by a mouth super-imposed over a number.

Hieroglyphics for some fractions[6]

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E. Chinese

13 22 66 or

143

Even as mathematical developments in

the ancient Greek world were beginning

to falter during the final centuries BCE,

the burgeoning trade empire of China

was leading Chinese mathematics to ever

greater heights.

The simple but efficient ancient Chinese numbering system, which dates back to

at least the 2nd millennium BCE, used

small bamboo rods arranged to represent

the numbers 1 to 9, which were then

places in columns representing units, tens, hundreds, thousands, etc. It was therefore a decimal place

value system, very similar to the one we use today - indeed it was the first such number system, adopted

by the Chinese over a thousand years before it was adopted in the West - and it made even quite

complex calculations very quick and easy.

Written numbers, however, employed the slightly less efficient system of using a different symbol for tens, hundreds, thousands, etc. This was largely because there was no concept or symbol of zero, and it

had the effect of limiting the usefulness of the written number in Chinese.

The use of the abacus is often thought of as a Chinese idea, although some type of abacus was in use in Mesopotamia, Egypt and Greece, probably much earlier than in China (the first Chinese abacus, or

“suanpan”, we know of dates to about

the 2nd Century BCE).

There was a pervasive fascination with

numbers and mathematical patterns in

ancient China, and different numbers

were believed to have cosmic

significance. In particular, magic squares

- squares of numbers where each row,

column and diagonal added up to the

same total - were regarded as having

great spiritual and religious significance.

The Lo Shu Square, an order three square where each row, column and

diagonal adds up to 15, is perhaps the

earliest of these, dating back to around

650 BCE (the legend of Emperor Yu’s

discovery of the square on the back of a turtle is set as taking place in about 2800 BCE). But soon,

bigger magic squares were being constructed, with even greater magical and mathematical powers,

Lo Shu magic square, with its traditional graphical representation

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culminating in the elaborate magic squares, circles and triangles of Yang Hui in the 13th Century (Yang

Hui also produced a triangular representation of binomial coefficients identical to the later Pascal’s

Triangle, and was perhaps the first to use

decimal fractions in the modern form).

But the main thrust of Chinese

mathematics developed in response to

the empire’s growing need for

mathematically competent

administrators. A textbook called

“Jiuzhang Suanshu” or “Nine Chapters

on the Mathematical Art” (written over a

period of time from about 200 BCE

onwards, probably by a variety of

authors) became an important tool in the

education of such a civil service,

covering hundreds of problems in

practical areas such as trade, taxation,

engineering and the payment of wages.

It was particularly important as a guide to how to solve equations - the deduction

of an unknown number from other

known information - using a

sophisticated matrix-based method

which did not appear in the West

until Carl Friedrich Gauss re-discovered

it at the beginning of the 19th Century

(and which is now known as Gaussian

elimination).

Early Chinese method of solving equations

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The Chinese went on to solve far more

complex equations using far larger

numbers than those outlined in the “Nine

Chapters”, though. They also started to

pursue more abstract mathematical

problems (although usually couched in

rather artificial practical terms),

including what has become known as the

Chinese Remainder Theorem. This uses

the remainders after dividing an

unknown number by a succession of

smaller numbers, such as 3, 5 and 7, in

order to calculate the smallest value of

the unknown number. A technique for

solving such problems, initially posed by

Sun Tzu in the 3rd Century CE and

considered one of the jewels of

mathematics, was being used to measure

planetary movements by Chinese

astronomers in the 6th Century AD, and

even today it has practical uses, such as

in Internet cryptography.

F. Indian

13 ١٣ 22 ٢٢ 66 ٦٦ 143 ١٤٣

١٠,٩,٨,٧,٦,٥,٤,٣,٢,١

Despite developing quite independently

of Chinese (and probably also

of Babylonian mathematics), some very

advanced mathematical discoveries were

made at a very early time in India.

Mantras from the early Vedic period (before 1000 BCE) invoke powers of ten

from a hundred all the way up to a

trillion, and provide evidence of the use

of arithmetic operations such as addition,

subtraction, multiplication, fractions,

squares, cubes and roots. A 4th Century

CE Sanskrit text reports Buddha

enumerating numbers up to 1053, as well

The Chinese Remainder Theorem

The evolution of Hindu-Arabic numerals

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as describing six more numbering systems over and above these, leading to a number equivalent to

10421. Given that there are an estimated 1080 atoms in the whole universe, this is as close to infinity as

any in the ancient world came. It also describes a series of iterations in decreasing size, in order to

demonstrate the size of an atom, which comes remarkably close to the actual size of a carbon atom

(about 70 trillionths of a metre).

The Sutras also contain geometric solutions of linear and quadratic equations in a single unknown, and

give a remarkably accurate figure for the square root of 2, obtained by adding 1 + 1⁄3 + 1⁄(3 x 4) - 1⁄(3 x 4 x 34),

which yields a value of 1.4142156, correct to 5 decimal places.

Ancient Buddhist literature also demonstrates a prescient awareness of indeterminate and infinite numbers, with numbers deemed to be of three types: countable, uncountable and infinite.

Like the Chinese, the Indians early discovered the benefits of a decimal place value number system, and were certainly using it before about the 3rd Century CE. They refined and perfected the system,

particularly the written representation of the numerals, creating the ancestors of the nine numerals that

(thanks to its dissemination by medieval Arabic mathematicians) we use across the world today,

sometimes considered one of the greatest

intellectual innovations of all time.

The Indians were also responsible for another hugely important development in

mathematics. The earliest recorded usage

of a circle character for the number zero

is usually attributed to a 9th Century

engraving in a temple in Gwalior in

central India. But the brilliant conceptual

leap to include zero as a number in its

own right (rather than merely as a

placeholder, a blank or empty space

within a number, as it had been treated

until that time) is usually credited to the

7th Century Indian

mathematicians Brahmagupta - or possibly another Indian, Bhaskara I - even though it may well have

been in practical use for centuries before that. The use of zero as a number which could be used in

calculations and mathematical investigations, would revolutionize mathematics.

Brahmagupta established the basic mathematical rules for dealing with zero: 1 + 0 = 1; 1 - 0 = 1; and 1 x

0 = 0 (the breakthrough which would make sense of the apparently non-sensical operation 1 ÷ 0 would

also fall to an Indian, the 12th Century mathematician Bhaskara II).

The so-called Golden Age of Indian mathematics can be said to extend from the 5th to 12th Centuries, and many of its mathematical discoveries predated similar discoveries in the West by several centuries,

which has led to some claims of plagiarism by later European mathematicians, at least some of whom

were probably aware of the earlier Indian work. Certainly, it seems that Indian contributions to

mathematics have not been given due acknowledgement until very recently in modern history.

Bhaskara II, who lived in the 12th Century, was one of the most accomplished of all India’s great

mathematicians. He is credited with explaining the previously misunderstood operation of division by

zero. He noticed that dividing one into two pieces yields a half, so 1 ÷ 1⁄2 = 2. Similarly, 1 ÷ 1⁄3 = 3. So,

dividing 1 by smaller and smaller factions yields a larger and larger number of pieces. Ultimately, therefore, dividing one into pieces of zero size would yield infinitely many pieces, indicating that 1 ÷ 0

= ∞ (the symbol for infinity).

The earliest use of a circle character for the number zero was in India

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G. Mayan/Incan mathematics

13 22 66 143

The Mayan civilization had settled in the region of Central America from about

2000 BCE, although the so-called

Classic Period stretches from about 250

CE to 900 CE. At its peak, it was one of

the most densely populated and

culturally dynamic societies in the world.

The importance of astronomy and

calendar calculations in Mayan society

required mathematics, and the Maya

constructed quite early a very

sophisticated number system, possibly

more advanced than any other in the

world at the time (although the dating of

developments is quite difficult).

The Mayan and other Mesoamerican cultures used a vigesimal number system

based on base 20 (and, to some extent,

base 5), probably originally developed from counting on fingers and toes. The numerals consisted of

only three symbols: zero, represented as a shell shape; one, a dot; and five, a bar. Thus, addition and

subtraction was a relatively simple matter of adding up dots and bars. After the number 19, larger

numbers were written in a kind of vertical place value format using powers of 20: 1, 20, 400, 8000,

160000, etc (see image above), although in their calendar calculations they gave the third position a

value of 360 instead of 400 (higher positions revert to multiples of 20).

The pre-classic Maya and their neighbors had independently developed the concept of zero by at least as early as 36 BCE, and we have evidence of their working with sums up to the hundreds of millions, and

with dates so large it took several lines just to represent them. Despite not possessing the concept of a

fraction, they produced extremely accurate astronomical observations using no instruments other than

sticks, and were able to measure the length of the solar year to a far higher degree of accuracy than that

used in Europe (their calculations produced 365.242 days, compared to the modern value of

365.242198), as well as the length of the lunar month (their estimate was 29.5308 days, compared to the

modern value of 29.53059).

However, due to the geographical disconnect, Mayan and Mesoamerican mathematics had absolutely no influence on Old World (European and Asian) numbering systems and mathematics.

H. Islamic mathematics

The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia and

parts of India from the 8th Century onwards made significant contributions towards mathematics. They

were able to draw on and fuse together the mathematical developments of both Greece and India.

Mayan numerals

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One consequence of the Islamic prohibition on depicting the human form was the extensive use of

complex geometric patterns to decorate their buildings, raising mathematics to the form of an art. In fact,

over time, Muslim artists discovered all the different forms of symmetry that can be depicted on a 2-

dimensional surface.

The Qu’ran itself encouraged the accumulation of knowledge, and a Golden Age of Islamic science and mathematics flourished throughout the medieval period from the 9th to 15th Centuries. The House of

Wisdom was set up in Baghdad around 810, and work started almost immediately on translating the

major Greek and Indian mathematical and astronomy works into Arabic.

Some complex symmetries in Islamic temple decorations:

II. Geometry

Pythagorean theorem/triples

s t st (s²-t²)/2 (s²+t²)/2

3 1 3 4 5

5 3 15 8 17

5 1 5 12 13

7 5 35 12 37

7 3 21 20 29

7 1 7 24 25

9 7 63 16 65

9 5 45 28 53

9 1 9 40 41

11 9 99 20 101

11 7 77 36 85

11 5 55 48 73

11 3 33 56 65

11 1 11 60 61

13 11 143 24 145

13 9 117 44 125

13 7 91 60 109

13 5 65 72 97

13 3 39 80 89

13 1 13 84 85

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Analytic geometry (conics) But Alexandria was not the only center 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 BCE 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.

Conic sections of Apollonius

Greek/Hellenistic: It is not known exactly when the great Library of Alexandria burned down, but

Alexandria remained an important intellectual center for some centuries. In the 1st century BCE, 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 CE, 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").

Euler's theorem for polyhedra, V - E + F = 2.

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Platonic solids

Tetrahedron

Cube Octahedron Dodecahedron Icosahedron

The Platonic solids have been known since antiquity. Carved stone balls created by the late Neolithic people of Scotland lie near ornamented models resembling them, but the Platonic solids do not

appear to have been preferred over less-symmetrical objects, and some of the Platonic solids may even

be absent. Dice go back to the dawn of civilization with shapes that predated formal charting of Platonic

solids.

The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus)

credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar

with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron

belong to Theaetetus, a contemporary of Plato.

The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus c.360 B.C. in which he associated each of the four classical

elements (earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the

octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification

for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the

octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron,

flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly

nonspherical solid, the hexahedron (cube) represents "earth". These clumsy little solids cause dirt to

crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's

being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the

Earth.

Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used [it] for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithēr (aether in Latin, "ether"

in English) and postulated that the heavens were made of this element, but he had no interest in

matching it with Plato's fifth solid.

Euclid completely mathematically described the Platonic solids in the Elements, the last book (Book

XIII) of which is devoted to their properties.

B. Indian

As early as the 8th Century BCE, long before Pythagoras, a text known as the “Sulba Sutras” (or "Sulva Sutras") listed several simple Pythagorean triples, as well as a statement of the simplified Pythagorean

theorem for the sides of a square and for a rectangle (indeed, it seems quite likely

that Pythagoras learned his basic geometry from the "Sulba Sutras").

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III. Algebra

By the 13th Century, the Golden Age of Chinese mathematics, there were over 30 prestigious

mathematics schools scattered across China. Perhaps the most brilliant Chinese mathematician of this

time was Qin Jiushao, a rather violent and corrupt imperial administrator and warrior, who explored

solutions to quadratic and even cubic equations using a method of repeated approximations very similar

to that later devised in the West by Sir Isaac Newton in the 17th Century. Qin even extended his

technique to solve (albeit approximately) equations involving numbers up to the power of ten,

extraordinarily complex mathematics for its time.

Brahmagupta also established rules for dealing with negative numbers, and pointed out that quadratic equations could in theory have two possible solutions, one of which could be negative. He even

attempted to write down these rather abstract concepts, using the initials of the names of colors to

represent unknowns in his equations, one of the earliest intimations of what we now know as algebra.

However, Bhaskara II also made important contributions to many different areas of mathematics from solutions of quadratic, cubic and quartic equations (including negative and irrational solutions) to

solutions of Diophantine equations of the second order to preliminary concepts of infinitesimal calculus

and mathematical analysis to spherical trigonometry and other aspects of trigonometry. Some of his

findings predate similar discoveries in Europe by several centuries, and he made important contributions

in terms of the systemization of (then) current knowledge and improved methods for known solutions.

In the 3rd Century CE, 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.

Al-Khwarizmi's other important contribution was algebra, and he introduced the fundamental algebraic

methods of “reduction” and “balancing” and provided an exhaustive account of solving polynomial

equations up to the second degree. In this way, he helped create the powerful abstract mathematical

language still used across the world today, and allowed a much more general way of analyzing problems

other than just the specific problems previously considered by the Indians and Chinese.

Among other things, Al-Karaji used mathematical induction to prove the binomial theorem. A binomial is a simple type of algebraic expression which has just two terms which are operated on only by

addition, subtraction, multiplication and positive whole-number exponents, such as (x +y)2. The co-

efficients needed when a binomial is expanded form a symmetrical triangle, usually referred to as

Pascal’s Triangle after the 17th Century French mathematician Blaise Pascal, although many other

mathematicians had studied it centuries before him in India, Persia, China and Italy, including Al-Karaji.

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Some hundred years after Al-Karaji, Omar Khayyam (perhaps better known as a poet and the writer of

the “Rubaiyat”, but an important mathematician and astronomer in his own right)

generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots

in the early 12th Century. He carried out a systematic analysis of cubic problems, revealing there were

actually several different sorts of cubic equations. Although he did in fact succeed in solving cubic

equations, and although he is usually credited with identifying the foundations of algebraic geometry, he

was held back from further advances by his inability to separate the algebra from the geometry, and a

purely algebraic method for the solution of cubic equations had to wait another 500 years and the Italian

mathematicians del Ferro and Tartaglia.

IV. Trigonometry

Golden Age Indian mathematicians made fundamental advances in the theory of trigonometry, a method of linking geometry and numbers first developed by the Greeks. They used ideas like the sine, cosine

and tangent functions (which relate the angles of a triangle to the relative lengths of its sides) to survey

the land around them, navigate the seas and even chart the heavens. For instance, Indian astronomers

used trigonometry to calculate the relative distances between the Earth and the Moon and the Earth and

the Sun. They realized that, when the Moon is half full and directly opposite the Sun, then the Sun,

Moon and Earth form a right angled triangle, and were able to accurately measure the angle as 1⁄7°. Their

sine tables gave a ratio for the sides of such a triangle as 400:1, indicating that the Sun is 400 times

further away from the Earth than the Moon.

Although the Greeks had been able to calculate the sine function of some angles, the Indian astronomers wanted to be able to calculate the sine function of any given angle. A text called the “Surya Siddhanta”,

by unknown authors and dating from around 400 CE, contains the roots of modern trigonometry,

including the first real use of sines, cosines, inverse sines, tangents and secants.

As early as the 6th Century CE, the great Indian mathematician and astronomer Aryabhata produced

categorical definitions of sine, cosine, versine and inverse sine, and specified complete sine and versine

tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. Aryabhata also

16

demonstrated solutions to simultaneous quadratic equations, and produced an approximation for the

value of π equivalent to 3.1416, correct to four decimal places. He used this to estimate the

circumference of the Earth, arriving at a figure of 24,835 miles, only 70 miles off its true value. But,

perhaps even more astonishing, he seems to have been aware that π is an irrational number, and that any

calculation can only ever be an approximation, something not proved in Europe until 1761.

The Kerala School of Astronomy and Mathematics was founded in the late 14th Century by Madhava of

Sangamagrama, sometimes called the greatest mathematician-astronomer of medieval India. He

developed infinite series approximations for a range of trigonometric functions, including π, sine, etc.

Some of his contributions to geometry and algebra and his early forms of differentiation and integration

for simple functions may have been transmitted to Europe via Jesuit missionaries, and it is possible that

the later European development of calculus was influenced by his work to some extent.

Hipparchus, who was also from Hellenistic Anatolia and who live in the 2nd Century BCE, 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 CE, 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.

X. Calculating pi

Among the greatest mathematicians of ancient China was Liu Hui, who produced a detailed commentary on the “Nine Chapters” in 263 CE, was one of the first mathematicians known to leave roots

unevaluated, giving more exact results instead of approximations. By an approximation using a regular

polygon with 192 sides, he also formulated an algorithm which calculated the value of π as 3.14159

(correct to five decimal places), as well as developing a very early forms of both integral and differential

calculus.

outside

diameter =

72(0.087156)

2 = 3.137607

sin 5 0.087156

outside

diameter =

24(0.258819)

2 = 3.105829

sin 15 0.258819

15

outside

diameter =

6

2 = 3

r = 1

36 triangles12 triangles6 triangles

About 250BC: Archimedes gives the formulae for calculating the volume of a sphere and a cylinder; he

gives an approximation of the value of π with a method which will allow improved approximations.

17

263: By using a regular polygon with 192 sides Liu Hui calculates the value of π as 3.14159 which is

correct to five decimal places.

About 460: Zu Chongzhi gives the approximation 355/113 to π which is correct to 6 decimal places.

499: Aryabhata I calculates π to be 3.1416. He produces his Aryabhatiya, a treatise on quadratic equations, the value of π, and other scientific problems.

1400: Madhava of Sangamagramma proves a number of results about infinite sums giving Taylor

expansions of trigonometric functions. He uses these to find an approximation for π correct to 11

decimal places.

1424: Al-Kashi writes Treatise on the Circumference giving a remarkably good approximation to π in both sexagesimal and decimal forms.

1593: Van Roomen calculates π to 16 decimal places.

1706: Jones introduces the Greek letter π to represent the ratio of the circumference of a circle to its diameter in his Synopsis palmariorum matheseos (A New Introduction to Mathematics).

1761: Lambert proves that π is irrational. He publishes a more general result in 1768.

1853: Shanks gives π to 707 places (in 1944 it was discovered that Shanks was wrong from the 528th place).

1882: Lindemann proves that π is transcendental. This proves that it is impossible to construct a square with the same area as a given circle using a ruler and compass. The classic mathematical problem of

squaring the circle dates back to ancient Greece and had proved a driving force for mathematical ideas

through many centuries.

VI. Medieval mathematics

During the centuries in which the Chinese, Indian and Islamic mathematicians had been in the

ascendancy, Europe had fallen into the Dark Ages, in which science, mathematics and almost all

intellectual endeavor stagnated. Scholastic scholars only valued studies in the humanities, such as

philosophy and literature, and spent much of their energies quarrelling over subtle subjects in

metaphysics and theology, such as "How many angels can stand on the point of a needle?"

From the 4th to 12th Centuries, European knowledge and study of arithmetic, geometry, astronomy and music was limited mainly to Boethius’ translations of some of the works of ancient Greek masters such

as Nicomachus and Euclid. All trade and calculation was made using the clumsy and

inefficient Roman numeral system, and with an abacus based on Greek and Roman models.

By the 12th Century, though, Europe, and particularly Italy, was beginning to trade with the East, and Eastern knowledge gradually began to spread to the West. Robert of Chester translated Al-Khwarizmi's

important book on algebra into Latin in the 12th Century, and the complete text of Euclid's “Elements”

was translated in various versions by Adelard of Bath, Herman of Carinthia and Gerard of Cremona. The

great expansion of trade and commerce in general created a growing practical need for mathematics, and

arithmetic entered much more into the lives of common people and was no longer limited to the

academic realm.

The advent of the printing press in the mid-15th Century also had a huge impact. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their

commercial needs and mathematics gradually began to acquire a more important position in education.

Europe’s first great medieval mathematician was the Italian Leonardo of Pisa, better known by his

nickname Fibonacci. Although best known for the so-called Fibonacci Sequence of numbers, perhaps

his most important contribution to European mathematics was his role in spreading the use of the Hindu-

Arabic numeral system throughout Europe early in the 13th Century, which soon made

the Roman numeral system obsolete, and opened the way for great advances in European mathematics.

18

An important (but largely unknown and underrated) mathematician and scholar of the 14th Century was

the Frenchman Nicole Oresme. He used a system of rectangular coordinates centuries before his

countryman René Descartes popularized the idea, as well as perhaps the first time-speed-distance graph.

Also, leading from his research into musicology, he was the first to use fractional exponents, and also

worked on infinite series, being the first to prove that the harmonic series 1⁄1 + 1⁄2 + 1⁄3 + 1⁄4 + 1⁄5... is a

divergent infinite series (i.e. not tending to a limit, other than infinity).

The German scholar Regiomontatus was perhaps the most capable mathematician of the 15th Century, his main contribution to mathematics being in the area of trigonometry. He helped separate trigonometry

from astronomy, and it was largely through his efforts that trigonometry came to be considered an

independent branch of mathematics. His book "De Triangulis", in which he described much of the basic

trigonometric knowledge which is now taught in high school and college, was the first great book on

trigonometry to appear in print.

Mention should also be made of Nicholas of Cusa (or Nicolaus Cusanus), a 15th Century German philosopher, mathematician and astronomer, whose prescient ideas on the infinite and the infinitesimal

directly influenced later mathematicians like Gottfried Leibniz and Georg Cantor. He also held some

distinctly non-standard intuitive ideas about the universe and the Earth's position in it, and about the

elliptical orbits of the planets and relative motion, which foreshadowed the later discoveries of

Copernicus and Kepler.

VII. 16th – 19th century mathematics

The cultural, intellectual and artistic movement of the Renaissance, which saw a resurgence of learning based on classical sources, began in Italy around the 14th Century, and gradually spread across most of

Europe over the next two centuries. Science and art were still very much interconnected and

intermingled at this time, as exemplified by the work of artist/scientists such as Leonardo da Vinci, and

it is no surprise that, just as in art, revolutionary work in the fields of philosophy and science was soon

taking place.

It is a tribute to the respect in which mathematics was held in Renaissance Europe that the famed

German artist Albrecht Dürer included an order-4 magic square in his engraving "Melencolia I". In fact,

it is a so-called "supermagic square" with many more lines of addition symmetry than a regular 4 x 4

magic square (see image at right). The year of the work, 1514, is shown in the two bottom central

squares.

An important figure in the late 15th and early 16th Centuries is an Italian Franciscan friar called Luca Pacioli, who published a book on arithmetic, geometry and book-keeping at the end of the 15th Century

which became quite popular for the mathematical puzzles it contained. It also introduced symbols for

plus and minus for the first time in a printed book (although this is also sometimes attributed to Giel

Vander Hoecke, Johannes Widmann and others), symbols that were to become standard notation. Pacioli

also investigated the Golden Ratio of 1 : 1.618... (see the section on Fibonacci) in his 1509 book "The

Divine Proportion", concluding that the number was a message from God and a source of secret

knowledge about the inner beauty of things.

b

a

a b b

b a

19

Finish rectangleDrop segment

length

width = 1.61963

Midpoint to vertexMidpointSquare

During the 16th and early 17th Century, the equals, multiplication, division, radical (root), decimal and

inequality symbols were gradually introduced and standardized. The use of decimal fractions and

decimal arithmetic is usually attributed to the Flemish mathematician Simon Stevin the late 16th

Century, although the decimal point notation was not popularized until early in the 17th Century. Stevin

20

was ahead of his time in enjoining that all types of numbers, whether fractions, negatives, real numbers

or surds (such as √2) should be treated equally as numbers in their own right.

In the Renaissance Italy of the early 16th Century, Bologna University in particular was famed for its

intense public mathematics competitions. It was in just such a competition that the unlikely figure of the

young, self-taught Niccolò Fontana Tartaglia revealed to the world the formula for solving first one

type, and later all types, of cubic equations (equations with terms including x3), an achievement hitherto

considered impossible and which had stumped the best mathematicians of China, India and the Islamic

world.

Building on Tartaglia’s work, another young Italian, Lodovico Ferrari, soon devised a similar method to solve quartic equations (equations with terms including x4) and both solutions were published

by Gerolamo Cardano. Despite a decade-long fight over the publication, Tartaglia, Cardano and

Ferrari between them demonstrated the first uses of what are now known as complex numbers,

combinations of real and imaginary numbers (although it fell to another Bologna resident, Rafael

Bombelli, to explain what imaginary numbers really were and how they could be used). Tartaglia went

on to produce other important (although largely ignored) formulas and methods, and Cardano published

perhaps the first systematic treatment of probability.

With Hindu-Arabic numerals, standardized notation and the new language of algebra at their disposal, the stage was set for the European mathematical revolution of the 17th Century.

The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculatory science with his discovery of logarithms. Cavalieri made progress towards the calculus with

his infinitesimal methods and Descartes added the power of algebraic methods to geometry.

Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical

study of probability. However the calculus was to be the topic of most significance to evolve in the 17th

Century.

Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new discoveries

showing the interaction between mathematics, physics and astronomy. Newton's theory of gravitation

and his theory of light take us into the 18th Century.

However we must also mention Leibniz, whose much more rigorous approach to the calculus (although still unsatisfactory) was to set the scene for the mathematical work of the 18th Century rather than that of

Newton. Leibniz's influence on the various members of the Bernoulli family was important in seeing the

calculus grow in power and variety of application.

The most important mathematician of the 18th Century was Euler who, in addition to work in a wide

range of mathematical areas, was to invent two new branches, namely the calculus of variations and

differential geometry. Euler was also important in pushing forward with research in number theory

begun so effectively by Fermat.

Toward the end of the 18th Century, Lagrange was to begin a rigorous theory of functions and of mechanics. The period around the turn of the century saw Laplace's great work on celestial mechanics as

well as major progress in synthetic geometry by Monge and Carnot.

The 19th Century saw rapid progress. Fourier's work on heat was of fundamental importance. In geometry Plücker produced fundamental work on analytic geometry and Steiner in synthetic geometry.

Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterization of geometry by Riemann. Gauss, thought by some to be the greatest mathematician of all time, studied quadratic

reciprocity and integer congruences. His work in differential geometry was to revolutionise the topic. He

also contributed in a major way to astronomy and magnetism.

The 19th Century saw the work of Galois on equations and his insight into the path that mathematics

would follow in studying fundamental operations. Galois' introduction of the group concept was to

herald in a new direction for mathematical research which has continued through the 20th Century.

21

Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of

the theory of functions of a complex variable. This work would continue through Weierstrass and

Riemann.

Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann. The end of the 19th Century saw Cantor invent set

theory almost single handedly while his analysis of the concept of number added to the major work of

Dedekind and Weierstrass on irrational numbers

Analysis was driven by the requirements of mathematical physics and astronomy. Lie's work on differential equations led to the study of topological groups and differential topology. Maxwell was to

revolutionise the application of analysis to mathematical physics. Statistical mechanics was developed

by Maxwell, Boltzmann and Gibbs. It led to ergodic theory.

The study of integral equations was driven by the study of electrostatics and potential theory. Fredholm's work led to Hilbert and the development of functional analysis.

VIII. Notation and communication

22

IX. Unlikely mathematicians

President Garfield’s Proof of the Pythagorean Theorem

23

A Mathematical Chronology

About 30,000BC

*Palaeolithic peoples in central Europe and France record numbers on bones.

About 5000BC

*A decimal number system is in use in Egypt.

About 4000BC

*Babylonian and Egyptian calendars in use.

About 3400BC

*The first symbols for numbers, simple straight lines, are used in Egypt.

About 3000BC

*The abacus is developed in the Middle East and in areas around the Mediterranean.

About 3000BC

*Hieroglyphic numerals in use in Egypt.

About 3000BC

*Babylonians begin to use a sexagesimal number system for recording financial transactions. It is a place-value

system without a zero place value.

About 2770BC

*Egyptian calendar used.

About 2000BC

*Harappans adopt a uniform decimal system of weights and measures.

About 1950BC

*Babylonians solve quadratic equations.

About 1850BC

*Babylonians know Pythagoras's Theorem.

About 1800BC

*Babylonians use multiplication tables.

About 1750BC

*The Babylonians solve linear and quadratic algebraic equations, compile tables of square and cube roots. They

use Pythagoras's theorem and use mathematics to extend knowledge of astronomy.

About 1700BC

*The Rhind papyrus (sometimes called the Ahmes papyrus) is written. It shows that Egyptian mathematics has

developed many techniques to solve problems. Multiplication is based on repeated doubling, and division uses

successive halving.

About 1400BC

*About this date a decimal number system with no zero starts to be used in China.

575BC

*Thales brings Babylonian mathematical knowledge to Greece. He uses geometry to solve problems such as

calculating the height of pyramids and the distance of ships from the shore.

530BC

*Pythagoras of Samos moves to Croton in Italy and teaches mathematics, geometry, music, and reincarnation.

About 500BC

*The Babylonian sexagesimal number system is used to record and predict the positions of the Sun, Moon and

planets.

About 465BC

Hippasus writes of a "sphere of 12 pentagons", which must refer to a dodecahedron.

About 440BC

*Hippocrates of Chios writes the Elements which is the first compilation of the elements of geometry.

24

About 400BC

*Babylonians use a symbol to indicate an empty place in their numbers recorded in cuneiform writing. There is

no indication that this was in any way thought of as a number.

387BC

Plato founds his Academy in Athens

About 250BC

*In On the Sphere and the Cylinder, Archimedes gives the formulae for calculating the volume of a sphere and

a cylinder. In Measurement of the Circle he gives an approximation of the value of π with a method which will

allow improved approximations. In Floating Bodies he presents what is now called "Archimedes' principle" and

begins the study of hydrostatics. He writes works on two- and three-dimensional geometry, studying circles,

spheres and spirals. His ideas are far ahead of his contemporaries and include applications of an early form of

integration.

About 230BC

*Eratosthenes of Cyrene develops his sieve method for finding all prime numbers.

About 225BC

Apollonius of Perga writes Conics in which he introduces the terms "parabola", "ellipse" and "hyperbola".

About 150BC

*Hypsicles writes On the Ascension of Stars. In this work he is the first to divide the Zodiac into 360 degrees.

127BC

*Hipparchus discovers the precession of the equinoxes and calculates the length of the year to within 6.5

minutes of the correct value. His astronomical work uses an early form of trigonometry.

About 1AD

*Chinese mathematician Liu Hsin uses decimal fractions.

About 60

*Heron of Alexandria writes Metrica (Measurements). It contains formulas for calculating areas and volumes.

About 150

Ptolemy produces many important geometrical results with applications in astronomy. His version of astronomy

will be the accepted one for well over one thousand years.

About 250

*The Maya civilization of Central America uses an almost place-value number system to base 20.

250

Diophantus of Alexandria writes Arithmetica, a study of number theory problems in which only rational

numbers are allowed as solutions.

263

*By using a regular polygon with 192 sides Liu Hui calculates the value of π as 3.14159 which is correct to five

decimal places.

About 400

*Hypatia writes commentaries on Diophantus and Apollonius. She is the first recorded female mathematician

and she distinguishes herself with remarkable scholarship. She becomes head of the Neo-Platonist school at

Alexandria.

About 460

*Zu Chongzhi gives the approximation 355/113 to π which is correct to 6 decimal places.

499

Aryabhata I calculates π to be 3.1416. He produces his Aryabhatiya, a treatise on quadratic equations, the value

of π, and other scientific problems.

594

*Decimal notation is used for numbers in India. This is the system on which our current notation is based.

25

628

*Brahmagupta writes Brahmasphutasiddanta (The Opening of the Universe), a work on astronomy; on

mathematics. He uses zero and negative numbers, gives methods to solve quadratic equations, sum series, and

compute square roots.

About 700

*Mathematicians in the Mayan civilization introduce a symbol for zero into their number system.

About 810

*Al-Khwarizmi writes important works on arithmetic, algebra, geography, and astronomy. In particular Hisab

al-jabr w'al-muqabala (Calculation by Completion and Balancing), gives us the word "algebra", from "al-jabr".

From al-Khwarizmi's name, as a consequence of his arithmetic book, comes the word "algorithm".

950

*Gerbert of Aurillac (later Pope Sylvester II) reintroduces the abacus into Europe. He uses Indian/Arabic

numerals without having a zero.

About 990

Al-Karaji writes Al-Fakhri in Baghdad which develops algebra. He gives Pascal's triangle.

1072

*Al-Khayyami (usually known as Omar Khayyam) writes Treatise on Demonstration of Problems of Algebra

which contains a complete classification of cubic equations with geometric solutions found by means of

intersecting conic sections. He measures the length of the year to be 365.24219858156 days, a remarkably

accurate result.

1149

Al-Samawal writes al-Bahir fi'l-jabr (The brilliant in algebra). He develops algebra with polynomials using

negative powers and zero. He solves quadratic equations, sums the squares of the first n natural numbers, and

looks at combinatorial problems.

About 1200

*Chinese start to use a symbol for zero.

1202

*Fibonacci writes Liber abaci (The Book of the Abacus), which sets out the arithmetic and algebra he had learnt

in Arab countries. It also introduces the famous sequence of numbers now called the "Fibonacci sequence".

1248

*Li Yeh writes a book which contains negative numbers, denoted by putting a diagonal stroke through the last

digit.

About 1260

*Campanus of Novara, chaplain to Pope Urban IV, writes on astronomy and publishes a Latin edition of

Euclid's Elements which became the standard Euclid for the next 200 years.

1400

Madhava of Sangamagramma proves a number of results about infinite sums giving Taylor expansions of

trigonometric functions. He uses these to find an approximation for π correct to 11 decimal places.

1424

Al-Kashi writes Treatise on the Circumference giving a remarkably good approximation to π in both

sexagesimal and decimal forms.

1482

*Campanus of Novara's edition of Euclid's Elements becomes the first mathematics book to be printed.

1489

*Widman writes an arithmetic book in German which contains the first appearance of + and - signs.

1514

*Vander Hoecke uses the + and - signs.

1515

*Del Ferro discovers a formula to solve cubic equations.

26

1525

*Rudolff introduces a symbol resembling √ for square roots in his Die Coss the first German algebra book. He

understands that x0 = 1.

1536

Hudalrichus Regius finds the fifth perfect number. The number 212(213 - 1) = 33550336 is the first perfect

number to be discovered since ancient times.

1555

J Scheybl gives the sixth perfect number 216(217 - 1) = 8589869056 but his work remains unknown until 1977.

1590

Cataldi uses continued fractions in finding square roots.

1591

*Viète writes In artem analyticam isagoge (Introduction to the analytical art), using letters as symbols for

quantities, both known and unknown. He uses vowels for the unknowns and consonants for known quantities.

Descartes, later, introduces the use of letters x, y ... at the end of the alphabet for unknowns.

1593

*Van Roomen calculates π to 16 decimal places.

1603

Cataldi finds the sixth and seventh perfect numbers, 216(217 - 1) =8589869056 and 218(219 - 1) = 137438691328.

1614

Napier publishes his work on logarithms in Mirifici logarithmorum canonis descriptio (Description of the

Marvellous Rule of Logarithms).

1615

Kepler publishes Nova stereometria doliorum vinarorum (Solid Geometry of a Wine Barrel), an investigation of

the capacity of casks, surface areas, and conic sections. He first had the idea at his marriage celebrations in

1613. His methods are early uses of the calculus.

1617

Briggs publishes Logarithmorum chilias prima (Logarithms of Numbers from 1 to 1,000) which introduces

logarithms to the base 10.

1617

*Napier invents Napier's bones, consisting of numbered sticks, as a mechanical calculator. He explains their

function in Rabdologiae (Study of Divining Rods) published in the year of his death.

1620

Gunter makes a mechanical device, Gunter's scale, to multiply numbers based on logarithms using a single

scale and a pair of dividers.

1626

*Albert Girard publishes a treatise on trigonometry containing the first use of the abbreviations sin, cos, tan. He

also gives formulas for the area of a spherical triangle.

1631

*Harriot's contributions are published ten years after his death in Artis analyticae praxis (Practice of the

Analytic Art). The book introduces the symbols > and < for "greater than" and "less than" but these symbols are

due to the editors of the work and not Harriot himself. His work on algebra is very impressive but the editors of

the book do not present it well.

635

Descartes discovers Euler's theorem for polyhedra, V - E + F = 2.

1636

Fermat discovers the pair of amicable numbers 17296, 18416 which were known to Thabit ibn Qurra 800 years

earlier.

1642

*Pascal builds a calculating machine to help his father with tax calculations. It performs only additions.

27

1647

*Fermat claims to have proved a theorem, but leaves no details of his proof since the margin in which he writes

it is too small. Later known as Fermat's last theorem, it states that the equation xn + yn = zn has no non-zero

1654

*Fermat and Pascal begin to work out the laws that govern chance and probability in five letters which they

exchange during the summer.

1659

*Rahn publishes Teutsche algebra which contains (the division sign) probably invented by Pell.

1662

*Graunt and Petty publish Natural and Political Observations made upon the Bills of Mortality. It is one of the

first statistics books.

1665

*Newton discovers the binomial theorem and begins work on the differential calculus.

1692

Leibniz introduces the term "coordinate".

1706

*Jones introduces the Greek letter π to represent the ratio of the circumference of a circle to its diameter in his

Synopsis palmariorum matheseos (A New Introduction to Mathematics).

1707

Newton publishes Arithmetica universalis (General Arithmetic) which contains a collection of his results in

algebra.

1710

*Arbuthnot publishes an important statistics paper in the Royal Society which discusses the slight excess of

male births over female births. This paper is the first application of probability to social statistics.

1727

Euler is appointed to St Petersburg. He introduces the symbol e for the base of natural logarithms in a

manuscript entitled Meditation upon Experiments made recently on firing of Cannon. The manuscript was not

published until 1862.

1736

*Euler solves the topographical problem known as the "Königsberg bridges problem". He proves

mathematically that it is impossible to design a walk which crosses each of the seven bridges exactly once.

1742

Goldbach conjectures, in a letter to Euler, that every even number ≥ 4 can be written as the sum of two primes.

It is not yet known whether Goldbach's conjecture is true.

1752

Euler states his theorem V - E + F = 2 for polyhedra.

1753

Simson notes that in the Fibonacci sequence the ratio between adjacent numbers approaches the golden ratio.

1770

*Lagrange proves that any integer can be written as the sum of four squares.

1777

*Euler introduces the symbol i to represent the square root of -1 in a manuscript which will not appear in print

until 1794.

1801

*Gauss proves Fermat's conjecture that every number can be written as the sum of three triangular numbers.

1809

Poinsot discovers two new regular polyhedra.

1814

Barlow produces Barlow's Tables which give factors, squares, cubes, square roots, reciprocals and hyperbolic

logs of all numbers from 1 to 10000.

28

1815

*Peter Roget (the author of Roget's Thesaurus) invents the "log-log" slide rule.

853

*Shanks gives π to 707 places (in 1944 it was discovered that Shanks was wrong from the 528th place).

1858

*Möbius describes a strip of paper that has only one side and only one edge. Now known as the "Möbius strip",

it has the surprising property that it remains in one piece when cut down the middle. Listing makes the same

discovery in the same year.

1881

*Venn introduces his "Venn diagrams" which become useful tools in set theory.

1905

*Einstein publishes the special theory of relativity.

1948

*Shannon invents information theory and applies mathematical methods to study errors in transmitted

information. This becomes of vital importance in computer science and communications.

1966

Lander and Parkin use a computer to find a counterexample to Euler's Conjecture. They find 275 + 845 + 1105 +

1335 = 1445.

1973

Chen Jingrun shows that every sufficiently large even integer is the sum of a prime and a number with at most

two prime factors. It makes a major contribution to the Goldbach Conjecture.

1982

*Mandelbrot publishes The fractal geometry of nature which develops his theory of fractal geometry more fully

than his work of 1975.

1994

*Wiles proves Fermat's Last Theorem.

1995

A large prize is offered by banker Andrew Beal for a solution to the Beal Conjecture: the equation xp + yq = zr

has no solutions for p, q, r > 2 and coprime integers x, y, z.

1999

*The Great Internet Mersenne Prime Search project finds the 38th Mersenne prime: 26972593 -1.

2016

Largest Mersenne Prime found (so far): 274,207,281 − 1, a number with 22,338,618 decimal digits. It’s the 49th prime.