Republic of the Philippines

LEYTE NORMAL UNIVERSITY

Graduate School
Tacloban City

A COMPENDIUM OF WRITTEN
OUTPUTS
in
Math 536 – History of Mathematics

Submitted By:

MARCO C. BERONGOY
HELEN GRACE A. ELLEMA
ROVINSON D. GAGANAO
ALLAN M. LUBIANO
DEDITH L. MARTINADA
MA. CRISTINA E. MEJARITO
BRIGITTE M. MORAL
NOEL S. PALOMAS
MARIA DIANA Y. TIU
MICHAEL DELL A. TUAZON

Submitted To:

PROF. NELSON D. BERNARDO

Summer 2017
1
 Republic of the Philippines
LEYTE NORMAL UNIVERSITY
Graduate School
Tacloban City

MARIA DIANA Y. TIU NELSON D. BERNARDINO

Student/Reporter Professor

Ancient Babylonian Math

The Babylonians lived in Mesopotamia, which is between the Tigris

and Euphrates rivers. They began a numbering system about 5,000 years

ago. It is one of the oldest numbering systems. The first mathematics can be

traced to the ancient country of Babylon, during the third millennium B.C.

Tables were the Babylonians most outstanding accomplishment which helped

them in calculating problems. One of the Babylonian tablets, Plimpton 322,

which is dated from between 1900 and 1600 BC, contains tables of

Pythagorean triples for the equation a2 + b2 = c2. It is currently in a British

museum. Nabu - rimanni and Kidinu are two of the only known

mathematicians from Babylonia. However, not much is known about them.

Historians believe Nabu - rimanni lived around 490 BC and Kidinu lived

around 480 BC.

The Babylonian number system began with tally marks just as most of

the ancient math systems did. The Babylonians developed a form of writing

based on cuneiform. Cuneiform means "wedge shape" in Latin. They wrote

these symbols on wet clay tablets which were baked in the hot sun. Many

thousands of these tablets are still around today. The Babylonians used a

stylist to imprint the symbols on the clay since curved lines could not be

2
drawn. The Babylonians had a very advanced number system even for

today's standards. It was a base 60 system (sexigesimal) rather than a base

ten (decimal). Base ten is what we use today. The Babylonians divided the

day into twenty-four hours, each hour into sixty minutes, and each minute to

sixty seconds. This form of counting has survived for four thousand years.

(http://www.math.wichita.edu/history/topics/num-sys.html#babylonian)

Babylonian math refers to mathematics developed in Mesopotamia,

from the days of the early Sumerians to the fall of Babylon in 539 B.C. and is

especially known for the development of the Babylonian System. With the

collapse of the Sumerian civilization in Mesopotamia Babylon is developed.

They inherited from the Sumerian cuneiform and hexagezimal number

system. After 3000 B.C. Babylonians developed a system of writing.

Pictograph – a kind of picture writing.

Sumerian early pictograph

Cuneiform – Latin word “cuneus” which means “wedge” The sharp edge of a

stylus made a vertical stroke (│) and the base made a more or less deep

impression (Δ), so that the combined effect was a head-and-tail figure

resembling a wedge, or nail.

3
 Cuneiform Script

Like the Egyptians, the Babylonians used two ones to represent two,

and so on, up to nine. However, they tended to arrange the symbols in the

neat piles. Once they got to ten, there were too many symbols, so they turned

the stylus on its side to make a different symbol. This is a unary system. The

symbol for sixty seems to be exactly the same as that for one.

The Babylonian numeration system was develop between 3000 and

2000 BCE. It uses only two numerals or symbols, a one and a ten to

represent numbers.

To represent numbers from 2 to 59, the system was simply additive.

Example #1.

5 is written as shown:

4
12 is written as shown:

Notice how the ones, in this case two ones are shown on the right just like

the Hindu-Arabic numeration system.

45 is written as shown:

For number bigger than 59, the babylonian used a place value system with a

base of 60.

62 is written as shown:

Notice this time the use of a big space to separate the space value

Without the big space, things look like this:

5
However, what is that number without this big space? Could it be 2 × 60 + 1

or 1× 602 + 1 × 60 + 1 or .....??? The babylonians introduced the big space

after they became aware of this ambiguity.

The number 4871 could be represented as follow: 3600 + 1260 + 11 = 4871

Even after the big space was introduced to separate place value, the

babylonians still faced a more serious problem? How would they represent

the number 60?

Since there was no zero to put in an empty position, the number 60 would

thus have the same representation as the number 1.

How did they make the difference? All we can say is that the context

must have helped them to establish such difference yet the Babylonian

numeration system was without a doubt a very ambiguous numeral system. If

this had become a major problem, no doubt the babylonians were smart

6
enough to come up with a working system (http://www.basic-

mathematics.com/babylonian-numeration-system.html).

THE DIGITS OF THE BABYLONIAN NUMBER SYSTEM

The Babylonian system of mathematics was sexagesimal (base 60)

numeral system. 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. Sexagesimal

(base 60) is a numeral system with sixty as its base. It originated with the

ancient Sumerians in the 3rd millennium BC, was passed down to the ancient

Babylonians, and is still used in a modified form for measuring time, angles,

and geographic coordinates. The Babylonians were able to make great

advances in mathematics for two reasons. 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 calculation with fractions.

Had a true place value system, where digits written in the left column

represented larger values (much in our base ten system

7
734=7x100+3x10+4x1. the Sumerian Babylonians were pioneers in this

respect.

Babylonians have a positional number system. Wherein the numbers

are arranged in columns. The Babylonians used powers of sixty rather than

ten. So the left-hand column were units, the second, multiples of 60, the third,

multiples of 3,600, and so on.

Positional number system

Babylonian Mathematics

. They used no zero.

. More general fractions, though not all fractions, were admitted.

. They could extract square roots.

. They could solve linear systems.

. They worked with Pythagorean triples.

. They solved cubic equations with the help of tables.

. They studied circular measurement.

. Their geometry was sometimes incorrect.

8
For enumeration the Babylonians used symbols for 1, 10, 60, 600,

3,600, 36,000, and 216,000, similar to the earlier period.

Republic of the Philippines

LEYTE NORMAL UNIVERSITY
Graduate School
Tacloban City

Math 536 – History of Mathematics

BRIGITTE M. MORAL NELSON D. BERNARDO

Student/Reporter Professor

************************************************************************************
Ancient Egyptian Mathematics
Ancient Mesopotamian Mathematics

Sub-Topics:
1. Ancient Egyptian Numerals
2. Ancient Egyptian Computation
3. Some example problems
4. Ancient Mesopotamian Mathematics
5. Ancient Mesopotamian Computation
6. Some example problems

************************************************************************************

Ancient Egyptian Numerals

The early Egyptians settled along the fertile Nile valley as early as

about 6000 BCE, and they began to record the patterns of lunar phases and

the seasons, both for agricultural and religious reasons. The Pharaoh’s

surveyors used measurements based on body parts (a palm was the width of

the hand, a cubit the measurement from elbow to fingertips) to measure land

and buildings very early in Egyptian history, and a decimal numeric system

was developed based on our ten fingers. The oldest mathematical text from

ancient Egypt discovered so far, though, is the Moscow Papyrus, which dates

from the Egyptian Middle Kingdom around 2000 - 1800 BCE.

9
 It is thought that the Egyptians introduced the earliest fully-developed

base 10 numeration system at least as early as 2700 BCE (and probably

much early). Written numbers used a stroke for units, a heel-bone symbol for

tens, a coil of rope for hundreds and a lotus plant for thousands, as well as

other hieroglyphic symbols for higher powers of ten up to a million. However,

there was no concept of place value, so larger numbers were rather unwieldy

(although a million required just one character, a million minus one required

fifty-four characters).

Ancient Egyptian Computation

The Rhind Papyrus, dating from around 1650 BCE, is a kind of

instruction manual in arithmetic and geometry, and it gives us explicit

demonstrations of how multiplication and division was carried out at that time.

It also contains evidence of other mathematical knowledge, including unit

fractions, composite and prime numbers, arithmetic, geometric and harmonic

means, and how to solve first order linear equations as well as arithmetic and

geometric series. The Berlin Papyrus, which dates from around 1300 BCE,

shows that ancient Egyptians could solve second-order algebraic (quadratic)

equations.

Some example problems

Multiplication, for example, was achieved by a process of repeated

doubling of the number to be multiplied on one side and of one on the other,

essentially a kind of multiplication of binary factors similar to that used by

modern computers (see the example at right). These corresponding blocks of

10
counters could then be used as a kind of multiplication reference table: first,

the combination of powers of two which add up to the number to be multiplied

by was isolated, and then the corresponding blocks of counters on the other

side yielded the answer. This effectively made use of the concept of binary

numbers, over 3,000 years before Leibniz introduced it into the west, and

many more years before the development of the computer was to fully explore

its potential.

Practical problems of trade and the market led to the development of a

notation for fractions. The papyri which have come down to us demonstrate

the use of unit fractions based on the symbol of the Eye of Horus, where each

part of the eye represented a different fraction, each half of the previous one

(i.e. half, quarter, eighth, sixteenth, thirty-second, sixty-fourth), so that the total

was one-sixty-fourth short of a whole, the first known example of a geometric

series.

Unit fractions could also be used for simple division sums. For

example, if they needed to divide 3 loaves among 5 people, they would first

divide two of the loaves

into thirds and the third

loaf into fifths, then they

would divide the left

over third from the

second loaf into five

pieces. Thus, each

person would receive

Ancient Egyptian method of division

11
one-third plus one-fifth plus one-fifteenth (which totals three-fifths, as we

would expect).

The Egyptians approximated the area of a circle by using shapes

whose area they did know. They observed that the area of a circle of diameter

9 units, for example, was very close to the area of a square with sides of 8

units, so that the area of circles of other diameters could be obtained by

multiplying the diameter by 8⁄9 and then squaring it. This gives an effective

approximation of π accurate to within less than one percent.

The pyramids themselves are another indication of the sophistication of

Egyptian mathematics. Setting aside claims that the pyramids are first known

structures to observe the golden ratio of 1: 1.618 (which may have occurred

for purely aesthetic, and not mathematical, reasons), there is certainly

evidence that they knew the formula for the volume of a pyramid - 1⁄3 times the

height times the length times the width - as well as of a truncated or clipped

pyramid. They were also aware, long before Pythagoras, of the rule that a

triangle with sides 3, 4 and 5 units yields a perfect right angle, and Egyptian

builders used ropes knotted at intervals of 3, 4 and 5 units in order to ensure

exact right angles for their stonework (in fact, the 3-4-5 right triangle is often

called "Egyptian").

Sumer (a region of Mesopotamia, modern-day Iraq) was the birthplace of

writing, the wheel, agriculture, the arch, the plow, irrigation and many other

innovations, and is often referred to as the Cradle of Civilization. The

Sumerians developed the earliest known writing system - a pictographic

writing system known as cuneiform script, using wedge-shaped characters

12
inscribed on baked clay tablets - and this has meant that we actually have

more knowledge of ancient Sumerian and Babylonian mathematics than of

early Egyptian mathematics. Indeed, we even have what appear to school

exercises in arithmetic and geometric problems.

As in Egypt, Sumerian mathematics initially developed largely as a

response to bureaucratic needs when their civilization settled and developed

agriculture (possibly as early as the 6th millennium BCE) for the measurement

of plots of land, the taxation of individuals, etc. In addition, the Sumerians and

Babylonians needed to describe quite large numbers as they attempted to

chart the course of the night sky and develop their sophisticated lunar

calendar.

They were perhaps the first people to assign symbols to groups of

objects in an attempt to make the description of larger numbers easier. They

moved from using separate tokens or symbols to represent sheaves of wheat,

jars of oil, etc, to the more abstract use of a symbol for specific numbers of

anything. Starting as early as the 4th millennium BCE, they began using a

small clay cone to represent one, a clay ball for ten, and a large cone for sixty.

Over the course of the third millennium, these objects were replaced by

cuneiform equivalents so that numbers could be written with the same stylus

that was being used for the words in the text. A rudimentary model of the

abacus was probably in use in Sumerian from as early as 2700 - 2300 BCE.

13
Ancient Mesopotamian Numerals

Sumerian and Babylonian

mathematics was based on a

sexagesimal, or base 60, numeric

system, which could be counted

physically using the twelve

knuckles on one hand the five

fingers on the other hand. Unlike

those of the Egyptians, Greeks

Sumerian Clay Cones
and Romans, Babylonian

numbers used a true place-value system, where digits written in the left

column represented larger values, much as in the modern decimal system,

although of course using base 60 not base 10. Thus, 1 1 1 in the Babylonian

system represented 3,600 plus 60 plus 1, or 3,661. Also, to represent the

numbers 1 - 59 within each place value, two distinct symbols were used, a

unit symbol (1) and a ten symbol (10) which were combined in a similar way

to the familiar system of Roman numerals (e.g. 23 would be shown as 23).

Thus, 1 23 represents 60 plus 23, or 83. However, the number 60 was

represented by the same symbol as the number 1 and, because they lacked

14
an equivalent of the decimal point, the actual place value of a symbol often

had to be inferred from the context.

It has been conjectured that Babylonian advances in mathematics were

probably facilitated by the fact that 60 has

many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15,

20, 30 and 60 - in fact, 60 is the smallest

integer divisible by all integers from 1 to

6), and the continued modern-day usage

of 60 seconds in a minute, 60 minutes in an Babylonian Numerals

hour, and 360 (60 x 6) degrees in a circle, are all testaments to the ancient

Babylonian system. It is for similar reasons that 12 (which has factors of 1, 2,

3, 4 and 6) has been such a popular multiple historically (e.g. 12 months, 12

inches, 12 pence, 2 x 12 hours, etc).

The Babylonians also developed another revolutionary mathematical

concept, something else that the Egyptians, Greeks and Romans did not

have, a circle character for zero, although its symbol was really still more of a

placeholder than a number in its own right.

We have evidence of the development of a complex system of

metrology in Sumer from about 3000 BCE, and multiplication and reciprocal

(division) tables, tables of squares, square roots and cube roots, geometrical

exercises and division problems from around 2600 BCE onwards. Later

Babylonian tablets dating from about 1800 to 1600 BCE cover topics as

varied as fractions, algebra, methods for solving linear, quadratic and even

some cubic equations, and the calculation of regular reciprocal pairs (pairs of

number which multiply together to give 60). One Babylonian tablet gives an

15
approximation to √2 accurate to an astonishing five decimal places. Others list

the squares of numbers up to 59, the cubes of numbers up to 32 as well as

tables of compound interest. Yet another gives an estimate for π of 3 1⁄8

(3.125, a reasonable approximation of the real value of 3.1416).

Ancient Mesopotamian Computation

The idea of square

numbers and quadratic

equations (where the

unknown quantity is Babylonian Clay tablets from c. 2100 BCE showing a

problem concerning the area of an irregular shape
multiplied by itself, e.g. x2)

naturally arose in the context Babylonian Clay tablets from c. 2100 BCE

showing a problem concerning the area of an irregular shape of the

measurement of land, and Babylonian mathematical tablets give us the first

ever evidence of the solution of quadratic equations. The Babylonian

approach to solving them usually revolved around a kind of geometric game

of slicing up and rearranging shapes, although the use of algebra and

quadratic equations also appears. At least some of the examples we have

appear to indicate problem-solving for its own sake rather than in order to

resolve a concrete practical problem.

The Babylonians used geometric shapes in their buildings and design

and in dice for the leisure games which were so popular in their society, such

as the ancient game of backgammon. Their geometry extended to the

calculation of the areas of rectangles, triangles and trapezoids, as well as the

16
volumes of simple shapes such as bricks and cylinders (although not

pyramids).

The famous and controversial Plimpton 322 clay tablet, believed to

date from around 1800 BCE, suggests that the Babylonians may well have

known the secret of right-angled triangles (that the square of the hypotenuse

equals the sum of the square of the other two sides) many centuries before

the Greek Pythagoras. The tablet appears to list 15 perfect Pythagorean

triangles with whole number sides, although some claim that they were merely

academic exercises, and not deliberate manifestations of Pythagorean triples.

Reference:

Cooke, Roger, 1942- The History of Mathematics: A brief course/Roger L.

Cooke, 3rd ed. p. cm.

17
 Republic of the Philippines
LEYTE NORMAL UNIVERSITY
Graduate School
Tacloban City

Math 536 – History of Mathematics

DEDITH L. MARTINADA NELSON D. BERNARDO
Student/Reporter Professor

************************************************************************************

Sub-Topics:
1. Ancient Math in India
2. Ancient Chinese Mathematics

************************************************************************************

History of Mathematics in India

In all early civilizations, the first expression of mathematical understanding

appears in the form of counting systems. Numbers in very early societies

were typically represented by groups of lines, though later different numbers

came to be assigned specific numeral names and symbols (as in India) or

were designated by alphabetic letters (such as in Rome). Although today, we

take our decimal system for granted, not all ancient civilizations based their

numbers on a ten-base system. In ancient Babylon, a sexagesimal (base 60)

system was in use.

The Decimal System in Harappa

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In India a decimal system was already in place during the Harappan period,

as indicated by an analysis of Harappan weights and measures. Weights

corresponding to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and

500 have been identified, as have scales with decimal divisions. A particularly

notable characteristic of Harappan weights and measures is their remarkable

accuracy. A bronze rod marked in units of 0.367 inches points to the degree

of precision demanded in those times. Such scales were particularly important

in ensuring proper implementation of town planning rules that required roads

of fixed widths to run at right angles to each other, for drains to be constructed

of precise measurements, and for homes to be constructed according to

specified guidelines. The existence of a gradated system of accurately

marked weights points to the development of trade and commerce in

Harappan society.

Mathematical Activity in the Vedic Period

In the Vedic period, records of mathematical activity are mostly to be found in

Vedic texts associated with ritual activities. However, as in many other early

agricultural civilizations, the study of arithmetic and geometry was also

impelled by secular considerations. Thus, to some extent early mathematical

developments in India mirrored the developments in Egypt, Babylon and

China . The system of land grants and agricultural tax assessments required

accurate measurement of cultivated areas. As land was redistributed or

consolidated, problems of mensuration came up that required solutions. In

order to ensure that all cultivators had equivalent amounts of irrigated and

non-irrigated lands and tracts of equivalent fertility - individual farmers in a

village often had their holdings broken up in several parcels to ensure

19
fairness. Since plots could not all be of the same shape - local administrators

were required to convert rectangular plots or triangular plots to squares of

equivalent sizes and so on. Tax assessments were based on fixed

proportions of annual or seasonal crop incomes, but could be adjusted

upwards or downwards based on a variety of factors. This meant that an

understanding of geometry and arithmetic was virtually essential for revenue

administrators. Mathematics was thus brought into the service of both the

secular and the ritual domains.

Arithmetic operations (Ganit) such as addition, subtraction, multiplication,

fractions, squares, cubes and roots are enumerated in the Narad Vishnu

Purana attributed to Ved Vyas (pre-1000 BC). Examples of geometric

knowledge (rekha-ganit) are to be found in the Sulva-

Sutras of Baudhayana (800 BC) and Apasthmaba (600 BC) which describe

techniques for the construction of ritual altars in use during the Vedic era. It is

likely that these texts tapped geometric knowledge that may have been

acquired much earlier, possibly in the Harappan period. Baudhayana's

Sutra displays an understanding of basic geometric shapes and techniques of

converting one geometric shape (such as a rectangle) to another of equivalent

(or multiple, or fractional) area (such as a square). While some of the

formulations are approximations, others are accurate and reveal a certain

degree of practical ingenuity as well as some theoretical understanding of

basic geometric principles. Modern methods of multiplication and addition

probably emerged from the techniques described in the Sulva-Sutras.

Pythagoras - the Greek mathematician and philosopher who lived in the 6th C

B.C was familiar with the Upanishads and learnt his basic geometry from the

20
Sulva Sutras. An early statement of what is commonly known as the

Pythagoras theorem is to be found in Baudhayana's Sutra: The chord which is

stretched across the diagonal of a square produces an area of double the

size. A similar observation pertaining to oblongs is also noted. His Sutra also

contains geometric solutions of a linear equation in a single unknown.

Examples of quadratic equations also appear. Apasthamba's sutra (an

expansion of Baudhayana's with several original contributions) provides a

value for the square root of 2 that is accurate to the fifth decimal

place. Apasthamba also looked at the problems of squaring a circle, dividing a

segment into seven equal parts, and a solution to the general linear equation.

Jain texts from the 6th C BC such as the Surya Pragyapti describe ellipses.

Modern-day commentators are divided on how some of the results were

generated. Some believe that these results came about through hit and trial -

as rules of thumb, or as generalizations of observed examples. Others believe

that once the scientific method came to be formalized in the Nyaya-Sutras -

proofs for such results must have been provided, but these have either been

lost or destroyed, or else were transmitted orally through the Gurukul system,

and only the final results were tabulated in the texts. In any case, the study

of Ganit i.e mathematics was given considerable importance in the Vedic

period. The Vedang Jyotish (1000 BC) includes the statement: "Just as the

feathers of a peacock and the jewel-stone of a snake are placed at the

highest point of the body (at the forehead), similarly, the position of Ganit is

the highest amongst all branches of the Vedas and the Shastras."

(Many centuries later, Jain mathematician from Mysore, Mahaviracharya

further emphasized the importance of mathematics: "Whatever object exists

21
in this moving and non-moving world, cannot be understood without the base

of Ganit (i.e. mathematics)".)

Panini and Formal Scientific Notation

A particularly important development in the history of Indian science that was

to have a profound impact on all mathematical treatises that followed was the

pioneering work by Panini (6th C BC) in the field of Sanskrit grammar and

linguistics. Besides expounding a comprehensive and scientific theory of

phonetics, phonology and morphology, Panini provided formal production

rules and definitions describing Sanskrit grammar in his treatise

called Asthadhyayi. Basic elements such as vowels and consonants, parts of

speech such as nouns and verbs were placed in classes. The construction of

compound words and sentences was elaborated through ordered rules

operating on underlying structures in a manner similar to formal language

theory.

Today, Panini's constructions can also be seen as comparable to modern

definitions of a mathematical function. G G Joseph, in The crest of the

peacock argues that the algebraic nature of Indian mathematics arises as a

consequence of the structure of the Sanskrit language. Ingerman in his paper

titled Panini-Backus form finds Panini's notation to be equivalent in its power

to that of Backus - inventor of the Backus Normal Form used to describe the

syntax of modern computer languages. Thus Panini's work provided an

example of a scientific notational model that could have propelled later

mathematicians to use abstract notations in characterizing algebraic

equations and presenting algebraic theorems and results in a scientific format.

22
Philosophy and Mathematics

Philosophical doctrines also had a profound influence on the development of

mathematical concepts and formulations. Like the Upanishadic world view,

space and time were considered limitless in Jain cosmology. This led to a

deep interest in very large numbers and definitions of infinite numbers. Infinite

numbers were created through recursive formulae, as in the Anuyoga Dwara

Sutra. Jain mathematicians recognized five different types of infinities: infinite

in one direction, in two directions, in area, infinite everywhere and perpetually

infinite. Permutations and combinations are listed in the Bhagvati Sutras (3rd

C BC) and Sathananga Sutra (2nd C BC).

Jain set theory probably arose in parallel with the Syadvada system of Jain

epistemology in which reality was described in terms of pairs of truth

conditions and state changes. The Anuyoga Dwara Sutra demonstrates an

understanding of the law of indeces and uses it to develop the notion of

logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached are

used to denote log base 2, log base 3 and log base 4 respectively.

In Satkhandagama various sets are operated upon by logarithmic functions to

base two, by squaring and extracting square roots, and by raising to finite or

infinite powers. The operations are repeated to produce new sets. In other

works the relation of the number of combinations to the coefficients occurring

in the binomial expansion is noted.

Since Jain epistemology allowed for a degree of indeterminacy in describing

reality, it probably helped in grappling with indeterminate equations and

finding numerical approximations to irrational numbers.

23
Buddhist literature also demonstrates an awareness of indeterminate and

infinite numbers. Buddhist mathematics was classified either

as Garna (Simple Mathematics) or Sankhyan (Higher Mathematics). Numbers

were deemed to be of three

types: Sankheya (countable), Asankheya (uncountable) and Anant (infinite).

Philosophical formulations concerning Shunya - i.e. emptiness or the void

may have facilitated in the introduction of the concept of zero. While the zero

(bindu) as an empty place holder in the place-value numeral system appears

much earlier, algebraic definitions of the zero and it's relationship to

mathematical functions appear in the mathematical treatises of Brahmagupta

in the 7th C AD. Although scholars are divided about how early the symbol for

zero came to be used in numeric notation in India, (Ifrah arguing that the use

of zero is already implied in Aryabhatta) tangible evidence for the use of the

zero begins to proliferate towards the end of the Gupta period. Between the

7th C and the 11th C, Indian numerals developed into their modern form, and

along with the symbols denoting various mathematical functions (such as

plus, minus, square root etc) eventually became the foundation stones of

modern mathematical notation.

The Indian Numeral System

Although the Chinese were also using a decimal based counting system, the

Chinese lacked a formal notational system that had the abstraction and

elegance of the Indian notational system, and it was the Indian notational

system that reached the Western world through the Arabs and has now been

accepted as universal. Several factors contributed to this development whose

24
significance is perhaps best stated by French mathematician, Laplace: "The

ingenious method of expressing every possible number using a set of ten

symbols (each symbol having a place value and an absolute value) emerged

in India. The idea seems so simple nowadays that its significance and

profound importance is no longer appreciated. It's simplicity lies in the way it

facilitated calculation and placed arithmetic foremost amongst useful

inventions."

Brilliant as it was, this invention was no accident. In the Western world, the

cumbersome roman numeral system posed as a major obstacle, and in China

the pictorial script posed as a hindrance. But in India, almost everything was

in place to favor such a development. There was already a long and

established history in the use of decimal numbers, and philosophical and

cosmological constructs encouraged a creative and expansive approach to

number theory. Panini's studies in linguistic theory and formal language and

the powerful role of symbolism and representational abstraction in art and

architecture may have also provided an impetus, as might have the rationalist

doctrines and the exacting epistemology of the Nyaya Sutras, and the

innovative abstractions of the Syadavada and Buddhist schools of learning.

Development of Mathematics in Ancient China

Chinese Math Texts

The history of Chinese math and mathematicians was mostly lost or

destroyed over the centuries. For example, the despotic emperor Shih Huang-

ti of the Ch'in dynasty (221-207 B.C.) ordered the burning of books in 213

B.C. Scholars in the following Han period (206 B.C. to 220 A.D.) had to

25
transcribe China's literary and scientific traditions from memory or remaining

fragments of scroll. Knowledge of astronomy and other areas was often

handed down from father to son, and only later recorded in texts.

Unfortunately, very few texts dedicated to mathematical astronomy have

survived.

Since the 16 century, Chinese math history has also been denied and

ignored in the Western dominance of science and technology, both inside and

outside China. However, there are several existing Chinese applied

mathematics texts, which are collections of problems and solutions organized

in chapters according to their practical applications. These texts prove that the

Chinese were the first society to use some of the most basic and advanced

mathematical principles and concepts utilized in modern times. Two of these

texts are the Chou Pei and Chiu Chang.

Chou Pei

The oldest existing Chinese texts containing formal mathematical

theories were produced during the Han period. The Arithmetic Classic of the

Gnomon and the Circular Paths of Heaven (Chou Pei Suan Ching) is dated

before the 3rd century B.C and contains various modern mathematical

principles such as working with fractions using a common denominator, and

proofs of many geometrical theories. The text contains an accurate process of

division for finding out the square root of numbers. In fact, the Chou

Pei presents the oldest known proof of the right-angle triangle theory in

the hsuan-thu diagram. This theory, commony known as the "Pythagorean

theorem," shows that the sum of the squares of the legs of a right triangle is

26
equal to the squares of the hypotenuse or a 2+ b2 = c2. The Chou Pei was not

an isolated academic text shared only by a few ancient Chinese

mathematicians. The principles in the text were reflected in the popular

approach known as chi-chu, or "the piling up of squares" which was a process

of using geometry to solve algebric problems.

Chiu Chang

Another 3rd century B.C. Han text, the Nine Chapters on the Mathematical Art

(Chiu Chang Suan Shu), was very influencial in asian mathematics. This text

was probably first written by Chang Tshang who made use of older works

then in existence.

The Chiu Chang focus on applied mathematics in engineering and

administration and include nine distinct chapters on impartial taxation (chun

shu), engineering works (shang kung), the surveying of land (fang thien), etc.

In total, 246 problem situations are presented, from those involving the

payment for livestock, weights and measures, currency and tax collection to

the construction of canals and simultaneous linear equations (fang chheng).

Other important Chinese math texts include the Mathematical Classic

of Sun Tzu (Sun Tzu Suan Ching) written in the 3rd century A.D., and The

Ten Mathematical Manuals (Suanjing Shi Shu). The 13 century text, Detailed

Analysis of the Mathematical Rules in the Nine Chapters (Hsiang Chieh Chiu

Chang Suan Fa), proved the theory known as "Pascal's Triangle" 300 years

before Pascal was born.

27
References:

http://www.storyofmathematics.com/chinese.html

http://britton.disted.camosun.bc.ca/china/development.htm

https://www.youtube.com/watch?v=i4BFJ6co4jM

https://www.youtube.com/watch?v=D_606_Dgyiw&t=5s

https://www.slideshare.net/angelmaelongakit/ancient-chinese-mathematics-

29685237

http://quatr.us/china/science/chinamath.htm

http://www.crystalinks.com/indiamathematics.html

http://archaeologyonline.net/artifacts/history-mathematics

https://www.youtube.com/watch?v=pElvQdcaGXE

https://www.youtube.com/watch?v=DeJbR_FdvFM

28
 Republic of the Philippines
LEYTE NORMAL UNIVERSITY
Graduate School
Tacloban City

MATH 536 – History of Mathematics

Rovinson D. Gaganao Nelson D. Bernardo

Student/Reporter Professor

****************************************************************************************
GREECE
Sub-Topics:
1. Political, Economic and Cultural Changes in the Iron Age
2. Famous Greek Mathematicians and Their Contributions to Science,
Philosophy, and Mathematics such as Pythagoras, Plato, Euclid,
Archimedes, Apolonuis, etc.

****************************************************************************************
1. Political, Economic and Cultural Changes in the Iron Age

The period of Human Culture was divided into Ancient, Post Classical, and

Modern Age. Ancient Period was re-classified into Stone Age, Bronze Age

and Iron Age. Generally Iron Age follows the Bronze Age, although some

societies went from the Stone Age straight into the Iron Age. Iron production

is known to have taken place as early as 1200 BC, though new archaeological

evidence suggests even earlier dates.

The adoption of iron and steel directly impacted changes in society,

affecting agricultural procedures and artistic expression, and also coincided

29
with the spread of written language. In historical archaeology, the earliest

preserved manuscripts are from the Iron Age. This is due to the introduction of

alphabetic characters, which allowed literature to flourish and for societies to

record historic texts. The beginning of the Iron Age differs from region to

region. It is characterized by the use of iron in tools, weapons, personal

ornaments, pottery and design. The differences from the preceding age of

bronze were due to more advanced ways of processing iron. Because iron is

softer than bronze, it could be forged, making design move from rectilinear

patterns to curvilinear, flowing designs.

Iron smelting is much more difficult than tin and copper smelting. These

metals and their alloys can be cold-worked, but smelted iron requires hot-

working and can be melted only in specially designed furnaces. Iron

fragments found in present day Turkey (c. 1800 BC) show the use of carbon

steel. These iron fragments are the earliest known evidence of steel

manufacturing. It is believed that a shortage of tin forced metalworkers to

seek an alternative to bronze. Many bronze objects were recycled into

weapons during this time. The widespread use of the more readily available

iron ore led to improved efficiency of steel-making technology. By the time tin

became available again, iron was cheaper, stronger and lighter, and forged

iron replaced bronze tools permanently. During the Iron Age, the best tools

and weapons were made from steel, particularly carbon alloys. Steel weapons

and tools were nearly the same weight as those of bronze, but much stronger.

Before the Industrial Revolution, majority of people lived an agrarian

lifestyle. Most people were farmers, and their lives revolved around the

farming seasons. Societies consisted of villages where communities of

30
families worked the land and made necessities for living by hand. All

essentials were made or grown locally. The production of iron tools helped

make the farming process easier and more efficient. Farmers could plow

tougher soil, making it possible to harvest new crops and freeing time for

more leisure. New varieties of crops and livestock were introduced at different

times over the span of the Iron Age. More time also meant that people could

make extra supplies to sell or exchange. Some farming families spent part of

their time making salt, quern stones or iron. Most settlements have evidence

of making clothes, woodworking and even blacksmithing. Iron has been

enhancing the quality of life for centuries. As more advanced technologies for

processing iron were discovered, the world would experience the most rapid

period of growth.

Social Impact

Beside the fact that people were now able to easily get everyday work

done they also could make better weapons and tools to get the work done.

Also they had differing agricultural practices, and religious beliefs, and artistic

styles.

Cultural Impact

The human achievements have now increased due to the new armor

and tools and buildings they can build from the new material called iron.

Political Impact

The Iron Age had a political impact on society by people finding out of

other places having this iron and then them wanting it so it gave to trade.

Economic Impact

31
 The impact economically was that the mass finding and having of iron

stimulated the economy from people selling amongst others.

The Iron Jewelry

Social Impact

The social impact of the iron jewelry being made during the Iron Age is

that the wealthy bunch of women could now show their wealth and power off

to everyone.

Cultural Impact

The cultural impact of iron jewelry is that people can show their wealth

and power in the culture and are they are living amongst their disciples.

Political Impact

The political impact on iron jewelry is that it shows your royalty.

2. Famous Greek Mathematicians and Their Contributions to Science,

Philosophy, and Mathematics such as Pythagoras, Plato, Euclid,
Archimedes, Apollonius, etc.

The early history of Greek geometry is unclear, because no original

sources of information remain and all of our knowledge is from secondary

sources written many years after the early period. However, we can still see a

decent overview and also start to look at some of the great names, the Greek

mathematicians who would shape the course of Greek geometry.

Thales of Miletus

The first, and one of the

greatest names, is Thales of Miletus, a

mathematician living in the 6th century

32
 BCE. He is regarded as the father of

geometry and began the process of

using deduction from first principles. It

is believed that he travelled to Egypt

and Babylon, picking up geometric

techniques from these cultures, and he

certainly would have had access to

their work.

Thales strongly believed that reasoning should supersede

experimentation and intuition, and began to look for solid principles upon

which he could build theorems. This introduced the idea of proof into

geometry and he proposed some axioms that he believed to be mathematical

truths.

 A circle is bisected by any of its diameters.

 The base angles of an isosceles triangle are equal.

 When two straight lines cross, the opposing angles are equal.

 An angle drawn in a semi-circle is a right angle.

 Two triangles with one equal side and two equal angles are

congruent.

Thales is credited with devising a method for finding the height of a

ship at sea, a technique that he used to measure the height of a pyramid,

much to the delight of the Egyptians. For this, he had to understand proportion

and possibly the rules governing similar triangles, one of the staples of

trigonometry and geometry. It is unclear exactly how Thales decided

33
that the above axioms were irrefutable proofs, but they were incorporated into

the body of Greek mathematics and the influence of Thales would influence

countless generations of mathematicians.

Pythagoras

Probably the most famous

name during the development of

Greek geometry is Pythagoras, even

if only for the famous law concerning

right angled triangles. This

mathematician lived in a secret

society which took on a semi-

religious mission. From this, the

Pythagoreans developed a number

of ideas and began to develop

trigonometry.

The Pythagoreans added a few new axioms to the store of geometrical

knowledge.

34
  The sum of the internal angles of a triangle equals two right angles

*(180o).

 The sum of the external angles of a triangle equals four right angles

(360o).

 The sum of the interior angles of any polygon equals 2n-4 right

angles, where n is the number of sides.

 The sum of the exterior angles of a polygon equals four right angles,

however many sides.

 The three polygons, the triangle, hexagon, and square completely fill

the space around a point on a plane - six triangles, four squares and

three hexagons. In other words, you can tile an area with these three

shapes, without leaving gaps or having overlaps.

 For a right angled triangle, the square of the hypotenuse is equal to

the sum of the squares of the other two sides.

Most of these rules are instantly familiar to most students, as basic principles

of geometry and trigonometry. One of his pupils, Hippocrates, took the

development of geometry further. He was the first to start using geometrical

techniques in other areas of maths, such as solving quadratic equations, and

he even began to study the process of integration. He studied the problem of

Squaring the Circle (which we now know to be impossible, simply because Pi

is an irrational number). He solved the problem of Squaring a Lune and

showed that the ratio of the areas of two circles equalled the ratio between the

squares of the radii of the circles.

Euclid

35
 Alongside Pythagoras, Euclid is a very famous name in the history of

Greek geometry. He gathered the work of all of the earlier mathematicians

and created his landmark work, 'The Elements,' surely one of the most

published books of all time. In this work, Euclid set out the approach for

geometry and pure mathematics generally,

proposing that all mathematical statements should be proved through

reasoning and that no empirical measurements were needed. This idea of

proof still dominates pure mathematics in the modern world.

Archimedes

Archimedes was a great mathematician and was a master at

visualising and manipulating space. He perfected the methods of integration

and devised formulae to calculate the areas of many shapes and the volumes

of many solids. He often used the method of exhaustion to uncover formulae.

For example, he found a way

to mathematically calculate the area

underneath a parabolic curve;

calculated a value for Pi more

accurately than any previous

mathematician; and proved that the

area of a circle is equal to Pi

multiplied by the square of its radius.

He also showed that the volume of a

sphere is two thirds the volume of a

cylinder with the same height and

36
 radius. This last discovery was

engraved into his tombstone.

Apollonius of Perga

Apollonius was a

mathematician and astronomer, and he

wrote a treatise called 'Conic Sections.'

Apollonius is credited with inventing the

words ellipse, parabola, and hyperbola,

and is often referred to as the Great

Geometer. He also wrote extensively on

the ideas of tangents to curves, and his

work on conics and parabolas would

influence the later Islamic scholars and

their work on optics.

Plato

Plato is arguably the single most influential philosopher of all time. So

significant is his work that the modern British philosopher Albert North

Whitehead quipped that "the safest general characterization of the European

philosophical tradition is that it consists of a series of footnotes to Plato." If

Whitehead overstated his case, it was not by much. Plato's work extends to

virtually every area of philosophy.

37
Political Philosophy and the Just City

Although Plato's writings extend to

every corner of academia, political

philosophy was his most common

subject. In "Republic," perhaps the

most famous of his dialogues, he

tackled the difficult questions of what

constitutes justice and how a just

state should function. Plato's

answers were controversial, and

"Republic" has encouraged

philosophical debate in every century

after its writing. Later philosophers

like Robert Nozick and John Rawls

drew heavily on Plato's work in

constructing their own conceptions of

justice and the state.

Philosophy of Rhetoric

38
Plato also studied and wrote extensively on rhetoric, the art of persuasion.

Two of Plato's most important dialogues, "Phaedrus" and "Gorgias," address

questions about the nature of rhetoric, according to the Stanford Encyclopedia

of Philosophy. For Plato, rhetoric and argument was a way to deduce truths

about the world through careful introspection. However, he also saw rhetoric

as a potentially dangerous weapon in the hands of sophists and demagogues.

Later philosophers like Aristotle, Cicero, and Augustine expanded on Plato's

philosophy of rhetoric.

Platonic Epistemology

Yet another of Plato's essential contribution to philosophy is his work on

epistemology. Epistemology is the study of knowledge, or how people come

to know things. According to the European Graduate School, Plato was

among the first philosophers to consider the idea of a priori knowledge,

defined as knowledge that exists independent of experience. In "Republic"

and other dialogues, Plato argued that human experience was always limited

and deceptive. The real truth, he said, emerged not from the real world but

from the world of ideas. Plato argued that idealized "forms" represented the

true, perfect version of everything or idea in the universe. This powerful

concept has puzzled and inspired great philosophers like David Hume, Martin

Heidegger and Jacques Derrida for generations.

Plato's Dialectic

Beyond his writings and findings, Plato also contributed to philosophy a new

method for answering philosophical questions. Nearly all of Plato's writings

took the form of dialogues between Socrates and various other characters.

The characters disagree and argue with each other. Plato's use of dialogue

39
pitted arguments and ideas against each other, allowing the best ideas to rise

to the surface. This dialectical method ensures rigorous scrutiny of every

premise and conclusion. Although few modern philosophers write in dialogue,

the dialectic has influenced subsequent methods of philosophical explication.

References

Famous Mathematicians (2013). Famous Mathematicians. Retrieved from

http://famous-mathematicians.org/

Martyn Shuttleworth (2010). Greek Geometry. Retrieved from Explorable.com:

https://explorable.com/greek-geometry

The Editors of Encyclopaedia Britannica (2017). Iron Age. Retrieved from

https://www.britannica.com/event/Iron-Age

The Story of Mathematics (2010). Greek Mathematics. Retrieved from

http://www.storyofmathematics.com/greek.html

Robinson, Nick (2001). What dis Plato Contribute to Philosophy? Retrieved

from http://classroom.synonym.com/did-plato-contribute-philosophy-
21831.html

40
 Republic of the Philippines
LEYTE NORMAL UNIVERSITY
Graduate School
Tacloban City

Math 536- History of Mathematics

HELEN GRACE A ELLEMA NELSON D. BERNARDO

Student/ Reporter Professor

THE ORIENT AFTER THE DECLINE OF GREEK SOCIETY

Subtopics:

1. The Growth of Islam Mathematics

2. Arabian Mathematicians and Their Work

-------------------------------------------------------------------------------------------------------

THE ORIENT AFTER THE DECLINE OF GREEK SOCIETY

Despite Hellenistic influence, Near Eastern thought remained intact, as is

evidenced by work in Alexandria, India, and Constantinople. The Byzantine

Empire served as a guardian for Greek culture while the Indus region and

41
Mesopotamia became independent. The sudden growth of Islam ended Greek

domination. Arabic administration and language competed with and

conquered Greek culture in much of the Mediterranean.

As the Roman Empire declined the center of math research shifted from

Alexandria to India and Mesopotamia. The Surya Siddhanta shows an

influence of Greek and Babylonian astronomy. Aryabhata (c. 500) and

Brahmagupta (c 625) were the best known. Mahavira considered rational

triangles and quadrilaterals. General solutions for indeterminate equations of

the first degree (ax+by =c) is found in Brahmagupta. Bhaskara admitted

negative roots of equations and his Lilavati became a standard text for

arithmetic and mensuration. Nilakantha (c. 1500) had already found the

Gregory Leibniz series for pi/4.

Our present decimal-position system first appeared in China and was used

increasingly in India (c. 595). The word sunya although the use of a dot

predates this in Babylonian texts. “0” probably comes from the

Greek ouden (nothing). However, in Hindu math, zero was equivalent to 1…9,

not just a holder as with the Babylonian dot. Translation of the Siddhantas into

Arabic introduced the Hindu system to the Islamic world, where permutations

of it (the gobar system made their way to Spain and to the West.

Persia and Baghdad were taken by Arabs, causing Arabic to be instated as

the official language, although other cultures remained. Islamic math was

influenced by the same factors as Alexandria and India. The caliphs promoted

astronomy and math, creating libraries and observatories. Muhammand ibn

Musa al-Khwarizmi (c. 825) wrote a book whose Latin translation (Algorithmi

de numero Indorum) spread the decimal position. The word “algorithms” is a

42
latinization of his name. Similarly his Hisab al-jabr wal-muqabala (science of

reduction and confrontation or science of equations) introduced al-habr or

algebra into the lexicon. Although lacking formalism and mostly geometric, his

examples (i.e. x^2+10x=39) were a thread appearing in algebras for several

centuries. He also included trigonometric tables. His geometry, while simple

can be traced to a Jewish text of 150 CE. His work lacked the axiomatic

foundation, but was important for the introduction of decimal position to the

West.

Arabic scholars also faithfully translated Greek classics into Arabic:

Apollonius, Archimedes, Euclid, Ptolemy (Almagest being the familiar name

for his Great Collection). Arabic math was particularly interested in

trigonometry (sinus is a latinization of the sanksirt jya). Sines were half a

chord, and were thought of as lines. Al-Battani provided extensive cotangent

tables (for every degree) Abu-l-Wafa inroduced secant and cosecant, and

derived the sign theorem of spherical trig. Al-Karkhi (d. c. 1029) was

monomaniacal interested in Greek and wrote an algebra inspired by

Diophantus and was interested in surds (sq roots).

Omar Khayyam (c. 1038-1123), who lived in northern Persia near Merv, was

notable for a reformed Persian calendar with an error of one day in 5000

years (compared to 330 years w/ the Gregorian). His Algebra examined cubic

equations and determined root as the intersection of two conic sections. He

also introduced a non-Euclidean geometry. Nasir al-din separated trig from

astronomy and attempted to prove Euclid’s parallel axiom, which was made

use of in Renaissance Europe by John Wallis. Nasir followed Khayyam’s

approach to theory of ratio and the irrational. Jemshid Al-Kashi (d. c. 1436)

43
was influenced by Chinese mathematics and knew of (what is now called)

Horner’s Method, iterative methods. He also provided the binomial formula for

a general positive integer exponent. Al-Kashi had pi to 16 decimals. Ibn Al-

Haitham, whose Optics was influential, solved the problem of Alhazen. He

also employed the exhaustion method. Abu Kamil (a follower of Al-

Khwarizmi) had influenced on Al-Karkhi and Leonardo of Pisa. Al-Zarqali, was

notable for the Toledan tables, which influenced the Alfosine tables, which

were authoritative trig tables for centuries.

Chinese mathematics was not isolated. It developed at least by the Han

Dynasty, the decimal position system was probably invented there. Pi was

found to many decimal places (Liu Hui had two digits, Tsu Ch’ung-Chih, had

seven [22/7]). During the Tang dynasty, imperial examinations made use of

math books, spurring the printing of Nine Chapters as early as 1084. The

Sung dynasty saw greater progress. Wang Hsio Tung exceeded the Nine

Chapters by looking at cubic equations of a higher complexity. Ch’in Chui -

Shao used successive approximation to solve higher degree polynomials

(similar to Horner’s work of a much later date). Yang Hui (c. 1261) used a

decimal notation similar to our modern style. He also provided the earliest

extant pascal’s triangle. Chi-Shih-Cieh, the most important Sung

mathematician, extended “matrix” methods to solve linear equations with

several unknowns and of a high degree. The post-Sung period saw a decline,

but diffusion of these developments westward.

References:

44
1. https://sublimated.wordpress.com/2002/07/08/struik-a-concise-history-

of-mathematics-the-orient-after-the-decline-of-the-greek-society/

2. https://books.google.com.ph/books?

id=V0RuCQAAQBAJ&pg=PA16&lpg=PA16&dq=mathematics+after+th

e+decline+of+Greek+Society&source=bl&ots=aqbcq-AGv3&sig=gVG-

35b4XW6YNZAZw-OXVEPL6-

M&hl=en&sa=X&ved=0ahUKEwip2KPjlenTAhWBNpQKHQHxDYYQ6A

EIPzAG#v=onepage&q=mathematics%20after%20the%20decline

%20of%20Greek%20Society&f=false

3. . https://www.geogebra.org/material/show/id/60209

Republic of the Philippines

LEYTE NORMAL UNIVERSITY
Graduate School
Tacloban City

Math 536 – History of Mathematics

MARCO C. BERONGOY NELSON D. BERNARDO

Student/Reporter Professor

************************************************************************************
THE BEGINNING OF WESTERN EUROPE

Sub-Topics:
1. Mathematics in the Western Roman Empire
2. Roman Numerals
a. History
b. Hypotheses about the Origin of Roman Numerals
3. Roman Numeral Counting System
4. Advantages & Disadvantages of Roman Numeral System
5. Addition of Roman Numerals
************************************************************************************

UNIT V

THE BEGINNING OF WESTERN EUROPE

45
MATHEMATICS IN THE WESTERN ROMAN EMPIRE

Laying the foundation of western thought and philosophy, few

civilizations in history have contributed to humanity more than the Greeks.

They built on the exceptional mathematics of the Egyptian and Babylonian

empires, and were without a doubt the best mathematicians on the planet at

that time.

Names like Pythagoras, Archimedes and Ptolemy are so synonymous

with mathematics that we recognize them two millennia later. Fast forward to

the first century B.C. and the Romans replaced Greek civilization.

The Romans carried a great deal of Greek culture with them. For

centuries they were the dominant empire on earth. However, there is one

missing element in the Romans contribution to humanity.

They had no notable contributions to mathematics, or mathematicians

for that matter.

The Romans built roads, invented the water wheel, and constructed

vast aqueducts that we marvel at today.

ROMAN NUMERALS

The Roman numeral system is an extremely complex (and irrational)

system of numbers. The system does not make sense, because ”zero” does

not exist, which makes all higher level mathematics impossible.

46
 Beyond that Roman numerals follow an additive system. When certain

values are reached they are added to a new symbol.

HISTORY

Pre-Roman times and ancient Rome

Although Roman numerals came to be written with letters of the

Roman alphabet, they were originally independent symbols. The Etruscans,

for example, used 𐌠, 𐌡, 𐌢, ⋔, 𐌚, and ⊕ for I, V, X, L, C, and M, of which

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

HYPOTHESES ABOUT THE ORIGIN OF ROMAN NUMERALS

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

47
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: 𐊌) came to predominate. It was written variously

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⟩. 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

48
into ∞, then ⋈, and eventually changed to M under the influence of the Latin

word mille "thousand".

Hand signals

Alfred Hooper has an alternative hypothesis for the origin of the Roman

numeral system, for small numbers. Hooper contends that the digits are

related to hand gestures for counting. For example, the

numbers I, II, III, IIII correspond to the number of fingers held up for another to

see. V, then represents that hand upright with fingers together and thumb

apart. Numbers 6–10, are represented with two hands as follows (left hand,

right hand) 6=(V,I), 7=(V,II), 8=(V,III), 9=(V,IIII), 10=(V,V) and X results from

either crossing of the thumbs, or holding both hands up in a cross. Another

possibility is that each I represents a finger and V represents the thumb of one

hand. This way the numbers between 1-10 can be counted on one hand using

the

order: I=P, II=PR, III=PRM, IV=IT, V=T, VI=TP, VII=TPR, VIII=TPRM, IX=IN,

X=N (P=Pinky, R=Ring, M=Middle, I=Index, T=Thumb N=No Fingers/Other

Hand). This pattern can also be continued using the other hand with the

fingers representing X and the thumb L.

Intermediate symbols deriving from few original symbols

A third hypothesis about the origins states that the basic ciphers

were I, X, C and Φ (or ⊕) and that the intermediary ones were derived from

taking half of those (half an X is V, half a C is L and half

a Φ/⊕ is D). The Φ was later replaced with M, the initial of Mille (the Latin

word for "thousand").

49
ROMAN NUMERAL COUNTING SYSTEM

Roman Counting System

Modern Roman Modern Roman

1 I 50 L

2 II 100 C

3 III 500 D

4 IV 1,000 M

5 V 10,000 X

6 VI 20,000 XX

7 VII 100,000 C

8 VIII

9 IX

10 X

ADVANTAGES AND DISADVANTAGES OF ROMAN NUMERAL SYSTEM

Advantages

o Easy to use and understand

o The most predominant representation of numbers in European culture

until the 14th century

o Familiarization of the system up to date

Disadvantages

o Completely unsuitable even for simple operation like addition, not to

mention multiplication

o Discourage serious calculations

50
 o Had no numeral “zero”

o In early European universities, algorithms for multiplication and division

using Roman numerals were doctoral research topics.

ADDITION OF ROMAN NUMERALS

ROMAN ARITHMETIC

Despite all of these deficiencies, Roman Numerals supposedly remained

the predominant representation of numbers in European culture until the 14 th

century.

SOURCES & REFERENCES:

The Story of Mathematics

http://www.storyofmathematics.com/roman.html

The Mathematics of Roman Empire by Sam Solomon

https://solomon.io/mathematicians-roman-empire/

Mathematics of the Past by Garry Kasparov

https://www.math.ualberta.ca/pi/issue5/page05-08.pdf

Menninger, Karl (1992). Number Words and Number Symbols: A Cultural

History of Numbers. Dover Publications. ISBN 978-0-486-27096-8.

Roman Numerals

51
 https://en.wikipedia.org/wiki/Roman_numerals

Republic of the Philippines

LEYTE NORMAL UNIVERSITY
Graduate School
Tacloban City

Math 536 – History of Mathematics

MA. CRISTINA E. MEJARITO NELSON D. BERNARDO
Student/Reporter Professor

************************************************************************************
SEVENTEENTH CENTURY
Sub-Topics:
1. Important Discoveries in the Field of Astronomy and Physics
and Mathematical Formula.

52
 ************************************************************************************

UNIT VI

SEVENTEENTH CENTURY

IMPORTANT DISCOVERIES IN THE FIELD OF ASTRONOMY AND

PHYSICS AND MATHEMATICAL FORMULAS

ASTRONOMY

The science that deals with the origin, evolution, composition,

distance, and motion of all bodies and scattered matter in the universe.

It includes astrophysics, which discusses the physical properties and

structure of all cosmic matter. Astronomy comprises the study of all

extra-terrestrial objects. Until the invention of the telescope and the

discovery of the laws of motion and gravity in the 17th century,

astronomy was primarily concerned with noting and predicting the

positions of the Sun, Moon, and planets, initially for calendrical and

astrological purposes and later for navigational applications and

scientific interest.

Astronomy is the most ancient of the sciences, having existed

since the dawn of recorded civilization. The 17th century witnessed

several momentous developments that led to major advances in

astronomy were:

53
Johaness Kepler – Where the discovery of principles of planetary

motion and Kepler also worked on and designed better cameras and

telescopes to use for observation. A mission aimed at detecting

planets around other stars has been named after Kepler.

Kepler’s Law of Planetary Motion:

 The planets orbit the sun in ellipses with the Sun at one

focus

 Equation of Time -the line joining the Sun and a planet

sweeps through equal areas in equal times

 The square of the period of the orbit of a planet is

proportional to the cube of its semi major axies (half the

longest dimension of the ellipse)

Hans Lippershey – Was born in Middelburg in Zeeland in

September 1608. The first known patent for a workable telescope.

Invented the first refracting telescope named spy-glass.

Harriot – A British Astronomer and was using X 6 power spyglass

for observing and mapping the surface of the moon.

Galileo Galilei - The Italian mathematician and physicist Galileo

Galilei develops an astronomical telescope powerful enough to

indentify moons orbiting Jupiter, sunspots on the Sun and the

different phases of Venus, all of which are instrumental in

convincing the scientific community of the day that the heliocentric

54
 Copernican model of the Solar System is superior to the geocentric

Ptolemiac model.

PHYSICS

1605

Johannes Kepler - The German mathematician and astronomer

Johannes Kepler establishes his three Laws of Planetary Motion,

mathematical laws that describe the motion of planets in the Solar

System, including the ground-breaking idea that the planets follow

elliptical, not circular, paths around the Sun. Newton later used

them to deduce his own Laws of Motion and his Law of Universal

Gravitation.

1610

Galileo Galilei - The Italian mathematician and physicist. Develops

an astronomical telescope powerful enough to indentify moons

orbiting Jupiter, sunspots on the Sun and the different phases of

Venus, all of which are instrumental in convincing the scientific

community of the day that the heliocentric Copernican model of the

Solar System is superior to the geocentric Ptolemiac model. On

1632 Galileo first describes the Principle of Relativity, the idea that

the fundamental laws of physics are the same in all inertial

frames and that, purely by observing the outcome of mechanical

55
 experiments, one cannot distinguish a state of rest from a state of

constant velocity.

1633

René Descartes – A French philosopher outlines a model of a static,

infinite universe made up of tiny “corpuscles” of matter, a viewpoint

not dissimilar to ancient Greek atomism. Descartes’ universe shares

many elements of Sir Isaac Newton’s later model, although

Descartes’ vacuum of space is not empty but composed of huge

swirling whirlpools of ethereal or fine matter, producing what would

later be called gravitational effects.

1638

Galileo Galilei - demonstrates that unequal weights would fall with

the same finite speed in a vacuum, and that their time of descent is

independent of their mass. Thus, freely falling bodies, heavy or light,

have the same constant acceleration, due to the force of gravity.

1675

Sir Isaac Newton - The English physicist Isaac argues that light is

composed of particles, which are refracted by acceleration toward a

denser medium, and posits the existence of “aether” to transmit

forces between the particles. On 1687 Sir Isaac Newton publishes

56
 his “Principia”, which describes an infinite, steady state,

static, universe, in which matter on the large scale is uniformly

distributed. In the work, he establishes the three Laws of Motion (“a

body persists its state of rest or of uniform motion unless acted

upon by an external unbalanced force”; “force equals mass times

acceleration”; and “to every action there is an equal and opposite

reaction”) and the Law of Universal Gravitation (that every particle in

the universe attracts every other particle according to an inverse-

square formula) that were not to be improved upon for more than

two hundred years. He is credited with introducing the idea that the

motion of objects in the heavens (such as planets, the Sun and the

Moon) can be described by the same set of physical laws as the

motion of objects on the ground (like cannon balls and falling

apples).

1734

Emanuel Swedenborg - The Swedish scientist and philosopher

Emanuel proposes a hierarchical universe, still generally based on a

Newtonian static universe, but with matter clustered on ever larger

scales of hierarchy, endlessly being recycled. This idea of a

hierarchical universe and the “nebular hypothesis” were developed

further (independently) by Thomas Wright (1750) and Immanuel

Kant (1775).

57
1761

Johann Heinrich Lambert - Supports Wright and Kant’s

hierarchical universe and nebular hypothesis, and also

hypothesizes that the stars near the Sun are part of a group which

travel together through the Milky Way, and that there are many such

groupings or star systems throughout the galaxy.

1783

John Michell - The amateur British astronomer John Michell

proposes the theoretical idea of an object massive enough that

its gravity would prevent even light from escaping (which has since

become known as a black hole). He realizes that such an object

would not be directly visible, but could be identified by the motions

of a companion star if it was part of a binary system. A similar idea

was independently proposed by the Frenchman Pierre-Simon

Laplace in 1795.

1789

Antoine-Laurent de Lavoisier – States the Law of Conservation of

Mass (although others had previously expressed similar ideas,

including the ancient Greek Epicurus, the medieval Persian Nasir al-

Din al-Tusi and the 18th Century scientists Mikhail Lomonosov,

Joseph Black, Henry Cavendish and Jean Rey), and identifies

58
 (albeit slightly incorrectly) 23 elements which he claims cannot be

broken down into simpler substances.

MATHEMATICS

In the wake of the Renaissance, the 17th Century saw an

unprecedented explosion of mathematical and scientific ideas across

Europe, a period sometimes called the Age of Reason. Hard on the

heels of the “Copernican Revolution” of Nicolaus Copernicus in the 16th

Century, scientists like Galileo Galilei, Tycho Brahe and Johannes

Kepler were making equally revolutionary discoveries in the exploration

of the Solar system, leading to Kepler’s formulation of mathematical

laws of planetary motion.

John Napier – The invention of logarithm in the early 17 th century,

contributed to the advance of science, astronomy and mathematics

by making some difficult calculations relative easy. It was one of the

most significant mathematical developments of the age. The

logarithm of a number is the exponent when that number is

expressed as a power of 10 (or any other base). It is effectively the

inverse of eponentation.

Marin Mersenne – Is largely remembered in mathematics in the

term Mersenne primes – prime numbers that are one less than a

power of 2, e.g. 3 (22-1), 7 (23-1), 31 (25-1), 127 (27-1), 8191 (213-1),

etc. In modern times, the largest known prime number has almost

59
always been a Mersenne prime, but in actual fact, Mersenne’s real

connection with the numbers was only to compile a none-too-

accurate list of the smaller ones (when Edouard Lucas devised a

method of checking them in the 19th Century, he pointed out that

Mersenne had incorrectly included 267-1 and left out 261-1, 289-1 and

2107-1 from his list).

Rene Decartes - Is sometimes considered the first of the modern

school of mathematics. His development of analytic geometry and

Cartesian coordinates in the mid-17th Century soon allowed the

orbits of the planets to be plotted on a graph, as well as laying the

foundations for the later development of calculus (and much later

multi-dimensional geometry). Descartes is also credited with the first

use of superscripts for powers or exponents.

Pierre de Fermat – Formulated several theorems which greatly

extended our knowledge of number theory as well as contributing

some early work on infinitesimal calculus.

Blaise Pascal – The Frenchman Blaise Pascal was a prominent

17th Century scientist, philosopher and mathematician. He is best

known, however, for Pascal’s Triangle, a convenient tabular

presentation of binomial co-efficients, where each number is the

sum of the two numbers directly above it. A binomial is a simple

type of algebraic expression which has just two terms operated on

only by addition, subtraction, multiplication and positive whole-

60
number exponents, such as (x +y)2. The co-efficients produced

when a binomial is expanded form a symmetrical triangle.

Republic of the Philippines

LEYTE NORMAL UNIVERSITY

61
 Graduate School
Tacloban City

Math 536 – History of Mathematics

NOEL S. PALOMAS NELSON D. BERNARDO

Student/Reporter Professor

************************************************************************************
SEVENTEENTH CENTURY
Sub-Topics:
1. Invention of Machines and their Effect on Theoretical
Mechanics and other Scientific Studies.
2. Resolution in Astronomy
************************************************************************************

UNIT VI

SEVENTEENTH CENTURY

INVENTION OF MACHINES AND THEIR EFFECT ON THEORITICAL

MECHANICS AND OTHER SCIENTIFIC STUDIES

1608

German-Dutch spectacle-maker HANS LIPPERSHEY invents the first

refracting telescope. Some telescopes and spyglasses may have been

created much earlier, but Lippershey is believed to be the first to apply for a

patent for his design, a few weeks before Jacob Metius (a Dutch instrument

maker and optician), and making it available for general use in 1608. Although

he failed to receive a patent, he was handsomely rewarded by the Dutch

government for copies of his design. The telescope invented by Lippershey

had a magnification of just 3x.

1620

62
Dutch builder CORNELIS DREBBEL invents the earliest human-powered

submarine. The world's first practical submarine was built in 1620 by Dutch

engineer Cornelis Jacobszoon Drebbel, under the patronage of James 1 of

England. Drebbel built three submarines according to the sketchy information

available from that time, each larger than the last and the third being capable

of carrying 16 people, of which 12 were the oarsmen.

1624

English mathematician WILLIAM OUGHTRED invents the slide rule. The slide

rule was invented in 1924 by William Oughtred, an English Mathematician

from Cambride University in England. The slide perform calculations even

faster than was previously possible and has been used by mathematicians

and engineers until the arrival of the pocket calculator around in 1974.

1629

Italian engineer and architect GIOVANNI BRANCA invents a steam turbine. A

steam turbine is a device that extracts thermal energy from pressurized steam

and uses it to do mechanical work on a rotating output shaft. It is a form of

heat engine that derives much of its improvement in thermodynamic efficiency

from the use of multiple stages in the expansion of the steam, which results in

a closer approach to the ideal reversible expansion process.

1636

English astronomer and mathematician W. GASCOIGNE invents the

micrometer. A micrometer, sometimes known as a micrometer screw gauge,

is a device incorporating a calibrated screw widely used for precise

measurement of components in mechanical engineering and machining as

63
well as most mechanical trades, along with other metrological instruments

such as dial, vernier, and digital calipers.

1642

French mathematician BLAISE PASCAL invents the adding machine. An

adding machine was a class of mechanical calculator, usually specialized for

bookkeeping calculations. Blaise Pascal and Wilhelm Schickard were the two

original inventors of the mechanical calculator in 1642. For Pascal this was an

adding machine that could perform additions and subtractions directly and

multiplication and divisions by repetitions.

1643

Italian mathematician and physicist EVANGELISTA TORRICELLI invents the

barometer. Torricelli filled a four-foot long glass tube with mercury and

inverted the tube into a dish. Some of the mercury did not escape from the

tube and Torricelli observed the vacuum that was created. He became the

first scientist to create a sustained vacuum and to discover the principle of a

barometer. Torricelli realized that the variation of the height of the mercury

from day to day was caused by changes in the atmospheric pressure.

1656

Dutch mathematician and scientist CHRISTIAN HUYGENS invents a

pendulum clock. In 1656 Dutch mathematician, astronomer, physicist and

horologist Christiaan Huygens invented the pendulum clock in 1656 and

patented it in 1657. This technology reduced the loss of time by clocks from

about 15 minutes to about 15 seconds per day.

64
1660

Cuckoo clocks were made in Furtwangen, Germany, in the Black Forest

region. A cuckoo clock is a typically pendulum-regulated clock that strikes the

hours with a sound like a common cuckoo's call and has an automated

cuckoo bird that moves with each note. Some move their wings and

open/close their beaks while leaning forward, whereas in others, only the

bird's body is leans forward.

1663

Mathematician and astronomer JAMES GREGORY invent the first reflecting

telescope. A reflecting telescope (also called a reflector) is an optical

telescope which uses a single or combination of curved mirrors that reflect

light and form an image. The reflecting telescope was invented in the 17th

century as an alternative to the refracting telescope which, at that time, was a

design that suffered from severe chromatic aberration.

1668

Mathematician and physicist ISAAC NEWTON invent a reflecting telescope.

Newton replaced the primary lens with a polished, rounded, metal mirror. He

experimented with different mixtures of metal and decided on one that was six

parts copper to two parts tin. It was almost as bright as expensive, quick-to-

corrode silver and would reflect a lot of light. The more light the mirror

reflected, the better view the telescope would provide of the sky.

1671

German mathematician and philosopher GOTTFRIED WILHELM LEIBNIZ

invents the calculating machine. Leibniz designed a calculating machine

called the Step Reckoner. (It was first built in 1673.) The Step Reckoner

65
expanded on Pascal's ideas and did multiplication by repeated addition and

shifting. Leibniz was prescient in seeing the appropriateness of the binary

system in calculating machines, but his machine did not use it. Instead, the

Step Reckoner represented numbers in decimal form, as positions on 10-

position dials.

1675

Dutch mathematician, astronomer and physicist CHRISTIAN HUYGENS

patents the pocket watch. On the 4 October, in 1675, Christian Huygens

patented a pocket watch. Huygens was a Dutch astronomer and physicist

who established the wave theory of light and made astronomical discoveries.

He also patented the first pendulum clock in 1656, which he had developed to

meet his need for exact time measurement while observing the heavens. In

1673, he studied the relation of the length of a pendulum to its period of

osciallation.

1676

English architect and natural philosopher ROBERT HOOKE invents the

universal joint. A universal joint (universal coupling, U-joint, Cardan joint,

Spicer or Hardy Spicer joint, or Hooke's joint) is a joint or coupling in a rigid

rod whose axis are inclined to each other, and is commonly used in shafts

that transmit rotary motion. It consists of a pair of hinges located close

together, oriented at 90° to each other, connected by a cross shaft. The

universal joint is not a constant-velocity joint.

1679

French physicist, mathematician, and inventor DENIS PAPIN invent the

pressure cooker. A pressure cooker is a vessel that uses steam under high

66
pressure for cooking food. It offers a number of benefits, including fast, often

low-fat cooking that preserves the minerals—and even the coloration—of

fruits, vegetables, and meats.

1698

English inventor and engineer THOMAS SAVERY invents a steam pump. The

first successful steam pump was patented by Thomas Savery in 1698, and in

his words provided an "engine to raise water by fire". The artist rendering here

is understood to use artistic liberty; it is unlikely the egg-shaped vessels

existed. The unit had two boilers, D and L, connected by pipe E.

RESOLUTION IN ASTRONOMY

Astronomy has always had a significant impact on our world view.

Early cultures identified celestial objects with the gods and took their

movements across the sky as prophecies of what was to come. We would

now call this astrology, far removed from the hard facts and expensive

instruments of today’s astronomy, but there are still hints of this history in

modern astronomy. Take, for example, the names of the constellations:

Andromeda, the chained maiden of Greek mythology, or Perseus, the demi-

god who saved her.

TOP 10 MOST IMPORTANT DISCOVERIES IN ASTRONOMY

1. EXTRASOLAR PLANETS

The Discovery

67
 An extra solar planet is one that’s outside of our solar system, and

astronomers believed in their existence for a long, long time. Yet, it

wasn’t until recently that the tools to actually spot one became available; it

was only in 1995 when Swiss astronomers Didier Queloz and Michel Mayor

discovered a planet in the constellation Pegasus they dubbed 51 Pegasi b.

Yeah, astronomers may be great at discovering things but they’re not great at

naming them.

2. COSMIC MICROWAVE BACKGROUND RADIATION

The Discovery

It was a pair of radio astronomers, Arno Penzias and Robert Wilson,

who discovered cosmic microwave background radiation in 1964. CMBR is a

type of radiation that’s present in very small quantities (hence the term

background) all throughout space, and is believed to be leftover from when

the universe was in a very early stage of growth.

3. RADIO ASTRONOMY

The Discovery

Remember when radio was all the rage in the entertainment world? Of

course you don’t, you’re not 80 years old. But in the world of astronomy radio

is still important today, thanks to a discovery by Karl Jansky in 1931. His

experiments with radio waves led him to find signals coming from the centre

of the galaxy, and he’s considered the founding father of radio astronomy as a

result.

4. THE EXPANDING UNIVERSE

68
 The Discovery

Edwin Hubble gave the astronomy world a one-two punch of knowledge

between 1924 and 1929. Not only was he the first to discover other galaxies,

but by tracking their movement he learned that they are moving away from us

(and the ones farther away are moving faster), which was the first evidence

we had to suggest that the universe is expanding.

5. THE THEORY OF RELATIVITY

The Discovery

Albert Einstein, a German scientist you may have heard of, proposed

his theory of relativity in 1915. Summed up, the theory states that mass can

warp both space and time, which allows large masses like stars to bend light.

It’s trippy stuff.

6. HERSCHEL’S MAP

The Discovery

From 1780 to 1834, telescope maker William Herschel and his sister

Caroline systematically mapped the heavens, charting thousands of stars and

nebulae in the process. He also discovered Uranus, and if astronomers had

stuck with his proposed name of Georgium Sidus (George’s Star) we would

have been saved centuries of terrible jokes.

7. THE MOONS OF JUPITER

The Discovery

Galileo, arguably the most important scientist ever, used a fancy

telescope he half invented and half stole the idea for to discover four moons

69
orbiting Jupiter in 1610. They were the first moons of another planet to be

spotted, making them a landmark discovery. More importantly, we recently

discovered that The Moons of Jupiter would make a sweet band name.

8. KEPLER’S LAWS

The Discovery

In 1609, a German astronomer named Johannes Kepler told the world

that planets moved around the sun on elliptical routes, not in perfect circles as

was commonly believed. Yeah, you know science can be boring when ellipses

instead of circles is one of its most important discoveries.

9. THE HELIOCENTRIC MODEL

The Discovery

Astronomers had speculated about heliocentrism (the idea that the Earth

revolves around the sun, not the other way around) since ancient times, but in

1543 Copernicus was the first person to actually demonstrate the math behind

the idea to prove it was a viable concept.

10. THE MOVEMENT OF THE STARS AND PLANETS

The Discovery

It’s tough to wade through a couple thousand years of ancient

Babylonian, Egyptian, Greek, Indian, Chinese, Mayan and Persian

astronomical history to pick out the highlights, so I’ going to cheat and roll all

of their achievements up into one entry. Maybe if their civilizations hadn’t died

out they would have got a better spot on this list, but because they couldn’t

70
keep their empires together the ancient world gets stuck with the number ten

spot.

Republic of the Philippines

LEYTE NORMAL UNIVERSITY
Graduate School
Tacloban City

Math 536 – History of Mathematics

MICHAEL DELL A. TUAZON NELSON D. BERNARDO
Student/Reporter Professor
*********************************************************************
*********
The Eighteenth Century

1. Concentrations on calculus and its application

2. The production of more mathematicians in the different parts of
European countries and their continued and improved on early
works

****************************************************************************************
Irrational numbers in Greek math
 Discovery of irrational numbers

 Greeks tried to avoid the use of irrationals

 The infinity was understood as potential for continuation of a process

but not as actual infinity (static and completed)

 Examples:

 1,2, 3,... but not the set {1,2,3,…}

 sequence x1, x2, x3,… but not the limit x = lim xn

 Paradoxes of Zeno (≈ 450 BCE): the Dichotomy

 there is no motion because that which is moved must arrive at

the middle before it arrives at the end

71
  Approximation of √2 by the sequence of rational number

4.2 Eudoxus’ Theory of Proportions

 Eudoxus (around 400 – 350 BCE)

 The theory was designed to deal with (irrational) lengths using only
rational numbers

 Length λ is determined by rational lengths less than and greater than λ

 Then λ1 = λ2 if for any rational r < λ1 we have r < λ2 and vice versa
(similarly λ1 < λ2 if there is rational r < λ2 but r > λ1 )

 Note: the theory of proportions can be used to define irrational

numbers: Dedekind (1872) defined √2 as the pair of two sets of positive
rationals L√2 = {r: r2< 2} and
U√2 = {r: r2>2} (Dedekind cut)

The Method of Exhaustion

 was designed to find areas and volumes of complicated objects
(circles, pyramids, spheres) using

 approximations by simple objects (rectangles, trianlges, prisms)

having known areas (or volumes)

 the Theory of Proportions

Examples

Example:
Area enclosed by a Circle
 Let C(R) denote area of the circle of radius R

 We show that C(R) is proportional to R2

72
 1) Inner polygons P1 < P2 < P3 <…

2) Outer polygons Q1 > Q2 > Q3 >…

3) Qi – Pi can be made arbitrary small

4) Hence Pi approximate C(R) arbitrarily closely

5) Elementary geometry shows that Pi is proportional to R2 . Therefore, for

two circles with radii R and R' we get:
Pi(R) : Ri (R’) = R2:R’2

6) Suppose that C(R):C(R’) < R2:R’2

7) Then (since Pi approximates C(R)) we can find i such that Pi (R) : Pi

(R’) < R2:R’2 which contradicts 5)

Thus Pi(R) : Ri (R’) = R2:R’2

4.4 The area of a Parabolic Segment

[Archimedes (287 – 212 BCE)]
73
  Triangles
Δ1 , Δ2 , Δ3 , Δ4,…

 Note that
Δ2 + Δ3 = 1/4 Δ1

 Similarly
Δ4 + Δ5 + Δ6 + Δ7
= 1/16 Δ1
and so on

Thus A = Δ1 (1+1/4 + (1/4)2+…) = 4/3 Δ1

What is Calculus?
 Calculus appeared in 17th century as a system of shortcuts to results
obtained by the method of exhaustion

 Calculus derives rules for calculations

 Problems, solved by calculus include finding areas, volumes (integral

calculus), tangents, normals and curvatures (differential calculus) and
summing of infinite series

 This makes calculus applicable in a wide variety of areas inside and

outside mathematics

 In traditional approach (method of exhaustions) areas and volumes

were computed using subtle geometric arguments

 In calculus this was replaced by the set of rules for calculations

74
17th century calculus
 Differentiation and integration of powers of x (including fractional
powers) and implicit differentiation of polynomials in x and y

 Together with analytic geometry this made possible to find tangents,

maxima and minima of all algebraic curves p (x,y) = 0

 Newton’s calculus of infinite series (1660s) allowed for differentiation

and integration of all functions expressible as power series

 Culmination of 17th century calculus: discovery of the Fundamental

Theorem of Calculus by Newton and Leibniz (independently)

 Features of 17th century calculus:

 the concept of limit was not introduced yet

 use of “indivisibles” or “infinitesimals”

 strong opposition of some well-known philosophers of that time

(e.g. Thomas Hobbes)

 very often new results were conjectured by analogy with

previously discovered formulas and were not rigorously proved

Early Results on Areas and Volumes

 Area ≈ [(1/n)k + (2/n)k + … + (n/n)k](1/n)

 → sum 1k + 2k + … + nk

Volume of the solid of revolution:

75
area of cross-section is π r2
and therefore it is required to compute sum
12k + 22k + 32k +… + n2k
 First results: Greek mathematicians (method of exhaustion,
Archimedes)

 Arab mathematician al-Haytham (10th -11th centuries) summed the

series
1k + 2k + … + nk for k = 1, 2, 3, 4 and used the result to find the volume
of the solid obtained by rotating the parabola about its base

 Cavalieri (1635): up to k = 9 and conjectured the formula for positive

integers k

 Another advance made by Cavalieri was introduction of “indivisibles”

which considered areas divided into infinitely thin strips and volumes
divided into infinitely thin slices

 It was preceded by the work of Kepler on the volumes of solids of

revolution (“New Stereometry of wine barrels”, 1615)

 Fermat, Descartes and Roberval (1630s) proved the formula for

integration of xk (even for fractional values of k)

 Torricelly: the solid obtained by rotating y = 1 / x about the x-axis from

1 to infinity has finite volume!

 Thomas Hobbes (1672): “to understand this [result] for sense, it is not
required that a man should be a geometrician or logician, but that he
should be mad”

Maxima, Minima, and Tangents

 The idea of differentiation appeared later than that one of integration

 First result: construction of tangent line to spiral r = aθ by Archimedes

 No other results until works of Fermat (1629)

“modern” approach:
lim f ( x+ ΔxΔx)−f ( x )
Fermat’s approach Δx →0

(tangent to y = x2) ( x + E )2 −x 2 2 xE+ E 2

slope= = =2 x+ E
E E
E – “small” or “infinitesimal” element which is set equal to zero at the end of all
computations
Thus at all steps E ≠ 0 and at the end E = 0
Philosophers of that time did not like such approach

76
  Fermat’s method worked well with all polynomials p(x)

 Moreover, Fermat extended this approach to curves given by p(x,y) = 0

 Completely the latter problem was solved by Sluse (1655) and Hudde
(1657)

 The formula is equivalent m,n

to the use of p( x , y )= ∑ aij x i y j
implicit differentiation i , j=1
⇓
dy
=−
∑ iaij xi−1 y j
dx ∑ jaij x i y j−1
The “Arithmetica Infinitorum” of Wallis (1655)

 An attempt to arithmetize the theory of areas and volumes

 Wallis found that ∫01 xpdx = 1/(p+1) for positive integers p (which was
already known)

 Another achievement: formula for ∫01 xm/ndx

 Wallis calculated ∫01 x1/2dx, ∫01 x1/3dx,…, using geometric arguments, and
conjectured the general formula for fractional p

 Note: observing a pattern for p = 1,2,3, Wallis claimed a formula for all
positive p “by induction” and for fractional p “by interpolation” (lack of
rigour but a great deal of analogy, intuition and ingenuity)

 Wallis’ formula:
π 2 4 4 6 6
= ⋅ ⋅ ⋅ ⋅ ⋅⋯
4 3 3 5 5 7

 Expansion of π as infinite product was known to Viète (before Wallis’

discovery):

77
 2
π
π
4
π
8
π
16
1 1
=cos ⋅cos ⋅cos ⋅⋯= ⋅
2 2
1+
√ √ ( √ ) √ [ √ ( √ )]
1 1
⋅
2 2
1+
1
2
1+
1
2
⋅⋯

Nevertheless Wallis’ formula relates π to the integers through a sequence of

rational operations

Moreover, basing on the formula for π Wallis’ found a sequence of fractions

he called “hypergeometric”, which as it had been found later occur as

coefficients in series expansions of many functions (which led to the class of

hypergeometric functions)

Other formulas for π related to Wallis’ formula

Continued fraction 4 12
=1+
π 32
(Brouncker): 2+
52
2+
72
2+
2+⋯

Series expansion discovered by 15th century Indian mathematicians and

rediscovered by Newton, Gregory and Leibniz: 4 12
=1+
π 32
2+
52
2+
72
2+
2+⋯

Euler
−1 x3 x5 x7 π 1 1 1
tan x=x− + − +⋯ =1− + − +⋯
3 5 7
sub. x = 4 3 5 7
1

Newton’s Calculus of Series

 Isaac Newton

78
  Most important discoveries in 1665/6

 Before he studied the works of Descartes, Viète and Wallis

 Contributions to differential calculus (e.g. the chain rule)

 Most significant contributions are related to the theory of

infinite series

 Newton used term-by-term integration and differentiation to find

power series representation of many of classical functions, such
as tan-1x or log (x+1)

 Moreover, Newton developed a method of inverting infinite

power series to find inverses of functions (e.g ex from log (x+1))

 Unfortunately, Newton’s works were rejected for publication by Royal

Society and Cambridge University Press

The Calculus of Leibniz

 The first published paper on calculus was by

Gottfried Wilhelm Leibniz (1684)

 Leibniz discovered calculus independently

 He had better notations than Newton’s

 Leibniz was a librarian, a philosopher and a diplomat

 “Nova methodus” (1864)

 sum, product and quotient rules

 notation dy / dx

 dy / dx was understood by Leibniz literally as a quotient of infinitesimals

dy and dx

 dy and dx were viewed as increments of x and y

The Fundamental Theorem of Calculus

 In “De geometria” (1686) Leibniz introduced
the integral sign ∫

 Note that ∫ f(x) dx meant (for Leibniz) a sum of terms representing

infinitesimal areas of height f(x) and width dx

 If one applies the difference operator d to such sum it yields the last
term f(x) dx

79
  Dividing by dx we obtain x
d
the Fundamental Theorem of Caculus ∫ f (t )dt=f ( x )
dx a
 Leibniz introduced the word “function”

 He preferred “closed-form” expressions to infinite series

 Evaluation of integral ∫ f(x) dx was for Leibniz the problem of finding a

known function whose derivative is f(x)

 The search for closed forms led to

 the problem of factorization of polynomials and eventually to the

Fundamental Theorem of Algebra (integration of rational
functions)

 the theory of elliptic functions (attempts to integrate 1/√1-x4 )

Biographical Notes
Archimedes

 Was born and worked in Syracuse (Greek city in

Sicily) 287 BCE and died in 212 BCE

 Friend of King Hieron II

 “Eureka!” (discovery of hydrostatic law)

 Invented many mechanisms, some of which were used for the defence
of Syracuse

 Other achievements in mechanics usually attributed to Archimedes (the

law of the lever, center of mass, equilibrium, hydrostatic pressure)

 Used the method of exhaustions to show that the volume of sphere is

2/3 that of the enveloping cylinder

 According to a legend, his last words were “Stay away from my

diagram!”, address to a soldier who was about to kill him

John Wallis

Born: 23 Nov 1616 (Ashford, Kent, England)

Died: 28 Oct 1703 (Oxford, England)

80
  went to school in Ashford

 Wallis’ academic talent was recognized very early

 14 years old he was sent to Felsted, Essex to attend the school

 He became proficient in Latin, Greek and Hebrew

 Mathematics was not considered important in the best schools

 Wallis learned rules of arithmetic from his brother

 That time mathematics was not consider as a “pure” science in the

Western culture

 In 1632 he entered Emmanuel College in Cambridge

 bachelor of arts degree (topics studied included ethics, metaphysics,

geography, astronomy, medicine and anatomy)

 Wallis received his Master's Degree in 1640

 Between 1642 and 1644 he was chaplain at Hedingham, Essex and in

London

 Wallis became a fellow of Queens College, Cambridge

 He relinquished the fellowship when he married in 1645

 Wallis was interested in cryptography

 Civil War between the Royalists and Parliamentarians began in 1642

 Wallis used his skills in cryptography in decoding Royalist messages

for the Parliamentarians

 Since the appointment to the Savilian Chair in Geometry of Oxford in

1649 by Cromwell Wallis actively worked in mathematics

Sir Isaac Newton

Born: 4 Jan 1643 (Woolsthorpe, Lincolnshire, England)

Died: 31 March 1727 (London, England)

81
 A family of farmers

 Newton’s father (also Isaac Newton) was a wealthy but completely

illiterate man who even could not sign his own name

 He died three months before his son was born

 Young Newton was abandoned by his mother at the age of three and
was left in the care of his grandmother

 Newton’s childhood was not happy at all

 Newton entered Trinity College (Cambridge) in 1661

 Newton entered Trinity College (Cambridge) in 1661 to pursue a law

degree

 Despite the fact that his mother was a wealthy lady he entered as a
sizar

 He studied philosophy of Aristotle

 Newton was impressed by works of Descartes

 In his notes “Quaestiones quaedam philosophicae” 1664 (Certain

philosophical questions) Newton recorded his thoughts related to
mechanics, optics, and the physiology of vision

 The years 1664 – 66 were the most important in Newton’s

mathematical development

 By 1664 he became familiar with mathematical works of Descartes,

Viète and Wallis and began his own investigations

 He received his bachelor's degree in 1665

 When the University was closed in the summer of 1665 because of the
plague in England, Newton had to return to Lincolnshire

 At that time Newton completely devoted himself to mathematics

82
  Newton’s fundamental works on calculus “A treatise of the methods of
series and fluxions” (1671) (or “De methodis”) and “On analysis by
equations unlimited in their number of terms” (1669) (or “De analysis”)
were rejected for publication

 Nevertheless some people recognized his genius

 Isaac Barrow resigned the Lucasian Chair (Cambridge) in 1669 and

recommended that Newton be appointed in his place

 Newton's first work as Lucasian Prof. was on optics

 In particular, using a glass prism Newton discovered the spectrum of

white light

 1665: Newton discovered inverse square law of gravitation

 1687: “Philosophiae naturalis principia mathematica” (Mathematical

principles of natural philosophy)

 In this work, Newton developed mathematical foundation of the theory

of gravitation

 This book was published by Royal Society (with the strong support
from Edmund Halley)

 In 1693 Newton had a nervous breakdown

 In 1696 he left Cambridge and accepted a government position in

London where he became master of the Mint in 1699

 In 1703 he was elected president of the Royal Society and was re-
elected each year until his death

 Newton was knighted in 1705 by Queen Anne

Gottfried Wilhelm von Leibniz

Born: 1 July 1646 (Leipzig, Saxony (now Germany)

Died: 14 Nov 1716 (Hannover, Hanover (now Germany)

83
 An academic family

 From the age of six Leibniz was given free access to his father’s library

 At the age of seven he entered school in Leipzig

 In school he studied Latin

 Leibniz had taught himself Latin and Greek by the age of 12

 He also studied Aristotle's logic at school

 In 1661 Leibniz entered the University of Leipzig

 He studied philosophy and mathematics

 In 1663 he received a bachelor of law degree for a thesis “De Principio

Individui” (“On the Principle of the Individual”)

 The beginning of the concept of “monad”

 He continued work towards doctorate

 Leibniz received a doctorate degree from University of Altdorf (1666)

 During his visit to the University of Jena (1663) Leibniz learned a little
of Euclid

 Leibniz idea was to create some “universal logic calculus”

 After receiving his degree Leibniz commenced a legal career

 From 1672 to 1676 Leibniz developed his ideas related to calculus and
obtained the fundamental theorem

 Leibniz was interested in summation of infinite series by investigation

of the differences between successive terms

 He also used term-by term integration to discover series representation

of π
[ ][ ][ ][ ]
∞ ∞
∑ 1n( n+1) =∑ 1 1
−
1 1 1 1 1
= 1− + − + − +…
n=1 n=1 n n+1 2 2 3 3 4

[ 1 1
2 2 ][
1 1
3 3 ][ ]
1 1
¿ 1+ − + + − + + − + +…=1
4 4 84
Reference: MATH 1037 by Alex Karassev

Republic of the Philippines

LEYTE NORMAL UNIVERSITY
Graduate School
Tacloban City

85
 Math 536 – History of Mathematics

ALLAN M. LUBIANO NELSON D. BERNARDO

Student/Reporter Professor

************************************************************************************
The History of Math in the Nineteenth Century

Sub-Topics:

1. Theory of functions of a complex variable

2. Algebraic functions and their integrals
3. Automorphic functions
4. Differential equations
5. Groups
6. Infinite Aggregates
7. Functions of real variables. The Critical Movement
8. Theory of Numbers. Algebraic Bodies
9. Projective Geometry
10. Differential Geometry
11. Other Branches of Geometry
12. Non-Euclidean Geometry
************************************************************************************

The extraordinary development of mathematics in the last century is

quite unparalleled in the long history of this most ancient of sciences. Not only

have those branches of mathematics which were taken over from the

eighteenth century steadily grown but entirely new ones have sprung up in

almost bewildering profusion, and many of these have promptly assumed

proportions of vast extent. As it is obviously impossible to trace in the short

time allotted to me the history of mathematics in the nineteenth century, even

in merest outline, I shall restrict myself to the consideration of some of its

leading theories.

86
1. Theory of Functions of a Complex Variable

Augustin Louis Cauchy

Born: 21 August 1789 in Paris, France

Died: 23 May 1857 in Sceaux (near Paris), France

Without doubt one of the most characteristic features of mathematics in

the last century is the systematic and universal use of the complex variable.

Most of the great mathematical theories received invaluable aid from it, and

many owe to it their very existence. What would the theory of differential

equations or elliptic functions be to-day without it, and is it probable that

Poncelet, Steiner, Chasles, and von Staudt would have developed synthetic

geometry with such elegance and perfection without its powerful stimulus?

Without doubt one of the most characteristic features of mathematics in the

last century is the systematic and universal use of the complex variable.

The theory of functions of a complex variable may be said to have had

its birth when Cauchy discovered his integral theorem published in 1825.:
❑

∫ f ( x ) dx=0
c

The history of functions as first developed is largely a theory of

algebraic functions and their integrals. A general theory of functions is only

slowly evolved. For a long time the methods of Cauchy, Riemann, and

Weierstrass were cultivated along distinct lines by their respective pupils. The

schools of Cauchy and Riemann were the first to coalesce. The entire rigor

87
which has recently been imparted to their methods has removed all reason for

founding, as Weierstrass and his school have urged, the theory of functions

on a single algorithm, viz., the power series. We may therefore say that at the

close of the century there is only one theory of functions, in which the ideas of

its three great creators are harmoniously united.

2. Algebraic Functions and their Integrals

A branch of the theory of functions has been developed to such an

extent that it may be regarded as an independent theory, we mean the theory

of algebraic functions and their integrals. The brilliant discoveries of Abel and

Jacobi in the elliptic functions from 1824 and 1829 prepared the way for a

similar treatment of the hyperelliptic case. Here a difficulty of gravest nature

was met. The corresponding integrals have 2p linearly independent periods;

but, as Jacobi had shown, a one-valued function having more than two

periods admits a period as small as we choose. It therefore looked as if the

elliptic functions admitted no further generalization. Guided by Abel’s theorem,

Jacobi at last discovered the solution to the difficulty, 1832; to get functions

analogous to the elliptic functions we must consider functions not of one but of

p independent variables, viz., the p independent integrals of the first species.

The great problem now before mathematicians, known as Jacobi’s problem of

inversion, was to extend this aper¸cu to the case of any algebraic

configuration and develop the consequences. The first to take up this

immense task were Weierstrass and Riemann, whose results belong to the

most brilliant achievements of the century. Among the important notions

88
hereby introduced we note the following: the birational transformation, rank of

an algebraic configuration, class invariants, prime functions, the theta and

multiply periodic functions in several variables. Of great importance is

Riemann’s method of proving existence theorems as also his representation

of algebraic functions by means of integrals of the second species. A new

direction was given to research in this field by Clebsch, who considered the

fundamental algebraic configuration as defining a curve. His aim was to bring

about a union of Riemann’s ideas and the theory of algebraic curves for their

mutual benefit. Clebsch’s labors were continued by Brill and Noether; in their

work the transcendental methods of Riemann are placed quite in the

background. More recently Klein and his school have sought to unite the

transcendental methods of Riemann with the geometric direction begun by

Clebsch, making systematic use of homogeneous co¨ordinates and the

invariant theory. Noteworthy also is Klein’s use of normal curves in (p − 1)-

way space to represent the given algebraic configuration. Dedekind and

Weber, Hensel and Landsberg have made use of the ideal theory with marked

success. Many of the difficulties of the older theory, for example the resolution

of singularities of the algebraic configuration, are treated with a truly

remarkable ease and generality. In the theory of multiply periodic functions

and the general θ functions we mention, besides those of Weierstrass, the

researches of Prym, Krazer, Frobenius, Poincar´e, and Wirtinger.

3. Automorphic Functions

89
 Closely connected with the elliptic functions is a class of functions

which has come into great prominence in the last quarter of a century, viz.:

the elliptic modular and automorphic functions. Let us consider first the

modular functions of which the modulus κ and the absolute invariant J are the

simplest types. The transformation theory of Jacobi gave algebraic relations

between such functions in endless number. Hermite, Fuchs, Dedekind and

Schwarz are forerunners, but the theory of modular functions as it stands to-

day is principally due to Klein and his school. Its goal is briefly stated thus: To

determine all subgroups of the linear

x’=
group

αx+ β
γx+ δ '

where α, β, γ, δ are integers and αδ − βγ 6= 0; to determine for each such

group associate modular functions and to investigate their relation to one

another and especially to J. Important features in this theory are the

congruence groups of (1); the fundamental polygon belonging to a given

subgroup, and its use as substitute for a Riemann surface; the principle of

reflection on a circle; the modular forms. The theory of automorphic functions

is due to Klein and Poincar´e. It is a generalization of the modular functions;

the coefficients in (1) being any real or imaginary numbers, with non-vanishing

determinant, such that the group is discontinuous. Both authors have

recourse to non-euclidean geometry to interpret the substitutions (1). Their

manner of showing the existence of functions belonging to a given group is

quite different. Poincar´e by a brilliant stroke of genius actually writes down

their arithmetic expressions in terms of his celebrated θ series. Klein employs

the existence methods of Riemann. The relation of automorphic functions to

90
differential equations is studied by Poincar´e in detail. In particular, he shows

that both variables of a linear differential equation with algebraic co¨efficients

can be expressed uniformly by their means.

4. Differential Equations

Lazarus Immanuel Fuchs

Born: 5 May 1833 in Moschin (near Posen), Prussia (now

Poznań, Poland)
Died: 26 April 1902 in Berlin, Germany

Let us turn now to another great field of mathematical activity, the

theory of differential equations. The introduction of the theory of functions has

completely revolutionized this subject. At the beginning of the nineteenth

century many important results had indeed been established, particularly by

Euler and Lagrange; but the methods employed were artificial, and broad

comprehensive principles were lacking. By various devices one tried to

express the solution in terms of the elementary functions and quadratures—a

vain attempt; for, as we know now, the goal they strove so laboriously to

reach was in general unattainable.

A new epoch began with Cauchy, who by means of his new theory of

functions first rigorously established the existence of the solution of certain

91
classes of equations in the vicinity of regular points. He also showed that

many of the properties of the elliptic functions might be deduced directly from

their differential equations.

One of the first to open up this new path was Fuchs, whose classic

memoirs (1866–68) gave the theory of linear differential equations its birth.

These equations enjoy a property which renders them particularly accessible,

viz., the absence of movable singular points. They may, however, possess

points of indetermination, to use Fuchs’s terminology, and little progress has

been made in this case. Noteworthy in this connection is the introduction by

von Koch of infinite determinants, whose importance was first shown by our

distinguished countryman, Hill; also the use of divergent series—that

invention of the devil, as Abel called them—by Poincar´e. A particular class of

linear differential equations of great importance is the hypergeometric

equation; the results obtained by Gauss, Kummer, Riemann, and Schwarz

relating to this equation have had the greatest influence on the development

of the general theory. The great extent of the theory of linear differential

equations may be estimated when we recall that within its borders it embraces

not only almost all the elementary functions, but also the modular and

automorphic functions. Too important to pass over in silence is the subject of

algebraic differential equations with uniform solutions. The brilliant researches

of Painlev´e deserve especial mention.

 Fuchs worked on differential equations and the theory of functions

 In 1865 Fuchs studied nth order linear ordinary differential equations

with complex functions as coefficients

92
  It was Fuchs' work on this inverse function which led Poincaré to

introduce what he called a Fuchsian group, and use this as a

fundamental concept in the development of the theory of automorphic

functions

 Fuchs also investigated how to find the matrix connecting two systems

of solutions of differential equations near two different points

Lazarus Fuchs' publications

1858. L Fuchs, De superficierum lineis curvaturae : dissertatio

inauguralis (Unger, Berlin, 1858).

1861. E B Christoffel and L Fuchs, Integration der partiellen

Differentialgleichung ..., J. Reine Angew. Math. 58 (1861), 80-89.

1863. L Fuchs, Über die Pieroden, welche aus den Wurzeln der Gleichung
wn= 1 gebildet sind, wenn n eine zusammengesetzte Zahl ist, J. Reine
Angew. Math. 61 (1863), 374-386.

1865. L Fuchs, Zur Theorie der linearen Differentialgleichungen mit

veränderlichen Coefficienten, Jahresbericht über die städtische
Gewerbeschule zu Berlin (Ostern 1865).

1866. L Fuchs, Über die aus Einheitswurzeln gebildeten complexen Zahlen

von periodischem Verhalten, insbesondere die Bestimmung der
Klassenanzahl derselben, J. Reine Angew. Math. 65 (1866), 74-111.

1866. L Fuchs, Zur Theorie der linearen Differentialgleichungen mit

veränderlichen Coefficienten, J. Reine Angew. Math. 66 (1866), 121-
160.

1868. L Fuchs, Zur Theorie der linearen Differentialgleichungen mit

veränderlichen Coefficienten (Ergänzungen zu der im 66 Bande dieses
Journals enthaltenen Abhandlung), J. Reine Angew. Math. 68(1868),
354- 385.

1870. L Fuchs, Über eine rationale Verbindung der Periodicitätsmoduln der

hyperelliptischen Integrale, J. Reine Angew. Math. 71 (1870), 128-136.

93
1870. L Fuchs, Sur le développement en séries des intégrales des équations
différentielles linéaires, Annali di matematica (2) 4 (1870), 36-49.

1870. L Fuchs, Die Periodicitätsmoduln der hyderelliptischen Integrale als

Funktionen eines Parameters aufgefasst, J. Reine Angew.
Math. 71 (1870), 91-127.

1870. L Fuchs, Bemerkungen zu der Abhandlung: "Über hypergeometrische

Funktionen nterOrdnung", in diesem Journal 71, 316, J. Reine Angew.
Math. 72 (1870), 255-262.

1871. L Fuchs, Über die Form der Argumente der Thetafunction und über die
Bestimmung von ¶(0, 0, ... 0) als Function der Klassenmoduln, J.
Reine Angew. Math. 73 (1871), 305-324.

1871. L Fuchs, Über die linearen Differentialgleichungen, welchen die

Periodictätsmoduln der Abel'schen Integrale genügen, und über
verschiedene Arten von Differentialgleichungen für ¶(0, 0, ... 0), J.
Reine Angew. Math. 73 (1871), 324-340.

1872. L Fuchs, Über die Darstellung der Funktionen complexer Variablen,

insbesondere der Integrale linearer Differentialgleichungen, J. Reine
Angew. Math. 75 (1872), 177-223.

1873. L Fuchs, Über die Darstellung der Funktionen complexer Variablen,

insbesondere der Integrale linearer Differentialgleichungen, J. Reine
Angew. Math. 76 (1873), 175-176.

1873. L Fuchs, Über Relationen, welche für die zwischen je zwei singulären
Punkten erstreckten Integrale der Lösungen linearer
Differentialgleichungen stattfinden, J. Reine Angew. Math. 76 (1873),
177- 214.

1873. L Fuchs, Über die Abbildung durch algebraische Functionen, J. Reine

Angew. Math. 77 (1873), 339-352.

1874. L Fuchs, Über die Abbildung durch algebraische Functionen, J. Reine

Angew. Math. 78 (1874), 338-339.

1875. L Fuchs, Über die linearen Differentialgleichungen zweiter Ordnung,

welche algebraische Integrale besitzen, und eine neue Anwendung der
Invariantentheorie, Göttingen Nachrichten 1875 (1875), 568-581; 612-
613.

1875. L Fuchs, Über die linearen Differentialgleichungen zweiter Ordnung,

welche algebraische Integrale besitzen, und eine neue Anwendung der
Invariantentheorie, J. Reine Angew. Math. 81 (1875), 97-142.

1876. L Fuchs, Extrait d'une lettre adressée à M. Hermite, Journal de

Mathématiques (3) 2 (1876), 158-160.

94
1876. L Fuchs, Sur les équations linéaires du second ordre, Comptes rendus
de l'Académie des Sciences 82 (1876), 1431-1437;

1876. L Fuchs, Sur les équations linéaires du second ordre, Comptes rendus
de l'Académie des Sciences 83 (1876), 46-47.

1877. L Fuchs, Sur quelques propriétés des intégrales des équations

différentielles auxquelles satisfont les modules de périodicité des
intégrales elliptiques des deux premières espèces, J. Reine Angew.
Math.83 (1877), 13-38.

1878. L Fuchs, Extrait d'une lettre adressée à M. Hermite, Comptes rendus

de l'Académie des Sciences 85 (1878), 947-950.

1878. L Fuchs, Sur les équations différentialles linéaires, qui admettent des
intégrales dont les différentielles logarithmiques sont des fonctions
doublement périodiques, Journal de Mathématiques (3) 4 (1878), 125-
141.

1878. L Fuchs, Über die linearen Diferentialgleichungen zweiter Ordnung,

welche algebraische Integrale bestizen, J. Reine Angew.
Math. 85 (1878), 1-26.

1878. L Fuchs, Über eine Klasse von Differentialgleichungen, welche durch

Abel'sche oder elliptische Functionen integrirbar sind, Göttingen
Nachrichten 1878 (1878), 19-33.

1878. L Fuchs, Über eine Klasse von Differentialgleichungen, welche durch

Abel'sche oder elliptische Functionen integrirbar sind, Annali di
matematica (2) 9 (1878), 25-35.

1880. L Fuchs, Über eine Classe von Functionen mehrerer Variablen, welche
durch Umkehrung der Integrale von Lösungen der linearen
Differentialgleichungen mit rationalen Coefficienten entstehen, J. Reine
Angew. Math. 89 (1880), 150-169.

1880. L Fuchs, Auszug aus einem Schreiben an Herrn Borchardt, J. Reine

Angew. Math. 90 (1880), 71-73.

1880. L Fuchs, Sur les fonctions provenant de l'inversion des intérales des
solutions des eequations différentielles lineeaires, Bulletin des
Sciences mathématique et astronomiques (2) 4 (1880), 328-336.

1880. L Fuchs, Über eine Classe von Functionen mehrerer Variablen, welche
durch Umkehrung der Integrale von Lösungen der linearen
Differentialgleichungen mit rationalen Coefficienten
entstehen, Göttingen Nachrichten 1880 (1880), 170-176.

1880. L Fuchs, Über eine Classe von Functionen mehrerer Variablen, welche
durch Umkehrung der Integrale von Lösungen der linearen

95
 Differentialgleichungen mit rationalen Coefficienten entstehen, J. Reine
Angew. Math. 89 (1880), 151-169.

1880. L Fuchs, Über eine Classe von Functionen mehrerer Variablen, welche
durch Umkehrung der Integrale von Lösungen der linearen
Differentialgleichungen mit rationalen Coefficienten
entstehen, Comptes rendus de l'Académie des Sciences 90 (1880),
678-680; 735-736.

1880. L Fuchs, Über die Functionen, welche durch Umkehrung der Integrale
von Lösungen der linearen Differentialgleichungen
entstehen, Göttingen Nachrichten 1880 (1880), 445-453,

1880. L Fuchs, Über die Functionen, welche durch Umkehrung der Integrale
von Lösungen der linearen Differentialgleichungen
entstehen, Bulletin des Sciences Mathématique (2) 4 (1880), 328-336.

1881. L Fuchs, Über Functionen zweier Variabeln, welche durch Umkehrung

der Integrale zweier gegebenen Functionen entstehen, Göttingen
Abhandlungen 27 (1881).

1881. L Fuchs, Sur les fonctions de deux variables, qui naissent de l'inversion
des intégrales de deux fonctions données, Comptes rendus de
l'Académie des Sciences 91 (1881), 1330-1332; 1401-1404.

1881. L Fuchs, Sur les fonctions de deux variables, qui naissent de l'inversion
des intégrales de deux fonctions données, Bulletin des Sciences
Mathématique (2) 5 (1881), 52-88.

1882. L Fuchs, Sur une équation différentielle de la forme f(u, du/dz) =

0, Comptes rendus de l'Académie des Sciences 93 (1882), 1063-1065.

1882. L Fuchs, Über lineare homogene Differentialgleichungen, zwischen

deren Ingegralen homogenen Relationen höheren als ersten Grades
bestehen, Sitzungsberichte der Königlich preussische Akademie der
Wissenschaften zu Berlin 1882 (1882), 703-710.

1882. L Fuchs, Über Functionen, welche durch lineare Substitutionen

unverändert bleiben, Göttingen Nachrichten 1882 (1882), 81-84.

1883. L Fuchs, Über Functionen einer beliebigen Anzahl unabhängiger

Variablen, die durch Umkehrung von Integralen
entstehen, Sitzungsberichte der Königlich preussische Akademie der
Wissenschaften zu Berlin 1883 (1883), 507-516.

1883. L Fuchs, Über lineare homogene Differentialgleichungen, zwischen

deren Integralen homogene Relationen höheren als ersten Grades
bestehen, Acta Mathematica 1 (1883), 321-362.

96
1884. Ch Hermite and L Fuchs, Sur un développement en fraction
continue, Acta Mathematica 4 (1884), 89-92.

1884. L Fuchs, Über eine Form, in welche sich das allgemeine Integral einer
Differentialgleichung erster Ordnung bringen lässt, wenn dasselbe
algebraisch ist, Sitzungsberichte der Königlich preussische Akademie
der Wissenschaften zu Berlin 1884 (1884), 1171-1177.

1884. L Fuchs, Über Differentialgleichungen, deren Integrale feste

Verzweigungspunkte besitzen, Sitzungsberichte der Königlich
preussische Akademie der Wissenschaften zu Berlin 1884 (1884),
699-710.

1885. L Fuchs, Über den Charakter der Integrale von Differentialgleichungen

zwischen complexen Variabeln, Sitzungsberichte der Königlich
preussische Akademie der Wissenschaften zu Berlin 1885 (1885), 5-
12.

1886. L Fuchs, Über diejenigen algebraischen Gebilde, welche eine Involution

zulassen, Sitzungsberichte der Königlich preussische Akademie der
Wissenschaften zu Berlin 1886 (1886), 797-804.

1886. L Fuchs, Über die Werte, welche die Integrale einer

Differentialgleichung erster Ordnung in singulären Punkten
annehmen können, Sitzungsberichte der Königlich preussische
Akademie der Wissenschaften zu Berlin 1886 (1886), 279-302.

1886. L Fuchs, Über eine Klasse linearer Differentialgleichungen zweiter

Ordnung, J. Reine Angew. Math. 100 (1886) 189-200.

1887. L Fuchs, Über einen Satz aus der Theorie der algebraischen
Functionen, und über eine Anwendung desselben auf die
Differentialgleichungen zweiter Ordnung, Sitzungsberichte der
Königlich preussische Akademie der Wissenschaften zu
Berlin 1887 (1887), 159-166.

1887. L Fuchs, Über Relationen zwischen den Integralen von

Differentialgleichungen, Sitzungsberichte der Königlich preussische
Akademie der Wissenschaften zu Berlin 1887 (1887), 1077-1094.

1887. L Fuchs, Über die Umkehrung von Functionen zweier

Veränderlichen, Sitzungsberichte der Königlich preussische Akademie
der Wissenschaften zu Berlin 1887 (1887), 99-108.

1887. L Fuchs, Bemerkungen zu einer Note des Herrn Hurwitz, Göttingen

Nachrichten 1887 (1887), 502-504.

1888. L Fuchs, Zur Theorie der linearen

Differentialgleichungen, Sitzungsberichte der Königlich preussische

97
 Akademie der Wissenschaften zu Berlin 1888 (1888), 1115-1126;
1273- 1290.

1889. L Fuchs, Zur Theorie der linearen Differentialgleichungen

(Fortsetzung), Sitzungsberichte der Königlich preussische Akademie
der Wissenschaften zu Berlin 1889 (1889), 713-726.

1890. L Fuchs, Bemerkung zu der Arbeit im Bande 75 Seite 177 dieses

Journals, J. Reine Angew. Math. 106 (1890), 1-4.

1890. L Fuchs, Bemerkung zu vorstehender Abhandlung des Herrn Heffter

zur Theorie der linearen Differentialgleichungen, J. Reine Angew.
Math. 106 (1890), 283-284.

1890. L Fuchs, Über algebraisch integrirbare lineare

Differentialgleichungen, Sitzungsberichte der Königlich preussische
Akademie der Wissenschaften zu Berlin 1890 (1890), 469-483.

1890. L Fuchs, Zur Theorie der linearen Differentialgleichungen.

(Schluss.), Sitzungsberichte der Königlich preussische Akademie der
Wissenschaften zu Berlin 1890 (1890), 21-38.

1891. L Fuchs, Über eine Abbildung durch eine rationale Function, J. Reine
Angew. Math. 108 (1891), 181-192.

1892. L Fuchs, Über die Relationen, welche die zwischen je zwei singulären
Punkten erstreckten Integrale der Lösungen linearer
Differentialgleichungen mit den Coefficienten der
Fundamentalsubstitutionen der Gruppe derselben
verbinden, Sitzungsberichte der Königlich preussische Akademie der
Wissenschaften zu Berlin 1892 (1892), 1113-1128.

1892. L Fuchs, Über lineare Differentialgleichungen, welche von Parametern

unabhängige Substitutionsgruppen besitzen, Sitzungsberichte der
Königlich preussische Akademie der Wissenschaften zu
Berlin1892 (1892), 157-176.

1893. L Fuchs, Note zu der im Bande 83 S. 13 fg. dieses Journals

enthaltenen Arbeit: Sur quelques propriétés etc; extrait d'une lettre
adressée à M. Hermite, J. Reine Angew. Math. 112 (1893), 156-
164.

1893. L Fuchs, Über lineare Differentialgleichungen, welche von Parametern

unabhängige Substitutionsgruppen besitzen, Sitzungsberichte der
Königlich preussische Akademie der Wissenschaften zu
Berlin1893 (1893), 975-988.

1894. L Fuchs, Über lineare Differentialgleichungen, welche von Parametern

unabhängige Substitutionsgruppen besitzen, Sitzungsberichte der

98
 Königlich preussische Akademie der Wissenschaften zu
Berlin1894 (1894), 1117-1127.

1895. L Fuchs, Remarques sur une note de M. Paul Vernier, J. Reine Angew.
Math. 114 (1895), 231-232.

1895. L Fuchs, Hermann von Helmholtz, J. Reine Angew. Math. 114 (1895),
353.

1895. L Fuchs, Nachruf für Arthur Cayley, J. Reine Angew. Math. 115 (1895),
349-350.

1895. L Fuchs, Nachruf für Josef Dienger, J. Reine Angew. Math. 115 (1895),
350.

1895. L Fuchs, Nachruf für Ludwig Schläfli, J. Reine Angew.

Math. 115 (1895), 350.

1895. L Fuchs, Über die Abhängigkeit der Lösungen einer linearen

Differentialgleichung von den in den Coefficienten auftretenden
Parametern, Sitzungsberichte der Königlich preussische Akademie der
Wissenschaften zu Berlin 1895 (1895), 905-920.

1896. L Fuchs, Remarques sur une note de M. Alfred Loewy, intitulée: "Sur
les formes quadratiques définies à indéterminées conjuguées de M.
Hermite", Comptes rendus de l'Académie des Sciences 123(1896),
289- 290.

1896. L Fuchs, Über eine Klasse linearer homogener

Differentialgleichungen, Sitzungsberichte der Königlich preussische
Akademie der Wissenschaften zu Berlin 1896 (1896), 753-769.

1897. L Fuchs, Karl Weierstrass, J. Reine Angew. Math. 117 (1897), 357.

1897. L Fuchs, Bemerkung zur vorstehenden Mitteilung des Herrn

Hamburger, J. Reine Angew. Math. 118 (1897), 354-355.

1897. L Fuchs, Zur Theorie der Abel'schen Functionen, Sitzungsberichte der

Königlich preussische Akademie der Wissenschaften zu
Berlin 1897 (1897), 608-621.

1898. L Fuchs, Ernst Christian Julius Schering, J. Reine Angew.

Math. 119 (1898), 86.

1898. L Fuchs, Francesco Brioschi, J. Reine Angew. Math. 119 (1898), 259.

1898. L Fuchs, Zur Theorie der Abel'schen Functionen, Sitzungsberichte der

Königlich preussische Akademie der Wissenschaften zu
Berlin 1898 (1898), 477-486.

99
1898. L Fuchs, Zur Theorie der simultanen linearen partiellen
Differentialgleichungen, Sitzungsberichte der Königlich preussische
Akademie der Wissenschaften zu Berlin 1898 (1898), 222-233.

1899. L Fuchs, Bemerkungen zur Theorie der associirten

Differentialgleichungen, Sitzungsberichte der Königlich preussische
Akademie der Wissenschaften zu Berlin 1899 (1899), 182-195.

1899. L Fuchs, Über das Verhältnis der exacten Naturwissenschaft zur

Praxis. Rede bei Antritt des Rectorates an der Universität Berlin,
geh. am 15. October 1899 (G. Schade (O. Francke), Berlin, 1899).

1900. L Fuchs, Über eine besondere Gattung von rationalen Curven mit
imaginären Doppelpunkten, Sitzungsberichte der Königlich
preussische Akademie der Wissenschaften zu Berlin 1900 (1900), 74-78.

1901. L Fuchs, Charles Hermite, J. Reine Angew. Math. 123 (1901), 174.

1901. L Fuchs, Zur Theorie der linearen

Differentialgleichungen, Sitzungsberichte der Königlich preussische
Akademie der Wissenschaften zu Berlin 1901 (1901), 34-48.

1902. L Fuchs, Über Grenzen, innerhalb deren gewisse bestimmte Integrale

vorgeschriebene Vorzeichen behalten, J. Reine Angew.
Math. 124 (1902), 278-291.

1902. L Fuchs, Über Grenzen, innerhalb deren gewisse bestimmte Integrale

vorgeschriebene Vorzeichen behalten, Sitzungsberichte der Königlich
preussische Akademie der Wissenschaften zu Berlin 1902(1902), 4-10.

1902. L Fuchs, Über zwei nachgelassene Arbeiten Abels und die sich daran
anschliessenden Untersuchungen in der Theorie der linearen
Differentialgleichungen, Acta Mathematica 26 (1902), 319-332.

1904. L Fuchs, Gesammelte mathematische Werke. Herausgegeben von R.

Fuchs und L. Schlesinger Erster Band: Abhandlungen (1858-1875)
redigiert von L. Schlesinger. Mit einem Bildnisse des Verfassers
(Mayer & Müller, Berlin, 1904).

1906. L Fuchs, Gesammelte mathematische Werke. Herausgegeben von

Richard Fuchs in Berlin und Ludwig Schlesinger in Klausenburg.
Zweiter Band: Abhandlungen (1875-1887) redigiert von L. Schlesinger
(Mayer & Müller, Berlin, 1906).

1909. L Fuchs, Gesammelte mathematische Werke. Herausgegeben von

Richard Fuchs und Ludwig Schlesinger. Dritter Band: Abhandlungen
(1888-1902) und Reden. Redigiert von Richard Fuchs (Mayer & Müller,
Berlin, 1909).

100
5. Groups

Évariste Galois

Born: 25 October 1811 in Bourg La Reine (near Paris),

France
Died: 31 May 1832 in Paris, France

We turn now to the second dominant idea of the century, the group

concept. Groups first became objects of study in algebra when Lagrange

1770, Ruffini 1799, and Abel 1826 employed substitution groups with great

advantage to their work on the quintic. The enormous importance of groups in

algebra was, however, first made clear by Galois, whose theory of the solution

of algebraic equations is one of the great achievements of the century. Its

influence has stretched far beyond the narrow bounds of algebra. With an

101
arbitrary but fixed domain of rationality, Galois observed that every algebraic

equation has attached to it a certain group of substitutions. The nature of the

auxiliary equations required to solve the given equation is completely revealed

by an inspection of this group. Galois’s theory showed the importance of

determining the sub-groups of a given substitution group, and this problem

was studied by Cauchy, Serret, Mathieu, Kirkman and others. At the tender

age of 17, he began making fundamental discoveries in the theory of

polynomial equations (equations constructed from variables and constants,

using only the operations of addition, subtraction, multiplication and non-

negative whole-number exponents, such as x2 - 4x + 7 = 0). The publication of

Jordan’s great treatise in 1870 is a noteworthy event. It collects and unifies

the results of his predecessors and contains an immense amount of new

matter. A new direction was given to the theory of groups by the introduction

by Cayley of abstract groups (1854, 1878). The work of Sylow, H¨older,

Frobenius, Burnside, Cole, and Miller deserves especial notice. Another line

of researches relates to the determination of the finite groups in the linear

group of any number of variables. These groups are important in the theory of

linear differential equations with algebraic solutions; in the study of certain

geometrical problems, as the points of inflection of a cubic, the 27 lines on a

surface of the third order; in crystallography, etc. They also enter prominently

in Klein’s Formenproblem. An especially important class of finite linear groups

are the congruence groups first considered by Galois. Among the laborers in

the field of linear groups we note Jordan, Klein, Moore, Maschke, Dickson,

Frobenius, and Wiman.

102
6. Infinite Aggregates

Georg Ferdinand Ludwig Philipp Cantor

Born: 3 March 1845 in St Petersburg, Russia

Died: 6 January 1918 in Halle, Germany

Leaving the subject of groups, we consider now briefly another

fundamental concept, viz., infinite aggregates. In the most diverse

mathematical investigations we are confronted with such aggregates. In

geometry the conceptions of a curve, surface, region, frontier, etc., when

examined carefully, lead us to a rich variety of aggregates. In analysis they

also appear, for example the domain of definition of an analytical function, the

points where a function of a real variable ceases to be continuous or to have a

differential coefficient, the points where a series of functions ceases to be

uniformly convergent, etc.

To say that an aggregate (not necessarily a point aggregate) is infinite

is often an important step; but often again only the first step. To penetrate

farther into the problem may require us to state how infinite. This requires us

to make distinctions in infinite aggregates, to discover fruitful principles of

classification, and to investigate the properties of such classes. The honor of

having done this belongs to Georg Cantor. The theory of aggregates is for the

most part his creation; it has enriched mathematical science with fundamental

103
and far reaching notions and results. The theory falls into two parts; a theory

of aggregates in general, and a theory of point aggregates. In the theory of

point aggregates the notion of limiting points gives rise to important classes of

aggregates, as discrete, dense, complete, perfect, connected, etc., which are

so important in the function theory. In the general theory two notions are

especially important, viz.: the one-to-one correspondence of the elements of

two aggregates, and well ordered aggregates. The first leads to cardinal

numbers and the idea of enumerable aggregates, the second to transfinite or

ordinal numbers. Two striking results of Cantor’s theory may be mentioned:

the algebraic and therefore the rational numbers, although everywhere dense,

are enumerable; and secondly, one-way and n-way space have the same

cardinal number. Cantor’s theory has already found many applications,

especially in the function theory, where it is today an indispensable instrument

of research.

7. Functions of Real Variables. The Critical Movement

Karl Theodor Wilhelm Weierstrass

Born: 31 October 1815 in Ostenfelde, Westphalia

(now Germany)
Died: 19 February 1897 in Berlin, Germany

104
Weierstrass gave definitions of continuity, limit and derivative of function ,

which are still used today.

This allowed him to demonstrate a set of theorems that were then

without demonstrating as the mean value theorem , the theorem of Bolzano-

Weierstrass and Heine-Borel Theorem .

He also made contributions in series convergence, in theory of periodic

functions, elliptic functions , convergence of infinite products, calculation of

variations , complex analysis , etc.

One of the most conspicuous and distinctive features of mathematical

thought in the nineteenth century is its critical spirit. Beginning with the

calculus, it soon permeates all analysis, and toward the close of the century it

overhauls and recasts the foundations of geometry and aspires to further

conquests in mechanics and in the immense domains of mathematical

physics. Ushered in with Lagrange and Gauss just at the close of the

eighteenth century, the critical movement receives its first decisive impulse

from the teachings of Cauchy, who in particular introduces our modern

definition of limit and makes it the foundation of the calculus. We must also

mention in this connection Abel, Bolzano, and Dirichlet. Especially Abel

adopted the reform ideas of Cauchy with enthusiasm and made important

contributions in infinite series. The figure, however, which towers above all

others in this movement, whose name has become a synonym of rigor, is

Weierstrass. Beginning at the very foundations, he creates an arithmetic of

real and complex numbers, assuming the theory of positive integers to be

given. The necessity of this is manifest when we recall that until then the

105
simplest properties of radicals and logarithms were utterly devoid of a rigorous

foundation; so for example √ 2 √ 5 = √ 10, log 2 + log 5 = log 10.

Characteristic of the pre-Weierstrassian era is the loose way in which

geometrical and other intuitional ideas were employed in the demonstration of

analytic theorems. Even Gauss is open to this criticism. The mathematical

world received a great shock when Weierstrass showed them an example of a

continuous function without a derivative, and Hankel and Cantor by means of

their principle of condensation of singularities could construct analytical

expressions for functions having in any interval, however small, an infinity of

points of oscillation, an infinity of points in which the differential coefficient is

altogether indeterminate, or an infinity of points of discontinuity. Another rude

surprise was Cantor’s discovery of the one-to-one correspondence between

the points of a unit segment and a unit square, followed up by Peano’s

example of a space filling curve.

8. Theory of Numbers. Algebraic Bodies

The theory of numbers as left by Fermat, Euler and Legendre was for

the most part concerned with the solution of diophantine equations, i. e., given

an equation f(x, y, z, . . .) = 0, whose coefficients are integers, to find all

rational, and especially all integral solutions. In this problem Lagrange had

shown the importance of considering the theory of forms. A new era begins

with the appearance of Gauss’s Disquisitiones Arithmeticae in 1801. This

great work is remarkable for three things; 1) The notion of divisibility in the

form of congruences is shown to be an instrument of wonderful power; 2) the

diophantine problem is thrown in the background and the theory of forms is

106
given a dominant role; 3) the introduction of algebraic numbers, viz., the roots

of unity.

The theory of forms has been further developed along the lines of the

Disquisitiones by Dirichlet, Eisenstein, Hermite, H. J. S. Smith, and

Minkowski. Another part of the theory of numbers also goes back to Gauss,

viz., algebraic numerical bodies. The law of reciprocity of quadratic residues,

one of the gems of the higher arithmetic, was first rigorously proved by Gauss.

His attempts to extend this theorem to cubic and biquadratic residues showed

that the elegant simplicity which prevailed in quadratic residues was

altogether missing in these higher residues until one passed from the domain

of real integers to the domain formed of the third and fourth roots of unity. In

these domains, as Gauss remarked, algebraic integers have essentially the

same properties as ordinary integers. Further exploration in this new and

promising field by Jacobi, Eisenstein, and others soon brought to light the fact

that already in the domain formed of the 23d roots of unity the laws of

divisibility were altogether different from those of ordinary integers; in

particular a number could be expressed as the product of prime factors in

more than one way. Further progress in this direction was therefore

apparently impossible.

It is Kummer’s immortal achievement to have made further progress

possible by the invention of his ideals. These he applied to Fermat’s

celebrated last theorem and the law of reciprocity of higher residues. The next

step in this direction was taken by Dedekind and Kronecker, who developed

the ideal theory for any algebraic domain. So arose the theory of algebraic

107
numerical bodies which has come into such prominence in the last decades of

the century through the researches of Hensel, Hurwitz, Minkowski, Weber,

and above all Hilbert.

9. Projective Geometry

Jean Victor Poncelet

Born: 1 July 1788 in Metz, Lorraine, France

Died: 22 December 1867 in Paris, France

The tendencies of the eighteenth century were predominantly analytic.

Mathematicians were absorbed for the most part in developing the wonderful

instrument of the calculus with its countless applications. Geometry made

relatively little progress. A new era begins with Monge. His numerous and

108
valuable contributions to analytic, descriptive, and differential geometry and

especially his brilliant and inspiring lectures at the Ecole polytechnique (1795–

1809) put fresh life in geometry and prepared it for a new and glorious

development in the nineteenth century.

When one passes in review the great achievements which have made

the nineteenth century memorable in the annals of our science, certainly

projective geometry will occupy a foremost place. Pascal, De la Hire, Monge,

and Carnot are forerunners, but Poncelet, a pupil of Monge, is its real creator.

The appearance of his Trait´e des propri´et´es projectives des figures in 1822

gives modern geometry its birth. In it we find the line at infinity, the

introduction of imaginaries, the circular points at infinity, polar reciprocation, a

discussion of homology, the systematic use of projection, section, and

anharmonic ratio.

While the countrymen of Poncelet, especially Chasles, do not fail to

make numerous and valuable contributions to the new geometry, the next

great steps in advance are made on German soil. In 1827 M¨obius publishes

the Barycentrische Calcul; Pl¨ucker’s Analytisch-geometrische

Entwickelungen appears in 1828–31; and Steiner’s Systematische

Entwickelung der Abh¨angigkeit geometrischer Gestalten von einander in

1832. In the ten years which embrace the publication of these immortal works

of Poncelet, Pl¨ucker, and Steiner, geometry has made more real progress

than in the 2,000 years which had elapsed since the time of Apollonius. The

ideas which had been slowly taking shape since the time of Descartes

109
suddenly crystallized and almost overwhelmed geometry with an abundance

of new ideas and principles.

To M¨obius we owe the introduction of homogeneous co¨ordinates, and

the far reaching conception of geometric transformation including collineation

and duality as special cases.

To Pl¨ucker we owe the use of the abbreviated notation which permits

us to study the properties of geometric figures without intervention of the

co¨ordinates, the introduction of line and plane co¨ordinates and the notion of

generalized space elements. Steiner, who has been called the greatest

geometer since Apollonius, besides enriching geometry in countless ways,

was the first to employ systematically the method of generating geometric

figures by means of projective pencils.

Other noteworthy works belonging to this period are Pl¨ucker’s System

der analytischen Geometrie (1835) and Chasles’s classic Aper¸cu (1837).

Already at this stage we notice a bifurcation in geometric methods.

Steiner and Chasles become eloquent champions of the synthetic school of

geometry, while Pl¨ucker and later Hesse and Cayley are leaders in the

analytic movement. The astonishing fruitfulness and beauty of synthetic

methods threatened for a short time to drive the analytic school out of

existence. The tendency of the synthetic school was to banish more and more

metrical methods. In effecting this the anharmonic ratio become constantly

more prominent. To define this fundamental ratio without reference to

measurement and so to free projective geometry from the galling bondage of

110
metric relations was thus a problem of fundamental importance. The glory of

this achievement, which has, as we shall see, a far wider significance,

belongs to von Staudt. Another equally important contribution of von Staudt to

synthetic geometry is his theory of imaginaries. Poncelet, Steiner, Chasles

operate with imaginary elements as if they were real. Their only justification is

recourse to the so-called principle of continuity or to some other equally vague

principle. Von Staudt gives this theory a rigorous foundation, defining the

imaginary points, lines and planes by means of involutions without ordinal

elements.

The next great advance made is the advent of the theory of algebraic

invariants. Since projective geometry is the study of those properties of

geometric figures which remain unaltered by projective transformations, and

since the theory of invariants is the study of those forms which remain

unaltered (except possibly for a numerical factor) by the group of linear

substitutions, these two subjects are inseparably related and in many respects

only different aspects of the same thing. It is no wonder then that geometers

speedily applied the new theory of invariants to geometrical problems. Among

the pioneers in this direction were Cayley, Salmon, Aronhold, Hesse, and

especially Clebsch.

10. Differential Geometry

During the first quarter of the century this important branch of geometry

was cultivated chiefly by the French. Monge and his school study with great

111
success the generation of surfaces in various ways, the properties of

envelopes, evolutes, lines of curvature, asymptotic lines, skew curves,

orthogonal systems, and especially the relation between the surface theory

and partial differential equations.

The appearance of Gauss’s Disquisitiones generales circa superficies

curvas in 1828 marks a new epoch. Its wealth of new ideas has furnished

material for countless memoirs and given geometry a new direction. We find

here the parametric representation of a surface, the introduction of curvilinear

co¨ordinates, the notion of spherical image, the gaussian measure of

curvature, and a study of geodesics. But by far the most important

contributions that Gauss makes in this work are the consideration of a surface

as a flexible inextensible film or membrane and the importance given

quadratic differential forms.

We consider now some of the lines along which differential geometry

has advanced. The most important is perhaps the theory of differential

quadratic forms with their associate invariants and parameters. We mention

here Lam´e, Beltrami, Mainardi, Codazzi, Christoffel, Weingarten, and

Maschke.

An especially beautiful application of this theory is the immense subject

of applicability and deformation of surfaces in which Minding, Bauer, Beltrami,

Weingarten, and Voss have made important contributions.

Intimately related with the theory of applicability of two surfaces is the

theory of surfaces of constant curvature which play so important a part in non-

112
euclidean geometry. We mention here the work of Minding, Bonnet, Beltrami,

Dini, B¨acklund, and Lie.

The theory of rectilinear congruences has also been the subject of

important researches from the standpoint of differential geometry. First

studied by Monge as a system of normals to a surface and then in connection

with optics by Malus, Dupin, and Hamilton, the general theory has since been

developed by Kummer, Ribaucour, Guichard, Darboux, Voss, and

Weingarten. An important application of this theory is the infinitesimal

deformation of a surface.

Minimum surfaces have been studied by Monge, Bonnet and Enneper.

The subject owes its present extensive development principally to

Weierstrass, Riemann, Schwarz, and Lie. In it we find harmoniously united the

theory of surfaces, the theory of functions, the calculus of variations, the

theory of groups, and mathematical physics.

Another extensive division of differential geometry is the theory of

orthogonal systems, of such importance in physics. We note especially the

investigations of Dupin, Jacobi, Lam´e, Darboux, Combescure, and Bianchi.

We have already mentioned the intimate relation between differential

geometry and differential equations developed by Monge and Lie. Among the

workers in this fruitful field Darboux deserves especial mention.

One of the most original and interesting contributions to geometry in

the last decades of the century is Lie’s sphere geometry. As a brilliant

application of it to differential geometry we may mention the relation

113
discovered by Lie between asymptotic lines and lines of curvature of a

surface. The subject of sphere geometry has been developed also by

Darboux, Reye, Laguerre, Loria, P. F. Smith, and E. M¨uller.

11. Other Branches of Geometry

Under this head we group a number of subjects too important to pass

over in silence, yet which cannot be considered at length for lack of time.

In the first place is the immense subject of algebraic curves and

surfaces. Adequately to develop all the important and elegant properties of

curves and surfaces of the second order alone, would require a bulky volume.

In this line of ideas would follow curves and surfaces of higher order and

class. Their theory is far less complete, but this lack it amply makes good by

offering an almost bewildering variety of configurations to classify and explore.

No single geometer has contributed more to this subject than Cayley.

A theory of great importance is the geometry on a curve or surface

inaugurated by Clebsch in 1863. Expressing the co¨ordinates of a plane cubic

by means of elliptic functions and employing their addition theorems, he

deduced with hardly any calculation Steiner’s theorem relating to the inscribed

polygons and various theorems concerning conics touching the curve.

Encouraged by such successes Clebsch proposed to make use of Riemann’s

theory of abelian functions in the study of algebraic curves of any order. The

most important result was a new classification of such curves. Instead of the

linear transformation Clebsch, in harmony with Riemann’s ideas, employs the

birational transformation as a principle of classification. From this standpoint

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we ask what are the properties of algebraic curves which remain invariant for

such transformation.

Brill and Noether follow Clebsch. Their method is, however, algebraical

and rests on their celebrated residual theorem, which in their hands takes the

place of Abel’s theorem. We mention further the investigations of

Castelnuovo, Weber, Krauss and Segre. An important division of this subject

is the theory of correspondences. First studied by Chasles for curves of

deficiency 0 in 1864, Cayley and immediately after Brill extended the theory to

the case of any p. The most important advance made in later years has been

made by Hurwitz, who considers the totality of possible correspondences on

an algebraic curve, making use of the abelian integrals of the first species.

Alongside the geometry on a curve is the vastly more difficult and

complicated geometry on a surface, or more generally, on any algebraic

spread in n-way space. Starting from a remark of Clebsch 1868, Noether

made the first great step in his famous memoirs of 1868–74. Further progress

has been due to the French and Italian mathematicians. Picard, Poincar´e,

and Humbert make use of transcendental methods in which figure prominently

double integrals which remain finite on the surface and single integrals of total

differentials. On the other hand Enriques and Castelnuovo have attacked the

subject from a more algebraic-geometric standpoint by means of linear

systems of algebraic curves on the surface.

The first invariants of a surface were discovered by Clebsch and

Noether; still others have been found by Castelnuovo and Enriques in

connection with irregular surfaces.

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 Leaving this subject let us consider briefly the geometry of n

dimensions. A characteristic of nineteenth century mathematics is the

generality of its methods and results. When such has been impossible with

the elements in hand, fresh ones have been invented; witness the introduction

of imaginary numbers in algebra and the function theory, the ideals of

Kummer in the theory of numbers, the line and plane at infinity in projective

geometry. The benefit that analysis derived from geometry was too great not

to tempt mathematicians to free the latter from the narrow limits of three

dimensions and so give it the generality that the former has long enjoyed. The

first pioneer in this abstract field was Grassmann (1844); we must, however,

consider Cayley as the real founder of n-dimensional geometry (1869).

Notable contributions have been made by the Italian school, Veronese, Segre,

and others.

12. Non-Euclidean Geometry

In about 300 BC Euclid wrote The Elements, a book which was

to become one of the most famous books ever written. Euclid stated

five postulates on which he based all his theorems:

1. To draw a straight line from any point to any other.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to each other.

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 5. That, if a straight line falling on two straight lines make the

interior angles on the same side less than two right angles, if

produced indefinitely, meet on that side on which are the

angles less than the two right angles.

It is clear that the fifth postulate is different from the other four.

It did not satisfy Euclid and he tried to avoid its use as long as

possible - in fact the first 28 propositions of The Elements are proved

without using it. Another comment worth making at this point is

that Euclid, and many that were to follow him, assumed that straight

lines were infinite.

Proclus (410-485) wrote a commentary on The

Elements where he comments on attempted proofs to deduce the

fifth postulate from the other four, in particular he notes

that Ptolemy had produced a false 'proof'. Proclus then goes on to

give a false proof of his own. However he did give the following

postulate which is equivalent to the fifth postulate.

Playfair's Axiom:- Given a line and a point not on the line, it is

possible to draw exactly one line through the given point parallel to

the line.

Although known from the time of Proclus, this became known

as Playfair's Axiom after John Playfair wrote a famous commentary

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on Euclid in 1795 in which he proposed replacing Euclid's fifth

postulate by this axiom.

Many attempts were made to prove the fifth postulate from the

other four, many of them being accepted as proofs for long periods of

time until the mistake was found. Invariably the mistake was

assuming some 'obvious' property which turned out to be equivalent

to the fifth postulate. One such 'proof' was given by Wallis in 1663

when he thought he had deduced the fifth postulate, but he had

actually shown it to be equivalent to:-

To each triangle, there exists a similar triangle of arbitrary

magnitude.

One of the attempted proofs turned out to be more important than

most others. It was produced in 1697 by Girolamo Saccheri. The

importance of Saccheri's work was that he assumed the fifth

postulate false and attempted to derive a contradiction.

Here is the Saccheri quadrilateral

In this figure Saccheri proved

that the summit angles

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at D and C were equal.The proof uses properties of congruent

triangles which Euclid proved in Propositions 4 and 8 which are

proved before the fifth postulate is used. Saccheri has shown:

a) The summit angles are > 90° (hypothesis of the obtuse angle).

b) The summit angles are < 90° (hypothesis of the acute angle).

c) The summit angles are = 90° (hypothesis of the right angle).

Euclid's fifth postulate is c). Saccheri proved that the

hypothesis of the obtuse angle implied the fifth postulate, so

obtaining a contradiction. Saccheri then studied the hypothesis of the

acute angle and derived many theorems of non-Euclidean geometry

without realising what he was doing. However he eventually 'proved'

that the hypothesis of the acute angle led to a contradiction by

assuming that there is a 'point at infinity' which lies on a plane.

In 1766 Lambert followed a similar line to Saccheri. However

he did not fall into the trap that Saccheri fell into and investigated the

hypothesis of the acute angle without obtaining a

contradiction. Lambert noticed that, in this new geometry, the angle

sum of a triangle increased as the area of the triangle decreased.

Legendre spent 40 years of his life working on the parallel

postulate and the work appears in appendices to various editions of

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his highly successful geometry book Eléments de

Géométrie. Legendre proved that Euclid's fifth postulate is equivalent

to:-

The sum of the angles of a triangle is equal to two right angles.

Legendre showed, as Saccheri had over 100 years earlier, that

the sum of the angles of a triangle cannot be greater than two right

angles. This, again like Saccheri, rested on the fact that straight lines

were infinite. In trying to show that the angle sum cannot be less than

180° Legendre assumed that through any point in the interior of an

angle it is always possible to draw a line which meets both sides of

the angle. This turns out to be another equivalent form of the fifth

postulate, but Legendre never realised his error himself.

Elementary geometry was by this time engulfed in the

problems of the parallel postulate. D'Alembert, in 1767, called it the

scandal of elementary geometry.

The first person to really come to understand the problem of

the parallels was Gauss. He began work on the fifth postulate in 1792

while only 15 years old, at first attempting to prove the parallels

postulate from the other four. By 1813 he had made little progress

and wrote:

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In the theory of parallels we are even now not further than Euclid.

This is a shameful part of mathematics...

However by 1817 Gauss had become convinced that the fifth

postulate was independent of the other four postulates. He began to

work out the consequences of a geometry in which more than one

line can be drawn through a given point parallel to a given line.

Perhaps most surprisingly of all Gauss never published this work but

kept it a secret. At this time thinking was dominated by Kant who had

stated that Euclidean geometry is the inevitable necessity of

thought and Gauss disliked controversy.

Gauss discussed the theory of parallels with his friend, the

mathematician Farkas Bolyai who made several false proofs of the

parallel postulate. Farkas Bolyai taught his son, János Bolyai,

mathematics but, despite advising his son not to waste one hour's

time on that problem of the problem of the fifth postulate, János

Bolyai did work on the problem.

In 1823 János Bolyai wrote to his father saying I have

discovered things so wonderful that I was astounded ... out of

nothing I have created a strange new world. However it took Bolyai a

further two years before it was all written down and he published

his strange new world as a 24 page appendix to his father's book,

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although just to confuse future generations the appendix was

published before the book itself.

Gauss, after reading the 24 pages, described János Bolyai in

these words while writing to a friend: I regard this young geometer

Bolyai as a genius of the first order . However in some

sense Bolyai only assumed that the new geometry was possible. He

then followed the consequences in a not too dissimilar fashion from

those who had chosen to assume the fifth postulate was false and

seek a contradiction. However the real breakthrough was the belief

that the new geometry was possible. Gauss, however impressed he

sounded in the above quote with Bolyai, rather devastated Bolyai by

telling him that he (Gauss) had discovered all this earlier but had not

published. Although this must undoubtedly have been true, it detracts

in no way from Bolyai's incredible breakthrough.

Nor is Bolyai's work diminished

because Lobachevsky published a work on non-Euclidean geometry

in 1829. Neither Bolyai nor Gauss knew of Lobachevsky's work,

mainly because it was only published in Russian in the Kazan

Messenger a local university publication. Lobachevsky's attempt to

reach a wider audience had failed when his paper was rejected

by Ostrogradski.

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 In fact Lobachevsky fared no better than Bolyai in gaining

public recognition for his momentous work. He

published Geometrical investigations on the theory of parallels in

1840 which, in its 61 pages, gives the clearest account

of Lobachevsky's work. The publication of an account in French

in Crelle's Journal in 1837 brought his work on non-Euclidean

geometry to a wide audience but the mathematical community was

not ready to accept ideas so revolutionary.

In Lobachevsky's 1840 booklet he explains clearly how his non-

Euclidean geometry works.

All straight lines which in a plane go out from a point can, with

reference to a given straight line in the same plane, be divided into

two classes - into cutting and non-cutting. The boundary lines of the

one and the other class of those lines will be called parallel to the

given line.

Here is

the Lobachevsky

's diagram

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Hence Lobachevsky has replaced the fifth postulate of Euclid by:-

Lobachevsky's Parallel Postulate. There exist two lines parallel to a

given line through a given point not on the line. Lobachevsky went

on to develop many trigonometric identities for triangles which held in

this geometry, showing that as the triangle became small the

identities tended to the usual trigonometric identities.

Riemann, who wrote his doctoral dissertation under Gauss's

supervision, gave an inaugural lecture on 10 June 1854 in which he

reformulated the whole concept of geometry which he saw as a

space with enough extra structure to be able to measure things like

length. This lecture was not published until 1868, two years

after Riemann's death but was to have a profound influence on the

development of a wealth of different geometries. Riemann briefly

discussed a 'spherical' geometry in which every line through a

point P not on a line AB meets the line AB. In this geometry no

parallels are possible.

It is important to realize that neither Bolyai's nor Lobachevsky's

description of their new geometry had been proved to be consistent.

In fact it was no different from Euclidean geometry in this respect

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although the many centuries of work with Euclidean geometry was

sufficient to convince mathematicians that no contradiction would

ever appear within it.

The first person to put the Bolyai - Lobachevsky non-Euclidean

geometry on the same footing as Euclidean geometry was Eugenio

Beltrami (1835-1900). In 1868 he wrote a paper Essay on the

interpretation of non-Euclidean geometry which produced a model for

2-dimensional non-Euclidean geometry within 3-dimensional

Euclidean geometry. The model was obtained on the surface of

revolution of a tractrix about its asymptote. This is sometimes called

a pseudo-sphere.

You can see the graph of a tractrix and what the top half of a

Pseudo-sphere looks like. In fact Beltrami's model was incomplete

but it certainly gave a final decision on the fifth postulate

of Euclid since the model provided a setting in which Euclid's first

four postulates held but the fifth did not hold. It reduced the problem

of consistency of the axioms of non-Euclidean geometry to that of the

consistency of the axioms of Euclidean geometry.

Beltrami's work on a model of Bolyai - Lobachevsky's non-

Euclidean geometry was completed by Klein in 1871. Klein went

further than this and gave models of other non-Euclidean geometries

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such as Riemann's spherical geometry. Klein's work was based on a

notion of distance defined by Cayley in 1859 when he proposed a

generalized definition for distance. Klein showed that there are three

basically different types of geometry. In

the Bolyai - Lobachevsky type of geometry, straight lines have two

infinitely distant points. In the Riemann type of spherical geometry,

lines have no (or more precisely two imaginary) infinitely distant

points. Euclidean geometry is a limiting case between the two where

for each line there are two coincident infinitely distant points.

References

The History Of Mathematics In The Nineteenth Century

http://www.ams.org/journals/bull/2000-37-01/S0273-0979-99-00804-
6/S0273-0979-99-00804-6.pdf

Augustin-Louis Cauchy-MacTutor History of Mathematics

http://www-history.mcs.st-andrews.ac.uk/Biographies/Cauchy.html

Lazarus Fuchs MacTutor History of Mathematics

http://www-history.mcs.st-and.ac.uk/Biographies/Fuchs.html
http://www-history.mcs.st-and.ac.uk/Extras/Fuchs_publications.html

Non-Eulidean Geometry – University of St Andrews

http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Non-
Euclidean_geometry.html

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