История математики

History of math. The most ancient mathematical activity was counting. The

counting was necessary to keep up a livestock of cattle and to do business.

Some primitive tribes counted up amount of subjects, comparing them various

parts of a body, mainly fingers of hands and foots. Some pictures on the

stone represents number 35 as a series of 35 sticks - fingers built in a

line. The first essential success in arithmetic was the invention of four

basic actions: additions, subtraction, multiplication and division. The

first achievements of geometry are connected to such simple concepts, as a

straight line and a circle. The further development of mathematics began

approximately in 3000 up to AD due to Babylonians and Egyptians.

BABYLONIA AND EGYPT

Babylonia. The source of our knowledge about the Babylon civilization are

well saved clay tablets covered with texts which are dated from 2000 AD and

up to 300 AD . The mathematics on tablets basically has been connected to

housekeeping. Arithmetic and simple algebra were used at an exchange of

money and calculations for the goods, calculation of simple and complex

percent, taxes and the share of a crop which are handed over for the

benefit of the state, a temple or the land owner. Numerous arithmetic and

geometrical problems arose in connection with construction of channels,

granaries and other public jobs. Very important problem of mathematics was

calculation of a calendar. A calendar was used to know the terms of

agricultural jobs and religious holidays. Division of a circle on 360 and

degree and minutes on 60 parts originates in the Babylon astronomy.

Babylonians have made tables of inverse numbers (which were used at

performance of division), tables of squares and square roots, and also

tables of cubes and cubic roots. They knew good approximation of a number

[pic]. The texts devoted to the solving algebraic and geometrical

problems, testify that they used the square-law formula for the solving

quadratics and could solve some special types of the problems, including up

to ten equations with ten unknown persons, and also separate versions of

the cubic equations and the equations of the fourth degree. On the clay

tablets problems and the basic steps of procedures of their decision are

embodied only. About 700 AD babylonians began to apply mathematics to

research of, motions of the Moon and planets. It has allowed them to

predict positions of planets that were important both for astrology, and

for astronomy.

In geometry babylonians knew about such parities, for example, as

proportionality of the corresponding parties of similar triangles,

Pythagoras’ theorem and that a corner entered in half-circle- was known for

a straight line. They had also rules of calculation of the areas of simple

flat figures, including correct polygons, and volumes of simple bodies.

Number [pic] babylonians equaled to 3.

Egypt. Our knowledge about ancient greek mathematics is based mainly on two

papyruses dated approximately 1700 AD. Mathematical data stated in these

papyruses go back to earlier period - around 3500 AD. Egyptians used

mathematics to calculate weight of bodies, the areas of crops and volumes

of granaries, the amount of taxes and the quantity of stones required to

build those or other constructions. In papyruses it is possible to find

also the problems connected to solving of amount of a grain, to set number

necessary to produce a beer, and also more the challenges connected to

distinction in grades of a grain; for these cases translation factors were

calculated.

But the main scope of mathematics was astronomy, the calculations connected

to a calendar are more exact. The calendar was used find out dates of

religious holidays and a prediction of annual floods of Nile. However the

level of development of astronomy in Ancient Egypt was much weaker than

development in Babylon.

Ancient greek writing was based on hieroglyphs. They used their alphabet. I

think it’s not efficient; It’s difficult to count using letters. Just think

how they could multiply such numbers as 146534 to 19870503 using alphabet.

May be they needn’t to count such numbers. Nevertheless they’ve built an

incredible things – pyramids. They had to count the quantity of the stones

that were used and these quantities sometimes reached to thousands of

stones. I imagine their papyruses like a paper with numbers ABC, that

equals, for example, to 3257.

The geometry at Egyptians was reduced to calculations of the areas of

rectangular, triangles, trapezes, a circle, and also formulas of

calculation of volumes of some bodies. It is necessary to say, that

mathematics which Egyptians used at construction of pyramids, was simple

and primitive. I suppose that simple and primitive geometry can not create

buildings that can stand for thousands of years but the author thinks

differently.

Problems and the solving resulted in papyruses, are formulated without any

explanations. Egyptians dealt only with the elementary types of quadratics

and arithmetic and geometrical progressions that is why also those common

rules which they could deduce, were also the most elementary kind. Neither

Babylon, nor Egyptian mathematics had no the common methods; the arch of

mathematical knowledge represented a congestion of empirical formulas and

rules.

THE GREEK MATHEMATICS

Classical Greece. From the point of view of 20 century ancestors of

mathematics were Greeks of the classical period (6-4 centuries AD). The

mathematics existing during earlier period, was a set of the empirical

conclusions. On the contrary, in a deductive reasoning the new statement is

deduced from the accepted parcels by the way excluding an opportunity of

its aversion.

Insisting of Greeks on the deductive proof was extraordinary step. Any

other civilization has not reached idea of reception of the conclusions

extremely on the basis of the deductive reasoning which is starting with

obviously formulated axioms. The reason is a greek society of the classical

period. Mathematics and philosophers (quite often it there were same

persons) belonged to the supreme layers of a society where any practical

activities were considered as unworthy employment. Mathematics preferred

abstract reasoning on numbers and spatial attitudes to the solving of

practical problems. The mathematics consisted of a arithmetic - theoretical

aspect and logistic - computing aspect. The lowest layers were engaged in

logistic.

Deductive character of the Greek mathematics was completely generated by

Plato’s and Eratosthenes’ time. Other great Greek, with whose name connect

development of mathematics, was Pythagoras. He could meet the Babylon and

Egyptian mathematics during the long wanderings. Pythagoras has based

movement which blossoming falls at the period around 550-300 AD.

Pythagoreans have created pure mathematics in the form of the theory of

numbers and geometry. They represented integers as configurations from

points or a little stones, classifying these numbers according to the form

of arising figures (« figured numbers »). The word "accounting" (counting,

calculation) originates from the Greek word meaning "a little stone".

Numbers 3, 6, 10, etc. Pythagoreans named triangular as the corresponding

number of the stones can be arranged as a triangle, numbers 4, 9, 16, etc.

- square as the corresponding number of the stones can be arranged as a

square, etc.

From simple geometrical configurations there were some properties of

integers. For example, Pythagoreans have found out, that the sum of two

consecutive triangular numbers is always equal to some square number. They

have opened, that if (in modern designations) n[pic] - square number,

n[pic] + 2n +1 = (n + 1)[pic]. The number equal to the sum of all own

dividers, except for most this number, Pythagoreans named accomplished. As

examples of the perfect numbers such integers, as 6, 28 and 496 can serve.

Two numbers Pythagoreans named friendly, if each of numbers equally to the

sum of dividers of another; for example, 220 and 284 - friendly numbers

(here again the number is excluded from own dividers).

For Pythagoreans any number represented something the greater, than

quantitative value. For example, number 2 according to their view meant

distinction and consequently was identified with opinion. The 4 represented

validity, as this first equal to product of two identical multipliers.

Pythagoreans also have opened, that the sum of some pairs of square numbers

is again square number. For example, the sum 9 and 16 is equal 25, and the

sum 25 and 144 is equal 169. Such three of numbers as 3, 4 and 5 or 5, 12

and 13, are called “Pythagorean” numbers. They have geometrical

interpretation: if two numbers from three to equate to lengths of

cathetuses of a rectangular triangle the third will be equal to length of

its hypotenuse. Such interpretation, apparently, has led Pythagoreans to

comprehension more common fact known nowadays under the name of a

pythagoras’ theorem, according to which the square of length of a

hypotenuse is equal the sum of squares of lengths of cathetuses.

Considering a rectangular triangle with cathetuses equaled to 1,

Pythagoreans have found out, that the length of its hypotenuse is equal to

[pic], and it made them confusion because they tried to present number

[pic]as the division of two integers that was extremely important for their

philosophy. Values, not representable as the division of integers,

Pythagoreans have named incommensurable; the modern term - « irrational

numbers ». About 300 AD Euclid has proved, that the number [pic]is

incommensurable. Pythagoreans dealt with irrational numbers, representing

all sizes in the geometrical images. If 1 and [pic]to count lengths of some

pieces distinction between rational and irrational numbers smoothes out.

Product of numbers [pic]also [pic]is the area of a rectangular with the

sides in length [pic]and [pic].Today sometimes we speak about number 25 as

about a square of 5, and about number 27 - as about a cube of 3.

Ancient Greeks solved the equations with unknown values by means of

geometrical constructions. Special constructions for performance of

addition, subtraction, multiplication and division of pieces, extraction of

square roots from lengths of pieces have been developed; nowadays this

method is called as geometrical algebra.

Reduction of problems to a geometrical kind had a number of the important

consequences. In particular, numbers began to be considered separately from

geometry because to work with incommensurable divisions it was possible

only with the help of geometrical methods. The geometry became a basis

almost all strict mathematics at least to 1600 AD. And even in 18[pic]

century when the algebra and the mathematical analysis have already been

advanced enough, the strict mathematics was treated as geometry, and the

word "geometer" was equivalent to a word "mathematician".

One of the most outstanding Pythagoreans was Plato. Plato has been

convinced, that the physical world is conceivable only by means of

mathematics. It is considered, that exactly to him belongs a merit of the

invention of an analytical method of the proof. (the Analytical method

begins with the statement which it is required to prove, and then from it

consequences, which are consistently deduced until any known fact will be

achieved; the proof turns out with the help of return procedure.) It is

considered to be, that Plato’s followers have invented the method of the

proof which have received the name "rule of contraries". The appreciable

place in a history of mathematics is occupied by Aristotle; he was the

Plato’s learner. Aristotle has put in pawn bases of a science of logic and

has stated a number of ideas concerning definitions, axioms, infinity and

opportunities of geometrical constructions.

About 300 AD results of many Greek mathematicians have been shown in the

one work by Euclid, who had written a mathematical masterpiece “the

Beginning”. From few selected axioms Euclid has deduced about 500 theorems

which have captured all most important results of the classical period.

Euclid’s Composition was begun from definition of such terms, as a straight

line, with a corner and a circle. Then he has formulated ten axiomatic

trues, such, as « the integer more than any of parts ». And from these ten

axioms Euclid managed to deduce all theorems.

Apollonius lived during the Alexandria period, but his basic work is

sustained in spirit of classical traditions. The analysis of conic sections

suggested by him - circles, an ellipse, a parabola and a hyperbole - was

the culmination of development of the Greek geometry. Apollonius also

became the founder of quantitative mathematical astronomy.

The Alexandria period. During this period which began about 300 AD, the

character of a Greek mathematics has changed. The Alexandria mathematics

has arisen as a result of merge of classical Greek mathematics to

mathematics of Babylonia and Egypt. Generally the mathematics of the

Alexandria period were more inclined to the solving technical problems,

than to philosophy. Great Alexandria mathematics - Eratosthenes, Archimedes

and Ptolemaist - have shown force of the Greek genius in theoretical

abstraction, but also willingly applied the talent for the solving of

practical problems and only quantitative problems.

Eratosthenes has found a simple method of exact calculation of length of a

circle of the Earth, he possesses a calendar in which each fourth year has

for one day more, than others. The astronomer the Aristarch has written the

composition “About the sizes and distances of the Sun and the Moon”,

containing one of the first attempts of definition of these sizes and

distances; the character of the Aristarch’s job was geometrical.

The greatest mathematician of an antiquity was Archimedes. He possesses

formulations of many theorems of the areas and volumes of complex figures

and the bodies. Archimedes always aspired to receive exact decisions and

found the top and bottom estimations for irrational numbers. For example,

working with a correct 96-square, he has irreproachably proved, that exact

value of number [pic] is between 3[pic] and 3[pic]. Архимед has proved also

some theorems, containing new results of geometrical algebra.

Archimedes also was the greatest mathematical physicist of an antiquity.

For the proof of theorems of mechanics he used geometrical reasons. His

composition “About floating bodies” has put in pawn bases of a

hydrostatics.

Decline of Greece. After a gain of Egypt Romans in 31 AD great Greek

Alexandria civilization has come to decline. Cicerones with pride approved,

that as against Greeks Romans not dreamers that is why put the mathematical

knowledge into practice, taking from them real advantage. However in

development of the mathematics the contribution of roman was insignificant.

INDIA AND ARABS

Successors of Greeks in a history of mathematics were Indians. Indian

mathematics were not engaged in proofs, but they have entered original

concepts and a number of effective methods. They have entered zero as

cardinal number and as a symbol of absence of units in the corresponding

category. Moravia (850 AD) has established rules of operations with zero,

believing, however, that division of number into zero leaves number

constant. The right answer for a case of division of number on zero has

been given by Bharskar (born In 1114 AD -?), he possesses rules of actions

above irrational numbers. Indians have entered concept of negative numbers

(for a designation of duties). We find their earliest use at Brahmagupta’s

(around 630). Ariabhata (born in 476 AD-?) has gone further in use of

continuous fractions at the decision of the uncertain equations.

Our modern notation based on an item principle of record of numbers and

zero as cardinal number and use of a designation of the empty category, is

called Indo-Arabian. On a wall of the temple constructed in India around

250 AD, some figures, reminding on the outlines our modern figures are

revealed.

About 800 Indian mathematics has achieved Baghdad. The term "algebra"

occurs from the beginning of the name of book Al-Jebr vah-l-mukabala

-Completion and opposition (Аль-джебр ва-л-мукабала), written in 830

astronomer and the mathematician Al-Horezmi. In the composition he did

justice to merits of the Indian mathematics. The algebra of Al-Horezmi has

been based on works of Brahmagupta, but in that work Babylon and Greek math

influences are clearly distinct. Other outstanding Arabian mathematician

Ibn Al-Haisam (around 965-1039) has developed a way of reception of

algebraic solvings of the square and cubic equations. Arabian mathematics,

among them and Omar Khayyam, were able to solve some cubic equations with

the help of geometrical methods, using conic sections. The Arabian

astronomers have entered into trigonometry concept of a tangent and

cotangent. Nasyreddin Tusy (1201-1274 AD) in the “Treatise about a full

quadrangle” has regularly stated flat and spherical to geometry and the

first has considered trigonometry separately from astronomy.

And still the most important contribution of arabs to mathematics of steel

their translations and comments to great creations of Greeks. Europe has

met these jobs after a gain arabs of Northern Africa and Spain, and later

works of Greeks have been translated to Latin.

MIDDLE AGES AND REVIVAL

Medieval Europe. The Roman civilization has not left an appreciable trace

in mathematics as was too involved in the solving of practical problems. A

civilization developed in Europe of the early Middle Ages (around 400-1100

AD), was not productive for the opposite reason: the intellectual life has

concentrated almost exclusively on theology and future life. The level of

mathematical knowledge did not rise above arithmetics and simple sections

from Euclid’s “Beginnings”. In Middle Ages the astrology was considered as

the most important section of mathematics; astrologists named

mathematicians.

About 1100 in the West-European mathematics began almost three-century

period of development saved by arabs and the Byzantian Greeks of a heritage

of the Ancient world and the East. Europe has received the extensive

mathematical literature because of arabs owned almost all works of ancient

Greeks. Translation of these works into Latin promoted rise of mathematical

researches. All great scientists of that time recognized, that scooped

inspiration in works of Greeks.

The first European mathematician deserving a mention became Leonardo

Byzantian (Fibonacci). In the composition “the Book Abaca” (1202) he has

acquainted Europeans with the Indо-Arabian figures and methods of

calculations and also with the Arabian algebra. Within the next several

centuries mathematical activity in Europe came down.

Revival. Among the best geometers of Renaissance there were the artists

developed idea of prospect which demanded geometry with converging parallel

straight lines. The artist Leon Batista Alberty (1404-1472) has entered

concepts of a projection and section. Rectilinear rays of light from an eye

of the observer to various points of a represented stage form a projection;

the section turns out at passage of a plane through a projection. That the

drawn picture looked realistic, it should be such section. Concepts of a

projection and section generated only mathematical questions. For example,

what general geometrical properties the section and an initial stage, what

properties of two various sections of the same projection, formed possess

two various planes crossing a projection under various corners? From such

questions also there was a projective geometry. Its founder - Z. Dezarg

(1593-1662 AD) with the help of the proofs based on a projection and

section, unified the approach to various types of conic sections which

great Greek geometer Apollonius considered separately.

I think that mathematics developed by attempts and mistakes. There is no

perfect science today. Also math has own mistakes, but it aspires to be

more accurate. A development of math goes thru a development of the

society. Starting from counting on fingers, finishing on solving difficult

problems, mathematics prolong it way of development. I suppose that it’s no

people who can say what will be in 100-200 or 500 years. But everybody

knows that math will get new level, higher one. It will be new high-tech

level and new methods of solving today’s problems. May in the future some

man will find mistakes in our thinking, but I think it’s good, it’s good

that math will not stop.

Bibliography:

Ван-дер-Варден Б.Л. «Пробуждающаяся наука». Математика древнего Египта,

Вавилона и Греции. МОСКВА, 1959

Юшкевич A.П. История математики в средние века. МОСКВА, 1961

Даан-Дальмедико А., Пейффер Ж. Пути и лабиринтыю Очерки по истории

математики МОСКВА, 1986

Клейн Ф. Лекции о развитии математики в XIX столетии. МОСКВА, 1989