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