# Реферат: История математики

<span Courier New"; color:black;mso-ansi-language:EN-US">History of math.

<span Courier New";color:black;mso-ansi-language:EN-US">The most ancient mathematical activity was counting. The counting was necessaryto keep up a livestock of cattle and to do business. Some primitive tribescounted up amount of subjects, comparing them various parts of a body, mainlyfingers of hands and foots. Some pictures on the stone represents number 35 asa series of 35 sticks — fingers built in a line. The first essential success inarithmetic was the invention of four basic actions: additions, subtraction,multiplication and division. The first achievements of geometry are connectedto such simple concepts, as a straight line and a circle. The furtherdevelopment of mathematics began approximately in 3000 up to AD due toBabylonians and Egyptians.

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<st1:place w:st=«on»><span Courier New";color:black;mso-ansi-language:EN-US">BABYLONIA

</st1:place><span Courier New";color:black;mso-ansi-language:EN-US">AND <st1:country-region w:st=«on»><st1:place w:st=«on»>EGYPT</st1:place></st1:country-region><span Courier New";color:black; mso-ansi-language:EN-US">

<st1:place w:st=«on»><span Courier New";color:black;mso-ansi-language:EN-US">Babylonia

</st1:place><span Courier New";color:black;mso-ansi-language:EN-US">.<span Courier New";color:black;mso-ansi-language:EN-US">The source of our knowledge about the Babylon civilization are well saved claytablets covered with texts which are dated from 2000 AD and up to 300 AD. Themathematics on tablets basically has been connected to housekeeping. Arithmeticand simple algebra were used at an exchange of money and calculations for thegoods, calculation of simple and complex percent, taxes and the share of a cropwhich are handed over for the benefit of the state, a temple or the land owner.Numerous arithmetic and geometrical problems arose in connection withconstruction of channels, granaries and other public jobs. Very importantproblem of mathematics was calculation of a calendar. A calendar was used toknow the terms of agricultural jobs and religious holidays. Division of a circleon 360 and degree and minutes on 60 parts originates in the <st1:City w:st=«on»><st1:place w:st=«on»>Babylon</st1:place></st1:City> astronomy.

<span Courier New"; color:black;mso-ansi-language:EN-US">Babylonians have made tables of inversenumbers (which were used at performance of division), tables of squares andsquare roots, and also tables of cubes and cubic roots. They knew goodapproximation of a number

<span Courier New"; color:black"><img src="/cache/referats/16352/image002.gif" v:shapes="_x0000_i1025"><span Courier New";color:black;mso-ansi-language:EN-US">. The texts devoted to the solvingalgebraic and geometrical problems, testify that they used the square-lawformula for the solving quadratics and could solve some special types of theproblems, including up to ten equations with ten unknown persons, and alsoseparate versions of the cubic equations and the equations of the fourthdegree. On the clay tablets problems and the basic steps of procedures of theirdecision are embodied only. About 700 AD babyloniansbegan to apply mathematics to research of, motions of the Moon and planets. Ithas allowed them to predict positions of planets that were important both for astrology,and for astronomy.

<span Courier New"; color:black;mso-ansi-language:EN-US">In geometry babyloniansknew about such parities, for example, as proportionality of the correspondingparties 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 areasof simple flat figures, including correct polygons, and volumes of simplebodies. Number <img src="/cache/referats/16352/image004.gif" v:shapes="_x0000_i1026">

<span Courier New";color:black">babylonians<span Courier New";color:black;mso-ansi-language:EN-US">equaled to 3.

<st1:country-region w:st=«on»><st1:place w:st=«on»><span Courier New";color:black;mso-ansi-language: EN-US">Egypt

</st1:place></st1:country-region><span Courier New";color:black;mso-ansi-language:EN-US">.<span Courier New";color:black;mso-ansi-language:EN-US">Our knowledge about ancient greek mathematics isbased mainly on two papyruses dated approximately 1700 AD. Mathematical datastated in these papyruses go back to earlier period — around 3500 AD. Egyptiansused mathematics to calculate weight of bodies, the areas of crops and volumesof granaries, the amount of taxes and the quantity of stones required to buildthose or other constructions. In papyruses it is possible to find also theproblems connected to solving of amount of a grain, to set number necessary toproduce a beer, and also more the challenges connected to distinction in gradesof a grain; for these cases translation factors were calculated.

<span Courier New"; color:black;mso-ansi-language:EN-US">But the main scope of mathematics wasastronomy, the calculations connected to a calendar are more exact. Thecalendar was used find out dates of religious holidays and a prediction ofannual floods of <st1:place w:st=«on»>Nile</st1:place>. However the level ofdevelopment of astronomy in Ancient Egypt was much weaker than development in <st1:City w:st=«on»><st1:place w:st=«on»>Babylon</st1:place></st1:City>.

<span Courier New"; color:black;mso-ansi-language:EN-US">Ancient greekwriting 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 suchnumbers as 146534 to 19870503 using alphabet. May be they needn’t to count suchnumbers. Nevertheless they’ve built an incredible things – pyramids. They hadto count the quantity of the stones that were used and these quantitiessometimes reached to thousands of stones. I imagine their papyruses like apaper with numbers ABC, that equals, for example, to 3257.

<span Courier New"; color:black;mso-ansi-language:EN-US">The geometry at Egyptians was reduced tocalculations of the areas of rectangular, triangles, trapezes, a circle, andalso formulas of calculation of volumes of some bodies. It is necessary to say,that mathematics which Egyptians used at construction of pyramids, was simpleand primitive. I suppose that simple and primitive geometry can not createbuildings that can stand for thousands of years but the author thinks differently.

<span Courier New"; color:black;mso-ansi-language:EN-US">Problems and the solving resulted inpapyruses, are formulated without any explanations. Egyptians dealt only withthe elementary types of quadratics and arithmetic and geometrical progressionsthat is why also those common rules which they could deduce, were also the mostelementary kind. Neither <st1:City w:st=«on»><st1:place w:st=«on»>Babylon</st1:place></st1:City>,nor Egyptian mathematics had no the common methods; the arch of mathematicalknowledge represented a congestion of empirical formulas and rules.

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<span Courier New"; color:black;mso-ansi-language:EN-US">THE GREEK MATHEMATICS

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<span Courier New"; color:black;mso-ansi-language:EN-US">Classical <st1:country-region w:st=«on»><st1:place w:st=«on»>Greece</st1:place></st1:country-region>.

<span Courier New";color:black;mso-ansi-language:EN-US"> From thepoint of view of 20 century ancestors of mathematics were Greeks of theclassical period (6-4 centuries AD). The mathematics existing during earlierperiod, was a set of the empirical conclusions. On the contrary, in a deductivereasoning the new statement is deduced from the accepted parcels by the wayexcluding an opportunity of its aversion.

<span Courier New"; color:black;mso-ansi-language:EN-US">Insisting of Greeks on the deductive proofwas extraordinary step. Any other civilization has not reached idea ofreception of the conclusions extremely on the basis of the deductive reasoningwhich is starting with obviously formulated axioms. The reason is a greek society of the classical period. Mathematics andphilosophers (quite often it there were same persons) belonged to the supremelayers of a society where any practical activities were considered as unworthyemployment. Mathematics preferred abstract reasoning on numbers and spatialattitudes to the solving of practical problems. The mathematics consisted of a arithmetic- theoretical aspect and logistic — computing aspect. The lowest layers wereengaged in logistic.

<span Courier New"; color:black;mso-ansi-language:EN-US">Deductive character of the Greekmathematics was completely generated by Plato’s and Eratosthenes’ time. Othergreat Greek, with whose name connect development of mathematics, was Pythagoras.He could meet the <st1:City w:st=«on»><st1:place w:st=«on»>Babylon</st1:place></st1:City>and Egyptian mathematics during the long wanderings. Pythagoras has basedmovement which blossoming falls at the period around 550-300 AD. Pythagoreanshave 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 (« figurednumbers »). The word «accounting» (counting, calculation) originatesfrom the Greek word meaning «a little stone». Numbers 3, 6, 10, etc.Pythagoreans named triangular as the corresponding number of the stones can bearranged as a triangle, numbers 4, 9, 16, etc. — square as the correspondingnumber of the stones can be arranged as a square, etc.

<span Courier New"; color:black;mso-ansi-language:EN-US">From simple geometrical configurationsthere were some properties of integers. For example, Pythagoreans have foundout, that the sum of two consecutive triangular numbers is always equal to somesquare number. They have opened, that if (in modern designations) n<img src="/cache/referats/16352/image006.gif" v:shapes="_x0000_i1027">  — square number,n<img src="/cache/referats/16352/image006.gif" v:shapes="_x0000_i1028"> + 2n +1 = (n+ 1)<img src="/cache/referats/16352/image006.gif" v:shapes="_x0000_i1029">. The number equal to the sum of all own dividers, exceptfor most this number, Pythagoreans named accomplished. As examples of theperfect numbers such integers, as 6, 28 and 496 can serve. Two numbersPythagoreans named friendly, if each of numbers equally to the sum of dividersof another; for example, 220 and 284 — friendly numbers (here again the numberis excluded from own dividers).

<span Courier New"; color:black;mso-ansi-language:EN-US">For Pythagoreans any number representedsomething the greater, than quantitative value. For example, number 2 accordingto their view meant distinction and consequently was identified with opinion.The 4 represented validity, as this first equal to product of two identicalmultipliers.

<span Courier New"; color:black;mso-ansi-language:EN-US">Pythagoreans also have opened, that thesum of some pairs of square numbers is again square number. For example, thesum 9 and 16 is equal 25, and the sum 25 and 144 is equal 169. Such three ofnumbers as 3, 4 and 5 or 5, 12 and 13, are called “Pythagorean” numbers. Theyhave geometrical interpretation: if two numbers from three to equate to lengthsof cathetuses of a rectangular triangle the thirdwill be equal to length of its hypotenuse. Such interpretation, apparently, hasled Pythagoreans to  comprehension morecommon fact known nowadays under the name of a pythagoras’theorem, according to which the square of length of a hypotenuse is equal thesum of squares of lengths of cathetuses.

<span Courier New"; color:black;mso-ansi-language:EN-US">Considering a rectangular triangle with cathetuses equaled to 1, Pythagoreans have found out, thatthe length of its hypotenuse is equal to

<span Courier New"; color:black"><img src="/cache/referats/16352/image002.gif" v:shapes="_x0000_i1030"><span Courier New";color:black;mso-ansi-language:EN-US">,and it made them confusion because they tried to present number <span Courier New";color:black"><img src="/cache/referats/16352/image002.gif" v:shapes="_x0000_i1031"><span Courier New";color:black;mso-ansi-language:EN-US">asthe division of two integers that was extremely important for their philosophy.Values, not representable as the division ofintegers, Pythagoreans have named incommensurable; the modern term — «irrational numbers ». About 300 AD <st1:City w:st=«on»><st1:place w:st=«on»>Euclid</st1:place></st1:City>has proved, that the number <span Courier New"; color:black"><img src="/cache/referats/16352/image002.gif" v:shapes="_x0000_i1032"><span Courier New";color:black;mso-ansi-language:EN-US">isincommensurable. Pythagoreans dealt with irrational numbers, representing allsizes in the geometrical images. If 1 and <span Courier New"; color:black"><img src="/cache/referats/16352/image002.gif" v:shapes="_x0000_i1033"><span Courier New";color:black;mso-ansi-language:EN-US">tocount lengths of some pieces distinction between rational and irrationalnumbers smoothes out. Product of numbers <span Courier New"; color:black"><img src="/cache/referats/16352/image009.gif" v:shapes="_x0000_i1034"><span Courier New";color:black;mso-ansi-language:EN-US">also<span Courier New";color:black"><img src="/cache/referats/16352/image011.gif" v:shapes="_x0000_i1035"><span Courier New";color:black;mso-ansi-language:EN-US">isthe area of a rectangular with the sides in length <span Courier New";color:black"><img src="/cache/referats/16352/image009.gif" v:shapes="_x0000_i1036"><span Courier New";color:black;mso-ansi-language:EN-US">and<span Courier New";color:black"><img src="/cache/referats/16352/image011.gif" v:shapes="_x0000_i1037"><span Courier New";color:black;mso-ansi-language:EN-US">.Todaysometimes we speak about number 25 as about a square of 5, and about number 27- as about a cube of 3.

<span Courier New"; color:black;mso-ansi-language:EN-US">Ancient Greeks solved the equations withunknown values by means of geometrical constructions. Special constructions forperformance of addition, subtraction, multiplication and division of pieces,extraction of square roots from lengths of pieces have been developed; nowadaysthis method is called as geometrical algebra.

<span Courier New"; color:black;mso-ansi-language:EN-US">Reduction of problems to a geometricalkind had a number of the important consequences. In particular, numbers beganto be considered separately from geometry because to work with incommensurable divisionsit was possible only with the help of geometrical methods. The geometry becamea basis almost all strict mathematics at least to 1600 AD. And even in 18<img src="/cache/referats/16352/image013.gif" v:shapes="_x0000_i1038"> century when thealgebra and the mathematical analysis have already been advanced enough, the strictmathematics was treated as geometry, and the word «geometer» wasequivalent to a word «mathematician».

<span Courier New"; color:black;mso-ansi-language:EN-US">One of the most outstanding Pythagoreanswas Plato. Plato has been convinced, that the physical world is conceivableonly by means of mathematics. It is considered, that exactly to him belongs amerit of the invention of an analytical method of the proof. (the Analyticalmethod begins with the statement which it is required to prove, and then fromit consequences, which are consistently deduced until any known fact will beachieved; the proof turns out with the help of return procedure.) It isconsidered to be, that Plato’s followers have invented the method of the proofwhich have received the name «rule of contraries». The appreciableplace in a history of mathematics is occupied by Aristotle; he was the    Plato’s learner. Aristotle has put in pawnbases of a science of logic and has stated a number of ideas concerningdefinitions, axioms, infinity and opportunities of geometrical constructions.

<span Courier New"; color:black;mso-ansi-language:EN-US">About 300 AD results of many Greekmathematicians have been shown in the one work by Euclid, who had written amathematical masterpiece “the Beginning”. From few selected axioms <st1:City w:st=«on»><st1:place w:st=«on»>Euclid</st1:place></st1:City> has deduced about500 theorems which have captured all most important results of the classicalperiod. <st1:City w:st=«on»><st1:place w:st=«on»>Euclid</st1:place></st1:City>’sComposition was begun from definition of such terms, as a straight line, with acorner and a circle. Then he has formulated ten axiomatic trues,such, as « the integer more than any of parts ». And from these ten axioms <st1:City w:st=«on»><st1:place w:st=«on»>Euclid</st1:place></st1:City> managed to deduce alltheorems.

<span Courier New"; color:black;mso-ansi-language:EN-US">Apollonius lived during the <st1:City w:st=«on»><st1:place w:st=«on»>Alexandria</st1:place></st1:City> period, buthis basic work  is sustained in spirit ofclassical traditions. The analysis of conic sections suggested by him — circles, an ellipse, a parabola and a hyperbole — was the culmination ofdevelopment of the Greek geometry. Apollonius also became the founder ofquantitative mathematical astronomy.

<span Courier New"; color:black;mso-ansi-language:EN-US">The <st1:City w:st=«on»><st1:place w:st=«on»>Alexandria</st1:place></st1:City>period.

<span Courier New";color:black; mso-ansi-language:EN-US"> During this period which began about 300 AD, thecharacter of a Greek mathematics has changed. The <st1:City w:st=«on»><st1:place w:st=«on»>Alexandria</st1:place></st1:City> mathematics has arisen as a resultof merge of classical Greek mathematics to mathematics of <st1:place w:st=«on»>Babylonia</st1:place>and <st1:country-region w:st=«on»><st1:place w:st=«on»>Egypt</st1:place></st1:country-region>.Generally the mathematics of the <st1:City w:st=«on»><st1:place w:st=«on»>Alexandria</st1:place></st1:City>period were more inclined to the solving technical problems, than tophilosophy. Great <st1:City w:st=«on»><st1:place w:st=«on»>Alexandria</st1:place></st1:City>mathematics — Eratosthenes, Archimedes and Ptolemaist — have shown force of theGreek genius in theoretical abstraction, but also willingly applied the talentfor the solving of practical problems and only quantitative problems.

<span Courier New"; color:black;mso-ansi-language:EN-US">Eratosthenes has found a simple method ofexact calculation of length of a circle of the Earth, he possesses a calendarin which each fourth year has for one day more, than others. The astronomer theAristarch has written the composition “About thesizes and distances of the Sun and the Moon”, containing one of the firstattempts of definition of these sizes and distances; the character of the Aristarch’s job was geometrical.

<span Courier New"; color:black;mso-ansi-language:EN-US">The greatest mathematician of an antiquitywas Archimedes. He possesses formulations of many theorems of the areas andvolumes of complex figures and the bodies. Archimedes always aspired to receiveexact decisions and found the top and bottom estimations for irrationalnumbers. For example, working with a correct 96-square, he has irreproachablyproved, that exact value of number <img src="/cache/referats/16352/image004.gif" v:shapes="_x0000_i1039"> is between3<img src="/cache/referats/16352/image016.gif" v:shapes="_x0000_i1040"> and 3<img src="/cache/referats/16352/image018.gif" v:shapes="_x0000_i1041">

<span Courier New";color:black">Архимед<span Courier New";color:black;mso-ansi-language:EN-US">has proved also some theorems, containing new results of geometrical algebra.

<span Courier New"; color:black;mso-ansi-language:EN-US">Archimedes also was the greatestmathematical physicist of an antiquity. For the proof of theorems of mechanicshe used geometrical reasons. His composition “About floating bodies” hasput in pawn bases of a hydrostatics.

<span Courier New"; color:black;mso-ansi-language:EN-US">Decline of <st1:country-region w:st=«on»><st1:place w:st=«on»>Greece</st1:place></st1:country-region>.

<span Courier New";color:black;mso-ansi-language:EN-US"> After again of Egypt Romans in 31 AD great Greek Alexandria civilization has come todecline. Cicerones with pride approved, that as against Greeks Romans notdreamers that is why put the mathematical knowledge into practice, taking fromthem real advantage. However in development of the mathematics the contributionof roman was insignificant.

<st1:country-region w:st=«on»><st1:place w:st=«on»><span Courier New";color:black;mso-ansi-language: EN-US">INDIA

</st1:place></st1:country-region><span Courier New";color:black;mso-ansi-language:EN-US"> ANDARABS <span Courier New";color:black; mso-ansi-language:EN-US">

<span Courier New"; color:black;mso-ansi-language:EN-US">Successors of Greeks in a history ofmathematics were Indians. Indian mathematics were not engaged in proofs, butthey have entered original concepts and a number of effective methods. Theyhave entered zero as cardinal number and as a symbol of absence of units in thecorresponding category. <st1:State w:st=«on»><st1:place w:st=«on»>Moravia</st1:place></st1:State>(850 AD) has established rules of operations with zero, believing, however,that division of number into zero leaves number constant. The right answer fora case of division of number on zero has been given by Bharskar(born In 1114 AD -?), he possesses rules of actions above irrational numbers. Indianshave entered concept of negative numbers (for a designation of duties). We findtheir earliest use at Brahmagupta’s (around 630). Ariabhata (born in 476 AD-?) has gone further in use ofcontinuous fractions at the decision of the uncertain equations.

<span Courier New"; color:black;mso-ansi-language:EN-US">Our modern notation based on an itemprinciple of record of numbers and zero as cardinal number and use of adesignation of the empty category, is called Indo-Arabian. On a wall of thetemple constructed in <st1:country-region w:st=«on»><st1:place w:st=«on»>India</st1:place></st1:country-region>around 250 AD, some figures, reminding on the outlines our modern figures arerevealed.

<span Courier New"; color:black;mso-ansi-language:EN-US">About 800 Indian mathematics has achieved <st1:City w:st=«on»><st1:place w:st=«on»>Baghdad</st1:place></st1:City>. The term«algebra» occurs from the beginning of the name of book Al-Jebr vah-l-mukabala -Completionand opposition (

<span Courier New"; color:black">Аль<span Courier New"; color:black;mso-ansi-language:EN-US">-<span Courier New";color:black">джебр<span Courier New";color:black;mso-ansi-language:EN-US"> <span Courier New";color:black">ва<span Courier New";color:black;mso-ansi-language:EN-US">-<span Courier New";color:black">л<span Courier New";color:black;mso-ansi-language:EN-US">-<span Courier New";color:black">мукабала<span Courier New";color:black;mso-ansi-language:EN-US">)<span Courier New";color:black;mso-ansi-language:EN-US">,written in 830 astronomer and the mathematician Al-Horezmi.In the composition he did justice to merits of the Indian mathematics. Thealgebra of Al-Horezmi has been based on works of Brahmagupta, but in that work <st1:City w:st=«on»><st1:place w:st=«on»>Babylon</st1:place></st1:City> and Greek math influences are clearlydistinct. Other outstanding Arabian mathematician IbnAl-Haisam (around 965-1039) has developed a way ofreception of algebraic solvings of the square andcubic 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 trigonometryconcept of a tangent and cotangent. Nasyreddin Tusy (1201-1274 AD) in the “Treatise about a fullquadrangle” has regularly stated flat and spherical to geometry and thefirst has considered trigonometry separately from astronomy.

<span Courier New"; color:black;mso-ansi-language:EN-US">And still the most important contributionof arabs to mathematics of steel their translationsand comments to great creations of Greeks. <st1:place w:st=«on»>Europe</st1:place>has met these jobs after a gain arabs of <st1:place w:st=«on»>Northern Africa</st1:place> and <st1:country-region w:st=«on»><st1:place w:st=«on»>Spain</st1:place></st1:country-region>, and later works of Greekshave been translated to Latin.

<span Courier New"; color:black;mso-ansi-language:EN-US">MIDDLE AGES AND REVIVAL

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<span Courier New"; color:black;mso-ansi-language:EN-US">Medieval <st1:place w:st=«on»>Europe</st1:place>.

<span Courier New";color:black;mso-ansi-language:EN-US">The Roman civilization has not left an appreciable trace in mathematics as wastoo involved in the solving of practical problems. A civilization developed in <st1:place w:st=«on»>Europe</st1:place> of the early Middle Ages (around 400-1100 AD), wasnot productive for the opposite reason: the intellectual life has concentratedalmost exclusively on theology and future life. The level of mathematicalknowledge did not rise above arithmetics and simplesections from <st1:City w:st=«on»><st1:place w:st=«on»>Euclid</st1:place></st1:City>’s“Beginnings”. In Middle Ages the astrology was considered as the mostimportant section of mathematics; astrologists named mathematicians.

<span Courier New"; color:black;mso-ansi-language:EN-US">About 1100 in the West-Europeanmathematics began almost three-century period of development saved by arabs and the Byzantian Greeks ofa heritage of the Ancient world and the East. <st1:place w:st=«on»>Europe</st1:place>has received the extensive mathematical literature because of arabs owned almost all works of ancient Greeks. Translationof these works into Latin promoted rise of mathematical researches. All greatscientists of that time recognized, that scooped inspiration in works ofGreeks.

<span Courier New"; color:black;mso-ansi-language:EN-US">The first European mathematician deservinga mention became Leonardo Byzantian (Fibonacci). Inthe composition “the Book Abaca” (1202) he has acquainted Europeans withthe Indо-Arabian figures and methods of calculations andalso with the Arabian algebra. Within the next several centuries mathematicalactivity in <st1:place w:st=«on»>Europe</st1:place> came down.

<span Courier New"; color:black;mso-ansi-language:EN-US">Revival.

<span Courier New";color:black;mso-ansi-language:EN-US"> Amongthe best geometers of Renaissance there were the artists developed idea ofprospect which demanded geometry with converging parallel straight lines. Theartist Leon Batista Alberty (1404-1472) has enteredconcepts of a projection and section. Rectilinear rays of light from an eye ofthe observer to various points of a represented stage form a projection; thesection turns out at passage of a plane through a projection. That the drawnpicture looked realistic, it should be such section. Concepts of a projectionand section generated only mathematical questions. For example, what general geometricalproperties the section and an initial stage, what properties of two varioussections of the same projection, formed possess two various planes crossing aprojection under various corners? From such questions also there was aprojective 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 Greekgeometer Apollonius considered separately. <span Courier New";color:black">

<span Courier New"; color:black;mso-ansi-language:EN-US">I think that mathematics developed byattempts and mistakes. There is no perfect science today. Also math has ownmistakes, but it aspires to be more accurate. A development of math goes thru adevelopment of the society. Starting from counting on fingers, finishing onsolving difficult problems, mathematics prolong it way of development. Isuppose 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 newhigh-tech level and new methods of solving today’s problems. May in the futuresome man will find mistakes in our thinking, but I think it’s good, it’s goodthat math will not stop.

<span Courier New";mso-ansi-language:EN-US">Bibliography

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<span Courier New";color:black">Ван-дер-Варден

<span Courier New";color:black"> Б.Л. «Пробуждающаяся наука». Математикадревнего Египта, Вавилона и Греции<span Courier New"; color:black">. МОСКВА, 1959

<span Courier New";color:black;mso-bidi-font-style:italic"> Юшкевич

<span Courier New";color:black;mso-ansi-language:EN-US">A<span Courier New";color:black">.П. История математики в средниевека<span Courier New";color:black">. <span Courier New";color:black;mso-ansi-language:EN-US">М<span Courier New";color:black">ОСКВА<span Courier New";color:black;mso-ansi-language:EN-US">, 1961

<span Courier New";color:black">Даан-Дальмедико

<span Courier New";color:black"> А., ПейфферЖ. Пути и лабиринтыю Очерки по истории математикиМОСКВА, 1986

<span Courier New";color:black">Клейн Ф. Лекции о развитииматематики в

<span Courier New"; color:black;mso-ansi-language:EN-US">XIX<span Courier New";color:black"> столетии<span Courier New"; color:black">. МОСКВА, 1989

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