Zakharova O.A.,Averyаnova O.V.

Pavlodar State university named after S.-M. Toraygyrov, , Republic of Kazakhstan

Pi: the history of its calculation

After the Athenians had won the Persians (479 B.C.), Athens consolidated its domination over the other Greek cities. That time Athens gained the greatest economic rising and incredible cultural flourishing. The democracy was established in VI century B.C. and stimulated the development of human knowledge in every science. The necessity of public speech led to the fact that the results of science exploration became common property. The first Athenian biggest school appeared to be the sophistical school, who were professional travelling teachers trying to exchange their knowledge for means of subsistence. The mathematical tasks studied by sophists were almost always connected with the one problem from the ancient cycle of the three famous problems of duplication of cube, trisection of angle and squaring the circle.

The problem of duplication of cube was linked with the legend where Apollo through an oracle’s mouth ordered to duplicate the temple altar in Delos. The population found themselves before the problem to deduce the cube side from the side A, i.e. to solve the equation or to find the cube root of 2. Trying to divide the given angle into three equal parts, the sophist Hippit from Enda contrived new curve, which was actually quadratissa. This curve was almost impossible to plot with the help of compasses and a ruler, but it worked well in some particular cases. Thus, even Pythagoreans could devide the right angle into three equal parts, predicating that every angle in equilateral triangle was of 60 degree. Let us divide the right angle into three equal parts. We should put on the ray segment , where we biuld equilateral triangle. As the angle =60^о, we will biuld bisector of the angle , in the result we will have the reqiured devision of angle into three parts.

When we set some other particular value of the angle, but not in common case, this problem can be solved, i.e. not every angle can be divided into three equal parts only with the help of ruler and compasses, and this fact was proved in the first half of the nineteenth century. If we use complementary tools, the problem of trisection of angle can be solved. Thus, the Alexandrine mathematician Nicomed (2^nd century B.C.) solved the problem of trisection of angle into three equal parts with the help of one curve named after Nicomed ‘the conchoid of Nicomed’. This scientist also described the device for drawing this curve. Archimed also gave an interesting problem solving, having used not compasses and ruler, but drawing scale which gave the length of definite segment. In the 2^nd thousand B.C. ancient Egyptian and Babylonian monuments saved the signs of problem of squaring the circle. However in the 5^th century B.C. Greek writings reveals firsthand target setting of squaring the circle. The historian Tristarchus narrated that the philosopher and astrologer Anaxogor (500-428 B.C.) being imprisoned dispelled sorrow by meditating on this problem. Hippocrate the Chiossian, one of the most famous geometrician of the 5^th century B.C., busied himself with the problem of squaring the circle. The contemporary of Socrate, sophist Antion considered that squaring the circle could be realised by inscribing the circle into the square, then bisecting the arches in corresponding sides, then constructing the right inscribed octagon, then biulding dioctagon etc., untill we could find polygon, which on account of sides frailness interflow with the circle. Since the square could be built equigraphic to polygon, the circle could be also squarred. However Aristotel explained that problem solving was approximate considering the polygon could never coincide with the circle.

Other different attempts to find the squaring the circle having endured thousand years failed. Only in the eigthies of the 19^th century the fact that area of squaring the circle with the help of compasses and ruler could not be calculated was definitely proved. Even in the 4^th century B.C. the Greek mathematician Dinatreum and Menechm for solving this problem used the curve found by Hippius from Elid in the 5^th century B.C.. However the ancient Greek scientists and their followers were unsatisfied with these solvings. Hence the problem being initially purely geometrical during centuries turned into important arithmetic-algebraic problem concerned with (pi). It also helped in development of new notions and ideas in mathematics.

The value of , i.e. 3,162… could be found in some Asian countries. The astronomer Van Fan (229-267 A.D.) asserted that , i.e. 3,155…, but Tszu Chun-Gi (428-499 A.D.) told ‘inexact’ value and ‘exact’ value showing that value of contained between 3,1415926 and 3,1415927.There are regulations in the Indian Sutra (7^th –5^th centuries B.C.) where it is readily apparent that . Aribhatta and Bhaskara set the value , i.e. 3,1416… .Al-Kashi (1421 A.D.) in his book ‘Measuring circumference’ found the value for , far excelling in precision all before known. Examining inscribed and attached polygons with the sides 800 335 168 he gained final result expressed in sexagesimal and denary fractions in the view of , i.e. – 16 exact signs.

If we mark the circle radius as , the matter concerns the construction of the circle, where the area equals , and the side is . It is a well-known fact that being in the ratio of the length of the circumference to its diameter is irrational number expressed by infinite decimal fraction of the value 3,141592… In 1766 mathematician Adrien Marie Legendre for the first time proved the irrationality of this number on the basis of the work of Johann Heinrich Lambert. Nevertheless this proof did not exclude the possibility of squaring the circle, since there were the irrational numbers, which could be built with the help of compasses and a ruler (e.g. ). Then in 1882 the German mathematician Ferdinand Lindemann found rigirous proof of the fact that was not only irrational, but this number cannot be the root of the algebraic equation . Consequently it was impossible to construct the interval equal to the length of the circemference with the help of compasses and a ruler.

In 1706 the English mathematician William Johnson firstly contrived the contemporary sign as a symbol of the ratio of the circle length to the circle diameter. As a symbol he took the first letter of the Greek word ‘periphery’, which meant the notion ‘circumference’. In 1736 the contrived sign became widespread after publication of Leonard Euler’s works. The number can be defined analytically. In the modern mathematics the number becomes not only the ratio of the circle length to the diameter but it also combines different formulas including the formulas of noneuclidean geometry and the famous formula of Leonard Euler. The last one establishes the connection of the number and the number е as following: , where . This interrelation and the other ones let mathematicians ascertain the nature of the number far deeper.

Bibliography cited:

1. Gleiser G.I. The history of mathematics at school (7-8 grades).-Moscow.: Prosveshchenie.-1982.

2. Stroyk D.Ya. The sketch of the history of mathematics.-Moscow.: Nauka.-1990.

3. Vileytner G. The history of mathematics from Descartes till the middle of 19^th century.-Мoscow.: The physical-mathematical publishing house.-1960.

4. Захарова О.А Математические концепции ученых Античности и Востока – Saarbrucken: LAP Lambert Academic Publishing, 2013.