Pythagoreans also opened that the sum of some couples 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 16 Such three of numbers as 3, 4 and 5 or 5, 12 and 13, are called as Pythagorean numbers. They have geometrical interpretation if to equate two numbers from the three to lengths of legs of a rectangular triangle, the third will be equal to length of its hypotenuse. Such interpretation, apparently, led Pythagoreans to understanding of more general fact known nowadays under the name of Pythagorean theorem according to which the square of length of a hypotenuse is equal in any rectangular triangle to the sum of squares of lengths of legs.
Ancient Greeks solved the equations with unknown by means of geometrical constructions. Special constructions were developed for performance of addition, subtraction, multiplication and division of pieces, extraction of square roots from lengths of pieces; nowadays this method is called as geometrical algebra.
The tasks and decisions provided in papyruses are formulated purely retsepturno, without any explanations. Egyptians dealt only with the simplest types of quadratic equations and arithmetic and geometrical progressions, and therefore those general rules which they could output, were also the simplest look. Neither the Babylon, nor Egyptian mathematics had no the general methods; all arch of mathematical knowledge represented a congestion of empirical formulas and rules.
Evdoks (apprx. 408–355 BC) was the greatest of the Greek mathematicians of the classical period conceding on the importance of the received results only to Archimedes. It entered concept of size for such objects as pieces of straight lines and corners. Having concept of size, Evdoks logically strictly proved a Pythagorean method of the address with irrational numbers.
Basis of all mathematical analysis is the concept of a limit. Speed in a timepoint is defined as a limit to which the average speed of d/t when value t everything approaches zero closer aspires. The differential calculus gives the convenient general method of finding of speed of change of function f in calculations (x) at any value x. This speed received the name of a derivative. From a community of record f (x) it is visible that the concept of a derivative is applicable not only in the tasks connected with need to find the speed or acceleration, but also in relation to any functional dependence, for example, to any ratio from the economic theory. One of the main appendices of differential calculus are so-called tasks on a maximum and a minimum; other important circle of tasks – finding of a tangent to this curve.
To Pythagoreans we are in many respects obliged to that by mathematics which then was is systematized is stated and proved at Euclid's Beginnings. There are bases to believe what exactly they opened that is nowadays known as theorems of triangles, parallel straight lines, polygons, circles, spheres and regular polyhedrons.
For Pythagoreans any number represented something bigger, than quantity. For example, number 2 according to their view meant distinction and therefore was identified with opinion. The four represented justice, as this first equal to work of two identical multipliers.
About 1100 in the West European mathematics almost three-century period of development of the heritage of the Ancient world and the East kept by Arabs and the Byzantine Greeks began. As Arabs owned almost all works of ancient Greeks, Europe received extensive mathematical literature. Transfer of these works to Latin promoted rise of mathematical researches. All great scientists of that time recognized that derived inspiration from works of Greeks.
Apollony (apprx. 262–200 BC) lived during the Alexandria period, but its main work is sustained in the spirit of classical traditions. The analysis of conic sections offered them – a circle, an ellipse, a parabola and a hyperbole – was the culmination of development of the Greek geometry. Apollony also became the founder of quantitative mathematical astronomy.