# Opening logarithms

**Opening logarithms** was based on the well-known by the end of the 16th century. properties of progressions. Many mathematicians noticed that in addition, subtraction, multiplication and division, arithmetic progression (in the same order) corresponds to multiplication, division, exponentiation, and extraction of a root in a geometric progression. The real triumph was the discovery of the logarithms as exponents. The main properties of logarithms allow you to replace the multiplication, division, erection in degree and extract the root with more simple addition, subtraction, multiplication and division.

Logarithms were invented independently of each other by Neper and Burgi in the early 16th century. In 1614, Naper published his "Description of the amazing table of logarithms", which contained the definition of logarithms (and their properties), which we now call Neperovye logarithms, and in 1620 the Swiss Iost Burgi (1552-1632) published the book "Tables of arithmetic and geometrical progressions, together with a thorough instruction, how they should be understood and used with advantage in all calculations. "However, the tables of Burg are not widely spread.

The discovery of the logarithms by Neper, in the first few years became extremely famous. With logarithms, many calculations have gone dozens of times faster and easier. No wonder the great French mathematician Pierre Simon Laplace said that "the invention of logarithms extended life".

The term "logarithmus" also belongs to Nepper. It originated from a combination of Greek words: logos - "relation" and arithmus - "number", that is, it meant the number of relations. However, neither Nepper nor Burgey had, strictly speaking, the base of the logarithms, since the logarithm of the unit differs from zero. Even much later, when we have already switched to decimal and natural logarithms, the definition of the logarithm as an indicator of the degree of this base has not yet been formulated.

The Napier tables, adapted to trigonometric calculations, were inconvenient for dealing with such numbers. In 1615, Naper met Henry Briggs (1561-1631), a professor of mathematics at Gresham College, who also thought about how to improve the tables of logarithms. During the conversation with Briggs, Nepper proposed to compile the tables of logarithms, taking one zero for the logarithm, and just one for the logarithm of ten, and thus eliminate the existing shortcomings. Nepper could not bring his ideas to life because of the shaken health, but he pointed to the idea of two computational techniques developed further by Briggs.

In 1617, Briggs published the first results of his painstaking calculations - "The First Thousand Logarithms". In these tables, eight-digit decimal logarithms of numbers from 1 to 1000 were given. Later (in 1624), after he became a professor at Oxford, Briggs released "Logarithmic arithmetic". The book contained fourteen-digit logarithms of numbers from 1 to 20000 and from 90000 to 100000.

The term "natural logarithm" was introduced in 1659 by Pietro Mengoli, an Italian mathematician who taught at the University of Bologna, and the sign *Log* was introduced in 1624 by Johannes Kepler (1571-1630), the famous German mathematician, an astronomer and an optometrist who discovered the laws of planetary motion.

It should be noted the enormous work done by the Dutch mathematician Andrian Vlakk. In 1628, he published ten-digit tables of logarithms from 1 to 100000. The Vlakka tables formed the basis for most of the subsequent tables, and their authors made many changes to the structure of the logarithmic tables and corrections.

For the base of the Brigg logarithms, as already noted, the number 10 was taken. In the case of Nperov logarithms, the constant itself (the base of the logarithms) is not explicitly defined. The first known use of this constant, where it was denoted by a letter, occurs in the letters of Gottfried Leibniz to Christian Huygens in 1690 and 1691. The letter *e* began to be used by Leonard Euler in 1727, and the first publication using this letter was his work "Mechanics, or Motion Science Imposed Analytically" (1736). Accordingly, *e* is sometimes called the Euler number. In 1874, the French mathematician S. Ermit proved that the base of the natural logarithms of e is transcendental (as number pi). The value of *e* = 2,71828182845904523536028747135266249.

The number *e* can be remembered by the following mnemonic reception: two and seven, then two times the year of Leo Tolstoy's birth (1828), and then the angles of an isosceles right-angled triangle (45, 90 and 45 degrees). And here is one more original way of memorizing: it is suggested to memorize the number e with an accuracy of three decimal places through the "number of the devil": divide 666 by a number composed of digits 6 - 4, 6 - 2, 6 - 1 (three sixes, from which in the reverse order remove the first three powers of the two): 666/245 = 2,718.