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MOST IMPORTANT SCIENTIFIC DISCOVERIES
Logarithms. History and essence of scientific discovery
Directory / The most important scientific discoveries Throughout the XNUMXth century, the number of approximate calculations increased rapidly, primarily in astronomy. The study of planetary motions required colossal calculations. Astronomers could simply drown in impossible calculations. Obvious difficulties arose in other areas, such as finance and insurance. The main difficulty was the multiplication and division of multi-digit numbers, especially trigonometric quantities. Sometimes, tables of sines and cosines were used to reduce multiplication to easier addition and subtraction. A table of squares up to 100 was also compiled, with the help of which multiplication could be carried out according to a certain rule. However, these methods did not provide a satisfactory solution to the problem. They brought him tables of logarithms. “The discovery of logarithms was based on the properties of progressions well known by the end of the XNUMXth century,” write M.V. Chirikov and A.P. Yushkevich. “The connection between the members of the geometric profession and the arithmetic progression has been noted more than once by mathematicians, it was mentioned in Psammit” Archimedes. Another prerequisite was the extension of the concept of degree to negative and fractional exponents, which made it possible to transfer the connection just mentioned to a more general case ... Many ... authors have pointed out that multiplication, division, raising to a power and extracting a root exponentially correspond in arithmetic - in the same order - addition, subtraction, multiplication and division. Here, the idea of the logarithm of a number as an indicator of the power to which a given base must be raised in order to obtain this number was already hidden. It remained to transfer the familiar properties of a progression with a common term to any real exponents. This would give a continuous exponential function that takes any positive value, as well as its reciprocal logarithmic function. But this idea of deep fundamental significance was developed after several decades. Logarithms were invented independently by Napier and Burgi about ten years later. Their goal was the same - the desire to provide a new convenient means of arithmetic calculations. The approach turned out to be different. Napier kinematically expressed the logarithmic function, which allowed him to essentially enter the almost unexplored field of function theory. Bürgi remained on the basis of consideration of discrete progressions. It should be noted that for both the definition of the logarithm was not like the modern one. The first inventor of logarithms, the Scottish Baron John Napier (1550–1617), was educated at home in Edinburgh. Then, after traveling through Germany, France and Spain, at the age of twenty-one, he settled permanently on the family estate near Edinburgh. Napier took up mainly theology and mathematics, which he studied from the works of Euclid, Archimedes, Regiomontanus, Copernicus. “To the discovery of logarithms,” Chirikov and Yushkevich note, “Neper came no later than 1594, but only twenty years later he published his “Description of the amazing table of logarithms” (1614), which contained the definition of Neper’s logarithms, their properties and tables of logarithms of sines and cosines from 0 to 90 degrees with an interval of 1 minute, as well as the differences of these logarithms, giving the logarithms of tangents.He set out the theoretical conclusions and explanations of the method for calculating the table in another work, prepared probably before the "Description", but published posthumously, in "Building an amazing tables of logarithms "(1619). Let us mention that in both works Napier also considers some issues of trigonometry. Especially known are the "analogies" convenient for logarithm, i.e. Napier's proportions used in solving spherical triangles on two sides and the angle between them, and also on two corners and the side adjacent to them. Napier from the very beginning introduced the concept of the logarithm for all values of continuously changing trigonometric quantities - sine and cosine. In the then state of mathematics, when there was still no analytical apparatus for the calculus of infinitesimals, the natural and only means for this was the kinematic definition of the logarithm. Perhaps the influence and traditions that dated back to the Oxford school of the XIV century were not left without influence here. Napier's definition of the logarithm is based on the kinematic idea, which generalizes to continuous quantities the connection between the geometric profession and the arithmetic progression of the indicators of its members. Napier presented the theory of logarithms in the work "Construction of amazing tables of logarithms", published posthumously in 1619 and republished in 1620 by his son Robert Napier. Here are excerpts from it: "The table of logarithms is a small table with which you can find out, by means of very easy calculations, all geometric dimensions and movements. It is justly called small, because it surpasses the tables of sines in volume, very easy, because with its help all complex multiplications, divisions and root extractions, and all figures and movements in general, are measured by doing the easier addition, subtraction, and division by XNUMX. It is made up of numbers in continuous proportion. 16. If you subtract its 10000000th part from the full sine with seven zeros added, and its 10000000th part from the number thus obtained, and so on, then this series can easily be continued up to one hundred numbers in the geometric ratio that exists between the full a sine and a sine less than one, namely between 10000000 and 9999999, and we will call this series of proportionals the First Table. 17. The second table follows from the full sine with six added zeros through fifty other numbers decreasing proportionally in a ratio that is the simplest and possibly closest to the ratio between the first and last numbers of the First Table. Since the first and last numbers of the First Table are 10000000.0000000 and 9999900.004950, it is difficult to form fifty proportional numbers in this respect. A close and at the same time simple ratio is 100000 to 99999, which can be continued with sufficient accuracy by adding six zeros to the full sine and successively subtracting its 100000th part from the previous one. This table contains, in addition to the full sine, which is the first number, another fifty proportional numbers, the last of which (if you are not mistaken) will be 9995001.222927. 18. The Third Table consists of sixty-nine columns, and in each column there are twenty-one numbers, following in a relation which is the simplest and as close as possible to the relation existing between the first and last members of the Second Table. Therefore, its first column can be very easily obtained from the full sine with five added zeros and from subsequent numbers by subtracting the 2000th part from them. 19. The first numbers of all columns follow from the full sine with four zeros added in a ratio that is the simplest and close to the ratio that exists between the first and last numbers of the first column ... 20. In the same ratio, a progression should be formed from the second number of the first column for the second numbers of all columns, and from the third for the third, and from the fourth for the fourth, and accordingly from the rest for the rest. Thus, from any number of the previous column, by subtracting its hundredth part, a number of the same order of the next column is obtained ... 21 .... these three tables (after they have been compiled) are sufficient to calculate the table of logarithms." In 1620, the Swiss Jost Burgi (1552–1632), a highly skilled mechanic and watchmaker, published the book "Tables of Arithmetic and Geometric Progressions, together with a thorough instruction on how they should be understood and used with benefit in all kinds of calculations" (1620). As Burgi himself wrote, he proceeded from considerations of the correspondence between multiplication in geometric progression and addition in arithmetic. The problem was to choose a progression with a denominator close enough to one, so that its terms follow each other at intervals small enough for practical calculations. However, Bürgi's tables did not receive significant distribution. They could not compete with Napier's tables, which were more convenient and by that time already widely known. Neither Napier nor Burga had, strictly speaking, a base of logarithms, since the logarithm of unity differs from zero. And much later, when we had already switched to decimal and natural logarithms, the definition of the logarithm as an indicator of the degree of a given base had not yet been formulated. It appears in manuals for the first time, probably in W. Gardiner (1742). However, Gardiner himself used the papers of the mathematics teacher W. Jones. The wide spread of the modern definition of the logarithm was more than others promoted by Euler, who used the term "foundation" in this regard. The term "logarithm" belongs to Napier, it originated from a combination of the Greek words "ratio" and "number", and means "number of ratio". Although initially Napier used a different term - "artificial numbers". Napier's tables, adapted to trigonometric calculations, were inconvenient for operations with given numbers. To eliminate these shortcomings, Napier proposed to compile tables of logarithms, taking zero for the logarithm of one, and just one for the logarithm of ten. He made this proposal during a discussion with Henry Briggs (1615–1561), a professor of mathematics at Gresh College in London, who visited him in 1631, and who himself thought about how to improve the tables of logarithms. Due to failing health, Napier could not engage in the implementation of his plan, but he indicated the idea of two computational methods developed further by Briggs. Briguet published the first results of his painstaking calculations - "The First Thousand Logarithms" (1617) in the year of Napier's death. Here were given the decimal logarithms of numbers from 1 to 1000 with fourteen digits. Most of the decimal logarithms of prime numbers Briguet found by extracting square roots. Later, already becoming a professor at Oxford, he published Logarithmic Arithmetic (1624). The book contained fourteen-digit logarithms of numbers from 1 to 20 and from 000 to 90. The remaining gap was filled by the Dutch bookseller and mathematician Andrian Flakk (1600–1667). Somewhat earlier, seven-digit decimal tables of logarithms of sines and tangents were calculated by Briggs' colleague at Gresham College, a graduate of Oxford University, Edmund Gunter (1581–1626), who published them in the Code of Triangles (1620). The discovery of Napier in the very first years gained exceptionally wide popularity. Many mathematicians have taken up the compilation of logarithmic tables and their improvement. So, Kepler in Marburg in 1624-1625 he applied logarithms to the construction of new tables of planetary motions. In the appendix to the second edition of Napier's Description (1618), several natural logarithms were also calculated. Here you can see the approach to the introduction of the limit. Most likely, this addition belongs to V. Ootred. Soon, the London mathematics teacher John Speydell published tables of natural logarithms of numbers from 1 to 1000. The term "natural logarithms" was introduced by P. Mengoli (1659), and somewhat later by N. Mercator (1668). The practical significance of the calculated tables was very great. But the discovery of logarithms was also of profound theoretical significance. It brought to life research that the first inventors could not even dream of, pursuing the goal only to facilitate and speed up arithmetic and trigonometric calculations with large numbers. Napier's discovery, in particular, opened the way to the realm of new transcendental functions and gave powerful stimuli to the development of analysis. Author: Samin D.K.
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