The history of trigonometry is inextricably linked with astronomy, because it was to solve the problems of this science that ancient scientists began to study the ratio of various quantities in a triangle.
Today, trigonometry is a microdivision of mathematics that studies the relationship between the values of the angles and lengths of the sides of triangles, as well as analyzing the algebraic identities of trigonometric functions.
The term "trigonometry"
The term itself, which gave the name to this section of mathematics, was first found in the title of the book, authored by the German scientist and mathematician Pitiscus in 1505. The word "trigonometry" is of Greek origin and means "measuring a triangle." To be more precise, we are not talking about the literal dimension of this figure, but about its solution, that is, determining the values of its unknown elements using known ones.
Trigonometry Overview
The history of trigonometry began more than two millennia ago. Initially, its occurrence was associated with the need to clarify the ratio of the angles and sides of the triangle. In the process of research, it turned out that the mathematical expression of these relations requires the introduction of special trigonometric functions, which were originally designed as numerical tables.
For many sciences related to mathematics, the trigonometry history was the impetus for development. The origin of the units of measurement of angles (degrees), associated with the research of scientists of Ancient Babylon, is based on a six-decimal system of calculus, which gave rise to modern decimal, used in many applied sciences.
It is assumed that initially trigonometry existed as part of astronomy. Then it began to be used in architecture. And over time, the expediency of applying this science in various fields of human activity arose. This, in particular, astronomy, sea and air navigation, acoustics, optics, electronics, architecture and others.
Early trigonometry
Guided by the data on preserved scientific relics, the researchers concluded that the history of trigonometry is connected with the work of the Greek astronomer Hipparchus, who first thought about finding ways to solve triangles (spherical). His works date back to the 2nd century BC.
Also, one of the most important achievements of those times was the determination of the relationship between legs and hypotenuse in right-angled triangles, which was later called the Pythagorean theorem.
The history of the development of trigonometry in ancient Greece is associated with the name of the astronomer Ptolemy - the author of the geocentric system of the world that prevailed before Copernicus.
Greek astronomers did not know the sines, cosines, and tangents. They used tables to find the value of the chord of a circle using a contractible arc. The units for measuring the chord were degrees, minutes, and seconds. One degree equated to the sixtieth of the radius.
Also, studies of the ancient Greeks promoted the development of spherical trigonometry. In particular, Euclid in his "Beginnings" gives a theorem on the laws of the ratios of the volumes of balls of different diameters. His works in this area became a kind of impetus in the development of related fields of knowledge. This, in particular, is the technology of astronomical instruments, the theory of cartographic projections, a system of celestial coordinates, etc.
Middle Ages: Research by Indian Scientists
Significant successes were achieved by Indian medieval astronomers. The death of ancient science in the IV century led to the displacement of the center of development of mathematics in India.
The history of trigonometry as a separate section of mathematical doctrine began in the Middle Ages. It was then that scientists replaced the chords with sinuses. This discovery allowed the introduction of functions relating to the study of the sides and angles of a right triangle. That is, it was then that trigonometry began to isolate itself from astronomy, turning into a branch of mathematics.
Ariabhata had the first sinus tables; they were drawn through 3 o , 4 o , 5 o . Later, detailed versions of the tables appeared: in particular, Bhaskara brought a table of sines through 1 about .
The first specialized treatise on trigonometry appeared in the X-XI century. The author was a Central Asian scientist Al-Biruni. And in his main work, The Canon of Mas'uda (Book III), the medieval author goes even deeper into trigonometry, giving a table of sines (with a step of 15 ') and a table of tangents (with a step of 1 °).
History of trigonometry in Europe
After the translation of Arabic treatises into Latin (XII-XIII c.), Most of the ideas of Indian and Persian scholars were borrowed by European science. The first mention of trigonometry in Europe dates back to the 12th century.
According to researchers, the history of trigonometry in Europe is connected with the name of the Englishman Richard Wallingford, who became the author of the essay “Four treatises on direct and inverted chords.” It was his work that became the first work, which is entirely devoted to trigonometry. By the 15th century, many authors in their writings mention trigonometric functions.
History of Trigonometry: New Time
In modern times, most scientists began to realize the extreme importance of trigonometry not only in astronomy and astrology, but also in other areas of life. This is, first of all, artillery, optics and navigation in long sea voyages. Therefore, in the second half of the XVI century, this topic interested many prominent people of that time, including Nicholas Copernicus, Johannes Kepler, Francois Viet. Copernicus devoted trigonometry to several chapters of his treatise On the Rotation of the Celestial Spheres (1543). A little later, in the 60s of the XVI century, Retik - a student of Copernicus - cites fifteen-digit trigonometric tables in his work “The Optical Part of Astronomy”.
Francois Viet in the “Mathematical Canon” (1579) gives a detailed and systematic, albeit unproven, characterization of flat and spherical trigonometry. And Albrecht Dürer became the one thanks to whom a sinusoid was born.
Merits of Leonard Euler
Giving trigonometry of modern content and appearance was the merit of Leonard Euler. His treatise "Introduction to the Analysis of the Infinite" (1748) contains a definition of the term "trigonometric functions", which is equivalent to the modern one. Thus, this scientist was able to determine the inverse functions. But that is not all.
The determination of trigonometric functions on the entire numerical line was made possible thanks to Euler's studies of not only permissible negative angles, but also angles of more than 360 °. It was he who in his works for the first time proved that the cosine and tangent of a right angle are negative. Decomposition of whole degrees of cosine and sine also became a merit of this scientist. The general theory of trigonometric series and the study of the convergence of the obtained series were not the objects of Euler's research. However, while working on related tasks, he made many discoveries in this area. It was thanks to his work that the history of trigonometry continued. Briefly in his writings, he also dealt with issues of spherical trigonometry.
Scopes of trigonometry
Trigonometry does not apply to applied sciences; in real everyday life, its tasks are rarely applied. However, this fact does not reduce its significance. Very important, for example, is the technique of triangulation, which allows astronomers to accurately measure the distance to nearby stars and to monitor satellite navigation systems.
Trigonometry is also used in navigation, music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medicine (for example, in decoding ultrasound examinations of ultrasound and computed tomography), pharmaceuticals, chemistry, number theory, seismology, meteorology , oceanology, cartography, many branches of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography, etc. The history of trigonometry and its role in science enii natural and mathematical sciences are studied to this day. Perhaps in the future there will be even more areas of its application.
The history of the origin of basic concepts
The history of the emergence and development of trigonometry dates back more than one century. The introduction of the concepts that form the basis of this section of mathematical science was also not instantaneous.
So, the concept of "sinus" has a very long history. Mention of various relationships of segments of triangles and circles is found in scientific works dating back to the III century BC. The works of such great ancient scholars as Euclid, Archimedes, Apollonius of Perga, already contain the first studies of these relationships. New discoveries required certain terminological clarifications. So, the Indian scientist Ariabhata gives the chord the name "jiva", meaning "bowstring." When Arabic mathematical texts were translated into Latin, the term was replaced by a sine with a similar meaning (that is, “bending”).
The word cosine appeared much later. This term is an abbreviation of the Latin phrase "additional sinus."
The emergence of tangents is associated with the decoding of the problem of determining the length of the shadow. The term “tangent” was introduced in the 10th century by the Arab mathematician Abul al-Wafa, who compiled the first tables for the definition of tangents and cotangents. But European scientists did not know about these achievements. The German mathematician and astronomer Regimontan rediscovered these concepts in 1467. The proof of the tangents theorem is his merit. And this term is translated as "relating".