The history of trigonometry: the emergence and development

The history of trigonometry is inextricably linked with astronomy, because it was to solve the problems of this science that ancient scientists began to investigate the relationships of various quantities in the triangle.

To date, trigonometry is a micro-division of mathematics that studies the relationship between the values of the angles and lengths of the sides of triangles, as well as dealing with the analysis of algebraic identities of trigonometric functions.

The term "trigonometry"

The term itself, which gave the name to this section of mathematics, was first discovered in the title of the book by the German scientist-mathematician Pitiscus in 1505. The word "trigonometry" has a Greek origin and means "I measure a triangle". To be more precise, we are not talking about the literal measurement of this figure, but about its solution, that is, determining the values of its unknown elements with the help of known ones.

General information about trigonometry

The history of trigonometry began more than two millennia ago. Initially, its emergence was associated with the need to clarify the relationship between the angles and sides of the triangle. In the process of research it was found out that the mathematical expression of these relations requires the introduction of special trigonometric functions, which were initially formalized as numerical tables.

For many mathematics adjacent to mathematics, the trigonometry story became the impetus for development. The origin of the units of measurement of angles (degrees), connected with the studies of the scientists of Ancient Babylon, is based on the sixty-tier system of calculus, which gave rise to the 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.

Trigonometry in the Early Ages

Guided by the data on the surviving scientific relics, the researchers concluded that the history of the origin of trigonometry is associated 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 is the definition of the ratio of the legs and the hypotenuse in right-angled triangles, which later became known as the Pythagorean theorem.

The history of the development of trigonometry in Ancient Greece is associated with the name of the astronomer Ptolomey - the author of the geocentric system of the world, which dominated Copernicus.

Greek astronomers were not aware of sinuses, cosines and tangents. They used tables to find the value of the chord of a circle by means of a shrinking arc. The units for measuring the chord were degrees, minutes and seconds. One degree was equal to the sixtieth part of the radius.

Also, studies of the ancient Greeks advanced the development of spherical trigonometry. In particular, Euclid in his "Beginnings" gives a theorem on the regularities of the ratio of the volumes of balls of different diameters. His works in this field have become a kind of impetus in the development of related fields of knowledge. This is, in particular, the technology of astronomical instruments, the theory of cartographic projections, the system of celestial coordinates, and so on.

Middle Ages: research of Indian scientists

Indian medieval astronomers have made significant progress. The destruction of ancient science in the IV century led to the movement of the center for the development of mathematics in India.

The history of the origin of trigonometry as an isolated section of mathematical theory began in the Middle Ages. It was then that scientists replaced chords with sinuses. This discovery made it possible to introduce functions relating to the study of the sides and angles of a right-angled triangle. That is, it was then that trigonometry began to become obsolete from astronomy, turning into a branch of mathematics.

The first tables of sines were in Aryabhata, they were conducted through 3 ^about , 4 ^about , 5 ^o . Later, detailed versions of the tables appeared: in particular, Bhaskara cited a table of sinuses at 1 ^o .

The first specialized treatise on trigonometry appeared in the 10th-11th century. The author was the Central Asian scholar Al-Biruni. And in his main work "The Canon of Mas'ud" (Book III), the medieval author is further deepened into trigonometry, giving a table of sines (in 15 'steps) and a table of tangents (in 1 ° increments).

History of the development of trigonometry in Europe

After the translation of the Arabic treatises into Latin (XII-XIII c), most of the ideas of Indian and Persian scholars were borrowed by European science. The first mentions of trigonometry in Europe date back to the 12th century.

According to researchers, the history of trigonometry in Europe is associated with the name of the Englishman Richard of Wallingford, who became the author of the work "Four treatises on direct and inverted chords". It was his work that became the first work, which is entirely devoted to trigonometry. By the XV century, many authors in their writings mention trigonometric functions.

The 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 cruises. Therefore, in the second half of the 16th century this topic interested many outstanding people of the time, including Nicholas Copernicus, Johannes Kepler, Francois Vieta. Copernicus selected trigonometry several chapters of his treatise "On the rotation of the heavenly spheres" (1543). A little later, in the 60s of the 16th century, Retik, a pupil of Copernicus, leads in his work "Optical part of astronomy" fifteen-digit trigonometric tables.

François Viet in the "Mathematical Canon" (1579) gives a detailed and systematic, though unproven, description of flat and spherical trigonometry. And Albrecht Durer became the one thanks to whom the sinusoid appeared.

Leonard Euler's merits

The creation of trigonometry of the modern content and kind became the merit of Leonhard Euler. His treatise "Introduction to the analysis of the infinite" (1748) contains the 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's not all.

The determination of trigonometric functions on the whole numerical line became possible due to Euler's research not only of permissible negative angles, but also angles of more than 360 °. It was in his works that he proved for the first time that the cosine and tangent of the right angle are negative. The decomposition of whole degrees of cosine and sinus also became the merit of this scientist. The general theory of trigonometric series and the study of the convergence of the series obtained were not objects of Euler research. However, working on solving related problems, he made many discoveries in this area. It was thanks to his work that the history of trigonometry continued. In his works he briefly touched upon the questions of spherical trigonometry.

Areas of application of trigonometry

Trigonometry does not apply to applied sciences, in its everyday life its tasks are rarely used. 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 control satellite navigation systems.

Trigonometry is also used in navigation, music theory, acoustics, optics, financial markets analysis, electronics, probability theory, statistics, biology, medicine (for example, in decoding ultrasound ultrasound and computed tomography), pharmacy, chemistry, number theory, seismology, meteorology , Oceanography, cartography, many sections of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, engineering, computer graphics, crystallography, etc. The history of trigonometry and its role in the study enii natural and mathematical sciences are studied to this day. Perhaps in the future areas of its application will become even greater.

History of the origin of the basic concepts

The history of the origin and development of trigonometry has more than one century. The introduction of concepts, which form the basis of this section of mathematical science, was also not instantaneous.

So, the concept of "sine" has a very long history. Mentions about the various relations of segments of triangles and circles are found even in scientific works dating back to the 3rd century BC. The works of such great ancient scholars as Euclid, Archimedes, Appolonius Pergsky, already contain the first studies of these relations. New discoveries required certain terminological refinements. So, the Indian scientist Aryabhata gives the chord the name "jiva", meaning "bow string". When the Arabic mathematical texts were translated into Latin, the term was replaced by a sine of similar value (i.e., "bend").

The word "cosine" appeared much later. This term is an abridged version of the Latin phrase "additional sine".

The origin of tangents is associated with the decoding of the problem of determining the length of a shadow. The term "tangent" was introduced in the 10th century by the Arab mathematician Abul-Wafa, who compiled the first tables for the determination of tangents and cotangents. But European scientists did not know about these achievements. The German mathematician and astronomer Regomontan re-opens these concepts in 1467. The proof of the theorem of tangents is his merit. And the term is translated as "relating".