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Basic formulas of spherical trigonometry conclusion. Spherical trigonometry

Spherical trigonometry

important a private section of trigonometry used in astronomy, geodesy, navigation and other industries is spherical trigonometry, which considers the properties of the angles between great circles on a sphere and the arcs of these great circles. The geometry of a sphere differs significantly from Euclidean planimetry; Thus, the sum of the angles of a spherical triangle, generally speaking, differs from 180°; a triangle can consist of three right angles. In spherical trigonometry, the lengths of the sides of a triangle (the arcs of the great circles of the sphere) are expressed in terms of the central angles corresponding to these arcs. Therefore, for example, the spherical sine theorem is expressed as:

and there are two cosine theorems dual to each other.

Application of trigonometric calculations

Trigonometric calculations are used in almost all areas of geometry, physics and engineering. Great importance has a triangulation technique that allows you to measure distances to nearby stars in astronomy, between landmarks in geography, and control satellite navigation systems. Also of note is the application of trigonometry in such areas as music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound and computed tomography), pharmaceuticals, chemistry, number theory ( and, as a result, cryptography), seismology, meteorology, oceanology, cartography, many branches of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography.

There are many areas where trigonometry and trigonometric functions are applied. For example, the triangulation method is used in astronomy to measure the distance to nearby stars, in geography to measure distances between objects, and in satellite navigation systems. The sine and cosine are of fundamental importance for the theory of periodic functions, for example, in the description of sound and light waves.

Trigonometry or trigonometric functions are used in astronomy (especially for calculating the position of celestial objects, when spherical trigonometry is required), in marine and air navigation, in music theory, in acoustics, in optics, in the analysis of financial markets, in electronics, in probability theory, in statistics, biology, medical imaging (e.g. computed tomography and ultrasound), pharmacies, chemistry, number theory (hence also cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, in architecture, in phonetics, in economics, in electrical engineering, in mechanical engineering, in civil engineering, in computer graphics, cartography, crystallography, game development and many other areas.

Spherical trigonometry

spherical triangles. On the surface of a ball, the shortest distance between two points is measured along the circumference of a great circle, that is, a circle whose plane passes through the center of the ball. Vertices of a spherical triangle are the intersection points of the three rays emerging from the center of the ball and the spherical surface. Parties a, b, c of a spherical triangle are those angles between the rays that are smaller (if one of these angles is equal, then the spherical triangle degenerates into a semicircle of a great circle). Each side of the triangle corresponds to an arc of a large circle on the surface of the ball (see figure).

corners A, B, C spherical triangle opposite sides a, b, c respectively, are, by definition, smaller than , the angles between the arcs of great circles corresponding to the sides of the triangle, or the angles between the planes defined by these rays.

Spherical trigonometry deals with the study of the relationship between the sides and angles of spherical triangles (for example, on the surface of the Earth and on the celestial sphere). However, physicists and engineers in many problems prefer to use rotation transformations rather than spherical trigonometry.

Properties of spherical triangles. Each side and angle of a spherical triangle is, by definition, less than .

The geometry on the surface of the ball is non-Euclidean; in every spherical triangle the sum of the sides is between 0 and , the sum of the angles is between and . In every spherical triangle, the larger angle lies opposite the larger side. The sum of any two sides is greater than the third side, the sum of any two angles is less than plus the third angle.

A spherical triangle is a part of the surface of the celestial sphere, bounded by three arcs of great circles (Fig. 7).

Rice. 7. Spherical triangle.

The arcs forming a spherical triangle intersect each other only at its vertices, and are called sides of the spherical triangle; they are measured by the corresponding central angles. The angles of a spherical triangle are measured by the dihedral angles formed by the planes of the corresponding great circles; they are equal to the angles between the tangents at the vertices drawn to the corresponding sides of the spherical triangle. Usually, the angles are indicated by capital letters of the Latin alphabet A, B, C ..... sides - by lowercase letters, and side a always lies against the corner (vertex) A, etc.

A spherical triangle with all sides less than 180° is called a simple triangle.

A spherical triangle is called a right triangle if one of its angles is right, and a quarter (quadrant) if one of its sides contains 90 °.

Let's call the pole of the great circle a point on the surface of the sphere, lying at an angular distance of 90 ° from any point of the circumference of this great circle, Then the spherical triangle,

A nickname formed by the poles of great circles whose arcs bound a given spherical triangle ABC (provided that the poles of the sides of this triangle are located in the direction of the corresponding vertices) is called a polar datum.

Rice. 8. Polar triangle.

Relationship between elements of a spherical triangle ABC and polar to it (Fig. 8) is given by the following relations:

Given a relation of the form

then for a spherical triangle polar to the given one, we have

i.e., the relation

Such a transformation is called a correlation.

1. The main systems of relations connecting the various elements of a spherical triangle. System I. Relations between three sides and one angle (cosine theorem)

System II. Relations between two sides and two opposite angles (sine theorem):

System III. Relations between three sides b two angles (formulas of five elements):

System IV. Relationship between two sides and two angles:

Correlating ratios (1.1.005) gives:

Each of the relations (1.1.010) connects three angles and one side.

Using the correlation of relations (1.1.008), we obtain the relations between three angles and two sides:

System V. Relations between the six elements (Cagnoli's formulas):

In the case of a right-angled spherical triangle, the following relations are valid (Fig. 9):

Formulas (1.1.013) can be obtained using Napier's rule, based on the Napier pentagon (Fig. 10) with the indicated order of designating the sides of this pentagon.

Rice. 9. Right-angled spherical triangle.

Rice. 10. The first scheme for Napier's rule.

The cosine of the side of the Napier pentagon is:

1) the product of the sines of opposite sides;

2) the product of the cotangents of the adjacent sides.

2. Quadrant (quarter) spherical triangle. Using the correlation of formulas (1.1.013), formulas for a quadrant spherical triangle are obtained

Formulas (1.1.014) can be derived from general relations by setting in them or according to Napier's rule (Fig. 11).

With the quadruple spherical triangle ABC, one can associate an attached spherical triangle (Fig. 12), the side of which is a continuation of side c and complements it to 90 °. Then in a spherical triangle there are two

sides are 90°, angle . In an attached spherical triangle

therefore, applying formulas (1.1.013) to it gives formulas (1.1.014).

Rice. 11. The second scheme for Napier's rule.

The above formulas allow you to determine any three elements of a spherical triangle, if the other three are known.

Main practical techniques calculations, as well as approximate formulas in the case of small angles, can be found in manuals on spherical astronomy, -.

Values trigonometric functions for arguments - angles expressed in various measures are taken from the corresponding tables - .

Complete information about tables of natural values ​​(and logarithms) of trigonometric functions and other mathematical tables that may be useful to the calculator is contained in special reference manuals -.

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G.P. Matvievskaya Spherics and spherical trigonometry in antiquity and in the medieval East / Development of methods astronomical research. Issue 8, Moscow-Leningrad, 1979

G.P. Matvievskaya

Spherics and spherical trigonometry in antiquity and in the medieval East

1. In antiquity and in the Middle Ages, the needs of astronomy served as the most important stimulus for the development of many branches, mathematics and, above all, spherical trigonometry, which was a mathematical apparatus for solving specific astronomical problems. With the development of astronomy, the complexity of its problems and the increase in the requirements for the accuracy of calculations, this apparatus was gradually improved and, accordingly, the content of spherical trigonometry was enriched. It was expounded both in astronomical treatises - as an introductory section of astronomy - and in special mathematical works.

Of particular importance for the history of spherical trigonometry are ancient Greek writings on the sphere - a science that included elements of astronomy, geometry on a sphere and trigonometry. By the 4th c. BC e. it was fully developed and regarded as an auxiliary astronomical discipline. The earliest known works on the sphere were written in the period of the 4th century BC. BC e. - I century. n. e. such outstanding scientists of antiquity as Autolik, Euclid, Theodosius, Hypsicles, Menelaus.

These works allow you to visually get acquainted with the initial stage in the development of spherical trigonometry.

All the results obtained by the Greeks in the field of astronomy and trigonometry were, as is known, generalized in the 2nd century BC. in Ptolemy's work entitled A Mathematical Collection in 13 Books. Later, probably in the 3rd century, it was called the “great” book, from which in the Middle Ages the name “Almagest”, which became generally accepted, came from: Latin the word "al-majisti" is an Arabized form of "megiste" (the greatest).

In contrast to the "great" book of Ptolemy, the writings of his predecessors, necessary for astronomical calculations and combined in the late Hellenistic period (no later than the 4th century) in one collection, were called "Small Astronomy". They had to be studied after Euclid's Elements, so that the Almagest could be understood. In Arabic literature, therefore, they appear under the name of "middle books" (kutub al-mutawasita).

This collection includes the works of Euclid "Data", "Optics", "Phenomena" and the pseudo-Euclidean "Katoptrik", the works of Archimedes ("On the ball and the cylinder", "Measurement of the circle", "Lemmas"), Aristarchus ("On quantities and distances Sun and Moon"), Hypsicles ("On the ascent of the constellations along the ecliptic"), Autolika ("On the moving sphere", "On the rising and setting of the fixed stars"), Theodosius ("Sphere", "On days and nights", "On dwellings") and Menelaus ("Sphere"). The work of Menelaus was added to the Minor Astronomy, possibly at a later time.

The Arabic translation of the “middle” books, including works on the sphere, appeared among the first translations of the works of the classics of Greek science. Later they were repeatedly commented on. Among the translators and commentators one can name such outstanding scientists as Kosta ibn Luka (IX century), al-Makhani (IX century), Sabit ibn Korra (X century), Ibn Iraq (X-XI centuries), Nasir ad -Din at-Tusi (XIII century) and others.

To the Greek “Minor Astronomy”, Eastern scholars later added the works “On the Measurement of Figures” by Banu Musa, “Data” and “The Book of the Complete Quadrilateral” by Sabit ibn Korra, “Treatise on the Complete Quadrilateral” by Nasir ad-Din at-Tusi.

The need for a deep acquaintance with "middle" books was well recognized by Eastern mathematicians and astronomers and was emphasized even in the 17th century. in the widely known bibliographic encyclopedia of Hajji Khalifa "Removing the Veil from the Titles of Books and Sciences". The text of these treatises, as well as commentaries to them, has been preserved in numerous Arabic manuscripts. These include, for example, a handwritten collection that has not yet been studied by anyone, stored in the State Public Library. M. E. Saltykov-Shchedrin in Leningrad (collection of Khanykov, No. 144).

Back in 1902, the well-known historian of mathematics A. Bjornbo noted with regret that too little attention was paid to that area of ​​ancient science, which can be defined as an "introduction to astronomy" and which is reflected in "average" books. In particular, he insisted on the need for a full-fledged critical edition of the text of the works and, in connection with this, raised the question of studying their Arabic versions. A great merit in the study of "small astronomy" belongs to A. Bjornbo himself, as well as F. Gulch, I.L. Geiberg, P. Tannery, A. Chvalina, J. Mozhene, and others. However, far from everything has been done in this direction so far. This applies especially to the "middle" books in the Arabic interpretation.

Scientists of the Eastern Middle Ages often made significant additions to Greek works, offered their own proofs of theorems, and sometimes introduced new ideas into the ancient theory. From this point of view, the Arabic versions of the works devoted to the sphere deserve great attention. Of particular importance is the study of the comments on the work of Menelaus, compiled by Abu Nasr ibn Iraq and Nasir ad-Din at-Tusi, who played a significant role in the history of spherical trigonometry.

2. The most ancient writings on the sphere that have come down to us - and, in general, from the mathematical writings of the Greeks - are the treatises of Autolik from Pitana (c. 310 BC) “On the Revolving Sphere” and “On Sunrises and Sunsets”. Both of them deal with questions of geometry on a sphere as applied to astronomy.

Autolik studies a sphere rotating around an axis and circular sections on it: large circles passing through both poles, small circles obtained by cutting the sphere with planes perpendicular to the axis, and large circles passing obliquely to it. The movement of the points of these circles is considered in relation to some fixed secant plane passing through the center. It is easy to see here a model of the celestial sphere with celestial meridians, parallels, equator, ecliptic and horizon. The presentation, however, is carried out in purely geometric language and astronomical terms do not apply.

In the 12-sentence essay “On a Moving Sphere,” Autolik introduces the concept of uniform motion (“a point moves uniformly if it travels equal paths in equal times”) and applies this concept to a rotating sphere. First of all, he shows that points of its surface that do not lie on the axis, during uniform rotation, describe parallel circles with the same poles as the sphere, and with planes perpendicular to the axis (Proposition 1). Further, it is proved that in equal time all points of the surface describe similar arcs (Proposition 2) and vice versa, i.e. if two arcs of parallel circles are traversed in equal time, then they are similar (Proposition 3).

Introducing the concept of the horizon - a large circle that separates the part of this sphere visible to an observer located in the center of the sphere from the invisible - Autolik considers the movement of surface points in relation to it. Various possible positions of the horizon are investigated, when it is perpendicular to the axis, passes through the poles and is inclined to the axis. In the first case (which takes place at the terrestrial pole), no point on the surface of the sphere, with uniform rotation, will be ascending or setting; all points of the visible part always remain visible, and all points of the invisible part remain invisible (Proposition 4).

In the second case, which takes place at the earth's equator, all points on the surface of the sphere rise and set, being the same time above and below the horizon (proposition 5).

Finally, in the last - general - case, the horizon touches two equal parallel circles, of which the one lying at the visible pole is always visible, and the other is always invisible (Proposition 6). Surface points between these circles rise and set, and always pass through the same points on the horizon, moving in circles perpendicular to the axis and inclined to the horizon at the same angle (Proposition 7). Each large circle fixed on the surface of the sphere, which touches the same parallel circles as the horizon, will coincide with the horizon when the sphere rotates (Proposition 8). In addition, it has been established that if the horizon is inclined to the axis, then of the two points ascending simultaneously, the one closer to the visible pole sets later; if two points set simultaneously, then the one located closer to the visible pole rises earlier.

Showing further that in the case when the horizon is inclined to the axis, the great circle passing through the poles of the sphere (i.e., the meridian) will twice be perpendicular to the horizon during its revolution (Proposition 10), Autolik formulates and proves the theorem (Proposition 11), which deals essentially with the ecliptic. It's about about how the rising and setting of the points lying on this great circle depend on its position relative to the horizon. It has been proven that if both of them are inclined to the axis, and the ecliptic touches two circles on the sphere parallel to each other and perpendicular to the axis, larger than those that the horizon touches, then the points of the ecliptic will always have their risings and sets on the segment of the horizon lying between parallel circles tangent to the ecliptic.

The last sentence states: If a fixed circle on the surface of a sphere always bisects another circle rotating with the sphere, both of which are not perpendicular to the axis and do not pass through the poles, then they are great circles.

Autolik's treatise "On Sunrises and Sunsets", consisting of two books, is based on the reviewed essay. It describes the movement of the fixed stars (book 1), with special attention to the twelve constellations located on; ecliptic (Book II). It turns out when the stars rise and set, having different positions on the celestial sphere, and under what circumstances they are visible or invisible.

Autolik's writings on the sphere, which were in the nature of elementary textbooks, did not lose their relevance either in antiquity or in the Middle Ages. The content of the treatise "On the Moving Sphere" was outlined in the 6th book of his "Mathematical Collection" by Pappus of Alexandria (3rd century AD). The significance of the role of Autolik in the development of science was written in the 6th century. Simplicius and John Philopon. The Greek text of both of his works has been fully preserved to this day.

On Arabic Autolik's writings were translated into IX-beginning 10th century among the first Greek writings that aroused the interest of Eastern scholars. The translation of the treatise "On the Moving Sphere" from the original Greek was carried out by the famous translator Ishaq ibn Hunayn (d. 910/911). His contemporary astronomer, philosopher and physician Kusta ibn Luka al-Baalbaki (d. 912) translated the treatise On Sunrises and Sunsets. These translations were then revised by the famous mathematician and astronomer Thabit ibn Korra (d. 901). Later, in the XIII century. the works of Autolik were commented on by the outstanding scientist, head of the Maraga observatory Nasir ad-Din at-Tusi (1201 - 1274) .

In Europe, Arabic versions of Autolik's works became known in the 12th century. By this time, the Latin translation of the treatise "On the Moving Sphere" was made by the largest medieval translator Gerardo of Cremona (1114-1187).

The Greek text of Autolik's writings, preserved in several manuscripts of the 10th-15th centuries, attracted the attention of scientists in the 16th century, when a careful study of the ancient scientific heritage began in Europe under the influence of humanist ideas. First time Latin; the translation of both treatises from the original Greek was published in the encyclopedia of the Italian educator George Balla (G. Valla, c. 1447-1500) in 1501, and then in the collection of ancient writings on the sphere, which was published in 1558 in Messina by Francesco Mavrolico (F. Maurolico, 1494-1575).

Active work on the publication of mathematical and astronomical works of ancient authors was carried out during this period in France, where it was initiated by one of the prominent figures of the French Renaissance, a passionate propagandist of ancient science P. Ramus (P. Ramus, Pierre de la Ramée, 1515-1572 ); He was dedicated to the first Greek edition of the works of Autolik, carried out by Conrad Dasipodius (Dasypodius, Conrad Rauchfuss, 1532-1600); it was published in 1572 in Strasbourg, together with a Latin translation. Another student of Ramus P. Forcadel (Pierre Forcadel, c. 1520-1574) published in 1572 a French translation of both Autolik's treatises.

In 1587-1588. another Latin edition appeared, made by I. Auria (I. Auria) on several Greek manuscripts from the Vatican library, and in 1644 M. Mersenne (M. Megsenn, 1588-1648) published an abridged Latin translation of the works of Autolik other Greek writings on mathematics and astronomy.

A complete critical edition of the Greek text of Autolik's treatises, together with a Latin translation, was carried out in 1855 by F. Gulch. It was the basis of the German translation by A. Chvalina, published in 1931.

Finally, a new edition of the Greek text, based on a thorough study of all surviving manuscripts, was undertaken by J. Maugenet in 1950; the text is preceded by a thorough study of the history of European editions of Autolik's works. In 1971 in Beirut was published English translation this text, which, however, caused serious criticism by O. Neugebauer.

Autolik's writings have attracted the attention of many historians of astronomy and mathematics. Both Autolik's theory and the text of his writings are studied. It is shown, for example, that the two books that make up "On Sunrise and Sunset" are, in all likelihood, two versions of the same work.

The Arabic versions of the Autolik treatises, which were among the "middle books", are still the least studied, although they exist in numerous manuscripts stored in different libraries Europe and Asia.

3. In the second half of the 4th c. BC e., another essay on the sphere appeared, close in content to the works of Autolik and written by his younger contemporary Euclid, the famous author of the Beginnings. In this treatise, entitled "Phenomena", Euclid largely repeats his predecessor, but the connection between the sphere and practical astronomy is much more clearly expressed in him.

Euclid's "Phenomena" consists of 18 sentences. The first formulates the statement underlying the geocentric system of the world that the Earth is taken as the center of the universe. Since the position of the observer on earth's surface should be considered arbitrary, it follows from this statement that, in relation to the entire universe, the Earth is considered as the point at which the observer is located.

Having repeated in the 2nd and 3rd sentences the seventh theorem of Autolik from the treatise “On the Moving Sphere”, Euclid proceeds to the study of the rising and setting of the signs of the zodiac - 12 constellations located on the ecliptic, that is, each of the twelve arcs, the ecliptic, equal to 30 ° and conditionally corresponding to these constellations. He proves (proposition 4) that if the ecliptic does not intersect with the largest of the always visible circles on the celestial sphere, i.e. if the latitude of the place of observation is less than 66 °, then the constellations that ascend first also set first; if it intersects with it, that is, if the latitude of the place of observation is greater than 66 °, then the constellations located to the north rise earlier and set later than those located to the south (proposition 5). Thus, the features of the rising and setting of the constellation depend on the latitude of the place of observation, that is, on the magnitude of the angle between the axis of the world and the horizon.

Having shown further that the rising and setting of stars located at opposite ends of the diameter of the ecliptic are opposite to each other (Proposition 6), Euclid explains the eleventh theorem from Autolik's treatise "On a Moving Sphere": stars located on the ecliptic, during their rising and setting, cross part of the horizon enclosed between the tropics, and this intersection occurs at constant points (Proposition 7).

Then he proves that equal arcs of the signs of the zodiac rise and set on unequal arcs of the horizon, the greater, the closer to the equinoxes they are located; at the same time, arcs equally distant from the equator ascend and set on equal arcs of the horizon (Proposition 8).

The following theorems concern the duration of the sunrises and sunsets of the various signs of the zodiac. First, it was established that the time required for the rising of the half of the ecliptic will be different depending on the position of the starting point of reference (Proposition 9). This corresponds to the statement about the different lengths of day and night in different seasons of the year, when the Sun is in different signs of the zodiac. Then the time required for the rising and setting of equal and opposite signs of the zodiac is considered.

The solution of the questions raised by Euclid was extremely important for the ancient astronomers, since it concerned methods for determining the hour of the day and night, establishing a calendar, etc.

4. Thus, in the considered works of Autolik and Euclid, the foundations of the ancient Greek spherics, both theoretical and practical, were outlined. Both authors, however, followed some earlier pattern, for they made a number of propositions about the sphere without proof, presumably considering them to be known. It is possible that the author of such a work on the sphere, generally recognized at that time, was the great mathematician and astronomer Eudoxus of Cnidus (c. 408-355 BC).

This lost work is now judged by Theodosius' Sphere, written later, but undoubtedly repeating its content in the main.

5. There are different opinions regarding the life and biography of Theodosius, based on the often conflicting reports of ancient historians, who mistakenly combined several figures who bore this name in one person. It has now been established that the author of The Sphere came from Bithynia, and not from Tripoli, as was previously believed and indicated in the titles of many editions of his works. He probably lived in the 2nd half of the 2nd century BC. BC e., although he was usually called a contemporary of Cicero (c. 50 BC).

In addition to the Spheres, two more writings by Theodosius, also included in the number of "middle books", have been preserved in the original Greek. The largest treatise "On dwellings" includes 12 sentences and is devoted to the description of the starry sky from the point of view of observers located on different geographical latitudes. The second treatise, entitled "On Days and Nights" and consisting of two books, considers the arc of the ecliptic through which the sun travels in one day, and examines the conditions necessary, for example, for day and night to really equal each other at the equinoxes.

These writings were studied and commented on by many Arab scholars, and attracted attention in Europe in the 16th century, when their Greek manuscripts were discovered. The first of them was published in Latin translation in 1558 by F. Mavroliko, along with a number of other works on the sphere, and then in 1572 by K. Dasipodius published the Greek and Latin formulations of the theorems from this treatise in the book mentioned above. In the same year, 1572, a French translation of Theodosius's work was published in the version of Dasipodius, made by P. Forcadel. The next Latin editions were made in 1587 (I. Auria) and in 1644 (M, Mersenne). The full Greek text of the treatise "On Dwellings" together with the Latin translation was published only in 1927 by R. Fecht. Also reproduced for the first time in the same edition original text writings "On Days and Nights" and its Latin translation. Previously, it was known thanks to the wording of sentences in Greek and Latin published in 1572 by K. Dasipodius and a complete Latin translation in the publication of I. Auria.

Theodosius' most famous work was his "Sphere", which occupies an important place in the history of astronomy, spherical trigonometry and non-Euclidean geometry.

Theodosius studies in detail the properties of lines on the surface of a sphere obtained by cutting it with different planes. It should be emphasized that the spherical triangle does not yet appear in him. The work is modeled after Euclid's "Beginnings" and consists of three books. The first book, which includes 23 sentences, begins with six definitions. The sphere is defined as "a solid figure bounded by one surface, so that all straight lines falling on it from one point lying inside the figure are equal to each other", i.e., similar to how the circle is defined in the "Principles" (book I, 15th definition) ; it is interesting to note that Euclid himself in book XI of the "Beginnings" defines the sphere in a different way - as a body formed by the rotation of a semicircle around a fixed diameter (book XI, 14th definition). Further, the definition of the center of the sphere, its axis and poles is given. The pole of a circle drawn on a sphere is defined as. a point on the surface of a sphere such that all lines drawn through it to the circumference of the circle are equal. Finally, the sixth definition concerns circles on the sphere equidistant from its center: according to Theodosius, these are circles such that the perpendiculars drawn from the center of the sphere to their planes are equal to each other.

The sentences of book 1 are quite elementary: proved; in particular, that any section of a sphere by a plane is a circle, that a straight line drawn from the center of the sphere to the center of a circular section is perpendicular to the plane of this section, that the sphere and the plane have one point of contact, etc.

Book 2 of Theodosius' Spheres begins with a definition of two circles on a sphere touching each other and contains 23 sentences about the properties of circles that are inclined to each other.

The third book consists of 14 sentences, more complex than the preceding ones, and dealing with systems of parallel and intersecting circles on a sphere. Here the service role of the sphere in relation to astronomy is clarified, although all the theorems are formulated and proved purely geometrically.

Theodosius' "Sphere" was carefully studied both in antiquity and in the Middle Ages. It was commented on by Pappus of Alexandria (3rd century) in the 6th book of his Mathematical Collection. In the VI century. John Philopon, considering the writings on the sphere of Euclid, Autolik and Theodosius, notes that the latter gives the most general abstract presentation of the subject, completely abstracting from real astronomical objects. Autolik, in his opinion, considers a more particular case, since "even if the author does not have in mind any specific object, then thanks to the combination of a spherical figure and movement, he approaches reality." The most special issue is treated in the "Phenomena" of Euclid, since the objects studied by astronomy - the sky, the sun, stars, planets - are quite real.

Theodosius first translated the "Sphere" into Arabic in the 9th century. Kusta ibn Luka al-Baalbaki; his translation, brought up to the 5th sentence of Book II, was completed by Thabit ibn Korra al-Harrani.

There are numerous comments on this, as well as on other writings of Theodosius, compiled by Eastern scholars of the 13th-15th centuries. , among which such prominent mathematicians and astronomers as Nasir ad-Din at-Tusi (1201 - 1274), Yahya ibn Muhammad ibn Abi Shukr Mukhi ad-Din al-Maghribi (d. c. 1285), Muhammad ibn Ma "ruf ibn Ahmad Taqi ad-Din (1525/1526-1585) and others.

Processing of Theodosius' Sphere, owned by a representative of the famous Maraga scientific school of the 13th century. Mukhi ad-Dinu al-Maghribi, was researched and partially translated into French B. kappa de Vaux. This treatise draws attention to astronomical terminology, which is used in the presentation and proof of Theodosius' theorems. Thus, here, even more clearly than in the Greek original, the connection of the sphere with astronomy appears, which explains its relevance to Eastern science.

In Europe, Theodosius' Sphere became known in the 12th century, when two Latin translations of this work from its Arabic version appeared. They were made by the eminent translators who worked in Spain, Gerardo of Cremona and Plato of Tivoli. The translation of the latter was published in 1518 in Venice, subsequently republished in 1529 in the edition of I. Voegelin (I. Voegelin, died in 1549), and in 1558 - the mentioned book by F. Mavroliko.

The Greek text of the "Spheres" was first published in 1558 by J. Pena along with a Latin translation. This edition made it possible to clarify the difference between the Arabic version of Theodosius's work and the original and to establish what additions and changes in the proof of theorems were made by Eastern scientists. However, the Greek manuscript used by Pena suffered from many shortcomings. Therefore, in 1707 at Oxford, I. Hunt undertook a new and improved edition, making some corrections on other manuscripts. Subsequently, the Greek text of the work (also with a Latin translation) was reprinted twice more: in 1862 by E. Nice and in 1927 by I. Geiberg.

Starting from the 2nd half of the 16th century, abridged and adapted editions of the Spheres began to appear in Latin, in which theorems were explained using new mathematical concepts and using spherical trigonometry. In 1586, an edition of X. Clavius ​​(Ch. Clavius) was published in Rome, and in the 17th century. it was followed by several others, including the editions of M. Mersenne (1644) and I. Barrow (1675). symbolism.

In 1826, The Sphere was published in a German translation by E. Nice. The second German edition of the work was carried out in 1931 by A. Chvalina (together with the treatises of Autolik). The first French translation of the "Spheres", made by D. Henrion, was published in 1615, the next, owned by J.B. Dugamel (J. V. Du Hamel), - in 1660; finally, in 1927, appeared modern translation P. Ver Eecke.

The works of many historians of mathematics (A. Knock, I. Geiberg, F. Gulch, P. Tannery, A. Bjornbo, etc.) are devoted to the study of the text and content of Theodosius' Sphere. in the III-VII centuries. and preserved in Greek manuscripts of a later time, the relationship between Theodosius's "Sphere" and Euclid's "Phenomena" and other works of ancient authors was considered. The results of these studies made it possible to clarify a number of questions concerning the history of mathematics and astronomy, as well as the biographies of Euclid, Autolik, Theodosius and some commentators on their works.

6. Close to the Greek works on the sphere in content short essay Hypsicles from Alexandria (lived between 200 and 100 BC), entitled "On the ascent of the constellations along the ecliptic" ("Anaphoric"). Hypsicles is best known as the author of a treatise on regular polyhedra, included in Euclid's Elements as Book XIV; another of his works, on polygonal numbers, which has not survived, is quoted in Diophantus' Arithmetic.

In the treatise "On the ascent of the constellations on the ecliptic", consisting of six sentences, the problem is solved of determining the time required for the rising or setting of each sign of the zodiac, which occupies 1/12 of the ecliptic, or "degree", i.e. 1/30 parts of the ecliptic. She played an important role in astrological reasoning and therefore enjoyed great popularity in antiquity and in the Middle Ages. The problem can be solved by means of spherical trigonometry, but Hypsicles, who did not yet have such means, solved it approximately, using the theorems on polygonal numbers known to him. In this work, for the first time, there is a division of the circumference of a circle into 360 parts, which was not the case with his predecessors and, in particular, with Autolik.

The treatise of Hypsicles was one of the "middle books" and was translated into Arabic in the 9th century. There are many manuscripts of this translation, but it remained unexplored for a long time and it was not precisely established whether Kusta ibn Luka, al-Kindi or Ishaq ibn Hunayn performed it. He translated the Arabic version of the work into Latin in the 12th century. Gerardo of Cremona.

A critical edition of the Greek original and the Latin translation by Gerardo of Cremona was carried out in 1888 by K. Manitius. The second edition, published in 1966, includes the Greek text, scholia and translation by V. De Falco, the Arabic text and German translation M. Krause, as well as an introductory article by O. Neugebauer.

7. Of all the ancient writings on the sphere, the greatest role in the history of science was played by the "Sphere" of Menelaus, who worked in Alexandria in the 1st century BC. n. e. and summarizing all the results that have been obtained in this area before him. In his work, not only the geometry on the sphere is stated, but the spherical triangle was first introduced, the theorems that served as the basis of spherical trigonometry were successively proved, and theoretical basis for trigonometric calculations.

Information about the life of Menelaus is extremely scarce. It is known that in 98 he produced astronomical observations in Rome. The Sphere, his main work, has not been preserved in the original Greek and is known only from medieval Arabic translations.

The Sphere consists of three books and is modeled after Euclid's Elements. First of all, definitions of basic concepts are introduced, including the concept of a spherical triangle, which is not found in earlier Greek works. A significant part of the work is devoted to the study of the properties of this figure.

When proving propositions about the properties of lines and figures on a sphere, he relies on definitions and theorems from Theodosius' Sphere. In the 2nd book, these theorems, as well as the propositions formulated in astronomical form in Euclid's Phenomena and Hypsicles' Anaphorica, are systematized and provided with new rigorous proofs.

A particularly important role in the history of trigonometry was played by the 1st sentence of book III, known as the “theorems of Menelaus” (as well as “theorems about the complete quadrilateral”, “rules of six quantities”, “theorems about transversals”). In the words of A. Braunmühl, it was "the foundation of the entire spherical trigonometry of the Greeks."

The theorem of Menelaus for the plane case is formulated as follows: let mutually intersecting lines AB, AC, BE and CD, forming the figure ACGB (Fig. 1), be given; then the following relations hold:

CE / AE = CG / DG * DB / AB, CA / AE = CD / DG * GB / BE

For the spherical case, in the theorem, as was customary in Greek trigonometry, the chords of doubled arcs appear. If the figure ACGB (Fig. 2) is given, formed by arcs of great circles on the surface of a sphere, then the relations are valid:

chord(2CE) / chord(2AE) = chord(2CG) / chord(2DG) * chord(2DB) / chord(2AB)

chord(2AC) / chord(2AE) = chord(2CD) / chord(2DG) * chord(2GB) / chord(2BE)

Menelaus also proved several other theorems of fundamental importance for the development of spherical trigonometry. These include the so-called "rule of four magnitudes" (2nd sentence of book III); if two spherical triangles ABC and DEG are given (Fig. 3), which respectively have equal (or add up to 180°) angles A and D, C and G, then

chord (2AB) / chord (2BC) = chord (2DE) / chord (2EG)

The third sentence of the III book of the "Spheres" of Menelaus, which later received the name "rules of tangents", reads; what if two right-angled spherical triangles ABC and DEG (Fig. 4) are given, for which

chord (2AB) / chord (2AC) = chord (2ED) / chord (2GD) * chord (2BH) / chord (2ET)

LITERATURE

1. Geiberg I.L. Natural science and mathematics in classical antiquity. Translation from him. S.P. Kondratiev, ed. with preface A.P. Yushkevich, M-L., ONTI, 1936.

2. Sarton G. Appreciation of ancient and medieval science during the Renaissance, Philadelphia, 1953.

3 Steinschneider M. Die "mittleren" Bücher der Araber und ihre Bearbeiter, "Zeitschr. für Math. u. Phys.", Bd 10, 1.865, 456-498.

4. Suter H. Die Mathematiker und Astronomen der Araber und ihre Werke, "Abhandl. zur Gesch. d. math. Wiss.", N. 10, Leipzig, 1900.

5. Björnbo A. Studienüber Menelaus Sphärik. Beiträge zur Geschichte der Sphärik und Trigonometrie der Griechen, "Abhandl. zur Gesch. d. math. Wiss.", H. 14, Leipzig, 1902.

6. Mogenet J. Autolycos de Pitane. Histoire du texte, suivie de l "édition critique des traités de la Sphère en mouvement et des levers et couchers, Louvain, 1950.

7. Theodosii Shpaericorum elementorum Libri III. Ex traditione Mauro-lyci... Menelai Sphaericorum lib. III. Ex traditione eiusdem. Maurolyci, Sphaericorum libri II. Autolyci. De sphaera quae movetur liber. Theodosii. De habitationibus. Euclidis Phaenomena brevissime demonstrata. Demonstratio et praxis trium tabellarum scilicet sinus recti, foecundae, et beneficae ad spheraiia triangula pertinentum. Compendium mathematicae mira brevitate ex clarissimis authoribus. Maurolyci de sphaera sermo. Messanae, 1558.

8. Mersenne M. Universae geometriae mixtaeque mathematicae synopsis, Parisiis, 1644.

9. Auto1yci. De Sphaera quae movetur liber. D.e ortibus et occasibus libri duo, willow cum scholiis antiquis o libris manuscriptis edidit, latina interpretatione et commentariis instruxit F. Hultsch, Leipzig, 1885.

10. Euclidis. Opera omnia. Ed. J. L. Heiberg et H. Menge, t. VIII. Phaenomena et scripta musica, Leipzig, 1916.

11. Tannery P. Recherches sur l "histoire sur l" astronomie ancienne, Paris, 1893.

12. Carra de Vaux B. Notice sur deux manuscrits arabes. I. Remaniement des sphériques de Théodose par labia ibn Muhammad ihn Abi Schukr Almaghribi Aiandalusî, "Journal asiatique", 8th sér., t. 17, 1894, 287-295.

13. Theodosius Tripolites. Sphaerica. Hrsg, von J. L. Heiberg, "Abhandl. d. G.es. d. Wissenschaften zu Göttihgen", phil. hist, Klasse, N. F., Bd 19, No 3, Berlin, 1927.

14. Hypsikles Die Aufgangszeiten der Gestirne, hrsg. und übers, von V. De Falco and M. Krause. Einführung von O. Neugebauer, "Abhandl. d. Akademie d. Wiss. zu Göttingen", phil-hist. Kl., F. 3, No 62, Göttingen, 1966.

15. Krause M. Die Sphärik von Menelaos von Alexandrien in der Verbesserung von Abu Nasr Mansur b. Ali b. Iraq mit Untersuchungen zur Geschichte des Textes bei den islamischen Mathematikern, Berlin, 1936.

Notes

A copy of this rare edition is available in the Library. IN AND. Lenin.

A copy is available in the Library of the Academy of Sciences of the USSR.

The task of spherical trigonometry is the solution of a spherical triangle, that is, the calculation of its unknown elements through the given (known).

It is known that in order to find any angle or side of a triangle, it is necessary that any three of its other elements be known (given).

Let us consider (without derivation) four main theorems of spherical trigonometry, establishing the necessary analytical dependence between the elements of a spherical triangle.

I. The formula for the cosine of a side.

This formula connects all three sides and one of the corners of a spherical triangle. For any combination of these four elements, a relationship is established that ...

"... the cosine of a side of a spherical triangle is equal to the product of the cosines of the other two sides plus the product of the sines of the same sides and the cosine of the angle between them ...".

Rice. 2.2. spherical triangle

Applied to the side A(Fig. 2.2) spherical triangle AVM, guided by the side cosine theorem, we can write:

cos a = cos b cos m + sin b sin m cos A

For parties b And m the relationship between the elements of the triangle is expressed by the formulas:

The sine formula is used to calculate one of the elements included in the written equalities, if the other three elements are known.

III. Cotangent formula links together four elements of a spherical triangle lying side by side.

"... the cotangent of the extreme angle, multiplied by the sine of the middle, is equal to the product of the cotangent of the extreme side by the sine of the middle without the product of the cosines of the middle elements ...".

AVM(Fig. 2.2) the dependence between the elements is established A, m, B And A, then the angle A and side A are extreme, and the angle IN and side m- middle elements, and then:

ctg A sin B = ctg a sin m - cos B cos m

In total, six such relations can be written for a triangle, namely:

These formulas are useful when calculating an angle from two other angles and the side between them, and also serve to find the side from three given angles.

Rice. 2.3. right spherical triangle

Solution rectangular triangles are simpler than oblique ones, since one of their elements (the angle of 90°) is always known, and to solve a triangle it is enough to know only two elements.

The same applies to quarter triangles in which one of the elements (the 90° side) is always known.

If in a spherical triangle AVM(Fig. 2.3) the angle is set IN= 90°, leg A and angle M , then to calculate the unknown angle A you can apply the formula for the cosine of the angle (6.4) → cos A = sin B sin M cos a - cos B cos M .

If we now replace all the functions of the angle IN= 90° by their values ​​( sin B = 1, cos B= 0), then we get

cos A = sin M cos a

2.3. Calculation of the horizontal coordinates of the luminaries according to the tables of logarithmic functions of the MT-75 nautical tables

When calculating the reckoning height ( h C) and azimuth ( A C) luminaries according to the formulas of spherical trigonometry, both according to the natural values ​​​​of trigonometric functions, and according to logarithms, the most convenient formulas are:

(2.6)

In the formula, the "~" sign means that when φ C And δ of the same name, the smaller value is subtracted from the larger value, and with opposite names → the values φ C And δ add up.

The values ​​of , and are tabulated so that the calculations do not need to divide the arguments Z C, φ C~δ And t M, and square the values ​​of trigonometric functions, → all these actions are performed in tables 5 A (5b) "MT-75" (there are no such tables in "MT-2000").

It is not required to study the formula for signs of trigonometric functions, since both terms of its right side are always positive.



We will consider the method of calculating the horizontal coordinates of the luminaries using the MT-75 using the example of solving a specific problem.

Task: Calculate the values ​​of the calculated heights ( h C) and azimuth ( A C) luminaries if:

φ C= 43°20.6′ N; δ = 17°36.7′ N; t M= 17°12.4′ W.