Keith Mack
The spherical law of sines deals with triangles on a sphere, whose sides are arcs of great circles. The first known use of the spherical law of sines can be traced back to the 10th century, credited to scholars Abu-Mahmud Khujandi and Abū al-Wafā. It was also presented in the 11th century by Ibn Muʿādh al-Jayyānī in his Book of Unknown Arcs of a Sphere. During the Renaissance, the law of sines was systematically presented in German mathematician and astronomer Regiomontanus's De triangulis omnimodis libri quinque (Five Books on Triangles of Every Kind), published in 1533.
| Characteristics | Values |
|---|---|
| First presented the spherical law of sines | The spherical law of sines was first presented by the 13th-century Persian mathematician and polymath Naṣīr al-Dīn al-Ṭūsī |
| First presented the law of sines | The law of sines was first presented by the 2nd-century astronomer Ptolemy |
| First presented the spherical law of cosines | The spherical law of cosines was first presented by Todhunter |
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What You'll Learn
- The spherical law of sines is sometimes credited to 10th-century scholars Abu-Mahmud Khujandi or Abū al-Wafā
- It is given prominence in Abū Naṣr Manṣūr's Treatise on the Determination of Spherical Arcs
- Nasir al-Din al-Tusi was the first to list six distinct cases of a right triangle in spherical trigonometry
- The law of sines was systematically presented in German mathematician Regiomontanus's Five Books on Triangles of Every Kind
- Brahmagupta's work contains an equation close to the modern law of sines
The spherical law of sines is sometimes credited to 10th-century scholars Abu-Mahmud Khujandi or Abū al-Wafā
The spherical law of sines is a mathematical principle of trigonometry that deals with triangles on a sphere, whose sides are arcs of great circles. The radius of the sphere is considered to be 1, and the angles at the centre of the sphere are represented in radians. The spherical law of sines is sometimes credited to the work of 10th-century scholars Abu-Mahmud Khujandi and Abū al-Wafāʾ.
Abu-Mahmud Khujandi, also known as Abu Mahmood Khujandi, al-khujandi or Khujandi, was a Persian Transoxanian astronomer and mathematician. He was born in Khujand, now part of Tajikistan, and lived from c. 940 to 1000. Khujandi helped build an observatory near the city of Ray, Iran, where he constructed the first huge mural sextant in 994 AD. This was used to determine the Earth's axial tilt, and he discovered that the tilt was not constant but decreasing.
Abū al-Wafāʾ was a prominent Persian mathematician and astronomer. He is known for his contributions to the fields of trigonometry, geometry, and astronomy. Al-Wafāʾ’s work, Almagest, contains the spherical law of sines. However, it was Abū Naṣr Manṣūr who gave the law of sines prominence in his Treatise on the Determination of Spherical Arcs, and it was credited to him by his student al-Bīrūnī.
The spherical law of sines was further developed by later scholars, including Naṣīr al-Dīn al-Ṭūsī, a 13th-century Persian mathematician and polymath. Al-Ṭūsī stated and proved the planar law of sines, which allowed for the solving of triangles when either two angles and a side or two sides and an angle were known. The law of sines was also systematically presented during the Renaissance by German astronomer-mathematician Regiomontanus in his work "De triangulis omnimodis libri quinque" ("Five Books on Triangles of Every Kind"), which helped establish it as a standard method for solving triangles.
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It is given prominence in Abū Naṣr Manṣūr's Treatise on the Determination of Spherical Arcs
The spherical law of sines is credited to 10th-century scholars Abu-Mahmud Khujandi or Abū al-Wafā. However, it is Abū Naṣr Manṣūr's Treatise on the Determination of Spherical Arcs that gives the law its prominence. Abū Naṣr's student, al-Bīrūnī, credits his teacher for the law of sines in his Keys to Astronomy.
Abū Naṣr Manṣūr ibn ‘Alī ibn ‘Irāq, or Abu Nasr Mansur, was an Islamic prince and mathematician who collaborated with al-Biruni on astronomy and mathematics. He was a native of Gilan and was born in 970 in what is now Afghanistan. He was a member of the Banu Iraq, the rulers of Khwarazm, the region adjoining the Aral Sea. He studied and became a disciple of Abu'l-Wafa in this region. Abu Nasr Mansur was teaching there when he first began his association with al-Biruni, whom he taught from about 990. This began an important collaboration that lasted many years.
Abu Nasr Mansur contributed to the development of spherical geometry and is considered the founder of spherical trigonometry. He formulated and proved the planar and spherical theorem of sines. He discovered the sine rule for triangles and improved upon the work of Arabic astronomers by using 1 as the value of the radius instead of 60. His proof of the sine law appears several times in his works, including his Almagest of the Shah, Book of the Azimuth, Treatise on the Determination of Spherical Arcs, and Treatise in Which Some Geometrical Questions Addressed to Him Are Answered.
Abu Nasr Mansur wrote at least 22 works, 17 of which remain, and 16 have been published. His other significant work includes the most complete Arabic version of the Spherics of Menelaus.
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Nasir al-Din al-Tusi was the first to list six distinct cases of a right triangle in spherical trigonometry
Nasir al-Din al-Tusi, also known as Muhammad ibn Muhammad ibn al-Hasan al-Tusi, was a Persian polymath, architect, philosopher, physician, scientist, theologian, and prolific writer. He is regarded as one of the greatest scientists of medieval Islam, and is often considered the creator of trigonometry as a distinct mathematical discipline.
Al-Tusi was the first to write about trigonometry independently of astronomy. In his 'Treatise on the Quadrilateral', he provided an extensive exposition of spherical trigonometry, which was previously linked to astronomy. In this work, al-Tusi listed the six distinct cases of a right triangle in spherical trigonometry. This was a significant contribution to the field, as it helped establish trigonometry as a branch of pure mathematics separate from astronomy.
Al-Tusi's work built upon earlier advancements by Greek mathematicians such as Menelaus of Alexandria, who wrote 'Sphaerica', a book on spherical trigonometry, and Muslim mathematicians like Abū al-Wafā' al-Būzjānī and Al-Jayyani. Al-Jayyani's 11th-century 'Book of Unknown Arcs of a Sphere' also contains the spherical law of sines.
Al-Tusi's contributions to spherical trigonometry were significant. In addition to listing the six distinct cases of a right triangle, he also stated the sine law for spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for these laws. He is also credited with establishing the important result that if the sum or difference of two arcs is provided along with the ratio of their sines, the arcs can be calculated.
Beyond trigonometry, al-Tusi made numerous scientific advancements. In astronomy, he created accurate tables of planetary motion, developed an updated planetary model, and critiqued Ptolemaic astronomy. He also made contributions to logic, mathematics, biology, and chemistry. His works, numbering over 150, include religious and secular topics, with 25 written in Persian and the rest in Arabic, and one treatise in Persian, Arabic, and Turkish.
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The law of sines was systematically presented in German mathematician Regiomontanus's Five Books on Triangles of Every Kind
The spherical law of sines is sometimes credited to 10th-century scholars Abu-Mahmud Khujandi or Abū al-Wafā. However, it was given greater prominence in Abū Naṣr Manṣūr's Treatise on the Determination of Spherical Arcs. The spherical law of sines also appears in the 11th-century Book of Unknown Arcs of a Sphere by Ibn Muʿādh al-Jayyānī.
During the Renaissance, the law of sines was systematically presented in German mathematician and astronomer Regiomontanus's De triangulis omnimodis libri quinque (1533; “Five Books on Triangles of Every Kind”). This work helped establish the law of sines as a standard method for solving triangles and is considered one of the earliest comprehensive works on trigonometry in Europe. Regiomontanus's work was based on the foundation laid by 13th-century Persian polymath Naṣīr al-Dīn al-Ṭūsī, who provided a general proof of the law of sines for plane triangles. Al-Tusi stated that in any triangle, the sides are proportional to the sines of their opposite angles.
The study of trigonometry has a long history, with early contributions from Egyptian and Babylonian mathematics during the 2nd millennium BC. Trigonometry was further developed in Hellenistic mathematics and spread to India, where it flourished during the Gupta period. In the Middle Ages, Islamic mathematicians such as al-Khwarizmi and Abu al-Wafa continued to advance trigonometry, and it became an independent discipline in the Islamic world. Trigonometry was later adopted in the Latin West during the Renaissance, with Regiomontanus playing a key role in its development as a distinct mathematical discipline in Europe.
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Brahmagupta's work contains an equation close to the modern law of sines
The spherical law of sines is sometimes credited to 10th-century scholars Abu-Mahmud Khujandi or Abū al-Wafā. However, statements related to the law of sines appear in the astronomical and trigonometric work of the 7th-century Indian mathematician, Brahmagupta.
Brahmagupta's best-known work, the Brahmasphutasiddhanta, was written in Bhinmal, a town in the Jalore district of Rajasthan, India. The work was written in 25 chapters and contains several unprecedented mathematical results. It includes a good understanding of the mathematical role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and some quadratic equations, and rules for summing series. Brahmagupta was the first to use zero as a number and gave rules for computing with zero.
In his Brahmasphutasiddhanta, Brahmagupta expresses the circumradius of a triangle as the product of two sides divided by twice the altitude. The law of sines can be derived by alternately expressing the altitude as the sine of one or the other base angle times its opposite side, then equating the two resulting variants. An equation even closer to the modern law of sines appears in Brahmagupta's Khandakhadyaka, in a method for finding the distance between the Earth and a planet following an epicycle.
Brahmagupta's work on interpolation theory has been thoroughly discussed by P.C. Sengupta. In Chapter IX of Khandakhadyaka, Brahmagupta introduces a new method of obtaining a given table of sines consisting of tabulated values of six angles at equal intervals of 15°. He takes the radius R of the circle as 150 instead of 3438 used by Suryasiddhanta and Aryabhata I. Brahmagupta's table of R sines is reasonably accurate.
In trigonometry, the law of sines is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, the ratio of the length of a side to the sine of its opposite angle is constant. The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known, a technique known as triangulation.
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Frequently asked questions
The spherical law of sines is credited to 10th-century scholars Abu-Mahmud Khujandi and Abū al-Wafā.
Yes, an early form of the law of sines appears in the work of the 2nd-century astronomer Ptolemy. He related chord lengths to angles in his studies.
Yes, the 13th-century Persian polymath Naṣīr al-Dīn al-Ṭūsī provided a general proof of the law of sines for plane triangles.
Yes, during the Renaissance, German astronomer-mathematician Regiomontanus systematically presented the law of sines in his work "De triangulis omnimodis libri quinque" (1533; “Five Books on Triangles of Every Kind”).
Yes, Ibn Muʿādh al-Jayyānī's 11th-century "Book of Unknown Arcs of a Sphere" also contains the spherical law of sines.
