Sources
Help build the largest human-edited directory on the web.
Submit a site - Become an editor
Wikipedia. Licensed under the GNU Free Documentation License.
Are you an expert in this subject? Join the discussion and share your knowledge at Wikipedia.org.
History of trigonometry
From Wikipedia, the free encyclopedia
The history of trigonometry and of trigonometric functions may span about 4000 years.
Contents |
Our modern word sine is derived from the Latin word sinus, which means "bay" or "fold", from a mistranslation (via Arabic) of the Sanskrit word jiva, alternatively called jya.[1] Aryabhata used the term ardha-jiva ("half-chord"), which was shortened to jiva and then transliterated by the Arabs as jiba (جب). European translators like Robert of Chester and Gherardo of Cremona in 12th-century Toledo confused jiba for jaib (جب), meaning "bay", probably because jiba (جب) and jaib (جب) are written the same in the Arabic script (this writing system, in one of its forms, does not provide the reader with complete information about the vowels). The words "minute" and "second" are derived from the Latin phrases partes minutae primar and partes minutae secundae.[2]
Trigonometry is not the work of any one man or nation. Its history spans thousands of years and has touched every major civilization. It should be noted that that from the time of Hipparchus until modern times there was no such thing as a trigonometric ratio. Instead, the Greeks and after them the Hindus and the Muslims used trigonometric lines. These lines first took the form of chords and later half chords, or sines. These chord and sine lines would then be associated with numerical values, possibly approximations, and listed in trigonometric tables.[2]
The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. But pre-Hellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were studied instead, a field that would be better called "trilaterometry".[3]
Based on one interpretation of the Plimpton 322 cuneiform tablet (circa 1900 BC), some have even asserted that the ancient Babylonians had a table of secants.[4] There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table.
Ancient Greek and Hellenistic mathematicians made use of the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of the bisected angle, that is, 
, and consequently the sine function is also known as the "half chord". Due to this relationship, many of the trigonometric identities and theorems that that are known today were also known to the ancient Greeks, but in their equivalent chord form.[5]
Although there is no trigonometry in the works of Euclid and Archimedes, in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas.[3] For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosine for obtuse and acute angles, respectively. Theorems on the lengths of chords are applications of the law of sines. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles.[3] To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the well known theorem that, in modern notation, sin α/ sin β < α/β < tan α/ tan β whenever 0° < β < α < 90°, among other theorems.[6]
The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 - 125 BC), who is now consequently known as "the father of trigonometry."[7] Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.[7][1]
Although it is not known when the systematic use of the 360° circle came into mathematics, it is known that the systematic introduction of the 360° circle came a little after Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon (ca. 260 B.C.), since he measured an angle in terms of a fraction of a quadrant.[6] It seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy.[8] In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts.[2] It is due to the Babylonian sexagesimal number system that each degree is divided into sixty minutes and each minute is divided into sixty seconds.[2]
Menelaus of Alexandria (ca. 100 A.D.) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles.[5] He establishes a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles.[5] Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.[5] Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".[5] He further gave his famous "rule of six quantities".[9]
Later, Claudius Ptolemy (ca. 90 - ca. 168 A.D.) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The thirteen book of the Almagest are the most influential and significant trigonometric work of all antiquity.[10] A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine. Ptolemy further derived the equivalent of the half-angle formula 
. Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.[10]
Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.[11]
The next significant developments of trigonometry were in India. The mathematician-astronomer Aryabhata (476–550 A.D.), in his work Aryabhata-Siddhanta, first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine. His works also contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. He used the words jya for sine, kojya for cosine, ukramajya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively after a mistranslation.
Other Indian mathematicians later expanded Aryabhata's works on trigonometry. In the 6th century A.D., Varahamihira used the formulas
In the 7th century A.D., Bhaskara I produced a formula for calculating the sine of an acute angle without the use of a table. He also gave the following approximation formula for sin(x), which had a relative error of less than 1.9%:
Later in the 7th century, Brahmagupta developed the formula 
as well as the Brahmagupta interpolation formula for computing sine values.
The Indian works were later translated and expanded in the Islamic world by Arab and Persian Muslim mathematicians. In the 9th century, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer of spherical trigonometry.
By the 10th century, in the work of Abū al-Wafā' al-Būzjānī, Muslim mathematicians were using all six trigonometric functions, after discovering the secant, cotangent and cosecant functions. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also developed the following trigonometric formula:
Also in the 10th century, Al-Battani was responsible for establishing a number of important trigometrical relationships such as:
In the 11th century, Omar Khayyam solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables.
All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by Bhaskara II and Nasir al-Din al-Tusi in the 13th century. Nasir al-Din al-Tusi stated the law of sines and provided a proof for it, and also listed the six distinct cases of a right angled triangle in spherical trigonometry.
In the 14th century, Ghiyath al-Kashi gave trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. Ulugh Beg (14th century) also gives accurate tables of sines and tangents correct to 8 decimal places.
In China, Aryabhata's table of sines were translated into the Chinese mathematical book of the Kaiyuan Zhan Jing, compiled in 718 AD during the Tang Dynasty.[12] Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the earlier Greek and then Indian and Islamic worlds.[13] Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.[12] However, this embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960-1279 AD), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations.[12] For instance, the polymath Chinese scientist Shen Kuo (1031-1095 AD) used trigonometric functions to solve mathematical problems of chords and arcs.[12] As the historians L. Gauchet and Joseph Needham state, the Chinese mathematician Guo Shoujing (1231-1316 AD) used spherical trigonometry in his calculations to improve Chinese astronomy and the calendar system.[14][12] Along with a later 17th century Chinese illustration of Guo's mathematical proofs, Needham states:
- Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).[15]
Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline, in his De triangulis omnimodus written in 1464, as well as his later Tabulae directionum which included the tangent function, unnamed.
The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.
In the 17th century, Isaac Newton and James Stirling developed the general Newton-Stirling interpolation formula for trigonometric functions.
Madhava (c. 1400) made early strides in the analysis of trigonometric functions and their infinite series expansions. He developed the concepts of the power series and Taylor series, and produced the trigonometric series expansions of sine, cosine, tangent and arctangent. Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the θ, radius, diameter and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century.[16][17]
Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, defining them as infinite series and presenting "Euler's formula" eix = cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.
Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of James Gregory and Colin Maclaurin were also very influential in the development of trigonometric series.
- ^ a b O'Connor (1996).
- ^ a b c d Boyer (1991). "Greek Trigonometry and Mensuration", , 166-167. “It should be recalled that form the days of Hipparchus until modern times there were no such things as trigonometric ratios. The Greeks, and after them the Hindus and the Arabs, used trigonometric lines. These at first took the form, as we have seen, of chords in a circle, and it became incumbent upon Ptolemy to associate numerical values (or approximations) with the chords. [...] It is not unlikely that the 260-degree measure was carried over from astronomy, where the zodiac had been divided into twelve "signs" or 36 "decans." A cycle of the seaons of roughly 360 days could readily be made to correspond to the system of zodiacal signs and decans by subdividing each sign into thirty parts and each decan into ten parts. Our common system of angle measure may stem from this correspondence. Moreover since the Babylonian position system for fractions was so obviously superior to the Egyptians unit fractions and the Greek common fractions, it was natural for Ptolemy to subdivide his degrees into sixty partes minutae primae, each of these latter into sixty partes minutae secundae, and so on. It is from the Latin phrases that translators used in this connection that our words "minute" and "second" have been derived. It undoubtedly was the sexagesimal system that led Ptolemy to subdivide the diameter of his trigonometric circle into 120 parts; each of these he further subdivided into sixty minutes and each minute of length sixty seconds.”
- ^ a b c Boyer (1991). "Greek Trigonometry and Mensuration", , 158-159. “Trigonometry, like other branches of mathematics, was not the work of any one man, or nation. Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians. In view of the pre-Hellenic lack of the concept of angle measure, such a study might better be called "trilaterometry," or the measure of three sided polygons (trilaterals), than "trigonometry," the measure of parts of a triangle. With the Greeks we first find a systematic study of relationships between angles (or arcs) in a circle and the lengths of chords subtending these. Properties of chords, as measures of central and inscribed angles in circles, were familiar to the Greeks of Hippocrates' day, and it is likely that Eudoxus had used ratios and angle measures in determining the size of the earth and the relative distances of the sun and the moon. In the works of Euclid there is no trigonometry in the strict sense of the word, but there are theorems equivalent to specific trigonometric laws or formulas. Propositions II.12 and 13 of the Elements, for example, are the laws of cosines for obtuse and acute angles respectively, stated in geometric rather than trigonometric language and proved by a method similar to that used by Euclid in connection with the Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the broken chord can readily be translated into trigonometric language analogous to formulas for sines of sums and differences of angles.”
- ^ Joseph, pp. 383–4.
- ^ a b c d e Boyer (1991). "Greek Trigonometry and Mensuration", , 163. “In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue - that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem A + B + C > 180° is established. The second book of the Sphaerica describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest. Book III, the last, contains the well known "theorem of Menelaus" as part of what is essentially spherical trigonometry in the typical Greek form - a geometry or trigonometry of chords in a circle. In the circle in Fig. 10.4 we should write that chord AB is twice the sine of half the central angle AOB (multiplied by the radius of the circle). Menelaus and his Greek successors instead referred to AB simply as the chord corresponding to the arc AB. If BOB' is a diameter of the circle, then chord A' is twice the cosine of half the angle AOB (multiplied by the radius of the circle).”
- ^ a b Boyer (1991). "Greek Trigonometry and Mensuration", , 159. “Instead we have an Aristarchan treatise, perhaps composed earlier (ca. 260 B.C.), On the Sizes and Distances of the Sun and Moon, which assumes a geocentric universe. In this work Aristarchus made the observation that when the moon is just half-full, the angle between the lines of sight to the sun and the moon is less than a right angle by one thirtieth of a quadrant. (The systematic introduction of the 360° circle came a little later. In trigonometric language of today this would mean that the ratio of the distance of the moon to that of the sun (the ration ME to SE in Fig. 10.1) is sin 3°. Trigonometric tables not having being developed yet, Aristarchus fell back upon a well-known geometric theorem of the time which now would be expressed in the inequalities sin α/ sin β < α/β < tan α/ tan β, where 0° < β < α < 90°.)”
- ^ a b Boyer (1991). "Greek Trigonometry and Mensuration", , 162. “For some two and a half centuries, from Hippocrates to Eratosthenes, Greek mathematicians had studied relationships between lines and circles and had applied these in a variety of astronomical problems, but no systematic trigonometry had resulted. Then, presumably during the second half of the second century B.C., the first trigonometric table apparently was compiled by the astronomer Hipparchus of Nicaea (ca. 180-ca. 125 B.C.), who thus earned the right to be known as "the father of trigonometry." Aristarchus had known that in a given circle the ratio of arc to chord decreases from 180° to 0°, tending toward a limit of 1. However, it appears that not until Hipparchus undertook the task had anyone tabulated corresponding values of arc and chord for a whole series of angles.”
- ^ Boyer (1991). "Greek Trigonometry and Mensuration", , 162. “It is not known just when the systematic use of the 360° circle came into mathematics, but it seems to be due largely to Hipparchus in connection with his table of chords. It is possible that he took over from Hypsicles, who earlier had divided the day into parts, a subdivision that may be been suggested by Babylonian astronomy.”
- ^ Needham, Volume 3, 108.
- ^ a b Boyer (1991). "Greek Trigonometry and Mensuration", , 164-166. “The theorem of Menelaus played a fundamental role in spherical trigonometry and astronomy, but by far the most influential and significant trigonometric work of all antiquity was composed by Ptolemy of Alexandria about half a century after Menelaus. [...] Of the life of the author we are as little informed as we are of that of the author of the Elements. We do not know when or where Euclid and Ptolemy were born. We know that Ptolemy made observations at Alexandria from A.D. 127 to 151 and, therefore, assume that he was born at the end of the first century. Suidas, a writer who lived in the tenth century, reported that Ptolemy was alive under Marcus Aurelius (emperor from A.D. 161 to 180).
Ptolemy's Almagest is presumed to be heavily indebted for its methods to the Chords in a Circle of Hipparchus, but the extent of the indebtedness cannot be reliably assessed. It is clear that in astronomy Ptolemy made use of the catalogue of star positions bequeathed by Hipparchus, but whether or not Ptolemy's trigonometric tables were derived in large part from his distinguioshed predecrssor cannot be determined. [...] Central to the calculation of Ptolemy's chords was a geometric proposition still known as "Ptolemy's theorem": [...] that is, the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. [...] A special case of Ptolemy's theorem had appeared in Euclid's Data (Proposition 93): [...] Ptolemy's theorem, therefore, leads to the result sin(α - β) = sin α cos β - cos α sin Β. Similar reasoning leads to the formula [...] These four sum-and-difference formulas consequently are often known today as Ptolemy's formulas.
It was the formula for sine of the difference - or, more accurately, chord of the difference - that Ptolemy found especially useful in building up his tables. Another formula that served him effectively was the equivalent of our half-angle formula.” - ^ Boyer, pp. 158–168.
- ^ a b c d e Needham, Volume 3, 109.
- ^ Needham, Volume 3, 108-109.
- ^ Gauchet, 151.
- ^ Needham, Volume 3, 109-110.
- ^ O'Connor and Robertson (2000).
- ^ Pearce (2002).
- Boyer, Carl B. (1991). A History of Mathematics, Second Edition, John Wiley & Sons, Inc.. ISBN 0471543977.
- Gauchet, L. (1917). Note Sur La Trigonométrie Sphérique de Kouo Cheou-King.
- Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.
- Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
- O'Connor, J.J., and E.F. Robertson, "Trigonometric functions", MacTutor History of Mathematics Archive. (1996).
- O'Connor, J.J., and E.F. Robertson, "Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2000).
- Pearce, Ian G., "Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2002).