History of trigonometry
Trigonometry |
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Reference |
Laws and theorems |
Calculus |
Mathematicians |
Early study of triangles can be traced to the
During the
The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748).
Etymology
The term "trigonometry" was derived from Greek τρίγωνον trigōnon, "triangle" and μέτρον metron, "measure".[3]
The modern words "sine" and "cosine" are derived from the
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans "cutting" since the line cuts the circle.[5]
The prefix "
The words "minute" and "second" are derived from the Latin phrases
Ancient
Ancient Near East
The ancient
The
The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC.[10] The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes (c. 1680–1620 BC), contains the following problem related to trigonometry:[10]
"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"
Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.[10]
Classical antiquity
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 one half the bisected angle, that is,[12]
and consequently the sine function is also known as the half-chord. Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to
Although there is no trigonometry in the works of
The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BCE), who is now consequently known as "the father of trigonometry."[16] Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.[4][16]
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
Menelaus of Alexandria (ca. 100 AD) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles.[14] He established 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.[14] Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.[14] Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".[14] He further gave his famous "rule of six quantities".[18]
Later, Claudius Ptolemy (ca. 90 – ca. 168 AD) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The Almagest is primarily a work on astronomy, and astronomy relies on trigonometry. Ptolemy's table of chords gives the lengths of chords of a circle of diameter 120 as a function of the number of degrees n in the corresponding arc of the circle, for n ranging from 1/2 to 180 by increments of 1/2.[19] The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity.[20] 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.[20] 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.[20]
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.[21]
Indian mathematics
Some of the early and very significant developments of trigonometry were in
In the 7th century,
Later in the 7th century, Brahmagupta redeveloped the formula
(also derived earlier, as mentioned above) and the
Another later Indian author on trigonometry was
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 power series expansions of sine, cosine, tangent, and arctangent.[26][27] 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 angle, 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.[26][27]
No. | Series | Name | Western discoverers of the series and approximate dates of discovery[28] |
---|---|---|---|
1 | sin x = x − x3 / 3! + x5 / 5! − x7 / 7! + ... | Madhava's sine series | Isaac Newton (1670) and Wilhelm Leibniz (1676) |
2 | cos x = 1 − x2 / 2! + x4 / 4! − x6 / 6! + ... | Madhava's cosine series | Isaac Newton (1670) and Wilhelm Leibniz (1676) |
3 | tan−1x = x − x3 / 3 + x5 / 5 − x7 / 7 + ... | Madhava's arctangent series
|
James Gregory (1671) and Wilhelm Leibniz (1676) |
The Indian text the
Chinese mathematics
In
Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).[32] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.[29][33] Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:
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).[34]
Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).[35]
Medieval Islamic world
Previous works were later translated and expanded in the
Methods dealing with spherical triangles were also known, particularly the method of
In the early 9th century AD,
By the 10th century AD, in the work of
- (a special case of Ptolemy's angle-addition formula; see above)
In his original text, Abū al-Wafā' states: "If we want that, we multiply the given sine by the cosine
For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the other minutes. If we want the sine of the sum, we add the products, if we want the sine of the difference, we take their difference".[43]
He also discovered the law of sines for spherical trigonometry:[40]
Also in the late 10th and early 11th centuries AD, the Egyptian astronomer
The method of
In the 15th century,
Modern
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European renaissance and afterwards
In 1342, Levi ben Gershon, known as
A simplified trigonometric table, the "
Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline,[53] in his De triangulis omnimodis written in 1464, as well as his later Tabulae directionum which included the tangent function, unnamed. The Opus palatinum de triangulis of
In the 17th century, Isaac Newton and James Stirling developed the general Newton–Stirling interpolation formula for trigonometric functions.
In the 18th century, Leonhard Euler's Introduction in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, deriving their 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. Prior to this, Roger Cotes had computed the derivative of sine in his Harmonia Mensurarum (1722).[54] Also in the 18th century,
See also
- Greek mathematics
- History of mathematics
- Trigonometric functions
- Trigonometry
- Ptolemy's table of chords
- Aryabhata's sine table
- Rational trigonometry
Citations and footnotes
- ISBN 978-3-540-06995-9.
- ^ Katz 1998, p. 212.
- ^ "trigonometry". Online Etymology Dictionary.
- ^ a b O'Connor, J.J.; Robertson, E.F. (1996). "Trigonometric functions". MacTutor History of Mathematics Archive. Archived from the original on 2007-06-04.
- ^ Oxford English Dictionary
- ^ Gunter, Edmund (1620). Canon triangulorum.
- ^ Roegel, Denis, ed. (6 December 2010). "A reconstruction of Gunter's Canon triangulorum (1620)" (Research report). HAL. inria-00543938. Archived from the original on 28 July 2017. Retrieved 28 July 2017.
- ^ 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, pp. 158–159, Greek Trigonometry and Mensuration: "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."
- ^ ISBN 978-0-691-09541-7.
- ^ a b Joseph 2000, pp. 383–384.
- ^ Katz 1998, p. 143.
- ^ As these historical calculations did not make use of a unit circle, the length of the radius was needed in the formula. Contrast this with the modern use of the crd function that assumes a unit circle in its definition.
- ^ a b c d e f Boyer 1991, p. 163, Greek Trigonometry and Mensuration: "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, p. 159, Greek Trigonometry and Mensuration: "Instead we have an treatise, perhaps composed earlier (ca. 260 BC), 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 been 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, for 0° < β < α < 90°.)"
- ^ a b Boyer 1991, p. 162, Greek Trigonometry and Mensuration: "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 2nd century BC, the first trigonometric table apparently was compiled by the astronomer Hipparchus of Nicaea (ca. 180–ca. 125 BC), 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 as the arc 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, p. 162, Greek Trigonometry and Mensuration: "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 have been suggested by Babylonian astronomy."
- ^ Needham 1986, p. 108.
- ISBN 978-0-691-00260-6.
- ^ a b c d Boyer 1991, pp. 164–166, Greek Trigonometry and Mensuration: "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 AD. 127 to 151 and, therefore, assume that he was born at the end of the 1st century. Suidas, a writer who lived in the 10th century, reported that Ptolemy was alive under Marcus Aurelius (emperor from AD 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 catalog of star positions bequeathed by Hipparchus, but whether or not Ptolemy's trigonometric tables were derived in large part from his distinguished predecessor 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 1991, pp. 158–168.
- ^ Boyer 1991, p. 208.
- ^ Boyer 1991, p. 209.
- ^ Boyer 1991, p. 210.
- ^ Boyer 1991, p. 215.
- ^ a b O'Connor, J.J.; Robertson, E.F. (2000). "Madhava of Sangamagramma". MacTutor History of Mathematics Archive.
- ^ a b Pearce, Ian G. (2002). "Madhava of Sangamagramma". MacTutor History of Mathematics Archive.
- ISBN 978-0-387-94313-8.
- ^ a b c d e Needham 1986, p. 109.
- ^ Needham 1986, pp. 108–109.
- ^ Katz 2007, p. 308.
- ^ Restivo 1992, p. 32.
- ^ Gauchet, L. (1917). Note Sur La Trigonométrie Sphérique de Kouo Cheou-King. p. 151.
- ^ Needham 1986, pp. 109–110.
- ^ Needham 1986, p. 110.
- ISBN 978-0-415-13159-9.)
- ^ O'Connor, John J.; Robertson, Edmund F., "Menelaus of Alexandria", MacTutor History of Mathematics Archive, University of St Andrews "Book 3 deals with spherical trigonometry and includes Menelaus's theorem".
- ISBN 978-0-415-13159-9.)
- doi:10.1038/scientificamerican0486-74. Archived from the originalon 2011-01-01. Retrieved 2008-05-18.
- ^ ISBN 978-1-4020-0260-1.
- ^ a b c "trigonometry". Encyclopædia Britannica. Retrieved 2008-07-21.
- ^ a b Boyer 1991, p. 238.
- ^ S2CID 171015175.
- ^ William Charles Brice, 'An Historical atlas of Islam', p.413
- ^ a b O'Connor, John J.; Robertson, Edmund F., "Abu Abd Allah Muhammad ibn Muadh Al-Jayyani", MacTutor History of Mathematics Archive, University of St Andrews
- Donald Routledge Hill(1996), "Engineering", in Roshdi Rashed, Encyclopedia of the History of Arabic Science, Vol. 3, p. 751–795 [769].
- ^ O'Connor, John J.; Robertson, Edmund F., "Abu Arrayhan Muhammad ibn Ahmad al-Biruni", MacTutor History of Mathematics Archive, University of St Andrews
- ISBN 978-0-691-11485-9.
- ^ "Al-Tusi_Nasir biography". www-history.mcs.st-andrews.ac.uk. Retrieved 2018-08-05.
One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.
- ISBN 978-0-511-97400-7.
- ^ electricpulp.com. "ṬUSI, NAṢIR-AL-DIN i. Biography – Encyclopaedia Iranica". www.iranicaonline.org. Retrieved 2018-08-05.
His major contribution in mathematics (Nasr, 1996, pp. 208-214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles.
- ^ Charles G. Simonson (Winter 2000). "The Mathematics of Levi ben Gershon, the Ralbag" (PDF). Bekhol Derakhekha Daehu. 10. Bar-Ilan University Press: 5–21.
- ^ Boyer 1991, p. 274.
- .. The proof of Cotes is mentioned on p. 315.
References
- ISBN 978-0-471-54397-8.
- Joseph, George G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd ed.). London: ISBN 978-0-691-00659-8.
- Katz, Victor J. (1998). A History of Mathematics / An Introduction (2nd ed.). Addison Wesley. ISBN 978-0-321-01618-8.
- Katz, Victor J. (2007). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton: Princeton University Press. ISBN 978-0-691-11485-9.
- Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
- Restivo, Sal (1992). Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers. ISBN 1-4020-0039-1.
Further reading
- Braunmühl, Anton von (1900–1903). Vorlesungen über Geschichte der Trigonometrie [Lectures on the History of Trigonometry] (in German). B. G. Teubner.
- Kennedy, Edward S. (1969). "The History of Trigonometry". Historical Topics for the Mathematics Classroom. NCTM Yearbooks. Vol. 31. National Council of Teachers of Mathematics. pp. 333–375.
- ISBN 0691057540. Archived from the originalon 2003-07-11.
- Ostermann, Alexander; Wanner, Gerhard (2012). "Trigonometry". Geometry by Its History. Undergraduate Texts in Mathematics. Springer. pp. 113–155. ISBN 978-3-642-29162-3.
- Van Brummelen, Glen (2009). The Mathematics of the Heavens and the Earth: The Early History of Trigonometry. Princeton University Press.
- Van Brummelen, Glen (2021). The Doctrine of Triangles: A History of Modern Trigonometry. Princeton University Press.