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History of trigonometry (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "History of trigonometry" in English language version.

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  • Katz 1998, p. 212. Katz, Victor J. (1998). A History of Mathematics / An Introduction (2nd ed.). Addison Wesley. ISBN 978-0-321-01618-8.
  • Boyer 1991, pp. 166–167, Greek Trigonometry and Mensuration: "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 360-degree measure was carried over from astronomy, where the zodiac had been divided into twelve "signs" or 36 "decans". A cycle of the seasons 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." Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • 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." Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • Maor, Eli (1998). Trigonometric Delights. Princeton University Press. p. 20. ISBN 978-0-691-09541-7.
  • Joseph 2000, pp. 383–384. Joseph, George G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd ed.). London: Penguin Books. ISBN 978-0-691-00659-8.
  • Katz 1998, p. 143. Katz, Victor J. (1998). A History of Mathematics / An Introduction (2nd ed.). Addison Wesley. ISBN 978-0-321-01618-8.
  • 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)." Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • 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°.)" Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • 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, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • 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." Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • 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, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • Boyer 1991, pp. 158–168. Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • Boyer 1991, p. 208. Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • Boyer 1991, p. 209. Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • Boyer 1991, p. 210. Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • Boyer 1991, p. 215. Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • Boyer 1991, p. 238. Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
  • Boyer 1991, p. 274. Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.

books.google.com

  • Otto Neugebauer (1975). A history of ancient mathematical astronomy. 1. Springer-Verlag. p. 744. ISBN 978-3-540-06995-9.
  • William Charles Brice, 'An Historical atlas of Islam', p.413
  • Naṣīr al-Dīn al-Ṭūsī (1891). "Ch. 3.2: Sur la manière de calculer les côtés et les angles d'un triangle les uns par les autres". Traité du quadrilatère attribué a Nassiruddinel-Toussy (in French). Translated by Caratheodory, Alexandre Pacha. Typographie et Lithographie Osmanié. p. 69. On donne deux côtés et un angle. [...] Que si l'angle donné est compris entre les deux côtés donnés, comme l'angle A est compris entre les deux côtés AB AC, abaissez de B sur AC la perpendiculaire BE. Vous aurez ainsi le triangle rectangle [BEA] dont nous connaissons le côté AB et l'angle A; on en tirera BE, EA, et l'on retombera ainsi dans un des cas précédents; c. à. d. dans le cas où BE, CE sont connus; on connaîtra dès lors BC et l'angle C, comme nous l'avons expliqué [Given [...] the angle A is included between the two sides AB AC, drop from B to AC the perpendicular BE. You will thus have the right triangle [BEA] of which we know the side AB and the angle A; in that triangle compute BE, EA, and the problem is reduced to one of the preceding cases; that is, to the case where BE, CE are known; we will thus know BC and the angle C, as we have explained.]

britannica.com

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handle.net

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harvard.edu

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hal.inria.fr

iranicaonline.org

  • 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.

kfupm.edu.sa

faculty.kfupm.edu.sa

projecteuclid.org

  • Azarian, Mohammad K. (2000). "Meftab Al-Hesab: A Summary" (PDF). Missouri Journal of Mathematical Sciences. 12 (2): 75–95. doi:10.35834/2000/1202075.
    Aydin, Nuh; Hammoudi, Lakhdar; Bakbouk, Ghada, eds. (2020). Al-Kashi's Miftah al-Hisab, Volume II: Geometry. Birkhäuser. p. 31. doi:10.1007/978-3-030-61330-3. ISBN 978-3-030-61329-7. Another case is when two sides and the angle between them are known and the rest are unknown. We multiply one of the sides by the sine of the [known] angle one time and by the sine of its complement the other time converted and we subtract the second result from the other side if the angle is acute and add it if the angle is obtuse. We then square the result and add to it the square of the first result. We take the square root of the sum to get the remaining side....

semanticscholar.org

api.semanticscholar.org

st-and.ac.uk

www-gap.dcs.st-and.ac.uk

www-groups.dcs.st-and.ac.uk

st-andrews.ac.uk

mathshistory.st-andrews.ac.uk

www-history.mcs.st-andrews.ac.uk

  • Pearce, Ian G. (2002). "Madhava of Sangamagramma". MacTutor History of Mathematics Archive.
  • "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.

stonehill.edu

web.stonehill.edu

tufts.edu

perseus.tufts.edu

  • Euclid. Thomas L. Heath (ed.). "Elements". Translated by Thomas L. Heath. Retrieved 24 January 2023. In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle.

web.archive.org