how did hipparchus discover trigonometry

"Le "Commentaire" d'Hipparque. . He did this by using the supplementary angle theorem, half angle formulas, and linear interpolation. He is best known for his discovery of the precession of the equinoxes and contributed significantly to the field of astronomy on every level. Ptolemy describes the details in the Almagest IV.11. Hipparchus concluded that the equinoxes were moving ("precessing") through the zodiac, and that the rate of precession was not less than 1 in a century. He is believed to have died on the island of Rhodes, where he seems to have spent most of his later life. [13] Eudoxus in the 4th century BC and Timocharis and Aristillus in the 3rd century BC already divided the ecliptic in 360 parts (our degrees, Greek: moira) of 60 arcminutes and Hipparchus continued this tradition. He had two methods of doing this. Hipparchus was perhaps the discoverer (or inventor?) How did Hipparchus influence? He communicated with observers at Alexandria in Egypt, who provided him with some times of equinoxes, and probably also with astronomers at Babylon. The Chaldeans took account of this arithmetically, and used a table giving the daily motion of the Moon according to the date within a long period. The three most important mathematicians involved in devising Greek trigonometry are Hipparchus, Menelaus, and Ptolemy. Ch. Similarly, Cleomedes quotes Hipparchus for the sizes of the Sun and Earth as 1050:1; this leads to a mean lunar distance of 61 radii. (He similarly found from the 345-year cycle the ratio 4,267 synodic months = 4,573 anomalistic months and divided by 17 to obtain the standard ratio 251 synodic months = 269 anomalistic months.) One of his two eclipse trios' solar longitudes are consistent with his having initially adopted inaccurate lengths for spring and summer of 95+34 and 91+14 days. Proofs of this inequality using only Ptolemaic tools are quite complicated. Hipparchus also tried to measure as precisely as possible the length of the tropical yearthe period for the Sun to complete one passage through the ecliptic. How did Hipparchus discover trigonometry? The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201BC, 19 March 200BC, and 11 September 200BC. That apparent diameter is, as he had observed, 360650 degrees. Some scholars do not believe ryabhaa's sine table has anything to do with Hipparchus's chord table. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Hipparchus may also have used other sets of observations, which would lead to different values. ", Toomer G.J. [citation needed] Ptolemy claims his solar observations were on a transit instrument set in the meridian. Hipparchus's only preserved work is ("Commentary on the Phaenomena of Eudoxus and Aratus"). Theon of Smyrna wrote that according to Hipparchus, the Sun is 1,880 times the size of the Earth, and the Earth twenty-seven times the size of the Moon; apparently this refers to volumes, not diameters. Trigonometry was probably invented by Hipparchus, who compiled a table of the chords of angles and made them available to other scholars. That would be the first known work of trigonometry. According to Pappus, he found a least distance of 62, a mean of 67+13, and consequently a greatest distance of 72+23 Earth radii. He observed the summer solstice in 146 and 135BC both accurate to a few hours, but observations of the moment of equinox were simpler, and he made twenty during his lifetime. Hipparchus's use of Babylonian sources has always been known in a general way, because of Ptolemy's statements, but the only text by Hipparchus that survives does not provide sufficient information to decide whether Hipparchus's knowledge (such as his usage of the units cubit and finger, degrees and minutes, or the concept of hour stars) was based on Babylonian practice. The papyrus also confirmed that Hipparchus had used Callippic solar motion in 158 BC, a new finding in 1991 but not attested directly until P. Fouad 267 A. A lunar eclipse is visible simultaneously on half of the Earth, and the difference in longitude between places can be computed from the difference in local time when the eclipse is observed. 2 - How did Hipparchus discover the wobble of Earth's. Ch. Author of. Rawlins D. (1982). Hipparchus produced a table of chords, an early example of a trigonometric table. That means, no further statement is allowed on these hundreds of stars. He didn't invent the sine and cosine functions, but instead he used the \chord" function, giving the length of the chord of the unit circle that subtends a given angle. Most of our knowledge of it comes from Strabo, according to whom Hipparchus thoroughly and often unfairly criticized Eratosthenes, mainly for internal contradictions and inaccuracy in determining positions of geographical localities. Another value for the year that is attributed to Hipparchus (by the astrologer Vettius Valens in the first century) is 365 + 1/4 + 1/288 days (= 365.25347 days = 365days 6hours 5min), but this may be a corruption of another value attributed to a Babylonian source: 365 + 1/4 + 1/144 days (= 365.25694 days = 365days 6hours 10min). It was disputed whether the star catalog in the Almagest is due to Hipparchus, but 19762002 statistical and spatial analyses (by R. R. Newton, Dennis Rawlins, Gerd Grasshoff,[44] Keith Pickering[45] and Dennis Duke[46]) have shown conclusively that the Almagest star catalog is almost entirely Hipparchan. According to Theon, Hipparchus wrote a 12-book work on chords in a circle, since lost. He computed this for a circle with a circumference of 21,600 units and a radius (rounded) of 3,438 units; this circle has a unit length of 1 arcminute along its perimeter. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses. Others do not agree that Hipparchus even constructed a chord table. "Associations between the ancient star catalogs". (1967). See [Toomer 1974] for a more detailed discussion. Before Hipparchus, astronomers knew that the lengths of the seasons are not equal. As shown in a 1991 The exact dates of his life are not known, but Ptolemy attributes astronomical observations to him in the period from 147 to 127BC, and some of these are stated as made in Rhodes; earlier observations since 162BC might also have been made by him. There are several indications that Hipparchus knew spherical trigonometry, but the first surviving text discussing it is by Menelaus of Alexandria in the first century, who now, on that basis, commonly is credited with its discovery. [41] This system was made more precise and extended by N. R. Pogson in 1856, who placed the magnitudes on a logarithmic scale, making magnitude 1 stars 100 times brighter than magnitude 6 stars, thus each magnitude is 5100 or 2.512 times brighter than the next faintest magnitude. Recent expert translation and analysis by Anne Tihon of papyrus P. Fouad 267 A has confirmed the 1991 finding cited above that Hipparchus obtained a summer solstice in 158 BC. 2 (1991) pp. Hipparchus: The birth of trigonometry occurred in the chord tables of Hipparchus (c 190 - 120 BCE) who was born shortly after Eratosthenes died. On this Wikipedia the language links are at the top of the page across from the article title. In essence, Ptolemy's work is an extended attempt to realize Hipparchus's vision of what geography ought to be. 2 - Why did Ptolemy have to introduce multiple circles. He was then in a position to calculate equinox and solstice dates for any year. Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans. It was only in Hipparchus's time (2nd century BC) when this division was introduced (probably by Hipparchus's contemporary Hypsikles) for all circles in mathematics. Some claim the table of Hipparchus may have survived in astronomical treatises in India, such as the Surya Siddhanta. Apparently it was well-known at the time. Earlier Greek astronomers and mathematicians were influenced by Babylonian astronomy to some extent, for instance the period relations of the Metonic cycle and Saros cycle may have come from Babylonian sources (see "Babylonian astronomical diaries"). Hipparchus observed (at lunar eclipses) that at the mean distance of the Moon, the diameter of the shadow cone is 2+12 lunar diameters. "Hipparchus and the Ancient Metrical Methods on the Sphere". Hipparchus adopted values for the Moons periodicities that were known to contemporary Babylonian astronomers, and he confirmed their accuracy by comparing recorded observations of lunar eclipses separated by intervals of several centuries. The result that two solar eclipses can occur one month apart is important, because this can not be based on observations: one is visible on the northern and the other on the southern hemisphereas Pliny indicatesand the latter was inaccessible to the Greek. (See animation.). Hipparchus was not only the founder of trigonometry but also the man who transformed Greek astronomy from a purely theoretical into a practical predictive science. He also introduced the division of a circle into 360 degrees into Greece. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) In fact, his astronomical writings were numerous enough that he published an annotated list of them. He knew that this is because in the then-current models the Moon circles the center of the Earth, but the observer is at the surfacethe Moon, Earth and observer form a triangle with a sharp angle that changes all the time. A rigorous treatment requires spherical trigonometry, thus those who remain certain that Hipparchus lacked it must speculate that he may have made do with planar approximations. His results were the best so far: the actual mean distance of the Moon is 60.3 Earth radii, within his limits from Hipparchus's second book. [54] These models, which assumed that the apparent irregular motion was produced by compounding two or more uniform circular motions, were probably familiar to Greek astronomers well before Hipparchus. The history of trigonometry and of trigonometric functions sticks to the general lines of the history of math. Get a Britannica Premium subscription and gain access to exclusive content. Hipparchus also analyzed the more complicated motion of the Moon in order to construct a theory of eclipses. In the first, the Moon would move uniformly along a circle, but the Earth would be eccentric, i.e., at some distance of the center of the circle. ", Toomer G.J. Emma Willard, Astronography, Or, Astronomical Geography, with the Use of Globes: Arranged Either for Simultaneous Reading and Study in Classes, Or for Study in the Common Method, pp 246, Denison Olmsted, Outlines of a Course of Lectures on Meteorology and Astronomy, pp 22, University of Toronto Quarterly, Volumes 1-3, pp 50, Histoire de l'astronomie ancienne, Jean Baptiste Joseph Delambre, Volume 1, p lxi; "Hipparque, le vrai pre de l'Astronomie"/"Hipparchus, the true father of Astronomy", Bowen A.C., Goldstein B.R. After Hipparchus the next Greek mathematician known to have made a contribution to trigonometry was Menelaus. Hipparchus also observed solar equinoxes, which may be done with an equatorial ring: its shadow falls on itself when the Sun is on the equator (i.e., in one of the equinoctial points on the ecliptic), but the shadow falls above or below the opposite side of the ring when the Sun is south or north of the equator. Therefore, his globe was mounted in a horizontal plane and had a meridian ring with a scale. Comparing both charts, Hipparchus calculated that the stars had shifted their apparent position by around two degrees. Chapront J., Touze M. Chapront, Francou G. (2002): Duke D.W. (2002). (Parallax is the apparent displacement of an object when viewed from different vantage points). [47] Although the Almagest star catalogue is based upon Hipparchus's one, it is not only a blind copy but enriched, enhanced, and thus (at least partially) re-observed.[15]. In modern terms, the chord subtended by a central angle in a circle of given radius equals the radius times twice the sine of half of the angle, i.e. "The Size of the Lunar Epicycle According to Hipparchus. One evening, Hipparchus noticed the appearance of a star where he was certain there had been none before. Russo L. (1994). how did hipparchus discover trigonometry 29 Jun. From the geometry of book 2 it follows that the Sun is at 2,550 Earth radii, and the mean distance of the Moon is 60+12 radii. [15], Nevertheless, this system certainly precedes Ptolemy, who used it extensively about AD 150. Alexander Jones "Ptolemy in Perspective: Use and Criticism of his Work from Antiquity to the Nineteenth Century, Springer, 2010, p.36. Hipparchus (/ h p r k s /; Greek: , Hipparkhos; c. 190 - c. 120 BC) was a Greek astronomer, geographer, and mathematician.He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Before Hipparchus, Meton, Euctemon, and their pupils at Athens had made a solstice observation (i.e., timed the moment of the summer solstice) on 27 June 432BC (proleptic Julian calendar). A solution that has produced the exact .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}5,4585,923 ratio is rejected by most historians although it uses the only anciently attested method of determining such ratios, and it automatically delivers the ratio's four-digit numerator and denominator.

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