He was equipped with a trigonometry table. I. However, this does not prove or disprove anything because the commentary might be an early work while the magnitude scale could have been introduced later. Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree. Thus, somebody has added further entries. This makes Hipparchus the founder of trigonometry. The two points at which the ecliptic and the equatorial plane intersect, known as the vernal and autumnal equinoxes, and the two points of the ecliptic farthest north and south from the equatorial plane, known as the summer and winter solstices, divide the ecliptic into four equal parts. Hipparchus's equinox observations gave varying results, but he points out (quoted in Almagest III.1(H195)) that the observation errors by him and his predecessors may have been as large as 14 day. In fact, his astronomical writings were numerous enough that he published an annotated list of them. Ch. This is a highly critical commentary in the form of two books on a popular poem by Aratus based on the work by Eudoxus. In, Wolff M. (1989). 2 - What two factors made it difficult, at first, for. Recalculating Toomer's reconstructions with a 3600' radiusi.e. That would be the first known work of trigonometry. Hipparchus produced a table of chords, an early example of a trigonometric table. His approach would give accurate results if it were correctly carried out but the limitations of timekeeping accuracy in his era made this method impractical. 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). THE EARTH-MOON DISTANCE [29] (The maximum angular deviation producible by this geometry is the arcsin of 5+14 divided by 60, or approximately 5 1', a figure that is sometimes therefore quoted as the equivalent of the Moon's equation of the center in the Hipparchan model.). In the second and third centuries, coins were made in his honour in Bithynia that bear his name and show him with a globe. 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. He tabulated the chords for angles with increments of 7.5. Hipparchus's celestial globe was an instrument similar to modern electronic computers. Lived c. 210 - c. 295 AD. Bianchetti S. (2001). With these values and simple geometry, Hipparchus could determine the mean distance; because it was computed for a minimum distance of the Sun, it is the maximum mean distance possible for the Moon. [26] Modern scholars agree that Hipparchus rounded the eclipse period to the nearest hour, and used it to confirm the validity of the traditional values, rather than to try to derive an improved value from his own observations. The history of celestial mechanics until Johannes Kepler (15711630) was mostly an elaboration of Hipparchuss model. "Hipparchus and the Ancient Metrical Methods on the Sphere". Many credit him as the founder of trigonometry. 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 origins of trigonometry occurred in Ancient Egypt and Babylon, where . Comparing both charts, Hipparchus calculated that the stars had shifted their apparent position by around two degrees. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses. [2] Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. Ptolemy cites more than 20 observations made there by Hipparchus on specific dates from 147 to 127, as well as three earlier observations from 162 to 158 that may be attributed to him. Hipparchus applied his knowledge of spherical angles to the problem of denoting locations on the Earth's surface. And the same individual attempted, what might seem presumptuous even in a deity, viz. A simpler alternate reconstruction[28] agrees with all four numbers. His two books on precession, 'On the Displacement of the Solsticial and Equinoctial Points' and 'On the Length of the Year', are both mentioned in the Almagest of Ptolemy. [48], Conclusion: Hipparchus's star catalogue is one of the sources of the Almagest star catalogue but not the only source.[47]. His theory influence is present on an advanced mechanical device with code name "pin & slot". You can observe all of the stars from the equator over the course of a year, although high- declination stars will be difficult to see so close to the horizon. how did hipparchus discover trigonometry 29 Jun. Calendars were often based on the phases of the moon (the origin of the word month) and the seasons. Hipparchus's long draconitic lunar period (5,458 months = 5,923 lunar nodal periods) also appears a few times in Babylonian records. [63], Jean Baptiste Joseph Delambre, historian of astronomy, mathematical astronomer and director of the Paris Observatory, in his history of astronomy in the 18th century (1821), considered Hipparchus along with Johannes Kepler and James Bradley the greatest astronomers of all time. Hipparchus of Nicaea was a Greek Mathematician, Astronomer, Geographer from 190 BC. Hipparchus wrote a commentary on the Arateiahis only preserved workwhich contains many stellar positions and times for rising, culmination, and setting of the constellations, and these are likely to have been based on his own measurements. Part 2 can be found here. In particular, he improved Eratosthenes' values for the latitudes of Athens, Sicily, and southern extremity of India. 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. He tabulated values for the chord function, which for a central angle in a circle gives the length of the straight line segment between the points where the angle intersects the circle. Bo C. Klintberg states, "With mathematical reconstructions and philosophical arguments I show that Toomer's 1973 paper never contained any conclusive evidence for his claims that Hipparchus had a 3438'-based chord table, and that the Indians used that table to compute their sine tables. The most ancient device found in all early civilisations, is a "shadow stick". Hipparchus was a famous ancient Greek astronomer who managed to simulate ellipse eccentricity by introducing his own theory known as "eccentric theory". That means, no further statement is allowed on these hundreds of stars. He also helped to lay the foundations of trigonometry.Although he is commonly ranked among the greatest scientists of antiquity, very little is known about his life, and only one of his many writings is still in existence. Later al-Biruni (Qanun VII.2.II) and Copernicus (de revolutionibus IV.4) noted that the period of 4,267 moons is approximately five minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. paper, in 158 BC Hipparchus computed a very erroneous summer solstice from Callippus's calendar. [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. Hipparchus introduced the full Babylonian sexigesimal notation for numbers including the measurement of angles using degrees, minutes, and seconds into Greek science. Hipparchus attempted to explain how the Sun could travel with uniform speed along a regular circular path and yet produce seasons of unequal length. Tracking and 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. It was also observed in Alexandria, where the Sun was reported to be obscured 4/5ths by the Moon. D. Rawlins noted that this implies a tropical year of 365.24579 days = 365days;14,44,51 (sexagesimal; = 365days + 14/60 + 44/602 + 51/603) and that this exact year length has been found on one of the few Babylonian clay tablets which explicitly specifies the System B month. . Hipparchus produced a table of chords, an early example of a trigonometric table. Alexandria and Nicaea are on the same meridian. Hipparchus must have used a better approximation for than the one from Archimedes of between 3+1071 (3.14085) and 3+17 (3.14286). [65], Johannes Kepler had great respect for Tycho Brahe's methods and the accuracy of his observations, and considered him to be the new Hipparchus, who would provide the foundation for a restoration of the science of astronomy.[66]. This is called its anomaly and it repeats with its own period; the anomalistic month. Ptolemy later used spherical trigonometry to compute things such as the rising and setting points of the ecliptic, or to take account of the lunar parallax. Hipparchus, also spelled Hipparchos, (born, Nicaea, Bithynia [now Iznik, Turkey]died after 127 bce, Rhodes? Aristarchus of Samos is said to have done so in 280BC, and Hipparchus also had an observation by Archimedes. He was one of the first Greek mathematicians to do this and, in this way, expanded the techniques available to astronomers and geographers. The purpose of this table of chords was to give a method for solving triangles which avoided solving each triangle from first principles. Apparently Hipparchus later refined his computations, and derived accurate single values that he could use for predictions of solar eclipses. ", Toomer G.J. He defined the chord function, derived some of its properties and constructed a table of chords for angles that are multiples of 7.5 using a circle of radius R = 60 360/ (2).This his motivation for choosing this value of R. In this circle, the circumference is 360 times 60. Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the . How did Hipparchus contribute to trigonometry? This opinion was confirmed by the careful investigation of Hoffmann[40] who independently studied the material, potential sources, techniques and results of Hipparchus and reconstructed his celestial globe and its making. However, by comparing his own observations of solstices with observations made in the 5th and 3rd centuries bce, Hipparchus succeeded in obtaining an estimate of the tropical year that was only six minutes too long. He also introduced the division of a circle into 360 degrees into Greece. 2 (1991) pp. Ptolemy gives an extensive discussion of Hipparchus's work on the length of the year in the Almagest III.1, and quotes many observations that Hipparchus made or used, spanning 162128BC. One method used an observation of a solar eclipse that had been total near the Hellespont (now called the Dardanelles) but only partial at Alexandria. Pappus of Alexandria described it (in his commentary on the Almagest of that chapter), as did Proclus (Hypotyposis IV). His contribution was to discover a method of using the . 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). Hipparchus could confirm his computations by comparing eclipses from his own time (presumably 27 January 141BC and 26 November 139BC according to [Toomer 1980]), with eclipses from Babylonian records 345 years earlier (Almagest IV.2; [A.Jones, 2001]). Not much is known about the life of Hipp archus. were probably familiar to Greek astronomers well before Hipparchus. It is known today that the planets, including the Earth, move in approximate ellipses around the Sun, but this was not discovered until Johannes Kepler published his first two laws of planetary motion in 1609. Etymology. After Hipparchus the next Greek mathematician known to have made a contribution to trigonometry was Menelaus. They write new content and verify and edit content received from contributors. Hipparchus obtained information from Alexandria as well as Babylon, but it is not known when or if he visited these places. common errors in the reconstructed Hipparchian star catalogue and the Almagest suggest a direct transfer without re-observation within 265 years. Hipparchus wrote a critique in three books on the work of the geographer Eratosthenes of Cyrene (3rd centuryBC), called Prs tn Eratosthnous geographan ("Against the Geography of Eratosthenes"). (The true value is about 60 times. Another table on the papyrus is perhaps for sidereal motion and a third table is for Metonic tropical motion, using a previously unknown year of 365+141309 days. Trigonometry Trigonometry simplifies the mathematics of triangles, making astronomy calculations easier. Alternate titles: Hipparchos, Hipparchus of Bithynia, Professor of Classics, University of Toronto. Ptolemy's catalog in the Almagest, which is derived from Hipparchus's catalog, is given in ecliptic coordinates. Chapront J., Touze M. Chapront, Francou G. (2002): Duke D.W. (2002). Therefore, Trigonometry started by studying the positions of the stars. "The Chord Table of Hipparchus and the Early History of Greek Trigonometry. He was an outspoken advocate of the truth, of scientific . He is also famous for his incidental discovery of the. Therefore, his globe was mounted in a horizontal plane and had a meridian ring with a scale. Dividing by 52 produces 5,458 synodic months = 5,923 precisely. [note 1] What was so exceptional and useful about the cycle was that all 345-year-interval eclipse pairs occur slightly more than 126,007 days apart within a tight range of only approximately 12 hour, guaranteeing (after division by 4,267) an estimate of the synodic month correct to one part in order of magnitude 10 million. [56] Actually, it has been even shown that the Farnese globe shows constellations in the Aratean tradition and deviates from the constellations in mathematical astronomy that is used by Hipparchus. With this method, as the parallax of the Sun decreases (i.e., its distance increases), the minimum limit for the mean distance is 59 Earth radiiexactly the mean distance that Ptolemy later derived. Hipparchus is sometimes called the "father of astronomy",[7][8] a title first conferred on him by Jean Baptiste Joseph Delambre.[9]. Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans. He had immense in geography and was one of the most famous astronomers in ancient times. Earth's precession means a change in direction of the axis of rotation of Earth. Hipparchus also wrote critical commentaries on some of his predecessors and contemporaries. Unclear how it may have first been discovered. Discovery of a Nova In 134 BC, observing the night sky from the island of Rhodes, Hipparchus discovered a new star. Hipparchus's treatise Against the Geography of Eratosthenes in three books is not preserved. Input the numbers into the arc-length formula, Enter 0.00977 radians for the radian measure and 2,160 for the arc length: 2,160 = 0.00977 x r. Divide each side by 0.00977. 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. Ptolemy quotes an equinox timing by Hipparchus (at 24 March 146BC at dawn) that differs by 5 hours from the observation made on Alexandria's large public equatorial ring that same day (at 1 hour before noon): Hipparchus may have visited Alexandria but he did not make his equinox observations there; presumably he was on Rhodes (at nearly the same geographical longitude). Hipparchus could draw a triangle formed by the two places and the Moon, and from simple geometry was able to establish a distance of the Moon, expressed in Earth radii. 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. He also might have developed and used the theorem called Ptolemy's theorem; this was proved by Ptolemy in his Almagest (I.10) (and later extended by Carnot). The Greeks were mostly concerned with the sky and the heavens. What is Aristarchus full name? 1:28 Solving an Ancient Tablet's Mathematical Mystery Isaac Newton and Euler contributed developments to bring trigonometry into the modern age. also Almagest, book VIII, chapter 3). How did Hipparchus discover trigonometry? During this period he may have invented the planispheric astrolabe, a device on which the celestial sphere is projected onto the plane of the equator." Did Hipparchus invent trigonometry? This was the basis for the astrolabe. He is considered the founder of trigonometry,[1] but is most famous for his incidental discovery of the precession of the equinoxes. This would correspond to a parallax of 7, which is apparently the greatest parallax that Hipparchus thought would not be noticed (for comparison: the typical resolution of the human eye is about 2; Tycho Brahe made naked eye observation with an accuracy down to 1). However, all this was theory and had not been put to practice. Eratosthenes (3rd century BC), in contrast, used a simpler sexagesimal system dividing a circle into 60 parts. In calculating latitudes of climata (latitudes correlated with the length of the longest solstitial day), Hipparchus used an unexpectedly accurate value for the obliquity of the ecliptic, 2340' (the actual value in the second half of the second centuryBC was approximately 2343'), whereas all other ancient authors knew only a roughly rounded value 24, and even Ptolemy used a less accurate value, 2351'.[53]. Hipparchus was an ancient Greek polymath whose wide-ranging interests include geography, astronomy, and mathematics. [33] His other triplet of solar positions is consistent with 94+14 and 92+12 days,[34] an improvement on the results (94+12 and 92+12 days) attributed to Hipparchus by Ptolemy, which a few scholars still question the authorship of. 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. His birth date (c.190BC) was calculated by Delambre based on clues in his work. Hipparchus assumed that the difference could be attributed entirely to the Moons observable parallax against the stars, which amounts to supposing that the Sun, like the stars, is indefinitely far away. The armillary sphere was probably invented only latermaybe by Ptolemy only 265 years after Hipparchus. Although Hipparchus strictly distinguishes between "signs" (30 section of the zodiac) and "constellations" in the zodiac, it is highly questionable whether or not he had an instrument to directly observe / measure units on the ecliptic. The eccentric model he fitted to these eclipses from his Babylonian eclipse list: 22/23 December 383BC, 18/19 June 382BC, and 12/13 December 382BC. This same Hipparchus, who can never be sufficiently commended, discovered a new star that was produced in his own age, and, by observing its motions on the day in which it shone, he was led to doubt whether it does not often happen, that those stars have motion which we suppose to be fixed. Delambre, in 1817, cast doubt on Ptolemy's work. Hipparchus thus calculated that the mean distance of the Moon from Earth is 77 times Earths radius. The Greek astronomer Hipparchus, who lived about 120 years BC, has long been regarded as the father of trigonometry, with his "table of chords" on a circle considered . Hipparchus is generally recognized as discoverer of the precession of the equinoxes in 127BC. There are a variety of mis-steps[55] in the more ambitious 2005 paper, thus no specialists in the area accept its widely publicized speculation. At the end of his career, Hipparchus wrote a book entitled Peri eniausou megthous ("On the Length of the Year") regarding his results. He . Hipparchus insists that a geographic map must be based only on astronomical measurements of latitudes and longitudes and triangulation for finding unknown distances. This was the basis for the astrolabe. (It has been contended that authors like Strabo and Ptolemy had fairly decent values for these geographical positions, so Hipparchus must have known them too. Others do not agree that Hipparchus even constructed a chord table. The distance to the moon is. Hipparchus had good reasons for believing that the Suns path, known as the ecliptic, is a great circle, i.e., that the plane of the ecliptic passes through Earths centre. The value for the eccentricity attributed to Hipparchus by Ptolemy is that the offset is 124 of the radius of the orbit (which is a little too large), and the direction of the apogee would be at longitude 65.5 from the vernal equinox. Get a Britannica Premium subscription and gain access to exclusive content. Ulugh Beg reobserved all the Hipparchus stars he could see from Samarkand in 1437 to about the same accuracy as Hipparchus's. Please refer to the appropriate style manual or other sources if you have any questions. He also discovered that the moon, the planets and the stars were more complex than anyone imagined. Hipparchus [51], He was the first to use the grade grid, to determine geographic latitude from star observations, and not only from the Sun's altitude, a method known long before him, and to suggest that geographic longitude could be determined by means of simultaneous observations of lunar eclipses in distant places. In this only work by his hand that has survived until today, he does not use the magnitude scale but estimates brightnesses unsystematically. [18] The obvious main objection is that the early eclipse is unattested, although that is not surprising in itself, and there is no consensus on whether Babylonian observations were recorded this remotely. Hipparchus also adopted the Babylonian astronomical cubit unit (Akkadian ammatu, Greek pchys) that was equivalent to 2 or 2.5 ('large cubit'). He did this by using the supplementary angle theorem, half angle formulas, and linear interpolation. Hipparchus and his predecessors used various instruments for astronomical calculations and observations, such as the gnomon, the astrolabe, and the armillary sphere. This makes Hipparchus the founder of trigonometry. In On Sizes and Distances (now lost), Hipparchus reportedly measured the Moons orbit in relation to the size of Earth. Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the equinoxes. "Associations between the ancient star catalogs". Prediction of a solar eclipse, i.e., exactly when and where it will be visible, requires a solid lunar theory and proper treatment of the lunar parallax. In any case, according to Pappus, Hipparchus found that the least distance is 71 (from this eclipse), and the greatest 81 Earth radii. [14], Hipparchus probably compiled a list of Babylonian astronomical observations; G. J. Toomer, a historian of astronomy, has suggested that Ptolemy's knowledge of eclipse records and other Babylonian observations in the Almagest came from a list made by Hipparchus. This is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. It was a four-foot rod with a scale, a sighting hole at one end, and a wedge that could be moved along the rod to exactly obscure the disk of Sun or Moon. Trigonometry (from Ancient Greek (trgnon) 'triangle', and (mtron) 'measure') [1] is a branch of mathematics concerned with relationships between angles and ratios of lengths. The term "trigonometry" was derived from Greek trignon, "triangle" and metron, "measure".. Hipparchus seems to have been the first to exploit Babylonian astronomical knowledge and techniques systematically. That apparent diameter is, as he had observed, 360650 degrees. 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. With an astrolabe Hipparchus was the first to be able to measure the geographical latitude and time by observing fixed stars. Vol. 103,049 is the tenth SchrderHipparchus number, which counts the number of ways of adding one or more pairs of parentheses around consecutive subsequences of two or more items in any sequence of ten symbols. 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. The first proof we have is that of Ptolemy. Chords are closely related to sines. According to Theon, Hipparchus wrote a 12-book work on chords in a circle, since lost. [40], Lucio Russo has said that Plutarch, in his work On the Face in the Moon, was reporting some physical theories that we consider to be Newtonian and that these may have come originally from Hipparchus;[57] he goes on to say that Newton may have been influenced by them. "Hipparchus' Treatment of Early Greek Astronomy: The Case of Eudoxus and the Length of Daytime Author(s)". [37][38], Hipparchus also constructed a celestial globe depicting the constellations, based on his observations. How did Hipparchus influence? Trigonometry developed in many parts of the world over thousands of years, but the mathematicians who are most credited with its discovery are Hipparchus, Menelaus and Ptolemy. "Hipparchus' Empirical Basis for his Lunar Mean Motions,", Toomer G.J. He did this by using the supplementary angle theorem, half angle formulas, and linear interpolation. Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the equinoxes. (1997). (1980). The geometry, and the limits of the positions of Sun and Moon when a solar or lunar eclipse is possible, are explained in Almagest VI.5. The 345-year periodicity is why[25] the ancients could conceive of a mean month and quantify it so accurately that it is correct, even today, to a fraction of a second of time. It had been known for a long time that the motion of the Moon is not uniform: its speed varies. Hipparchus (/hprks/; Greek: , Hipparkhos; c.190 c.120BC) was a Greek astronomer, geographer, and mathematician. The globe was virtually reconstructed by a historian of science. A new study claims the tablet could be one of the oldest contributions to the the study of trigonometry, but some remain skeptical. But a few things are known from various mentions of it in other sources including another of his own. to number the stars for posterity and to express their relations by appropriate names; having previously devised instruments, by which he might mark the places and the magnitudes of each individual star. 2 He is called . The branch called "Trigonometry" basically deals with the study of the relationship between the sides and angles of the right-angle triangle. Hipparchus initially used (Almagest 6.9) his 141 BC eclipse with a Babylonian eclipse of 720 BC to find the less accurate ratio 7,160 synodic months = 7,770 draconitic months, simplified by him to 716 = 777 through division by 10. Before Hipparchus, astronomers knew that the lengths of the seasons are not equal. Mott Greene, "The birth of modern science?" Like others before and after him, he found that the Moon's size varies as it moves on its (eccentric) orbit, but he found no perceptible variation in the apparent diameter of the Sun. An Investigation of the Ancient Star Catalog. Hipparchus apparently made similar calculations. 1 This dating accords with Plutarch's choice of him as a character in a dialogue supposed to have taken place at or near Rome some lime after a.d.75. 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. 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. Alexander Jones "Ptolemy in Perspective: Use and Criticism of his Work from Antiquity to the Nineteenth Century, Springer, 2010, p.36. [17] But the only such tablet explicitly dated, is post-Hipparchus so the direction of transmission is not settled by the tablets. His contribution was to discover a method of using the observed dates of two equinoxes and a solstice to calculate the size and direction of the displacement of the Suns orbit. Hipparchus compiled a table of the chords of angles and made them available to other scholars. Hipparchus used the multiple of this period by a factor of 17, because that interval is also an eclipse period, and is also close to an integer number of years (4,267 moons: 4,573 anomalistic periods: 4,630.53 nodal periods: 4,611.98 lunar orbits: 344.996 years: 344.982 solar orbits: 126,007.003 days: 126,351.985 rotations).