Bowen A.C., Goldstein B.R. Hipparchus apparently made similar calculations. According to Ptolemy, Hipparchus measured the longitude of Spica and Regulus and other bright stars. Hipparchus's celestial globe was an instrument similar to modern electronic computers. Hipparchus discovered the precessions of equinoxes by comparing his notes with earlier observers; his realization that the points of solstice and equinox moved slowly from east to west against the . "Hipparchus and Babylonian Astronomy." (1973). [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. [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. According to Roman sources, Hipparchus made his measurements with a scientific instrument and he obtained the positions of roughly 850 stars. On this Wikipedia the language links are at the top of the page across from the article title. (1974). how did hipparchus discover trigonometry 29 Jun. Trigonometry was probably invented by Hipparchus, who compiled a table of the chords of angles and made them available to other scholars. 2 - Why did Copernicus want to develop a completely. Hence, it helps to find the missing or unknown angles or sides of a right triangle using the trigonometric formulas, functions or trigonometric identities. They write new content and verify and edit content received from contributors. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses. He was equipped with a trigonometry table. It is believed that he was born at Nicaea in Bithynia. It had been known for a long time that the motion of the Moon is not uniform: its speed varies. Ptolemy established a ratio of 60: 5+14. 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). Review of, "Hipparchus Table of Climata and Ptolemys Geography", "Hipparchos' Eclipse-Based Longitudes: Spica & Regulus", "Five Millennium Catalog of Solar Eclipses", "New evidence for Hipparchus' Star Catalog revealed by multispectral imaging", "First known map of night sky found hidden in Medieval parchment", "Magnitudes of Thirty-six of the Minor Planets for the first day of each month of the year 1857", "The Measurement Method of the Almagest Stars", "The Genesis of Hipparchus' Celestial Globe", Hipparchus "Table of Climata and Ptolemys Geography", "Hipparchus on the Latitude of Southern India", Eratosthenes' Parallel of Rhodes and the History of the System of Climata, "Ptolemys Latitude of Thule and the Map Projection in the Pre-Ptolemaic Geography", "Hipparchus, Plutarch, Schrder, and Hough", "On the shoulders of Hipparchus: A reappraisal of ancient Greek combinatorics", "X-Prize Group Founder to Speak at Induction", "A new determination of lunar orbital parameters, precession constant, and tidal acceleration from LLR measurements", "The Epoch of the Constellations on the Farnese Atlas and their Origin in Hipparchus's Lost Catalogue", Eratosthenes Parallel of Rhodes and the History of the System of Climata, "The accuracy of eclipse times measured by the Babylonians", "Lunar Eclipse Times Recorded in Babylonian History", Learn how and when to remove this template message, Biography of Hipparchus on Fermat's Last Theorem Blog, Os Eclipses, AsterDomus website, portuguese, Ancient Astronomy, Integers, Great Ratios, and Aristarchus, David Ulansey about Hipparchus's understanding of the precession, A brief view by Carmen Rush on Hipparchus' stellar catalog, "New evidence for Hipparchus' Star Catalogue revealed by multispectral imaging", Ancient Greek and Hellenistic mathematics, https://en.wikipedia.org/w/index.php?title=Hipparchus&oldid=1141264401, Short description is different from Wikidata, Articles with unsourced statements from September 2022, Articles with unsourced statements from March 2021, Articles containing Ancient Greek (to 1453)-language text, Wikipedia articles incorporating a citation from the 1911 Encyclopaedia Britannica with Wikisource reference, Wikipedia external links cleanup from May 2017, Creative Commons Attribution-ShareAlike License 3.0. Bianchetti S. (2001). 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. Hipparchus was born in Nicaea, Bithynia (now Iznik, Turkey) and most likely died on the island of Rhodes. 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. Ch. Omissions? As shown in a 1991 Once again you must zoom in using the Page Up key. Hipparchus insists that a geographic map must be based only on astronomical measurements of latitudes and longitudes and triangulation for finding unknown distances. This is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. He had immense in geography and was one of the most famous astronomers in ancient times. However, Strabo's Hipparchus dependent latitudes for this region are at least 1 too high, and Ptolemy appears to copy them, placing Byzantium 2 high in latitude.) If he did not use spherical trigonometry, Hipparchus may have used a globe for these tasks, reading values off coordinate grids drawn on it, or he may have made approximations from planar geometry, or perhaps used arithmetical approximations developed by the Chaldeans. Hipparchus applied his knowledge of spherical angles to the problem of denoting locations on the Earth's surface. [58] According to one book review, both of these claims have been rejected by other scholars. According to Synesius of Ptolemais (4th century) he made the first astrolabion: this may have been an armillary sphere (which Ptolemy however says he constructed, in Almagest V.1); or the predecessor of the planar instrument called astrolabe (also mentioned by Theon of Alexandria). 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. 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. Hipparchus thus had the problematic result that his minimum distance (from book 1) was greater than his maximum mean distance (from book 2). He did this by using the supplementary angle theorem, half angle formulas, and linear . As a young man in Bithynia, Hipparchus compiled records of local weather patterns throughout the year. In fact, he did this separately for the eccentric and the epicycle model. [42], It is disputed which coordinate system(s) he used. Hipparchus and his predecessors used various instruments for astronomical calculations and observations, such as the gnomon, the astrolabe, and the armillary sphere. Hipparchus In, This page was last edited on 24 February 2023, at 05:19. However, the Suns passage through each section of the ecliptic, or season, is not symmetrical. This is called its anomaly and it repeats with its own period; the anomalistic month. "Geographical Latitudes in Eratosthenes, Hipparchus and Posidonius". [60][61], He may be depicted opposite Ptolemy in Raphael's 15091511 painting The School of Athens, although this figure is usually identified as Zoroaster.[62]. Hipparchus, the mathematician and astronomer, was born around the year 190 BCE in Nicaea, in what is present-day Turkey. the inhabited part of the land, up to the equator and the Arctic Circle. This is the first of three articles on the History of Trigonometry. Hipparchus: The birth of trigonometry occurred in the chord tables of Hipparchus (c 190 - 120 BCE) who was born shortly after Eratosthenes died. Pliny also remarks that "he also discovered for what exact reason, although the shadow causing the eclipse must from sunrise onward be below the earth, it happened once in the past that the Moon was eclipsed in the west while both luminaries were visible above the earth" (translation H. Rackham (1938), Loeb Classical Library 330 p.207). Sidoli N. (2004). So the apparent angular speed of the Moon (and its distance) would vary. Hipparchus also adopted the Babylonian astronomical cubit unit (Akkadian ammatu, Greek pchys) that was equivalent to 2 or 2.5 ('large cubit'). (1991). The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Nadal R., Brunet J.P. (1984). Hipparchus must have lived some time after 127BC because he analyzed and published his observations from that year. Scholars have been searching for it for centuries. [40] He used it to determine risings, settings and culminations (cf. The historian of science S. Hoffmann found proof that Hipparchus observed the "longitudes" and "latitudes" in different coordinate systems and, thus, with different instrumentation. He was an outspoken advocate of the truth, of scientific . It remained, however, for Ptolemy (127145 ce) to finish fashioning a fully predictive lunar model. 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. Hipparchus discovered the Earth's precession by following and measuring the movements of the stars, specifically Spica and Regulus, two of the brightest stars in our night sky. 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. Before Hipparchus, astronomers knew that the lengths of the seasons are not equal. He found that at the mean distance of the Moon, the Sun and Moon had the same apparent diameter; at that distance, the Moon's diameter fits 650 times into the circle, i.e., the mean apparent diameters are 360650 = 03314. 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. Most of Hipparchuss adult life, however, seems to have been spent carrying out a program of astronomical observation and research on the island of Rhodes. [10], Relatively little of Hipparchus's direct work survives into modern times. Alexander Jones "Ptolemy in Perspective: Use and Criticism of his Work from Antiquity to the Nineteenth Century, Springer, 2010, p.36. We know very little about the life of Menelaus. He considered every triangle as being inscribed in a circle, so that each side became a chord. In this way it might be easily discovered, not only whether they were destroyed or produced, but whether they changed their relative positions, and likewise, whether they were increased or diminished; the heavens being thus left as an inheritance to any one, who might be found competent to complete his plan. Chords are nearly related to sines. Hipparchus, also spelled Hipparchos, (born, Nicaea, Bithynia [now Iznik, Turkey]died after 127 bce, Rhodes? ", Toomer G.J. In, Wolff M. (1989). Not much is known about the life of Hipp archus. Hipparchus (190 120 BCE) Hipparchus lived in Nicaea. and for the epicycle model, the ratio between the radius of the deferent and the epicycle: Hipparchus was inspired by a newly emerging star, he doubts on the stability of stellar brightnesses, he observed with appropriate instruments (pluralit is not said that he observed everything with the same instrument). 2 He is called . G J Toomer's chapter "Ptolemy and his Greek Predecessors" in "Astronomy before the Telescope", British Museum Press, 1996, p.81. [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. Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the . (In fact, modern calculations show that the size of the 189BC solar eclipse at Alexandria must have been closer to 910ths and not the reported 45ths, a fraction more closely matched by the degree of totality at Alexandria of eclipses occurring in 310 and 129BC which were also nearly total in the Hellespont and are thought by many to be more likely possibilities for the eclipse Hipparchus used for his computations.). In fact, his astronomical writings were numerous enough that he published an annotated list of them. Aubrey Diller has shown that the clima calculations that Strabo preserved from Hipparchus could have been performed by spherical trigonometry using the only accurate obliquity known to have been used by ancient astronomers, 2340. He knew the . This was presumably found[30] by dividing the 274 years from 432 to 158 BC, into the corresponding interval of 100,077 days and 14+34 hours between Meton's sunrise and Hipparchus's sunset solstices. [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]. One evening, Hipparchus noticed the appearance of a star where he was certain there had been none before. Using the visually identical sizes of the solar and lunar discs, and observations of Earths shadow during lunar eclipses, Hipparchus found a relationship between the lunar and solar distances that enabled him to calculate that the Moons mean distance from Earth is approximately 63 times Earths radius. Thus, somebody has added further entries. Hipparchus discovered the wobble of Earth's axis by comparing previous star charts to the charts he created during his study of the stars. He was also the inventor of trigonometry. While every effort has been made to follow citation style rules, there may be some discrepancies. In geographic theory and methods Hipparchus introduced three main innovations. 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. At the end of the third century BC, Apollonius of Perga had proposed two models for lunar and planetary motion: Apollonius demonstrated that these two models were in fact mathematically equivalent. Recalculating Toomer's reconstructions with a 3600' radiusi.e. This makes Hipparchus the founder of trigonometry. "Dallastronomia alla cartografia: Ipparco di Nicea". His contribution was to discover a method of using the . (1997). It is not clear whether this would be a value for the sidereal year at his time or the modern estimate of approximately 365.2565 days, but the difference with Hipparchus's value for the tropical year is consistent with his rate of precession (see below). In this case, the shadow of the Earth is a cone rather than a cylinder as under the first assumption. This is where the birthplace of Hipparchus (the ancient city of Nicaea) stood on the Hellespont strait. ???? Trigonometry was probably invented by Hipparchus, who compiled a table of the chords of angles and made them available to other scholars. [2] 43, No. [50] That apparent diameter is, as he had observed, 360650 degrees. With his value for the eccentricity of the orbit, he could compute the least and greatest distances of the Moon too. Parallax lowers the altitude of the luminaries; refraction raises them, and from a high point of view the horizon is lowered. paper, in 158 BC Hipparchus computed a very erroneous summer solstice from Callippus's calendar. Most of what is known about Hipparchus comes from Strabo's Geography and Pliny's Natural History in the first century; Ptolemy's second-century Almagest; and additional references to him in the fourth century by Pappus and Theon of Alexandria in their commentaries on the Almagest.[11]. The distance to the moon is. Trigonometry (from Ancient Greek (trgnon) 'triangle', and (mtron) 'measure') [1] is a branch of mathematics concerned with relationships between angles and ratios of lengths. He was one of the first Greek mathematicians to do this and, in this way, expanded the techniques available to astronomers and geographers. At the end of his career, Hipparchus wrote a book entitled Peri eniausou megthous ("On the Length of the Year") regarding his results. 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. 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. Trigonometry Trigonometry simplifies the mathematics of triangles, making astronomy calculations easier. He was intellectually honest about this discrepancy, and probably realized that especially the first method is very sensitive to the accuracy of the observations and parameters. 3550jl1016a Vs 3550jl1017a . This would be the second eclipse of the 345-year interval that Hipparchus used to verify the traditional Babylonian periods: this puts a late date to the development of Hipparchus's lunar theory. How did Hipparchus discover a Nova? Like most of his predecessorsAristarchus of Samos was an exceptionHipparchus assumed a spherical, stationary Earth at the centre of the universe (the geocentric cosmology). : The now-lost work in which Hipparchus is said to have developed his chord table, is called Tn en kukli euthein (Of Lines Inside a Circle) in Theon of Alexandria's fourth-century commentary on section I.10 of the Almagest. 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. [citation needed] Ptolemy claims his solar observations were on a transit instrument set in the meridian. Ancient Trigonometry & Astronomy Astronomy was hugely important to ancient cultures and became one of the most important drivers of mathematical development, particularly Trigonometry (literally triangle-measure). He then analyzed a solar eclipse, which Toomer (against the opinion of over a century of astronomers) presumes to be the eclipse of 14 March 190BC. This makes Hipparchus the founder of trigonometry. He was able to solve the geometry Previously this was done at daytime by measuring the shadow cast by a gnomon, by recording the length of the longest day of the year or with the portable instrument known as a scaphe. UNSW scientists have discovered the purpose of a famous 3700-year-old Babylonian clay tablet, revealing it is the world's oldest and most accurate trigonometric table. Roughly five centuries after Euclid's era, he solved hundreds of algebraic equations in his great work Arithmetica, and was the first person to use algebraic notation and symbolism. Some claim the table of Hipparchus may have survived in astronomical treatises in India, such as the Surya Siddhanta. Hipparchus may also have used other sets of observations, which would lead to different values. But a few things are known from various mentions of it in other sources including another of his own. Hipparchus is credited with the invention or improvement of several astronomical instruments, which were used for a long time for naked-eye observations. Hipparchus's catalogue is reported in Roman times to have enlisted about 850 stars but Ptolemy's catalogue has 1025 stars. Analysis of Hipparchus's seventeen equinox observations made at Rhodes shows that the mean error in declination is positive seven arc minutes, nearly agreeing with the sum of refraction by air and Swerdlow's parallax. This was the basis for the astrolabe. . Let us know if you have suggestions to improve this article (requires login). Alexandria and Nicaea are on the same meridian. [22] Further confirming his contention is the finding that the big errors in Hipparchus's longitude of Regulus and both longitudes of Spica, agree to a few minutes in all three instances with a theory that he took the wrong sign for his correction for parallax when using eclipses for determining stars' positions.[23]. 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. 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. was a Greek astronomer, geographer, and mathematician of the Hellenistic period. Proofs of this inequality using only Ptolemaic tools are quite complicated. 1. "Hipparchus and the Stoic Theory of Motion". How did Hipparchus discover and measure the precession of the equinoxes? One evening, Hipparchus noticed the appearance of a star where he was certain there had been none before. How did Hipparchus discover trigonometry? Hipparchus was recognized as the first mathematician known to have possessed a trigonometric table, which he needed when computing the eccentricity of the orbits of the Moon and Sun. [15], Nevertheless, this system certainly precedes Ptolemy, who used it extensively about AD 150. [3], Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity. ), Italian philosopher, astronomer and mathematician. Not only did he make extensive observations of star positions, Hipparchus also computed lunar and solar eclipses, primarily by using trigonometry. Toomer, "The Chord Table of Hipparchus" (1973). Delambre in his Histoire de l'Astronomie Ancienne (1817) concluded that Hipparchus knew and used the equatorial coordinate system, a conclusion challenged by Otto Neugebauer in his A History of Ancient Mathematical Astronomy (1975). Ptolemy discovered the table of arcs. It was based on a circle in which the circumference was divided, in the normal (Babylonian) manner, into 360 degrees of 60 minutes, and the radius was measured in the same units; thus R, the radius, expressed in minutes, is This function is related to the modern sine function (for in degrees) by Hipparchus obtained information from Alexandria as well as Babylon, but it is not known when or if he visited these places. He is known for discovering the change in the orientation of the Earth's axis and the axis of other planets with respect to the center of the Sun. Get a Britannica Premium subscription and gain access to exclusive content. 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. Hipparchus produced a table of chords, an early example of a trigonometric table. 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. It was also observed in Alexandria, where the Sun was reported to be obscured 4/5ths by the Moon. Therefore, Trigonometry started by studying the positions of the stars. The somewhat weird numbers are due to the cumbersome unit he used in his chord table according to one group of historians, who explain their reconstruction's inability to agree with these four numbers as partly due to some sloppy rounding and calculation errors by Hipparchus, for which Ptolemy criticised him while also making rounding errors. It is known to us from Strabo of Amaseia, who in his turn criticised Hipparchus in his own Geographia. Hipparchus could have constructed his chord table using the Pythagorean theorem and a theorem known to Archimedes. (Parallax is the apparent displacement of an object when viewed from different vantage points). Ulugh Beg reobserved all the Hipparchus stars he could see from Samarkand in 1437 to about the same accuracy as Hipparchus's. Aristarchus of Samos is said to have done so in 280BC, and Hipparchus also had an observation by Archimedes. 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). 2 (1991) pp. Hipparchus was the first to show that the stereographic projection is conformal, and that it transforms circles on the sphere that do not pass through the center of projection to circles on the plane. How did Hipparchus discover and measure the precession of the equinoxes? 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. ", Toomer G.J. 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. The lunar crater Hipparchus and the asteroid 4000 Hipparchus are named after him. Therefore, his globe was mounted in a horizontal plane and had a meridian ring with a scale. In the practical part of his work, the so-called "table of climata", Hipparchus listed latitudes for several tens of localities. The Greeks were mostly concerned with the sky and the heavens. He is also famous for his incidental discovery of the. 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. 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. [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.). What fraction of the sky can be seen from the North Pole. 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. At school we are told that the shape of a right-angled triangle depends upon the other two angles. Ptolemy has even (since Brahe, 1598) been accused by astronomers of fraud for stating (Syntaxis, book 7, chapter 4) that he observed all 1025 stars: for almost every star he used Hipparchus's data and precessed it to his own epoch 2+23 centuries later by adding 240' to the longitude, using an erroneously small precession constant of 1 per century. Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. The history of celestial mechanics until Johannes Kepler (15711630) was mostly an elaboration of Hipparchuss model. Astronomy test. [35] It was total in the region of the Hellespont (and in his birthplace, Nicaea); at the time Toomer proposes the Romans were preparing for war with Antiochus III in the area, and the eclipse is mentioned by Livy in his Ab Urbe Condita Libri VIII.2. 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? 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. How did Hipparchus discover trigonometry? Hipparchus was a Greek astronomer and mathematician. Hipparchus's ideas found their reflection in the Geography of Ptolemy. The earlier study's M found that Hipparchus did not adopt 26 June solstices until 146 BC, when he founded the orbit of the Sun which Ptolemy later adopted. 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. Delambre, in 1817, cast doubt on Ptolemy's work. Hipparchus made observations of equinox and solstice, and according to Ptolemy (Almagest III.4) determined that spring (from spring equinox to summer solstice) lasted 9412 days, and summer (from summer solstice to autumn equinox) 92+12 days. Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans.