how did hipparchus discover trigonometry how did hipparchus discover trigonometry

[31] Speculating a Babylonian origin for the Callippic year is difficult to defend, since Babylon did not observe solstices thus the only extant System B year length was based on Greek solstices (see below). He is considered the founder of trigonometry,[1] but is most famous for his incidental discovery of the precession of the equinoxes. [49] His two books on precession, On the Displacement of the Solstitial and Equinoctial Points and On the Length of the Year, are both mentioned in the Almagest of Claudius Ptolemy. (1988). A new study claims the tablet could be one of the oldest contributions to the the study of trigonometry, but some remain skeptical. In fact, his astronomical writings were numerous enough that he published an annotated list of them. 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. In On Sizes and Distances (now lost), Hipparchus reportedly measured the Moons orbit in relation to the size of Earth. 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. For other uses, see, Geometry, trigonometry and other mathematical techniques, Distance, parallax, size of the Moon and the Sun, Arguments for and against Hipparchus's star catalog in the Almagest. Alexandria and Nicaea are on the same meridian. "Hipparchus' Treatment of Early Greek Astronomy: The Case of Eudoxus and the Length of Daytime Author(s)". He is considered the founder of trigonometry. So the apparent angular speed of the Moon (and its distance) would vary. Hipparchus insists that a geographic map must be based only on astronomical measurements of latitudes and longitudes and triangulation for finding unknown distances. 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. A simpler alternate reconstruction[28] agrees with all four numbers. 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. In the second method he hypothesized that the distance from the centre of Earth to the Sun is 490 times Earths radiusperhaps chosen because that is the shortest distance consistent with a parallax that is too small for detection by the unaided eye. . The angle is related to the circumference of a circle, which is divided into 360 parts or degrees.. Hipparchus attempted to explain how the Sun could travel with uniform speed along a regular circular path and yet produce seasons of unequal length. [48], Conclusion: Hipparchus's star catalogue is one of the sources of the Almagest star catalogue but not the only source.[47]. Mott Greene, "The birth of modern science?" 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? Hipparchus opposed the view generally accepted in the Hellenistic period that the Atlantic and Indian Oceans and the Caspian Sea are parts of a single ocean. "Le "Commentaire" d'Hipparque. But a few things are known from various mentions of it in other sources including another of his own. [15][40] He probably marked them as a unit on his celestial globe but the instrumentation for his observations is unknown.[15]. how did hipparchus discover trigonometry. How did Hipparchus contribute to trigonometry? How did Hipparchus discover and measure the precession of the equinoxes? The ecliptic was marked and divided in 12 sections of equal length (the "signs", which he called zodion or dodekatemoria in order to distinguish them from constellations (astron). Trigonometry was probably invented by Hipparchus, who compiled a table of the chords of angles and made them available to other scholars. 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. "The Size of the Lunar Epicycle According to Hipparchus. As shown in a 1991 ), Greek astronomer and mathematician who made fundamental contributions to the advancement of astronomy as a mathematical science and to the foundations of trigonometry. But Galileo was more than a scientist. Because of a slight gravitational effect, the axis is slowly rotating with a 26,000 year period, and Hipparchus discovers this because he notices that the position of the equinoxes along the celestial equator were slowly moving. 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. Posted at 20:22h in chesapeake bay crater size by code radio police gta city rp. Some of the terms used in this article are described in more detail here. 2 - Why did Ptolemy have to introduce multiple circles. Thus it is believed that he was born around 70 AD (History of Mathematics). [40] He used it to determine risings, settings and culminations (cf. [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 the first of three articles on the History of Trigonometry. of trigonometry. 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. It had been known for a long time that the motion of the Moon is not uniform: its speed varies. 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. 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. Our editors will review what youve submitted and determine whether to revise the article. Ptolemy quotes (in Almagest III.1 (H195)) a description by Hipparchus of an equatorial ring in Alexandria; a little further he describes two such instruments present in Alexandria in his own time. 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. 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. He . This has led to speculation that Hipparchus knew about enumerative combinatorics, a field of mathematics that developed independently in modern mathematics. legacy nightclub boston Likes. "Hipparchus' Empirical Basis for his Lunar Mean Motions,", Toomer G.J. These must have been only a tiny fraction of Hipparchuss recorded observations. Trigonometry Trigonometry simplifies the mathematics of triangles, making astronomy calculations easier. 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. Hipparchus was perhaps the discoverer (or inventor?) Thus, by all the reworking within scientific progress in 265 years, not all of Hipparchus's stars made it into the Almagest version of the star catalogue. From where on Earth could you observe all of the stars during the course of a year? Others do not agree that Hipparchus even constructed a chord 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. THE EARTH-MOON DISTANCE 3550jl1016a Vs 3550jl1017a . Discovery of a Nova In 134 BC, observing the night sky from the island of Rhodes, Hipparchus discovered a new star. All thirteen clima figures agree with Diller's proposal. (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. The term "trigonometry" was derived from Greek trignon, "triangle" and metron, "measure".. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. One evening, Hipparchus noticed the appearance of a star where he was certain there had been none before. Hipparchus and his predecessors used various instruments for astronomical calculations and observations, such as the gnomon, the astrolabe, and the armillary sphere. As the first person to look at the heavens with the newly invented telescope, he discovered evidence supporting the sun-centered theory of Copernicus. ?rk?s/; Greek: ????? Rawlins D. (1982). ", Toomer G.J. Russo L. (1994). Therefore, it is possible that the radius of Hipparchus's chord table was 3600, and that the Indians independently constructed their 3438-based sine table."[21]. In Raphael's painting The School of Athens, Hipparchus is depicted holding his celestial globe, as the representative figure for astronomy.[39]. Some scholars do not believe ryabhaa's sine table has anything to do with Hipparchus's chord table. Aristarchus of Samos (/?r??st? 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. Knowledge of the rest of his work relies on second-hand reports, especially in the great astronomical compendium the Almagest, written by Ptolemy in the 2nd century ce. Even if he did not invent it, Hipparchus is the first person whose systematic use of trigonometry we have documentary evidence. Hipparchus also wrote critical commentaries on some of his predecessors and contemporaries. How did Hipparchus contribute to trigonometry? Hipparchus was born in Nicaea (Greek ), in Bithynia. 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. For his astronomical work Hipparchus needed a table of trigonometric ratios. 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. Hipparchus produced a table of chords, an early example of a trigonometric table. ), Greek astronomer and mathematician who made fundamental contributions to the advancement of astronomy as a mathematical science and to the foundations of trigonometry. One evening, Hipparchus noticed the appearance of a star where he was certain there had been none before. [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. 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. 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. He had immense in geography and was one of the most famous astronomers in ancient times. 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]. 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. 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. The shadow cast from a shadow stick was used to . This claim is highly exaggerated because it applies modern standards of citation to an ancient author. However, the Suns passage through each section of the ecliptic, or season, is not symmetrical. 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. "Hipparchus recorded astronomical observations from 147 to 127 BC, all apparently from the island of Rhodes. ), Italian philosopher, astronomer and mathematician. Hipparchus thus had the problematic result that his minimum distance (from book 1) was greater than his maximum mean distance (from book 2). Corrections? As a young man in Bithynia, Hipparchus compiled records of local weather patterns throughout the year. Unclear how it may have first been discovered. While every effort has been made to follow citation style rules, there may be some discrepancies. 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. Ulugh Beg reobserved all the Hipparchus stars he could see from Samarkand in 1437 to about the same accuracy as Hipparchus's. Hipparchus was a Greek astronomer and mathematician. He is also famous for his incidental discovery of the. 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. 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. 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. The history of trigonometry and of trigonometric functions sticks to the general lines of the history of math. He actively worked in astronomy between 162 BCE and 127 BCE, dying around. 2 - What two factors made it difficult, at first, for. Updates? Ancient Instruments and Measuring the Stars. Hipparchus must have used a better approximation for than the one from Archimedes of between 3+1071 (3.14085) and 3+17 (3.14286). Hipparchus is said to be the founder of Trigonometry, and Ptolemy wrote the Almagest, an important work on the subject [4]. Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans. Articles from Britannica Encyclopedias for elementary and high school students. His birth date (c.190BC) was calculated by Delambre based on clues in his work. Ptolemy's catalog in the Almagest, which is derived from Hipparchus's catalog, is given in ecliptic coordinates. It seems he did not introduce many improvements in methods, but he did propose a means to determine the geographical longitudes of different cities at lunar eclipses (Strabo Geographia 1 January 2012). 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. "Hipparchus and Babylonian Astronomy." The armillary sphere was probably invented only latermaybe by Ptolemy only 265 years after Hipparchus. 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. Ptolemy mentions (Almagest V.14) that he used a similar instrument as Hipparchus, called dioptra, to measure the apparent diameter of the Sun and Moon. He did this by using the supplementary angle theorem, half angle formulas, and linear interpolation. He tabulated the chords for angles with increments of 7.5. 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. 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. Perhaps he had the one later used by Ptolemy: 3;8,30 (sexagesimal)(3.1417) (Almagest VI.7), but it is not known whether he computed an improved value. 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. Pliny the Elder writes in book II, 2426 of his Natural History:[40]. What fraction of the sky can be seen from the North Pole. [15], Nevertheless, this system certainly precedes Ptolemy, who used it extensively about AD 150. According to Theon, Hipparchus wrote a 12-book work on chords in a circle, since lost. In, Wolff M. (1989). 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. This is called its anomaly and it repeats with its own period; the anomalistic month. 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. In Tn Aratou kai Eudoxou Phainomenn exgses biblia tria (Commentary on the Phaenomena of Aratus and Eudoxus), his only surviving book, he ruthlessly exposed errors in Phaenomena, a popular poem written by Aratus and based on a now-lost treatise of Eudoxus of Cnidus that named and described the constellations. 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. 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. Ptolemy mentions that Menelaus observed in Rome in the year 98 AD (Toomer). 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. Therefore, Trigonometry started by studying the positions of the stars. "Hipparchus on the distance of the sun. How to Measure the Distance to the Moon Using Trigonometry First, change 0.56 degrees to radians. [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. The system is so convenient that we still use it today! The map segment, which was found beneath the text on a sheet of medieval parchment, is thought to be a copy of the long-lost star catalog of the second century B.C. The Chaldeans also knew that 251 synodic months 269 anomalistic months. Many credit him as the founder of trigonometry. 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. What is Aristarchus full name? 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. Not much is known about the life of Hipp archus. According to Roman sources, Hipparchus made his measurements with a scientific instrument and he obtained the positions of roughly 850 stars. Did Hipparchus invent trigonometry? Though Hipparchus's tables formally went back only to 747 BC, 600 years before his era, the tables were good back to before the eclipse in question because as only recently noted,[19] their use in reverse is no more difficult than forward. . 2 - What are two ways in which Aristotle deduced that. Set the local time to around 7:25 am. Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the equinoxes. 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"). 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. [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. Etymology. Thus, somebody has added further entries. 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. Hipparchus produced a table of chords, an early example of a trigonometric table. Hipparchus, also spelled Hipparchos, (born, Nicaea, Bithynia [now Iznik, Turkey]died after 127 bce, Rhodes? Apparently it was well-known at the time. 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. Hipparchus adopted the Babylonian system of dividing a circle into 360 degrees and dividing each degree into 60 arc minutes. He was then in a position to calculate equinox and solstice dates for any year. Every year the Sun traces out a circular path in a west-to-east direction relative to the stars (this is in addition to the apparent daily east-to-west rotation of the celestial sphere around Earth). Ptolemy established a ratio of 60: 5+14. He is believed to have died on the island of Rhodes, where he seems to have spent most of his later life. Galileo was the greatest astronomer of his time. 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). 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). Ch. Hipparchuss most important astronomical work concerned the orbits of the Sun and Moon, a determination of their sizes and distances from Earth, and the study of eclipses. Hipparchus introduced the full Babylonian sexigesimal notation for numbers including the measurement of angles using degrees, minutes, and seconds into Greek science. That apparent diameter is, as he had observed, 360650 degrees. Tracking and 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. Let the time run and verify that a total solar eclipse did occur on this day and could be viewed from the Hellespont. 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). Sidoli N. (2004). The first trigonometric table was apparently compiled by Hipparchus, who is consequently now known as "the father of trigonometry". An Australian mathematician has discovered that Babylonians may have used applied geometry roughly 1,500 years before the Greeks supposedly invented its foundations, according to a new study. That would be the first known work of trigonometry. (1997). [15] However, Franz Xaver Kugler demonstrated that the synodic and anomalistic periods that Ptolemy attributes to Hipparchus had already been used in Babylonian ephemerides, specifically the collection of texts nowadays called "System B" (sometimes attributed to Kidinnu).[16]. 1. Applying this information to recorded observations from about 150 years before his time, Hipparchus made the unexpected discovery that certain stars near the ecliptic had moved about 2 relative to the equinoxes. Since Nicolaus Copernicus (14731543) established his heliocentric model of the universe, the stars have provided a fixed frame of reference, relative to which the plane of the equator slowly shiftsa phenomenon referred to as the precession of the equinoxes, a wobbling of Earths axis of rotation caused by the gravitational influence of the Sun and Moon on Earths equatorial bulge that follows a 25,772-year cycle. Earth's precession means a change in direction of the axis of rotation of Earth. Hipparchus is sometimes called the "father of astronomy",[7][8] a title first conferred on him by Jean Baptiste Joseph Delambre.[9]. 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. Hipparchus is conjectured to have ranked the apparent magnitudes of stars on a numerical scale from 1, the brightest, to 6, the faintest. He may have discussed these things in Per ts kat pltos mniaas ts selns kinses ("On the monthly motion of the Moon in latitude"), a work mentioned in the Suda. [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.