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40 north 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 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 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 because the eclipse occurred in the morning the moon was not in the meridian and it has been proposed that as a consequence the distance found by hipparchus was a lower limit in any case according to pappus hipparchus found that the least distance is 71 from this eclipse and the greatest 83 earth radii in the second book hipparchus starts from the opposite extreme assumption he assigns a minimum distance to the sun of 490 earth radii 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 in this case the shadow of the earth is a cone rather than a cylinder as under the first assumption hipparchus observed at lunar eclipses that at the mean distance of the moon the diameter of the shadow cone is 2 1 2 lunar diameters that apparent diameter is as he had observed 360 650 degrees 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 with his value for the eccentricity of the orbit he could compute the least and greatest distances of the moon too according to pappus he found a least distance of 62 a mean of 67 1 3 and consequently a greatest distance of 72 2 3 earth radii 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 radii exactly the mean distance that ptolemy later derived hipparchus thus had the problematic result that his minimum distance from book 1 was greater than his maximum mean distance from book 2 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 in fact modern calculations show that the size of the 14 03 190 bc solar eclipse at alexandria must have been closer to 9 10 ths and not the reported 4 5 ths a fraction more closely matched by the degree of totality at alexandria of eclipses occurring on 15 08 310 and 20 11 129 bc 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 ptolemy later measured the lunar parallax directly almagest v 13 and used the second method of hipparchus with lunar eclipses to compute the distance of the sun almagest v 15 he criticizes hipparchus for making contradictory assumptions and obtaining conflicting results almagest v 11 but apparently he failed to understand hipparchus s strategy to establish limits consistent with the observations rather than a single value for the distance 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 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 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 1 2 radii 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 apparently hipparchus later refined his computations and derived accurate single values that he could use for predictions of solar eclipses see toomer 1974 for a more detailed discussion 35 eclipses edit pliny naturalis historia ii x tells us that hipparchus demonstrated that lunar eclipses can occur five months apart and solar eclipses seven months instead of the usual six months and the sun can be hidden twice in thirty days but as seen by different nations ptolemy discussed this a century later at length in almagest vi 6 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 hipparchus apparently made similar calculations 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 hemisphere and the other on the southern hemisphere as pliny indicates and the latter was inaccessible to the greek 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 hipparchus must have been the first to be able to do this 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 may have discussed these things in perí tēs katá plátos mēniaías tēs selēnēs kinēseōs on the monthly motion of the moon in latitude a work mentioned in the suda 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 toomer argued that this must refer to the large total lunar eclipse of 26 november 139 bc when over a clean sea horizon as seen from rhodes the moon was eclipsed in the northwest just after the sun rose in the southeast 24 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 it is not known what exact reason hipparchus found for seeing the moon eclipsed while apparently it was not in exact opposition to the sun parallax lowers the altitude of the luminaries refraction raises them and from a high point of view the horizon is lowered astronomical instruments and astrometry edit hipparchus and his predecessors used various instruments for astronomical calculations and observations such as the gnomon the astrolabe and the armillary sphere hipparchus is credited with the invention or improvement of several astronomical instruments which were used for a long time for naked eye observations 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 with an astrolabe hipparchus was the first to be able to measure the geographical latitude and time by observing fixed stars 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 equatorial ring of hipparchus s time 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 pappus of alexandria described it in his commentary on the almagest of that chapter as did proclus hypotyposis iv 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 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 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 hipparchus applied his knowledge of spherical angles to the problem of denoting locations on the earth s surface before him a grid system had been used by dicaearchus of messana but hipparchus was the first to apply mathematical rigor to the determination of the latitude and longitude of places on the earth hipparchus wrote a critique in three books on the work of the geographer eratosthenes of cyrene 3rd century bc called pròs tèn eratosthénous geographían against the geography of eratosthenes it is known to us from strabo of amaseia who in his turn criticised hipparchus in his own geographia hipparchus apparently made many detailed corrections to the locations and distances mentioned by eratosthenes 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 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 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 star catalog edit late in his career possibly about 135 bc hipparchus compiled his star catalog scholars have been searching for it for centuries 36 in 2022 it was announced that a part of it was discovered in a medieval parchment manuscript codex climaci rescriptus from saint catherine s monastery in the sinai peninsula egypt as hidden text palimpsest 37 38 this finding was questioned in 2024 39 with a rebuttal by the discoverers in 2025 40 there is ongoing scholarly debate about the interpretation of the discovery 41 the figure on the left may be hipparchus from raphael s fresco the school of athens hipparchus also constructed a celestial globe depicting the constellations based on his observations his interest in the fixed stars may have been inspired by the observation of a supernova according to pliny or by his discovery of precession according to ptolemy who says that hipparchus could not reconcile his data with earlier observations made by timocharis and aristillus for more information see discovery of precession in raphael s painting the school of athens hipparchus may be depicted holding his celestial globe as the representative figure for astronomy it is not certain that the figure is meant to represent him 36 previously eudoxus of cnidus in the fourth century bc had described the stars and constellations in two books called phaenomena and entropon aratus wrote a poem called phaenomena or arateia based on eudoxus s work hipparchus wrote a commentary on the arateia his only preserved work which 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 a 19th century artist s impression of hipparchus 42 according to roman sources hipparchus made his measurements with a scientific instrument and he obtained the positions of roughly 850 stars pliny the elder writes in book ii 24 26 of his natural history 43 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 and the same individual attempted what might seem presumptuous even in a deity viz 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 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 this passage reports that hipparchus was inspired by a newly emerging star he doubts on the stability of stellar brightnesses he observed with appropriate instruments plural it is not said that he observed everything with the same instrument he made a catalogue of stars it is unknown what instrument he used the armillary sphere was probably invented only later maybe by ptolemy 265 years after hipparchus the historian of science s hoffmann found clues that hipparchus may have observed the longitudes and latitudes in different coordinate systems and thus with different instrumentation 16 right ascensions for instance could have been observed with a clock while angular separations could have been measured with another device stellar magnitude edit hipparchus is conjectured to have ranked the apparent magnitudes of stars on a numerical scale from 1 the brightest to 6 the faintest 44 this hypothesis is based on the vague statement by pliny the elder but cannot be proven by the data in hipparchus s commentary on aratus s poem in this only work by his hand that has survived until today he does not use the magnitude scale but estimates brightnesses unsystematically 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 16 yet it was proven that the error bars of magnitudes in ancient star catalogue is 1 5 mag which suggests that these numbers are not based on measurements 45 there were several suggestions on measurement methodologies and feasibility studies 46 in all cases the error bars would be smaller hence hoffmann 2022 suggested that the magnitudes were not measured at all but mere estimates for globe makers to improve pattern recognition on globes as astronomer s computing machines 47 nevertheless this system certainly precedes ptolemy who used it extensively about ad 150 44 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 5 100 or 2 512 times brighter than the next faintest magnitude 48 coordinate system edit it is disputed which coordinate system s he used ptolemy s catalog in the almagest which is derived from hipparchus s catalog is given in ecliptic coordinates 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 16 43 he probably m...