• Mathematician and Geographer Eratosthenes of Cyrene (Ερατοσθένης ο Κυρηναίος), 276 B.C.
• Astronomer and Geographer Hipparchos of Nicaeae (Ίππαρχος ο Νικαεύς), 190 B.C.

Eratosthenes of Cyrene (Ancient Greek: Ἐρατοσθένης; c. 276 BC – c. 195 BC) was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.

He was the first person to use the word "geography" and invented the discipline of geography as we understand it. He invented a system of latitude and longitude.

He was the first person to calculate the circumference of the earth by using a measuring system using stades, or the length of stadiums during that time period (with remarkable accuracy). He was the first to calculate the tilt of the Earth's axis (also with remarkable accuracy). He may also have accurately calculated the distance from the earth to the sun and invented the leap day. He also created a map of the world based on the available geographical knowledge of the era. In addition, Eratosthenes was the founder of scientific chronology; he endeavored to fix the dates of the chief literary and political events from the conquest of Troy.

According to an entry in the Suda (a 10th century reference), his contemporaries nicknamed him beta, from the second letter of the Greek alphabet, because he supposedly proved himself to be the second best in the world in almost every field.

Eratosthenes was born in Cyrene (in modern day Libya). He was the third chief librarian of the Great Library of Alexandria, the center of science and learning in the ancient world, and died in Alexandria, then the capital of Ptolemaic Egypt.

Eratosthenes studied in Alexandria, and claimed to have also studied for some years in Athens. In 236 BC he was appointed by Ptolemy III Euergetes I as librarian of the Alexandrian library, succeeding the second librarian, Apollonius of Rhodes, in that post. He made several important contributions to mathematics and science, and was a good friend to Archimedes. Around 255 BC he invented the armillary sphere. In On the Circular Motions of the Celestial Bodies, Cleomedes credited him with having calculated the Earth's circumference around 240 BC, using knowledge of the angle of elevation of the sun at noon on the summer solstice in Alexandria and on Elephantine Island near Syene (now Aswan, Egypt).

Eratosthenes criticized Aristotle for arguing that humanity was divided into Greeks and barbarians, and that the Greeks should keep themselves racially pure, believing there was good and bad in every nation. By 195 BC, Eratosthenes became blind. He died in 194 BC at the age of 82.

Eratosthenes calculated the circumference of the Earth without leaving Egypt. Eratosthenes knew that on the summer solstice at local noon in the Ancient Egyptian city of Swenet (known in Greek as Syene, and in the modern day as Aswan) on the Tropic of Cancer, the sun would appear at the zenith, directly overhead (he had been told that the shadow of someone looking down a deep well would block the reflection of the Sun at noon). He also knew, from measurement, that in his hometown of Alexandria, the angle of elevation of the sun was 1/50th of a circle (7°12') south of the zenith on the solstice noon. Assuming that the Earth was spherical (360°), and that Alexandria was due north of Syene, he concluded that the meridian arc distance from Alexandria to Syene must therefore be 1/50 = 7°12'/360°, and was therefore 1/50 of the total circumference of the Earth. His knowledge of the size of Egypt after many generations of surveying trips for the Pharaonic bookkeepers gave a distance between the cities of 5000 stadia (about 500 geographical miles or 800 km). This distance was corroborated by inquiring about the time that it takes to travel from Syene to Alexandria by camel. He rounded the result to a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia. The exact size of the stadion he used is frequently argued. The common Attic stadion was about 185 m, which would imply a circumference of 46,620 km, i.e. 16.3% too large. However, if we assume that Eratosthenes used the "Egyptian stadion" of about 157.5 m, his measurement turns out to be 39,690 km, an error of less than 2%.

Eusebius of Caesarea in his Preparatio Evangelica includes a brief chapter of three sentences on celestial distances. He states simply that Eratosthenes found the distance to the sun to be "σταδίων μυριάδας τετρακοσίας και οκτωκισμυρίας" (literally "of stadia myriads 400 and 80,000") and the distance to the moon to be 780,000 stadia. The expression for the distance to the sun has been translated either as 4,080,000 stadia (1903 translation by E.H. Gifford), or as 804,000,000 stadia (edition of Edouard des Places, dated 1974 – 1991). The meaning depends on whether Eusebius meant 400 myriad plus 80,000 or "400 and 80,000" myriad.

With a stadium of 185 meters, 804,000,000 stadia is 149,000,000 kilometers, the modern distance from the earth to the sun.

The NASA site gives the modern distance from the earth to the moon as "English: 238855 miles. Scientific Notation: 3.84400 x 10^5 km".
Eratosthenes also proposed a simple algorithm for finding prime numbers. This algorithm is known in mathematics as the Sieve of Eratosthenes.

Hipparchus, or more correctly Hipparchos (Greek: Ἵππαρχος; c. 190 BC – c. 120 BC), was a Greek astrologer, astronomer, geographer, and mathematician of the Hellenistic period. He is considered the founder of trigonometry.

Hipparchus was born in Nicaea (now Iznik, Turkey), and probably died on the island of Rhodes. He is known to have been a working astronomer at least from 162 to 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity. He was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Chaldeans from Babylonia. He developed trigonometry and constructed trigonometric tables, and he solved several problems of spherical trigonometry. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses. His other reputed achievements include the discovery of Earth's precession, the compilation of the first comprehensive star catalog of the western world, and possibly the invention of the astrolabe, also of the armillary sphere, which he used during the creation of much of the star catalog. It would be three centuries before Claudius Ptolemaeus' synthesis of astronomy would supersede the work of Hipparchus; it is heavily dependent on it in many areas.

Relatively little of Hipparchus' direct work survives into modern times. Although he wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratus was preserved by later copyists. Most of what is known about Hipparchus comes from Ptolemy's (2nd century) Almagest, with additional references to him by Pappus of Alexandria and Theon of Alexandria (c. 4th century AD) in their commentaries on the Almagest; from Strabo's Geographia ("Geography"), and from Pliny the Elder's Naturalis historia ("Natural history") (1st century AD).

There is a strong tradition that Hipparchus was born in Nicaea (Greek Νίκαια), in the ancient district of Bithynia (modern day Iznik in province Bursa), in today's Turkey.

The exact dates of his life are not known, but Ptolemy attributes to him astronomical observations in the period from 147 BC to 127 BC, and some of these are stated as made in Rhodes; earlier observations since 162 BC might also have been made by him. His birth date (c. 190 BC) was calculated by Delambre based on clues in his work. Hipparchus must have lived some time after 127 BC because he analyzed and published his observations from that year. Hipparchus obtained information from Alexandria as well as Babylon, but it is not known when or if he visited these places. He is believed to have died on the island of Rhodes, where he seems to have spent most of his later life.

It is not known what Hipparchus' economic means were nor how he supported his scientific activities. His appearance is likewise unknown: there are no contemporary portraits. In the 2nd and 3rd centuries coins were made in his honor in Bithynia that bear his name and show him with a globe; this supports the tradition that he was born there.

Hipparchus is thought to be the first to calculate a heliocentric system, but he abandoned his work because the calculations showed the orbits were not perfectly circular as believed to be mandatory by the science of the time. As an astronomer of antiquity his influence, supported by Aristotle, held sway for nearly 2000 years, until the heliocentric model of Copernicus.

Hipparchus' only preserved work is Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις ("Commentary on the Phaenomena of Eudoxus and Aratus"). This is a highly critical commentary in the form of two books on a popular poem by Aratus based on the work by Eudoxus. Hipparchus also made a list of his major works, which apparently mentioned about fourteen books, but which is only known from references by later authors. His famous star catalog was incorporated into the one by Ptolemy, and may be almost perfectly reconstructed by subtraction of two and two thirds degrees from the longitudes of Ptolemy's stars. The first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry".

Hipparchus was in the international news in 2005, when it was again proposed (as in 1898) that the data on the celestial globe of Hipparchus or in his star catalog may have been preserved in the only surviving large ancient celestial globe which depicts the constellations with moderate accuracy, the globe carried by the Farnese Atlas. There are a variety of mis-steps  in the more ambitious 2005 paper, thus no specialists in the area accept its widely publicized speculation.

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; he goes on to say that Newton may have been influenced by them. Both of these claims have been rejected by other scholars.

A line in Plutarch's Table Talk states that Hipparchus counted 103049 compound propositions that can be formed from ten simple propositions; 103049 is the tenth Schröder – Hipparchus number and this line has led to speculation that Hipparchus knew about enumerative combinatorics, a field of mathematics that developed independently in modern mathematics.

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. Hipparchus seems to have been the first to exploit Babylonian astronomical knowledge and techniques systematically. Except for Timocharis and Aristillus, he was the first Greek known to divide the circle in 360 degrees of 60 arc minutes (Eratosthenes before him used a simpler sexagesimal system dividing a circle into 60 parts). He also used the Babylonian unit pechus ("cubit") of about 2° or 2.5°.

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. Hipparchus' use of Babylonian sources has always been known in a general way, because of Ptolemy's statements. 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).

Hipparchus's long draconitic lunar period (5458 months = 5923 lunar nodal periods) also appears a few times in Babylonian records. But the only such tablet explicitly dated is post - Hipparchus so the direction of transmission is not secured.

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. He tabulated values for the chord function, which gives the length of the chord for each angle. He did this for a circle with a circumference of 21,600 and a radius (rounded) of 3438 units: this circle has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord of an angle equals twice the sine of half of the angle, i.e.:

chord(A) = 2 sin(A/2).

He described the chord table in a work, now lost, called Tōn en kuklō_i eutheiōn (Of Lines Inside a Circle) by Theon of Alexandria (4th century) in his commentary on the Almagest I.10; some claim his table may have survived in astronomical treatises in India, for instance the Surya Siddhanta. Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques.

For his chord table Hipparchus must have used a better approximation for π than the one from Archimedes of between 3 + 1/7 and 3 + 10/71; perhaps he had the one later used by Ptolemy: 3;8:30 (sexagesimal) (Almagest VI.7); but it is not known if he computed an improved value himself.

Hipparchus could construct his chord table using the Pythagorean theorem and a theorem known to Archimedes. He also might have developed and used the theorem in plane geometry called Ptolemy's theorem, because it was proved by Ptolemy in his Almagest (I.10) (later elaborated on by Carnot).

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. This was the basis for the astrolabe.

Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans. He was one of the first Greek mathematicians to do this, and in this way expanded the techniques available to astronomers and geographers.

There are several indications that Hipparchus knew spherical trigonometry, but the first surviving text of it is that of Menelaus of Alexandria in the 1st century, who on that basis is now commonly credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute things like the rising and setting points of the ecliptic, or to take account of the lunar parallax. 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. He might have used spherical trigonometry.

Aubrey Diller has shown that the clima calculations which Strabo preserved from Hipparchus were performed by spherical trigonometry with the sole accurate obliquity known to have been used by ancient astronomers, 23°40'. All thirteen clima figures agree with Diller's proposal. 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.

Hipparchus also studied the motion of the Moon and confirmed the accurate values for two periods of its motion that Chaldean astronomers certainly possessed before him, whatever their ultimate origin. The traditional value (from Babylonian System B) for the mean synodic monthis 29 days; 31,50,8,20 (sexagesimal) = 29.5305941... days. Expressed as 29 days + 12 hours + 793/1080 hours this value has been used later in the Hebrew calendar (possibly from Babylonian sources). The Chaldeans also knew that 251 synodic months = 269 anomalistic months. Hipparchus used a multiple of this period by a factor of 17, because that interval is also an eclipse period. The Moon also is close to an integer number of years (4267 moons : 4573 anomalistic periods : 4630.53 nodal periods : 4611.98 lunar orbits : 344.996 years : 344.982 solar orbits : 126,007.003 days : 126,351.985 rotations). The 345 year eclipses reoccur with almost identical time of day, elevation, and celestial position.

Hipparchus could confirm his computations by comparing eclipses from his own time (presumably 27 January 141 BC and 26 November 139 BC), with eclipses from Babylonian records 345 years earlier (Almagest IV.2). Already al-Biruni (Qanun VII.2.II) and Copernicus (de revolutionibus IV.4) noted that the period of 4,267 moons is actually about 5 minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. However, the timing methods of the Babylonians had an error of no less than 8 minutes. 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 try to derive an improved value from his own observations. From modern ephemerides and taking account of the change in the length of the day we estimate that the error in the assumed length of the synodic month was less than 0.2 seconds in the 4th century BC and less than 0.1 seconds in Hipparchus' time.

It had been known for a long time that the motion of the Moon is not uniform: its speed varies. This is called its anomaly, and it repeats with its own period; the anomalistic month. 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 Greeks however preferred to think in geometrical models of the sky. Apollonius of Perga had at the end of the 3rd century BC proposed two models for lunar and planetary motion:

1. 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. So the apparent angular speed of the Moon (and its distance) would vary.
2. The Moon itself would move uniformly (with some mean motion in anomaly) on a secondary circular orbit, called an epicycle, that itself would move uniformly (with some mean motion in longitude) over the main circular orbit around the Earth, called deferent.

Apollonius demonstrated that these two models were in fact mathematically equivalent. However, all this was theory and had not been put to practice. Hipparchus was the first astronomer we know attempted to determine the relative proportions and actual sizes of these orbits.

Hipparchus devised a geometrical method to find the parameters from three positions of the Moon, at particular phases of its anomaly. In fact, he did this separately for the eccentric and the epicycle model. Ptolemy describes the details in the Almagest IV.11. Hipparchus used two sets of three lunar eclipse observations, which he carefully selected to satisfy the requirements. The eccentric model he fitted to these eclipses from his Babylonian eclipse list: 22/23 December 383 BC, 18/19 June 382 BC, and 12/13 December 382 BC. The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201 BC, 19 March 200 BC, and 11 September 200 BC.

• For the eccentric model, Hipparchus found for the ratio between the radius of the eccenter and the distance between the center of the eccenter and the center of the ecliptic (i.e., the observer on Earth): 3144 : 327+2/3 ;
• and for the epicycle model, the ratio between the radius of the deferent and the epicycle: 3122+1/2 : 247+1/2 .

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 criticized him (he himself made rounding errors too). A simpler alternate reconstruction agrees with all four numbers. Anyway, Hipparchus found inconsistent results; he later used the ratio of the epicycle model (3122+1/2 : 247+1/2), which is too small (60 : 4;45 sexagesimal). Ptolemy established a ratio of 60 : 5+1/4. (The maximum angular deviation producible by this geometry is arcsin(5.25 / 60), or about 5° 1', a figure that is sometimes therefore quoted as the equivalent of the Moon's equation of the center in the Hipparchan model.)

Before Hipparchus, Meton, Euctemon, and their pupils at Athens had made a solstice observation (i.e., timed the moment of the summer solstice) on June 27, 432 BC (proleptic Julian calendar). Aristarchus of Samos is said to have done so in 280 BC, and Hipparchus also had an observation by Archimedes. Hipparchus himself observed the summer solstice in 135 BC, but he found observations of the moment of equinox more accurate, and he made many during his lifetime. Ptolemy gives an extensive discussion of Hipparchus' work on the length of the year in the Almagest III.1, and quotes many observations that Hipparchus made or used, spanning 162 BC to 128 BC.

Ptolemy quotes an equinox timing by Hipparchus (at 24 March 146 BC 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). He could have used the equatorial ring of his armillary sphere or another equatorial ring for these observations, but Hipparchus (and Ptolemy) knew that observations with these instruments are sensitive to a precise alignment with the equator, so if he were restricted to an armillary, it would make more sense to use its meridian ring as a transit instrument. The problem with an equatorial ring (if an observer is naive enough to trust it very near dawn or dusk) is that atmospheric refraction lifts the Sun significantly above the horizon: so for a northern hemisphere observer its apparent declination is too high, which changes the observed time when the Sun crosses the equator. (Worse, the refraction decreases as the Sun rises and increases as it sets, so it may appear to move in the wrong direction with respect to the equator in the course of the day – as Ptolemy mentions. Ptolemy and Hipparchus apparently did not realize that refraction is the cause.) However, such details have doubtful relation to the data of either man, since there is no textual, scientific, or statistical ground for believing that their equinoxes were taken on an equatorial ring, which is useless for solstices in any case. Not one of two centuries of mathematical investigations of their solar errors has claimed to have traced them to the effect of refraction on use of an equatorial ring. Ptolemy claims his solar observations were on a transit instrument set in the meridian.

At the end of his career, Hipparchus wrote a book called Peri eniausíou megéthous ("On the Length of the Year") about his results. The established value for the tropical year, introduced by Callippus in or before 330 BC was 365 + 1/4 days. (Possibly from Babylonian sources.) Speculating a Babylonian origin for the Callippic year is hard to defend, since Babylon did not observe solstices thus the only extant System B year length was based on Greek solstices. Hipparchus' equinox observations gave varying results, but he himself points out (quoted in Almagest III.1 (H195)) that the observation errors by himself and his predecessors may have been as large as 1/4 day. He used old solstice observations, and determined a difference of about one day in about 300 years. So he set the length of the tropical year to 365 + 1/4 - 1/300 days (= 365.24666... days = 365 days 5 hours 55 min, which differs from the actual value (modern estimate) of 365.24219... days = 365 days 5 hours 48 min 45 s by only about 6 min).

Between the solstice observation of Meton and his own, there were 297 years spanning 108,478 days. D.Rawlins noted that this implies a tropical year of 365.24579... days = 365 days;14,44,51 (sexagesimal; = 365 days + 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. This is an indication that Hipparchus' work was known to Chaldeans.

Another value for the year that is attributed to Hipparchus (by the astrologer Vettius Valens in the 1st century) is 365 + 1/4 + 1/288 days (= 365.25347... days = 365 days 6 hours 5 min), but this may be a corruption of another value attributed to a Babylonian source: 365 + 1/4 + 1/144 days (= 365.25694... days = 365 days 6 hours 10 min). It is not clear if this would be a value for the sidereal year (actual value at his time (modern estimate) about 365.2565 days), but the difference with Hipparchus' value for the tropical year is consistent with his rate of precession.

Before Hipparchus, astronomers knew that the lengths of the seasons are not equal. Hipparchus made observations of equinox and solstice, and according to Ptolemy (Almagest III.4) determined that spring (from spring equinox to summer solstice) lasted 94½ days, and summer (from summer solstice to autumn equinox) 92½ days. This is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. Hipparchus' solution was to place the Earth not at the center of the Sun's motion, but at some distance from the center. This model described the apparent motion of the Sun fairly well. 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 value for the eccentricity attributed to Hipparchus by Ptolemy is that the offset is 1/24 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 may also have used other sets of observations, which would lead to different values. One of his two eclipse trios' solar longitudes are consistent with his having initially adopted inaccurate lengths for spring and summer of 95¾ and 91¼ days. His other triplet of solar positions is consistent with 94¼ and 92½ days, an improvement on the results (94½ and 92½ days) attributed to Hipparchus by Ptolemy, which a few scholars still question the authorship of. Ptolemy made no change three centuries later, and expressed lengths for the autumn and winter seasons which were already implicit.

Hipparchus also undertook to find the distances and sizes of the Sun and the Moon. He published his results in a work of two books called Perí megethōn kaí apostēmátōn ("On Sizes and Distances") by Pappus in his commentary on the Almagest V.11; Theon of Smyrna (2nd century) mentions the work with the addition "of the Sun and Moon".

Hipparchus measured the apparent diameters of the Sun and Moon with his diopter. 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. 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 360/650 = 0°33'14".

Like others before and after him, he also noticed that the Moon has a noticeable parallax, i.e., that it appears displaced from its calculated position (compared to the Sun or stars), and the difference is greater when closer to the horizon. 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 surface — the Moon, Earth and observer form a triangle with a sharp angle that changes all the time. From the size of this parallax, the distance of the Moon as measured in Earth radii can be determined. For the Sun however, there was no observable parallax (we now know that it is about 8.8", several times smaller than the resolution of the unaided eye).

In the first book, Hipparchus assumes that the parallax of the Sun is 0, as if it is at infinite distance. 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 190 BC. It was total in the region of the Hellespont (and in fact in his birth place 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 VIII.2. It was also observed in Alexandria, where the Sun was reported to be obscured 4/5ths by the Moon. Alexandria and Nicaea are on the same meridian. Alexandria is at about 31° North, and the region of the Hellespont about 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 81 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+½ 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 190 BC solar eclipse at Alexandria must have been closer to 9/10ths and not the reported 4/5ths, a fraction more closely matched by the degree of totality at Alexandria of eclipses occurring in 310 BC and 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' 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' 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 60½ 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.

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 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 (1980) 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. 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' lunar theory. We do not know 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.

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 stars. Previously this was done at daytime by measuring the shadow cast by a gnomon, or with the portable instrument known as a scaphe.

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 4-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 geografían ("Against the Geography of Eratosthenes"). It is known to us from Strabo of Amaseia, who in his turn criticized Hipparchus in his own Geografia. 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 Geografia 1.1.12). 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.

Late in his career (possibly about 135 BC) Hipparchus compiled his star catalog, the original of which does not survive. He 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.

Previously, Eudoxus of Cnidus in the 4th 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' 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.

Hipparchus made his measurements with an armillary sphere, and obtained the positions of at least 850 stars. It is disputed which coordinate system(s) he used. Ptolemy's catalog in the Almagest, which is derived from Hipparchus' catalog, is given in ecliptic coordinates. However 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). Hipparchus seems to have used a mix of ecliptic coordinates and equatorial coordinates: in his commentary on Eudoxos he provides stars' polar distance (equivalent to the declination in the equatorial system), right ascension (equatorial), longitude (ecliptical), polar longitude (hybrid), but not celestial latitude.

As with most of his work, Hipparchus' star catalog was adopted and perhaps expanded by Ptolemy. Delambre, in 1817, cast doubt on Ptolemy's work. It was disputed whether the star catalog in the Almagest is due to Hipparchus, but 1976 – 2002 statistical and spatial analyses (by R. R. Newton, Dennis Rawlins, Gerd Grasshoff, Keith Pickering and Dennis Duke) have shown conclusively that the Almagest star catalog is almost entirely Hipparchan. 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' data and precessed it to his own epoch 2⅔ centuries later by adding 2°40' to the longitude, using an erroneously small precession constant of 1° per century.

In any case the work started by Hipparchus has had a lasting heritage, and was much later updated by Al Sufi (964) and Copernicus (1543). Ulugh Beg reobserved all the Hipparchus stars he could see from Samarkand in 1437 to about the same accuracy as Hipparchus's. The catalog was superseded only in the late 16th century by Brahe and Wilhelm IV of Kassel via superior ruled instruments and spherical trigonometry, which improved accuracy by an order of magnitude even before the invention of the telescope.

Hipparchus ranked stars in six magnitude classes according to their brightness: he assigned the value of one to the twenty brightest stars, to fainter ones a value of two, and so forth to the stars with a class of six, which can be barely seen with the naked eye. A similar system is still used today.

Hipparchus is known for being almost universally recognized as discoverer of the precession of the equinoxes. 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 Claudius Ptolemy. According to Ptolemy, Hipparchus measured the longitude of Spica and Regulus and other bright stars. Comparing his measurements with data from his predecessors, Timocharis and Aristillus, he concluded that Spica had moved 2° relative to the autumnal equinox. He also compared the lengths of the tropical year (the time it takes the Sun to return to an equinox) and the sidereal year (the time it takes the Sun to return to a fixed star), and found a slight discrepancy. 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.

The rather cumbersome formal name for the ESA's Hipparcos Space Astrometry Mission was High Precision Parallax Collecting Satellite; it was deliberately named in this way to give an acronym, HiPParCoS, that echoed and commemorated the name of Hipparchus. The lunar crater Hipparchus and the asteroid 4000 Hipparchus are more directly named after him.

The Astronomer's Monument at the Griffith Observatory in Los Angeles, California, USA, features a relief of Hipparchus as one of six of the greatest astronomers of all time and the only one from Antiquity.