Hipparchus

How to use this course

Hipparchus is the astronomer whom later Greek, Arabic and European scientists regarded as the founder of quantitative astronomy. He built the first known table of chords, fitted geometric models to the Sun and Moon from dated observations, estimated the Moon’s distance from parallax, compiled a star catalogue, and discovered that the equinoxes slowly move against the stars. He also used Babylonian eclipse records and period relations more systematically than any Greek before him.

Only one of his books survives, and it is not one of the important ones. Almost everything else is known from Claudius Ptolemy’s Almagest, written about 280 years later, and from scattered reports by Strabo, Pliny, Theon of Smyrna and Pappus. This course keeps that problem in view. Each lesson says where the evidence comes from, and separates what Hipparchus is reported to have done from what modern historians have reconstructed.

Numbers in base 60 are written as in the Ptolemy course: 29;31,50 means \(29+\tfrac{31}{60}+\tfrac{50}{3600}\). This course is designed to be read alongside the Ptolemy course, which treats in detail the parts of Hipparchus’s work that Ptolemy kept unchanged, and the courses on Aristarchus of Samos, Eratosthenes and Eudoxus of Cnidus, whose work he criticized and extended.

Timeline

All dates for Hipparchus’s own life are inferred. Observation dates come from the Almagest; the birth year is a modern estimate going back to Jean-Baptiste Delambre.

Nicaea and the problem of sources

No ancient life of Hipparchus exists. The facts of his biography fit in a paragraph, and each of them rests on a different kind of evidence.

Birthplace

The geographer Strabo, writing around the start of the Christian era, lists Hipparchus among the notable men of Bithynia, a kingdom in north-west Asia Minor. The Byzantine lexicon called the Suda and other later sources call him “of Nicaea”. Coins struck at Nicaea in the Roman period, under five emperors between AD 138 and 253, show a seated man contemplating a globe with his name beside him. A city honoured him this way three centuries after his death, which is strong evidence that it claimed him as its own. Ptolemy reports that some of his early observations were made in Bithynia.

Dates

The Almagest preserves observations that Ptolemy attributes to Hipparchus, with dates. Gerald Toomer, whose account in the Dictionary of Scientific Biography remains the standard, identifies the earliest as the autumnal equinox of 26/27 September 147 BC and the latest as a lunar observation of 7 July 127 BC. Many other records he used, such as Babylonian eclipses and some Alexandrian equinoxes, were made by others. Delambre estimated his birth at about 190 BC by working backwards from an active career of some thirty-five years. The death date is unknown.

What the later reports add

• Ptolemy is the main witness. He praises Hipparchus as a lover of work and of truth, adopts his solar theory unchanged, and corrects his lunar theory. He reports his methods in detail but often only the results Ptolemy himself needed.
• Strabo preserves much of Hipparchus’s attack on the geography of Eratosthenes.
• Pliny the Elder (first century AD) says Hipparchus predicted eclipses for six hundred years ahead and catalogued the stars after seeing a new one. Toomer treats the first claim as exaggerated and suggests the second may refer to a comet.
• Pappus and Theon of Smyrna preserve numbers from On Sizes and Distances.
• Plutarch preserves a single combinatorial result (Lesson 10).

Claims that Hipparchus worked at Alexandria rest, in Toomer’s view, on misreadings of the Almagest: he used Alexandrian observations, but nothing places him there. Whether he travelled to Babylon is also unknown.

Rhodes: an observing career

Rhodes in the second century BC was a rich, independent maritime republic, a centre of trade and of learning. The Stoic philosopher Posidonius later taught there. Toomer concludes that Hipparchus moved from Bithynia to Rhodes before 141 BC and observed there until at least 127 BC. Nothing is known of a patron, a school, pupils or a family.

What he observed

Ptolemy reports summer solstices of 146 and 135 BC, about twenty equinox observations, many lunar positions, lunar eclipses, and star positions. Hipparchus also collected observations by others, and he seems to have been unusually concerned with their reliability. Ptolemy reports his discussion of an equatorial ring at Alexandria that sometimes showed the equinox twice in one day, and his estimate that solstice times could be uncertain by a quarter of a day.

Instruments

Ptolemy credits him with a four-cubit dioptra, a sighting rod with a movable plate, used to compare the apparent diameter of the Moon with the Sun’s. He probably used an armillary sphere for star positions and a solid celestial globe for working out risings and settings; the coins of Nicaea show him with a globe. The claim, made by the fifth-century writer Synesius, that he invented the plane astrolabe is late and generally doubted.

Writing

Hipparchus wrote many specialized monographs rather than one systematic treatise. Titles known from Ptolemy, the Suda and others include On Sizes and Distances (two books), On the Length of the Year, On the Displacement of the Solstitial and Equinoctial Points, On Intercalary Months and Days, On the Monthly Motion of the Moon in Latitude, On the Rising of the Twelve Signs, a work on chords, Against the Geography of Eratosthenes (three books), and works on astrology, weather signs, optics and falling bodies. The one survivor is the Commentary on the Phenomena of Aratus and Eudoxus, in three books, written early in his career.

Two inheritances: Greek and Babylonian

Hipparchus stood where two astronomical traditions met. The Greeks had geometric models of the heavens with almost no numbers. The Babylonians had centuries of dated records and powerful arithmetic schemes for prediction, with no geometry. He was the first Greek known to combine them on a large scale.

The Greek side

• Eudoxus (fourth century BC) had modelled the planets with nested spheres and described the constellations. Hipparchus’s surviving Commentary criticizes Eudoxus and the poet Aratus, who versified his star descriptions, for errors in risings, settings and positions, and corrects the more recent commentator Attalus of Rhodes.
• Apollonius of Perga (c. 200 BC) had shown that eccentric and epicyclic motions are interchangeable, the basis of Hipparchus’s solar and lunar models (Lesson 05).
• Aristarchus had measured the relative distances of the Sun and Moon by geometry, and his solstice of 280 BC gave Hipparchus a long baseline for the year.
• Eratosthenes had measured the Earth and drawn a map of the inhabited world, which Hipparchus subjected to sharp mathematical criticism (Lesson 09).
• Timocharis and Aristillus had recorded star positions at Alexandria early in the third century BC, which made the discovery of precession possible (Lesson 08).

The Babylonian side

Ptolemy says that Hipparchus used eclipse records from Babylon going back to the eighth century BC. In 1900 Franz Xaver Kugler showed that the lunar periods Ptolemy attributes to Hipparchus already appear in Babylonian cuneiform tablets of the type now called System B. Hipparchus also adopted the Babylonian division of the circle into 360 degrees with sexagesimal fractions, which the Greek mathematician Hypsicles had used a little earlier in a work on rising times. From Hipparchus on, Greek astronomy measured angles in degrees and minutes, and wrote fractions in base 60.

What Hipparchus added was a habit of testing: he checked Babylonian periods against eclipses of his own time, derived geometric parameters from carefully chosen observations, and estimated the uncertainty of his results.

The first table of chords

For the general reader. Every geometric model of the sky must convert angles into lengths: how far the Earth sits from the centre of the Sun’s circle, how large an angle that offset produces. Hipparchus seems to have been the first to compute a table for this purpose. It is lost, but its scale and step size can be recovered, and it may survive in disguise in the earliest Indian tables of sines.

What the sources say

Theon of Alexandria, commenting on the Almagest in the fourth century AD, says that Hipparchus wrote a treatise on chords in a circle in twelve books. Neugebauer thought the number of books implausible, but the existence of the work is not in doubt. The table itself is lost. Toomer reconstructed it in 1973 from numbers that Hipparchus computed in lunar problems reported by Ptolemy. His reconstruction has the following features.

• Radius 3438. Measuring the circumference in minutes of arc, \(360\times60=21\,600\), gives radius \(21\,600/2\pi\approx3437.75\), rounded to 3438. Then a small arc and its chord have the same numerical value, which simplifies conversions.
• Step \(7\tfrac12^\circ\), a forty-eighth of the circle, with linear interpolation between entries.

The chord of a central angle \(\alpha\) in a circle of radius \(R\) is

Proof. The supplement rule is Pythagoras’ theorem in a triangle inscribed in a semicircle (Thales). For the half-angle rule, \(\operatorname{crd}^2(\alpha/2)=4R^2\sin^2(\alpha/4)=2R^2(1-\cos(\alpha/2))\), while \(\operatorname{crd}(180^\circ-\alpha)=2R\cos(\alpha/2)\). Both identities can be proved from Euclid’s Elements without trigonometric functions, and Hipparchus probably did so.

Starting from \(\operatorname{crd}60^\circ=R\) (hexagon) and \(\operatorname{crd}180^\circ=2R\), these two rules reach every multiple of \(7\tfrac12^\circ\) (Exercise 4.1). They cannot reach \(1^\circ\), which needs the inequality used later by Ptolemy (Ptolemy course, Lesson 04). For Hipparchus’s purposes, \(7\tfrac12^\circ\) steps with interpolation were sufficient.

The Indian connection

The Sanskrit astronomer Āryabhaṭa (AD 499) tabulates the half-chord, our sine, \(R\sin\theta\) with \(R=3438\), at steps of \(3\tfrac34^\circ\). These are half the chords of \(7\tfrac12^\circ,15^\circ,\dots\), exactly the entries of a Hipparchan table. The first entry is 225. Toomer and others argue that Greek astronomy of the pre-Ptolemaic period reached India and carried Hipparchus’s table with it. The identical radius is the main evidence; the route of transmission is not documented.

The Sun: seasons, eccentric, year

For the general reader. Spring and summer together are about a week longer than autumn and winter. Hipparchus turned this observation into a geometric model. If the Sun moves uniformly on a circle, but the Earth is not at the circle’s centre, the Sun takes longer to cover the part of its circle farther from us. Two measured season lengths fix how far off-centre the Earth is and in which direction. Ptolemy kept the model and its numbers for the Almagest.

Eccentric or epicycle

Proof. Put the observer at the origin of the complex plane. In the eccentric model the position is \(z=ee^{i\lambda_A}+Re^{i\bar\lambda}\), with \(\bar\lambda\) increasing uniformly. In the epicycle model it is \(z=Re^{i\bar\lambda}+ee^{i\lambda_A}\). Addition of complex numbers is commutative.

Ptolemy reports that Hipparchus knew both forms and chose the eccentric for the Sun. The angle between the true and mean directions is the equation of centre \(q\), with

where \(M\) is the mean anomaly measured from apogee.

From seasons to geometry

Hipparchus took spring (vernal equinox to summer solstice) as \(94\tfrac12\) days and summer (solstice to autumnal equinox) as \(92\tfrac12\) days. The equinox and solstice lines are perpendicular and pass through the Earth. If the Sun took exactly a quarter year for each season the Earth would be at the centre. The excess of spring and summer over a half year shows how far the centre is displaced from the equinox line; the excess of spring over summer shows how far it is displaced from the solstice line.

The derivation is given in Lesson 06 of the Ptolemy course. With Hipparchus’s numbers the result is \(e/R\approx1/24\) and \(\lambda_A\approx65\tfrac12^\circ\), that is, apogee at \(5\tfrac12^\circ\) of Gemini (Exercise 5.1). The greatest equation of centre is then \(\arcsin(1/24)\approx2;23^\circ\). The true eccentricity of the Earth’s orbit in his time gives a greatest equation near \(2^\circ\); the seasons Hipparchus used were each off by a fraction of a day.

The length of the year

Ptolemy reports that Hipparchus compared the summer solstice observed by Aristarchus in 280 BC with his own of 135 BC. After 145 years the solstice came about half a day earlier than 145 years of \(365\tfrac14\) days would predict. Comparing equinox observations as well, he concluded that the tropical year is shorter than \(365\tfrac14\) days by about \(\tfrac1{300}\) of a day:

The modern value for his epoch is about \(365\) d \(5\) h \(48.8\) min, so his year is about 6 minutes too long. He also used a sidereal year (return to the same star) of \(365\tfrac14+\tfrac1{144}\) days, a value of Babylonian origin. Ptolemy adopted Hipparchus’s tropical year unchanged.

The Moon: periods, eclipses, anomaly

For the general reader. The Moon speeds up and slows down in a cycle that does not match its cycle of phases. To predict eclipses an astronomer must know both cycles precisely, and must know how large the speed variation is. Hipparchus took the periods from Babylonian astronomy, checked them against eclipses centuries apart, and measured the size of the variation by fitting a geometric model to three eclipses.

Period relations

The Moon has several periods: the synodic month (new moon to new moon), the anomalistic month (return to the same speed), the draconitic month (return to the same node) and the sidereal month. A period relation states that whole numbers of two periods are nearly equal. Ptolemy (Almagest IV.2) reports that Hipparchus showed, from Babylonian and his own observations, that

Both numbers are 17 times the Babylonian relation 251 synodic months = 269 anomalistic months. After such an interval the Moon has the same phase and the same speed, so two eclipses separated by it are nearly identical. Hipparchus tested the relation by finding pairs of eclipses separated by exactly this interval in which both phase and anomaly agree.

Fitting the anomaly

In a lunar eclipse the Moon is exactly opposite the Sun, so its true longitude is known from the solar theory of Lesson 05. The period relations give the Moon’s mean longitude and mean anomaly at any time. For three eclipses one therefore knows three true longitudes and the changes in mean longitude and anomaly between them. The differences between true and mean motion over the two intervals give two equations, enough to determine the epicycle radius and the Moon’s position on the epicycle at the first eclipse.

Ptolemy reports that Hipparchus carried out this computation twice, once with an eccentric model and once with an epicycle model, using different eclipse trios. He obtained ratios of

By Apollonius’s theorem these models should give the same amplitude, so the discrepancy is not caused by the choice of model. Ptolemy attributes it to errors in Hipparchus’s calculations and in his choice of observations. Hipparchus used the epicycle value in On Sizes and Distances. Ptolemy’s value, from his own trios, is \(5;15\) parts of 60.

Hipparchus also found the greatest lunar latitude, the inclination of the Moon’s orbit to the ecliptic, to be \(5^\circ\), a value Ptolemy kept. Ptolemy says that some of Hipparchus’s observations away from syzygy did not fit his model, and that Hipparchus noted this without finding the second anomaly (evection) that Ptolemy later modelled.

On Sizes and Distances

For the general reader. A total solar eclipse seen at one place is only partial a few hundred kilometres away, because the Moon is close enough for its position to shift with the observer. Hipparchus turned this into a measurement of the Moon’s distance. In a second book he used a different method, based on lunar eclipses, and showed honestly how the answer depends on what one assumes about the Sun.

Parallax

An observer on the surface of an Earth of radius 1 sees a body at distance \(D\) displaced from its direction as seen from the centre. When the body is on the horizon the displacement is the horizontal parallax \(p\), with \(\sin p=1/D\). At zenith distance \(z\) the displacement is approximately \(p\sin z\).

Book 1: a solar eclipse

Pappus reports that Hipparchus used a solar eclipse that was total in the region of the Hellespont and covered about four-fifths of the Sun at Alexandria. Toomer identified it as the eclipse of 14 March 190 BC. Hipparchus assumed the Sun is so distant that its parallax is zero. Then the whole difference between the two places, a fifth of the Sun’s diameter, is a difference in lunar parallax. With the apparent diameter of the Sun and Moon as \(\tfrac{1}{650}\) of a circle, measured with his dioptra, and the latitudes of the two places, he found the Moon’s least distance to be 71 Earth radii and its greatest 83.

Book 2: lunar eclipses

The second method uses the Earth’s shadow. During a lunar eclipse the Moon crosses the shadow, and the width of the shadow at the Moon’s distance can be measured in Moon diameters.

Proof. Measure lengths in Earth radii, and let the Sun have radius \(R_S\) at distance \(D_S\), so \(s\approx R_S/D_S\) and \(p_S\approx1/D_S\). The shadow is a cone tangent to the Sun and the Earth, with half-angle \(\beta\approx(R_S-1)/D_S=s-p_S\). At the Moon’s distance \(D_M\) the shadow’s radius is \(1-D_M(s-p_S)\) Earth radii, which subtends \(\sigma=1/D_M-(s-p_S)=p_M-s+p_S\) from the Earth. Rearranging gives the relation.

Ptolemy reports that Hipparchus took the shadow’s diameter as \(2\tfrac12\) times the Moon’s, and the Moon’s diameter as \(\tfrac1{650}\) of its circle. Then \(s=\tfrac{1}{1300}\cdot360^\circ\approx16;37'\) and \(\sigma=2\tfrac12s\approx41;32'\). The Moon’s distance depends on the unknown solar parallax:

With \(p_S=0\) this gives the least possible lunar distance, about 59 Earth radii. Pappus reports that Hipparchus then took \(p_S=7'\), the largest parallax that he thought could escape detection, and obtained a mean lunar distance of \(67\tfrac13\) Earth radii with the Sun at 490. The true mean distance is about 60.3. Hipparchus’s bounds, 59 to \(67\tfrac13\), enclose it, and the method is sound.

Hipparchus’s lesson for later astronomers was that the Sun’s distance cannot be found this way: the solar parallax is too small. The values 490 and the later Ptolemaic 1,210 Earth radii are both far below the true 23,500.

Precession and the star catalogue

For the general reader. The stars seem fixed, and the equinoxes seem fixed against them. Hipparchus discovered that they are not: the point where the Sun crosses the equator in spring drifts slowly westward along the zodiac, about one degree in a lifetime. He found it by comparing his own star positions with those taken at Alexandria some 160 years earlier. The discovery made a reliable star catalogue essential, and he made one.

The discovery

Ptolemy (Almagest VII.2–3) reports that Hipparchus found the bright star Spica \(6^\circ\) west of the autumnal equinox, whereas Timocharis had found it \(8^\circ\) west, roughly 160 years earlier. Similar comparisons for other stars, and lunar eclipses near Spica, showed the same shift for stars near the zodiac. In his book On the Displacement of the Solstitial and Equinoctial Points he concluded that the equinoxes move westward relative to the stars, or equivalently that the sphere of the stars moves slowly eastward about the poles of the ecliptic, at a rate of at least \(1^\circ\) per century. Because the observations were uncertain, he stated this only as a lower bound.

Proof. In one tropical year the Sun returns to the moving equinox. Because the equinox has moved \(k\) degrees to meet it, the Sun has travelled \(360^\circ-k\) relative to the stars. At the sidereal rate \(360^\circ/Y_s\), this takes \(Y_t=Y_s(1-k/360^\circ)\).

With Hipparchus’s values, \(Y_s-Y_t=\tfrac1{144}+\tfrac1{300}\) day, which gives about \(36''\) per year, \(1^\circ\) per century, consistent with his lower bound. The modern rate is \(50.3''\) per year. Ptolemy adopted \(1^\circ\) per century as the actual rate, which, as the Ptolemy course shows, introduced a systematic error into his catalogue. The physical cause, the torque of the Sun and Moon on the Earth’s equatorial bulge, was explained by Isaac Newton.

The catalogue

Pliny says Hipparchus, after seeing a new star, set out to record the stars so that later generations could detect changes. Roman sources give about 850 stars; the catalogue was compiled by modern estimates between about 135 and 129 BC. It is lost as a whole. Whether Ptolemy’s catalogue of 1,022 stars is Hipparchus’s with a constant added to every longitude has been argued since Tycho Brahe (Ptolemy course, Lesson 09). Toomer argued that Hipparchus did not use ecliptic latitude and longitude, and the surviving Commentary gives positions in a mixture of equatorial measures.

In 2022 Victor Gysembergh, Peter Williams and Emmanuel Zingg published Greek text read by multispectral imaging beneath a Syriac manuscript from Saint Catherine’s Monastery in Sinai, the Codex Climaci Rescriptus. It gives the extent of the constellation Corona Borealis and the coordinates of four of its stars in equatorial coordinates. The precession-implied epoch fits Hipparchus’s time, and the positions are accurate to about a degree. The identification as Hipparchus’s catalogue was challenged in 2024 and defended by its authors in 2025.

The sphere, the climata and geography

For the general reader. Hipparchus insisted that a map of the world must rest on astronomy: latitudes from the Sun’s height or the length of the longest day, longitudes from the timing of eclipses. He applied this standard to the most respected map of his time, that of Eratosthenes, and found it wanting. His surviving Commentary shows the same habit applied to the sky.

Latitude and the longest day

The derivation is in the Ptolemy course, Lesson 05. Hipparchus is reported by Strabo to have arranged the inhabited world in zones, the climata, by the length of the longest day, and to have given the celestial phenomena for each. Toomer describes seven climata, with longest days from 13 to 16 hours, and notes that he used arithmetic schemes of Babylonian type for rising times as well as geometry. Whether he had spherical trigonometry in the form Menelaus later gave it is doubtful.

Longitude from eclipses

Strabo reports that Hipparchus said longitude differences could only be found by comparing observations of the same lunar eclipse at different places. A lunar eclipse begins at the same instant for every observer. If it begins at local time \(t_1\) at one place and \(t_2\) at another, their longitude difference is

The method was correct, but few simultaneous observations were ever made in antiquity, and local times were measured poorly. It remained the standard method until the seventeenth century.

Against Eratosthenes

Eratosthenes had built his map from travellers’ distances, a few latitudes and some straight reference lines. In three books, known mainly through Strabo, Hipparchus checked his positions against astronomy and against each other, showing by computation that several of Eratosthenes’ sections were inconsistent with the latitudes they implied. He objected to mixing road distances with positions fixed by the stars. He also rejected the view that the outer seas, from the Atlantic to the Indian Ocean, formed one continuous ocean surrounding the inhabited world.

The Commentary on Aratus and Eudoxus

Aratus’s poem Phaenomena (c. 275 BC) put Eudoxus’ description of the constellations into verse. Hipparchus’s commentary corrects both, and the commentator Attalus, and then gives his own systematic data: for each constellation, which stars rise and set first and last, and which stars culminate with each sign. He noted that no star stands at the celestial pole, contrary to Eudoxus, and that Eudoxus’ risings and settings fit a latitude with a longest day of about 15 hours, not Athens or Cnidus. The work is the oldest surviving Greek text to give numerical star positions.

Counting compound propositions

For the general reader. Plutarch records an argument between philosophers. The Stoic Chrysippus had claimed that ten simple statements can be combined into more than a million compound ones. Hipparchus answered with exact figures: 103,049 affirmative combinations and 310,952 negative ones. For two thousand years nobody knew where the numbers came from. In 1994 a graduate student noticed that 103,049 is the number of ways to bracket a sequence of ten symbols.

The report

Plutarch (Table Talk VIII.9 and On Stoic Self-Contradictions) says that arithmeticians, Hipparchus among them, refuted Chrysippus by showing that the affirmative compounds number 103,049 and the negative ones 310,952. The logic behind compound propositions in Stoic theory, with conjunctions grouped in nested clauses, is not fully known.

In 1994 David Hough, a graduate student at George Washington University, observed that 103,049 is the tenth little Schröder number. Richard Stanley published the observation in 1997. Fabio Acerbi later showed that 310,952 can be explained by a related count, since it is within 2 of the mean of the tenth and eleventh Schröder numbers.

Proof sketch. Wrap the whole word in an outer pair. A bracketed word is then either a single letter or a sequence of \(k\ge2\) bracketed words placed side by side. Lengths add, so \(S=x+\sum_{k\ge2}S^k=x+S^2/(1-S)\). Clearing denominators, \(2S^2-(1+x)S+x=0\), and the root with \(S(0)=0\) is the one shown. Writing \(P=\sqrt{1-6x+x^2}\), one checks \((1-6x+x^2)P'=(x-3)P\), and translating this differential equation into coefficients gives the recurrence.

Bracketings of \(n\) letters correspond to dissections of a convex \((n+1)\)-gon by non-crossing diagonals, and to the faces of the associahedron. Whether Hipparchus counted them by a recurrence or by another route is unknown, but the match with a ten-digit sequence is unlikely to be chance.

Planets, last years and lost books

Hipparchus did not produce a planetary theory, and Ptolemy says why. Hipparchus collected planetary observations and showed that the models of his predecessors, built from a single epicycle on a concentric deferent, could not reproduce the observed variation in the size of retrograde arcs. He judged that the data available were insufficient to determine better models, and left the problem to his successors. Ptolemy treats this restraint as admirable.

Ptolemy also credits Hipparchus with period relations for the planets, such as that Saturn completes 57 returns in anomaly and two revolutions of the zodiac, plus a little, in 59 years. These relations appear in Babylonian texts too, and were probably taken from them.

The last observations

The latest dated observation attributed to him is a lunar position of 7 July 127 BC, at Rhodes. His work on lunar theory in these years suggests he suspected the model failed away from new and full moon. After 127 BC the record falls silent. The common statement that he died around 120 BC is an estimate, and no source records where he died, though Rhodes is the natural presumption.

Why the books were lost

Hipparchus wrote monographs on separate problems. Ptolemy absorbed their results into a single systematic treatise with tables, and once the Almagest existed there was little reason to copy the older books. The Commentary on Aratus survived because it was attached to a popular poem, not because of its astronomy. Fragments of other works survive in astrological and geographical writers, and new fragments, like the Sinai palimpsest, may still appear.

Hipparchus after Hipparchus

Hipparchus’s influence came almost entirely through Ptolemy. The Almagest kept his solar model, his tropical year, his lunar periods and maximum latitude, his eclipse method for distances, and his rate of precession as Ptolemy understood it. Through the Almagest these numbers passed to the astronomers of Baghdad and Damascus, and from them to Latin Europe and to Nicolaus Copernicus.

Assessment

• Method: he made astronomy a science in which geometric models are fitted to dated observations, with parameters derived from chosen data and errors estimated.
• Trigonometry: the first known chord table, and probably, through India, the ancestor of the sine table.
• Discovery: precession of the equinoxes, the first secular change found in the heavens.
• Distances: the first sound bounds on the Moon’s distance, which enclose the true value.
• Transmission: the systematic use of Babylonian records and periods in Greek astronomy, together with degrees and sexagesimal fractions.

Reputation

Ptolemy’s praise set the tone. In 1817 Delambre, a severe critic of Ptolemy, called Hipparchus the father of astronomy and ranked him with Kepler and James Bradley among the greatest astronomers. The debate over Ptolemy’s star catalogue (Ptolemy course, Lessons 09 and 12) repeatedly turned on how much of the Almagest was really Hipparchus’s; the 2022 palimpsest has reopened the question with new evidence. Historians of Babylonian astronomy, beginning with Kugler, have shown how much he took from Mesopotamia, without diminishing what he did with it.

Names

The European Space Agency’s Hipparcos satellite (High Precision Parallax Collecting Satellite), launched in 1989, measured the positions and parallaxes of about 118,000 stars, a modern successor to his catalogue. A lunar crater, a Martian crater and the asteroid 4000 Hipparchus bear his name.

Connections

People linked to Hipparchus across this collection. A name without a link has no page here yet. See Hipparchus in the network of all 337 lives →

Influenced by

• Apollonius of Perga
• Aristarchus of Samos
• Eratosthenes
• Eudoxus of Cnidus
• Aratus of Soli
• Aristillus
• Timocharis

Influenced

• Aryabhata
• Claudius Ptolemy
• Nicolaus Copernicus
• Tycho Brahe
• Menelaus of Alexandria

Rivals

• Eratosthenes
• Chrysippus

Also mentioned in this life

• James Bradley
• Euclid
• Johannes Kepler
• Isaac Newton
• Pappus of Alexandria
• Yakov Sinai

Fields

• astronomy 50 lives
• trigonometry 9 lives
• geography 5 lives
• combinatorics 9 lives

On CoursesHub

• Astrophysics for Beginners

Sources and further reading

Biography and context

1. J. J. O’Connor and E. F. Robertson, Hipparchus of Rhodes, MacTutor History of Mathematics.
2. G. J. Toomer, Hipparchus, Dictionary of Scientific Biography, vol. 15, Scribner, 1978.
3. Hipparchus, Wikipedia (with extensive references).
4. O. Neugebauer, A History of Ancient Mathematical Astronomy, Springer, 1975.
5. J. Evans, The History and Practice of Ancient Astronomy, Oxford University Press, 1998.
6. D. R. Dicks, The Geographical Fragments of Hipparchus, Athlone Press, 1960.

Astronomy and mathematics

1. G. J. Toomer, Ptolemy’s Almagest, translation and notes, Duckworth, 1984; Princeton University Press, 1998.
2. G. J. Toomer, The chord table of Hipparchus and the early history of Greek trigonometry, Centaurus 18 (1973).
3. G. J. Toomer, Hipparchus on the distances of the Sun and Moon, Archive for History of Exact Sciences 14 (1974).
4. N. M. Swerdlow, Hipparchus on the distance of the Sun, Centaurus 14 (1969).
5. On Sizes and Distances (Hipparchus), Wikipedia.
6. V. Gysembergh, P. J. Williams and E. Zingg, New evidence for Hipparchus’ star catalogue revealed by multispectral imaging, Journal for the History of Astronomy 53 (2022). Report: Sky & Telescope.
7. R. P. Stanley, Hipparchus, Plutarch, Schröder, and Hough, American Mathematical Monthly 104 (1997). See also Schröder–Hipparchus number, Wikipedia.
8. Timocharis, Wikipedia, on the Spica observations.