Hipparchus On Sizes and Distances
Encyclopedia
On Sizes and Distances [of the Sun and Moon] (Περὶ μεγεθῶν καὶ ἀποστημάτων – Peri megethon kai apostematon) is a text by the ancient Greek
astronomer Hipparchus
. It is not extant
, but some of its contents have been preserved in the works of Ptolemy
and his commentator Pappus of Alexandria
. Several modern historians have attempted to reconstruct the methods of Hipparchus using the available texts.
and others, but their accounts have proven less useful in reconstructing the procedures of Hipparchus.
This passage gives a general outline of what Hipparchus did, but provides no details. Ptolemy clearly did not agree with the methods employed by Hipparchus, and thus did not go into any detail.
Then later,
This passage provides enough details to make a reconstruction feasible. In particular, it makes clear that there were two separate procedures, and it gives the precise results of each. It provides clues with which to identify the eclipse, and says that Hipparchus used a formula "as in Theorem 12," a theorem of Ptolemy's which is extant.
in 1969. G. J. Toomer expanded on his efforts in 1974.
His analysis was based on a text by Theon of Smyrna
which states that Hipparchus found the sun to be 1880 times the size of the earth, and the earth 27 times the size of the moon. Assuming that this refers to volumes, it follows that
and
Assuming that the sun and moon have the same apparent size in the sky, and that the moon is 67 1/3 earth radii distant, it follows that
This result was generally accepted for the next seventy years, until Noel Swerdlow
reinvestigated the case.
Swerdlow determined that Hipparchus relates the distances to the sun and moon using a construction found in Ptolemy
. It would not be surprising if this calculation had been originally developed by Hipparchus himself, as he was a primary source for the Almagest
.
Using this calculation, Swerdlow was able to relate the two results of Hipparchus (67 1/3 for the moon and 490 for the sun). Obtaining this relationship exactly requires following a very precise set of approximations.
Using simple trigonometric identities gives
and
By parallel lines and taking t = 1, we get
By similarity of triangles,
Combining these equations gives
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
astronomer Hipparchus
Hipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...
. It is not extant
Extant literature
Extant literature refers to texts that have survived from the past to the present time. Extant literature can be divided into extant original manuscripts, copies of original manuscripts, quotations and paraphrases of passages of non-extant texts contained in other works, translations of non-extant...
, but some of its contents have been preserved in the works of Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
and his commentator Pappus of Alexandria
Pappus of Alexandria
Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
. Several modern historians have attempted to reconstruct the methods of Hipparchus using the available texts.
Sources
Most of what is known about Hipparchus' text comes from two ancient sources: Ptolemy and Pappus. The work is also mentioned by Theon of SmyrnaTheon of Smyrna
Theon of Smyrna was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. His surviving On Mathematics Useful for the Understanding of Plato is an introductory survey of Greek mathematics.-Life:Little is known about the life of Theon of...
and others, but their accounts have proven less useful in reconstructing the procedures of Hipparchus.
Ptolemy
In Almagest V, 11, Ptolemy writes:- Now Hipparchus made such an examination principally from the sun. Since from other properties of the sun and moon (of which a study will be made below) it follows that if the distance of one of the two luminaries is given, the distance of the other is also given, he tries by conjecturing the distance of the sun to demonstrate the distance of the moon. First, he assumes the sun to show the least perceptible parallax in order to find its distance. After this, he makes use of the solar eclipse adduced by him, first as if the sun shows no perceptible parallax, and for exactly that reason the ratios of the moon's distances appeared different to him for each of the hypotheses he set out. But with respect to the sun, not only the amount of its parallax, but also whether it shows any parallax at all is altogether doubtful.
This passage gives a general outline of what Hipparchus did, but provides no details. Ptolemy clearly did not agree with the methods employed by Hipparchus, and thus did not go into any detail.
Pappus of Alexandria
The works of Hipparchus were still extant when Pappus wrote his commentary on the Almagest in the 4th century. He fills in some of the details that Ptolemy omits:- Now, Hipparchus made such an examination principally from the sun, and not accurately. For since the moon in the syzygies and near greatest distance appears equal to the sun, and since the size of the diameters of the sun and moon is given (of which a study will be made below), it follows that if the distance of one of the two luminaries is given, the distance of the other is also given, as in Theorem 12, if the distance of the moon is given and the diameters of the sun and moon, the distance of the sun is given. Hipparchus tries by conjecturing the parallax and the distance of the sun to demonstrate the distance of the moon, but with respect to the sun, not only the amount of its parallax, but also whether it shows any parallax at all is altogether doubtful. For in this way Hipparchus was in doubt about the sun, not only about the amount of its parallax but also about whether it shows any parallax at all. In the first book "On Sizes and Distances" it is assumed that the earth has the ratio of a point and center to the sun. And by means of the eclipse adduced by him...
Then later,
- For in Book 1 of "On Sizes and Distances" he takes the following observation: an eclipse of the sun, which in the regions around the Hellespont was an exact eclipse of the whole solar disc, such that no part of it was visible, but at Alexandria by Egypt approximately four-fifths of it was eclipsed. By means of this he shows in Book 1 that, in units of which the radius of the earth is one, the least distance of the moon is 71, and the greatest 83. Hence the mean is 77... Then again he himself in Book 2 of "On Sizes and Distances" shows from many considerations that, in units of which the radius of the earth is one, the least distance of the moon is 62, and the mean 67 1/3, and the distance of the sun 490. It is clear that the greatest distance of the moon is 72 2/3.
This passage provides enough details to make a reconstruction feasible. In particular, it makes clear that there were two separate procedures, and it gives the precise results of each. It provides clues with which to identify the eclipse, and says that Hipparchus used a formula "as in Theorem 12," a theorem of Ptolemy's which is extant.
Modern reconstructions
Several historians of science have attempted to reconstruct the calculations involved in On Sizes and Distances. The first attempt was made by Friedrich Hultsch in 1900, but it was later rejected by Noel SwerdlowNoel Swerdlow
Noel M. Swerdlow is a professor emeritus of history, astronomy and astrophysics at the University of Chicago. He is currently a visiting professor at the California Institute of Technology.-Career:...
in 1969. G. J. Toomer expanded on his efforts in 1974.
Hultsch
Friedrich Hultsch determined in a 1900 paper that the Pappus source had been miscopied, and that the actual distance to the sun, as calculated by Hipparchus, had been 2490 earth radii (not 490). As in English, there is only a single character difference between these two results in Greek.His analysis was based on a text by Theon of Smyrna
Theon of Smyrna
Theon of Smyrna was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. His surviving On Mathematics Useful for the Understanding of Plato is an introductory survey of Greek mathematics.-Life:Little is known about the life of Theon of...
which states that Hipparchus found the sun to be 1880 times the size of the earth, and the earth 27 times the size of the moon. Assuming that this refers to volumes, it follows that
and
Assuming that the sun and moon have the same apparent size in the sky, and that the moon is 67 1/3 earth radii distant, it follows that
This result was generally accepted for the next seventy years, until Noel Swerdlow
Noel Swerdlow
Noel M. Swerdlow is a professor emeritus of history, astronomy and astrophysics at the University of Chicago. He is currently a visiting professor at the California Institute of Technology.-Career:...
reinvestigated the case.
Book 2 reconstruction (Swerdlow)
Swerdlow determined that Hipparchus relates the distances to the sun and moon using a construction found in Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
. It would not be surprising if this calculation had been originally developed by Hipparchus himself, as he was a primary source for the Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
.
Using this calculation, Swerdlow was able to relate the two results of Hipparchus (67 1/3 for the moon and 490 for the sun). Obtaining this relationship exactly requires following a very precise set of approximations.
Using simple trigonometric identities gives
and
By parallel lines and taking t = 1, we get
By similarity of triangles,
Combining these equations gives
-
The values which Hipparchus took for these variables can be found in Ptolemy's Almagest IV, 9. He says Hipparchus found that the moon measured its own circle close to 650 times, and that the angular diameter of earth's shadow is 2.5 times that of the moon. Pappus tells us that Hipparchus took the mean distance to the moon to be 67 1/3. This gives:
Quantity Value 
67 1/3
According to Swerdlow, Hipparchus now evaluated this expression with the following roundings (the values are in sexagesimal):
and
Then, because
-
it follows that
Swerdlow used this result to argue that 490 was the correct reading of the Pappus text, thus invalidating Hultsch' interpretation. While this result is highly dependent on the particular approximations and roundings used, it has generally been accepted. It leaves open, however, the question of where the lunar distance 67 1/3 came from.
Following Pappus and Ptolemy, Swerdlow suggested that Hipparchus had estimated 490 earth radii as a minimum possible distance to the sun. This distance corresponds to a solar parallax of 7', which may have been the maximum that he thought would have gone unnoticed (the typical resolution of the human eye is 2'). The formula obtained above for the distance to the sun can be inverted to determine the distance to the moon:
-
Using the same values as above for each angle, and using 490 earth radii as the minimum solar distance, it follows that the maximum mean lunar distance is
-
Toomer expanded on this by observing that as the distance to the sun increases without bound, the formula approaches a minimum mean lunar distance:
This is close to the value later claimed by Ptolemy.
Book 1 reconstruction (Toomer)
In addition to explaining the minimum lunar distance that Hipparchus achieved, Toomer was able to explain the method of the first book, which employed a solar eclipseSolar eclipseAs seen from the Earth, a solar eclipse occurs when the Moon passes between the Sun and the Earth, and the Moon fully or partially blocks the Sun as viewed from a location on Earth. This can happen only during a new moon, when the Sun and the Moon are in conjunction as seen from Earth. At least...
. Pappus states that this eclipse was total in the region of the Hellespont, but was observed to be 4/5 of total in Alexandria.
If Hipparchus assumed that the sun was infinitely distant (i.e. that "the earth has the ratio of a point and center to the sun"), then the difference in magnitude of the solar eclipse must be due entirely to the parallax of the moon. By using observational data, he would be able to determine this parallax, and hence the distance of the moon.
Hipparchus would have known
and 
, the latitudes of AlexandriaAlexandriaAlexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
and the Hellespontine region, respectively. He also would have known
, the declinationDeclinationIn astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...
of the moon during the eclipse, and
, which is related to the difference in totality of the eclipse between the two regions.
Crd here refers to the chordChord (geometry)A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
function. Since the moon is very distant, it follows that
. Using this approximation, we have
Hence,
With
and 
, we only need 
to get 
. Because the eclipse was total at H, and 4/5 total at A, it follows that 
is 1/5 of the apparent diameter of the sun. This quantity was well known by Hipparchus—he took it to be 1/650 of a full circle. The distance from the center of the earth to the moon then follows from 
.
Toomer determined how Hipparchus determined the chord for small angles (see Chord (geometry)Chord (geometry)A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
). His values for the latitudes of the Hellespont (41 degrees) and Alexandria (31 degrees) are known from Strabo'sStraboStrabo, also written Strabon was a Greek historian, geographer and philosopher.-Life:Strabo was born to an affluent family from Amaseia in Pontus , a city which he said was situated the approximate equivalent of 75 km from the Black Sea...
work on GeographyGeographyGeography is the science that studies the lands, features, inhabitants, and phenomena of Earth. A literal translation would be "to describe or write about the Earth". The first person to use the word "geography" was Eratosthenes...
. To determine the declination, it is necessary to know which eclipse Hipparchus used.
Because he knew the value which Hipparchus eventually gave for the distance to the moon (71 earth radii) and the rough region of the eclipse, Toomer was able to determine that Hipparchus used the eclipse of March 14, 190 BC. This eclipse fits all the mathematical parameters very well, and also makes sense from a historical point of view. The eclipse was total in NicaeaIznikİznik is a city in Turkey which is primarily known as the site of the First and Second Councils of Nicaea, the first and seventh Ecumenical councils in the early history of the Church, the Nicene Creed, and as the capital city of the Empire of Nicaea...
, Hipparchus' birthplace, so he may have heard stories of it. There is also an account of it in Strabo'sStraboStrabo, also written Strabon was a Greek historian, geographer and philosopher.-Life:Strabo was born to an affluent family from Amaseia in Pontus , a city which he said was situated the approximate equivalent of 75 km from the Black Sea...
Ab Urbe Condita VIII.2. The declination of the moon at this time was
. Hence, using chord trigonometry, we have
Now using Hipparchus' chord tables,
and hence
This agrees very well with the value of 71 earth radii that Pappus reports.
This analysis assumed that the eclipse took place in the middle of the day, with the sun and moon on the meridian. But this was not the case for the eclipse of 190 BC190 BCYear 190 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Asiaticus and Laelius...
, which took place
Conclusion
Assuming that these reconstructions accurately reflect what Hipparchus wrote in On Sizes and Distances, then this work was a remarkable accomplishment. This approach of setting limits on an unknown physical quantity was not new to Hipparchus (see Aristarchus of SamosAristarchus of SamosAristarchus, or more correctly Aristarchos , was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He presented the first known heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe...
. Archimedes also did the same with piPi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
), but in those cases, the bounds reflected the inability to determine a mathematical constant to an arbitrary precision, not uncertainty in physical observations.
Hipparchus appears to have eventually resolved the contradiction between his two results. His aim in calculating the distance to the moon was to obtain an accurate value for the lunar parallax, so that he might predict eclipses with more precision. To this, he had to settle on a particular value for the distance/parallax, not a range of values.
There is some evidence that he did this. Combining the calculations of Book 2 and the account of Theon of SmyrnaTheon of SmyrnaTheon of Smyrna was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. His surviving On Mathematics Useful for the Understanding of Plato is an introductory survey of Greek mathematics.-Life:Little is known about the life of Theon of...
yields a lunar distance of 60.5 earth radii. Doing the same with the account of CleomedesCleomedesCleomedes was a Greek astronomer who is known chiefly for his book On the Circular Motions of the Celestial Bodies.-Placing his work chronologically:...
yields a distance of 61 earth radii. These are remarkably close to both Ptolemy's value and the modern one.
According to Toomer,
- This procedure, if I have constructed it correctly, is very remarkable... What is astonishing is the sophistication of approaching the problem by two quite different methods, and also the complete honesty with which Hipparchus reveals his discrepant results... which are nevertheless of the same order of magnitude and (for the first time in the history of astronomy) in the right region.
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