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''On the Sizes and Distances (of the Sun and Moon)'' ( grc, Περὶ μεγεθῶν καὶ ἀποστημάτων ��λίου καὶ σελήνης}) is widely accepted as the only extant work written by
Aristarchus of Samos Aristarchus of Samos (; grc-gre, Ἀρίσταρχος ὁ Σάμιος, ''Aristarkhos ho Samios''; ) was an ancient Greek astronomer and mathematician who presented the first known heliocentric model that placed the Sun at the center of the ...
, an ancient Greek astronomer who lived circa 310–230 BCE. This work calculates the sizes of the Sun and
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
, as well as their distances from the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
in terms of Earth's radius. The book was presumably preserved by students of
Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
's course in mathematics, although there is no evidence of this. The ''
editio princeps In classical scholarship, the ''editio princeps'' (plural: ''editiones principes'') of a work is the first printed edition of the work, that previously had existed only in manuscripts, which could be circulated only after being copied by hand. For ...
'' was published by
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
in 1688, using several medieval manuscripts compiled by Sir
Henry Savile Henry Savile may refer to: *Henry Savile (died 1558) (1498–1558), MP for Yorkshire *Henry Savile (died 1569) (1518–1569), MP for Yorkshire and Grantham * Henry Savile (Bible translator) (1549–1622), English scholar and Member of the Parliamen ...
. The earliest Latin translation was made by Giorgio Valla in 1488. There is also
1572 Latin translation and commentary
by Frederico Commandino.


Symbols

The work's method relied on several observations: * The apparent size of the Sun and the Moon in the sky. * The size of the Earth's shadow in relation to the Moon during a
lunar eclipse A lunar eclipse occurs when the Moon moves into the Earth's shadow. Such alignment occurs during an eclipse season, approximately every six months, during the full moon phase, when the Moon's orbital plane is closest to the plane of the Ear ...
* The angle between the Sun and Moon during a half moon is very close to 90°. The rest of the article details a reconstruction of Aristarchus' method and results.A video on reconstruction of Aristarchus' method
(in Turkish, no subtitles) The reconstruction uses the following variables:


Half Moon

Aristarchus began with the premise that, during a half moon, the moon forms a
right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
with the Sun and Earth. By observing the angle between the Sun and Moon, ''φ'', the ratio of the distances to the Sun and Moon could be deduced using a form of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
. From the diagram and trigonometry, we can calculate that : \frac = \frac = \sec \varphi. The diagram is greatly exaggerated, because in reality, ''S = 390 L'', and ''φ'' is extremely close to 90°. Aristarchus determined ''φ'' to be a thirtieth of a quadrant (in modern terms, 3°) less than a right angle: in current terminology, 87°. Trigonometric functions had not yet been invented, but using geometrical analysis in the style of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
, Aristarchus determined that : 18 < \frac < 20. In other words, the distance to the Sun was somewhere between 18 and 20 times greater than the distance to the Moon. This value (or values close to it) was accepted by astronomers for the next two thousand years, until the invention of the telescope permitted a more precise estimate of
solar parallax Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby object ...
. Aristarchus also reasoned that as the
angular size The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it ...
of the Sun and the Moon were the same, but the distance to the Sun was between 18 and 20 times further than the Moon, the Sun must therefore be 18–20 times larger.


Lunar eclipse

Aristarchus then used another construction based on a lunar eclipse: By similarity of the triangles, \frac = \frac \quad and \quad \frac = \frac. Dividing these two equations and using the observation that the apparent sizes of the Sun and Moon are the same, \frac = \frac, yields : \frac = \frac \ \ \Rightarrow \ \ \frac = \frac \ \ \Rightarrow \ \ 1 - \frac = \frac - \frac \ \ \Rightarrow \ \ \frac + \frac = 1 + \frac. The rightmost equation can either be solved for ''ℓ/t'' : \frac(1+\frac) = 1 + \frac \ \ \Rightarrow \ \ \frac = \frac. or ''s/t'' : \frac(1+\frac) = 1 + \frac \ \ \Rightarrow \ \ \frac = \frac. The appearance of these equations can be simplified using ''n'' = ''d/ℓ'' and ''x'' = ''s/ℓ''. : \frac = \frac : \frac = \frac The above equations give the radii of the Moon and Sun entirely in terms of observable quantities. The following formulae give the distances to the Sun and Moon in terrestrial units: : \frac = \left( \frac \right) \left( \frac \right) : \frac = \left( \frac \right) \left( \frac \right) where ''θ'' is the apparent radius of the Moon and Sun measured in degrees. It is unlikely that Aristarchus used these exact formulae, yet these formulae are likely a good approximation for those of Aristarchus.


Results

The above formulae can be used to reconstruct the results of Aristarchus. The following table shows the results of a long-standing (but dubious) reconstruction using ''n'' = 2, ''x'' = 19.1 (''φ'' = 87°) and ''θ'' = 1°, alongside the modern day accepted values. The error in this calculation comes primarily from the poor values for ''x'' and ''θ''. The poor value for ''θ'' is especially surprising, since
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
writes that Aristarchus was the first to determine that the Sun and Moon had an apparent diameter of half a degree. This would give a value of ''θ'' = 0.25, and a corresponding distance to the Moon of 80 Earth radii, a much better estimate. The disagreement of the work with Archimedes seems to be due to its taking an Aristarchus statement that the lunisolar diameter is 1/15 of a "meros" of the zodiac to mean 1/15 of a zodiacal sign (30°), unaware that the Greek word "meros" meant either "portion" or 7°1/2; and 1/15 of the latter amount is 1°/2, in agreement with Archimedes' testimony. A similar procedure was later used by
Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the e ...
, who estimated the mean distance to the Moon as 67 Earth radii, and
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
, who took 59 Earth radii for this value.


Illustrations

Some interactive illustrations of the propositions in ''On Sizes'' can be found here:
Hypothesis 4
states that when the Moon appears to us halved, its distance from the Sun is then less than a quadrant by one-thirtieth of a quadrant hat is, it is less than 90° by 1/30th of 90° or 3°, and is therefore equal to 87°(Heath 1913:353).
Proposition 1
states that two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres (Heath 1913:354).
Proposition 2
states that if a sphere be illuminated by a sphere greater than itself, the illuminated portion of the former sphere will be greater than a hemisphere (Heath 1913:358).
Proposition 3
states that the circle in the Moon which divides the dark and the bright portions is least when the cone comprehending both the Sun and the Moon has its vertex at our eye (Heath 1913:362).
Proposition 4
states that the circle which divides the dark and the bright portions in the Moon is not perceptibly different from a great circle in the Moon (Heath 1913:365).
Proposition 6
states that the Moon moves n an orbitlower than hat ofthe Sun, and, when it is halved, is distant less than a quadrant from the Sun (Heath 1913:372).
Proposition 7
states that the distance of the Sun from the Earth is greater than 18 times, but less than 20 times, the distance of the Moon from the Earth (Heath 1913:377). In other words, the Sun is 18 to 20 times farther away and wider than the Moon.
Proposition 13
states that the straight line subtending the portion intercepted within the earth's shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the Moon move is less than double of the diameter of the Moon, but has to it a ratio greater than that which 88 has to 45; and it is less than 1/9th part of the diameter of the Sun, but has to it a ratio greater than that which 21 has to 225. But it has to the straight line drawn from the centre of the Sun at right angles to the axis and meeting the sides of the cone a ratio greater than that which 979 has to 10 125 (Heath 1913:394).
Proposition 14
states that the straight line joined from the centre of the Earth to the centre of the Moon has to the straight line cut off from the axis towards the centre of the Moon by the straight line subtending the ircumferencewithin the Earth's shadow a ratio greater than that which 675 has to 1 (Heath 1913:400).
Proposition 15
states that the diameter of the Sun has to the diameter of the Earth a ratio greater than 19/3, but less than 43/6 (Heath 1913:403). This means that the Sun is (a mean of) 6¾ times wider than the Earth, or that the Sun is 13½ Earth-radii wide. The Moon and Sun must then be 20¼ and 387 Earth-radii away from us in order to subtend an angular size of 2º.
Proposition 17a
in al-Tusi's medieval Arabic version of the book ''On Sizes'' states that the ratio of the distance of the vertex of the shadow cone from the center of the Moon (when the Moon is on the axis hat is, at the middle of an eclipseof the cone containing the Earth and the Sun) to the distance of the center of the Moon from the center of the Earth is greater than the ratio 71 to 37 and less than the ratio 3 to one (Berggren & Sidoli 2007:218).Berggren, J. L. & N. Sidoli (2007) . In other words, that the tip of the Earth's shadow cone is between 108/37 and four times farther away than the Moon.


Known copies

* Library of Congress Vatican Exhibit.


See also

*
Aristarchus of Samos Aristarchus of Samos (; grc-gre, Ἀρίσταρχος ὁ Σάμιος, ''Aristarkhos ho Samios''; ) was an ancient Greek astronomer and mathematician who presented the first known heliocentric model that placed the Sun at the center of the ...
*
Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ;  – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandr ...
(), a Greek mathematician who calculated the circumference of the Earth and also the distance from the Earth to the Sun. *
Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the e ...
(), a Greek mathematician who measured the radii of the Sun and the Moon as well as their distances from the Earth. * ''On the Sizes and Distances'' (Hipparchus) *
Posidonius Posidonius (; grc-gre, Ποσειδώνιος , "of Poseidon") "of Apameia" (ὁ Ἀπαμεύς) or "of Rhodes" (ὁ Ῥόδιος) (), was a Greek politician, astronomer, astrologer, geographer, historian, mathematician, and teacher nativ ...
(), a Greek astronomer and mathematician who calculated the circumference of the Earth.


Notes


Bibliography

* This was later reprinted, see (). * van Helden, A. ''Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley''. Chicago: Univ. of Chicago Pr., 1985. . {{Greek mathematics Ancient Greek astronomical works Astronomy books