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Astrolabe

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For other pages with a similar name, see Astrolabe (disambiguation). Not to be confused with Cosmolabe.
Planispheric Astrolabe made of brass, cast, with fretwork rete and surface engraving
North African, 9th century AD, Planispheric Astrolabe. Khalili Collection.
A modern astrolabe made in Tabriz, Iran in 2013.

An astrolabe (Greek: ἀστρολάβος astrolábos, 'star-taker'; Arabic: ٱلأَسْطُرلاب al-Asṭurlāb; Persian: ستاره‌یاب Setāreyāb) is an astronomical instrument dating to ancient times. It serves as a star chart and physical model of visible heavenly bodies. Its various functions also make it an elaborate inclinometer and an analog calculation device capable of working out several kinds of problems in astronomy. In its simplest form it is a metal disc with a pattern of wires, cutouts, and perforations that allows a user to calculate astronomical positions precisely. Historically used by astronomers, it is able to measure the altitude above the horizon of a celestial body, day or night; it can be used to identify stars or planets, to determine local latitude given local time (and vice versa), to survey, or to triangulate. It was used in classical antiquity, the Islamic Golden Age, the European Middle Ages and the Age of Discovery for all these purposes.

The astrolabe is effective for determining latitude on land or calm seas. Although it is less reliable on the heaving deck of a ship in rough seas, the mariner's astrolabe was developed to solve that problem.

Applications

16th-century woodcut of measurement of a building's height with an astrolabe

A 10th-century astronomer deduced that there were around 1000 applications for the astrolabe's various functions,[1][better source needed] and these ranged from the astrological, the astronomical and the religious, to seasonal and daily time-keeping and tide tables. At the time of their use, astrology was widely considered as much of a serious science as astronomy, and study of the two went hand-in-hand. The astronomical interest varied between folk astronomy (of the pre-Islamic tradition in Arabia) which was concerned with celestial and seasonal observations, and mathematical astronomy, which would inform intellectual practices and precise calculations based on astronomical observations. In regard to the astrolabe's religious function, the demands of Islamic prayer times were to be astronomically determined to ensure precise daily timings, and the qibla, the direction of Mecca towards which Muslims must pray, could also be determined by this device. In addition to this, the lunar calendar that was informed by the calculations of the astrolabe was of great significance to the religion of Islam, given that it determines the dates of important religious observances such as Ramadan.

Etymology

The Oxford English Dictionary gives the translation "star-taker" for the English word astrolabe and traces it through medieval Latin to the Greek word ἀστρολάβος : astrolábos,[2][3] from ἄστρον : astron "star" and λαμβάνειν : lambanein "to take".[4]

In the medieval Islamic world the Arabic word al-Asturlāb (i.e., astrolabe) was given various etymologies. In Arabic texts, the word is translated as ākhidhu al-Nujūm (Arabic: آخِذُ ٱلنُّجُومْ, lit. 'star-taker'), a direct translation of the Greek word.[5]

Al-Biruni quotes and criticises medieval scientist Hamza al-Isfahani who stated:[5] "asturlab is an arabisation of this Persian phrase" (sitara yab, meaning "taker of the stars").[6] In medieval Islamic sources, there is also a folk etymology of the word as "lines of lab", where "Lab" refers to a certain son of Idris (Enoch). This etymology is mentioned by a 10th-century scientist named al-Qummi but rejected by al-Khwarizmi.[7]

History

Ancient world

Armillary Sphere[8][9] mosaic from the Casa di Leada at Soluntum in Sicily.

An early astrolabe was invented in the Hellenistic civilization by Apollonius of Perga between 220 and 150 BC, often attributed to Hipparchus. The astrolabe was a marriage of the planisphere and dioptra, effectively an analog calculator capable of working out several different kinds of problems in astronomy. The earliest known depiction of an astrolabe is a mosaic in the Casa di Leada at Soluntum in Sicily, which may have been imported from Alexandria in Egypt.[10][dubious – discuss]

Theon of Alexandria (c. 335 – c. 405) wrote a detailed treatise on the astrolabe, and Lewis[11] argues that Ptolemy used an astrolabe to make the astronomical observations recorded in the Tetrabiblos. The invention of the plane astrolabe is sometimes wrongly attributed to Theon's daughter Hypatia (c. 350–370; died 415 AD),[12][13][14][15] but it is, in fact, known to have already been in use at least 500 years before Hypatia was born.[13][14][15] The misattribution comes from a misinterpretation of a statement in a letter written by Hypatia's pupil Synesius (c. 373 – c. 414),[13][14][15] which mentions that Hypatia had taught him how to construct a plane astrolabe, but does not state anything about her having invented it herself.[13][14][15]

Astrolabes continued in use in the Greek-speaking world throughout the Byzantine period. About 550 AD, Christian philosopher John Philoponus wrote a treatise on the astrolabe in Greek, which is the earliest extant treatise on the instrument.[a] Mesopotamian bishop Severus Sebokht also wrote a treatise on the astrolabe in the Syriac language in the mid-7th century.[b] Sebokht refers to the astrolabe as being made of brass in the introduction of his treatise, indicating that metal astrolabes were known in the Christian East well before they were developed in the Islamic world or in the Latin West.[16]

The first Renaissance treatises dealing with scientific problems were based on earlier classical works and were often concerned with Ptolemaic doctrines.[citation needed]

Medieval era

Astrolabes were further developed in the medieval Islamic world, where Muslim astronomers introduced angular scales to the design,[17] adding circles indicating azimuths on the horizon.[18] It was widely used throughout the Muslim world, chiefly as an aid to navigation and as a way of finding the Qibla, the direction of Mecca. Eighth-century mathematician Muhammad al-Fazari is the first person credited with building the astrolabe in the Islamic world.[19]

The mathematical background was established by Muslim astronomer Albatenius in his treatise Kitab az-Zij (c. 920 AD), which was translated into Latin by Plato Tiburtinus (De Motu Stellarum). The earliest surviving astrolabe is dated AH 315 (927–928 AD). In the Islamic world, astrolabes were used to find the times of sunrise and the rising of fixed stars, to help schedule morning prayers (salat). In the 10th century, al-Sufi first described over 1,000 different uses of an astrolabe, in areas as diverse as astronomy, astrology, navigation, surveying, timekeeping, prayer, Salat, Qibla, etc.[20][21]

The spherical astrolabe was a variation of both the astrolabe and the armillary sphere, invented during the Middle Ages by astronomers and inventors in the Islamic world.[c] The earliest description of the spherical astrolabe dates to Al-Nayrizi (fl. 892–902). In the 12th century, Sharaf al-Dīn al-Tūsī invented the linear astrolabe, sometimes called the "staff of al-Tusi", which was "a simple wooden rod with graduated markings but without sights. It was furnished with a plumb line and a double chord for making angular measurements and bore a perforated pointer".[22] The geared mechanical astrolabe was invented by Abi Bakr of Isfahan in 1235.[23]

The first known metal astrolabe in Western Europe is the Destombes astrolabe made from brass in the eleventh century in Portugal.[24][25] Metal astrolabes avoided the warping that large wooden ones were prone to, allowing the construction of larger and therefore more accurate instruments. Metal astrolabes were heavier than wooden instruments of the same size, making it difficult to use them in navigation.[26]

Herman Contractus of Reichenau Abbey, examined the use of the astrolabe in Mensura Astrolai during the 11th century.[27] Peter of Maricourt wrote a treatise on the construction and use of a universal astrolabe in the last half of the 13th century entitled Nova compositio astrolabii particularis. Universal astrolabes can be found at the History of Science Museum in Oxford.[28] David A. King, historian of Islamic instrumentation, describes the universal astrolobe designed by Ibn al-Sarraj of Aleppo (aka Ahmad bin Abi Bakr; fl. 1328) as "the most sophisticated astronomical instrument from the entire Medieval and Renaissance periods".[29]

English author Geoffrey Chaucer (c. 1343–1400) compiled A Treatise on the Astrolabe for his son, mainly based on a work by Messahalla or Ibn al-Saffar.[30][31] The same source was translated by French astronomer and astrologer Pélerin de Prusse and others. The first printed book on the astrolabe was Composition and Use of Astrolabe by Christian of Prachatice, also using Messahalla, but relatively original.

In 1370, the first Indian treatise on the astrolabe was written by the Jain astronomer Mahendra Suri, titled Yantrarāja.[32]

A simplified astrolabe, known as a balesilha, was used by sailors to get an accurate reading of latitude while at sea. The use of the balesilha was promoted by Prince Henry (1394–1460) while navigating for Portugal.[33]

The astrolabe was almost certainly first brought north of the Pyrenees by Gerbert of Aurillac (future Pope Sylvester II), where it was integrated into the quadrivium at the school in Reims, France, sometime before the turn of the 11th century.[34] In the 15th century, French instrument maker Jean Fusoris (c. 1365–1436) also started remaking and selling astrolabes in his shop in Paris, along with portable sundials and other popular scientific devices of the day.

Astronomical Instrument Detail by Ieremias Palladas 1612

Thirteen of his astrolabes survive to this day.[35] One more special example of craftsmanship in early 15th-century Europe is the astrolabe designed by Antonius de Pacento and made by Dominicus de Lanzano, dated 1420.[36]

In the 16th century, Johannes Stöffler published Elucidatio fabricae ususque astrolabii, a manual of the construction and use of the astrolabe. Four identical 16th-century astrolabes made by Georg Hartmann provide some of the earliest evidence for batch production by division of labor. In 1612, Greek painter Ieremias Palladas incorporated a sophisticated astrolabe in his painting depicting Catherine of Alexandria. The painting was entitled Catherine of Alexandria and featured a device called the System of the Universe (Σύστημα τοῦ Παντός). The device featured the planets with the names in Greek: Selene (moon), Hermes (Mercury), Aphrodite (Venus), Helios (Sun), Ares (Mars), Zeus (Jupiter), and Chronos (Saturn). The device also featured celestial spheres following the Ptolemaic model and Earth was depicted as a blue sphere with circles of geographic coordinates. A complex line representing the axis of the Earth covered the entire instrument.[37]

Medieval astrolabes

Astrolabes and clocks

Amerigo Vespucci observing the Southern Cross by looking over the top of an Armillary Sphere bizarrely held from the top like it's an astrolabe, however an astrolabe cannot be used by looking over the top of it. The page inexplicably contains the word "Astrolabium," by Jan Collaert II. Museum Plantin-Moretus, Antwerp, Belgium.

Mechanical astronomical clocks were initially influenced by the astrolabe; they could be seen in many ways as clockwork astrolabes designed to produce a continual display of the current position of the sun, stars, and planets. For example, Richard of Wallingford's clock (c. 1330) consisted essentially of a star map rotating behind a fixed rete, similar to that of an astrolabe.[38]

Many astronomical clocks use an astrolabe-style display, such as the famous clock at Prague, adopting a stereographic projection (see below) of the ecliptic plane. In recent times, astrolabe watches have become popular. For example, Swiss watchmaker Dr. Ludwig Oechslin designed and built an astrolabe wristwatch in conjunction with Ulysse Nardin in 1985.[39] Dutch watchmaker Christaan van der Klauuw also manufactures astrolabe watches today.[40]

Construction

An astrolabe consists of a disk, called the mater (mother), which is deep enough to hold one or more flat plates called tympans, or climates. A tympan is made for a specific latitude and is engraved with a stereographic projection of circles denoting azimuth and altitude and representing the portion of the celestial sphere above the local horizon. The rim of the mater is typically graduated into hours of time, degrees of arc, or both.[41]

Above the mater and tympan, the rete, a framework bearing a projection of the ecliptic plane and several pointers indicating the positions of the brightest stars, is free to rotate. These pointers are often just simple points, but depending on the skill of the craftsman can be very elaborate and artistic. There are examples of astrolabes with artistic pointers in the shape of balls, stars, snakes, hands, dogs' heads, and leaves, among others.[41] The names of the indicated stars were often engraved on the pointers in Arabic or Latin.[42] Some astrolabes have a narrow rule or label which rotates over the rete, and may be marked with a scale of declinations.

The rete, representing the sky, functions as a star chart. When it is rotated, the stars and the ecliptic move over the projection of the coordinates on the tympan. One complete rotation corresponds to the passage of a day. The astrolabe is, therefore, a predecessor of the modern planisphere.

On the back of the mater, there is often engraved a number of scales that are useful in the astrolabe's various applications. These vary from designer to designer, but might include curves for time conversions, a calendar for converting the day of the month to the sun's position on the ecliptic, trigonometric scales, and graduation of 360 degrees around the back edge. The alidade is attached to the back face. An alidade can be seen in the lower right illustration of the Persian astrolabe above. When the astrolabe is held vertically, the alidade can be rotated and the sun or a star sighted along its length, so that its altitude in degrees can be read ("taken") from the graduated edge of the astrolabe; hence the word's Greek roots: "astron" (ἄστρον) = star + "lab-" (λαβ-) = to take.

A shadow square also appears on the back of some astrolabes, developed by Muslim astrologists in the 9th Century, whereas devices of the Ancient Greek tradition featured only altitude scales on the back of the devices.[43] This was used to convert shadow lengths and the altitude of the sun, the uses of which were various from surveying to measuring inaccessible heights.[44]

Devices were usually signed by their maker with an inscription appearing on the back of the astrolabe, and if there was a patron of the object, their name would appear inscribed on the front, or in some cases, the name of the reigning sultan or the teacher of the astrolabist has also been found to appear inscribed in this place.[45] The date of the astrolabe's construction was often also signed, which has allowed historians to determine that these devices are the second oldest scientific instrument in the world. The inscriptions on astrolabes also allowed historians to conclude that astronomers tended to make their own astrolabes, but that many were also made to order and kept in stock to sell, suggesting there was some contemporary market for the devices.[45]

Parts of the astrolabe consisted of a circular disk, a sighting tube, an alidade, and arms that held graduated scales. The circular disk is the main part of the astrolabe. It was mainly used to view figures in the sky. The sighting tube is placed on top of the circular disk and was used to observe stars or planets. The alidade had vertical and horizontal cross-hairs which plots locations on an azimuthal ring called an almucantar (altitude-distance circle). An arm called a radius connects from the center of the astrolabe to the optical axis which is parallel with another arm also called a radius. The other radius contains graduations of altitude and distance measurements.

Construction of astrolabes
  • The Hartmann astrolabe in Yale collection. This instrument shows its rete and rule.
    The Hartmann astrolabe in Yale collection. This instrument shows its rete and rule.
  • Celestial Globe, Isfahan (?), Iran 1144. Shown at the Louvre Museum, this globe is the third oldest surviving in the world.
    Celestial Globe, Isfahan (?), Iran 1144. Shown at the Louvre Museum, this globe is the third oldest surviving in the world.
  • Computer-generated planispheric astrolabe
    Computer-generated planispheric astrolabe

Mathematical basis

The construction and design of astrolabes are based on the application of the stereographic projection of the celestial sphere. The point from which the projection is usually made is the South Pole. The plane onto which the projection is made is that of the Equator.[46]

Designing a tympanum through stereographic projection

Parts of an Astrolabe tympanum

The tympanum captures the celestial coordinate axes upon which the rete will rotate. It is the component that will enable the precise determination of a star's position at a specific time of day and year.

Therefore, it should project:

  1. The zenith, which will vary depending on the latitude of the astrolabe user.
  2. The horizon line and almucantar or circles parallel to the horizon, which will allow for the determination of a celestial body's altitude (from the horizon to the zenith).
  3. The celestial meridian (north-south meridian, passing through the zenith) and secondary meridians (circles intersecting the north-south meridian at the zenith), which will enable the measurement of azimuth for a celestial body.
  4. The three main circles of latitude (Capricorn, Equator, and Cancer) to determine the exact moments of solstices and equinoxes throughout the year.

The tropics and the equator define the tympanum

Stereographic projection of Earth's tropics and equator from the South Pole.

On the right side of the image:

  1.   The blue sphere represents the celestial sphere.
  2.   The blue arrow indicates the direction of true north (the North Star).
  3.   The central blue point represents Earth (the observer's location).
  4.   The geographic south of the celestial sphere acts as the projection pole.
  5.   The equatorial plane serves as the projection plane.
  6. Three parallel circles represent the projection in the sky of Earth's main circles of latitude:

When projecting onto the equatorial plane, three concentric circles correspond to Earth's three circles of latitude (left side of the image). The largest of these, the Tropic of Capricorn, defines the size of the astrolabe's tympanum. The center of the tympanum (and the center of the three circles) is actually the north-south axis around which Earth rotates, and therefore, the rete of the astrolabe will rotate around this point as the hours of the day pass (due to Earth's rotational motion).

The three concentric circles on the tympanum are useful for determining the exact moments of solstices and equinoxes throughout the year: if one knows the sun's position on the rete and its position coincides with the outer circle of the tympanum (Tropic of Capricorn), it signifies the winter solstice (the sun will be at the zenith for an observer at the Tropic of Capricorn, meaning summer in the southern hemisphere and winter in the northern hemisphere). If, on the other hand, its position coincides with the inner circle (Tropic of Cancer), it indicates the summer solstice. If its position is on the middle circle (equator), it corresponds to one of the two equinoxes.

The horizon and the measurement of altitude

Stereographic projection of an observer's horizon at a specific latitude

On the right side of the first image:

  1.   The blue arrow indicates the direction of true north (the North Star).
  2.   The central blue point represents Earth (the observer's location).
  3.   The black arrow represents the zenith direction for the observer (which would vary depending on the observer's latitude).
  4.   The black circle represents the horizon surrounding the observer, which is perpendicular to the zenith vector and defines the portion of the celestial sphere visible to the observer.
  5.   The geographic south of the celestial sphere acts as the projection pole.
  6.   The equatorial plane serves as the projection plane.
Stereographic projection of the horizon and an almucantar.

When projecting the horizon onto the equatorial plane, it transforms into an upward-shifted ellipse relative to the north-south axis (the center of the tympanum). This implies that a portion of the celestial sphere will fall outside the outer circle of the tympanum (the Tropic of Capricorn) and, therefore, won't be represented.

Additionally, when drawing circles parallel to the horizon up to the zenith (almucantar), a grid of consecutive ellipses is constructed, allowing for the determination of a star's altitude when the rete overlaps with the designed tympanum (as observed in the second image).

The meridians and the measurement of azimuth

Stereographic projection of the north-south meridian and a meridian 40° E on the tympanum of an astrolabe

On the right side of the image:

  1.   The blue arrow indicates the direction of true north (the North Star).
  2.   The central blue point represents Earth (the observer's location).
  3.   The black arrow represents the zenith direction for the observer (which would vary depending on the observer's latitude).
  4.   The black circle represents the horizon surrounding the observer, which is perpendicular to the zenith vector and defines the portion of the celestial sphere visible to the observer.
  5.   The red dots represent the zenith and nadir (the point on the celestial sphere opposite the zenith with respect to the observer).
  6.   The orange circle represents the celestial meridian (or meridian that goes, for the observer, from the north of the horizon to the south of the horizon passing through the zenith).
  7.   The red circle represents a secondary meridian with an azimuth of 40° East relative to the observer's horizon, and like all secondary meridians, it intersects the principal meridian at the zenith and nadir.
  8.   The geographic south of the celestial sphere acts as the projection pole.
  9.   The equatorial plane serves as the projection plane.

When projecting the celestial meridian, it results in a straight line that overlaps with the vertical axis of the tympanum, where the zenith and nadir are located. However, when projecting the 40° E meridian, another circle is obtained that passes through both the zenith and nadir projections, so its center is located on the perpendicular bisection of the segment connecting both points. In deed, the projection of the celestial meridian can be considered as a circle with an infinite radius (a straight line) whose center is on this bisection and at an infinite distance from these two points.

If successive meridians that divide the celestial sphere into equal sectors (like "orange slices" radiating from the zenith) are projected, a family of curves passing through the zenith projection on the tympanum is obtained. These curves, once overlaid with the rete containing the major stars, allow for determining the azimuth of a star located on the rete and rotated for a specific time of day.

See also

References

Footnotes
  1. ^ Modern editions of John Philoponus' treatise on the astrolabe are De usu astrolabii eiusque constructione libellus (On the Use and Construction of the Astrolabe), ed. Heinrich Hase, Bonn: E. Weber, 1839, OCLC 165707441 (or id. Rheinisches Museum für Philologie 6 (1839): 127–71); repr. and translated into French by Alain Philippe Segonds, Jean Philopon, traité de l'astrolabe, Paris: Librairie Alain Brieux, 1981, OCLC 10467740; and translated into English by H.W. Green in R.T. Gunther, The Astrolabes of the World, Vol. 1/2, Oxford, 1932, OL 18840299M repr. London: Holland Press, 1976, OL 14132393M pp. 61–81.
  2. ^ O'Leary, De Lacy (1948). How Greek Science Passed to the Arabs. Routledge and Kegan Paul. "The most distinguished Syriac scholar of this later period was Severus Sebokht (d. 666–7), Bishop of Kennesrin. [...] In addition to these works [...] he also wrote on astronomical subjects (Brit. Mus. Add. 14538), and composed a treatise on the astronomical instrument known as the astrolabe, which has been edited and published by F. Nau (Paris, 1899)."
    Severus' treatise was translated by Jessie Payne Smith Margoliouth in R.T. Gunther, Astrolabes of the World, Oxford, 1932, pp. 82–103.
  3. ^ Savage-Smith, Emilie (1993). "Book Reviews". Journal of Islamic Studies. 4 (2): 296–299. doi:10.1093/jis/4.2.296. There is no evidence for the Hellenistic origin of the spherical astrolabe, but rather evidence so far available suggests that it may have been an early but distinctly Islamic development with no Greek antecedents.
Notes
  1. ^ Greenwood, William (January 28, 2018). "The British Museum: How to Use an Astrolabe". YouTube. Archived from the original on 2021-11-07.
  2. ^ "Astrolabe". Oxford English Dictionary (2nd ed.). 1989.
  3. ^ "Astrolabe". Oxford Dictionaries. Archived from the original on October 22, 2013.
  4. ^ "Online Etymology Dictionary". Etymonline.com. Retrieved 2013-11-07.
  5. ^ a b King 1981, p. 44.
  6. ^ King 1981, p. 51.
  7. ^ King 1981, p. 45.
  8. ^ Berlin, Nicole (June 2019). "Mixed-Media Domestic Ensembles in Roman Sicily: The House of Leda at Soluntum". Arts. 8 (2): 62. doi:10.3390/arts8020062. ISSN 2076-0752.
  9. ^ Westgate, Ruth (2000). "Pavimenta atque emblemata vermiculata: Regional Styles in Hellenistic Mosaic and the First Mosaics at Pompeii". American Journal of Archaeology. 104 (2): 255–275. doi:10.2307/507451. ISSN 0002-9114. JSTOR 507451. S2CID 194101486.
  10. ^ Wolf 2003, pp. 64–69.
  11. ^ Lewis 2001.
  12. ^ Michael Deakin (August 3, 1997). "Ockham's Razor: Hypatia of Alexandria". ABC Radio. Retrieved July 10, 2014.
  13. ^ a b c d Theodore, Jonathan (2016). The Modern Cultural Myth of the Decline and Fall of the Roman Empire. Manchester, England: Palgrave, Macmillan. p. 183. ISBN 978-1-137-56997-4.
  14. ^ a b c d Deakin, Michael A. B. (2007). Hypatia of Alexandria: Mathematician and Martyr. Amherst, New York: Prometheus Books. pp. 102–104. ISBN 978-1-59102-520-7.
  15. ^ a b c d Bradley, Michael John (2006). The Birth of Mathematics: Ancient Times to 1300. New York City, New York: Infobase Publishing. p. 63. ISBN 9780816054237.
  16. ^ Sebokht, Severus. "Description of the astrolabe". Tertullian.org.
  17. ^ See p. 289 of Martin, L. C. (1923), "Surveying and navigational instruments from the historical standpoint", Transactions of the Optical Society, 24 (5): 289–303, Bibcode:1923TrOS...24..289M, doi:10.1088/1475-4878/24/5/302, ISSN 1475-4878.
  18. ^ Berggren, J. Lennart (2007), "Mathematics in Medieval Islam", in Katz, Victor J. (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: a Sourcebook, Princeton University Press, p. 519, ISBN 978-0-691-11485-9
  19. ^ Richard Nelson Frye: Golden Age of Persia. p. 163
  20. ^ Nizamoglu, Cem (2005-08-10). "Using an Astrolabe". Muslim Heritage. Retrieved 2023-10-16.
  21. ^ Lachièz-Rey, Marc; Luminet, Jean-Pierre (2001). Celestial Treasury: From the Music of Spheres to the Conquest of Space. Translated by Joe Laredo. Cambridge, United Kingdom: Cambridge University Press. p. 74. ISBN 978-0-521-80040-2.
  22. ^ O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics Archive, University of St Andrews
  23. ^ Bedini, Silvio A.; Maddison, Francis R. (1966). "Mechanical Universe: The Astrarium of Giovanni de' Dondi". Transactions of the American Philosophical Society. 56 (5): 1–69. doi:10.2307/1006002. JSTOR 1006002.
  24. ^ "Qantara – 'Carolingian' astrolabe". Qantara-med.org. Retrieved 2013-11-07.
  25. ^ Nancy Marie Brown (2010), "The Abacus and the Cross". p. 140. Basic Books. ISBN 978-0-465-00950-3
  26. ^ Boyle, David (2011). Toward the Setting Sun: Columbus, Cabot, Vespucci, and the Race for America. Bloomsbury Publishing USA. p. 253. ISBN 9780802779786..
  27. ^ Northrup, Cynthia Clark, ed. (2015). Encyclopedia of world trade: from ancient times to the present (Enhanced Credo ed.). Armonk, New York: Routledge. pp. 72. ISBN 978-0765680587. OCLC 889717964.
  28. ^ "Introduction". The Astrolabe: an Online Resource. 2006. Retrieved 2020-05-15.
  29. ^ Harley, J. B.; Woodward, David (1992). The history of cartography. Chicago, Illinois: University of Chicago Press. p. 31. ISBN 0-226-31635-1.
  30. ^ Kunitzsch, Paul (1981). "On the authenticity of the treatise on the composition and use of the astrolabe ascribed to Messahalla". Archives Internationales d'Histoire des Sciences Oxford. 31 (106): 42–62.
  31. ^ Selin, Helaine (2008-03-12). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer Science & Business Media. p. 1335. ISBN 978-1-4020-4559-2. Paul Kunitzsch has recently established that the Latin treatise on the astrolabe long ascribed to Ma'sh'allah and translated by John of Seville is in fact by Ibn al-Saffar, a disciple of Maslama al-Majriti.
  32. ^ Glick, Thomas; et al., eds. (2005), Medieval Science, Technology, and Medicine: An Encyclopedia, Routledge, p. 464, ISBN 0-415-96930-1
  33. ^ Northrup, Cynthia Clark, ed. (2015). Encyclopedia of world trade : from ancient times to the present ([Enhanced Credo edition] ed.). Armonk, New York: Routledge. pp. 460. ISBN 978-0765680587. OCLC 889717964.
  34. ^ Nancy Marie Brown (2010), "The Abacus and the Cross". p. 143. basic Books. ISBN 978-0-465-00950-3
  35. ^ Hockey, Thomas (2009). The Biographical Encyclopedia of Astronomers. Springer Publishing. ISBN 978-0-387-31022-0. Retrieved August 22, 2012.
  36. ^ Ralf Kern (2010), Wissenschaftliche Instrumente in ihrer Zeit. Band 1: Vom Astrolab zum mathematischen Besteck. Cologne, S. 204. ISBN 978-3-86560-865-9
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