Ḥabash Al-Ḥāsib, Aḥmad Ibn ‘Abdallāh Al-Marwazī

(b. Marw, Turkestan [now Mary, Turkmen S.S.R.]; d.864–874)

trignometry, astronomy.

Little is known of Ḥabash’s life and family. He worked at Baghdad as astronomer under the ’Abbāsid caliphs al-Ma’mūn and al-Mu’taṣim, but he may not have belonged to the small group that collaborated in the Mumtaḥan observations. He made observations from 825 to 835 in Baghdad. Abū Ja’far ibn Ḥabash, the son of Ḥabash, was also a distinguished astronomer and an instrument maker.

Works . The biographers Ibn al-Nadīm, Ibn al-Qifṭī, and Ḥājjī Khalīfa ascribe the following works to Ḥabash:

1. A reworking of the Sindhind.
2. The Mumtahan Zīj, the best known of his works, which relies on Ptolemy and is based on his own observations. Ibn Yūnus called it al-Qānūn (“The Canon”).
3. The Shāh Zīg, the shortest of his ziyajāt.
4. The Damascene Zīj.
5. The Ma’mūnī Zīj (or Arabic Zīj). This and the Damascene Zīj are based on the Hijra calendar rather than on the Yazdigird r Seleucid eras.
6. On the Rukhāmā and Measurements.
7. On the Celestial Spheres.
8. On Astrolabes.
9. On the Oblique and Perpendicular Planes.
10. On the Distances of the Stars.

Since not all of these works are extant, it is almost impossible to determine how many ziyajāt Ḥabash wrote and their titles. Two manuscripts on the tables of Ḥabash are preserved, one in Istanbul (Yeni Cami, no. 784) and the other in Berlin (no. 5750). These are not copies of his original works. There has been criticism of the Yeni Cami copy, suggesting that it is a revision of Ḥabash’s zīj by Kūshyār ibn Labban. In one way or another the introduction and the passages have come to us in their original forms and can be used, as can the Berlin manuscript, as the sources on Ḥabash.

Trigonometry . Ḥabash’s trigonometric contributions are very important.

Sines. In the Sūrya-Siddhānta(a.d. 400) a table of half chords is given. A special name for the function which we call the sine is first found in the works of Āryabhaṭa I (a.d. 500). Besides half chord he also uses the term jya or jiva. In the Islamic world this word was transcribed as jayb. Al-Khwārizmī(ca. 825) was the first to prepare a table of sines. Ḥabash followed him by constructing such a table for

θ = 0; 0°,0; 15°,0; 30°,0; 45°, 1;0° … 90; 0°.

Versed sine. Among the trigonometric functions the versed sine (versine) also attracted attention. We know that it was mentioned in the Sūrya-Siddhānta, and a table for the versed sine is given in Āryabhaṭa. In Islam astronomers used special names to distinguish the versed sine, such as jayb ma’kūs (used by Ḥabash), jayb mankus (used by al-Khwāirizmī).and sahm. Ḥabash may be the first who clearly defined the sine and the versed sine as follows: “A perpendicular from the circumference to the diameter is the sine (jayb mabsūṭ) of the arc between the diameter and the perpendicular; the distance between the circumference and the perpendicular upon the diameter is the versed sine (jayb ma´kūs) of the above mentioned arc;” He showed that if A<90°, the versed sine = 60^P– cos A = 1 – cos A; and if A>90°, the Versed sine = 60^p + cos A= 1 + cos A. Also, if A< 90°, the versed sine < sine; if A > 90°, the versed sine>sine: and if A = 90°, the versed sine = sine.

Tangent. The Sūrya-Siddhānta and other Hindu works mention the shadows, particularly in connection with astronomy. Ḥabash seems to have been the first to compile a table of tangents for

θ = 0; 0°,0; 30°, 1;0° … 90; 0°.

The function of umbra extensa (the length of shadow) is defined as

p = the length of gnomon. For the computation of the umbra extensa from the altitude of the sun, he gives the following steps (see Figure 1):

In addition to finding the umbra extensa from the altitude of the sun, Ḥabash presents the following equations:

Spherical Astonomy, For the solution of problems in spherical astronomy, transformations of coordinates, time measurements, and many other problems, Ḥabash gives astronomical tables of functions which are standard for all ziyajāt.

He gives the general rule for calculating the declination of the sun (the first declination, al-mayl alawwal (see Figure 2);

The declination depends not only on λ but also on the value of the obliquity of the ecliptic.

The culmination of the sun is defined. (See Figure 3.) If the declination of the sun is northern, h = (90 – φ) + δ⊙. If the declination of the sun is southern, h = (90 – φ) – δ⊙

The time of day, measured from sunrise, is proportional to the altitude of the sun, i.e., the “are of revolution” (al-dā’ir min al-falak). Islamic astronomers gave many trigonometric functions showing the relations between the time and the altitude of the sun. The first exact solution was given by Ḥabash and proved by Abu’l-Wafā’ and al-Bīrūnī. This function

is equivalent to the function given by Brahmagupta in his Khaṇḍakhādyaka:

where P = half of the length of daylight

h = altitude of the sun

t = time

vers P = day sin (jayb al-nahār; Sanskrit, antyā).

Then he computes the altitude of the sun from the time:

Ḥabash calculates the length of daylight, i.e., equation of daylight (ta ‘dīl al-nahār), which al-khwārizmī calls the ascensional difference. (See Figure 4).

He shows that if the declination of the sun is northern, the length of daylight = the equation of daylight + 90°; and if the declination of the sun is southern, the length of daylight = 90° – the equation of daylight. The equation of daylight is tabulated for the planets, the sun, and the moon. With the aid of this table one can easily find the are of daylight.

The ascensions (maṭāli‘ al-burūj) or rising times in the right sphere (al-falak al-mustanqīm), i.e., right ascension, is defined (see Figure 5) as

Ḥabash prepared such tables because of the right ascension’s astrological importance.

The ascension for a particular latitude is called the oblique ascension. Ḥabash showed that if the arbitrary point P on the ecliptic is between the vernal and autumnal equinoxes, the right ascension – ½ equation of daylight = oblique ascension; and if it is between the autumnal and vernal equinoxes, the right ascension + ½ equation of daylight = oblique ascension.

Ḥabash prepared tables for the seven climates. According to him the first climate (iqlīm) was that portion of the northern hemisphere in which

13 ≤ max. D ≥ 13 – 0.50, ie., a band of a half-hour advance in length of daylight.

For finding the ortive amplitude (jayb al-mashriq) Ḥabash gives the following function (see Figure 6);

Astronomy. Ḥabash generally follows Ptolemy, but some sections of his work are distinctly non-Ptolemaic.

Theory of the Sun. Ḥabash compiled tables of mean motion of the sun for 1, 31, 61, 91, … 691 and 1, 2, 3 , 4, 5,. . . 30 Hijra years; for 10, 20, 30, . . . 60 minutes. The mean motion of the sun is 384;55, 14° per Hijra year and 0;59, 8° per day (the value given in the Almagest). He computed the eccentricity of the sun 2^p 5¹ (2; 1°).

Ḥabash divided half of the ecliptic into eighteen parts, each part called Kardaja. The Arabic-Persian term Kardaja (pl., Kardajāt) is usually derived from the Sanskrit Kramajya. It seems to have stood for a unit length of arc. He also prepared equation tables of the sun (ta´dīl al-shams) for each degree of anomaly.

Methods for the calculation of the equation of the sun were given by Ḥabash. This classical procedure was given in the Almagest and followed by the Islamic astronomers. If the mean motion or anomaly is given, to find λe, i.e., the true motion of the sun (see Figure 7):

The converse of the problem, i.e., given λ, determine , is set out. This equation gives the following approximate solution (see Figure 8);

Since the angle λe is small, Ḥabash supposed BD = BC, i.e.., 60^p:

Ḥabash gave another rule to solve the above problem:

This function is correct, but the independent variable is λ rather than . If replaces λ, this Will lead to the equation λ = λ – e sin A, which is known as Kepler’s equation. The equivalent of this equation is found in Tamil astronomy in south India.

From the mean position Ḥabash found the true position of the sun (see Figure 9)

If ⊙ > λ A

and ⊙ – λA < 90°,

anomaly ā = ⊙ – λA

where ⊙ = mean longitude of the sun

and λ A = longitude of the apogee.

Thus,

and ⊙ – λe = ture position of the sun.

He also computed from the true position the mean position of the sun (see Figure 10). The true position B, i.e., λB, is given:

By applying these methods, Ḥabash calculated the

entrance of the sun into the zodiacal signs and compiled tables for them.

Lunar Theory. Ḥabash constructed several tables for the longitudinal and latitudinal motions of the moon fat’ periods of thirty lunar years, for years, for months, for days (13;10.35°, the value given in the Almagest), for hours (0; 32, 27°), and for fractions of hours. He also drew up tables for general lunar anomaly and for the equation of the moon (ta‘ dīl al-qamar) in four columns.

Ḥabash’s technique for computing the true longitude of the moon was based on the model of the lunar motion given by Ptolemy in book V of the Almagest. The most essential deviation from the previous tables consisted in the arrangement of all corrections of the mean positions so that they are never negative. As Neugebauer remarks, this constituted a great practical advantage over the Ptolemaic method. (See Figure 11.)

The true place of the moon is determined as follows:

M = movable center of eccenter

δ₀ = apparent radius of epicyle when in apogee of eccenter

G = “true apogee” of epicycle

F = “mean apogee” of epicycle

W₀ = maximum angular distance possible between F and G

F₀ = point on the epicycle such that F = W₀)

ā = “mean anomaly” of moon counted from F₀

W = angular distance between G and E₀

a = ā+ W₁ “true anomaly” counted from G

λ = “true longitude” of the moon counted from γ₀

The “first correction,” W₁:

The function of K is tabulated in the first column of the table, called “equation of the moon (ta‘ dīl al-qamar).” It gives the distance from G to F₀, the value given in Almagest V, except that Ḥabash had added W₀, which makes W nonnegative.

ā + the first equation = a.

The “second correction,” W₂, is a function of a, tabulated in the third column. It corresponds in value to Almagest V, 8, col. 4, but the maximum equation δ₀) is added to it. It is assumed that the epicycle is located at the apogee of the eccenter. When it is in the perigee of the eccenter, the amount of the excess of the epicyclic equation is tabulated in the fourth column of the table. This is the function of K (corresponding to Almagest V, 8, col. 5); the result will be

obtained by multiplying the value of the fourth column by the second: δ=W₂ +μγ

If a< 180°, the result will be subtracted from or added to the true center. Finally Ḥabash obtains

Latitude of the Moon. The latitude of the moon for a given moment is determined by means of a table prepared for one degree. The true place of the moon is added to the mean position of the ascending node . Because of the longitude of the ascending node, the distance of the node from γ₀ is counted in a negative direction. This total, A, is the argument with which the table of the latitude of the moon is entered.

Theory of the Planets. For finding the longitudes of the planets Ḥabash prepared several tables for mean motions, in longitude and latitude, and the equations. His procedure for finding the true longitude of a planet for a given moment t is based on the Ptolemaic method (Almagest XI).

For outer planets (see Figure 13):

where λ⊙ = mean longitude of the sun

λ = mean longitude of the planet

ā = anomaly or argumentum.

It implies that the radius of the planet on the epicycle is always parallel to the direction from 0 to the mean sun.

Inner Planets. For inner planets and the anomaly can be found from the tables. (See Figure 14.)

For the outer planets he first found the mean longitude , mean anomaly a, and the longitude of the apogee λ A of the planet (see Figure 15):

He found the distance of the center C of the epicycle from the apogee for the given moment. According to the Ptolemaic planetary theory, the planet makes its regular motion not around the point O but around E. i.e., the “equant.” The center of the deferent

falls between O and E. Then he found the epicyclic equation as seen from O. Thus the true anomaly, a, counted from true apogee, is found. Ḥabash tabulated this difference. W₁ = a-ā, as function , in the first column. This is called the “first correction”: a = W₁ + ā.

Then he computed the distance of C from A, i.e., K, as seen from A. This difference is also equal to the angle W₁:

If

Then comes the “second correction,” W₂. This depends not only on the true anomaly a but also on the position of the epicycle. If it is exactly in the apogee, this amount of correction will be less than W₂ by the amount μ A tabulated in column 4 as the function a:

γ is found as a function of K in the second column.

All these procedures are correct only if the value found in the second column is negative. If it is positive, the second column is multiplied by the value found in the fifth column, then subtracted from the fourth. The true longitude of the planet is

Latitude of the Planets. The procedure for calculating the latitude of the superior planets is based on Ptolemaic method (Almagest XIII, 6). The table of latitudes was prepared in three columns. Ḥabash used the same numeric values as Ptolemy (Almagest XIII, table 5). According to him, the latitudes can be found by the addition of two components: the inclination of the epicycle about its second diameter (β₁) and the angle at the line of nodes between the deferent and ecliptic planes (β₂).

The first and second columns are functions of anomaly a : b₁(a) and b²(a). The third column is the function θ.

The latitude of the planet β = β₁ + β₂

For the inferior planets (based on Ptolemaic method Almagest XIII, 6), one enters the latitude table with the truly determined anomaly and records the corresponding numbers in the first and the second columns. These are the functions of a : b₁(a) and b₂(a).

One finds the determined true longitude of the planets. For Venus, A = λ – the longitude of apogee. For Mercury, if the determined true anomaly is in the first fifteen rows,

A = λ – 10°;

if it is in those that follow,

A = λ + 10°.

Next, A + 90° = θ for Venus

A + 270° = θ for Mercury

One enters the table with that value and finds the corresponding number in the third column. This is the function c (θ)

Then b₁.c = the first latitude = β₁.

If θ is in the first fifteen rows, the planet is northern. If it is in those that follow, it is southern. If θ is after the first fifteen rows at the same time that a is in the first fifteen rows, the planet is northern.

Next, one enters the latitude table with

θ for Venus

θ + 180° for Mercury

and finds the corresponding value in the third column. This is the function of θ or θ+180°: c (θ) or c (θ+180°). Then b₁c = the second latitude = β₂. If (θ) or (θ+180°) is in the first fifteen rows and a< 180° the planet has a northern latitude. If a> 180°, it has a southern latitude. If (θ) or (θ +180°) is below the first fifteen rows and a< 180° the planet has a southern latitude; and if a> 180° the planet has a northern latitude. Then

c² + C^2/6 = β₃ for Venus. If the planet has northern latitude, β = β₂ + β₃ and C² = 3C^2/4 = β₃ for Mercury. If the planet has southern latitude β = β₂ + β₃.

Parallax (Ikhtilāf al-Manẓar) Theory. Ḥabash had two entirely different methods for determining the parallax, i.e., parallax in longitude P_λ and parallax in latitude P_β. One of them may seem a transition between that of Ptolemy and the later Islamic astronomers. This solution depends on the first sine (al-jayb al-awwal) that can be formulated (see Figure 16):

equal cos B and the second sine, which is equal to sin B. Without proof he states that

P_am is measured according to Ptolemy (see Figure 17):

The other method seems to derive from the Sūrya Siddhanta.

The technique for determining the longitude component is of great interest. Ḥabash first determined t (see Figure 18), then used it as argument in the parallax table. The result was called the first parallax. He added this to t and with that value entered the parallax table. The result was the second parallax. These operations were repeated until the fifth parallax—a quarter of the parallax in longitude, expressed in hours.

For finding the lunar parallax in latitude Ḥabash

used A as an argument, and the corresponding value of the function was to be doubled (see Figure 18). This would be the lunar parallax in latitude.

Visibility Theory (Ru’yat al-Hilāl). Ḥabash may have been the first astronomer to engage in the computation of the new crescent. Like the ancient Babylonians and the Jews, the Muslims depended on visual observation of the new crescent for their religious and secular calendars. This led the Muslim astronomers to realize that the knowledge of the visibility of the new crescent is an essential task of astronomy. Ḥabash used the following method for the determination of the visibility of the new crescent. He added twenty or thirty minutes to the time of sunset, thus obtaining the mean position of the moon at the time when the new crescent becomes visible. Then the true position of the sun, the moon, and the head were needed for the above-mentioned time (see Figure 19). Thus. λ-λ⊙ =λ₁, At which Maimonides called the first elongation.

For an observer on the surface of the earth, the moon M would appear in a lower position M₁ because of the parallax.

Then the parallax in latitude P_β and longitude P_λ can be obtained (see Figure 20). Thus, λ_l- Pλ=λ₂, which Maimonides called the second elongation. The

true latitude of the moon (Maimonides called it the first latitude) was subtracted from or added to the parallax in latitude, depending on the variable position of the moon:

From this second latitude one can derive half of the day arc of the moon, and from that the equation or the day of the moon is obtained, This equation of the moon is added to or subtracted from the longitude of the moon. Thus the point of the ecliptic O (see Figure 21), which sets simultaneously with the moon, is obtained: λ₂ -C=λ₃O.

Then the arc of the equator QA, which sets simultaneously with the arc of the ecliptic, λ₃, is calculated

(see Figure 22). This is the difference between the rising times of the moon and the sun. This time difference is multiplied by the surplus of the moon in one hour and divided by fifteen. The result K is

added to the true longitude of the moon, i.e., the distance cut by the “moon during that time is added to get the distance between the moon and the sun at sunset. Then (see Figure 23)

If QR > 10°, the moon will be visible on that day. If QR < 10°, the moon will not be visible.

BIBLIOGRAPHY

The following works may be consulted for further information: A. Braunmühl. Vorlesungen über Geschichte der Trigonometrie (Leipzig. 1900); G. Caussin. Le livre de la grande Hakèmite, vol. VII of Notices et Extraits des MSS (Paris, 1804); S. Gaudz, J. Obemann, and O. Neugebauer. The Code of Maimonides. Book Three. Treatise Eight; Sanctification of the New Moon (New Haven, 1956); J. Hamadanizadeh, “A Medieval Interpolation Scheme for Oblique Ascensions,” in Centaurus, 9 (1963). 257–265; E. S. Kennedy, “An Islamic Computer for Planetary Latitudes,” in Journal of the American Oriental Society, 71 (1951), 12–21: and “Parallax Theory in Islamic Astronomy,” in Isis, 47 (1956), 33–53; E. S. Kennedy and M. Agna, “Planetary Visibility Tables in Islamic Astronomy,” in Centaurus, 7 (1960), 134–140; E. S. Kennedy and Janjanian, “The Crescent Visibility Table in Al-Khwālrizmīs Zij,” ibid., 11 (1965), 73–78; E. S. Kennedy and Ahmad Muruwwa, “Bīrūnī the Solar Equation,” in Journal of Near Eastern Studies, 17 (1958), 112–121; E. S. Kennedy and Sharkas, “Two Medieval Methods for Determining the Obliquity of the Ecliptic,” in Mathematics Teacher, 55 (1962), 286–290; E. S. Kennedy and W. R. Transue, “A Medieval Iterative Algorism.” in American Mathematical Monthly, 63 , no. 2 (1956), 80–83; Hājjī Khalīfa, Kashf al-Zunūn, S. Yaltkaya, ed., 2 vols. (Istanbul, 1941–1943): E. Kramer, The Main Stream of Mathematics (New York, 1951); Ibn al-Nadīm, Fihrist, FIūgel, ed., I (1871): N. Nadir, “Abūl-Waāfā’ on the Solar Altitude,” in Mathematics Teacher, 53 (1960), 460–463; C. A. Nallino, Al-Battānī Opus astronomicum, 3 vols. (Brera, 1899–1907). see vols; I and III; O. Neugebauer, “The Transmission of Planetary Theories in Ancient and Medieval Astronomy.” in Scripta mathematica, 22 (l956); “Studies in Byzantine Astronomical Terminology,” in Transactions of the American Philosophical Society, 50 (1960); “The Astronomical Tables of Al-Khwātizmī,” in Hist. Filos. Skrifter. Danske Videnskabernes Selskab, 4 , no. 2 (1962); and “Thābit ben Qurra ’On the Solar Year’ and ’On the Motion of the Eighth Sphere,’” in Transaction of the American Philosophical Society, 106 (1962), 264–299: Ibnal-Qiftī, Ta’rīkhalhukamā, Lippert, ed. (Berlin. 1903); G. Sarton, Introduction to the History Science, I (Baltimore, 1927), 545, 550, 565, 667; A. Sayili, “Habeş el Hasib’in ’El Dimişki’ adiyla MarufZîci’nin Mukaddemesi,” in Ankara üniversitesi dil ve tarih-coğrafya fakültesi dergisi, 13 (1955), 133–151; C. Schoy, “Beitrāge zur arabischen Trigonometrie,” in Isis, 5 (1923), 364–399: D. E. Smith, History of Mathematics, II (London, 1925); and H. Suter, Die Mathematiker and Asrtranomen der Araber and ihre Werke (Leipzig, 1900).

S. Tekeli