## Muḥyi

## Muḥyi

# MUḥYI ’L-DīN AL-MAGHRIBī (MUḥYI ’I-MILLA WA ’L-DīN YAḥYā IBN MUḥAMMAD IBN ABI ’I-SHUKR AL-MAGHRIBī’ AL-ANDALUSī)

(*fl*. Syria, and later Marāgha, *ca*. 1260–1265)

*trigonometry, astronomy, astrology*.

Al-Maghribī was a Hispano-Muslim mathematician and astronomer, whose time and place of birth and death cannot be determined. Little is known about his life except that he was born in the Islamic West and flourished for a time in Syria and later in Marāgha, where he joined the astronomers of the Marāgha directed by Nạṣir al-Dīn al-Ṭūsī. He made observations in 1264–1265. It has been said that he was a guest of Hūlāgū Khān (īl-khān of Persia, 1256–1265) and met Abu ’1 Faraj (Bar Hebraeus, 1226–1286).

Suter and Brockelmann ascribe quite a long list of writings to al-Maghribī.

Trigonometry

1. *Kit̄b shakl al-qạ̣̄’* (“Book on the Theorem of Menelaus”).

2.*Ma yanfari’u ’an shakl al-qạ̣̣̄* (“Consequences Deduced From shakl al-qạ̣̄’”).

3.Ris̄la f̄ kayfiyyat istikhr̄j al-juȳb al-w̄qi’a fi͗l-d̄’ira (“Treatise on the Calculation of Sines”). Astronomy

4. Khul̄ṣat al-Majiṣṭ̄(“Essence of the *Almagest*”). It contains a new determination of the obliquity of the ecliptic made at Mar̄gha in 1264, 23; 30° (the real value in 1250 was 23; 32, 19°).

5. Maq̄la f̄ istikhr̄j ta ’d̄l al-nah̄r wa sa ’at al-mashriq wa ͗l-d̄͗ir min al-falak (“Treatise on Finding the Meridian, Ortive Amplitude, and Revolution of the Sphere”).

6. Muqaddam̄t tata’allaqu b̄-̣arak̄t al-kaw̄kib (“Premises on the Motions of the Stars”).

7. Taṣ̣̄ al-ạ̣url̄b (“The Flattening of the Astrolabe”).

Editions of the Greek classics; they are called recensions (sing. *taḥrīir*).

8. Euclid’s *Elements*.

9. Apollonius’ *Conics*.

10. Theodosius’ *Spherics*.

11. Menelaus’ *Spherics*.

He also wrote more than six books on astrology and a memoir on chronology.

Al-Maghrib̄’s writings on trigonometry contain original developments. For example, two proofs are given of the sine theory for right-angled spherical triangles, and one of them is different from those given by Ṇ̄ir al-D̄n al-̣̄s̄ this theorem is generalized for other triangles. He also worked in several other branches of trigonometry.

Ptolemy (A.D 150) used an ingenious method of interpolation in the calculation of chord 1°. This is of course approximately equivalent to chord 1°. The same method was used for sines in Islam. To find the exact value, one must solve a cubic equation. This was done later by the Persian astronomer al-Kāshī (*d*. 1429/1430), Al-Maghribī, and before him Abu ’l-Wafā’ (940–997/998),tried to find the value of the sine of one-third of an arc. For that purpose Abu ’l-Wafā’ laid down a preliminary theorem that the differences of sines of arcs having the same origin and equal differences become smaller as the arcs become larger.

Using this preliminary theorem, al-Maghribī calculated sin 1° in the following way (see Fig. 1):

*VF* = 1; 7, 30° and sin *VF* = *FK* = 1; 10, 40, 12, 34p

*AV* = 0; 45° and sin *AV* = *AI* = 0; 44, 8, 21, 8, 38^{p}.

The arc *AF* is divided into six equal parts and each part = 0; 3, 45° therefore,

arc *DV* + arc *DH* = 1° and sin *HV*(=1°) = *HZ*.

The perpendiculars *AT*, *BY*, and *CK* divide *DT* into three unequal parts: *TY* > *YK* > *DK*; *TD/3* > *HL*; *DQ* + *TD/3* (=1; 2, 49, 43, 36, 9^{p}) > *HZ*(=sin 1°). *FM* is divided into three unequal parts: *MN* > *NS* > *SF*; *DQ* + *FM/3* (=1; 2, 49, 42, 50, 40, 40^{p}) *NK* = *HZ* (=sin 1°). Then he found sin 1° =1; 2, 49, 43, 24, 55p.

Al-Maghribī calculated sin 1° by using another method of interpolation based on the ratio of arcs greater than the ratios of sines. He found sin 1° = 1; 2, 49, 42, 17, 15, 12p and said that the difference between two values of sines found by using different methods is 0; 0, 0, 0, 56p, which is correct to four places.

Using these methods, al-Maghribī calculated the ratio of the circumference to its diameter (that is, π).

*AC*(=2*AT*)< are *ABC* < *RF*

sin *AB*(=3/4°) = *AT* = 0; 47, 7, 21, 7, 37p

Δ*RFH* ~ Δ*AHC*, *RF*/*AC* = *BH*/*TH*.

*RF* = 1; 34, 15, 11, 19, 25p

arc *ABC* =
= 1; 34, 14, 16, 47, 19, 30p.

The circumference = 240. Arc *AB* = 6; 16, 59, 47, 18p, the diameter being 2p. The diameter being lp, the circumference 3; 8, 29, 53, 34, 39p < 3*R* + 1/7, since 1/7 = 0; 8, 34, 17, 8, 34, 17p.

Al-Maghribī compared the latter and Archimedes’ value, 3*R* + 1/7 < the circumference < 3*R* + 10/71, found by computing the lengths of inscribed and circumscribed regular polygons of ninety-six sides. Half of the difference between 10/71 and 10/70 is equal to 0; 8, 30, 40p.

Al-Maghribī determined two mean proportionals between two lines, that is, the duplication of the cube (the problem of Delos). In antiquity many solutions were produced for this problem. It was thought that in terms of solving this problem the mathematicians of Islam stood strangely apart from those of antiquity; but recently many examples have been discovered, thus altering this opinion. The following example of Al-Maghribī’s is of interest in this respect. He finds two values (see Fig. 3):

*AB* and *BC* are given and *AB* > *BC*, and *AB* ∧ *BC*. *AC* are joined. Triangle *ABC* is circumscribed by a circle. The perpendicular *DH* is drawn so that *DC* must pass through point *R*.

*HR* = *AB*, *RH*/*DH* = *BA*/*DH*

*AH* = *BR*, *RH*/*HD* = *DH*/*HA*, Since angle *D* = 90°

*BA*/*DH* = *DH*/*HA*

But

*RH*/*DH* = *RB*(=*HA*)/*BC*

*BA*/*DH* = *DH*/*HA* = *HA*/*BC*.

## BIBLIOGRAPHY

The following works should be consulted: A. Aaboe, “Al-Kāshī’s Iteration Method for Determination of sin 1° in *Scripta mathematica*, **20** (1954), 24–29; and *Episodes From the Early History of Mathematics* (New Haven, 1964), 120; C. Brockelmann, *Geschichte des arabischen Literatur*, I (Leiden, 1943), 626, and supp. I (Leiden, 1937), 868–869; P. Brunet and A. Mieli, *Histoire des sciences d’antiquité* (Paris, 1935), 333–415; H. Bürger and K. Kohl, “Zur Geschichte des Transversalenzatzes,” in *Abhandlungen zur Geschichte des Naturwissenschaften und der Medizin*, no. 7 (1924), 55–57, 67, 70, 71, 73–75, 89; Carra de Vaux, “Remaniement des sphériques de Théodose par lahia ibn Muhammed ibn Abi Schukr Almaghrabi Alandalusi,” in *Journal asiatique*, **17** (1891), 287–295; T. Heath, *A History of Greek Mathematics*, I (Oxford, 1921), 244–270; P. Luckey, “Der Lehrbrief über Kreis
Umfang (ar-Risāla al-Muhītīya) von Čamšid b. Mas’ūd al-kāši,” *Abhandlungen der Deutschen Akademie der Wissenschaften zu Berlin*, Math.-naturwiss. Kl., no. 5 (1950); and G. Sarton, *Introduction to the History of Science*, II, pt. 2 (Baltimore, 1931), 1015–1017.

H. Suter, *Die Mathematiker und Astronomen der Araber und ihre Werke* (Leipzig, 1900), 155; S. Tekeli, “Taqī al-Din’s Work on Extracting the Chord 2° and sin 1°,” in *Arastirma*, **3** (1965), 123–131; and “The Works on the Duplication of the Cube in the Islamic World,” *ibid*., **4** (1966), 87–105; F. Woepcke, “Sur une mesure de la circonférence du cercle due anx astronomes arabes et fondée sur un calcul d’Aboul Wafa,” in *Journal asiatique*, **15** (1860), 281–320; and S. Zeki, *Asari bakiye*, I (Istanbul, 1913), 106–120.

S. Tekeli

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