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Inspired science: methods for finding the qibla inthe medieval Islamic tradition.

Jan P. Hogendijk

Department of Mathematics, Utrecht, Netherlands / KFUPM

April 17, 2012

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Introduction

1. Non-mathematical ways to find the qibla

2. Simple methods using geographical coordinates

3. Distance along a great circle; spherical trigonometry

4. Two examples of my own research

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Prayer facing the Kaaba

Since the second year of the Hijra (624 CE), the qibla (direction ofprayer) in Islam has been towards the Kaaba in Makkah.

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We know a lot about Islamic methods for finding the qiblathanks to two Western scholars:

E.S. Kennedy (1912-2009)David A. King (Frankfurt,Germany; born 1941)

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Three types of Islamic methods for finding the qibla.

1. Traditional non-mathematical methods (from early Islam untilrecently)

2. Simple approximation methods using geographical coordinates(since the time of Caliph al-Ma’mun, ca. 200 H./ 810 CE untilrecently)

3. Exact mathematical computation along great circle on thespherical earth (from the early 4th century H./900 CE onwards)

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Traditional non-mathematical methods

Method A: Use the same qibla as the Prophet Mohammad (saws)and his Companions in Al-Medina: due south.

Method B: Use the direction of the road leaving your city towardsMecca.

Method C: Use a symbolic world map with the Kaaba als its center(second century H. and later).

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Traditional non-mathematical methods

Method A: Use the same qibla as the Prophet Mohammad (saws)and his Companions in Al-Medina: due south.

Method B: Use the direction of the road leaving your city towardsMecca.

Method C: Use a symbolic world map with the Kaaba als its center(second century H. and later).

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Traditional non-mathematical methods

Method A: Use the same qibla as the Prophet Mohammad (saws)and his Companions in Al-Medina: due south.

Method B: Use the direction of the road leaving your city towardsMecca.

Method C: Use a symbolic world map with the Kaaba als its center(second century H. and later).

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A symbolic world map (second century H.)

The qibla can be a direction on the horizon, for example wherecertain stars rise or set. No numbers are involved.

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Another symbolic world map (10th c. Hijra / 16th c. CE)

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2. Geographical coordinates: the general idea

Obtain geographical coordinates of your locality and of Makkah,and draw them on a map with a rectangular grid.

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Geographical coordinates: the general idea

Then join your locality and Makkah by a straight line. This willindicate the direction of the qibla

Output: an angle, usually expressed in degrees, either the samtal-qibla (azimuth of the qibla, deviation from east or west) or theinhiraf al-qibla: its deviation from the south.

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Geographical coordinates were known to Caliph al-Ma’mun(ca. 200 H./820 CE)

Zero meridian: usually CanaryIslands (west of inhabitedworld)

1 degree on the sphere is 562

3miles where 1 mile = 4000ancient Babylonian cubits (49,3cm).

so the circumference of theearth is ca. 40200 km

Globe owned by Ma’mun(reconstruction by Fuat Sezgin)

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Finding the qibla of Baghdad by the astronomers of Caliphal-Ma’mun (as reconstructed from defective sources)

Geographical latitude of MakkahφM = 21o40′, Baghdad φ = 33o25′,difference ∆φ = 11o45′. Differencebetween geographical longitudes∆λ = 3o (found from lunar eclipseobserved at Baghdad and Makkah)On the grid, longitudes are shortenedby cos 21o42′ ≈ 56

60 .

The inh. irafof the qibla was fonud as13o West of South. Formula:tan q = ∆λ cos φM/∆φ.

11 45o

Makkah

Baghdad

o

q

3 cos (21 42 )o

’

’

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According to the mathematicians from the 4th c. H/10thc. CE onwards

The qibla is along an arc of agreat circle on earth (shortestdistance to Makkah as a birdwould fly)The center of any great circlecoincides with the center of theearth.

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Modern formula

Notations: ∆λ difference between the geographical longitudes ofone’s city and Makkah, φ geographical latitude of one’s city,φM geographical latitude of Makkah,q inh. iraf of the qibla (angle with due South), Then:

tan q =sin ∆λ

cos ∆λ sin φ− cos φ tanφM.

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Muslim scientists used equivalent methods such as thefollowiing

Spherical triangle with threegreat circle arcs North pole -one’s city - Makkah.Red quantities are given. Wewant to compute the bluequantity.From your city draw a greatcircle arc perpendicular to thearc North Pole - Makkah.Then we obtain tworight-angled spherical triangleswith a right angle X .

90−φ ∆λ

q

North pole

X

Makkah

90−φcity

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Muslim scientists used equivalent methods such as thefollowing (by Abu Rayh. an al-Bırunı, ca. 410 H. / 1020 CE)

We know ∆λ, φ, φM .Then compute:(1) sin a = cos φ sin ∆λ(2) sin φ1 = sinφ/ cos a(3) f = φ1 − φM

(4) cos d = cos f cos aand finally(5):sin q = cos φM sin ∆λ/ sin d(sine rule in spherical triangle).

90−φ ∆λ

q

North pole

X

90−φ1

f

a

d

Makkah

city

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Al-Birunı expressed this as follows (his jayb is modern 60times the modern sine, al-jayb kulluhu = 60)

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Al-Birunı’s numerical example: Ghazni, Afghanistan.

We know ∆λ, φ, φM .Then compute:(1) sin a = cos φ sin ∆λ(2) sin φ1 = sinφ/ cos a(3) f = φ1 − φM

(4) cos d = cos f cos aand finally(5):sin q = cos φM sin ∆λ/ sin d(sine rule in spherical triangle).

∆λ = 27o22′24′′,φ = 33o35′, φM = 21o40′.(1) a = 22o31′19′′

(2) φ1 = 36o46′48′′

(3) f = 15o6′48′′

(4) d = 26o54′6′′

(5) q = 70o47′6′′

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Reflections by medieval Islamic scientists on the qiblaproblem

Most Muslim scientists give only the mathematics, nothing else.Al-Bırunı is an exception.

He says that the qibla can only be found by “astronomers” if thedistance to Makkah is large, but the computation is difficult andtakes time. In order to find the correct qibla, correct geographicalcoordinates are necessary.

He makes fun of the traditional non-mathematical solutions, andabout one person who tried to prove that the qibla is always duesouth.He says that for the qibla, “exactness can only exist in thought; inreality approximations are unavoidable” and relates this to Qur’an2:144, 155.

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Reflections by medieval Islamic scientists on the qiblaproblem

Most Muslim scientists give only the mathematics, nothing else.Al-Bırunı is an exception.

He says that the qibla can only be found by “astronomers” if thedistance to Makkah is large, but the computation is difficult andtakes time. In order to find the correct qibla, correct geographicalcoordinates are necessary.He makes fun of the traditional non-mathematical solutions, andabout one person who tried to prove that the qibla is always duesouth.

He says that for the qibla, “exactness can only exist in thought; inreality approximations are unavoidable” and relates this to Qur’an2:144, 155.

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Reflections by medieval Islamic scientists on the qiblaproblem

Most Muslim scientists give only the mathematics, nothing else.Al-Bırunı is an exception.

He says that the qibla can only be found by “astronomers” if thedistance to Makkah is large, but the computation is difficult andtakes time. In order to find the correct qibla, correct geographicalcoordinates are necessary.He makes fun of the traditional non-mathematical solutions, andabout one person who tried to prove that the qibla is always duesouth.He says that for the qibla, “exactness can only exist in thought; inreality approximations are unavoidable” and relates this to Qur’an2:144, 155.

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4. Two examples of historical research on the qiblaproblem

1. Islamic table for the inh. iraf of the qibla computed in degreesand minutes of arc for longitude and latitude differencesDeltaλ = 1 . . . 30 and ∆phi = 1 . . . 30. Source: Ashrafı Zıj(astronomical handbook, ca. 700 H/ 1300 CE).

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Ashrafi Zıj: part of the table

It is necessary to identifyand correct scribal errorsin the manuscript.

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Ashrafi Zıj: all values for multiples of 3 degrees werecomputed by an exact method, rest by linear interpolation

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Second example. Instrument for qibla finding, constructedin the 11th/17th c. in Isfahan, discovered in 1990 byDavid A. King.

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How to use this instrument? Example: Istanbul

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Finding the qibla in Istanbul

N

W E

S

Ist

distance

qibla

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Questions

1. Is this instrument mathematically correct?

2. When and by whom was it invented?

A Dutch historian of science Elly Dekker answered in 2000;1. yes, the curves on the instrument are ellipses.2. Probably in Europe (France), and presented to the Shah ofEsfahan in the 17th century.

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My own research: this instrument is Islamic (not French)and related to an Islamic geometric construction of theqibla by al-Bırunı.

Only preserved in an Arabic manuscript in Leiden, Holland.

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Essentially the same construction, also by al-BırunıStarting with a circle with center C and the four directions, andinputting the geographical coordinates of a city (for exampleIstanbul) and Makkah, the construction produces a point M insuch a way that CM is the direction of the qibla. See figure; noteLM = KJ.

φ

∆λ

90 − φM

= 11o

= 69o

= 41o

N

Mist

latitudeIst.

qibla

C

P

A

B

L K

J

JK=LMS = South

E = EastW = West

B’

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Point M depends on the geographical coordinates. Fordifferent cities, we obtain different points M .

ist

qibla

CEast

South

M konya meshed

West

North

isfahan

MM

M

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It turns out that the mathematics is exactly the same as inthe instrument (if North/South and East/West areinterchanged)

ist

C

konya meshedisfahan

EastWest

North

Southqibla

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Modern proof (I will skip this in the lecture but you candownload the presentation from the internet):

Referring to the previous figure, put x = -LM and y = LC , anddrop perpendicular BB ′ to CN. Put the radius of the circle equal to1. Then we have ∠NCP = φ, ∠PCA = 90o − φM so AB = cos φM ,∠ABJ = ∆λ so x = −LM = −KJ = − sin ∆λ cos φM (1).Also CB = sinφM so CB ′ = sinφM cos φ and since∠LKB = ∠BCN = φ, LB ′ = KB sin φ = cos ∆λ cos φM sin φ.Therefore y = LB ′ − CB ′ = cos ∆λ cos φM sin φ− sin φM cos φ,and ( x

cos φM)2 + ( y+sin φM cos φ

cos φM sin φ )2 = 1. (2).By (1), the meridians (φ variable, ∆λ constant) are represented bylines x = − sin ∆λ cos φM parallel to the y -axis, and by (2), theparallels (∆λ variable, φ constant) by ellipses( xcos φM

)2 + ( y+sin φM cos φcos φM sin φ )2 = 1. Note that the modern formula for

the inh. iraf q = ∠MCL can be found from tan q = |x/y |, and thatby Al-BIruni’s formula (5), MC = |x | sin q = sin d .

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Modern version of the instrument (programmed by mystudent Eelco Nederkoorn, joint project of Utrechtuniversity and KFUPM)

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Instructions for use

Makkah is the center of the instrument.Turn the ruler in such a way that the side which passes throughthe center (Makkah) also passes through the city where you wantto know the qibla.

Then the tip of the ruler indicates the qibla on the circular scale.On the ruler itself you can read the (shortest) distance to Makkahalong the great circle arc.

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What happens to the rest of the world?

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A proposal

Perhaps one can make, in the city of Makkah, a large copy of theinstrument (ca. 3 meters of diameter), with the grid but withmodern cities inscribed on it, and mount it horizontally so thatpeople can walk around it and turn the ruler.This will be a splendid example of the achievements of Islamicscience, and will illustrate the central position of Makkah in Islam.

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Further literature:

See the webpagehttp://www.jphogendijk.nl/talks/main.html

There you can download this presentation with detailed references,and there are some links to internet sources.

THANK YOU!

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Further literature:

See the webpagehttp://www.jphogendijk.nl/talks/main.html

There you can download this presentation with detailed references,and there are some links to internet sources.

THANK YOU!