Principle and Use of Ottoman Sundials
By Atilla Bir*
1. The definition of the day, the hour and the prayer times 
In the Islamic world the new day begins with the sunset. When the sun is lost in the horizon it is 12 or 0 hours. The interval until the next sunset is divided into 2 x 12 hours. As defined in the "ezanic hour", the start of the day is changing but the duration of the hour-intervals remains the same throughout the day (7).
Besides this, another concept of the hour remaining from the Hellenistic age, the "unequal hour" is also used. In this concept of the hour, the periods of night and daylight are separately divided into 12 equal parts in themselves. As the length of the day remains constant by definition, the lengths of the night and daylight hours change according to the season.
- The prayer times are defined as stated below (Fig. 1) (3):
- 1.1. The Evening Prayer: Is called at the moment of sunset as observed from a point with an elevation of 625 metres. In this situation it is 12 hours according to the ezanic hour and the new day begins.
- 1.2. The Night Prayer: Is called at the moment when the sun is 17° below the horizon. Subjectively, this is defined as the moment in which two objects in black and white standing together can not be differentiated.
- 1.3. The Morning Prayer: Since the Morning Prayer must have ended at the moment of sunrise, its beginning is arranged accordingly.
- 1.4. The Midday Prayer: Is called at the moment when the shadow of a stick placed perpendicular to the horizon begins to get longer (Fig. 2).
- 1.5. The Afternoon Prayer: Is called between the moments named "Asr-i evvel" and "Asr-i sani" (Fig. 2).
The "Asr-i evvel": Is defined as the moment at which the length of the shadow of the stick is equal to the length of the shortest shadow of the stick at midday of the same day plus the length of the stick itself.
The "Asr-i sani": Is defined as the moment at which the length of the shadow of the stick is equal to the length of the shortest shadow at midday of the same day plus twice the length of the stick itself.
Figure 1: The determination of the prayer times according to the positions of the sun.
Besides these definitions, during the month of Ramadan, fasting begins at down (Imsak): the moment at which the sun is 19° beneath the horizon and there is still 1 hour and 16 minutes to the sunrise (1º 4 minutes). This time is also subjectively defined as the moment at which one can begin to differentiate between two objects in black and white which are standing together. Fasting continues until the sunset. The Bayram prayer (iyd) is conducted in mosques when the sun is 5° above the horizon or 20 minutes after sunrise.
Since the exact determination of the time is only possible for the midday and afternoon prayers, a tolerance limit of 10 minutes is recognised for the times of the other prayers and the time of commencing the fast. Accordingly the time of the Morning Prayer and the commencement of the fasting can begin 10 minutes earlier, the evening and night prayers alongside the ending of the fast may be 10 minutes later.
Figure 2: The determination of the "asr-i awwal" and asr-i sani" time according to the shadow of the stick.
The "Eid al-Fitr" begins with the first observation of the crescent at the end of the month of Ramadan which lasts 29 days. The "Eid al-Adha" is celebrated 68 days after the former bayram. Thus the Eid al-fitr is celebrated in the first day of Shawwal, whereas the Eid al-Adha is celebrated in the 10th day of zilhicce, which is 2 months and 10 days later.
Although it is known that sundials were used extensively for the establishment of time in the Islamic world, in our day the original examples of sundials can rarely be seen except the ones embodied into the walls of the mosques (2, 3, 4). The sundials which had lost their importance with the extensive use of mechanical clocks after the 17th century have quickly vanished due to the wear and tear of the external effects.
2.1. Horizontal Sundials
Picture 1: The sundial near the library of Ahmet III in the 3rd courtyard of the Topkapi Palace.(2)
The sundial near the Library of Ahmet III in the 3rd courtyard of the Topkapi Palace can be given as an example of a horizontal sundial which has been protected due to its location (Picture 1). One has to climb four steps on a marble stairway in order to take a glance at the dial settled above an elevated base. Under the scale at the eastern side of the dial is seen the inscription "Ameli Suleyman Katib-i evvel" meaning that it has been built by Suleyman, the first secretary of the Treasury. At the western side the inscription "This sundial has been established during the reign of Fatih Sultan Mehmet, let the rule of Sultan Selim, the son of Sultan Mustafa, extend to the world to come, it has been renovated by Seyyid Abdullah, his guard, year 1208,-the month of Shaban (February/April 1794)" can be read (Picture 2).
Figure 3: The drawings of the horizontal sundial near the Library of Ahmet III in the Topkapi Palace.
According to these registrations this sundial has been drawn by Suleyman Bey, the secretary of the Treasury in the era of Sultan Mehmet the Conqueror (1453-1481 ) and has been restored by the guard Seyyid Abdullah in the era of Sultan Selim III in the Hijra year of 1208 (1794): Indeed, according to an expense account found in the archives of the Topkapi Palace remaining from the year 1794, it is clearly seen that in the month of March of that year 1000 kuruş's (one gold lira) has been spent for the repair of the mentioned sundial (2).
The dial has been placed on a north-south axis (Fig. 3). The face of the dial is made of marble and has a hole at the middle which drains the rainwater. There exist two sundials on this quadrant with dimensions 65 x 44 cm2. In these dials, the time is determined by the shadow of a short vertical stick of 5 cm at the south, and the shadow of a wire stretched tightly from an ornamented pole at the north with an inclination of 41° to a point at the south.
Picture 2: The plate and inscriptions on the horizontal sundial of the Topkapi Palace.
Of these two dials, let us first examine the one which has a shape like a butterfly and on which measurement is done through a vertically placed stick. These horizontal dials are named "basita" as stated on the legend. This name means "simple horizontal dial" in the terminology of sundials. The northern and southern sections of the dial are limited by a hyperbolic curve Of these curves, the one at the north is for the Tropic of Capricorn at December 22, and the one at the south is for the Tropic of Cancer at June 21. As it will be shown in the following sections, the shadow of a stick generally draws a hyperbole throughout the day in the horizontal sundials. This curve turns into a line only two times in a year, these being the dates of March 21 (the constellation of Aries) and September 23 (the constellation of Libra). In these special dates, the time is read on a line which extends in the east-west direction at the middle of the sundial.
This sundial with a shape of a butterfly is used in establishing the ezanic hours. According to the definition given in section 1 it is 0 or 12 hours at sunset. The shadow must be reflected on the dial in order to read the time. Since the shadow is infinitely long at sunset and sunrise, a healthy determination can only be made after 1 hour from the sunrise and this last until 1 hour before the sunset. The numerals begins with 10 at the southwest end of the dial and following the west direction reaches the northwest end of the dial of the numerals 11, 12, ,1,2,3 and 4 consecutively. From this point onwards, the numerals are lined from 5 to 11 following the winter hyperbole (December 22) and thus reach the northeast end of the dial. Since in the ezanic hyperbole the sunset is at 12, each day 11 is the last numeral that can be read before the sunset.
Figure 4: The placement of the Cartesian system of coordinates on the vertical sundial.
Due to the inclination of the turning axis of the world, every day the sun rises at a different hour according to the season. Only in the dates of March 21 and September 23, when the days and nights are equal, the sunrise and sunset is at 12 hours according to the ezanic time. In this system, the shadow of the stick follows the west-east axis in the summer and winter tropics. One hour after sunrise the shadow of the stick falls on the numeral 1 which lies at the western end of the axis. If we wish to establish the position of the shadow at 2 o'clock we follow the line extending in the south-east direction from the point on which lies the numeral 2 and we find the point at which this line intersects the east-west axis. Since each hour interval is divided into 4 sections, we find the points indicating the quarter hours, as well as the values in between, which can easily be found by extrapolation. Similarly when it is 6 o'clock it we follow the line extending in the southeast direction from the numeral 6, we see that the north-south axis also passes from the point at which the line intersects the west-east axis. In this position, it is midday and the shadow of the stick has reached its shortest length in summer or winter tropics. During the afternoon, the shadow of the stick symmetrically follows the east half of the dial and 1 hour before sunset the shadow reaches the point on which lies the numeral 11 and intersects the line which forms the eastern end of the dial.
In summer, on the date of June 21, the shadow of the stick follows the southern hyperbole. The sun rises at 9 and one hour later the shadow falls on the-numeral 10 lying on the western side of the dial. The time is 4.50 in the midday hour-at which the shadow is shortest and the time is again 11 before the sunset.
In winter, on the date of December 22, the shadow of the stick follows the northern hyperbole. On this date, the sun rises at 3 o'clock and one hour later it is 4 o'clock on the dial, while the midday at which the shadow is shortest it is 7,30 and the hour is again 11 one hour before the sunset.
To determine the related hour belonging to a point outside of the mentional hyperboles it must be kept in mind that the line of the tropic of spring in March 21 and that of the tropic of autumn in September 23 are divided into 12 equal, parts over and below the east-west axis which determines the lines uniting the same ezanic hours. Accordingly, the shadow of the stick for a- certain hour moves in the clockwise direction in this .process completes its cycle of 12x4=48 sections in a year. In each section, the shadow rests for approximately one week (365/48=7,6 days ≈ l week). If we wish to determine it more correctly it must be evaluated that:between March 22 and June 21 it is 92 days,between June 21 and September 23 it is 94 days,between September 23 and December 22 it is 90 days,between December 22 and March 22 it is 89 (88) days.
If, for example, we take point A (Fig. 3), this point lies between the lines 10 and 10° 15' and is 5 sections away from the hyperbole of December 22. If we add (89x5)/12 = 37 days to December 22 due to the differences in the number of days as stated above we find January 28, or if we add 90 x (12 – 5)/12 = 52 days to September 23, we find November 14. According to this, the shadow of the stick in January 26 and November 14 falls on point A exactly at 10° 07' 30" in ezanic hours (10+1/8 sections =10+15'/2 = 10° 07' 30").
With this clock we can easily determine the sunrise, the time elapsed since sunrise and the time left until sunset for every position of the shadow. In the day in which the shadow of the stick had fallen on point A the sun had risen at 2° 45' (the fifth section beneath the numeral 4), the time elapsed after sunrise is (10° 07' 30") – (2° 45') = 7° 22' 30" and the time left until sunset is 12 – (10° 07'30") = 1° 52' 30".
In the eastern side of the cadran, there is the curve of "asr" which is functional in determining the time of the afternoon prayer. According to the detiniton (see Fig.2) the afternoon prayer begins when the length of the stick is equal to its own length plus the shortest length of the shadow in the midday of the same day (asr-i awwal) and ends when the length of the shadow is twice its own length plus the shortest length of the shadow at midday of the same day (asr-i thani). Thus, according to the ezanic time the afternoon prayer (asr-i awwal) must be conducted at 8° 22' in June 22, at 9° 48' in December 22, and at 9° 25' in March 21 and September 23. In the date at which the shadow falls on point A, the afternoon prayer must be conducted at 9° 40' (the fifth section under the northern hyperbole).
In order to examine analytically the curves drawn on the cadran, let us place the centre of coordinates to the point at which the stick lies with the OX axis on the northerly direction and the OY axis on the westerly direction (Fig. 4). According to the position of the sun with the angle of inclination h and the angle of azimuth a the coordinates x and y of a stick shadow with the length q is found out to be:
(1) x=q. cotg h. cos a
(2) y=q.cotg h.sin a =x.tg a.
Figure 5: The determination of the time angle s with respect to the latitude of the location j the declination d, the angle of azimuth a and the height of the sun h.
Let us try to determine the movement of the sun on the sky according to our location 0 on the earth (Fig. 5). Since the latitude of our location is (p, the North Pole makes an angle of φ=41° with respect to the horizon. The sun S rotates on a plane parallel to the equator around the axis KOG of the world throughout the day. Yet, since the axis of the rotation of the world has an inclination of e = 23° 27' in relation to the solar rotation plane, the sun illuminates the earth on the equatorial plane in the dates of March 21 and September 23, at the angle of +23° 27' in June 21 and at the angle of –23° 27' in December 22. The values in between are defined by the angle of declination: e £ d £ e.
The time is determined by the angle of time s which falls across the rotation axis KOG of the sun. In the spherical triangle of ZKS, since the angle of K is s, the angle of Z is (180- a), the arch of KZ is (90- φ), the arch of KS is (90- d), and the arch of ZS is (90-h), if we apply the theorem of spherical sinus:
(3) sin a. cos h = sin s. cos d
the theorem of cosines
(4) sin h = sin φ. sin d + cos φ. cos d . cos s
and finally the theorem of cotangent
(5) cotg a. sin s = sin φ. cos s - cos φ. tg d
can be written.
If the angles of h and a in the equation (2) are eliminated with the help of (3) and (4), then:
(6) y = (q.cos d. sin s) / (sin j.sin d + cos j.cos d.cos s)
(7) x = y.cot a = q.(cos d.sin j.cos s - sin d.sin j)/(sin a.sin d+cos j.cos d.cos s)
together with (5) we found the relation
Since our intention is to eliminate the angle of time s, if we solve cos s from the equation (7) and replace it into the equation (6) we arrive to the analytical expression of the hyperbole which will be followed by the shadow of the stick throughout the day:
(8) cotg a.sin s = sin φ. cos s - cos φ. tg d
y2.sin2 d - x2. cos (φ+d). cos (φ+d) - q2.sin (φ+d). sin (φ - d) + x.q.sin (2φ) = 0
This curve for φ = 41° (the latitude of Istanbul) and d = £ = + 23° 27' gives the summer hyperbole of June 21, for d = -e = - 23° 27' gives the winter hyperbole of December 22, and for φ = 0 gives the east-west line in March 21 and September 23. From (8) we find the equation of the east-west line as
(9) x=q.tg φ.
Since the ezanic hour is defined by the sunset, the starting time of the day varies according to the location (the angle of φ) and the height of the sun level with respect to the equator (the angle of d).
After the sunset, the coordinates of x and y, as given in equations (6) and (7) goes to infinity. The denominator becomes zero for the time angle s = s0:
(10) s0 = arccos (-tg φ. tg d)
where s0 defines the starting of the day. In a certain ezani hour s1, for a certain latitude φ and the solar declination of 8, if we wish to find the coordinates of the stick on the sundial we have to replace the time of s = so+ s1, into the relations of (6) and (7). If we cancel the angle in the relations we find the analytic expression of the lines on the cadran. These expressions are not given here since they are extremely complicated.
On the second sundial of the cadran, time is determined with the help of a scale running through the west, north and east edges of the cadran and the shadow of a wire stretched at an angle of 41° with respect to the horizon (Picture 2 and Fig. 3). The scale on the cadran begins from 0 at the west, reaches 6 on the north-south axis and ends on 12 on the east. Since the earth turns 360° each 24 hours, each hour the sun makes a. turn of 15°. Again, accordingly 1° equals (1 hour º 60') /15 = 4' = 4 minutes. Since each hour on the cadran is divided into intervals of 30', the short lines on the scale corresponds to 2', the long ones to 4', longer ones to 20' and the longest to 60' of the hour marks. The hours are inscribed by numerals alongside of these longest lines.
Figure 6: The position of the wire parallel to the 41° latitude.
The wire whose shadow will determine the time shows north with 41° angle on the horizon is parallel to the axis of the world in Istanbul where the latitude is φ = 41° (Fig. 6). Since the sun runs around the axis of the world through the day, it also runs around the axis of the wire. On a surface lying perpendicular to the wire the shadow of the wire gives the angles of hours s. Yet since the cadran is not perpendicular to the wire and is parallel to the horizon, the angles of the hour stated as s have been projected on a surface whose inclination is 41°. On the second sundial which is scaled by a wire, time has been given with respect to a characteristic date and "the real date is found by adding the differences to the value which is read. The scale on the cadran belongs to the dates of March 21 and September 21 where days and nights are equal. In those dates the sun rises on the point D lying on the horizon, reaches the highest point S at noon and sets at the evening on the point B (Fig. 7a). If the hourly angles of 15° are taken on the arch of DSB and projected on the horizon we find the scale as seen on the cadran (Fig. 7b). This scale is more compressed on the axis of KG as compared to the axis of DB. Thus, only at the spring and autumn equinox the hour can be read directly from the shadow of the wire.
Figure 7a: The trajectory of the sun depending on the season and the latitude of the location.
Figure 7b: The projection of the sun trajectory on the plane of the horizon and the scaling of the horizontal sundials.
Two methods are utilised in order to determine the ezani hours outside the time of equinox of spring and autumn. The first method is reading from a table the time difference for each day of the year with respect to the equinox. This table lies on the eastern end of the sundial (Table 1). The date in which the reading is to be done is determined by the sign of the zodiac. The signs of zodiac are named according to the 12 constellations on the ecliptic belt on which the sun moves. Their Latin, old Turkish, English names, symbol and commencing dates are given in Table 2.
Table 1: The table related to the time difference which must be added or subtracted to the value read on the cadran of the sundial scaled according to the topic of cancer, in order to obtain the real time.(2)
Table 2: The Latin, old Turkish, English names and signs of the Zodiac and their commencing dates.
For example, let us take the date of February 27. According to the list above, the sun enters the constellation of the Fishes in February 19. The date given is the 8th day of Fishes. We read the value of 7° 2' at the side of Fishes (Hūt) 8 in Table 1. Accordingly, the midday value of 6 as read on the cadran should be taken as 6 30', i.e., (7° x 4') +2'= 30 minutes have to be added.
The sun is in the Crab (Sertan) at the date of July 1 for 10 days. The value for Sertan 10 is read as 21° 3' in Table 1. Accordingly, the value of (21° x 4')+3' = 87' = 1 27' must be subtracted from the value indicated. For example, the ezanic time is 6-(l 27')=(5 60')-(l 27')= 4 33' instead of the value of 6 as read at the midday. The second method used in determining the time difference is to find the differences between the times of sunrise and sunset in the tropics of spring and autumn. As seen in Fig. 7a, in June 21 the sun rises at east on the point D2, rises at midday to S2, and in the evening sets on the point of B2 at the west. On December 22nd the sun rises at the point of D1, at the east, rises at midday to S1, and sets on the point B1, at the west in the evening. On figure 7b, the movement of the sun is shown as projected on the plane of horizon. It is seen that the angular difference between the shortest and the longest day is a2 - a1, = 4 Δa. If we take OO2 = x and OB2 = R in the right angular triangle of OO2B2 (Fig. 7b) we can write the relation
(11) sin Δa = x /R
Similarly, in the right angular triangle of OMS2 (Fig. 7a) since, OS2 = R and OM= q, the equation
(12) sin e = q/R
and finally in the right angular triangle of OMO2 (Fig. 7a) since, OO2 =x, the equation
(13) cos φ = q/ x
can be written.
If we take the ratio of the relations (12) and (11) we find
(14) cos φ = sin e / sin Δa
For the latitude of Istanbul φ =41° and the ecliptic angle e = 23° 27' we find Δa = 32°.
The calculated angle of difference Ace is marked on the dial in the form of two arches, one extending towards south-east and the other extending towards south-west, taking the fixing point of the wire to the cadran as centre (Fig. 3). The direction of the angle Δa is parallel to the asymptotes of the northern winter and southern summer hyperboles. The scale of 1 hour and 30 minutes as inscribed on the arches lies against the time difference between the longest (June 21) and the shortest (December 21) days and the difference between the days of March 21 and September 23.
When using this sundial, the time difference for the day in which the measurement is made must be read on the scale after determining the angle of sunrise and sunset for the same day. This difference must be added to the readings done during the summer months and must be subtracted in the winter months.
2.2. Vertical Sundials
The vertical sundials are generally found on the south-west sides of the mosques for the purpose of determining the prayer times (2,5,6) . In the mosques, since the south side is facing toward Mecca, the walls on which the sundials are placed run parallel to the direction of qibla (south-west) or vertical to the direction of qibla (south- east). In order to determine of direction of qibla correctly, we have to draw a great circle uniting the two cities and the longitudinal lines passing from Istanbul and Mecca and find the spherial triangle IKM. In this triangle, the angle K shows the differences of longitudes (l2- l1), the angle I indicates the angle of qibla with respect to north (180 - a1), the arch IK can be expressed as the angle of latitude for Istanbul (90° - (φ1) and the arch KM as the angle of longitude for Mecca (90° - φ 2) (Fig. 8). If we apply the theorem of cotangent for the triangle IKM and arrange it for the corresponding angles we find the following relation
(15) tg a1 = sin (l2 - l1) / sin j1.cos (l2 - l1) - cos j1. tg j2
Figure 8: The determination of the qibla direction.
Since the longitude and latitude of Istanbul is l1= 29°, and φ = 41° the longitude and the latitude of Mecca l2 = 40° and φ2 = 21° 30' respectively the direction of qibla in Istanbul is calculated as a1 = 29°.
The south east walls of the mosques in Istanbul are situated at angle of 180° + 29° towards the east and at an angle of 29° towards the south (Fig. 9). Since it is important to determine the time of the midday and afternoon prayers, the sundials at the mosques are situated on the south east walls. In this position, the angle of the sun lies between 360°- a1 = 331° and 331°-180°=151° .If the point where the stick of length q lies vertically on the wall is taken as centre of coordinates of the vertical sundial, y-axis vertical to the earth, x-axis extending towards the west, φ being the latitude of the location, IdI< e is the angle of declination and finally a is the angle made by the stick with respect to south, the shadow of the stick follows the hyperbole of
Figure 9: The position of the mosques with respect to the qibla direction.
(16) y2. (sin2 φ - sin2 d) + x2 (cos2 φ. sin2a - sin2 d) + x.y (sin a. sin 2 φ) -x.q.sin 2a. cos2 φ - y.q.cos a. sin2φ + q2 (cos2 a.cos2 φ - sin2 d)= 0
If we take d = 0° for the spring equinox we obtain the line
(17) y.sin φ +x.sin a .cos φ =q.cos a.cos φ
In the above equations the latitude angle for Istanbul must be taken as φ =41° and the angle with respect to south as a = -(90° - a1) = - 61°.The shadow of the stick at midday, when the sun lies directly at the south
(18) x = x1 = q. Cotg a1.
is the line of midday and in the evening at sunset it falls on the line y=0. The position of the shadow with respect to the season can be calculated from the relation (16).
On the cadrans of the vertical sundials, in order to determine the time difference between the midday and evening hours which is constantly changing, as it is generally the case with horizontal sundial, there exists a second stick which is placed parallel to the rotation axis of the world on the north-south axis (Fig. 10). In order to match the time which is determined by the stick parallel to the rotation axis of the world with the time which is read from the stick parallel to the wall, the stick which is parallel to the rotation axis must be placed on the cadran on the midday plane. As it can easily be calculated from the construction in figure 11, the point P (x1, y1) can be determined from (18) and below relation
(19) y1 = P. tg j = tg j. √ (x12 + q2) = q. tg j / sin a1
Figure 10: The drawings on the vertical sundial.
Figure 11: The relation between the stick parallel to the wall and the stick parallel to the world axis in vertical sundial.
The vertical sundials in Istanbul can be seen in the mosques of Murat Pasa, Fatih, Sultan Selim, Sultan Ahmet, Beyazit, Suleymaniye and Ayasofya. On most of these sundials, the marks directly made on the wall have been erased, the sticks having being lost, fallen or refracted. The most beautiful vertical sundial of Istanbul is in the Mihrimah Mosque in Uskudar (Picture 3, Fig. 10). At the lower right edge of the dial there exists an inscription on a marble plate fixed with iron claps, indicating that this dial was ordered in the year 1183 H (1770 CE) by a certain muwaqqit (timekeeper) with the name of Dervis Yahya Muhyittin in order to be used in the muvaqqithane (time keepers office) of the Beylerbeyi Mosque built by Sultan Abdulhamit I, but was later carried to the Mihrimah Mosque, which is also named as the iskele Mosque. The vertical stick was replaced in 1970 but the stick parallel to the polar axis has not been placed although its location is marked.
Picture 3: The vertical sundial at the Mihrimah Mosque in Uskudar.(2) (© Salim Aydüz).
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- 3.Meyer,Wolfgang, "Instrumente zur Bestimmung des Gebetszeiten im Islam", I. Uluslararasi Turk-Islam ve Teknoloji Tarihi Kongresi, 14-18 Eylul 1981, Voll, pp. 9-32.
- 4. Rohr, Rene, "Sonnenuhr und Astrolabium in Dienste der Moschee" Centauris, Vol 18, No.l, Copenhagen, 1975, s.44-45.
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 We attract the attention of our readers on two editorial features in the article: the figures accompanying the text were supplied by the author; the numbers between brackets in the body of the text refer to the sources quoted at the end of the article. (Chief Editor).
* Prof. Dr., Istanbul Technical University, Electric and Electronic Faculty. This article first published at Prof. Dr. Kazim Çeçen Anisina (Editor: Aslan Terzîoǧlu and Mehmetçik Beyazit, Istanbul 1998, pp. 83-92). Originally published as Atilla Bir, "Principle and Use of Ottoman Sundials", Prof. Dr. Kazim Çeçen Anisina, Edited by Aslan Terzîoǧlu and Mehmetçik Beyazit, Istanbul 1998, pp. 83-92.
by: Atilla Bir, Fri 30 May, 2008