During the reign of King
Sejong
(世宗, 14181450) in the Joseon Dynasty, there were lots of astronomical instruments, including miniaturized ones. Those instruments utilized the technical knowhow acquired through building contemporary astronomical instruments previously developed in the
Song
(宋),
Jin
(金), and
Yuan
(元) dynasties of China. In those days, many astronomical instruments had circles, rings, and spheres carved with a scale of 365.25, 100, and 24 parts, respectively, on their circumference. These were called the celestialcircumference degree, hundredinterval (
Baekgak
), and 24 direction, respectively. These scales are marked by the angular distance, not by the angle. Therefore, these circles, rings, and spheres had to be optimized in size to accomodate proper scales. Assuming that the scale system is composed of integer multiples of unit length, we studied the sizes of circles by referring to old articles and investigating existing artifacts. We discovered that the star chart of
Cheonsang yeolcha bunyajido
was drawn with a royal standard ruler (周尺) based on the unit length of 207 mm. Interestingly, its circumference was marked by the unit scale of 3
puns
per 1
du
(or degree) like
Honsang
(a celestial globe). We also found that
Hyeonju ilgu
(a equatorial sundial) has a Baekgak disk on a scale of 1
pu
n per 1
gak
(that is an interval of time similar to a quarter). This study contributes to the analysis of specifications of numerous circular elements from old Korean astronomical instruments.
1. INTRODUCTION
In the 15
^{th}
century of the Joseon Dynasty, during the reign of King
Sejong
(14181450), various astronomical instruments were developed. In
Sejong sillok
(世宗實錄, the veritable record of King Sejong), there is a summary of astronomical instruments made from the period of July 1432 (the 14
^{th}
year of
Sejong
) to January 1438 (the 20
^{th }
year of
Sejong
) by the scholars of
Jiphyeonjeon
(Hall of Worthies) and the astronomical officials of Gwansanggam (Royal Observatory). Lee Cheon(李蕆), the
Ganuidae jejo
(簡儀 臺提調) took a leading role in making those astronomical instruments in the
Ganuidae Project
, which lasted for about 5.5 years (
Nha et al. 1992
,
Jeon 2011
). Those instruments comprised
Ganui
(簡儀, a simplified armillary sphere),
Dae gyupyo(
大圭表, a large measuringscale and gnomon), the waterpowered
Honcheonui
(渾天儀, an armillary sphere, in other words,
Honui
) and
Honsang
(渾象, a celestial globe),
Angbu ilgu
(仰釜日晷, a scaphe sundial),
Ilseong jeongsiui
(日 星定時儀, a sunandstars timedetermining instrument),
So ganui
(小簡儀, a small, simplified armillary sphere), and
Hyeonju ilgu
(懸珠日晷, a plummet sundial). Most of these astromical instruments also had markings of angle, time, and direction on their spheres, circles, and rings.
There have been several attempts to make replicas of the astronomical instruments from the Joseon Dynasty (
Nam 1989
,
Nha et al. 1992
,
Lee 1996
,
Kim 1997
,
Lee et al. 2006
,
Lee & Kim 2011
,
Lee at al. 2011
,
Lee & Kim 2012
). However, the markings on the spheres or circles of these replicas were engraved using modern technology and knowledge. In this paper, some specifications of early Joseonese astronomical instruments are summarized. A scale marking method regarding the circumference of spheres or circles was examined using a stonecarved astronomical chart,
Honsang
, and
Hyeonju ilgu
.
2. ASTRONOMICAL INSTRUMENTS IN THE REIGN OF KINGSEJONG
 2.1 Classification
Astronomical instruments made in the early Joseon Dynasty are categorized into two types: restored astronomical instruments and creative astronomical instruments. The restored astronomical instruments were quite large usually and built based on the astronomical literatures of old Chinese dynasties. In contrast, the creative astronomical ones, which were relatively small in scale, were made using technical knowhow acquired during the process of building the restored astronomical instruments.
Typical restored astronomical instruments are
Honsang
,
Honui
, and
Dae gyupyo
. In
Sejong sillok
, there is no description on the detailed structures or specifications of those instruments. Instead, there are only descriptions of the Chinese literatures referenced and the places where the instruments were installed. Meanwhile, although there is currently a debate on the replica of the original
Cheonsang Yeolcha Bunyaji do
(
Rufus 1913
,
Koo 2007
,
Ahn 2010
,
Ahn 2011
),
Cheonsang Yeolcha Bunyajido
can be classified as one of the restored astronomical instruments because it was made in January 1396 during the early Joseon Dynasty. However, for
So ganui,
Ilseong jeongsiui
, and
Hyeonju ilgu
, the detailed description of the shapes and dimensions is given in
Sejong sillok
. These astronomical instruments are not mentioned in the Chinese articles from those days and were downsized while being built. Thus, it is speculated that these are creative astronomical instruments.
 2.2 Composition and Specification
In the Joseon Dynasty, the celestialcircumference degree, also known as the great circle of celestial sphere, was divided into 365.25 dus while the time scale of a day was divided into 100
gaks
(a time interval where 1
gak
corresponds to 14.4 minutes today) prior to the introduction of the
Shixian
calendar. Also, the 24 direction systems were used. However, after the introduction of the
Shixian
calendar, the metrics of degree and time were changed to 360˚ and 96 quarterofanhour(so called 96
gaks
), respectively. Moreover, regardless of the calendar system, the twentyeight lunar lodge system of irregular intervals and the twelve doublehour system of uniform scale were traditionally kept for the celestialcircumference degree and time, respectively. These systems were marked on the astronomical instruments of those days.
For example, in
Cheonsang Yeolcha Bunyajido
, there is a circular star chart with a diameter of 761 mm (
Park 1998
). The diameter of the circle can be divided into a coordinate circle and an outermost circle . While the degree of celestialcircumference is inscribed on the coordinate circle, twelve zodiac and 12 directions are inscribed on the outermost circle. According to the replica of the
Cheonsang Yeolcha Bunyajido
housed in Korea Astronomy and Space Science Institute (KASI), the longitudinal diameters of the coordinate circle and outermost circle are about 723 mm and 760 mm, respectively (
Table 3
). Whereas the circular star chart of the
Cheonsang Yeolcha Bunyajido
shows old Chinese constellations on a 2dimensional plane,
Honsang
shows them on a 3dimensional spherical surface. Also,
Sejong sillok
describes the circumference of Honsang to be 10.86
ja
, while there is an assertion that it is a typographical error of 10.96
ja
(
Han et al. 2001
).
Most Joseonese astronomical instruments had various sizes of rings corresponding to the great circle of the celestial sphere. These rings were used to measure the position, time, and direction of celestial bodies, and those were known as
Jucheon dobun hwan
(a celestialcircumference degreeandfractions ring; abbreviated to celestialcircumference ring), Baekgak hwan (a hundredinterval ring), and
Jipyeong hwan
(a horizontal ring). The
Jucheon dobun hwan
divides the celestial sphere into 365.25 uniform parts according to the total days of a year. Occasionally, twentyeight lunar lodges are marked together. This ring is either marked by a scale of 365.25 equal parts or marked twice by a scale of 182.6 equal parts. For example, the
Jekdo hwan
(an equatorial ring) of Ganui has twentyeight lunar lodges with 365.25 ticks and its
Sa

yu
whan
(a declination ring) marks the scale of 182.6 on the northern and southern half circles, respectively.
Baekgak
hwan is a ring carved with a twelve doublehour system and 100
gaks
. However, after the introduction of the
Shixian
calendar, the time scale of a day was modified into 96
gaks
. Also, 24 directions were marked on
Jipyeong hwan
.
Honcheonui
is an astronomical instrument containing the most number of rings. It is believed that
Honcheonui
was made during the reign of King
Sejong
(世宗, 14191450) and is referred to in the “Compiled Description on the
Shujing
(書纂言)” by
Wu Cheng
(吳澄, 12491333) in the
Yuan
Dynasty.
Honcheonui
has a 3layered structure of
Yukhapui
(a fixed celestialcoordinate part),
Samsinui
(a rotating three celestial body part), and
Sayuui
(a observational part). Each layer is composed of several single or double rings (
Lee et al., 2010
). The structrure of
Honcheonui
is complicated, as described above. However,
Ganui
(a simplified armillary sphere) was developed for a simpler structure. Moreover, So ganui was developed for a much simpler and smaller structure. The records of
Ganui
and
Yangyi
(仰儀, a scaphe sundial) are listed in the Yuan History’s (元史) section of
Jega ryeoksangjib
(諸家曆象集, collections of astronomy from all Chinese dynasties). According to this record,
Ganui
has a
Baekgak hwan
,
Jekdo hwan
, and
Sa

yu
ssanghwan
. In particular, the
Jekdo hwan
rotates on the inner cross section of the Lshaped
Baekgak hwan
(
Lee 1996
). Most of the existing
Angbu

ilgus
(仰釜日晷, a scaphe sundial) from Joseon originated from the 17
^{th}
century or later. However, the prototype of the Angbuilgus from the early Joseon Dynasty is presumed to be a downsized Yangyi according to the description of
Xuanjiban
(旋璣板, a pinpoint plate) in the inscription of
Angbu

ilgus
written by
Kim Don
(金墩, 13851440).
As mentioned above,
So ganui
in the
Sejong
era is a simplified form of
Ganui
and it has three rings of
Jekdo hwan
,
Baekgak hwan
, and
Sa

yu
hwan
. Even though their sizes are not known, according to the inscription of
So ganui
in the King
Seongjong
era, the diameters of
Jekdo hwan
and
Sa

yu
hwan
are the same as those of 2
ja
, which are based on the So
ganui
of the
Sejong
era (
Lee & Moon 2004
).
Lastly,
Ilseong jeongsiui
is a typical creative astronomical instrument and detailed dimensions are shown in the
Sejong sillok
. According to this record, the rings of
Ilseong jeongsiui
consist of a
Jucheon dobun hwan
(a celestialcircumference ring) and two
Baekgak hwans
(hundredinterval rings). Also, the record shows that
So Ilseongjeongsiui
was made to be almost similar to
Ilseongjeongsiui
. We can presume that the So
Ilseongjeongsiui
has three rings and the same dimensions.
Various small scale sundials were made in King
Sejong
’s era, for example,
Hyeonju ilgu
,
Cheonpyeong ilgu
(a ruletheworld sundial or a plummet sundial), and Jeongnam ilgu. According to S
ejong sillok
, there is a
Baekgak disk
with a diameter of 0.32
ja
in the
Hyeonju ilgu
and the structure and dimension of
Cheonpyeong ilgu
are similar to
Hyeonju ilgu
.
Cheonpyeong ilgu
, which was developed to determine the time in the middle of horse riding, has a
Baekgak disk
similar to
Hyeonju ilgu
. Unlike
Hyeonju ilgu
, it has irrigable pools at both the south and north sides and a rope attached to the top of the post in order to grip it while winding a rope around a wrist.
Jeongnam ilgu
measures the time using
Sa

yu
ssanghwan
(a declination double ring). However, the exact dimensions of
Sa

yu
ssanghwan
are not known.
Table 1
shows the dimensions of astronomical instruments according to the category of restored astronomical instruments (I) and creative astronomical instruments (II). The rings of each astronomical instrument are divided into
Jucheon dobun hwan
(celestialcircumference), Baekgak hwan (hundredinterval rings), and
Jipyeong hwan
(horizontal ring). In this table, the unit length of specifications is
ja
(refer to Section 3.2). The star charts of
Cheonsang yeolcha bunyajido
and
Honsang
are classified as celestialcircumference rings and the
Baekgak
disks of
Hyeonju ilgu
and
Cheonpyeong ilgu
are classified as hundredinterval rings for convenience.
Specifications of astronomical instruments developed in the early Joseon Dynasty (1 ja = 207 mm).
1. The unit length of specifications is ja; c : circumference, d: diameter, w: width, t: thickness. 2. I: Restored Astronomical Instruments, II: Creative Astronomical Instruments 3. Measurments from the Relica of the Korea Astronomy and Space Science Institute. 4. Sejong sillok (世宗實錄, Veritable Record of the King Sejong) 5. Shu zuanyan(書纂言, Compiled Description on the Shujing) 6. 8 ja, recorded as the diameter of two heukssanghwan; would be a typographical error of 8.64 ja. 7. Jega ryeoksangjib (諸家曆象集, Collections of Astronomy from all Chinese Dynasties) 8. Gukjo ryeoksanggo (國朝曆象考, Reference for Astronomy and its Calendar of Our Country) † obvious value known through the context of Sejong sillok.
3. METHOD FOR DRAWING SCALE
 3.1 Circumference and scale
In ancient China, the
du
in the celestialcircumference degree is a unit of length rather than of angle. This unit has been used in astronomical instruments (
Zhang 2000
). According to
Zhoubi suanjing
(周脾算經, arithmetic classic of circles and gnomons), the circle of diameter, 121.75
ja
, has a circumference of 365.25
ja
which makes 1
du
equal 1
ja
. According to this method, the circle is equally divided by using the celestialcircumference degree, and the ratio of the circumference used was 3. Thus, the scales of a circle or a sphere were marked as angular distance.
While the ratio of circumference was 3 in
Zhoubi suanjing
,
Liu Hu
i (劉徽) from the Wei Dynasty (魏, 220~265) of China used 157/50(=3.14) and
Zhao Chong Zhi
(趙冲之, 429~500) of the
Liu’s Song
(劉宋, 420~479) used 3.1415926 (
Needham 1959
,
Kim 2006
). It seems that the ratio ocircumference varies depending on the precision of the scales and the convenience of calculation. However, as astronomical instruments require precise measurement of the position of the celestial body, it is believed that the value of 3.14 was used in marking the scales on the circumferences.
Similar to China, the units of length used in Joseon are
jang
,
ja
,
chi
,
pun
,
li
, and
ho
, where, 0.1
jang
= 1
ja
= 10
chi
= 100
pun
= 1000
li
= 10000
ho
. In particular,
Ju cheok
(周尺, a royal standard ruler) was used in Joseon and 1
ja
of this ruler was 207 mm (
Nam 1995
). In contrast, 1
ja
of the astronomical instruments from China was 245 mm (
Kim 1993
,
Lee et al. 2011
)
 3.2 Diameter of the Circle, Sphere, and Ring
As shown in
Fig. 1
, it is assumed that the scales of a circle, sphere, or ring in the astronomical instruments of China or Joseon were inscribed by dividing the circumference equally with the integer multiple of unit length. First, if we designate the circumference as
l
, the ratio of circumference as
π
, and the ratio of the division as
ĸ
, the diameter of a ring can be expressed as follows:
where
a
is the size of the unit scale.
The scale by dividing a circumferance into equal angular distances.
Although it is possible to express
l
or
d
in the unit of
ja
,
a
can have the unit of chi or pun. If
n
is a positive number and
c
is a conversion constant, it is possible to express
n
as
n
=
n
/
c
. For example, if
c
= 1, the unit of
n
is
ja
, and if
c
= 100, the unit of
n
is
pun
. If the unit length of a
du
from
Jucheon du
(the celestialcircumference degree) or a
gak
from
Baekgak
(the hundredinterval) is 1
chi
(
c
=10), the diameter and circumference of a circle can be obtained by multiplying 10 to the value of
Table 2
.
Diameter and circumference of a circle with the integer multiple scale of unit length in Jucheondu, 100gak, 24 direction (where, π= 3.14).
Diameter and circumference of a circle with the integer multiple scale of unit length in Jucheon du, 100 gak, 24 direction (where, π= 3.14).
4. COMPARISON WITH THE RECORDS AND RELICS
 4.1Cheonsang yeolcha bunyajido(天象列次分野之圖) andHonsang
Although there is no record on this in
Taejo sillok
(Veritable Record of the King
Taejo
), the original
Cheonsang yeolcha bunyajido
appeared to have been made at the end of the 4
^{th}
year of
Taejo
(1395). Using the replica in KASI, the diameter of the coordinate circle of
Jucheon dobun
was measured to find the northsouth diameter of 723 mm. When this value is converted using the royal standard ruler of the
Sejong
era, the diameter is about 3.49
ja
. Furthermore, the
Chunyou
star chart (淳祐天文圖) in China, which has a circular star chart on the top, was made 150 years earlier than the
Cheonsang yeolcha bunyajido
. According to the rubbing of the Chunyou star chart at KASI, the measurements of the northsouth diameter from the coordinate circle, on which
Jucheon du
is marked, is 844 mm, which is 3.44
ja
in units from old China.
The value of 3.490
ja
in
Table 3
is quite similar to the diameter of the coordinate cicle with
n
= 3 in Jucheon
du
’s column of
Table 2
. Although the physical sizes of both star charts are different from each other, it is clear that the sizes of those charts in units of
ja
are similar and that the diameter of the star chart in
Cheonsang yeolcha bunyajido
is 3.49
ja
based on the royal standard ruler of the
Sejong
era. In the two star charts mentioned above, the total degree of
Xu
lunar lodge (虛宿) is about 9 1/4
du
. However, it is actually 9
du
. Therefore, the circumference can be uniformly divided into 365 parts rather than 365.25 parts and the circumference is calculated to be about 10.95
ja
based on the scale of 3
pun
per 1
du
.
Specifications of the outer circles of Cheonsang yeolcha bunyaji do and Chunyou star chart
Specifications of the outer circles of Cheonsang yeolcha bunyaji do and Chunyou star chart
In
Cheonsang yeolcha bunyajido
, the center of the chart is the north pole and the radius of the coordinate circle is the boundary of seeing a star at an observatory. Thus, that boundary depends on the latitude (
φ
, the altitude of north pole) of the observer. The boundary circle indicates – (91.3125 –
φ
)
du
in the declination and (182.625 –
φ
)
du
in polar distance. In
Table 3
, the radius of equatorial circle is about 1.08
ja
and the radius of the coordinate circle is 1.745
ja
. Hence,
φ
is calculated to be (182.625 – 91.3125)
du
ㆍ(1.745/1.08) = 35.09
du
. Converting this value into a 360˚system gives us 34.58˚, which is close to the latitude of
Yangcheng
(陽城) of the
Zhuo
Dynasty (周) or the current
Dengfeng
(登封) where the
Zhougong cejing tai
(周公測景 臺) and
Guanxing tai
(觀星臺) are located. In
Cheonsang yeolcha bunyajido
, 91.3125
du
corresponds to 1.08
ja
, so 1
du
in declination is about 1.2 pun for scale marking. On the contrary, if we fix 1 du as 1.2
pun
in declination, the radius of the equatorial circle is about 1.096
ja
while that of the coordinate circle is 1.758
pun
, which shows a difference of about 1.5
pun
each.
Ahn (2010)
suggested that the
Cheonsang yeolcha bunyajido
was made according to the method described in
Tianwen zhi
(天文志, the treatise on astronomy) of
Xin tangshu
(新唐書, the new book of Tang history). If the star chart followed the method of
Xin tangshu
, the right ascension should have been marked by 3
pun
per 1
du
on the basis of the circumference of the coordinate circle and polar distance being 1.2
pun
per 1
du
or 6
pun
per 5
du
, which is like a polar coordinate with the origin as the north pole. According to
Xin Tangshu
, the ruler made the skin of a bamboo tree have 147 scales from a hole at one end of it (
Ahn 2010
). 147 scales of that ruler agrees well with those of the radius of the coordinate circle of
Cheonsang yeolcha bunyajido
(181.625
du
– 35 du=146.625
du
).
According to
Sejong sillok
, the circumference of
Honsang
is 10.86
ja
, as shown in
Table 1
. However, the circumference of a circle with
n
=3, as shown in the Jucheon
du
column of
Table 2
, is 10.958 ja. If we assume that the record of
Sejong sillok
is a typo of 10.96
ja
(
Han et al. 2001
), the diameter of the coordinate circle in
Cheonsang yeolcha bunyajido
is the same as that of
Honsang
. The difference from
Cheonsang yeolcha bunyajido
is that
Honsang
is 3
pun
per 1
du
in the declination scales, which are similar to the right ascention ones. If the bamboo ruler described in
Xin Tanshu
(新唐書) was used to put stars on the surface of
Honsang
, it would have had 147 scales of 3
pun
and not 1.2
pun
.
Sejong sillok
described in
Honsang
as being installed with
Honui
in the pavilion and powered by water force. The diameter of
Honui
is more than twice of the diameter of the
Honsang
shown in
Table 1
, so the volumes differ by 8 times or more.
The royal standard ruler did not exist in the Goryeo Dynasty and it was newly introduced in the
Sejong
era of the Joseon Dynasty (
Lee 2001
). This does not agree with the date of
Cheonsang yeolcha bunyajido
of
Taejo
era, the 228
^{th}
Korea national treasure. In order to find out whether this treasure is from the
Taejo
era or not, or whether a ruler similar to the royal standard ruler was used or not, archeological study is necessary. It is wellknown that stone carved astronomical charts were built in the
Sejong
era (
Koo 2007
). The epilogue of
Jega ryeoksangjib
said that an astronomical chart in the
Sejong
era was inscribed to a stone based on the
Shoushi
calendar. If so, at that time, the new engraving of
Cheonsang yeolcha bunyajido
might be reproduced by the originals. In
Cheonsang yeolcha bunyajido
, the width of stele in the
Taejo
era was 122, which is 8 cm wider than that of the
Sukjong
era (the 837
^{th}
Korea Treasure). Its configuration of the planisphere and the inscription are not in the middle of the stele and leftward drawing(
Jeon et al. 1984
).
 4.2.Hyeonju ilguandJeongnam ilgu
Hyeonju ilgu
and
Cheonpyeong ilgu
are downscaled astronomical instruments from the
Sejong
era and could be classified as equatorial sundials. As indicated in
Table 1
, there is a
Baekgak
disk with the diameter of
Hyeonju ilgu
being 0.32
ja
. The diameter of this disk corresponds with
n
=1 in the
Baekgak
column shown in
Table 2
where 1
gak
is 1
pun
. There is an artefact known as
Hyeonju ilgu
and the diameter of its
Baekgak
disk is 7.1 cm (
Song et al. 1994
), which is 0.34
ja
in the unit of the royal standard ruler from the
Sejong
era. Assuming the ratio of circumference as 3 rather than 3.14, this can be interpreted to be the diameter of a circle with 1
gak
per 1
pun
.
Applying Equation (1), we can predict whether the ratio of circumference applied to astronomical instruments is 3.14 or 3. If we let
π
_{1}
and
π
_{2}
be 3 and 3.14, respectively, and the corresponding diameters based on Equation (1) are
d
_{1}
and
d
_{2}
, the difference of diameters is calculated as follows:
Where,
π
_{2}
^{1}
−
π
_{1}
^{1}
=14.86×10
^{−3}
. If we mark the scale of 100 gak (
κ
=100), Δ
d
≈ −1.49
_{n⋅ c}
^{−1}
. In the case
c
is 100, Δ
d
becomes 1.49
pun
(about 3 mm) per unit scale of 1
pun
. However, when the unit of 1
gak
is
chi
or
ja
( c is 10 or 1), becomes 1.49
chi
(about 3 cm) or 1.49
ja
(about 30 cm). Thus, it is difficult to build astromical instruments based on the circular constant of 3.
If we let
π
_{1}
and
π
_{2}
be 3.14 and 3.1415926, respectively, and the corresponding diameters are
d
_{1}
and
d
_{2}
, assuming that
Jucheon du
is marked on the circumference, Δ
d
≈ −5.89×10
^{−2}
n⋅c
^{−1}
according to Equation (2). Although
c
is 1, Δ
d
has little discrepancy of 5.89
pun
(about 12.2 mm) per unit scale of 1
ja
. Since the difference is excessively small, it is speculated that astronomical instruments are built based on the circular constant of 3.14.
Jeongnam ilgu
is also one of the downscaled sundials and has a special shape of sundial, which was described in
Sejong sillok
.
Sa

yu
hwan
, half
Baekgak hwan
, and
Jipyeong hwan
, which are rings of
Jeongnam ilgu
, have a scale of the celestialcircumference degree, a hundredinterval, and 24 directions, repectively. Though the dimension of these rings was not recorded, it is estimated that the diameter of
Jipyeong hwan
is greater than 1.0
ja
and the diameter of half
Baekgak hwan
is between 1.0
ja
and 0.9
ja,
while the diameter of
Sa

yu
whan is between 0.9
ja
and 0.67
ja
.
The length of the bottom plate of
Jeongnam ilgu
is 1.25
ja
and the distance between the two posts in the south and north is about 1.0
ja
. Because the summer solstice is the longest day, half
Baekgak hwan
is aligned along the line of the summer solstice and the scales of 100
gak
are marked on it. The heights of the northern post and southern post are 1.1
ja
and 0.59
ja
, respectively. The inclined axis of
Sa

yu
whan passes through two posts, and the intersection occurs at the height of 0.99
ja
in the northern post while that occurs at 0.21
ja
in the southern post. The intersection point of the northern post is 0.78
ja
higher than that of the southern post. The latitude of Hanyang (the old name of Seoul), the capital of Joseon, was set as 38
du
(37.45°) while the tan (37.45°) was about 0.766. In other words, the distance between the two posts and the height difference between the two intersections of the axis and post were designed to closely correspond to the latitude of Hanyang. While it is not certain how they determined 0.766 in those days, it can be estimated from the table in
Lizhi
(曆志, Treatise of the Calender) of
Mingshi
(明史, History of Ming Dynasty). According to
Mingshi
, values corresponding to the adjacent leg and opposite leg of a right triangle are obtained using
Hushi geyuan
(弧矢割圓, Calculation of the arc and sagitta in the secant of the circle ) and
Zhehui
(折會術, Trapezium Method of the secant of the circle ) of
Shen Gua
(沈括, 1031  1095). Based on these, the ecliptic opposite leg (黃道半弧弦) is 36.7486 du and the ecliptic adjacent leg(黃道大股) is 48.5316 du. Thus, the ecliptic tangent is 0.7572.
The arm of the
Baekgak
disk of
Hyeonju ilgu
is slantly fixed to the right triangle shaped socket which soars from the bottom (Song et al. 1994). If the lengths of the adjacent leg and opposite leg in the right triangle are 3 and 4, respectively, the slanting angle will be tan
^{1}
(4/3)=53.13° (53.9
du
). Fortunately, it is similar to the intersection angle between the horizontal plane and the equatorial plane, which is about 52.55° (53.3
du
) at Hanyang. For example, if we set the adjacent leg and the opposite leg of the socket to 0.12
ja
and 0.16
ja
, respectively, the
Baekgak
disk of
Hyeonju ilgu
could be installed to fit the latitude of Hanyang.
5. CONCLUSIONS
Various astronomical instruments were made through the construction of an observatory during the
Sejong
era. Some of these instruments were replicas of astronomical instruments from old Chinese dynasties while others were newly developed instruments. There are circular components in these instruments indicating the great circle of the celestial sphere, and there are scale inscriptions of the celestialcircumference degree, a hundredinterval, and 24 directions. Circles, spheres, and rings having these scales do not have arbitrary sizes, but specific sizes of circles, which could be divided into equal parts of 365.25, 100, or 24. That is, the angular distance on the circumference of a circle is subdivided instead of the angle itself in order to mark scales.
In this paper, simple cases of marking scales in integer multiple unit intervals were analyzed through the articles from the Joseon dynasty and the extant artifacts of astronomical instruments. The star chart of
Cheonsang yeolcha bunyajido
in the early Joseon Dynasty and the
Honsang
of
Sejong
era drew the celestialcircumference degrees on a circle whose circumference is 10.96
ja
in a unit scale of 3
pun
per 1
du
. While the declination is on a scale of 1.2
pun
per 1
du
in the star chart of
Cheonsang yeolcha bunyajido
, the declination of
Honsang
is on a scale of 3
pun
per 1
du
. In this analysis, it was found that the planisphere of
Cheonsang yeolcha bunyajido
was designed with a royal standard ruler. If the roral standard ruler did not exist in the
Taejo
era of Joseon, the 228
^{th}
national treasure of
Cheonsang yeolcha bunyajido
might be a stone copy of the original made after the
Sejong
era.
According to the record,
Hyeonju ilgu
and
Cheonpyeong ilgu
had a
Baekgak
disk with a diameter of 0.32
ja
, of whose circumference divided hundred ticks by 1
gak
per 1
pun
and corresponds to the size of the existing artifacts. Although there is no record about the dimension of rings in
Jeongnam ilgu
, it is estimated that the diameter of
Jipyeong hwan
is greater than 1.0
ja
, the diameter of half
Baekgak hwan
is between 0.9
ja
and 1.0
ja
, and the diameter of
Sa

yu
whan
is between 0.67
ja
and 0.9
ja
. There is a right triangle socket for the
Baekgak
disk in the existing artifact of
Hyeonju ilgu
, making it possible to build the
Baekgak
disk to fit the latitude of Seoul by using the Pythagorean’s theorem.
Acknowledgements
KiWon Lee was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2013747). Yong Sam Lee was supported by the research grant of Chungbuk National University in 2011.
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