PART THREE THE APPENDICES

PART THREE
THE APPENDICES
NOTE: The spreadsheets as printed here cover limited ranges of latitude, however the actual
spreadsheet cell formulae are shown so readers may develop spreadsheets for their own latitude.
Errors due to rounding and significant digits and actual dial latitude cause the author to
strongly recommend downloading the tables so they may be tailored appropriately.
Appendix 1
Trigonometric functions and geometrical rules
Tables of trig values – sin, cos, and tan
Appendix 2
Tables independent of location
The equation of time, and 5 day values, and averages, and more
Sun's apparent hour angle
Longitude to time
Julian day
Polar and Meridian dial tables
Declination of the sun by the day
Appendix 3
Tables for horizontal and vertical dial hour line angles
Hour line angles for horizontal and vertical dials latitude 30 to 60
Declination by day and altitude by day and hour latitude 30 – 40
Appendix 4
Tables for sun declination and hourly altitude - considers latitude
Tables for sun declination and hourly azimuth - considers latitude
Table for analemmatic dials – angle to hour point, and analemma
Appendix 5
South facing decliner hour angles and gnomon adjustment
East or west facing decliner hour angles
Tables for the great decliner (vertical), east/west gnomon adjustment
Appendix 6
Sunrises and sunsets for some longitudes and latitudes.
Ways of finding sunrise and sunset.
Appendix 7
Miscellaneous proofs, and oddities
Proof that the sun is at 90 degrees to the style on the equinox.
Proof that cotan (angle) = defined as tan (90-angle) is = 1/tan(angle)
Proof that a vertical dial design is the same as a horizontal design for the
co-latitude
The Southern Hemisphere – Australia and the like
Interesting trivia for the dinner table
Appendix 8
Collection of formulae
Spreadsheet implementations of tables in these appendices
Appendix 9
Books, software, and cross references to them
Index
APPENDIX 7
MISCELLANEOUS PROOFS AND ODDITIES
Proof that the sun is at 90 degrees to the style on the equinox.
Many of the geometric projections project a line from the style which is at 90 degrees from it.
Why? Because that is where the sun comes from at the equinox. In other words when the day
and night time are equal, March 21 and September 21 or thereabouts.
e
c
b
d
a
f
b
a
c
Fig a7.1
Fig a7.2
The top left figure is a very common projection. "ab" is the sub style, the style or gnomon is
rotated to the left around "ab" and winds up at "c". A 90 degree line is drawn to intercept the
extension of "ab", and the line length, "cd" is used for a half circle, whose radius "de" is the same
as "cd". An angle is drawn from "e" being an hour angle, it intercepts the horizontal line touching
the circle and the orthogonal line (namely at "d"). And from thence, an hour line is drawn "fa".
This is a very common projection.
But why is angle acd 90 degrees? Look at the diagram on the top right, it is for the equinox. To
make life simple, the earth and sun are rotated as if the earth axis was not at an angle of 23.5
degrees. When the sun is at the equinox position, it is on the equator. So it is 90 degrees above
the equator. And its angle is 90 degrees to the earths polar axis. Since the gnomon parallels the
earth axis, that is the whole point of the style being set at an angle equal to the latitude, then the
sun's rays are also at 90 degrees to the style of the gnomon.
Proof that cotan (angle) = defined as tan (90-angle) is = 1/tan(angle)
tan(x) = o/a
x
h
by definition cotan(x) = tan(90-x)
a
90 – x
thus a/o = 1/tan(x)
90
tan(90-x) = a/o which = 1/tan(x)
thus:
cotan(x) = tan(90-x) = 1/tan(x)
o
Fig a7.3
2
Proof that a vertical dial design is the same as a horizontal design for the
co-latitude
Consider the following diagram of the planet earth. At latitude 20 degrees for example there is a
vertical dial plate.
A horizontal dial
A vertical dial
w=x
d
e
v=x
latitude
a
c
b
Earth's rotation
Celestial
Polar
Axis
Fig a7.4
Angle ecb is the latitude, and we set angle dce to 90 degrees, so angle acd is the co-latitude.
The angle x is the vertical dial's dial plate angle with the sun. In the horizontal dial, angle w is also
by definition equal to x. Thus the hour lines for the vertical dial at latitude x match those of a
horizontal dial of latitude 90-x. By definition the reverse is also true.
This is why a horizontal dial may be used for a vertical dial at the co-latitude.
Some further research:R. Sagot and D. Savoie of the sundial section of the French astronomical society developed a
method, documented in Chapter 58 of Jean Meeus's work called "Astronomical Algorithms". An
avid sundial enthusiast might choose to study their general methodology.
Frank Cousins in his "A Simplified Approach by Means of the Equatorial Dial" has towards the
end a lucid explanation of the spherical triangle, and from that he develops the various formulae
used in sun dial construction.
The terms SH (style angular height) and DL (difference in longitude) relate among other things to
how a vertical decliner has a horizontal dial to be found at latitude "sh" and at a longitude
displaced from the dial's longitude by "dl". This, and Atkinson's Theorem discussed in this
volume, all combine and are fruitful subjects for further research.
These and other expanded topics of sundials in interesting poses are discussed in the sequel to
this book, "Illustrating More Shadows".
3
Proof of the Great Decliner gnomon angles and distances (Tables A5.3a, b)
In the proof and pictorial that follows, please note where the right angle is. Also, in the math that
follows, "dec" (wall declination) is actually 90-wall.declination if you use the SxxxW
terminology. The style of the gnomon for a great decliner is tilted to match the latitude and twisted
to align with true north. However, the style's tilt on the surface is less than or equal to the latitude,
and the twist is less than or equal to the walls declination.
g
SUBSTYLE
STYLE
f
f
sl
sa
g
nodus
h
Fig A7.5
lat
dec
e
sl
Fig A7.6
SUB STYLE ANGLE WITH THE HORIZONTAL (less than or equal to latitude).
Angle "sl" equates to style distance (SD) discussed elsewhere and provides the
same figures.
tan (sl) =
f/h
sl
=
=
and
tan (lat) = f/e
thus
f = e . tan(lat)
and
cos (dec) = e/h thus
h = e / cos(dec)
tan(lat) . cos(dec)
thus
atan ( tan(lat) . cos(dec) )
f
g
= VERTICAL RISE IN SUB STYLE
= HORIZONTAL DISTANCE OF STYLE TO SUB STYLE
sa
ANGLE BETWEEN STYLE AND SUBSTYLE (less than or equal to declination). Angle
"sa" does not equate to style height (SH) discussed elsewhere and does not provide the
same values.
sin(sa)
sa
=
f = e . tan(lat)
g = e . tan(dec)
g/k
=
=
and
g = e . tan(dec)
and
k = f / sin(sl)
and
f = e . tan(lat)
so
k = e . tan(lat) / sin(sl)
e . tan(dec).sin(sl) / (e . tan(lat)/sin(sl))
tan(dec) . sin(sl) / tan(lat)
and sl was given above
=
asin
tan(dec) * sin(atan( tan(lat) . cos(dec) ))
tan(lat)
4
Proof of Decliner/Great Decliner SD & SH angle (Tables A5.1… and A5.2...)
In the proof and pictorial that follows, please note where the right angle is. Also, in the math that
follows, "dec" (wall declination) is the normally defined wall declination using the SxxxW
terminology (unlike the definition used in (Tables A5.3a, b)
T
SD
SH
N
E
lat
Q
dec
B
C
dec
W
S
[SD]
tan (SD)
=
thus
SD = style distance = atan (CQ / TC)
and
tan (lat) = TC / CB
thus
TC = CB * tan (lat)
and
sin (dec) = CQ / CB
thus
CQ = CB * sin (dec)
and given
SD = style distance = atan (CQ / TC)
then
SD = atan ( CB * sin(dec) /
so
SD = atan ( sin(dec) / tan(lat) )
[SH]
sin (SH)
thus
SH = style height = asin ( QB / TB )
and
cos (lat) = CB / TB
thus
TB = CB / cos(lat)
and
cos (dec) = QB / CB
thus
QB = CB * cos (dec)
and given
SH = style height = asin ( QB / TB )
then
SH = asin ( CB * cos (dec) / (CB / cos(lat)) )
so
SH = asin ( cos (dec) * cos(lat) )
=
CQ / TC
CB * tan (lat) )
(formula A.22)
QB / TB
(formula A.23)
5
COLLECTION OF FORMULAE
INCONSISTENCIES AND APPARENT INCONSISTENCIES
FORMULAE CONVERTED FOR A SPREADSHEET
Published works referenced here are listed in appendix 9.
SUNS DECLINATION FOR ANY GIVEN DAY OF THE YEAR
Source:
http://eande.lbl.gov/Task21/C2/algo1/1-11.html
Day number, J J=1 on 1 January, J=365 on 31 December. February is taken to have 28 days.
Jan
0
Jly
181
Feb
31
Aug
212
Mar
59
Sep
243
Apr
90
Oct
273
May
120
Nov
304
Jun
151
Dec
334
A8.1
Day angle:
da = 2 * pi * ( j-1 ) / 365 (in radians, is an intermediate figure)
Sun Declination:
decl = degrees (0.006918 – 0.399912*cos(da) + 0.070257*sin(da)
– 0.006758*cos(2*da) + 0.000907*sin(2*da)
– 0.002697*cos(3*da) + 0.001480*sin(3*da)
A8.2
SUNS ALTITUDE AND AZIMUTH ON ANY GIVEN HOUR GIVEN THE SUNS DECLINATION
Source:
The sun's declination is the value "decl" in the calculations above.
Useful for the shepherd's dial.
Waugh p92, Waugh ch15 p139, Mayall p243
Sundials Australia, Folkard and Ward page 74
ALTITUDE:
The sun's altitude is its angle when looked at face on in degrees
alt = degrees( ASIN( SIN(decl) * SIN(lat) + COS(decl) *
COS(lat)*COS(lha) ) )
A8.3
n = atan ( tan(dec) / cos(ha) )
[intermediate figure]
lha = radians(0.15*(1200-h))
[local hour angle]
azi = degrees( atan( tan(lha) *cos(n) / sin(radians(lat-n)) )
A8.4
Waugh ch15 p139
AZIMUTH:
The suns azimuth:
Waugh ch15 p92
Mayall ap1 p243
The suns azimuth:
Waugh ch15 p139
Folkard and Ward page 74
azi = ATAN( SIN(lha) / ( (SIN(lat) * COS(lha))
– (COS(lat) * TAN(decl)))
[human]
A8.5
azi = degrees(atan(sin(RADIANS(15*(12-(hour/100))))/
(sin(radians(lat))*cos(RADIANS(15*(12-(hour/100))))tan(radians(decl))*cos(radians(lat)))))
[spreadsheet]
note: Waugh presents two different formulae. The one on page 139 agrees with Folkard and
Ward, and the one on page 92 does not, but agrees in all aspects except for 6am and thus also
6pm, when using the author's spreadsheet. Similarly, Mayall's formula on page 243 has the same
6am and 6pm anomaly. The Waugh page 139, and Folkard & Ward page 74 seems to work best
with the author's spreadsheet.
6
SUNRISE AND SUNSET TIME FORMULA
Source:
Mayall appendix 1 page 243, sun's declination is "decl" above
Azimuth of rising/setting sun:
A = 180 – arccos ( sin(decl) / cos(lat) )
Hour angle of rising/setting sun: hsr = arccos( tan(lat) * tan(decl) ) from noon
A8.6
A8.7
HORIZONTAL DIAL HOUR LINE ANGLE
Proof contained in this book
A8.8
H = atan ( sin(lat) * tan (lha) )
POLAR DIAL
Proof contained in this book
sh = style linear height, times are from noon
from style to hour line = sh * tan(lha from noon)
distance up an hour line to a calendar line = sh * tan (declination) / cos (time )
A8.9
A8.10
EAST OR WEST VERTICAL HOUR LINES NON DECLINER (MERIDIAN DIAL)
Proof contained in this book
sh = style linear height, times are from 6am or 6pm
from style to hour line = sh * tan(lha from 6pm or 6am)
distance up an hour line to a calendar line = sh * tan (declination) / cos (time )
A8.11
A8.12
SOUTH VERTICAL NON DECLINER HOUR LINE ANGLE
Proof contained in this book
x = atan ( tan ( lha ) * sin ( 90 - lat )
x = atan ( tan ( lha ) * cos ( lat )
)
) angle hour line makes with
12 o'clock line
A8.13
proof contained in this book
ANALEMMATIC DIAL
Source:
m = M sin ( Ø )
Waugh chapter 13, and Rohr chapter 6
semi minor axis is sin of latitude times
major axis
Ø means latitude
to get an hour point's horizontal distance:Oh' = M * sin ( 15 * hours)
to get an hour point's vertical distance:hh' = M * sin ( Ø ) * cos ( 15 * hours )
or to get an hour point by angle from O assuming an ellipse has been drawn
x = arctan ( tan( t ) / sin ( Ø ) )
to get the analemmatic points for the gnomon
z = M * tan (dec) * cos ( Ø )
A8.14
A8.15
A8.16
A8.17
A8.18
7
STANDARD TIME FROM LAT (Local Apparent Time)
legal time
or standard
time
= LAT
+ EOT.corr + west.long.corr + 1 if summer
– east.long.corr
A8.19
SOLAR TIME FROM STANDARD TIME (as in calibrating hour lines) (empirical dialing)
dial hour point = clock time for that hour point + EOT
This is not inconsistent. If the EOT were –10 minutes, then when the dial reads
1400, the legal time would be 1350. So, at 1350, with an EOT of – 10, the sun's
shadow will indicate the 1400 hour point. This may appear inconsistent with the
rules of algebra, however it is correct. Because of this apparent inconsistency,
the dialist is advised to draft a table of times and the hour point they would thus
indicate before marking a dial empirically.
SOLAR TIME FROM STANDARD TIME (as in finding true north)
Legal time
= LAT + (
+ west.long.corr + EOT.corr + 1 if summer)
– east.long.corr
Solar noon indicates true north because the sun is at its highest point. Thus the shadow
produced at solar noon will point to true north. Solar noon happens at the standard time
adjusted as follows:legal time
for solar
noon
= 12:00:00 + EOT +1 (if summer) + longitude correction
A8.20
This is not inconsistent. It may appear that signs should be reversed, however we are in
fact achieving the correct arithmetic rules.
Time when the sun will be on the
standard noon meridian
Equation of tim e
20
15
12:10
12:05
12:00
11:55
11:50
11:45
11:40
Plus is sun slow, minus is sun fast
12:15
10
5
0
1
15 29
43 57 71 85
99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323 337 351 365
-5
-10
-15
-20
Day of the ye ar
Fig a8.1
8
VERTICAL DECLINER FACING GENERALLY SOUTH ~ HOUR ANGLES & GNOMON ANGLES
The hour line angles are based on:
z = atan(cos(lat)/(cos(dec) cot(ha) + sin(dec) sin(lat))
Gnomon rotation or slewing is optional and if used employs the following formula:Gnomon offset from vertical is: sd = atan( sin(dec) / tan(lat) )
Style Distance
Style and sub style angle is:
sh = asin( cos(lat) * cos(dec) ) Style Height
Style to horizontal angle is:
sh" = 90 – sh
Difference in longitude is:
dl = atan ( tan(dec) / sin(lat) ) Diff in Longitude
A8.21
A8.22
A8.23
A8.24
gnomon
sd
sd
z
sh
sh"
11am
noon
These formulae are tabulated in the appendices for several latitudes.
Source:
Rohr ch 3, page 56, 62, Waugh ch 10 p78, Mayall ap 1 p 237
VERTICAL DECLINER DIALS FACING GENERALLY EAST OR WEST ~ GNOMON ANGLES
Please note where the right angle is. Also, in the math that follows, "dec" (wall declination) is
actually 90-wall.declination if you use the SxxxW terminology. The style of the gnomon is tilted to
match the latitude and twisted to align with true north. However, the style's tilt on the surface is
less than or equal to the latitude, and the twist is less than or equal to the walls declination.
g
SUBSTYLE
STYLE
f
f
sl
Fig A8.2
sa
g
h
lat
dec
Fig A8.3
e
sl
sa
f
g
= SUB STYLE ANGLE WITH THE HORIZONTAL (less than or equal to latitude)
= atan ( tan(lat) . cos(dec) )
ANGLE BETWEEN STYLE AND SUBSTYLE (less than or equal to declination)
= asin ( tan(dec) * sin( atan( tan(lat) . cos(dec) ) ) / tan(lat) )
= VERTICAL RISE IN SUB STYLE
=
e . tan(lat)
= HORIZONTAL DISTANCE OF STYLE TO SUB STYLE =
e . tan(dec)
A8.25
A8.26
source: Proof contained in this book
9
BIFILAR SUNDIAL
There are variations on this dial design that make it universal. However, the following two formula
are for the bifilar dial designed for a specific latitude.
The north south wire can be any height, the east west wire height is equal to:-
A8.27
east west wire height = height of the north south wire * sin(latitude)
While the north south wire is placed over the noon line, the east west wire is placed at a distance
from the dial center that is equal to:A8.28
dist from dial center = height of the north south wire * cos(latitude)
reference: http://www.de-zonnewijzerkring.nl/eng/index-bif-zonw.htm
A caution is that the shadow of the cross-hair can be off the dial plate for early hours or winter
hours.
EQUATION OF TIME
Approximation within 1minute 30 seconds (sum of three sine waves)-
A8.29
-1*(9.84*SIN(RADIANS(2*(360*(mm1+dd-81)/365))) - 7.53*COS(RADIANS(360*(mm1+dd-81)/365)) 1.5*SIN(RADIANS(360*(mm1+dd-81)/365)))-0.3
where:
mm1
dd
is the number of days prior to this month's day 1, So Jan is 0, Feb is 31, Mar
is 59, April is 90, etc, assuming a non leap year.
is the day of the month, being 1 to 31
The formula above is used by the author for some spreadsheets. Some other approximations
worth playing with might include variations on the following:A formula derived from Frank Cousins works produces the EOT in seconds and uses seven sine
wavesE=-(-97.8*SIN(SL)-431.3*COS(SL)+596.6*SIN(2*SL)-1.9*COS(2*SL)+4*SIN(3*SL)+19.3*COS(3*SL)12.7*SIN(4*SL))
where "SL" is the solar longitude, being SL=(-1*((356/365.2422)*360-270)) + julian day of year
Even established published tables vary by almost a minute. Part of this is explained by the year
within a leap year cycle, part by the decade the table was printed, and so on.
The most accurate formula used the astronomical Julian day. The book, "Astronomical Formulae
For Calculators" by Jean Meeus has a formulae, should further research be desired, in
chapter21, which in turn uses formulae in chapter 18, which uses the Julian day in chapter 3.
10
A DRAFTING SHEET FOR HORIZONTAL DIALS
A8.30
11
A DRAFTING SHEET FOR VERTICAL DIALS
A8.31
12
FORMULA OR GEOMETRIC INCONSISTENCIES OR APPARENT INCONSISTENCES
SUNS ALTITUDE AND AZIMUTH
ACTUAL INCONSISTENCY WHEN USING THE AUTHOR'S SPREADSHEET:
Waugh presents two different formulae. The one on page 139 agrees with Folkard and Ward, and
the one on page 92 does not, but agrees in all aspects except for 6am and thus also 6pm using
my spreadsheet version. Similarly, Mayall's formula on page 243 has the same 6am and 6pm
anomaly with my spreadsheet version. The Waugh page 139, and Folkard & Ward page 74 works
best with my spreadsheet.
DECLINING VERTICAL DIALS
ACTUALLY NOT INCONSISTENT:
The method presented by Rohr (chapter three, page 59) and by Dolan (pages 104 of chapter five)
is simple and uses an auxiliary horizontal dial's hour lines, and the angle between the vertical and
projected horizontal dial is the wall's declination. Waugh (chapter nine page 76) or Mayall (page
112 of chapter seven) use a very similar appearing model, but they use 15 degree lines for the
horizontal dial and a calculated angle for the separation of the vertical dial and the horizontal
projection. This is not a mistake. Their method differs from Rohr's and Dolan's in that they first
develop the gnomon's rotation (SH style height, SD style distance), deriving an angle that
separates the real vertical dial from the horizontal construction dial, and that angle is not the
wall's declination. Because their method derives a new angle, it also provides for using 15 degree
distances on that construction horizontal dial. The choice of method is up to the dialist. Also,
Mayall and Waugh differ slightly in explaining their geometric method for the gnomon rotation.
HOUR ANGLE, LATITUDE, ELEVATION, AND DECLINATION
ACTUAL INCONSISTENCY:
Rohr chapter eight page 109 on the left side of an equation simplification, converts "cos(90-h)" to
"cos(h)" which is probably an error. On the right side of the simplification however, the
conversions appear correct.
BASIC ARITHMETIC AND THE EQUATION OF TIME
Sometimes we go from clock time to solar, so why do we not reverse the signs for the EOT, etc.
Because we are still using the solar indicated local apparent time and the EOT to deduce a clock
time needed to get the sun's shadow in the right place. Thus signs are not reversed because we
are using the standard formula as is, the EOT is still operating on solar apparent time.
EAST OR WEST VERTICAL (MERIDIAN) DIAL FORMULA APPARENT INCONSISTENCY
This book uses 6am or 6pm as the baseline for calculating the time difference for hour lines,
many other books use noon, which results in radically different looking formulae. This is not an
error, it is the result of selecting 6 o'clock versus noon as the base from which hour lines are
calculated.
13