Horizontal Sundials
R.L. Kellogg, PhD
Reference: Sundials, Their Theory and Construction
by Albert Waugh, Dover Paperback, 1957
Identifying The Parts of a Sundial
Gnomon
Base
Dial Plate
Hour Lines
Gnomon Top Edge Points North
Gnomon
North
Dial Plate
Points to the
North Celestial
Pole (NCP)
in the Sky
Top View
Points To The
Celestial Equator
 = latitude of dial
(e.g. Los Angeles 34°)

Gnomon
North
South
Dial Plate
South
Point
Sun Through the Seasons
Summer Solstice
~ 21 June
Equinox ~ 21Mar and ~ 21 Oct
 = 23.5°
NCP
Winter Solstice
~ 21 Dec
 = 0°
 = - 23.5°
The angle of the sun from the celestial
equator is called the sun’s “declination”

Gnomon
North
South
Dial Plate
South
Point
6 pm
Noon and 6am/6pm Lines
South
Point
12 pm
At 6am local sun time
the shadow is due west
At 6pm local sun time
the shadow is due east
At noon local sun time
the shadow is due north
6 am
Note: cheap dials may not
have a straight line between
the 6am and 6pm hours. And
if the construction is hasty,
the line does not meet the
South Point of the gnomon!
6 pm

Gnomon
12 pm
Dial Plate
South
Point
6 am
Finding the Sundial Equation
O
D
Shadow
Triangle
Shadow
Dial Plate
West

O
D
East
Sundial
Shadow
Angle
A
P

Latitude
Angle
A

O
Gnomon
A
Finding the Sundial Equation
P
E
Shadow
Triangle
To MeridianOE
sin( ) 
And
OA
Celestial
Equator
O
Latitude
Angle


O
Gnomon
OE
sin( ) 
OA
D
A
Sundial
Shadow
Angle
A
OD
tan( ) 
OA
Finding the Sundial Equation
Sun’s
Local
Meridian Meridian
Gnomon
Plane of
Celestial
Equator
E
E
D
Shadow
Line
O
Plane of
Celestial
Equator
H
O

A
D
H is the
Sun’s
Hour Angle
OD
tan( H ) 
OE
Finding the Sundial Equation
•
From the dial plate and the celestial equatorial plane, we can
obtain the tangents of the shadow angle  and sun’s hour angle off
the meridian
tan( H ) 
•
OD
OE
OD
tan( ) 
OA
Take their ratio
tan( )  OD OD   OD OE  OE





tan(H )  OA OE   OA OD  OA
•
But from the gnomon triangle that has the latitude angle , we
recognize that
OE
sin( ) 
OA
hence
tan( )
 sin( )
tan(H )
or
tan( )  sin( ) tan(H )
6 pm
Dial Lines – The Math
2 pm
12 pm

tan() = sin() tan(H)
 = dial hour angle measured from 12 pm noon
H = sun “hour angle” is the distance of the sun
away from the noon meridian.
The sun moves 15° per hour, so
9 am gives H = - 45° (morning )
2 pm gives H = +30° (afternoon)
12 pm 2 pm

6 am
Example:
6 am
South
Point
 = 40° (latitude)
H = 30° (hour angle of sun = 2 pm)
gives
tan() = sin(40°) x tan(30°)
tan() = .6428 x .5774
tan() = .3711
 = atan(.3711)
 = 20.36°
“sin” is the sine trigonometric function
“tan” is the tangent trig function
“atan” is the arctangent (arctan) trig function
These functions can be found on scientific
calculators, Excel spreadsheet functions, etc.
6 pm
Draw A Sundial Hour Line
12pm
gnomon
2 pm
6 am

South
Point
tan() = sin() tan(H)
 = 20.36°
for H = 2pm (30°)
and latitude  = 40°
6 pm
Here’s what the
2pm shadow might
look like
A Complete Dial
9 am 10 am 11 am noon 1 pm
8 am
2 pm
3 pm
4 pm
15 cm
12 cm
7 am
6 am
5 pm
6 pm
For a dial with a 6 cm high gnomon cut at an
angle of 40°, it’s base is about 12 cm long
and the dial fits nicely on a 15 x 17 cm plate.
Measuring The Latitude of A Sundial
If you have a sundial, then you
can use a protractor to measure
the gnomon’s angle and determine
the dial’s latitude.
Commercial dials usually have
a “one size fits all” approach,
using a generic latitude of 40 or 45
degrees.
 = latitude
Specially built sundials have
a gnomon tailor made for their
placed location. If the dial is
moved to a different latitude, the
dial no longer keeps precise solar
time.
Some dials have “reworked” gnomons for their new, displaced homes. The
owner mistakenly things that by just altering the angle of the gnomon, the
dial will tell correct time at its new latitude. But as you now know (see previous
vugraphs for the math), the dial plate is also made for a specific latitude.
Measuring the Latitude from a Dial Plate
Although we could measure the various dial hour line angles and work
our mathematics backward, there is a simple way both to test dials
and to create new ones.
The tool is called Serle’s Ruler. A copy of the ruler reproduced by the
North American Sundial Society (NASS) is shown below.
Make a copy of this page and cut out the ruler for your use.
Serle’s Ruler – Step One
Start with the Dial Plate (or a copy
transferred to paper). Align the
ruler so that the ends always
lie on the noon line and the 6pm
hour lines (arrows)
Noon
Carefully tilt and slide the ruler
keeping the end points on the
noon and 6pm hour lines until
the hour line scale marks from
1pm to 5pm match up with the
corresponding 1pm to 5pm
dial hour lines.
When aligned, mark the point
where the ruler touches the
6pm hour line (red X).
Noon
x
6pm
6pm
Serle’s Ruler – Step Two
Noon
Now place Serle’s Ruler
along the dial’s 6pm line, with
the latitude scale starting at
the dial’s south point.
At the mark on the 6pm line
read the dial’s latitude (this
dial here has a reading of
about 34°).
The measurements of the
gnomon angle and the dial
plate latitude should agree.
If not, it could be a “generic”
dial that was commercially
assembled for quick and low
cost sale;
x
6pm
Or the dial could have been moved from its original
site and the gnomon refitted (under the false
assumption that reshaping corrects the dial’s ability
to tell time … there are a number these “discordant” dials with non-matching gnomon and
dial plate, and usually an interesting story behind the dial and its owners.