Quadratum Horarium Generale
is a portable sundial for all latitudes, developed by
Regiomontanus (1436-1476). It also indicates the time
of sunrise and sunset.
It is equipped with a simple Sun sight on the upper edge. A thread with a sliding bead is hanging from the point of suspension (at the end of a brachiolus) which is adjustable in two dimensions (declination, latitude).
into text field and hit "Apply input".
(Gregorian Calendar only, later than 1582)
(decimal degrees) into the text field and hit "Apply
The latitude is indicated in the text field.
interactive regions (light gray scales) are changing
the cursor to cross hair.
|Click into the
degree scale (light gray) on the lower and left limb to
direct the quadrant to the Sun. The thread will follow
the elevation angle.
|Use the "Today"
button to set the thread to the current date. The bead
is set to the current Sun's declination.
|- Click into
the light gray calendar (date scale,
upper part for winter and
spring, or lower part for summer and autumn) to set
the thread to the date.
- To bead is set to the declination (by the declination scale at right) automatically.
the date, the declination, and the time of sunrise and
sunset (neglecting refraction on the horizon), the
equation of time, the current time, and elevation as
computed by astronomical algorithms.
from the "Display Options" menu.
red frame of the applet area is a square (753 x 753
pix, same size as for Gunter's quadrant).
|The dial is
obeying the equation of the nautic spherical triangle
(h = elevation angle,
φ = latitude, AH =
sin h = sin φ sin δ + cos φ cos δ cos AH
Spherical triangles are the subject of the 4th and 5the book "De Triangulis" by Regiomontanus.
The formula is symmetric with respect to φ and δ, thus the dials of Regiomontanus and Apian are equivalent.
A very short proof (by E. Guyot) can be found in the book of Rohr:
φ = Latitude, h = Altitude, δ = Declination, τ = Local Hour Angle
The thread is suspended at P and the bead is set to R.
The radius r = MO = OS = 1 is set to unity.
OQ = tan φ
SR = tan δ
PQ = tan φ tan δ
The angles POQ and ROS are equal to the declination δ.
The triangle ∆POR is rectangular:
PR2 = OR2 + OP2 = OS2 + SR2 + OQ2 + PQ2 = 1 + tan2 δ + tan2 φ + tan2 φ tan2 δ
PR2 = (1 + tan2 φ) (1 + tan2 δ) = 1 / (cos2 φ cos2 δ)
PR = 1 / (cos φ cos δ)
Directing the dial to the Sun the bead is at C and the altitude angle is h.
The hour angle is τ and BC = sin (90°-τ) = cos τ
AC = PR sin h = sin h / (cos φ cos δ) = AB + BC = tan φ tan δ + cos τ
sin h / (cos φ cos δ) = tan φ tan δ + cos τ
sin h = sin φ sin δ + cos φ cos δ cos τ
monument in Königsberg (Bavaria)
birthplace in Königsberg
René R. J.: Die
Sonnenuhr. Geschichte, Theorie, Funktion.
Callwey, München 1982.
Meyer, Jörg: Die Sonnenuhr und ihre Theorie.
Harry Deutsch, Frankfurt 2008.
2009-2013 J. Giesen