Quadratum Horarium Generale (Regiomontanus Dial)


This instrument 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).

See instructions for interactive use below

Capuchin Dial (single latitude)

Apian Dial

Gunter's quadrant

Visit my Applet Collection

Details for interactive use:

year   Enter the year into text field and hit "Apply input".
(Gregorian Calendar only, later than 1582)

Enter the latitude (decimal degrees) into the text field and hit "Apply input".
The latitude is indicated in the text field.

The interactive regions (light gray scales) are changing the cursor to cross hair.
elevation angle
                altitude Sun
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.

                sunrise sunset
Read 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.

Select from the "Display Options" menu.
The 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 = hour angle):
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
SR = tan
PQ =
tan φ tan δ
The angles POQ and ROS are equal to the declination

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

Regiomontanus Koenigsberg monument memorial Johannes

Regiomontanus monument in Königsberg (Bavaria)

            Johannes Muller Königsberg

            Johannes Muller Königsberg Bavaria

Regiomontanus' birthplace in Königsberg

Rohr, René R. J.: Die Sonnenuhr. Geschichte, Theorie, Funktion.
Callwey, München 1982.

Meyer, Jörg: Die Sonnenuhr und ihre Theorie.
Harry Deutsch, Frankfurt 2008.
Web Links

Regiomontanus: De Triangulis Planis Et Sphaericis Libri Cinque (1561)

Regiomontanus, Apian and Capuchin Sundials

Johannes Regiomontanus: Calendar

Das Allgemeine Uhrentäfelchen von Regiomontan

Tragbare Sonnenuhren in Europa ab 1400 (PDF)

Uhrentäfelchen von P. Aegid Everard

Pourquoi le cadran de Regiompntanus fonctionne-t-il ?

Cadran de Hauteur de Regiomontanus

Les cadrans de hauteur à lignes horaires rectilignes

Horoscopion Apiani Generale Dignoscendis Horis cuiuscumque generis aptissimum

Quadratum Horarium Generale Georgius Hartman

Quadratum horarium generale (1496)

A Mediaeval Portable Sundial

A ship-shaped sundial, dated 1620

Regiomontanus (Wikipedia)

Regiomontanus (Mac Tutor)

© 2009-2023 J. Giesen

Last modified: 2023, Oct 05