How it works
Vitruvius, On Architecture IX.8.8–11Vitruvius, writing in the 20s BC, calls it the anaphoric clock, the “rising” clock. Behind a face of fixed bronze rods turns a bronze drum painted with the sky. Water runs at a steady rate into a vessel and lifts a cork float. A chain from the float passes over the drum’s axle to a bag of sand of equal weight, so as the water rises the sky turns, once a day. A gilded boss, the image of the Sun, sits in a hole on the zodiac; it is moved one hole each day, against the turning. Wherever the boss stands among the rods is the hour.
- Steady water, steady turning. Because the drip never changes, the float rises the same height every hour and the disc turns at an even pace, once a day.
- A map of the sky. The disc is a projection of the heavens with the north pole at its centre. On it, the Sun’s boss rises over the horizon rod when the real Sun rises, and sets when it sets.
- Hours by geometry. A Roman hour was a twelfth of the daylight, so it was long in summer and short in winter. The rods divide each day into twelve, whatever its length. The clock needs no seasonal adjustment of the water.
- The boss is moved by hand. “In every one of the signs there are as many holes as the corresponding month has days” (IX.8.9). The attendant moved the Sun on through the year.
How long was an hour?
At the latitude the clock is set up forThe rods on the face are the same seasonal hours a sundial shows. Here is how long one of them lasted through the year where the clock is set up. The dot is today on the clock.
What the two bronzes show
Measured from their 3D models in October 2026, checked against Rehm (1903, 1921)Two fragments of anaphoric star discs survive. Alexander Jones (Institute for the Study of the Ancient World, NYU) photographed both and published 3D models under an open licence. We flattened the models to scale, found the peg holes and engraved lines, and fitted them against Ptolemy’s star catalogue. Alfred Rehm had already measured most of this on the bronzes themselves, in 1903 and 1921, and our numbers agree with his. One thing is new: an explanation for why the Grand dial is drawn the way it is.
1. Grand draws the Sun’s path by a plainer rule than an astrolabe’s
Rehm saw in 1921 that the Grand dial does not behave like a true astrolabe drawing. An angle at the equinox that should equal the tilt of the Earth’s axis, about 24°, measures 17°, and he put it down to careless work. We find the same thing on the 3D model, and a second thing he did not report: the holes are spaced more evenly from August to November than an astrolabe drawing allows. A plainer rule explains both at once. In it, distance from the centre is simply proportional to distance from the pole.
| Measured on the plate | Grand | Astrolabe rule | Plain rule |
|---|---|---|---|
| How far the ring sits off-centre, as a share of its radius | 0.28 | 0.40 | 0.27 |
| Hole spacing in November against August | 1.13 ± 0.05 | 1.48 | 1.26 |
Read with the plain rule, the off-centre ring gives the tilt of the Earth’s axis as about 25°, close to the round 24° Greek astronomers used before Ptolemy. Read as an astrolabe, it gives Rehm’s 17° (our 16°), a value no ancient source uses. Switch “How the sky is drawn” on the clock above to see the two rules side by side. We have not been able to read Nordon’s 1994 study of the Grand plate, which may already make this argument.
2. Both dials have one hole for every two days
Vitruvius describes a hole for every day. Both surviving dials have one for every two, 182 or 183 to the year, as Rehm counted (Salzburg in 1903, Grand in 1921). The Grand model agrees: from the Nones of August to the Kalends of November is 88 days, across about 42 holes. The labels are Roman: the Kalends, Nones and Ides of each month, the eighth day before the Kalends where the Sun enters a new sign, and a line marked AEQVINOCTIVM.
3. The Salzburg disc was about 1.2 metres across
Rehm worked this out from the plate’s geometry in 1903: about 1.2 m. Price later gave 1.7 m (1959), and his larger figure is the one usually repeated. We came at it another way, by laying four of the figures on Ptolemy’s stars: Andromeda’s head, the Charioteer’s head, the Ram’s horns and the Bull’s eye. On a sky seen from outside they land within 2–3 cm of their stars (the mirror image misses by 11–15 cm), and the fit gives 1.0–1.15 m to the winter circle, in line with Rehm. Weiss noted in 1903 that the figures are drawn freely, not star by star, and the misses are of that kind.
4. The two dials run opposite ways round
On Salzburg the zodiac runs counter-clockwise, the sky as seen from outside, like an astrolabe’s star map. On Grand the calendar runs clockwise. Rehm argued in 1921 that the Grand plate did not turn at all, and that the hour net turned in front of a fixed date plate. If so, the two were different kinds of clock face, and the difference in direction would follow from that.
How sure are we?
These are measurements of 3D models built from phone photographs, made by people who are not specialists in ancient instruments.
- The Grand plate is in two pieces and its holes were punched by hand.
- The Salzburg size rests on four figures drawn by an artist; it agrees with Rehm’s geometry.
- We have read Rehm (1903, 1921), Weiss (1903) and Price (1959). We have not read Drachmann (1954) or Nordon (1994).
How the measurements were made
- Each textured model was laid flat on its best-fitting plane and rendered to scale, with a height map.
- On Grand, the peg holes were found as pits in the height map. Three holes the detector missed were counted by eye. A circle was fitted through the holes.
- The engraved equinox line was fitted from its deepest point on 96 short cross-sections.
- In both projections, the line through the equinoxes passes through the pole. So the foot of the perpendicular from the ring’s centre to that line is the pole.
- Model dials drawn both ways were put through the same steps, so the comparison is like for like.
- On Salzburg, four figure landmarks were fitted to Ptolemy’s catalogue in the machine-readable edition of Verbunt and van Gent.
In Vitruvius’ words
De architectura IX.8.8–10, with M. H. Morgan’s translation (1914)Sources
- 3D models: Alexander Jones, Salzburg water clock dial plate fragment (2019) and Anaphoric clock dial fragment from Grand (2021), NYU Faculty Digital Archive, hdl 2451/60019 and 2451/63553, CC BY 4.0. The objects are in the Salzburg Museum (SM 3985) and the Musée d’Archéologie nationale, Saint-Germain-en-Laye (MAN 31433).
- The Grand plate before further loss: Collection H. Hoffmann sale catalogue (1888), pl. XLIII, public domain, via the same archive.
- Stars: Ptolemy, Almagest VII–VIII, in the machine-readable edition of F. Verbunt and R. H. van Gent, Astronomy & Astrophysics 544 (2012), A31. The stick figures joining them are a modern drawing aid.
- Constellation drawings: made with ChatGPT for Skirnir Studios, then fitted to Ptolemy’s stars using his own descriptions of where each star sits on the figure (“on the horn”, “at the root of the tail”), from the Greek text of the Almagest in the First1KGreek edition.
- Sun and obliquity: Ptolemy, Almagest III and I.12.
- Earlier measurements of the plates: O. Benndorf, E. Weiss and A. Rehm, “Zur Salzburger Bronzescheibe mit Sternbildern”, Jahreshefte des Österreichischen Archäologischen Institutes 6 (1903) 32–49; A. Rehm, “Neue Beiträge zur Kenntnis der antiken Wasseruhren”, Sitzungsberichte der Bayerischen Akademie der Wissenschaften 1920, Abh. 17 (1921); D. J. de Solla Price, On the Origin of Clockwork, Perpetual Motion Devices, and the Compass (1959).
- Vitruvius: Latin text and M. H. Morgan’s translation, Perseus Digital Library, CC BY-SA 4.0.