In the Decoding Time article, we laboriously but successfully reverse-engineered the mystery of reading time from an analog clock.
Here, I'll provide a few examples of timepieces that add extra complexity to the process of reading the time.
The Museum Watch
The "Museum Dial" was created by Nathan George Horwitt, an American Designer, in 1947.
Influenced by the Bauhaus movement and inspired by sundials, his goal was to remove all unnecessary complications on a dial, by which he meant any markings that help you actually read the time. In 1960, it became the first watch to be accepted into the permanent collection of New York City’s Museum of Modern Art.
The Movado Group stole Mr. Horwitt's design to create the Museum Watch, eventually acknowledging his contribution and paying him $29k in 1975.
The only marking on the dial is a big circle at the 12 o'clock position, requiring the wearer to use that as the primary reference for orienting the watch. This requires the user to mentally estimate the value of the minute based on the angle from the top of the y-axis, in a clockwise direction:
\[minute = \alpha \div 6\]
Where \(\alpha\) is the current angle of the minute hand in degrees. Then the user computes the hour based on the angle of the hour hand, and accounting for the fact that the hand moves continuously toward the next hour.
On an analog clock, the hour hand does not simply jump from one hour mark to the next. It moves continuously as minutes progress. The correct formula accounts for both the hour and the fraction of the hour passed.
Let:
- \(\theta\) = observed angle of the hour hand, measured clockwise from 12 o’clock (in degrees, with 12 = 0°)
- \(m\) = current minute value (0–59)
The hour hand completes a full circle (360°) in 12 hours, so it moves at:
\[\frac{360^\circ}{12 \,\text{hours}} = 30^\circ \text{ per hour}\]
It also moves an additional fraction each minute:
\[\frac{30^\circ}{60 \,\text{minutes}} = 0.5^\circ \text{ per minute}\]
So the angle of the hour hand at time \((h, m)\) is:
\[\theta = 30h + 0.5m\]
To solve for the hour \(h\), rearrange:
\[h = \frac{\theta - 0.5m}{30}\]
This gives the exact fractional hour (e.g., 3.25 means quarter past 3).
If you want the integer clock hour (1–12), take the floor modulo 12:
\[\text{Hour} = \Big\lfloor \frac{\theta - 0.5m}{30} \Big\rfloor \bmod 12\]
It's amazing that we have to do that mental math just to tell the time.
Let's take a look at some of the more difficult-to-read clocks I've come across.
Sundials
On average, Amsterdam has sunlight for 35.8% of its daylight hours, with the remaining being cloudy or overcast. That didn't stop the Dutch from installing a sundial on the Nieuwe Kerk at Dam Square. This sundial marked the official time in Amsterdam until the end of the 19th century.
The Ghent Town Hall (Stadhuis) has a more intricate sundial, featuring a cube above the rooftop with two sundials to account for different orientations of the sun.
Sundials and temporal hours
Romans used temporal hours, also called unequal hours.
A day was always divided into 12 hours of daylight and 12 hours of night, regardless of the season. This meant that the length of an "hour" varied:
In summer, daytime hours were longer than 60 minutes and nighttime hours were shorter.
In winter, daytime hours were shorter than 60 minutes and nighttime hours were longer.
The Romans measured time primarily using sundials during the day and water clocks (clepsydrae) when sunlight was unavailable. Noon (meridies) was marked when the sun was at its highest point, and the hours were counted outward from dawn and dusk.
This system remained common in Europe through the Middle Ages until the adoption of mechanical clocks in the late Middle Ages, which introduced the concept of equal hours (fixed 60-minute divisions).
Hours only
The clock on the Torre del Mangia in Siena, Italy, looks conventional at first glance. It even has a modern-looking feature of a window that displays the current date, which may be surprising considering that it was installed in 1360. What it doesn't have is a minute hand.
Orientationally challenged
At least the hour markings on the Torre del Mangia clock are in their conventional positions, which is more than I can say for the clock at St. Mark's Clock Tower (Torre dell'Orologio). Built in the late 15th century, it features a 24-hour scale, and displays time, the moon phases, and the zodiac, but is also missing a minute hand. You can argue that placing 24:00 along the horizontal axis makes sense since we typically measure angles from the x-axis, but it's unconventional these days:
Directionally challenged
Some of the clocks in the Doge's Palace in Venice are not only missing a minute hand but run backward on a dial that shows all 24 hours:
In addition, the observer has to read some roman numerals upside-down: IIIIXX is really XXIIII, or 24. Incidentally, the story that clocks use IIII for the number four instead of IV because of symmetry doesn't make a lot of sense when looking at this clock face.
Here's another backward-running, hour-hand-only, 24-hour-dial clock in the Doge's Palace:
Reporting zodiac months
Doge's Palace also has this clock, which only indicates zodiac signs. The Venetians were into astrology, which influenced their daily life, governance, and their belief in a structured celestial order.
Six-hour clock (quarti di giornata)
Finally, the Chamber of the State Advocacies (i.e., the lawyers' room) in the Doge's Palace has this clock in the room for lawyers. Like the others, it has only an hour hand, moving in the expected direction for a change. However, this clock only indicates six hours, so you have to know what quartile of a day you were in.
This is an example of the Italian six‑hour clock system (sistema orario a sei ore, also known as the Roman or Italian system).
The six-hour system was a bridge between Roman temporal hours and equal hours.
In this system, the day officially began half an hour after sunset, a practice rooted in both Roman law and Christian liturgy. Cocks, whether in towers or palaces, were adjusted each evening to align the “zero point” of the dial with local sunset. After that reset, the single hand would make four revolutions of six hours each, covering the full 24-hour cycle
This meant that across the year the clock’s relationship to solar time shifted daily. Timekeeping remained solar-anchored but mechanically simplified, unlike the Roman temporal hours where the length of the hour itself changed with the season.
A six-hour clock simplifies the mechanism and reduces the amount of information needed on the dial.
For calibration, sunsets were determined via sundials, astronomical tables and almanacs, as well as observation of the horizon in places (like venice) that offered a clear western horizon.
The six-hour system was used from the late Middle Ages and gradually disappeared in the 18th and 19th centuries. By the 18th century, major Italian cities began shifting toward the 12-hour dial aligned with European practice. By the 1860s and 1870s, After Italian unification, the 24-hour railway timetable and later standard time zones finished the transition
Surprisingly, this way of indicating time did not catch on in other places.
