## 2024-07-30

### 12:60

I understand a clock face showing 12 at the top and numbers around 1 to 11, and then back to 12.

As a programmer, the issue of starting counting from zero or one is a massive issue.

But for time of day, we understand the clock goes 11:59, 12:00, 12:01, and that 12:00 is both midnight (and ongoing minute) and midday (and ongoing minute). I won't even start on the concept of "pre" and "post" meridiem and what 12:00 exactly means.

But what has always puzzled me is the use of minute marks that include a "60" at the top. I have always considered minutes as 00 to 59. But clearly some old clock faces considered that minutes, like hours, got 1 to 60, or 60, 1, to 59.

This leads me to wonder. Does the time, for those that created and used such clock faces, go (at around midday) :-

• 11:59
• 11:60 (midday)
• 12:01

Or does it go

• 11:59
• 12:60 (midday)
• 12:01

And then I wonder on seconds, using the same logic?

• 11:59:59
• 11:59:60 (midday)
• 11:60:01
• ...
• 11:60:59
• 11:60:60 (one minute after midday)
• 12:01:01
• 12:01:02

Or what?

Does anyone know what the actual logic was historically for the use of "60" as a minute, and why "60" was ever on a clock face?

And I know, leap seconds 23:59:60 exist, but that is a different matter!

1. It's modular arithmetic. 60 is congruent to 00, so 11:59:60 is congruent to 11:60:00 is congruent to 12:00:00. Same value, but represented by different symbols. (I'm not a mathematician, so that might not be quite right).

2. Excellent question. There's a lot of strange stuff on clock faces that actually has more reason to it than you'd think: IIII instead of IV is because otherwise the IV and VI might get confusing (more so than IX and XI, since at least IX and XI are on the same side of the dial). Not sure why 60 on faces might happen though (presumably never in Roman numerals, only in Arabic?)

3. Don't forget leap seconds. When a leap second is added, the seconds go 59, 60, 00, 01 in one second increments.

4. https://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/

5. I know what it is 24/60/60 but not why "60" is on a clock face.

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