**The Optimal Base, Divisions and Hands for Analogue Clocks**

For a clock showing the time in base twelve, some have proposed dividing the day successively by powers of twelve each time, starting for the first division with a division of twelve giving twelve double hours or duors, and that the number of tick marks should be either twelve or the square of twelve. However, these proposals give rise to potential problems in mechanical analogue clocks. So few as only twelve tick marks wastes the angular space that the hands traverse between them and requires the hands to move quickly, which wastes energy. So many as twelve squared tick marks poses potential for error in reading the time by decreased distinguishability of adjacent tick marks, especially in small watch faces, and made worse by parallax. Ideally, the number of tick marks should be not too many more than the number of a conventional clock, so that the dozenal clock cannot be accused of being too much harder to read, and not too many fewer so that the dozenal clock cannot be accused of being less efficient than existing clocks. At the same time, ideally we seek to make the clock exhibit greater accuracy and range than conventional clocks, so that the proposal may be used as an argument favoring the adoption of twelve as the best base in the measurement of time.

The following are a set of conditions sought in the design of the optimal mechanical analogue timepiece:

- The clock should allow measurement of the greatest practical range of durations, from the quickly elapsing to the long lasting.
- The instrument should be achievable by the minimum number of moving parts or hands, as extra hands produce extra complication in the mechanism, increased cost of manufacture and decreased facility of being read.
- The hands should move as slowly as practically possible, to conserve energy and permit accuracy of the time being read.
- The tick marks ought to be as numerous as can be easily distinguishable, even under conditions such as of mild parallax and on the small faces typically found on ladies wristwatches.

Under these conditions, the proposal for an optimal watch ought to be no worse than a conventional clock.

As for the first listed condition, the longest time measured is set around the day, which is then subdivided by the tick marks and the hands to produce the finer increments of time. This standard allows comparison of the different numerical bases and their schemes of hands and tick marks.

As for the number of hands, this should be no more than three, and also capable of being just two in emulation of those watches that do not require a second hand.

Slow movement of the hands is promoted when the base of division is small. However, small bases of division lead to less refinement of discernment among brief times at the quick end of the range. The conditions sought of slow movement and fine discrimination counterbalance each other and imply an optimum base for this application.

Regarding the number of tick marks, it is apparent that the optimum number is somewhere between twelve and its square. Since a base larger than three times the dozen is better represented in figures by only as many symbols as its square root, the square root of the maximum permissible number of tick marks should be the base of numeration for the clock.

There is only a small number of candidate bases with practical attributes for measurement to check, and these include bases eight, ten, twelve, and their squares. Certain bases that may be practical for metrology have been omitted from the table of temporal resolutions as it seemed obvious to me from the evidence with base four in this table that this base and other bases less than eight were too small to offer any advantage for this application without being used to encode their squares. In fact, as can be seen from the table, even octal was too small to offer any advantage in itself, but its square was not too large. On the other hand, decimal was also so small that it offered no advantage over conventional temporal notation yet its square is probably too large. The quartic power of four is certainly too large for a wristwatch and would lead to an enormous instrument that could not be read except face-on in the hands of the observer and not affixed to any immovable structure at an inclined or oblique angle from the line of sight in the distance.

Table of Analogue Clock Hand Resolutions

Base | Pseudo-Sexagesimal | Four | Octal | Decimal | Dozenal | Pseudo-Dozenal | Four-Squared | Eight-Squared | ||||||||||

# of Gradations | 60 | ⑤④⁏ | ⑤④⁏ | 100 | ①⓪⓪⁏ | ⑥⓪⁏ | ⑤④⁏ | ①⑨④⁏ | ⑤④⁏ | |||||||||

Hand Fraction of Day & Resolution | Slow | twice/day | ①⓪⁏ minutes | daily | 22.5 minutes | daily | 22.5 minutes | daily | 14.4 minutes | daily | 10 minutes | twice/day | 10 minutes | daily | 22.5 minutes | 5.625 minutes | daily | 22.5 minutes |

Medium | hourly | ⑤⓪⁏ seconds | 4/day | 5.625 minutes | 8/day | 2.8125 minutes | 10/day | 86.4 seconds | ①⓪⁏/day | 50 seconds | hourly | 50 seconds | 4^2/day | 84.375 seconds | 21.09375 seconds | 8^2/day | 21.09375 seconds | |

Fast | 60/hour | second | 4/quarter day | 84.375 seconds | 8/8th day | 21.09375 seconds | 10/10th day | 8.64 seconds | ①⓪⁏/①⓪⁏th day | ④⁏② seconds | ⑥⓪⁏/hour | ⓪⁏⑧④ seconds | ①④⁏/①④⁏th day | ~ 5.27 seconds | ~ 1.32 seconds | ⑤④⁏/⑤④⁏th day | ~ 0.33 seconds |

The decimal scheme is less refined than the conventional system and as well as that would be more difficult to read, and therefore has no merit as a proposal for replacing the conventional system. Pure dozenal, in contrast, is more refined than the current system, but this comes at the cost of lesser distinguishability of the tick marks. The best rigourous base for the application of the clock appears to be the square of eight, with the bonus of a number of advantages in that resolutions it represents are the same as are found in other binary bases such as the square of four and octal, and thus it can be used for a clock providing measurement in those bases as well. With base eight squared, in the hand of medium motion the same resolution of time is provided as by the square of four as base yet with a quarter as many or four times fewer tick marks, leading to the same resolution and readability in a face sixteen times smaller in area. This is an enormous advantage. As well as that, base eight squared achieves this same resolution by fewer hands than octal, without the need for any more tick marks.

The "pseudo-dozenal" scheme is my original dozenal proposal. It achieves the same resolution in the first two hands as a regular dozenal scheme yet with half as many tick marks, enabling as much ease of being read from a face of four times smaller area. These two hands are the same as on existing conventional clocks, which would only need to be repainted on their surfaces without alteration of their mechanisms, and it has this advantage over other bases. It could be supplemented with a third hand, which if counting the tick marks as even numbers would be the same as the fourth hand of the regular dozenal scheme. So, my proposal is obviously far superior from the viewpoint of practical accuracy in measurement and simplicity of the mechanism in requiring fewer hands and smaller size to get the same resolution.

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