[maxbond]

I think it is different in the continuous case though, because you can average two (reasonably accurate) chronometers and get a better measurement. But we can't average true and false, at least not in the context of this problem definition.

But the chronometers are will sync with each other if you don't store them apart, which would result correlated noise that an average won't fix.

[yccs27]

The saying probably assumes that each chronometer has a certain small probability of malfunctioning, resulting in a significant error (basically a fat-tailed error distribution). With three chronometers, you can use a robust estimator of the true value (consensus value or median). With two, there's no robust estimator and if you use the mean, you have twice the probability of being significantly wrong (though only by half as much).

[jethro_tell]

You can definitely average two relatively accurate chronometers but you if you only have two it’s difficult to tell if one is way fast or way slow.

In a perfect world they drift less than a minute per day and you’re relatively close to the time with an average or just by picking one and knowing that you don’t have massive time skew.

I believe this saying was first made about compasses which also had mechanical failures. Having three lets you know which one failed. The same goes for mechanical watches, which can fail in inconsistent ways, slow one day and fast the next is problematic the same goes for a compass that is wildly off, how do you know which one of the two is off?

[maxbond]

> In a perfect world they drift less than a minute per day...

A minute per day would be far too much drift for navigation, wouldn't it?

From Wikipedia [1]:

> For every four seconds that the time source is in error, the east–west position may be off by up to just over one nautical mile as the angular speed of Earth is latitude dependent.

That makes me think a minute might be your budget for an entire voyage? But I don't know much about navigation. And it is beside the point, your argument isn't changed if we put in a different constant, so I only mention out of interest.

> Having three lets you know which one failed.

I guess I hadn't considered when it stops for a minute and then continues ticking steadily, and you would want to discard the measurement from the faulty watch.

But if I just bring one watch as the expression councils, isn't that even worse? I don't even know it malfunctioned and if it failed entirely I don't have any reference for the time at the port.

My interpretation had been that you look back and forth between the watches unable to make a decision, which doesn't matter if you always split the difference, but I see your point.

[1] https://en.wikipedia.org/wiki/Marine_chronometer

[bandrami]

> A minute per day would be far too much drift for navigation, wouldn't it?

Even that was much better than the dead-reckoning they had to do in bluewater before working chronometers were invented. Your ship's "position" would be a triangle that might have sides ten miles long at lower latitudes.

[jethro_tell]

I’ve never heard the bring one or three, I’ve always just heard three. I think that exact saying implies that if you have two and one isn’t working out you’ll go crazy but if you have one you’ll be oblivious until it’s too late.

A well serviced rolex in 2026 with laser cut gears drifts +/- 15sec per day.

One with hand filed gears is going to be +/- a minute on a good day, and that’s what early navigation was using. I have watches with hand filed gears and they can be a bit rough.

Prior to that, it was dead reckoning, dragging a string every now and again to calculate speed and heading and the current and then guesstimating your location on a twice daily basis.

Those two wildly inaccurate systems mapped most of the world for us.

[manwe150]

Reading the comment thread here made me realize the idea seems to be that having 2 just means double the probability of one of them failing in some undetectable way. The resulting error magnitude is reduced by half, but the probability of that error is doubled. So it gains you nothing to expected value to have 2. Unlike with 3, where the probability of undetectable failure and the error rate from partial failure are both reduced by the ability to make comparative measurements (eg pick the middle number not the average)

[Yodel0914]

> A well serviced rolex in 2026 with laser cut gears drifts +/- 15sec per day.

Modern Rolex (and Omega et al) are more like +/-2s.

[fainpul]

You would probably use a chronometer:

"The precision of a COSC-certified chronometer must be between -4 and +6 seconds per day."

https://www.cosc.swiss/cosc-faq

[maxbond]

Interesting, thanks for sharing.

Though not without significant errors, the most amusing to me being that islands had a tendency to multiply because different maps would be combined and the cartographer would mistake the same island on two maps as being separate islands due to errors. A weird case of aliasing I suppose.

[iberator]

15s per day sounds EXTREME. Are you even serious? Even the cheapest Chinese hand watch has 50x better precision.

[fainpul]

Cheap watches are almost always quartz watches (electronic). Those are much more accurate than even high end mechanical watches.

15s per day is not unusual for a good mechanical watch.

[jethro_tell]

I am serious, mechanical watches are fascinating but they are much less accurate that a cheap quartz watch.

[jtbayly]

The book “Longitude” is fascinating, and discusses the challenges prior to chronometers (many people died), as well as the rewards offered for a precise chronometer, the attempts, etc.