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Apparently Ptolemy was aware of the fact that the duration of time from noon to noon varied by many seconds throughout the course of a year. In modern times this fluctuation in length of day leads to what is referred to as the equation of time. Now, if you can make sightings of meridian crossings of the sun and compare the times of those crossings to the time of day you can make this measurement yourself. But I guess Ptolemy only had water clocks. I don't know if these clocks were accurate enough to measure fluctuations of 10s of seconds over the course of 24 hours.

How, then, did Ptolemy come to the conclusion that the length of day varies throughout the course of the year? Did he use a water clock which was in fact precise enough for this measurement? Or did he perform some other sort of astronomical observation to draw this conclusion?

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    $\begingroup$ This is one of my favorite topics: how people thought of time "in the abstract" in ancient Greece and elsewhere. One inventor came up with a device to signify the end of periods in court and he made the device take into account the difference in daylight during winter vs summer. This implies that they understood unequal length of daylight in winter (who would not notice this) but the important thing was one twelfth of the daylight, not some abstract hour. I have also wondered about speed being defined as distance divided by time -- a sophisticated concept I think. $\endgroup$
    – releseabe
    Mar 24, 2023 at 18:39
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    $\begingroup$ @RonJohn If "noon" is defined as the solar noon -- the moment when the sun transits the local meridian (i.e. when it is most directly overhead) -- then the time interval between two successive noons varies throughout the year, just as the time between successive sunrises varies. $\endgroup$
    – David
    Mar 24, 2023 at 20:50
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    $\begingroup$ The duration of time from solar noon to solar noon only varies by +-30 seconds. See apparent solar day. The accumulated effect produces seasonal deviations of up to 18 minutes from the mean (the equation of time). There's no apparent solar day which lasts 24h+18min, though. $\endgroup$ Mar 24, 2023 at 21:55
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    $\begingroup$ Yes, I had a typo in the question. I corrected it in one place not the other, should be all fixed now. The solar noon to solar noon duration varies by 10s of seconds throughout the year. If an accurate watch is started at the vernal equinox then this duration variation will result in the sun clock and the watch being out of sync by up to 10s of minutes throughout the year. $\endgroup$
    – Jagerber48
    Mar 24, 2023 at 22:18
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    $\begingroup$ @Jagerber48 Sorry, but equation of time refers to the variation between "clock noon" and "astronomical noon", not to variation in the length of the day. Apart from that though I think it's a good question. $\endgroup$ Mar 25, 2023 at 8:07

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You are right: at the time of Ptolemy they could not measure the length of a day directly. Actually Ptolemy never discusses any clocks in his book, he probably used some crude devices record the approximate times of observations, which he never gives with accuracy better than 1/4 of an hour.

The main reason for the equation of time is that ecliptic is inclined to the equator. So the hour angle of the Sun (the angle measured along the equator) is not a linear function of Sun's latitude (position on the ecliptic). In addition to this, Sun moves on the ecliptic not uniformly.

The primary observation for this phenomenon is that the seasons have unequal length (measured in days). This was known long before Ptolemy. This implies that the Sun moves on the ecliptic with varying speed. The speed depends on its position on the ecliptic. The equation of time reflects the difference between the actual speed on a given day and the average speed over the year. Ptolemy introduces an imaginary object, the "mean Sun", which moves on the ecliptic with constant speed (=the average speed of the Sun), and uses it as a reference for the time of all other astronomical phenomena.

To be specific, all solar theory of Ptolemy is derived from 15 observations of solstices and equinoxes which are recorded with accuracy of 1 hour, and spread over 570 years.

So for example, the length of the year can be obtained as the number of hours between the first and last observation, divided by 570. If the error of the observations is $E$, then the error in the length of the year is $E/570$. This shows the principle, how great precision is achieved.

Remarks. 1.This brings a more general question: what is time, really? Which time intervals are equal, and how do we know (or decide) this? Nowadays we use all kinds of clocks, but this was not the case before accurate clocks were invented.

To define time scale one has to choose some periodic (repetitive) natural process and postulate that it happens "at the constant speed".

At the time of Ptolemy, there were essentially two choices: a) to assume that the diurnal rotation of the sky is uniform (that is to choose a day as the basic unit), or b) to assume that Sun moves at a constant speed on the ecliptic (that is to choose the year as the basic unit). Equation of time shows that these two choices are somewhat inconsistent. Once this is discovered, the choice is dictated by simplicity of the mathematical model. In Ptolemy's own words:

And in general, we consider it a good principle to explain the phenomena by the simplest hypotheses possible, in so far as there is nothing in the observations to provide a significant objection to such a procedure.

(Chap. III, section 1, 201 of Toomer's translation).

Ptolemy chooses b), that is the motion of the "mean Sun" as the standard for time measurement. The result is that days are of unequal length.

(There was actually a third possibility: to choose the diurnal rotation of the sky (="fixed" stars) for the time scale. It was actually a great insight of Hipparchus that Sun gives the "correct" choice of the time scale).

A good general philosophical discussion of the meaning of time is an article of Poincare Time measurement in his book The value of science.

At the time of Poincare, time was defined as "the variable which makes the Second Law of Newton true". Nowadays we use a more fundamental physical theory (quantum mechanics) to define time scale.

  1. Almagest is a difficult reading for a modern reader. There are two excellent modern expositions in English: O. Neugebauer, A history of ancient mathematical astronomy, and O. Pedersen, A survey of Almagest.
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