Until the invention of the pendulum and the development of reliable clocks during the 17th century, the equation of time as defined by Ptolemy remained a curiosity, of importance only to astronomers. However, when mechanical clocks started to take over timekeeping from sundials, which had served humanity for centuries, the difference between clock time and solar time became an issue for everyday life. Apparent solar time (or true or real solar time) is the time indicated by the Sun on a sundial (or measured by its transit over the local meridian), while mean solar time is the average as indicated by well-regulated clocks. The first tables for the equation of time which accounted for its annual variations in an essentially correct way were published in 1665 by Christiaan Huygens. Huygens set his values for the equation of time so as to make all values positive throughout the year. This meant that a clock set by Huygens' tables would be consistently about 15 minutes slow on mean time.

Another set of tables was published in 1672/73 by John Flamsteed, who later became the first royal astronomer of the new Greenwich Observatory. These appear to have been the first essentially correct tables which also led to mean time without an offset. Flamsteed adopted the convention of tabulating and naming the correction in the sense that it was to be applied to the apparent time to give mean time.

The equation of time, correctly based on the two major components of the Sun's irregularity of apparent motion, i.e. the effect of the obliquity of the ecliptic and the effect of the Earth's orbital eccentricity, was not generally adopted until after Flamsteed's tables of 1672/3, published with the posthumous edition of the works of Jeremiah Horrocks.

Eighteenth and early nineteenth centuries

The corrections in Flamsteed's tables of 1672/3 and 1680 led to mean time computed essentially correctly and without an offset, i.e. in principle as we now know it. But the numerical values in tables of the equation of time have somewhat changed since then, owing to three kinds of factors:

General improvements in accuracy that came from refinements in astronomical measurement techniques, slow intrinsic changes in the equation of time, occurring as a result of very slow long-term changes in the Earth's obliquity and eccentricity and the position of its perihelion (or, equivalently, of the Sun's perigee), and the inclusion of small sources of additional variation in the apparent motion of the Sun, unknown in the 17th century, but discovered from the eighteenth century onwards, including the effects of the Moon, Venus and Jupiter.

Until 1833, the equation of time was tabulated in the sense 'mean minus apparent solar time' in the British Nautical Almanac and Astronomical Ephemeris published for the years 1767 onwards. Before the issue for 1834, all times in the almanac were in apparent solar time, because time aboard ship was most often determined by observing the Sun. In the unusual case that the mean solar time of an observation was needed, the extra step of adding the equation of time to apparent solar time was needed. In the Nautical Almanac issues for 1834 onwards, all times have been in mean solar time, because by then the time aboard ship was increasingly often determined by marine chronometers. In the unusual case that the apparent solar time of an observation was needed, the extra step of applying the equation of time to mean solar time was needed, requiring all differences in the equation of time to have the opposite sign than before.

As the apparent daily movement of the Sun is one revolution per day, that is 360? every 24 hours, and the Sun itself appears as a disc of about 0.5? in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections due to one's distance from the local time zone meridian and summer time, if any.

The tiny increase of the mean solar day itself due to the slowing down of the Earth's rotation, by about 2 ms per day per century, which currently accumulates up to about 1 second every year, is not taken into account in traditional definitions of the equation of time, as it is imperceptible at the accuracy level of sundials.

I do not know if the bells were heard in New York at midnight, March 4th, 1789.