Editing Equation of time (section)

== History ==

The phrase "equation of time" is derived from the [[medieval Latin]] ''aequātiō diērum'', meaning "equation of days" or "difference of days".  The word ''equation'' is used in the medieval sense of "reconciliation of a difference".  The word ''[[:wikt:aequatio|aequātiō]]'' (and [[Middle English]] ''[[:wikt:equation|equation]]'') was used in medieval astronomy to tabulate the difference between an observed value and the expected value (as in the equation of the centre, the equation of the equinoxes, the equation of the epicycle). [[Gerald J. Toomer]] uses the medieval term "equation", from the Latin ''aequātiō'' (equalization or adjustment), for Ptolemy's difference between the mean solar time and the apparent solar time. [[Johannes Kepler]]'s definition of the equation is "the difference between the number of degrees and minutes of the mean anomaly and the degrees and minutes of the corrected anomaly."{{r|Kepler|p=155}}

The difference between apparent solar time and mean time was recognized by astronomers since antiquity, but prior to the invention of accurate mechanical clocks in the mid-17th century, [[sundial]]s were the only reliable timepieces, and apparent solar time was the generally accepted standard. Mean time did not supplant apparent time in national almanacs and ephemerides until the early 19th century.{{sfn| McCarthy| Seidelmann| 2009| page=9}}

=== Early astronomy ===
The irregular daily movement of the Sun was known to the Babylonians.{{citation needed|date=May 2020}}

Book III of [[Ptolemy]]'s ''[[Almagest]]'' (2nd century) is primarily concerned with the Sun's anomaly, and he tabulated the equation of time in his ''Handy Tables''.<ref>{{Citation | last = Neugebauer | first = Otto | author-link = Otto Neugebauer | date = 1975 | title = A History of Ancient Mathematical Astronomy | publisher = Springer-Verlag | location = New York / Heidelberg / Berlin | pages = 984–986 | isbn = 978-0-387-06995-1 |series=Studies in the History of Mathematics and Physical Sciences |volume=1 |doi=10.1007/978-3-642-61910-6}}</ref> Ptolemy discusses the correction needed to convert the meridian crossing of the Sun to mean solar time and takes into consideration the nonuniform motion of the Sun along the ecliptic and the meridian correction for the Sun's ecliptic longitude. He states the maximum correction is {{frac|8|1|3}}&nbsp;time-degrees or {{frac|5|9}} of an hour (Book III, chapter 9).<ref name="Toomer"/> However he did not consider the effect to be relevant for most calculations since it was negligible for the slow-moving luminaries and only applied it for the fastest-moving luminary, the Moon.

Based on Ptolemy's discussion in the ''[[Almagest]]'', values for the equation of time (Arabic ''taʿdīl al-ayyām bi layālayhā'') were standard for the tables (''zij'') in the works of [[medieval Islamic astronomy]].<ref>{{cite journal |last1=Kennedy |first1=E. S. |title=A Survey of Islamic Astronomical Tables |journal=Transactions of the American Philosophical Society |date=1956 |volume=46 |issue=2 |page=141 |doi=10.2307/1005726 |jstor=1005726|hdl=2027/mdp.39076006359272 |hdl-access=free }}<br/>Reprinted in: {{cite book |last1=Kennedy |first1=E. S. |title=A survey of Islamic astronomical tables |date=1989 |publisher=American Philosophical Society |location=Philadelphia, PA |isbn=9780871694621 |page=19 |edition=2nd |url=https://books.google.com/books?id=EywLAAAAIAAJ&pg=PA19}}</ref>

=== Early modern period ===
{{See also|Equation clock}}

A description of apparent and mean time was given by [[Nevil Maskelyne]] in the ''Nautical Almanac'' for 1767: "Apparent Time is that deduced immediately from the Sun, whether from the Observation of his passing the Meridian, or from his observed [[Sunrise|Rising]] or [[Sunset|Setting]].  This Time is different from that shewn by Clocks and Watches well regulated at Land, which is called equated or mean Time." He went on to say that, at sea, the apparent time found from observation of the Sun must be corrected by the equation of time, if the observer requires the mean time.<ref name=Maskelyne67/>

The right time was originally considered to be that which was shown by a sundial. When good mechanical clocks were introduced, they agreed with sundials only near four dates each year, so the equation of time was used to "correct" their readings to obtain sundial time. Some clocks, called [[equation clock]]s, included an internal mechanism to perform this "correction". Later, as clocks became the dominant good timepieces, uncorrected clock time, i.e., "mean time", became the accepted standard. The readings of sundials, when they were used, were then, and often still are, corrected with the equation of time, used in the reverse direction from previously, to obtain clock time. Many sundials, therefore, have tables or graphs of the equation of time engraved on them to allow the user to make this correction.{{r|Waugh|p=123}}

The equation of time was used historically to [[wikt:clocksetter|set clocks]]. Between the invention of accurate clocks in 1656 and the advent of commercial time distribution services around 1900, there were several common land-based ways to set clocks. A sundial was read and corrected with the table or graph of the equation of time.

If a [[transit instrument]] was available or accuracy was important, the sun's transit across the [[meridian (astronomy)|meridian]] (the moment the sun appears to be due south or north of the observer, known as its [[culmination]]) was noted; the clock was then set to noon and offset by the number of minutes given by the equation of time for that date. A third method did not use the equation of time; instead, it used [[wikt:star|stellar]] observations to give [[sidereal time]], exploiting the relationship between sidereal time and [[mean solar time]].{{r|Olmstead|p=57–58}} The more accurate methods were also precursors to finding the observer's [[longitude]] in relation to a [[prime meridian]], such as in [[geodesy]] on land and [[celestial navigation]] on the sea.

The first tables to give the equation of time in an essentially correct way were published in 1665 by [[Christiaan Huygens]].<ref name="Huygens"/> Huygens, following the tradition of Ptolemy and medieval astronomers in general, set his values for the equation of time so as to make all values positive throughout the year.<ref name="Huygens"/> This meant that any clock being set to mean time by Huygens's tables was consistently about 15&nbsp;minutes slow compared to today's mean time.

Another set of tables was published in 1672–73 by [[John Flamsteed]], who later became the first [[Astronomer Royal]] of the new [[Royal Observatory, Greenwich|Royal Greenwich Observatory]]. These appear to have been the first essentially correct tables that gave today's meaning of Mean Time (previously, as noted above, the sign of the equation was always positive and it was set at zero when the apparent time of sunrise was earliest relative to the clock time of sunrise). 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.<ref name="Flamsteed"/>

The equation of time, correctly based on the two major components of the Sun's irregularity of apparent motion, was not generally adopted until after Flamsteed's tables of 1672–73, published with the posthumous edition of the works of [[Jeremiah Horrocks]].{{r|Vince|p=49}}

[[Robert Hooke]] (1635–1703), who mathematically analyzed the [[universal joint]], was the first to note that the geometry and mathematical description of the (non-secular) equation of time and the universal joint were identical, and proposed the use of a universal joint in the construction of a "mechanical sundial".{{r|Mills|p=219}}

=== 18th and early 19th centuries ===

The corrections in Flamsteed's tables of 1672–1673 and 1680 gave mean time computed essentially correctly and without need for further offset. But the numerical values in tables of the equation of time have somewhat changed since then, owing to three 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 small long-term changes in the Earth's obliquity and eccentricity (affecting, for instance, the distance and dates of [[perihelion]]), 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 18th century onwards, including the effects of the Moon (See [[barycentre]]), Venus and Jupiter.<ref name="Maskelyne64"/>

[[File:Derby Sundial C 5810.JPG|thumb|A sundial made in 1812 by [[Whitehurst & Son sundial (1812)|Whitehurst & Son]], with a circular scale showing the equation of time correction. This is now on display in [[Derby Museum and Art Gallery]].]]

From 1767 to 1833, the British ''[[Nautical Almanac|Nautical Almanac and Astronomical Ephemeris]]'' tabulated the equation of time in the sense 'add or subtract (as directed) the number of minutes and seconds stated to or from the apparent time to obtain the mean time'.  Times in the Almanac were in apparent solar time, because time aboard ship was most often determined by observing the Sun. This operation would be performed in the unusual case that the mean solar time of an observation was needed. In the issues since 1834, all times have been in mean solar time, because by then the time aboard ship was increasingly often determined by [[marine chronometer]]s. The instructions were consequently to add or subtract (as directed) the number of minutes stated to or from the mean time to obtain the apparent time. So now addition corresponded to the equation being positive and subtraction corresponded to it being negative.

As the apparent daily movement of the Sun is one revolution per day, that is 360° every 24&nbsp;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 33&nbsp;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 [[Daylight saving time|summer time]], if any.

The tiny increase of the mean solar day due to the slowing down of the Earth's rotation, by about 2 [[Millisecond|ms]] per day per century, which currently accumulates up to about 1&nbsp;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.