NavList:

Doug MacPherson, in your original post on this (in your PDF), you wrote:
"It can be seen that entry 23 is the Equation of Time which is to be subtracted or added to Mean Time. I had thought that this correction would have been subtracted from line 9 [the Greenwich Civil Time (G.C.T.) to obtain Greenwich Apparent Time (G.A.T.)] I guess it does not matter whether you subtract it from t or G.C.T. Maybe someone can shed light on this for me. It’s clear that the math ends up being the same."

As you note, it's all the same in the end, so in a way it's nothing to worry about. Nonetheless, it reveals something about the way that people were thinking about time in this era. The equation of time is a correction that is applied here to the solar time. So if you treat the process as a literal reflection of the way they (they who?) thought about time, then solar time by this date is seen as the inferior product that needs to be adjusted to bring it into line with the pure time of machines and physics --in other words, sun time must be "fixed" to compare it with mean time.

The Sun is wrong. The clock is right. That was how they saw it (again, they who?) by the 1890s, and that is also the modern perspective on this. If you visit a sundial in a public setting, and if it is not purely ornamental, it will usually include a correction graph or table that tells us how much the sundial is "wrong". The correction displays the equation of time usually with a location-specific time zone offset added in. Of course, from a certain perspective, a common sundial is always right when it is properly adjusted, functioning, and read properly. It reads local apparent time or local sun time, and it does that faithfully. So the correction tables connected with sundials could just as well be interpreted in reverse: the table tells us how to take our clock time and convert it into Nature's "true time" or solar time.

Time by the Sun was formerly called "true time". Back at the beginning of the 19th century, the perspective was reversed. Time read from machines was a special product. It was time "of the clock" or o'clock as if it was tainted by its mechanical source. It was the clocks that had to be corrected. And in that earlier era the equation of time was seen as the corrective for the machines --the quantity that converted mean time into "true time", which we know today as "apparent time". And notice what this latter name is saying to us: time by the Sun is the apparent time... it only appears to be true. 

This attitude towards time, and the perception of what counted as the more proper time also had a significant influence on lunars. In that era the lunar distances given in the almanacs for comparison against observations were listed in Greenwich Apparent Time rather than Greenwich Mean Time. This meant that a navigator could directly compare the time derived from a lunar observation with the "true time" taken directly from a sundial or, more likely, from that sight which turns a sextant into a sundial --known to us today as a "time sight". There was no equation of time involved in it at all! This standard of providing the predicted distances in G.A.T. was the practice from the beginning of the almanacs in 1767 (once could argue another four years before that since Maskelyne's British Mariner's Guide was available for those with lots of time and an inclination to calculate their own ephemeris data) right up until 1834. That's quite a long time...  nearly seventy years (and more than that if we count those extra years). There's almost nothing in today's celestial navigation that has lasted so long. For that long period sun time or apparent time was, from a certain perspective, primary while mean time, the time read by common clocks and chronometers was a second class creation. On the other hand, it should be said that the Nautical Almanac by the 1830s was frequently criticized as old-fashioned and perhaps even an embarrassment to British navigation and astronomy.

Frank Reed
Conanicut Island USA

PS: a small "correction": you mentioned "G.C.T." in your comments. It's just G.M.T. in this period.