For common purposes we may without sensible error suppose the earth's motion equable and neglect the corrections, and then the rule for the parallax … - James Bradley

My Instrument being fixed, I immediately began to observe such Stars as I judged most proper to give me light into the Cause of the Motion... There was Variety enough of small ones; and not less than twelve, that I could observe through all the Seasons of the Year; they being bright enough to be seen in the Day-time, when nearest the Sun. I had not been long observing, before I perceived, that the Notion we had before entertained of the Stars being farthest North and South, when the Sun was about the Equinoxes, was only true of those that were near the solstitial Colure: And after I had continued my Observations a few Months, I discovered what I then apprehended to be a general Law, observed by all the Stars, viz. That each of them became stationary, or was farthest North or South, when they passed over my Zenith at six of the Clock, either in the Morning or Evening. I perceived likewise, that whatever Situation the Stars were in with respect to the cardinal Points of the Ecliptick, the apparent Motion of every one tended the same Way, when they passed my Instrument about the same Hour of the Day or Night; for they all moved Southward, while they passed in the Day, and Northward in the Night; so that each was farthest North, when it came about Six of the Clock in the Evening, and farthest South when it came about Six in the Morning.

But before I proceed farther it may be proper to take notice, that since the time when I gave their lordships an account of the near agreement of Mr. Professor Mayer's lunar tables with the observations that had been then made at the Royal Observatory, I have compared several others, which concurred to prove that the difference between the observed and computed places nowhere amounted to more than about one minute and a half; and I find that the difference (small as it is) may yet be diminished by making alterations in some of the equations, whose true quantity could not be determined without proper observations; after making the needful corrections it appeared, by the comparison of above eleven hundred observations taken here since the new instruments were fixed up, that the difference did nowhere amount to more than a minute: it may therefore be reasonably concluded, that so far as it will depend upon the lunar tables the true longitude of a ship at sea may in all cases be found within about half a degree, and generally much nearer.

When the year was completed, I began to examine and compare my observations, and having pretty well satisfied myself as to the general laws of the phenomena, I then endeavored to find out the cause of them. I was already convinced that the apparent motion of the stars was not owing to the of the earth's axis. The next thing that offered itself was an alteration in the direction of the plumb-line with which the instrument was constantly rectified; but this upon trial proved insufficient. Then I considered what refraction might do, but here also nothing satisfactory occurred. At length I conjectured that all the phenomena hitherto mentioned, proceeded from the progressive motion of light and the earth's annual motion in its orbit. For I perceived that, if light was propagated in time, the apparent place of a fixed object would not be the same when the eye is at rest, as when it is moving in any other direction than that of the line passing through the eye and the object; and that, when the eye is moving in different directions, the apparent place of the object would be different.