Sir, Having long deferred to make any report relating to the observations that were taken at sea by captain Campbell, in the year 1757, which you tra… - James Bradley

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 of will be this:
Sine lat. : rad. : : cotang. A : cotang. C, or rad : sine lat. : : tang. A : tang. C; then long. star <math>\mp</math> C = long. of λ.
Cosine C : cosine A : : semi transverse axis : z.
And cosine decl. cosine (ʘ - λ) :: z : x = parallax of right ascension.

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.

This fundamental and most important article being established upon such full evidence, it remained to be examined within what limits the errors arising from observations actually taken at sea could be contained.
In order to determine this point, I computed the ship's longitude from each of the observations made by captain Campbell, and, upon comparing the results of several that were taken near the same time and under the like circumstances, it appeared that in general the observer was not liable to err more than one minute in judging of the apparent contact of the moon's limb and the object with which it was compared. Now this being nearly the same error that would be found to obtain, if the like observations were to be made with the same instruments on land, it may hence be inferred, that in moderate weather the motion of the ship is no otherwise an impediment in this sort of observations, than as it renders the repetition of them more tedious and troublesome to the observer, which yet ought by no means to be omitted; because if each single observation be liable to an error of a minute only, by taking the mean of five or six, the error on this head may be so far diminished as to be of small moment.