"[Vol. LIV, Anno 1764] XLVIII. Concise Rules for Computing the Effects of Refraction and Parallax in Varying the Apparent Distance of the Moon from the Sun or a Star; also an Easy Rule of Approximation for Computing the Distance of the Moon from a Star, the Longitudes and Latitudes of both being given. By the Rev. , A.M. F.R.S. p. 263. The following rules, excepting one, are the same which Mr. M. before communicated to the R.S., but without demonstration, in a letter from St. Helena, containing the results of his observations of the distance of the moon from the sun and fixed stars, taken in his voyage thither, for finding the longitude of the ship from time to time; since printed in vol. lii. of the Phil. Trans. The two rules for the correction of refraction and parallax, he had also communicated to the public in his British Mariner's Guide to the discovery of longitude from like observations of the moon; and added in the preface a rule for computing a second but smaller correction of parallax, necessary on account of a small imperfection lying in the first rule derived from the fluxions of a spherical triangle. To the rules he has here subjoined their demonstrations. With respect to the usefulness of these rules, he entertains hopes that they will appear more simple and easy than any yet proposed; for the same purpose, the last rule, for computing the distance of the moon from a star, though only an approximation, being so very exact seems particularly adapted for the construction of a nautical , containing the distances of the moon from the sun and proper fixed stars, ready calculated for the purpose of finding the longitude from observations of the moon at sea; an assistance which, in an age abounding with so many able computers, mariners need not doubt they will be provided with, as soon as they manifest a proper disposition to make use of it. A RULE. To compute the contraction of the apparent distance of any two heavenly bodies by refraction; the zenith distances of both, and their distance from each other being given nearly. Add together the tangents of half the sum, and half the difference of the zenith distances; their sum, abating 10 from the index, is the tangent of arc the first. To the tangent of arc the first, just found, add the co-tangent of half the distance of the stars; the sum, abating 10 from the index, is the tangent of arc the second. Then add together the tangent of double the first arc, the co-secant of double the second arch, and the constant logarithm of 114″ or 2.0569: the sum, abating 20 from the index, is the logarithm of the number of seconds required, by which the distance of the stars is contracted by refraction: which therefore added to the observed distance gives the true distance cleared from the effect of refraction. This rule is founded on an hypothesis, that the refraction in altitude is as the tangent of the zenith distance: and the refraction at the altitude of 45 degrees being 57″, according to Dr. Bradley's observations, therefore the refraction at any altitude, calling the radius unity, is 57″ × tangent of the zenith distance. This rule is exact enough for the purpose of the calculation of the longitude from observations of the distance of the moon from stars at sea as low down as the altitude of 10°, for there the error is only 10″ from the truth. But if the altitude of the moon or star be less than 10°, the rule may be still made to answer sufficiently, by only first correcting the observed zenith distances by subtracting from them 3 times the refraction corresponding to them, taken out of any common table of refraction, and making the computation with the zenith distances thus corrected. This correction depends on Dr. Bradley's rule for refraction, which he found to answer, in a manner exactly, from the zenith quite down to the horizon, namely that the refraction is = 57″ × tangent of the apparent zenith distance lessened by 3 times the corresponding refraction taken out of any common table."