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, with Demonstrations of the Same: By the Rev. Nevil Maskelyne, A. M. Fellow of Trinity College, in the University of Cambridge, and F. R. S.

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, with Demonstrations of the same: By the Rev. Nevil Maskelyne, A. M. Fellow of Trinity College, in the University of Cambridge, and F. R. S. Read Nov. 15, 1764. The following rules, excepting one, are the same which I have already communicated to the Royal Society, but without demonstration, in a letter to the reverend Dr. Birch from St. Helena, containing the results of my observations of the distance of the Moon from the Sun and fixed stars, taken in my voyage thither, for finding the longitude of the ship from time to time; since printed in Part II. Vol. LII. of the Philosophical Transactions for 1762. The two rules for the correction of refraction and parallax I have also since communicated to the public in my British Mariner's Guide to the discovery of longitude from like observations of the Moon; and have added in the Preface a rule for computing a second but smaller correction of parallax, necessary on account of a small imperfection. imperfection lying in the first rule derived from the fluxions of a spherical triangle. To the rules I have here subjoined their demonstrations. With respect to the usefulness of these rules, I cannot but entertain 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 Ephemeris, 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 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. Explication of the foundation of the preceding rule. This rule is founded upon 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 $= 57'' \times$ 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 $10^\circ$, for there the error is only $10''$ from the truth. But, if the altitude of the Moon or star be less than $10^\circ$, the rule may be still made to answer sufficiently, by only first correcting the observed zenith distances by subtracting from them three 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 upon 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 $= 57'' \times$ tangent of the apparent zenith distance. lessened by three times the corresponding refraction taken out of any common table. Demonstration of the preceding rule. Let $ZXY$ Tab. XVII. Fig. 1. represent a spherical triangle formed by great circles joining the zenith $Z$ and the stars $X$ and $Y$. Refraction acting in the vertical circles $ZX$ and $ZY$ will carry the star $X$ nearer the zenith by a quantity $Xb = 57'' \times$ tangent of $ZX$, and the stars $Y$ towards $Z$ by the quantity $Yd = 57'' \times$ tangent of $ZY$; so that the apparent distance of the two stars will be $bd$ instead of $XY$ or less than $XY$, the true distance, by the Sun of the two little spaces $Xa$, $Ye$, terminated by the perpendiculars $ba$ and $de$. The little space $Xa = Xb \times$ cosine of the angle $ZXY$ (calling radius unity) $= 57'' \times$ tang. of $ZX \times$ cosine of angle $ZXY$, or, by spherics, $= 57'' \times$ tang. of $XP$. ($ZP$ being an arch drawn from $Z$ perpendicular to the arch $XY$). In like manner the little space $Ye = 57'' \times$ tang. of $YP$; and therefore $Xa + Ye$ or the total effect of refraction $= 57'' \times$ tang. $XP +$ tang. $YP$. Let $M$ be the middle of the arch $XY$, and put the tangent of $XM$ or $YM$ or $\frac{1}{2} XY = t$, and the tangent of $MP$, or the distance of the perpendicular $ZP$ from $M$ the middle of the arch $XY = n$. By trigonometry, tang. of $XP$ or $XM + MP = \frac{t+n}{1-tn}$, and tang. of $YP$ or $YM - MP = \frac{t-n}{1+tn}$; the sum of which, or tang. $XP +$ tang. $YP = \frac{t+n}{1-tn} + \frac{t-n}{1+tn} = \frac{2t+2tn^2}{1-t^2n^2} = \frac{2tn}{1-t^2n^2} \times \frac{1+n^2}{2n} \times 2$. Now $2tn$ \[ \frac{2tn}{1-t^2n^2} \] is the tangent of double the angle whose tangent is \( tn \), and \( tn \) or the product of the tangents of \( \frac{1}{2} XY \) and \( MP \), by spherics, is equal to the product of the tangents of half the sum and half the difference of the zenith distances \( ZX \) and \( ZY \); whence \( \frac{2tn}{1-t^2n^2} \) is equal to the tangent of double arch the first found by the rule. Also arch the second found by the rule being by spherics \( = MP \), whose tangent is represented by \( n \), and \( \frac{1+n^2}{2n} \) being by trigonometry equal to the cosecant of double the arch whose tangent is \( n \), therefore \( \frac{1+n^2}{2n} = \) cosecant of \( 2MP \) or double arch the second. Whence the rule is manifest; namely, that \( Xa + Ye \), the total effect of refraction in contracting the apparent distance of the two stars \( = 57'' \times \tan. XP + \tan. YP = \) double \( 57'' \) or \( 114'' \times \) tang. of double the first arch \( \times \) cosecant of double the second arch. Q.E.D. **Remark.** When the perpendicular arch \( ZP \), let fall from the Zenith on the arch \( XY \), falls without the triangle \( ZXY \), the effect of refraction in diminishing the apparent distance of the stars \( X, Y \) is the difference of \( Xa \) and \( Ye \): but the rule being general, gives always the sum or difference, which-ever it be, which is a great advantage, and removes all grounds of ambiguity in the correction of refraction; as the the total effect thereof always diminishes the distance of two stars from each other, however they are posited. A RULE To compute the contraction or augmentation of the apparent distance of the Moon from a star, on account of the Moon's parallax; the zenith distances of the Moon and star, 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 of the Moon and star, and the cotangent of half the distance of the Moon from the star; the sum, abating 20 from the index, is the tangent of an arch, which call A. Then, if the zenith distance of the Moon is greater than that of the star, take the Sum of the arch A, just found, and half the distance of the Moon from the star; but, if the zenith distance of the Moon be less than that of the star, take the difference of the said arch A and half the distance of the Moon from the star; and the sum, or difference call B. To the tangent of B, thus found, add the cosine of the Moon's zenith distance, and the logarithm of the Moon's horizontal parallax, expressed in minutes and decimals; the sum, abating 20 from the index, is the logarithm of the effect of parallax, tending always to augment the apparent distance of the Moon from the star; except the zenith distance of the Moon be less than that of the star, and, at the same time, the arch A be greater than half the distance of the Moon from the star, in which case the effect of parallax diminishes the apparent distance of the Moon from the star. DEMONT DEMONSTRATION. In the spherical triangle ZLS, see Fig. 2, 3, 4, and 5th, Z represents the zenith, L the Moon, and S the star; the effect of parallax depressing the Moon from L to r, r is the apparent place of the Moon, and rS the apparent distance of the Moon from the star; let fall the perpendicular Lt upon rS, produced if necessary, and rt will be the difference of LS and rS, or the effect of parallax. Draw the arch ZP perpendicular to rS, and let M be the middle of rS. The Moon's parallax in altitude, being to her horizontal parallax, as the sine of her apparent zenith distance, to the radius, \( Lr = \text{Moon's horizontal parallax} \times \text{sine of } Zr \); and rt the effect of parallax upon the apparent distance of the Moon from the star will be \( Lr \times \cos ZrS = \text{horizontal parallax} \times \sin Zr \times \cos ZrS \) (or, because tan. rP : cos. ZrP :: tan Zr : rad :: fin Zr : cos Zr; and therefore fin. Zr × cos ZrS = cos. Zr × tan rP) = horizontal parallax × cos. Zr × tan. rP agreeably to the rule. For it is evident by spherics that the arch A, found by the rule, is the same with MP the distance of the perpendicular from the middle of the arch rS: and it is evident, by the inspection of the figures, that the arch B or rP is equal to the sum of rM and MP, if the zenith distance of the Moon be greater than that of the star, as in Fig. 2d and 4th; but is the difference of rM and MP, if the zenith distance of the Moon be less than that of the star, as in Fig. 3d and 5th. Lastly, it may appear from the consideration consideration of the figures, that, as the effect of parallax depresses the Moon directly towards the horizon, so it will always encrease her apparent distance from a star, except in the case represented by Fig. 5th; that is to say, unless the zenith distance of the Moon be less than that of the star, and, at the same time, the arch MP be greater than \( rM \) or half the distance of the Moon from the star. Q.E.D. Remarks on the use of the two foregoing rules. It has been remarked, after the rule for refraction above, that if the altitudes of the Moon or star are under 10 degrees, the zenith distances must be first lessened by 3 times the refractions corresponding to their respective altitudes before the effect of refraction be computed. But in order to compute the effect of parallax from the second rule, the observed distance of the Moon from the star must be first corrected by adding the effect of refraction to it found by rule the first; as must the observed altitudes of the Moon and star be also corrected by taking from them their respective refraction in altitude, and the corrected arches thus found must be made use of in computing the parallax. Only, if the altitudes of the Moon and star are both 10 degrees or more, part of the calculation of rule the second may be saved, and arch the second, found by rule the first, taken for arch A in the second rule without any sensible error. In this case, it will be most convenient to observe the following order of computation instead of that before prescribed. prescribed to be used when the altitudes are under 10 degrees. 1st. Making use of the apparent altitudes of the Moon and star uncorrected, compute arches the first and second by the directions contained in the rule of refraction. 2dly. Taking arch the second for arch A in the rule of parallax, compute the effect of parallax according to rule the second. 3dly. With arches the first and second compute the effect of refraction by rule the first. 4thly, and lastly. Applying the two corrections of parallax and refraction duly, according to the rules, to the observed distance of the Moon from the star, you will have the true and correct distance of the Moon from the star, cleared both of refraction and parallax. A RULE For computing a second, but smaller correction than the first, necessary to be applied to the observations of the distance of the Moon from a star on account of parallax. Call the principal effect of parallax, found by the preceding rule, the parallax in distance; and find the parallax answering to the Moon's altitude. Then to the constant logarithm 0.941 add the logarithm of the sum of the parallax in altitude and the parallax in distance, the logarithm of the difference of the same parallaxes, and the cotangent of the observed distance of the Moon from the star (corrected for refraction, and the principal effect of parallax), the sum, abating 13 from the index, is the logarithm logarithm of the number of seconds required, being the second correction of parallax; and is always to be added to the distance of the Moon from the star, first corrected for refraction, and the principal effect of parallax found above, in order to obtain the true distance; unless the distance exceeds 90 degrees, in which case it is to be subtracted. DEMONSTRATION. Let L Fig. 6. represent the Moon's true place in the sphere, and \( r \) her apparent place as depressed by parallax, \( S \) the place of the star, and \( Lt \) a perpendicular let fall from the true place of the Moon \( L \) upon the great circle \( rS \) joining the star \( S \) and the apparent place of the Moon \( r \); (all as in the four figures belonging to the preceding rule). Let \( La \) be the arch of a parallel circle described from the star \( S \) as a pole through the true place of the Moon \( L \). \( Sa \) terminated by the parallel circle \( La \), and not \( St \) terminated by the perpendicular \( Lt \), as was supposed in the former demonstration, is equal to \( SL \) or the true distance of the Moon from the star, which was therefore computed too small from the former rule by the little space \( at \). Let \( LT \) and \( aT \) be the equal tangents of the equal arches \( LS \) and \( aS \) in \( L \) and \( a \), meeting in the radius \( CS \), drawn from the centre of the sphere \( C \) and produced, in \( T \). The space \( Lat \), on account of its smallness, may be looked upon as lying all in one plane namely \( LaT \), and \( La \) as the small arch of a circle described from the point \( T \) as a centre with the line \( LT \) as a radius, thro' through L and a, Lt as the sine, and at as the verse-sine of the arch La; and consequently at equal to the square of Lt divided by 2 LT. But, the triangle Lrt being right-angled in t, the square of Lt is equal to the difference of the squares of Lr and rt, and consequently to the product of their sum and difference; that is to say, \( at = \frac{Lr + rt \times Lr - rt}{2 \times LT} \) or (because the tangent TL is equal to the square of the radius CS divided by the cotangent of LS) \[ = \frac{Lr + rt \times Lr - rt \times \text{cotangent of LS}}{2 \times \text{square of CS}}. \] Now suppose the spaces Lr, rt to be expressed in minutes, which will be most convenient in practice, then the radius of the sphere CS must be taken equal to 3437\(\frac{3}{4}\), for so many minutes are contained in an arch of a circle equal to its radius: and at will be \[ = \frac{Lr + rt + Lr - rt \times \text{cotan. of LS}}{2 \times 3437\frac{3}{4} \times 3437\frac{3}{4}}. \] But, the cotangents of similar arches of circles of different radii being directly as the radii, therefore the cotangent of LS to the radius CS or 3437\(\frac{3}{4}\), is to the cotangent of the same arch to 1000000000, which is the radius to which the logarithmic tables are adapted, its logarithm being 10; as 3437\(\frac{3}{4}\) to 1000000000. Therefore the cotangent of LS = tabular cotangent of LS \(\times\) 3437\(\frac{3}{4}\) \(\times\) 3437\(\frac{3}{4}\) \(\times\) 1000000000, which, being substituted in the value of at above, gives at, expressed in minutes \[ = \frac{Lr + rt + Lr - rt \times \text{tabular cotangent LS}}{687550000000}. \] or, multiplying by 60, the value of at will come out in seconds. conds = \frac{Lr + rt \times Lr - rt \times \text{tabular cotangent of LS}}{114600000000} The logarithm of the denominator 114600000000 is 12.059, instead of subtracting which, when the operation is performed by logarithms, add 0.941 (its compliment to 13) to the value of the numerator found in logarithms, and subtract 13 from the index: the remainder will be the value of at in seconds. Q.E.D. A concise rule to find the distance of the Moon from a zodiacal star, very nearly; the difference of the longitudes of the Moon and star, and the latitudes of both being given. To the cosine of the difference of the longitudes add the cosine of the difference of the latitudes, if both of the same denomination, or sum; if of contrary denominations, the sum of the two logarithms, abating 10 from the index, is the cosine of the approximate distance. This gives the true distance of the Moon from the Sun, being then nothing more than the common rule for finding the hypotenuse of a right-angled spherical triangle from the two sides given. But in the case of a zodiacal star apply the following correction to the approximate distance thus found. To the constant logarithm 5.3144 add the sine of the Moon's latitude, the sine of the star's latitude, the verse-sine of the difference of longitude, and the cosecant of the approximate distance; the sum of these 5 logarithms, abating 40 from the index, is the logarithm of a number of seconds, which sub- tracted from the approximate distance, found before, if the latitudes of the moon and star are of the same denomination, or added thereto, if they are of different denominations, gives the true distance of the Moon from the star. N. B. This rule, though only an approximation, is so very exact, that even, if the latitude of the Moon was $5^\circ$, and that of the star $15^\circ$, the error would be only $10''$; and if the latitude of the Moon be $5^\circ$, and that of the star $10^\circ$, the error is only $4''\frac{1}{2}$; and if the latitudes be less, will be less in proportion as the squares of the sines of the latitudes decrease. **DEMONSTRATION.** Let $P$ [Fig. 7.] represent one of the poles of the ecliptic and $Q$, $R$ the places of the Moon and star. From $R$ let the arch of a great circle $RD$ be drawn perpendicular to $PQ$. By spherics, the tangent of $PD = \text{tangent of } PR \times \text{cosine of the angle } RPD$. And, by trigonometry, cosine of $QD$ or $(QP - PD) = \text{cos. } QP \times \text{cos. } PD + \text{sin. } QP \times \text{sin. } PD = \text{cos. } QP \times \text{cos. } PD + \text{sin. } QP \times \text{tan. } PD = \text{cos. } PD \times \text{cos. } QP + \text{sin. } QP \times \text{tan. } PD = \text{cos. } PD \times \text{cos. } QP + \text{sin. } QP \times \text{tan. } PR \times \text{cos. } P : 1$. But, by spherics, $\text{cos. } QD : \text{cos. } PD :: \text{cos. } RQ : \text{cos. } PR$. $\therefore \text{cos. } RQ : \text{cos. } PR :: \text{cos. } QP + \text{sin. } QP \times \text{tan. } PR \times \text{cos. } P : 1$. Whence $\text{cos. } RQ = \text{cos. } PR \times \text{cos. } QP + \text{sin. } QP \times \text{sin. } PR \times \text{cos. } P : \text{Now, by trigonometry, cos. } (QP - PR) = \text{cos. } Nn2$. \[ = \cos QP \times \cos PR + \sin QP \times \sin PR; \] whence \( \sin QP \times \sin PR = \cos (QP - PR) - \cos QP \times \cos PR; \) which being substituted above, gives \( \cos RQ = \cos (QP - PR) \times \cos P - \cos PR \times \cos QP \times \cos P + \cos PR \times \cos QP = \cos (QP - PR) \times \cos P + \text{verse-fine} P \times \cos PR \times \cos QP. \) Now put \( \cos (QP - PR) \times \cos P = \cos G, \) or the approximate distance, then \( \cos RQ - \cos G, \) or (because the difference of \( RQ \) and \( G \) is but small) \( G - RQ \times \sin \left( \frac{G + RQ}{2} \right) \) \( = \text{verse-fine} P \times \cos PR \times \cos PQ \) nearly. Whence \( RQ = G - \frac{\text{verse-fine} P \times \cos PR \times \cos QP}{\sin G + RQ} \) nearly \( = G - \frac{\text{v. f.} P \times \cos PR \times \cos QP}{\sin G} \) nearly. Q. E. D. Note, the error of this formula arises from taking \( G = \frac{G + RQ}{2} \) by which means it will always give \( RQ \) too great, nearly by the following quantity, \( \frac{1}{2} S q \cos PR \times S q \cos QP \times \cot G \times S q \tan \frac{1}{2} G. \) This comes to a maximum when \( G \) is \( 60^\circ, \) and is then \( = \frac{1}{8} \sqrt{\frac{1}{3}} \times S q \cos RP \times S q \cos PQ. \) If the latitudes of the Moon and star are both \( 5^\circ \) it is \( = 1''. \) If the Moon's latitude be \( 5^\circ, \) and that of the star \( 10^\circ, \) it is \( = 4'' \frac{1}{2}; \) and if the latitude of the star be \( 15^\circ \) it is \( = 10''. \) XLIX.