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the mouth was surrounded by a radiated substance, about the size of a silver penny, thicker and more callous than the coats of any other part. The internal aperture of the anus was composed of fibres interwoven with one another. From the apex to the base, on each side, descended obliquely, and winding, a smooth, solid body, in width about one fifth part of an inch, part of which separated in the examining, so that it is but imperfectly represented in the drawing. We cannot give a clearer idea of this body, than by saying, that it had greatly the appearance (except in size) of one of the small intestines, and was attached to the interior surface of the main body, much in the way as they are to the mesentery.
XCIII. Results of Observations of the Distance of the Moon from the Sun and fixed Stars, made in a voyage from England to the island of St. Helena, in order to determine the Longitude of the Ship, from Time to Time; with the whole Process of Computation used on this Occasion. By the Rev. Nevil Maskelyne, M.A., F.R.S. Dated St. Helena. Sept. 9, 1761. p. 558.
During the course of his voyage from England to St. Helena, Mr. M. made frequent observations of the distance of the moon from the sun and fixed stars, in order to determine the ship's longitude: and, as from their agreement with each other, he conceived it will be allowed, that the longitude may in general be ascertained by this method to sufficient exactness for nautical purposes, he thus communicated to the R. s. the results of his observations. He also delivers the whole process of computation, which he used in deducing the longitude from an observation, wherein he included several useful rules of his own investigation, which he apprehends render the calculation not only much shorter, but also much less intricate than it was before.
The time being determined by an altitude of the sun or a star, and the distance of a proper star from the moon's limb, or the distance of the sun and moon's nearest limbs in the first and last quarter, being carefully observed, the longitude may thence be found without any other observations; and this is the method proposed by the late Dr. Halley, which certainly deserves to be highly esteemed for its great simplicity, and the small number of observations which it requires. Yet Mr. M. owns himself of opinion with the Abbé De la Caille, that it will be more convenient at sea to require the aid of more observations, which is the method Mr. M. constantly practised during his voyage, having always two observers, who were ready, one to take the altitude of the star, and the other of the moon's upper or lower limb, at the instant he spoke when he had made the observation of the distance of the star from the moon.
Mr. M. can therefore answer from his own experience, both that the method
is practicable at sea, and also that so far from being less simple, it is more so than the other method; for the additional observations that it requires are very easily made, and even the error of a degree in the altitudes would seldom be of more consequence than an error of a minute, in taking the distance of the star from the moon; so that an error of 10 or 15' in the altitudes would be of no great prejudice: but with respect to the facility of the calculations, there is no comparison between the methods, the latter being much less intricate, and much more concise. The Abbe de la Caille requires the altitude of that part of the moon's limb from which the distance of the star is taken; but as at sea we can only take the altitude of the moon's upper or lower limb, an allowance might be made near enough, by estimation of the eye, for the difference of altitude between the moon's upper or lower limb, and that part of the limb from which the distance of the star is taken, he generally added the semidiameter of the moon to, or subtracted it from the observed altitude of the lower or upper limb, in order to have the apparent altitude of the centre, and he found the apparent distance of the star from the moon's centre, by adding or subtracting the moon's horizontal semidiameter, augmented according to her height, to or from the observed distance of the star from the moon's nearest or remotest limb.
This method will be exact enough, if the altitude of the moon or star be not less than 5o. Having thus got 3 sides of the spherical triangle formed by the moon, the star, and the zenith; namely, the apparent zenith distance of the moon, the apparent zenith distance of the star, and the distance of the star from the moon, he finds the effect of refraction and parallax, in altering the apparent distance of the star from the moon, by the two following rules.
Rule 1. To find the effect of refraction in contracting the apparent distance of two stars, or of the moon and a star.
Add together the logarithm-tangents of half the sum, and half the difference of the two zenith distances, the sum abating 10 from the index is the tangent of arc the first. To the logarithm-tangent just found, add the logarithm-cotangent of half the distance of the two stars, the sum abating 10 from the index is the tangent of arc the 2d. Then add together into one sum the logarithm-tangent of double the first arc, the co-secant of double the 2d arc, and the constant logarithm 2.0569; the sum abating 20 from the index is the logarithm of the num ber of seconds required; by which the distance of the stars, or of the moon and stars, is contracted by refraction: which therefore added to the observed distance, gives the true distance cleared from refraction.
This rule may be made universal, so as to serve with equal exactness almost down to the horizon, if the apparent zenith distances be diminished by 3 times the refraction belonging to them, found from any common table of refraction, and the computation be made with the zenith distances thus corrected. But if
the altitudes of the moon and star be not less than 10°, this correction will not be necessary. It will not be proper to make the observations, if the Ititudes of the star and moon are either of them less than 4° or 5°, on account of the variableness of refraction near the horizon.
Rule 2. To find how much the distance of the moon and a star is increased. or diminished, on account of the moon's parallax.
Add together into one sum the logarithm-tangents of half the sum, and half the difference of the zenith distances, and the cotangent of half the distance of the moon and star, all corrected for refraction; the sum, abating 20 from the index, is the tangent of arc the 3d, for which arc the 2d, found by the first rule, may be taken, without any sensible error.
Then if the zenith distance of the moon is greater than that of the star, take the sum of this arc and half the distance of the moon and star; but if the zenith distance of the moon is less than that of the star, take the difference of the said arcs; the tangent of the sum or difference, which may be called the parallactic arc, added to the cosine of the moon's zenith distance, and the logarathm of the moon's horizontal parallax in minutes abating 20 from the index, is the logarithm of the number of minutes required, by which the apparent distance of the moon from the star is always augmented by parallax, unless the zenith distance of the star be greater than that of the moon, and at the same time, arc the 3d be greater than half the distance of the moon and star, in which case the apparent distance of the moon and star is diminished by the parallax.
Therefore the number of minutes found by this rule is always to be subtracted from the observed distance of the moon and star, first corrected for refraction, inorder to find the true distance, cleared from the effect of parallax likewise; except in the case specified, when the zenith distance of the star is greater than that of the moon, and arc the 3d is at the same time greater than half the distance of the moon and star, when the correction is to be added. In computing these corrections, 4 places of figures beside the index will be sufficient.
It remains to be found by calculation, at what hour under a known meridian, the distance of the moon from the star will be the same as results from the observation, cleared of refraction and parallax. For this purpose it is necessary to compute the moon's longitude and latitude, and horary motion both in longitude and latitude, from the most exact tables, for the time under the known meridian, which is judged to correspond nearly to the given time of observation under the unknown meridian. The mean motions of the sun and moon he took from very exact tables, which he received as a present from the ingenious Mr. Gael Morris. composed by himself, from the comparison of a great number of Dr. Bradley's observations; to which he applied the lunar equations, as they stand in Mr. Mayer's printed tables. After finding the mean longitude of the star at the pre
sent time, he always allowed for its aberration in longitude, which will sometimes amount to 20", without considering the aberration in latitude, which can be of no consequence in a zodiacal star, such as those are which are always to be used in these observations. The distance of the star from the moon he computed from their longitudes and latitudes, by the two following rules:
Rule 1. Add together the logarithmic cosine of the difference of the computed longitudes of the moon and star, and logarithmic cosine of the difference of their latitudes, if they are of the same denomination; or sum if they are of different denominations; the sum abating 10 from the index is the cosine of the approximate distance.-This gives the absolute distance of the moon from the sun, without further calculation. But in case of a star, it is necessary to apply another rule also. Seven places of logarithms, besides the index, must be used in computing from this rule, and the calculation must be carried to seconds.
Rule 2. To the constant logarithm 3.5363, add the sines of the moon and star's latitudes, the versed-sine of the difference of longitude, and the co-secant of the approximate distance just found; the sum abating 40 from the index is the logarithm of a number of minutes, to be subtracted from the approximate distance, to find the true distance, if the latitudes of the moon and star are of the same denomination; but to be added if they are of contrary denominations. The 2d
of these two rules, though only an approximation, is so exact, that if the latitude of the moon was 5o, and that of the star 15°, the error resuiting would be only 10" in the distance. Four places of figures will be sufficient in computing from this rule.
If the distance of the moon from the star thus computed, at the assumed time under a known meridian, suppose Greenwich, agrees with the distance observed, corrected for refraction and parallax, the time at Greenwich was assumed right; and the difference between this time and the time of observation under the unknown meridian, is the difference of longitude in time between the said meridian and Greenwich; which is turned into degrees and minutes of the equator, by allowing 15° for every hour, and 1° for every 4 minutes of time.
But if the distance computed differs from the distance inferred from the observation, it must be found by proportion from the moon's horary motion to or from the star, how long time she will take to run over that difference; whence the time will be found at Greenwich, when the true distance of the moon from the star was the same with that resulting from the observation; which, compared with the time of the observation by the meridian of the ship, gives the difference of longitude from Greenwich as before. If the distance of the moon from the star computed, agrees with that resulting from observation within 10′ or 12′, and the distance of the moon from the star be not less than 20 or 30', the horary motion of the moon in the ecliptic may be taken for the horary motion of the moon to or from the star; but otherwise the moon's longitude and latitude must
be found at an hour's interval after the time assumed at Greenwich, by adding the horary motions to the longitude and latitude computed; and by the applica tion of the rules, the distance of the star from the moon must be found again at the end of that hour; which gives the horary motion to or from the star as required.
It is to be observed that the longitude thus found is that of the ship, at the instant when the altitude of the sun or star was taken, by which the watch was regulated, and not at the time of the observation of the distance of the star from the moon; for the watch being supposed not to vary considerably during that interval of time, must continue to indicate the time according to the meridian by which it was corrected; and the observation of the distance of the moon from the star showing the time at Greenwich, the difference must show the difference of longitude between that meridian and Greenwich.
Perhaps the following method of deducing the longitude from the observations may be least liable to mistake;-Find what the longitude by account was, at the instant of taking the sun or star's altitude, for the regulation of the watch; which being turned into time, at the rate of one hour for every 15o, and 4 minutes for every degree, add to the correct time from noon, when the distance of the star from the moon was taken, if the ship is to the west of Greenwich, or subtract from it, if it be to the east; this gives the apparent time at Greenwich by account; and the mean time is found, by applying the equation of time; to which time, compute the moon's longitude and latitude from the tables, and the distance of the star from the moon by the rules, and find by proportion as before, what time the moon will take to run over the difference between the distance computed, and that resulting from the observation; this turned into degrees and minutes of the equator, will show the error of the ship's account; and the following rules will show whether the ship is to the east or west of its account.
If the distance of the moon observed east of a star (or the sun in the first quarter) is greater than that computed, the ship is west of the longitude by account; but if the distance observed is less than that computed, it is east of account. If the distance of the moon observed west of a star (or the sun in the last quarter) is greater than computed, the ship is east of account; but if the distance observed is less than computed, it is west of account.
The horary motion of the moon in the ecliptic, may be thus made out very expeditiously from Mayer's equations, by the help of the principal arguments used in the computations of the moon's place. Call A, B, C, and D, the differences of the equations of the centre, evection, and variation, and reduction to the ecliptic, for 1° addition to their arguments; where it must be noted, that they must have the same sign as the equation, if it is increasing; but a contrary sign, if it is decreasing. Compute the value of 32′ 56′′ + A × 1 ×+ BX + X 10, which putн, and the true horary motion of the moon in her orbit