the base draw a perpendicular, which of course will be a parallel to that drawn by the other edge of the paper. Number the degrees of longitude properly and subdivide them into halves, quarters, sixths or tenths, as their size will admit. Next take the meridional parts corresponding to the extreme degrees of latitude which you intend the chart to contain, and dividing the meridional difference of latitude by sixty, to obtain the corresponding space in degrees and minutes on the longitude scale; take it from the bottom line, and lay it on each of the vertical ones, and join the extreme points of those lines, and you have then the boundary of the chart. In a similar way take from the longitude line the meridional difference of latitude between the lowest latitude, or that at the bottom of the chart, and every degree of latitude upwards to the extremity of the chart, and lay them upon each of the vertical lines, and you will have the points at which the degrees of latitude must be marked on the vertical lines. Subdivide the degrees of latitude in the same manner as the degrees of longitude are subdivided; and, as the degrees of latitude differ in length, this subdivision must be made for each separately. Next proceed to lay down the principal places on the chart according to their latitudes and longitudes, thus:-Place the edge of a parallel ruler on the parallel of latitude of the place which it is proposed to lay down, and draw a fine line in pencil through that part of the chart which contains the given longitude. Again place the edge of a parallel ruler on a meridian near the given longitude, and move it parallel to itself to the given longitude, and draw another fine pencil line across the former, and the point of section will be the required point on the chart. In the same way determine the positions of all the principal points on the chart, and then sketch in the coast by the eye, in a fine, free, distinct line, and shade it slightly on the land side. If necessary, mark by the shading whether the coast is rocky or sandy; and note in their proper places all rocky shoals, sand banks, depth of water, nature of bottom, and places for anchoring. Lastly, in one or more convenient places draw a compass at the intersection of a meridian and parallel. In charts constructed in this manner, the relative situations of places are properly represented, and the course from one place to another is correctly represented by the angle which a straight line drawn through both places makes with the meridian. The distance may in any case be correctly obtained from the chart from the application of the following principles. The difference of longitude is to the departure as the enlarged distance on the Mercator's chart is to the true distance; and radius is to the cosine of the middle latitude as the difference of longitude is to the departure. Hence as radius is to cosine of the true middle latitude, so is the enlarged distance to the true distance. If therefore from the top of the chart on any meridian there be laid a scale of natural cosines, from 0°, onwards to 90°; the distance may be found thus: take the distance on the chart between the proposed places in your compasses, and lay it on the graduated parallel, at the top of the chart ex actly from the end of the meridian on which the graduated line of cosines is laid down, and draw a line diagonally from the point determined by the compasses to 90° on the line of cosines. On the line of cosines find the true middle latitude, and through it draw a parallel of latitude cutting the diagonal line; the distance between the graduated meridian and the diagonal in this middle latitude, measured on the graduated parallel at the top or bottom of the chart, is the degrees and minutes of the true distance. The method of finding the true middle latitude has been already shown, but if the diff. lat. do not exceed two or three degrees the mean middle latitude may be used instead of it. NAUTICAL ASTRONOMY. If the course of a ship and the distance sailed could always be correctly ascertained, the principles of navigation which we have already explained would be sufficient for conducting a ship to any part of the world. But there are many circumstances which often render the real course and distance very uncertain, and it becomes therefore of the utmost importance to be able to find the place of the ship from time to time by observation on the heavenly bodies. This application of the principles of astronomy is called nautical astronomy. The earth is a spherical body which revolves on an imaginary line called its axis from west to east in twenty-four hours; and in consequence of this rotation the heavenly bodies appear to revolve from east to west in the same time. While the earth is thus daily performing its rotation on its axis, it is carried round the sun from west to east in a year, the axis of rotation continuing during the whole year parallel to itself; and in consequence of the comparative sinallness of the orbit which it describes, when compared with the distance of the fixed stars, the axis appears to be always directed towards the same points in the heavens; and these points are called the poles of the celestial sphere. The moon accompanies the earth in its annual revolution round the sun, and it also revolves round the earth once in a month, moving apparently among the stars, towards those that are eastward of her, and from those that are westward of her. There are ten planets besides the earth, which like it revolve round the sun, and several of these are accompanied by smaller ones called satellites, which revolve round them as the moon the earth's satellite revolves round it. All these planets shine only by reflecting the light of the sun.. Those which are further from the earth than the sun are called superior, and those which are nearer to the earth than the sun are called inferior planets. For a full explanation of the celestial motions see the article ASTRONOMY. But it is only the apparent motions of the heavenly bodies that are the objects of consideration in nautical astronomy; and these motions would be the same as they are, if the earth were stationary and the sun revolved annually among the fixed stars in the plane of the earth's orbit; the planets at the same time performing their apparent evolutions round him on the immeasu rably distant concavity of the celestial sphere, while that sphere, with the sun, stars, and planets, revolved daily round the earth from east to west. The plane of the earth's orbit produced to the heavens, or that circle in the heavens in which the sun appears to move among the stars, is called the ecliptic, and circles perpendicular to it are called circles of celestial latitude. The terrestrial equator produced to the heavens is called the celestial equator; the meridians produced in like manner are called celestial meridians, and the parallels of latitude produced in the same. way are called parallels of declination. The ecliptic intersects the equator in two points called equinoctial points; that at which the sun passes from the south to the north side of the equator is called the first point of Aries; that being the first of the twelve equal parts into which astronomers divide the ecliptic. The ecliptic and equator are inclined to each other in an angle which at present is about 23° 28', and the point in which they intersect has a progressive motion from west to east. See ASTRONOMY, PRECISSION. The inclination of the equator to the ecliptic is called the obliquity of the ecliptic. The latitude of a celestial object is its distance from the ecliptic measured on a perpendicular to that circle, and the longitude of a celestial object is the arc of the ecliptic between the first point of Aries and the perpendicular to the ecliptic passing over the object; or it is the angle at the pole of the ecliptic, intercepted by two great circles, one passing through the first point of Aries, and the other over the object. The declination of a celestial object is its dis-, tance from the equator measured on the meridian passing over the object, and the right ascension of a celestial object is the arc of the equator between the first point of Aries and the meridian passing over the object; or it is the angle at the pole of the equator intercepted between the meridian passing through the first point of Aries, and that passing over the object. The sensible horizon is a plane touching the earth at the point at which the observer is situated, and the rational horizon is a plane passing through the centre of the earth parallel to the sensible one. The pole of the horizon over the head of the observer is called the zenith, and the opposite point the nadir. Great circles passing through the zenith, and of course cutting the horizon perpendicularly, are called vertical circles, azimuth circles, or altitude circles. The angle which any vertical circle makes with the meridian is called the azimuth of that circle or of any object over which it passes. The vertical circle perpendicular to the meridian, or that which cuts the horizon in the east and west points, is called the prime vertical. The amplitude of an object is its distance from the east or west at rising or setting; or the angle which the vertical circle passing over it at those times makes with the prime vertical. Circles parallel to the horizon are called parallels of altitude, and that which is 18° below the horizon is called the twilight circle, because in a mean state of the atmosphere twilight begins in the morning and ends in the evening when the sun is on that circle. The polar distance of an object is its distance from the celestial pole nearest the observer. A sidereal day is the interval between the two successive times in which a fixed star attains the same situation; and a solar day is the interval between the two successive times in which the sun arrives at the same meridian. The sidereal day at any place commences when the first point of Aries is on the meridian of that place, and the solar or apparent day when the sun is on the meridian of the place. Mean time is that which would be shown by the sun if it revolved on the plane of the equator with a uniform angular motion equal to the mean motion of the sun in the ecliptic. The sidereal time of day is the angle at the pole betweeen the meridian of the place and the meridian passing over the first point of Aries, and the apparent time of day is the angle at the pole included between the meridian of the place, and the meridian passing over the true place of the sun; and the inean time of day is the angle included between the meridian of the place and the mean place of the sun reckoned on the equator: the angles in each case being reckoned from the meridian towards the west, or in the direction of the apparent daily revolution of the heavenly bodies. The angle included between the meridian of any place, and the meridian passing over any celestial object, is called the meridian distance of that object. A mean solar day is longer than a siderial one; for the sun daily advances in the ecliptic eastward so far that the mean interval between his transits is about 3'. 55'9". greater than the interval between the transits of a star. The apparent altitude of a celestial object is its distance from the sensible horizon measured on a vertical circle, and the true altitude of a celestial object is its distance from the rational horizon measured also upon a vertical circle The true and apparent zenith distances are the complements of the true and apparent altitudes. and, when an object is on the meridian of the place of observation, its altitude and zenith disance are termed its meridian altitude and meridian zenith distance. When the altitude of an object is spoken of, the altitude of its centre is generally understood. But altitudes observed on or above the surface of the earth require several corrections before the true altitude is obtained. First for semidiameter. The semidiameter of a celestial object is the angle which the radius of its apparent circular disk subtends at the eye of the observer. If the altitude of the lower edge (or lower limb as it is called) is observed, the semidiameter is added; but, if the upper limb is observed, the semidiameter is subtracted from the observed altitude to obtain the apparent altitude of the centre. If A B (fig. 3. plate II.) be the horizon, C the centre of the object, D its lower and E its upper limb, and A the place of the observer, then DAB is the altitude of the lower limb, EA B that of the upper limb, and CAD or CAE the semidiameter. Second, the parallax. The angle which the radius of the earth on which the observer is situated subtends at the centre of the object is called its parallax. If the object is in the horizon it is called the horizontal parallax; if above the horizon it is called the parallax in altitude. The sines of the horizontal parallaxes of different objects are inversely as their distances from the centre of the earth. For fig. 4, plate II., ADB is the horizontal parallax of D, and ACB the horizontal parallax of C; and ACAD:: sin. A DC: sin. AC B. Further with respect to the same object, rad. : sin. hor. par. :: cos. app. alt. sin. par. in alt. For fig. 5, plate II., Cis the horizontal parallax, and D the parallax in altitude; and, AC and A D being equal, we have AC: AB:: AD: AB; but AC: AB :: rad. cos. C; and AD: AB :: sin. ABD cos. CBD : sin. D; whence rad. : sin. ACB :: cos. CBD : cos. D. The angle DBC is the apparent, and the angle DAF DEC the true altitude. But DEC DBE + D; hence the parallax must be added to the apparent altitude to obtain the true altitude. Third, the dip. We have hitherto considered the observations as made on the surface of the earth, but it will seldom, especially at sea, be possible so to make the observations. The allowance that must be made for the apparent depression of the horizon arising from the elevation of the eye above the surface of the earth is called the dip. Let A C, fig. 6, plate II., be the height of the eye, A B the section of a plane parallel to the horizon, A D a line touching the earth at D, and AS a line in the plane of AB and AD drawn to the centre of the object; then BAD is the dip of the horizon, SAD the observed altitude, and SA B (the difference) the true altitude. But, in what is called a back observation, the depression of the point opposite to the sun is measured and in this case the dip is added to the observed depression, to obtain the true depression, which is the same as the true altitude. For (fig. 7, plate II.) if S be the celestial object, and S the point diametrically opposite to it below the horizon; then S A B, the altitude, is equal to S' A B' the depression; and B' A D' the dip, added to S' A D' the depression below the visible horizon at D', is equal to the altitude. The dip is computed from the following formula, where D= dip, h height of the eye, and a the earth's diameter. D Cot = h+ h ५ = But from D so com puted one-twelfth of itself is generally deducted for the effect of horizontal refraction. The following TABLE of DIP was so com puted The horizontal parallax of the sun being always nearly 9", its parallax in altitude may conveniently be entered in a table. Altitude in degrees 0° 12° 15° 30° 33° 42° 51° 60° 69° 75° 81° 90°. The difference between mean and apparent time is called the equation of time. The parallax of the moon being exceedingly tracted from its right ascension, will leave the variable, tables of her parallax in altitude corres- sidereal time; and, when west of the meridian, ponding to all the variations in her horizontal its meridian distance added to its right ascension, parallax must necessarily be extensive. Such will give the sidereal time. Further, it appears, tables, however, are given in many practical that the sun's right ascension, subtracted from the works on navigation.-See Inman's, Norie's, Ria's sidereal time, leaves the apparent time; and and Riddle's Nautical Tables. It may, however, conversely that the sun's right ascension, added be readily computed thus:-Add the secant of to the apparent time, gives the sidereal time. the apparent altitude to the proportional logarithm of the horizontal parallax, and the sum will be the proportional logarithm of the parallax in altitude. The refraction is the last correction of apparent altitudes; and, whether the altitudes are observed by a back or a fore observation, this correction is subtractive. See an extensive table of refractions under the article ASTRONOMY. From what has been said it may be inferred that, when a star is on the meridian of any place, the sidereal time at that place is equal to the star's right ascension; and that, when a star is east of the meridian, its meridian distance sub In the Nautical Almanac, the sun's longitude, right ascension, declination, and the equation of time, are given for the instant at which the sun is on the meridian of Greenwich every day, and the moon's longitude, latitude, right ascension, declination, horizontal parallax, and semidiameter, for noon and midnight, apparent time at Greenwich, for every day. As 360° of longitude correspond to one revolution of the earth, which is measured by twentyfour hours of time, 15° of longitude correspond to an hour of time, 1° of longitude to four minutes of time, 1' of longitude to four seconds of time, &c. The equation of time as given in the Nautical Almanac is intended to be applied to apparent to obtain mean time: when mean time is given, and apparent time required, the equation of time must be applied with a sign contrary to that given in the almanac. It is evident that the difference between the times at Greenwich, whether mean, sidereal, or apparent, and the corresponding times at any other meridian is the longitude of that meridian from Greenwich west, when the Greenwich time is the greater, or more forward, and east when the Greenwich time is the less or behind. See LONGITUDE. Chronometers for finding the longitude at sea, as well as clocks and watches for use in civil life, are regulated by mean time; but observatory clocks by sidereal time, for the convenience of determining the right ascensions of celestial objects by the times of their passing the meridian. On land, when nautical instruments are used for taking altitudes, the distance of the object from its image, as reflected from a fluid or polished horizontal surface, is taken, and its half is the measure of the altitude. The difference between the parallax of any object and the refraction corresponding to its altitude, is called the correction of altitude; and, the correction for semi-diameter and dip being first applied, this correction is additive in the case of the moon, but subtractive with respect to all other celestial objects, as the moon's parallax is greater than the refraction at any altitude; but the parallax of any other object is less than the refraction. The fixed stars have no sensible parallax. The astronomical day commences at the noon of the civil day, and the hours are reckoned straight forward to twenty-four. Therefore in the afternoon of the civil day the hours of the astronomical and civil day are the same, but in the forenoon of the civil day they differ twelve hours. Thus April 4th at 6 h. 8 m. 5 s. civil time, is also April 4th 6h. 2 m. astronomical time; but April 5th, 6 h. 2 m. A. M. civil time, is April 4th, 18 h. 2 m. astronomical time. NTRODUCTORY PRACTICAL PROBLEMS. PROB. I. To convert longitude into time.Multiply the longitude by 4, divide the degrees of the product by 60, and the quotient will be the hours, the remainder the minutes, and the other parts of the product the seconds, &c., in the required time. Example.-Required the time corresponding to 83° 12′ 9′′ of longitude? 83° 12′ 9′ 60) 33-2 48 36 4 Hours 5.32 48 36 Ans. PROB. II.-To convert time into longitude. Reduce the hours into minutes, and divide the whole by 4, and the quotient will be the degrees, minutes, &c., of the corresponding longitude. PROB. III.-Given the time at any place, the longitude of that place, to find the corresponde Greenwich time. Reduce the longitude into time, and addan the time at the place if west, but subtract east, and the sum in the remainder will be t Greenwich time. If in adding, the sum should exceed tw four hours, the excess will be the Greenwies re past noon on the follow day; and, if in s tracting, the longitude in time should excee astronomical time at the place, increase the gre time by 24 hours, before subtracting the le tude in time, and the remainder then will be time at Greenwich past noon of the precedy day. Example 1.-In longitude 18° 4′ E. Sept ber 3, 8h. 5m. 10s., A. M., what is the astrale mical time at Greenwich? Astronom. time, Sept. 2d 20h. 5m. 10s. Long. in time E. 1 12 16 PROB. IV. To take the right ascension, deelnation, &c., of the sun and moon, from the Near cal Almanac, for any time. Find the Greenwich time corresponding to the time at the place, and its longitude." Then, if th object be the sun, take from the Nautical Al ! nac the required number for the noon preceding the instant for which it is wanted, and the change of the number in twenty-four hours. The say, As twenty-four hours is to this daily charge, so is the Greenwich time to the correction to te added to, or subtracted from, the number at the preceding noon, according as it is increasing decreasing, to obtain its value at the required Example. What are the right ascension and declination of the sun, the equation of time, the right ascension, declination, horizontal parallax, and semidiameter of the moon, March 3d, 1828, at 9h. 48m. 20s. P. M. in long. 65° W. ? 5° 21 14 5 22 14 12 1.8 As the Greenwich time is 14h. 8m. 20s. past noon, it is 2h. 8m. 20s. past midnight. Therefore we take out the numbers for the moon for midnight of March 3d, and then change from that time till the following noon. Per Nautical Almanac, March 3d, 1828, midnight. 1 15 19 Declin. S. Var in 12h. > Hor. par. Var. in 12h. 15" Sem. Var. 12h. 15′ 18" 4" 56 10 PROB. V. To find the true altitude of a lestial object from its observed altitude. ce From the observed altitude subtract the dip; and, if the lower limb is observed, add the semidiameter; if the upper, subtract it. Then take the difference between the refraction and the parallax in altitude, and add it if the object be the moon; but subtract it if the object be the sun or a star, and the result will be the true altitude. Example. What is the true altitude of a fixed star, whose observed altitude is 28° 19′, the height of the eye being sixteen feet? Here there is no parallax and no semidiameter to allow for Observed altitude 28° 19′ 0′′ Dip Correction of O's R. A. O's declin >'s R. A. 's hor. par. >'s semid. 3 56 2753 31' 38" Prop. long. 7552 35 True altitude 31 3 PROB. VI. To find the sun's declination when he is on a given meridian. Take the declination for the noon of the given day from the Nautical Almanac, and its daily change, noting whether it is increasing or decreasing. Then say, as 360°: the daily change :: the longitude of the given meridian: the correction to be added, if the declination is increasing, and the longitude west, or if the declination is decreasing, and the longitude east; otherwise to be subtracted from the declination at noon, before taken out. Example. Required the sun's declination, when on the meridian of 40° W., March 4th, 1828? |