middle latitude at the side, and the nearest degree of the difference of latitude at the top, and the correction is found under the latter in a line with the former, which correction, added to the mean middle latitude, gives the true middle latitude. If there are odd minutes, a proportional part may be allowed for them. We shall presently illustrate these tables by an example. (53.) Before we proceed to solutions by actual computation, we shall restate the four proportions we have obtained. In addition to these, we must remember that the three right angled triangles of By the plane triangle, R: d. d: l= or dist.log. c', or cos. course, 2,845098 log. dist. 9,582840 log. cos. of 6 points. 12,427938 10,000000 log. rad. c' R d log. R, = the two latitudes = 47 43 47 the mean middle lat. Now in Workman's Table, (Table II.) in a line with 48° and under 4° is found 3', and under 5° is found 4'; therefore we will call the correction + 3′ 30′′, but 47° 43' 47" + 3′ 30′′ = 47° 47′ 17′′ the true middle latitude, which in this case, as the course is so nearly west, differs but little from the mean. ΤΖ and therefore log. L с 10,382776 log. tan. 6 points. = = 21 13 33 W = longitude in. = 16° 2′ 28′′. ing for this current: it is now required to find the latitude and longitude in, when the proper allowance is made for the effect of this current. And as by the d d By Mercator triangle, proportion B, we have R:T::m: L: 9,298662 log. tan. 1 point. 2,149219 log. mer. diff. lat. = 141. 11,447881 10,000000 log. rad. 1,447881 log. diff. long. = 28',05 = 28′ 3′′. = 47 7 55 = the true latitude in. With the true diff. lat. and long. above obtained by resolving the traverse, the true course is found as in Example V. The true course S 74° 37' W is nearly WSW W; supposing the variation 24 points westerly, this will give a compass course WbN; but the course by compass from the Lizard to the Southernmost of the Scilly Islands is WNW, consequently the ship will pass one point clear of these islands, so far as our current affects her; but should the tides cooperate with the current, and produce a more rapid drift to the Northward, the ship might be lost on these islands, or the rocks around them; which illustrates our observation in Article 22, with respect to the judgment and skill requisite in a pilot, to whom the difficult task belongs of estimating and allowing for the probable effects of currents and tides, London 51° 30′ 49′′ N latitude, Naples 40 50 15 5° 11' 5" W + 15 34 25 = 20 45 30 W = the true longitude in. which are ever varying in their power and direction, and therefore embarrass, and often destroy, the most experienced mariners. In steering across a river, for instance, from point A to point B on the opposite shore in a tideway, steer for a point on that side of B from which the tide flows, and as much above or below B, as in your judgment you conceive the tide would have carried the boat on the side of B towards which the tide flows, during the passage across, had it been constantly steered in a direction parallel to the line joining A and B. and by the plane triangle, d R 10,1345808 log. of s'. 2,8065598 log. of l. 12,9411406 10,0000000 log. of R. 2,9411406 log. of d. = or d nautical miles, and 1003,63 statute miles, 69,04 being reckoned to a de 873,25 nautical miles. gree: this is, however, what is termed in common conversation the distance as the crow flies. CHAPTER V. On Leeway, and Plying to Windward. The only method that ought to be relied on in practice of ascertaining the amount of this correction of the apparent course of a ship, is that of actually measuring the angle formed as before mentioned; which angle is in fact the bearing of the wake by the fixed compass: means therefore should be devised for enabling the proper officer to take such a bearing with correctness, and in this respect practical navigation seems to be deficient. (56.) There are few large vessels that can lie within less than six points of the wind; and therefore, when the wind blows from any point within six points of the bearing of a port for which a vessel is bound, she must tack or ply to windward; that is, she must steer a course as near to the bearing of the port as the wind and other circumstances will admit of, and she must steer that course, until the bearing of her port is altered so as to become a course, which the direction of the wind will allow her to steer. Suppose, for instance, a ship is bound to a place bearing E, but the wind is ENE, that is, two points from the bearing of her port; the ship's course must be SE or N; for these courses are respectively six points from the direction of the wind, and are, therefore, nearer to the bearing of the port than any other courses which the ship can describe; and the ship, supposing the wind to remain in the same quarter, must sail SE, until the bearing of her port becomes from the alteration of her place due N, or N till it becomes SE; and then she must tack and steer N or SE, according as her first course has been SE or N; but if it should be more convenient so to do, the ship may make a great number of tacks, or SE and N courses, before she arrives at her port; and the whole distance sailed will not be greater in this case than in the other. PART III. On Nautical Astronomy. (57.) THE two first Parts of this treatise We have sup The object of the third Part is to explain, how the relative angular posi tions of the celestial bodies, with respect to each other, and the horizon and meridian of any place, enable the mariner to determine the situation of that place, and thus correct the errors of the reckoning. So that these two different methods serve as a check upon each other, and have together been found amply sufficient for all the purposes of the practical navigator. That part of the heavens which is visible to us, and in which the celestial bodies appear, is in the shape of a hollow hemisphere; and it will facilitate greatly the comprehension of nautical astronomy, if we imagine the earth placed in the centre of a hollow glass sphere, which has the heavenly bodies exhibited on its surface;-that the earth, ing through a heavenly body and the celestial meridian opposite to a place, is called its azimuth. The sensible horizon is that line in the heavens which is the intersection of a circular plane touching the earth at the point at which the spectator is placed and extended to the heavens, or hollow sphere; and the rational horizon is the intersection with the heavens of a circular plane parallel to the former, similarly extended, and passing through the centre of the earth. When a heavenly body, the sun for example, by the revolution of the glass sphere, is brought opposite to the meridian of any place, it is said to be on the meridian of that place; and when it is at a distance from that meridian, the angle contained at the pole between a celestial meridian passing through that object, and the celestial meridian then opposite to the place is called its hour angle; for it expresses, or rather is proportional to, the number of hours which must elapse before the sun is upon the meridian. When the sun is on the meridian of any place, it is twelve o'clock at that place: and, as the sun is supposed to revolve round the earth uniformly in twenty-four hours, if his hour-angle can be ascertained, the time at the place will be ascertained; for the whole hour-angle made in the course of his revolution is equal to 360°, which is described in twenty-four hours; hence, 15° of an hour-angle will be performed in an hour, &c.; and by the rule of proportion any number of degrees of an hour-angle may be converted into time, and this time will express the time before, or the time after apparent noon at the place of observation. instead of revolving round its axis every twenty-four hours from West to East, remains at rest; and that the hollow glass sphere with the heavenly bodies upon it revolves uniformly round the earth in the same time from East to West: the suppositions we have made will accurately represent the diurnal motions of those heavenly bodies: the axis round which the hollow sphere revolves is the axis of the earth produced from each of its extremities till it meets the sphere: let us imagine great circles traced out on the hollow sphere corresponding with and opposite to the great circles of the earth already described; corresponding to the equator, let there be a celestial equator; corresponding to the meridians, celestial meridians; corresponding to the poles, poles of the heavens at the extremities of the axis of the sphere; let the parallels also be represented in the same manner if the circular plane of the equator of the earth were to be enlarged and extended so as to reach the glass sphere, then should its circumference coincide with the celestial equator traced out upon that sphere, and the same observation applies to all the other circles above described; and it is in this sense, therefore, that we use the term corresponding. If a straight line joining the earth's centre, and any place on the earth's surface, be produced until it meet the hollow glass sphere, the point at which it meets that sphere is called the zenith of that place; and as the sphere revolves, every point of it which successively touches the extremity of that line will successively become a zenith to the place; and stars on the sphere, which in succession pass that line, will be said to pass over the zenith of the place. The celestial meridians are called circles of declination: for the arc of a celestial meridian intercepted between any heavenly body and the celestial equator is not called its latitude, but its declination. As each celestial meridian is brought in succession opposite to the terrestrial meridian of any place, by the revolution of the sphere, it acquires the name of the celestial meridian of that place. Great circles drawn on the hollow sphere passing through the zenith of any place, are called vertical circles; and that particular vertical circle which passes through the East and West points is called the prime vertical; the angle contained between a vertical circle pass But though the sun may not have reached the meridian of a place a, (fig. 7,) it may be on the meridian of a place e to the east of a, for the glass sphere revolves from the east towards the west; it is therefore noon at the place e, before it is noon at the place a: but if a mariner placed at e could, when the sun was on his meridian, ascertain the hour angle of the sun from the meridian of a, or how much it wanted of being noon at a at that time, he would then know his longitude from a; for the longitude is measured by the arc EA, which arc measures likewise the angle contained between the meridians of a and e at the pole P, (Art. 35 ;) and the celestial meridians, as they correspond with those meridians, must contain the same angle at the celestial pole, which those meridians contain at the terrestrial; but those celestial meridians contain the hour angle of the sun from the meridian of a, as appears from the definition of an hour angle; therefore the two angles are one and the same angle: if the mariner at e, therefore, be possessed of a watch, which shows the hour angle at a, or what o'clock it is at a, and he have the means also of ascertaining at the same instant of time what o'clock it is at e, he can thus determine the arc A E, or the longitude of e; and it is evident also that any other means of ascertaining what o'clock it is at a, besides a watch, would answer the same purpose. Suppose that the mariner knew that at twelve o'clock at night at a,two heavenly bodies, whose distance from each other is continually varying, would be found at a certain distance from each other, say 30°, on a particular day; suppose that on that day at the spot e he watches those heavenly bodies, and ascertains them to be 30° of the arc of a great circle, drawn on the hollow sphere, distant from each other; then let him ascertain the time at e, if he finds that time to be three o'clock in the morning, as three hours difference in time answer to 45° of an hour angle, he may thence conclude, that he is situated at a spot 45° east of Greenwich. The moon is one of those heavenly bodies, which is continually changing its place on the surface of our sphere, moving from some fixed stars and towards others; it seems to revolve rapidly round the sphere from west to east in a great circle, which makes an angle of about 28° with the equator: the sun seems to move also, but not so rapidly, from west to east, and the great circle he seems to trace out is called the ecliptic, making an angle of about 23° with the equator. Now the distances of the moon from the sun and nine principal fixed stars near to her apparent path, are, in fact, computed and set down in the Nautical Almanac, as they would appear at the centre of the earth every three hours of Greenwich time, on those days when the moon is visible; and these distances therefore may be found for any other time at Greenwich by allowing a proportional difference. This mode of ascertaining the longitude by measuring these distances is called the lunar method. (58.) Let the circle HEZPHQ(fig. 12.) represent one of the great circles of our glass sphere; let it represent the celestial meridian of the place e in fig. 7, that is, it represents in succession all those circles of declination, which by the revolution of the sphere come in succession opposite to the meridian of e. Now as zenith of a place passes through its the straight line which determines the meridian, and lies also in the circular plane of its meridian, the circular plane itself extended, or the celestial meridian, must pass through its zenith. Let Z be the zenith of e, let P be the north pole of the sphere, and ECQ the celestial equator, then Z D and Z d are arcs of vertical circles; ifs be the sun, then the angle H Zs represents his azimuth, and s PE his hour angle from the meridian of e, or the time before noon at e; if m be the moon, ms may represent the distance between the moon and the sun measured on the arc of a great circle m s, also HDCH' represents the rational horizon of e. Let M and S, also, be two places of the moon and sun; then Sd and MD are called the altitudes of the sun and moon; they are the arcs of vertical circles intercepted between those heavenly bodies and the rational horizon: how high a celestial object is above that but we can have no means of measuring horizon; we can ascertain it, however, by measuring its height above the sensible horizon, and adding a correction to that altitude for the difference between its heights above the two horizons; this correction is called parallax. corrections to be applied to the observed Besides parallax, there are two more altitudes of heavenly bodies; these are first refraction, which is to be subtracted from those altitudes; for it is the raised above the sensible horizon, in quantity which heavenly bodies appear consequence of their light having to pass When the eye is in the plane of a circle it appears a straight line. |