268 NAVIGATION one angle previously known, and the side PH common to both is to be found. First, to find the side (or segment) HC. The angle at C is obviously the "middle part," as it is between the known side and the one sought ; they are joined or combined. Therefore rad. +log (co)sin C=log (co)tan PC 4- log tan HC ; rad. xcosC TTO . . ~, or 1T ^ =tan HC 4/ 42 . Similarly cot PC rad. x cosV cot PV tanVH 42 30 . The sum of the two segments will be 90 12 , equal to the side VC, representing the distance of 5412 miles, which is 543 miles less than the rhumb line. Similar calculations must be made every day at sea to find the new course and remaining distance. It is necessary, before deciding upon a voyage following the arc of a great circle, to ascertain the shape of the curve and the highest latitude it will attain ; for which purpose the side PH must be found (fig. 15) by either of the segments as before explained ; it will be 32 43 30", which is the complement of the latitude 57 16 30". To find the longitude of that point : by opposite sides and angles ^ sin PC which added to the longitude of position off the Cape, 18 24 , gives 82 12 30" E. A glance at the chart will show that siich a course would run the ship among the icebergs ; therefore a position three or four hundred miles farther from the pole should be chosen, according to the time of year, and the course divided in two parts. It is easy to make a pencil curve on the chart in a lower latitude : the saving of distance would still be great. The arc of a great circle from a position off Rio de Janeiro, 23 5 S. and 43 4 W., to the vicinity of Perth in Western Australia, 32 2 S. and 115 25 E., would measure 7270 miles. The rhumb line by Mercator (if it were possible) would be 8437 miles, or 1167 more. The highest latitude would be 70 28 , therefore impracticable. Such a curve as this could not be laid down on Mercator s chart. Soundings. When approaching land in thick weather the pre caution of taking frequent soundings should not be neglected, especially when the depth and nature of the bottom is clearly defined on the chart, as it is at the entrance of the English Channel. Loss of time should not be made an excuse for incur ring a serious additional risk. Several ingenious devices have been provided by which the depth of 80 fathoms can be tested without stopping the ship, but the result is not so certain as the old plan of getting the line " up and down" and feeling the lead touch the bottom. If soundings are taken at ten or more miles apart, and do not coincide with those on the chart, it is a good plan to write them up the edge of a strip of paper, preserving the distances the ship has run, according to the scale of the chart; when two or three are thus marked the slip should be moved about the chart, with the edge of the paper parallel to the course steered, till it coincides. Observations. The quadrant and sextant are practically the same instrument: the first is the eighth part of a circle, and by reflexion measures an angle of 90 ; the second is the sixth part of a circle, and measures an angle of 120. Both are fitted with verniers graduated in such a manner that the angle on the arc of the quad rant can be read to half a minute, and that on the sextant to ten or even six seconds. The handling of the instruments for five minutes will be better than a long description. The errors and adjustments are similar. The large movable mirror must stand perpendicular to the plane of the instrument. This is tested by placing the radius bar near the centre of the arc, and looking into the mirror at the reflexion of the uncovered part, which if all be right will continue in a straight line from the arc itself. If it appear broken, the screws must be moved. The error is difficult to remedy; therefore when practicable it is better to send the instrument to a maker. The fixed reflector, or horizon glass, has two adjustments. To make it perpendicular to the plane of the instrument, let the radius bar be placed near zero, and while the instrument is held quite vertical make the direct and reflected view of the horizon coincide, by moving the tangent screw. Then, if by sloping the instrument on one side or the other that line becomes broken, the glass is not vertical, and the screw for that purpose must be slightly moved till the lines coincide even when the instrument is held sixty degrees on one side or the other of the perpendicular. Also the reflected image of the sun or a star will cover the direct rays from them if the glass be vertical. If an horizon be not available, a distant hill or any sharp outline will do as well. The second adjustment is to make it parallel with the movable reflector. If at the end of the adjustment or test just described the radius stands at zero, or within two minutes, no adjustment is necessary. If it be much beyond that amount the radius should be set exactly at zero ; and whil looking at the horizon, which will appear broken, let the screw under the horizon glass be turned gently till the two parts coincide. To find the amount of index error accurately, the best method is that of measuring the diameter of the sun many times, which will give a number of small angles on and off the arc, as it is commonly called: that is, on the positive and negative side of zero, the mean of which will be the correction, h if it be off the arc, and - if on. Thus 32 20" on and 30 40" off would give a mean of -50". The observer can at the same time test his own accuracy: thus, in the above example, the sun s diameter appears to be 31 30"; the Nautical Almanac will show if it were so on that day. There is one error to which all sextants are liable that is seldom mentioned or attended to. It arises from the great difficulty of placing the centre of motion given to the radius bar and movable reflector exactly in the centre of the arc, or from the contraction or expansion of the metal. It has no connexion with the index error, and admits of no adjustment. Its existence and amount are not easily ascertained, but demand both time and patience. As it has no appreciable effect on small angles, it is advisable to use the artificial horizon, and take a set of altitudes, say ten, which will form a mean of about 100 on the arc, noting the time of each accurately by a trustworthy chronometer. Take similar altitudes in the afternoon, and work each set independ ently, as though to find the error of the chronometer (see below). Should the time so found coincide with the known rate of the chronometer, there is no error. Should the results differ several seconds of time, it may be assumed that the error of the instrument combined with personal error has caused it. By the rate at which the sun was rising or going down during the observations the amount of angle due to those seconds is easily found. Half that amount will be the error of the sextant upon that angle. As an example, suppose the true reflected altitude to be 100 while the instrument made it 100 1 , the calculation would make it about three seconds later than the truth ; in the afternoon a similar error would make it three seconds earlier. Thus a disagreement of six seconds arises for about one minute of altitude. By four or five such sets of altitudes at different parts of the arc sufficient data will be procured from which to form a table of corrections for all altitudes. This can be done by calculation, but a simple graphic method will be found sufficiently correct. Describe an arc of a circle with a radius of 6 or 8 inches. Let a line from the centre of projection indicate zero, from which lay off half the altitudes which have been observed, but give them the full numbers, e.g., mark 45 as 90, to correspond with the motion of the radius bar and numbers on the sextant. From the points representing the altitudes lay off the errors towards the centre, from any convenient scale of equal parts, perhaps one hundred seconds to half an inch. Between zero and the centre of the arc will be found the centre of a second and smaller arc which will pass through all the points representing the corrections, or nearly so. The space between the two arcs may be remeasured at every 5 or 10 for the table of corrections (plus or minus, as the case may be). Meridian Altitudes. The first astronomical observation at sea- is usually a meridian altitude of the sun for the purpose of obtain ing the latitude. The reflexion of the sun s lower limb having, been brought by a sextant or quadrant to the edge of the visible horizontal on May 10, 1882, commencing a few minutes before noon, the greatest altitude that could be obtained was 58 20 15". The index correction was + 1 30", and the eccentric error due to 58 degrees was -40". The Nautical A Imanac shows the sun s; semidiameter on that day to be 15 52" ; as the lower limb was observed it is plus. The height of the observer s eye above the sea was 20 feet, which on account of the spherical figure of the earth makes the altitude appear too great. The correction is taken from a small table for the dip of the sea horizon, " which opposite 20 feet gives 4 24" always minus. The result will be the apparent altitude of the sun s centre, which must still be corrected for refraction and parallax, the former on account of the rays of light from any object beyond the earth s atmosphere being deflected upwards, and the latter because the sun does not appear to a person on the earth s surface so high as it is in reference to the centre of the earth ; the horizontal parallax is only nine seconds in the winter of the northern hemisphere, when the earth is nearest the sun. A table is usually arranged for refraction, minus the sun s parallax; in one such table the altitude 58 32 33" requires a correction of 31 seconds always minus. The result will be the sun s true altitude, which taken from 90 gives the sun s zenith distance 31 27 58", the sun being thus much south of the observer when on the meridian. By the Nautical! Almanac it will be seen that the sun s declination at apparent noon at Greenwich on May 10th was 17 39 57" N. , increasing- 39" hourly. As the ship is in 7 W. long, nearly, the meridian passage was 28 m later, and about 18" must be added. This is of importance when the difference of longitude is great, as Greenwich time must always be found within a few minutes. The observer having found the distance he is north of the sun, and also that the sun is 17 40 15" north of the equator, the sum, or 49 8 13", will obviously be his distance from the equator, which is the latitude.
