meridians of oo and 180°, and if the semicircular arcs of the largest circle on each side of this meridian be divided into 18 equal parts, and lines be drawn from the centre to the points of division, these lines will represent meridians 10° apart, and the longitudes E. and W. 10°, 20°, 30°, etc., can be marked on this outer circle. With a sufficiently large circle these divisions may easily be divided into single degrees. The following table gives the co-tangent for every 5° from 10° to 80°. The co-tangents are natural co-tangents and may be found by taking the L co-tangent from Table of Log. Sines, etc. (rejecting to from the index) and finding from Log. table the natural number corresponding to it. cot, 10° = 5.671 cot. 30° 1.732 cot. 50° .839 cot. 70° •364 cot. 15° 3.732 cot. 35° 1.428 cot. 55° 7 cot. 75° .268 cot. 20° 2.747 cot. 40° I`192 cot. 60° •577 cot. 80° *176 cot. 25° 2.145 cot. 45° cot. 65o = -466 = I To exemplify this Suppose the paper to be cut square and a side to measure one foot, then by choosing r = 1 inch, the radius of the parallel of 10° will be 5.671 inches, which can be included in the paper. The radius of the parallel of 20° will be 2.747 inches, and so on. But if 20° be considered the lowest latitude, then make v = 2 inches, and the radius of the parallel of 20° will be 2.747 X 2 = 5.494 inches; the radius of the parallel of 30° will be 3.464 inches, etc., etc. NEW NAVIGATION It is claimed for the New Navigation that it is the best method of finding a ship's position, and is equally accurate whether the observations are taken near the prime vertical or near the meridian ; and in all cases it gives a more accurate line of position than is given by our ordinary methods when the dead reckoning differs from the true position. The method is due to Capt. Marcq St. Hilaire, of the French Navy, and is based on the difference between the altitude observed and the altitude calculated from the dead reckoning position of the ship. The computation, therefore, consists in finding the altitude by one of the methods given. The last method given has been specially proposed, as it finds not only the altitude but also the azimuth in the same computation. Computation and Projection Combined.—If the sun is the object observed, take from the Nautical Almanac the declination and equation of time and correct them for the Greenwich time in the usual way. If a star is the object observed take out the right ascension, the declination, and the mean sun's right ascension, correcting the last mentioned. Then compute the altitude of the object, using the latitude by dead reckoning, the declination and the hour-angle. Find also the true altitude from the observed altitude. Take the difference between the computed altitude and the true altitude from the observation, and mark it + when the true altitude from the observation is greater than the computed altitude, but mark it - when the true altitude from the observation is less than the computed altitude. Next find the azimuth from A, B and C Azimuth Tables in Norie's Tables. Enter the Traverse Table with the azimuth as course and the difference of altitudes in the distance column and take out the difference of latitude and departure and convert the departure into difference of longitude. Apply this difference of latitude and difference of longitude to the latitude and longitude by dead reckoning in the direction of the azimuth if the difference of altitudes is marked +, but in the opposite direction if the difference of altitudes is marked and the result is a new point from which the course and distance in the interval must be reckoned to give the estimated position at the time of taking the second observation. Again, take the required data from the Nautical Almanac and correct them for the Greenwich time of the second observation. Compute the altitude, using the latitude and longitude just found. Find also the true altitude from the observed altitude and take the difference between the computed true altitude and the true altitude from the observation. Mark it + when the true altitude from observation is greater than the computed true altitude, but mark it — when the true altitude from observation is less than the computed true altitude. Next find the azimuth by Burdwood's or Davis's Time Azimuth Tables; but if the object used is not within the limits of those tables, the azimuth can be found by inspection from the A, B and C Azimuth Tables in Norie's Tables. On the chart lay down the position given by the dead reckoning at the time of taking the first observation, and from it draw a line in the direction of the azimuth. Set off on this line from the dead reckoning position a distance equal to the difference of altitudes, in the direction of the azimuth if the difference of altitudes is marked t, but in the opposite direction if marked, and through the point thus obtained draw a line at. right angles to the direction of the azimuth; this line is the first line of position. Next, from this latter point (not the dead reckoning position) set off the true course and the distance made good in the interval between the observations, thus giving an approximate position at the time of the second observation. From this point set off the azimuth and the difference of altitudes at the second observation, in the direction of the azimuth if it is marked +, but in the opposite direction if it is marked —, and through the point thus obtained draw a line at right angles to the direction of the azimuth; this is the second line of position. Lastly through the approximate position at the time of the second observation draw a line parallel to the first line of position ; the intersection of this line with the second line of position gives the true position of the ship. P2 Z2 Z, z Sr The accompanying sketch will show this plainly. Let Z, be the dead reckoning position at the first observation, ZS, the direction of the azimuth. Suppose the difference of altitudes to be + 20'. Set off 20' from Z, in the direction of S, because the sign is + ; let this be Z, P, From P, Pi Direction of Az.at /stops." draw P, A at right angles to Z, S1, P, A is the first line of position. From P, draw P, Z, to represent the course and distance in the interval, thus placing the ship at Z, when the second observation was taken. From Z, draw Z, S2 the direction of the azimuth. Suppose the difference of altitudes to be — 10'. Set off 10' from Z, in the opposite direction to S, because the sign is, that is on S, Z, produced, let this be Z, Pg. From P, draw P, Z at right angles to Z, S2, P, Z is the second line of position. Lastly through Z, draw Z, Z parallel to P, A, the first line of position, and its intersection with the second line of position at Z is the true position of the ship. Direction of Az.at 2ndObs! By calculation.—To obtain the position by calculation, it is necessary to reduce the altitude taken at one position to what it would have been if it had been taken at the other position, by the rule on pp. 259-61. In this problem it will be more convenient to reduce the altitude taken at the second observation, and thus determine the ship's position at the time of taking the first observation; the position at the second observation being then found by using the course and distance in the ordinary way. By doing this, all the calculations belonging to the first observation can be made during the interval, and a line of position found if considered necessary. Take from the Nautical Almanac the required data and correct them for the Greenwich time at the first observation. For the sun, take the declination and equation of time ; for a star, the right ascension, the declination, and the mean sun's right ascension. Compute the altitude of the object, using the latitude and longitude by dead reckoning. Find also the true altitude from the observed altitude, and take the difference between the computed altitude and the true altitude from the observation. Mark it + when the true altitude from the observation is greater than the computed altitude, but mark it - when the true altitude from the observation is less than the computed altitude. Next find the azimuth from the A. B and C Azimuth Tables. For simplicity call the difference of altitudes di Again take the required data from the Nautical Almanac and correct them for the Greenwich time at the second observation. Compute the altitude of the object, using the latitude and longitude by dead reckoning at the first position. Find the azimuth from A, B and C Azimuth Tables. Find the true altitude from the observed altitude and apply the correction to reduce it to what it would have been if taken at the first position. Take the difference between the reduced true altitude and the computed altitude, and mark it + when the reduced true altitude is greater than the computed altitude, but mark it - when the reduced true altitude is less than the computed altitude. For simplicity call the difference of altitudes d2. Enter the Traverse Table with the angle between the directions on which d, and d, are measured, that is, the angle between the azimuths (remembering that a - sign represents a direction opposite to the azimuthi, as course and d, in the departure column, take out the distance and call it Enter again with d, in the departure column, and take out the difference of latitude and call it b. If the angle is less than 90° subtract b from 2, which is (a - b); but if the angle is more than 90° add b to a, which is (a + b). a. Again enter the Traverse Table with the azimuth on which d, is measured as course and d, in the distance column, take out the difference of latitude and departure and mark them according to the direction of the line. Enter also with the complement of the azimuth as course and with (a - b) or (a + b) in the distance column, and take out the difference of latitude and departure, placing them underneath those first taken out, and name them with the names at right angles to the azimuth used. (The line at right angles to the azimuth trends both northerly and southerly; it is necessary to discover which of the directions must be taken. If (a - b) is positive, that is, b can be subtracted from a in the ordinary way, that direction must be taken which would make an acute angle with the direction of dą. But if (a – b) is negative, that is, a has to be subtracted from b, then that direction must be taken which would make an obtuse angle with the direction of , ; (a + b) is always positive. Add the differences of latitude together it of the same name, but subtract them if of different names, marking the sum or difference with the name of the greater. Also add or subtract the departures by the same rule. This gives the total difference of latitude and departure. Convert the departure into difference of longitude and apply the difference of latitude and difference of longitude to the latitude and longitude by dead reckoning; the result is the position of the ship at the first observation. Then from the rule d, will be reckoned in the direction S. 33° W., but d, will be reckoned in the opposite direction to the azimuth, that is, N. 61o W.; the angle between these is 86°. Enter the Traverse Table with 86° as course and d, 4:55 in the departure column; the distance will be 4:56, call this a. Enter again with d, 21.07 in the departure column ; the difference of latitude will be 1.47, call this b; then the angle being less than 90°, (a — b) is 3.09. Now enter the table with S. 33° W. as course, that is the azimuth on which d, is measured, and with d, 21.07 in the distance column, the difference of latitude is 17.67 S. and the departure 11:47 W. Also with 57° (the complement of 339) as course and (a - b) 3.09 in the distance column, the difference of latitude is 1.68 N. and the departure 2:59 W. (Since d, is measured in a direction N. 61° W. and (a - b) is positive, the line at right angles to S. 33° W. must be N. and W. to make an acute angle with the direction of dz.) The formulæ area = 2 X cosec. A? A= angle between the lines on which di and d2 are measured. do x cot. The azimuth being Ist d. lat. di X cos. az. Ist dep. =d; X sin. az. that on which di is 2nd = (a - b) x sin. az. 2nd (a - b) x cos. az. measured. b= A } When the second altitude is reduced to the first position the graphic asthod is slightly modified, and becomes similar to that of simultaneous |