latitude. Supposing R to be the north point of the horizon, then P is the north pole (Fig. 3), and the latitude is north. For south latitude, the south pole P1 must be above the horizon (Fig. 4). Hour circles are represented by arcs from P to P1, and vertical or altitude circles by arcs from Z to N. N E (c) Projection on the Plane of the Horizon (Fig. 5.)-Here the primitive circle is the horizon, N., E., S., W. being the cardinal points. The zenith (Z) is the projecting point, hence all vertical circles appear as straight lines passing through Z, NZS being the celestial meridian, and WZE the prime vertical. P is the north pole, at a distance from N. equal to the latitude. The arc ZQ is also equal to the latitude, and PZ is the co-latitude; WQE is the equator, cutting the horizon in the E. and W. points; Ppi, Ppa, etc., are hour circles, Wk S p3 FIG. 5. The projection on the plane of the horizon has some advantages. over that on the plane of the meridian for illustrating most of the problems of nautical astronomy, and is therefore more commonly employed. To draw a figure to scale, a "Gunter" rule is necessary; the following illustration explains the connection between the lines marked Rum., Cho., Sin., Tan., and S.T. on the rule (Fig. 6). With centre O, distance OA, describe a semicircle. AOB is a vertical diameter; OX, a horizontal line through centre; BY, a tangent at B. OX divides the figure into two quadrants; the upper one is divided into eight equal parts, and with A as centre and distance the different chords from A to each point; the distances are brought back on the chord of 90°, forming the line of Rhumbs (RUM.). A3 is thus the chord of a circle described with radius OA, which subtends an angle of three points at centre O. The line of Chords (CHO.) is similarly formed on the lower quadrant, but the quadrant is here divided into nine equal parts instead of eight, and thus gives divisions to 10°, which can be further subdivided. We can now see that the chord of 60° is the radius that must be used to describe the circle. For the line of Tangents (TAN.) through O and points of division in lower quadrant, draw straight lines meeting BY in required points. For the line of Semi-Tangents (S.T.) through A, and points of division in lower quadrant, and also corresponding points in upper quadrant, draw lines to cut OX in required points. Example.-Draw a figure from following data: Latitude 40° N.; western hour angle 2 48; declination 10° N.; and take off the Amplitude at setting, Azimuth and Altitude (Fig. 7). With centre Z, and radius the chord of 60°, describe a circle N.E.S.W.; draw SZNB vertically, and lay off E. and W. as the extremities of a horizontal diameter through Z. Make ZM = 40° (Lat.) from scale of tangents, and through M draw a line perpendicular to MZ; in this line are the centres of all hour circles. Make NO = chord of 40° (Lat.), where the line OW cuts the vertical line is the elevated pole P. At P to the right make an angle = complement hour angle that is 48°; this line cuts the line through M in D, and is the centre for describing the hour circle required PXR. For the equinoctial, add the colat to 90° = 140°; subtract the colat from 90° 40°, and from scale of S.T. make ZB = 140°, and ZQ = 40°; bisect BQ in 1, this is centre for describing equinoctial WQE. = = For the declination parallel, add colat to polar distance 130°, subtract colat from polar distance 30°; from S.T. scale make ZC = 130°, and ZL = 30°; bisect CL in 2; this is centre for describing the parallel TXL, where the parallel cuts the hour circle is X, the position of the object. Draw ZXA the vertical circle through the object. = Measure WT on the scale of chords for the Amplitude 15°, or W. 15° N. Measure SA on the scale of chords for the Azimuth = 63°, or S. 63° W. Measure ZX on the S.T. scale for the Zenith Distance = 49°, or Altitude = 41°. For South Latitude set off ZM to the north, and construct the figure downwards instead of upwards. The only Artificial Projection to be considered is Mercator's (Fig. 8). Mercator's Projection.-This is the kind of projection used in the construction of Navigating Charts, and represents the Earth's surface as a plane. The meridians of longitude are drawn as parallel straight lines, and the divisions of latitude are expanded in the same proportion as the meridian distances are lengthened. The effect of this is to make the rhumb course a straight line, which is of great advantage to the navigator. The distance from the equator of any given parallel of latitude, and the amount of expansion of a degree of latitude, can be found from the table of Meridional Parts; for example, the parallel of 30° is 1800 geographical miles from the equator on the globe, but the meridional parts for 30° 1888, FIG. 8. which means that on a Mercator's chart drawn to the same scale this parallel would be 1888 geographical miles from the equator. Again, the meridional parts for 31° 1958, and 1958 1888 70 miles, which would be the length of the expanded degree from 30° to 31°. = = The meridional parts for any given latitude may be calculated by adding together the secants of all minutes of latitude up to the given latitude; thus sec 0' + sec l' + sec 2' + sec 3', etc. or they may be calculated independently from the formulalog of mer. parts = 3.8984895 + log (log cot co-lat. — 10) CHAPTER II. DEFINITIONS AND INSTRUMENTS. 1. Great Circles.-Circles whose planes pass through the centre of a sphere or globe. 2. Vertex of a Great Circle.-That point in it which is farthest from the Equator. 3. Small Circles.-Circles whose planes do not pass through the centre of the sphere. 4. Right Anglo.-An angle measuring 90°, or one quarter of a circle. 5. Oblique Angle.-An angle not a right angle. 6. Obtuse Angle.-An angle greater than a right angle. 8. Spherical Angle.-An angle included between two great circles of a sphere (Fig. 13). B AOC and BOC are spherical angles. A FIG. 13. |