2. Line of Chords —The line of chords, and the other circular lines on the same side of the scale, have a refer ence to one another, and are all adapted to the same radius, which is commonly two inches. Let CA. (Fig. 3,) represent that radius, and a circle being described with it about the centre C, draw the diameters AB and DE at right angles to each other. Divide one of the quadrants as DB into 9 equal parts, and having joined DB, transfer the chords of the several arcs reckoned from D, to tlie straight line DB, and DB will be a line of chords, exhibiting every 10th division of the scale. The intermediate divisions are obtained by subdividing each of the ptincipal divisions of the quadrant into 10 equal parts, and transferring their chords as before. The scale of chords is employed for laying down or measuring angles, the chord of 60°, which is equal to the radius, being always assumed as the radius of the measuring arc.
3. Line of Rhumbs.—The line of rhumbs, which is employed to lay down a ship's course when it is express ed in points, is described in the same manner as the scale of chords, only thc quadrant is divided into 8 equal parts in place of 9. Each of the principal divisions is again subdivided into 4 equal parts, and the chords of the resulting arcs being transferred to the straight line .AD, the scale of rhumbs is completed.
4. Line of Sines.—The scS1e of sines is constructed by dividing one °Utile quadrants, as BE, into 90 equal parts, as was done for the scale of chords. Perpendicu lars to the radius CE from every division in the qua tlrant give the seale of sines reckoned from C towards E. This scale is chiefly used in the °idiographic pro jection of the sphere.
5. Line of Secants.—Produce CE indefinitely to F, and through B draw BG parallel CF, and also of unli mited length. Through the centre C, and each division of the quadrant 13E, draw straight lines intersecting BG in the divisions 10, 20, 30, Sm. The distance of each of these divisions from C being transferred to the line CF, will form the line of sccants, which comprehends in it the line of sines, the secant of 0 being equal to the radi us, or sine of 90°.
6. Line of Tangents.—The construction by which the line of secants is obtained, gives at the same time the line of tangents, along the line 13G. This line, and the line of secants, are chiefly etnployed in the stereogra phie projection of the sphere.
7. Line of Semitangents.—The semitangents of arcs are not, as the designation of these lines would imply, the halves of the tangents, but the tangents of hall' the corresponding arcs. Thus the semitangcnt of 40° isnot
half the tangent of the 40°, but the tangent of 20°. Hence the scale of semitangents may easily be obtained from the scale of tangents. It may also be obtained by join ing the point A, and each of the divisions in the quadrant BD ; the intersections of the lines thus drawn, with the line CD, will give along that line from C, the scale of semitangents. This line, as well as the two preceding, is used in the stereographic projection of the sphere.
8. Line of Longitudes.—To construct the line of lon gitudes, divide the radius AC into 60 equal parts, and through each of the divisions draw straight lines parallel to CE, and intersecting the quadrantal arc AE. The chords of the resulting arcs, reckoned from A, being transferred to the line AE, will give the line of longi tudes. This line being applied, in an inverted order, to a corresponding line of chords, 60 in the line of longi tude being against 0° on the line of chords, and 0 on the former line against 90° on the latter, if the divisions on the line of chords be considered a line of latitude, the opposite divisions on the line of longitude will exhibit the corresponding number of miles belonging to a degree of longitude, in cach particular latitude. This compound line is used to show the convel Kelley of the terrestrial meridians, and to perform instrumentally questions in parallel sailing.
The lines on the other side of the scale, which we have denominated Artificial or Logarithmic, are the fol lowing : 1. The Line of .Anumbers.—We begin with the con struction of this logarithmic line, because all the other lines are censtruetecl with reference to it. The pur poses to which this line is usually applied, do not re quire that it should have numbers represented on it, such that thc greatest should excced the least more than 100 times; and accordingly, as the common logarithm of 100 is 2, and that of 10 is 1, the line of numbers is made to consist of two equal parts, the subordinate divisions in one of which have a reference to an order of units, ten times greater than those of the other. Lct an accu rate scale of equal parts, therefore, be constructed, of half the length of the proposed line of numbers, and for the sake of greater precision, let it be a diagonal scale, with ten primary divisions. Then, by the common lo garithms, the logarithm of 10 being 1, and that of 1, nothing.