So gial! if, to suit our diagonal scale, the logarithm of 10 be increased a thousand times, In like manner, the distance of the intermediate di visions, on the first half of the scale, reckoned from the beginning of it, will be obtained by taking the three first figures or the logarithms of 1.1, 1 2, 1.3, Szc. which will he found to be 41, 79, 114, Exc. These numbers, there fore, being taken from the diagonal scale, and applied successively from the beginning of the logarithmic scale, will give the points on the latter, corresponding to 1.1, 1.2, 1.3, Sze. All the other intermediate divisions are obtained in the same way ; and the construction of the other half of the scale is in no respect different from the first. It is scarcely necessary to observe, that if the first unit, at the beginning of the scale, be reckoned 1, the second unit is 10, and the third 100 ; whereas, if thc first unit be accounted 10, Che second is 100, and the third 1000 ; and so on with regard to any oilicr value that may be attached to the first unit.
2. The Line of Sine Rhambs.—From the same scale of equal parts which was employed in the construction of the line of numbers, take the distances expressed ,by the three first figures of the arithmetical complement,* of the logarithmic sines of 7, 6, 5, points, Etc. or the secants of 1, 2, 3, points, Ecc. rejecting the indices, and lay them off, towards the left hand successively, from the point corresponding to the sine of 90°, which is usually placed immediately below the last unit on the right hand of the line of numbers; and thus the scale of rhumbs will be obtained for the several points. The quarter points are laid down alter the same manner.
3. The Line of Sines.—The construction of the scale of logarithmic sines differs in no respect from that of the rhumbs, only degrees are employed in place of points.
4. The Line of Tangent Rhumbs.—This scale, as far as 4 points, is constructed in the sante manner as the scale of log. sines, by using the three first figures of arithmetical complements of the logarithmic tangents of 3, 2, and 1 points, or of the logarithms of the tangents of 5, 6, and 7 points, rejecting the index. 'rile tangent of 4 points being equal to the radius, terminates the scale on the right hand. The points above 4, if the scale were extended from that point towards the right hand, would be at the same distance from 4, or the point cor responding to the radius, as the points as much below 4 are from that point. Hence, the points corresponding to 3 and 5 points, 2 and 6 points, and 1 and 7 points, are coincident on the scale, only the points above 4, though actually laid down to the left hand of the point corres ponding to the radius, must be conceived, in the per formance of problems, to extend to the right hand of it.
5. 7'he Line of Tangents.—'1The scale of logarithmic tangents is constructed, like that of logarithmic tangent rhumbs, by employing the three first figures of the arithmetical complements of the log. tangents of 40°, 30°, Sze. or of the logarithms of the tangents of 50°, 60°, rejecting their indices.
6. The Line of Versed Sines.—Froin the scale of equal parts, employed in the construction of all the preceding scales, take the distances expressed by the three first figures or the arithmetical complements of the logarith mic cosines of 5, 10, 15, Ste. degrees, rejecting the in dices, and lay off the double of these distances, respec tively, from right to left of the extended scale. The divisions, thus obtained, will correspond to the log. tersed sines of 10, 20, 30, &c. degrees.
7. Meridional Line.—'rake the meridional parts for every 10° from Table II. and having divided them by 60, lay off, by means of some convenient scale of equal parts, the distance expressed by the several quotients. The line of equal parts, which is placed immediately above or below the meridional line, is divided into 19 principal divisions, marked frotn right to left, 0, 10, 20, Ste. each of these divisions representing 10 degrees of the equator, or 600 nautical miles. The first division on the right front 0 to 10, is divided into 10 equal parts, each of which, therefore, corresponds to a degree of the equator, or 60 miles; and consequently, the bisection of each of these equal parts reduces the ultimate divisions to a distance representing 30 miles. The extent from the right hand extremity or the meridional line to any particular division being applied, in like manner, to the scale of equal parts, will indicate, in degrees, the meri dional parts belonging to that division, reckoned from the equator. Also, the extent between any two divisions in the meridional line being applied to the line of equal parts will give, in degrees, the meridional difference of latitude between the two latitudes, expressed by the numbers at the extreme points of extent. Hence this compound scale may serve, instrumentally, the purposes of a table of meridional parts, the use and construction of which will afterwards be explained.