The lines of chords, sines, tangents, and secants exhibit the relative lengths of these lines to every degree of the quadrant; and the chord of 60° is the radius of the circle to which each of the scales is adapted. The line of semitangents contains merely the tangents of half the arcs marked on the line. Thus the semitangent of 20° is the tangent of 10°, &c.

On another part of the scale a line of chords is generally given to a longer radius, accompanied with a line of rhumbs to the same radius. To this line of chords a line marked M. LON. is also adapted, shewing the number of miles on any parallel of latitude, which corresponds to a degree of the terrestrial equator. Thus opposite 40° on the line of chords stands nearly 54 on the line M. LoN.; which denotes that on the parallel of 40°, 54 miles correspond to a degree of longitude.

The other parts of this side of the scale are generally filled up with lines of equal parts of various lengths, marked L, P, Lea, and C; but the principal and most useful of these lines are two diagonal scales, of which the lengths of the largest divisions are an inch and half an inch respectively.

The manner of using these diagonal scales will be readily understood from an example.

Let it be required to take the number 438 from a diagonal scale. Extend the compasses from 4 on the larger divisions to the third of the smaller ones; move one point of the compasses along the vertical line 4, and the other on the oblique line running from the third smaller division till they both come to the eighth parallel line; and the distance between the points of the compasses will be the measure of the required number on that scale.

The application of logarithms to a scale was first made by Mr. Edmund Gunter, and hence that side which is logarithmically divided is called Gunter's Scale. The first two lines marked S. RHUMB and T. RHUMB, one divided, according to the logarithmic relations of the sines and tangents, to every point and quarter-point of the compass; and the lines marked SIN and TAN are in fact the same lines as those marked S. RHUмв and T. RHUмв, but divided to degrees.

The tangents of arcs greater than half a right angle are placed and numbered backward on the same line with the tangents of their

complements. For as

log cot log rad. Now the tangent of half a right angle is equal to the radius; therefore the distance on the logarithmic scale from the tangent of any less angle than half a right angle to the tangent of half a right angle, is the same as the distance from the tangent of half a right angle to the tangent of the complement of the proposed angle.

The tangents therefore which increase towards the left must be

used as if they were placed on the scale continued towards the right from tangent 45°, or to points. The line marked Num. exhibits the relations between the logarithms of all numbers, of which the greatest does not exceed 100 times the least. As the difference between the logarithms of 1 and 10 is the same as the difference between the logarithms 10 and 100, this scale is divided into two parts, which are perfectly equal, and similarly divided. Thus log 21 log 1 = log 21 log 10; therefore the distance from the beginning to 2.1 is the same as that from 10, the middle of the scale, to 2, and one smaller division towards the right, or 21. If the one on the left hand be considered as 10, the 10 in the middle will represent 100, and the 10 on the right 1000. If the 1 on the right be considered as 1, the ten in the middle will represent 1, and the 10 on the right will represent 10; and in each case the intermediate divisions and parts of divisions will have corresponding values.

The line of VERSED SINES is not exactly what its title might import; it contains the logarithms of half the versed sines of the supplements of the arcs marked on it. Its chief use is to determine, by an instrumental operation, the angles of a plane or spherical triangle when the sides of it are given.

In the construction of the line of the logarithms of numbers, a line equal to the distance from 1 to 10 is taken, and decimally divided; and this whole line being considered as representing the logarithm of 10, those parts of it from I which correspond to the logarithms of the increasing integers, and the leading decimal parts, are taken and transferred to the scale, which is thus completed. The various other logarithmic lines on this side of the scale are taken from the same decimal scale of equal parts that the line of the logarithms of numbers is formed from.

In constructing the line of sines, the difference between 10, the log sine of 90, and the logarithmic sine of any other angle is taken from the decimal scale and applied from 90° towards the left; and in constructing the line of tangents, the difference between 10, the log tangent of 45° and the log tangent of any other angle, is taken from the decimal scale, and applied from 45° towards the left; the degrees being then marked at the points corresponding to them, the scales are finished.

The line marked versed sines is formed by subtracting the logarithm of 2 from the log suversed sine of the given arc, and taking the remainder, for the parts of the required scale, from the same line of equal parts that the other scales are formed from.

The line marked Mer. exhibits the length of the meridian from the equator, according to Mercator's projection of the globe, in which projection the meridians are all parallel lines. The line below it, marked E. parts, or equal parts, shews the parts of the meridian, whose projection is shewn in the line above. These two lines

are used jointly in forming a Mercator's chart; the line of equal parts being used to lay down the equator and the parallels of latitude, and the line of Mer. parts to project the meridians; but the lines are too small to be of much use in the practice of drawing charts.

The lines on the scale afford a ready method of obtaining, in å rough way, a fourth proportional to three given numbers. And when, through the uncertainty of the data, a rough estimate of the result is all that can be obtained; or, when, from the nature of the case, à very accurate result would be of no practical benefit, such results as can be obtained from the scale may be as useful as those deduced by methods which admit of greater precision.

The following general rule may be given for finding a fourth proportional by the scale.

If the first and the second terms are of the same denomination, extend from the first term to the second, and that extent applied on the same direction will reach on the proper line from the third term to the required fourth proportional; or if the first term and the third are of the same denomination, extend from the first term to the third, and that extent applied in the same direction will reach, on the appro✶ priate line, from the second term to the fourth proportional.

Nole. The tangents of arcs above 45°, which increase on the scale towards the left, must, in estimating the direction, be considered as continuing to increase from 45° towards the right.

Let it be required, by way of example, to find a fourth proportional to 34, 58, and 49.

Extend towards the right on the line of numbers from 34 to 58, and that extent applied from 49, also toward the right, on the same line, will reach to 836, the required fourth proportional.

Again, let it be required to find the arc whose sine is a fourth proportional to 484, 367, and sine of 51°. Extend towards the left from 484 to 367 on the line of numbers, and that extent applied on the line of sines towards the left, will reach from 51° to 36o 6′, the arc whose sine is the required fourth proportional.

Let us again inquire what that are is whose tangent is a fourth proportional to the sine of 26°, the sine of 78°, and the tangent of 40°.

Extend towards the right on the line of sines from 26° to 78°, and that extent applied from 40° on the line of tangents towards the right, will reach beyond the limits of the line. To find how much it reaches beyond the line, with the same extent in the compasses place one foot of them at 45°, then the other will extend towards the left as much beyond 40° as in the former case the extent went beyond the end of the line, and the distance between this point and 40° applied from 45° towards the left will reach to 61° 54', the arc whose tangent is the required fourth proportional.

The use of the line of versed sines may be thus explained. We have found in plane trigonometry that S being half the sum of the three sides of a triangle, of which the three sides are a, b, and e, and the angles opposite those sides A, B, and C, respectively, then

And, adopting a like notation, we have in any spherical triangle, 1 + cos A suvers A rad. sin S. sin S

a

Hence to find, by the scale, any angle of a plane triangle whose sides are given, we have this rule.

Extend on the line of numbers from half the sum of the three sides to one of the sides containing the required angle, and that extent will reach, in the same direction, from the other side about the required angle to a fourth number. Extend from this fourth number to the difference between half the sum of the sides and the side opposite the required angle, and the extent will reach from 90° on the line of sines to a point on the same line, below which on the line of versed sines will be found the required angle.

And to find any angle of a spherical triangle whose three sides are given, we have this rule.

Extend on the line of sines from 90° to either of the sides about the required angle, and this extent will reach, in the same direction, on the line of sines, from the other side about the required angle to a

fourth one. Extend on the same line from this fourth arc to half the sum of the three sides of the triangle, and this extent applied in the same direction and on the same line, will reach from the difference between half the sum of the three sides and the side opposite the required angle, to another point on the line, below which, on the line of versed sines, will stand the required angle.

Example 1. Given the three sides of a plane triangle 1267, 849, and 729, to find the angle opposite the greatest side.

1267
849

729

2)2845

1422 Half the sum of the three sides.

155 Difference between the half sum and the greatest side.

Extend then from 1422 to 849, and that extent will reach in the same direction from 729 to 435. Extend from 435 to 155, and that extent will reach from 90° on the line of sines to nearly 29° on the same line; below which, on the line of versed sines, stands about 106+, the required angle.

Example 2. Given the three sides of a spherical triangle 43° 14′, 37° 28′, and 51° 20′ to find the angle opposite the least side.

28 33

Difference between the half sum and the least side.

Extend on the line of sines from 90° to 51° 20′, and that extent will reach in the same direction from 43° 14′ to 32° 20′ on the line of sines.

Extend from 32° 20′ to 66° 1′, and that extent will reach in the same direction from 28° 33' to about 54° on the line of sines, below which on the line of versed sines stands nearly 5030, the required angle.

Example 3. Let it be required to find the greatest angle of a spherical triangle whose three sides are 114° 28', 121o 6', and 88o 4'.

114° 28'

121 6

88 4

2) 323 38

161 49 Half the sum of the three sides.

40 43 Difference between half the sum and the greatest side.