correspond to the logarithms of the tangents of every point and quarter-point of the compass. This line is numbered from near the middle of the scale, thus, 1, 2, 3, 4, towards the right hand, and back again, thus, 5, 6, 7, from the right towards the left. In order to take off any number of points below four, we must begin at 1 and count towards the right hand; but when we require to take off any number of points above four, we must begin at 4, and count towards the left hand.

3. Line of numbers, sometimes marked Num., is numbered from the left hand end of the scale towards the right; thus, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, which last stands exactly in the middle of the scale; the numbers then go on 2, 3, 4, 5, 6, 7, 8, 9, 10, which last stands at the right hand end of the scale. These two equal parts of the scale are divided equally, the distance between the left hand 1, and the first 2, 3, 4, 5, &c. being exactly equal to the distance between the middle 1, and the numbers 2, 3, 4, 5, &c. which come after it.

The subdivisions of these scales are likewise similar, viz. each being one-tenth of the primary divisions, distinguished by lines of about half the length of the primary divisions.

When the extent of the scale will admit, these divisions are again divided into ten parts; and when the length of the scale will not permit of such subdivision, the units must only be estimated, or guessed at by the eye, which is not very easy without considerable practice. We estimate the primary divisions on the second part of the scale, according to the value see the unit on the left hand of the scale. If we call

upon

it 1, then the first 1, 2, 3, 4, 5, &c. stand for 1, 2, 3, 4, 5, &c. the middle 1 is 10, and the 2, 3, 4, 5, &c. following stand for 20, 30, 40, 50, &c., and the 10 at the right hand is 100. If we make the first 1 stand for 10, the first 2, 3, 4, 5, &c. are to be counted 20, 30, 40, 50, &c., the middle 1 will be 100, and the second 2, 3, 4, 5, &c. will be 200, 300, 400, 500, &c. and the 10 at the right hand will be 1000. If we consider the first 1 as of an unit, the 2, 3, 4, 5, &c. following will be fo, fo, fo, fo, &c.; the middle 1 will stand for 1 unit, and the 2, 3, 4, 5, &c. following will stand for 2, 3, 4, 5, &c., and the division at the right hand end will stand for 10, and the values of the small divisions must be estimated according to the value set upon the primary ones.

4. The line of sines, marked Sin., is numbered from the left hand of the scale towards the right, 1, 2, 3, 4, 5, &c. to 10, then 20, 30, 40, 50, &c. to 90, where it terminates just opposite 10 on the line of numbers.

5. The line of versed sines, marked V. Sines, is placed under the line of sines, and numbered in a contrary direction, viz. from the right hand towards the left, 10, 20, 30, 40, 50, &c. to about 169; the smaller divisions here are to be estimated according to the number to a degree.

6. The line of tangents, marked Tan. begins at the left hand, and is numbered 1, 2, 3, 4, 5, &c. to 10; then 20, 30, 40, 45, where there is a small brass pin just opposite 90 on the line of sines; because the sine of 90° has been shewn to be equal to the tangent of 45°. It is numbered from 45 towards the left hand,

50, 60, 70, 80, &c. Therefore the tangents of arcs above 45° are counted backward on the line, and are found at the same points of the line as the tangents of their complements. Thus, the division at 40 represents both 40 and 50; the division at 30 represents both 30 and 60, &c.

7. Meridianal parts. This line stands above the line of equal parts, with which it must always be compared, when used. The extent from the brass pin on the scale of meridianal parts to any division on that scale, applied to the line of equal parts, will give, in degrees, the meridianal parts answering to the latitude of that division. Or, the extent from any division to another on the line of meridianal parts, applied to the line of equal parts, will give the meridianal difference of latitude between the two places denoted by the divisions.

8. The line of equal parts is marked from the right hand towards the left; thus, 0, 10, 20, 30, &c.; each of these large divisions represents 10 degrees of the equator, or 600 miles.

The first of these divisions is sometimes divided into 40 equal parts, each representing 15 miles. The line of numbers, sometimes called the line of lines, is that upon which most of the others depend. It is usually graduated upon scales, but sometimes upon sectors, &c. It consists of the logarithms of numbers transferred upon a ruler, &c. from the tables, by means of a scale of equal parts, which therefore serves to resolve problems instrumentally, in the same manner as logarithms do arithmetically. For, as logarithms resolve problems, or perform multiplication and division

by addition and subtraction, the same is performed on this line by only turning a pair of compasses in a certain way, or by sliding a slip of wood by the side of another, &c.

A line of this description has been contrived various ways-Mr. Gunter on the two feet ruler or scale, which is generally used at sea; and Wingate on two separate rulers, standing against each other in order to dispense with the use of compasses. Oughtred solved proportions by means of concentric circles. The same was performed by Mr. Milburne, by means of a spiral. Seth Patridge effected the same on the common sliding ruler, for a description of which, see my Mensuration for the National Schools. Mr. William Nicholson has proposed another on concentric circles.

The Construction of the Logarithmical lines on Gunter's Scale.

1. The line of numbers, on which most of the others depend, is constructed thus:-Let half the length of the line of numbers be divided into 1000 equal parts; then, since the logarithm of 1 is 0, (see my Philosophy and Practice of Arithmetic,) the distance from the end to 1 is 0; that is, 1 stands at the beginning of the line. And as the logarithm of 2 is 301, the distance between 1 and 2 is 301 equal parts, taken from a scale of equal parts; the distance 477, the logarithm of 3, is to be set off from 1 to 3; and 602, the logarithm of 4, is to be set off from 1 to 4, &c. These are all the primary divisions on this line, and the intermediate

divisions are formed in a similar manner, by taking the logarithms of the intermediate numbers.

2. The line of sines is constructed by taking the arithmetical complements of the logarithmic sines from the same scale of equal parts from which the line of numbers was constructed, and setting them off from 90 backwards, or towards the left hand; thus the arithmetical complement of the logarithmic sines of 80°, 70°, 60°, 50°, 40°, 30°, 20°, 10°, are 7, 27, 116, 192, 301, 466, 760; and these taken from the same scale of equal parts from which the line of numbers was constructed, and set off from 90 towards the left hand, gives the sines of the above degrees respectively.

As the sine of 90° is equal to radius, and as the several arithmetical complements are what the sines of the arcs want of radius, the reason of setting off the numbers as above is obvious.

3. The line of Rhumbs is constructed in a similar manner to the line of sines, by taking the arithmetical complements of the logarithmic sines of the degrees and minutes which are contained in the several points and quarter-points of the compass, and setting them off from the right-hand end of the scale towards the left.

4. The line of tangents is constructed in the same way as the line of sines, by setting off the arithmetical complements of the tangents under 45° backwards, from 45° towards the left hand. For the tangent of 45° is equal to the radius, and the arithmetical complement of any logarithmic tangent under 45° is what the tangent wants of radius.