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From the trigonometrical definitions and the nature of logarithms, the construction of the plane scale will appear obvious.

1. Describe a circle with any radius, in which draw the two diameters AB, DE, at right angles to each other, and draw the chords BD, BE, AE, AD. Then for the line of chords, divide the quadrant BE into 90 equal parts; from B, as a centre, transfer, with the compasses, these several divisions to the chord line EB, which mark with the corresponding numbers, as in the figure, and it will become a line of chords to be transferred to the ruler.

2. For the line of rhumbs, divide the quadrant AD into 8 equal parts, then with the centre A transfer the divisions to the chord AD, for the line of rhumbs.

3. For the line of sines, parallel to the radius BC, and through each of the divisions of the quadrant BE, draw right lines, which will divide the radius CE into sines and versed sines, numbering it from C to E for the sines, and from E to C for the versed sines.

4. For the line of tangents, lay the ruler on C and the several divisions of the quadrant BE, and it will intersect the line BG, which will become a line of tangents, and numbered from B to G with 10, 20, 30, &c.

5. For the line of secants, transfer the distances between the centre C, and the divisions on the line of tangents, to the line EF, from the centre C, and these

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will give the divisions of the line of secants, which are to be numbered from E to F, with 10, 20, 30, &c.

6. For the line of semitangents, lay a ruler on A, and the several divisions of the quadrant BD, which will intersect the radius CD in the divisions of the semitangents, which are to be marked with the corresponding figures on the quadrant BD.

7. For the line of longitude, divide the radius AC into 60 equal parts, through each of these, parallels to the radius CE will intersect the arc AE in as many points. From A as a centre, the divisions of the arc AE being transferred to the chord AE, will give the division of the line of longitude.

For a description of the diagonal scale, see my Mensuration for the Irish National Schools.

These are some of the principal lines on the plane scale, which are always transferred from the general figure to a ruler of convenient length for practice.

Mr. Edmond Gunter, a respectable English mathematician, born in Hertfordshire, 1581, was the first who applied the logarithms of numbers, and of sines and tangents to straight lines, drawn on a scale or ruler, by means of which and a pair of compasses, proportions in common numbers, and trigonometry may be solved.

The eight following are the lines on Gunter's scale: 1. The line of rhumbs, generally marked S. Rhumb, is a line on which are the logarithms of the natural sines of every point and quarter-point of the compass, numbered from a brass pin on the right hand towards the left thus, 8, 7, 6, 5, 4, 3, 2, 1.

2. Tangent Rhumbs, generally marked T. Rhumbs,

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 upon the unit on the left hand of the scale. If we call

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, &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,

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