Let ABFE be a circle, of which the diameters AF

and CE are at right angles; having taken any arc AB, erect the perpendicular AP, meeting OB produced; also draw CG and DB perpendicular to OC, and BS perpendicular to OA. Then CB is the complement of the assumed arc AB; its supplement is BCF; BS is the sine; BD or OS is the cosine; SA is the versed sine; DC the coversed sine; the supplementary versed sine is FS; the tangent is AP, and cotangent CG; the secant of AB is OP; cosecant OG; and the chord of the arc FkE is EF.

In naming the sine, tangent, or secant of the complement of an arc, we generally use the abbreviated terms, cos. cot. co-sec.; we also use the letter R for radius.

From these definitions numerous obvious consequences flow :

1. The sine of a quadrant, or 90°, is equal to the radius, OC being the sine of ABC; and as the chord

of 60° is the side of a hexigon inscribed in a circle, it is equal to radius also, (Prop. 15, 4.) And if the angle AOB, measured by the arc AB, be 45°, the angle APO is 45°, (32.1); therefore (6.1) AO=AP; that is, the tangent of 45° is equal to the radius: hence the sine of 90°, the chord of 60°, the tangent of 45°, and radius, are all equal.

2. As the diameter which bisects an arc, bisects also the chord of that arc at right angles, it necessarily follows that half the chord of an arc is the sine of half that arc. For, as Fk is equal to kE, the angles FOx, EOx are equal (27.3), and FO being equal to OE, and Ox common, Fr is equal to Ex, and the angle OxF equal to OxE (4.1): hence, Fr or Ex is the sine of the arc Fk or kE; that is, of half the arc FkE.

6

SECTION II.

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

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,