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 are. For, as Fk is equal to kE, the angles Fox, Ex are equal (27.3), and FO being equal to OE, and Ox common, Fx is equal to Ex, and the angle OxF equal to OxE (4.1): hence, Fx or Ex is the sine of the arc Fk or kE; that is, of half the are FkE.

3

fres de prezenticnar 43. meecmg 03 zrodced; list frew CG mi perpendicular CC, and BS perpendicar a CA. Then CB is the complement of the assumed are AB: is smquemen is FCF; BS is the sine; BD cr OS is the ansie: SA is the versed sine;

DC the coversed size: the supplementary versed sine is PS; 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

is equal to radius also, (Prop. 15, 4.) And if the ang! AOB, measured by the arc AB, be 45°, the angh APO is 45°, (32.1); therefore (6.1) AO=AP; tha is, the tangent of 45° is equal to the radius: hence th sine of 90°, the chord of 60°, the tangent of 45°, an radius, are all equal.

2. As the diameter which bisects an arc, bisects als the chord of that arc at right angles, it necessarily follow that half the chord of an arc is the sine of half that are For, as Fk is equal to kE, the angles FOX, EO are equal (27.3), and FO being equal to OE, and Ox common, Fx is equal to Ex, and the angle OxF equal to OrE (4.1): hence, Fx or Er is the sine of the arc Fk or kE; that is, of half the are

FkE.

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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