Proportional COMPASSES with the sector lines. The structure of these is so like that of the common proportional compasses, only a little nicer, that it needs no particular description.
The lines on the first face are the line of lines, marked lines; it is divided into 100 equal parts, every tenth numbered; and the line of chords, which goes to 60°, is marked chords.
On the other face are a line of sines to 90°, and a line of tangents to 45°. On the other side are the tangents from 45° to 71° 34'; on the other, secants from 0° to 70° 30'.
For the use of these compasses. 1. To divide a line into any number of equal parts less than 100: Divide 100 by the number of parts required; slip the cursor till the line on the sliding dovetail be against the quotient on the line of lines; then, the whole line being taken between the points of the compasses most remote from the centre, the aperture of the other will show the division required. 2. A right line given, supposed to be divided into 100 parts, to take any number of those parts: slip the line on the sliding dovetail to the number of parts required; the whole line being taken between the points farthest from the centre, the aperture of the other two will include the number of divisions required. 3. The radius being given, to find the chord of any arch under 60°: slip the line on the sliding dovetail to the degrees required on the line of chords; the radius being taken between the points farthest from the centre of the cursor, the aperture of the other line will be the chord required, provided the number of degrees be greater than 29: if it be less, the aperture taken from the radius will leave the chord required. 4. If the chord of an arch under 60° be given, and the radius required; slip the line on the dovetail to the degrees given on the line of chords; the given chord being taken between the two points next the cursor, the aperture of the other will be the radius required. 5. The radius being given, to find the sine of any number of degrees: slip the line on the dovetail to the degree on the line of sines whose sine is required; the radius taken between the points fur-
thest from the cursor, the aperture of the other will give the sine of the angle required: but if the sine sought be less than 30°, the difference of the apertures of the opposite points will be the sine required. 6. The radius being given, to find the tangent of any number of degrees under 71: if the tangent required be under 26° 30', then slip the line on the dovetail to the degree proposed on the tangent line; the radius taken between the points farthest from the cursor, the aperture of the other will be the tangent of the degrees required: if the tangent required be above 26° 30', but under 45°, the line on the cursor must be slipped to the degrees given on the tangent line; then the radius being taken between the points furthest from the cursor, the aperture of the other will be the tangent. If the tangent required be greater than 45°, but less than 56° 20', slip the notch on the tangent side of the turned cheek to the degree on the tangent line on the side of the compass; the radius taken between the points farthest from the cursor, the difference between the aperture of the other and these added together, will be the tangent required. Thus, for the tangents of other degrees under 71. After the like manner may the secant of any number of degrees under 71 be found.
Mr Heath, a mathematical-instrument maker in London, constructed a pair of proportional compasses in 1746, with a curious and useful contrivance for preventing the shorter legs from changing their position when these compasses were used. It consisted of a small beam folded to a screw, and running parallel to the leg of the compasses, nearly of the length of the groove: in this beam a slit was made, which admitted of a sliding-nut, the other end of which fell into a hole in the bottom of the screw belonging to the great nut of the compasses. The screw-pin of the beam passed through an adjuster, by means of which the mark on the slider might be brought exactly to any division. But the proportional compasses have been much out of use since the invention of the sector.