a tracer may be put on the pin D, and are retained there by a screw G; the pen is furnished with a moveable arm EF, having a small ball of brass F at the end, whose use is to cause the pen to press with due force upon the paper, the degree of which can be regulated by placing the arm in different po

sitions.

The ruler AB being set to any given line, by rolling it along, other lines may be drawn, all of which will tend to some one point in the given line, or a continuation of it, whose distance will be greater, as the distance between the rollers A and C is increased; and as the diameter of C approaches that of A; all which is evident from prini ciple 2, page 130.

It also appears from the said principle, that during the motion of the ruler, any point in its axis will accurately describe the arc of a circle, having the said point of intersection for its centre; and, consequently, the pen or tracer, put on the pin D, will describe such arcs.

The rollers, as C, which screw upon the end of the inner tube are numbered, 1, 2, 3, &c. and as many scales are drawn on that tube as there are rollers, one belonging to each, and numbered accordingly. These scales shew the distance in inches, of the centre or point of intersection, reckoned from the middle of the pin D, (agreeing to the point of the pen or tracer ;) thus ;

No. 1, will describe circles, or serve for drawing lines tending to a point, whose radius or distance from D, is from 1200 inches to 600 inches, according as the tube is drawn out.

No. 2

P

from 600 inches to 300 inches

If it should be required to extend the radius or distance farther than 1200 inches, by using another ruler, it might be carried to 2400 inches; but nes in any common sized drawing,' which tend to point above 100 feet distance, may be esteemed as parallel.

3. ANOTHER RULER OF THE SAME kind.

Fig. 18, plate 10. This is nothing more than the ast instrument applied to a flat ruler, in the manner the rolling parallel rulers are made.

CD is an hexagonal axis, moveable on pivots in the heads A and F fixed upon a flat ruler; on this axis the smaller roller B, is made to slide through one half of its length; the larger roller A, is serewed on the other end of the axis, and can be changed occasionally for others of different diameters. Scales adapted to each of the rollers at A, are either put on the flat sides of the axis from C to E, or drawn on the corresponding part of the flat ruler; and the scales and rollers distinguished by the same number: at F is a screw to raise or lower the end C of the axis, till the ruler goes parallel to the paper on which the drawing is made; and at G there is a socket, to which a drawing pen and tracer is adapted for describing arcs.

In using these instruments, the fingers should be placed about the middle' part between the rollers; and the ruler drawn, or pushed at right angles to its length. The tube AB, fig. 17, and one, or both of the edges of the flat ruler, fig. 18, are divided into inches and tenths.

4. THE CYCLOGRAPH.

This instrument is constructed upon the third principle mentioned in page 130 of this Essay. Fig. 16, plate 10, is composed of five rulers; four

L

of them DE, DF, G E and G F, forming a trapezium, are moveable on the joints D, E, F and G; the fifth ruler D I, passes under the joint D, and through a socket carrying the opposite joint G. The distances from the centre of the joint D, to that of the joints E and F, are exactly equal, as are the distances from G to the same joints. The rulers DE and DF pass beyond the joints E and F, where a roller is fixed to each; the rollers are fixed upon their axes, which move freely, but steadily on pivots, so as to admit of no shake by which the inclination of the axes can be varied. The ruler ID passing beyond the joint D, carries a third roller A, like the others, whose axis lies precisely in the direction of that ruler; the axes of B and C extend to K and L.

A scale is put on the ruler DI, from H to G, shewing, by the position of the socket G thereon, the length of the radius of the arc in inches, that would be described by the end I, in that position of the trapezium. When the socket G is brought to the end of the scale near I, the axes of the two rollers B and C, the ruler DI, and the axis of the roller A, are precisely parallel; and in this position, the end I, or any other point in DI, will describe straight lines at right angles to DI; but on sliding the socket G towards H, an inclination is given to the axes of B and C, so as to tend to some point in the line ID, continued beyond D, whose distance from I is shewn by the scale.

A proper socket, for holding a pen or tracer, is made to put on the end I, for the purpose of describing arcs; and another is made for fixing on any part of the ruler D I, for the more convenient description of concentric arcs, where a number are wanted.

It is plain from this description, that the middle ruler DI in this instrument, is a true oblique ruler, by which lines may be drawn tending to a point, whose distance from I is shewn by the position of the

socket G on the scale; and the instrument is made sufficiently large, so as to answer this purpose as well as the other.

5. A DIFFERENT CONSTRUCTION OF THE SAME INSTRUMENT.

In fig. 16, plate 10, the part intended to be used in drawing lines, lies within the trapezium, which is made large on that account; but this is not necessary; and fig. 15, plate 10, will give an idea of a like instrument, where the trapezium may be made much smaller, and consequently less cumbersome.

DBEC represents such a trapezium, rollers, socket, and scale as above described, but much maller. Here the ruler E D is continued a sufficient length beyond D, as to A, where the third roller is fixed; a pen or tracer may be fitted to the end E, or made to slide between Ď and A, for the purpose of drawing arcs.

METHODS OF DESCRIBING AN ELLIPSE, AND SOME OTHER CURVES.

To describe an ellipse, the transverse and conjugate axes being given.

Let AB be the given transverse, and CD the conjugate axes, fig. 13, plate 13.

Method 1. By the line of sines on the sector, open the sector with the extent AG of the semi-transverse axis in the terms of 90 and 90; take out the transverse distance of 70 and 70, 60 and 60, and so for every tenth sine, and set them off from G to A, and from G to B; then draw lines through these points perpendicular to AB. Make GC a transverse distance between 90 and 90, and set off each tenth sine from G towards C, and from G towards D, and through these points draw lines parallel to AB,

which will intersect the perpendiculars to AB in the points A, a, b, c, d, e, f, g, h, i, k, l, m, n, o, p, q, B, for half the ellipse, through which points and the intersections of the other half, a curve being drawn with a steady hand, will complete the ellipse.

Method 2. With the elliptical compasses, fig. 3, plate 11, apply the transverse axis of the elliptical compasses to the line A B; and discharge the screws of both the sliders; set the beam over the transverse axis A B, and slide it backwards and forwards until the pencil or ink point coincide with the point A, and tighten the screw of that slider which moves on the conjugate axis; now turn the beam so as to lay over the conjugate axis C D, and make the pencil or ink point coincide with the point C, and then fix the screw, which is over the slider of the transverse axis of the compasses; the compasses being thus adjusted, move the ink point gently from A, through C to B, and it will describe the semi-ellipse A CB; reverse the elliptical compasses, and describe the other semiellipse B D A. These compasses were contrived by my Father in 1748; they are superior to the trammel which describes the whole ellipse, as these will describe an ellipse of any eccentricity, which the others will not.

Through any given point F to describe an ellipse, the transverse axis A B being given.

Apply the transverse axis of the elliptical compasses to the given line A B, and adjust it to the point A; fix the conjugate screw, and turn the beam to F, sliding it till it coincide therewith, and proceed as in the preceding problem. '

Fig. 2, plate 11, represents another kind of elliptical apparatus, acting upon the principle of the oval lathes; the paper is fixed upon the board A B, the pencil C is set to the transverse diameter by sliding it on the bar D E, and is adjusted to the conjugate diameter by the screw G; by turning the board A B,