front, fig. 7, plate 3: the principle on which they both act is exactly the same; those with an adjusting screw are inore easily set to any given division or line, and are also more firmly fixed, when adjusted. There is a groove in each shank of these compasses, and the centre is moveable, being constructed to slide with regularity in these grooves, and when properly placed, is fixed by a nut and screw; on one site of these grooves are placed two scales, one for lines, the other for circles. By the scale of lines, a right line may be divided into any number of equal parts expressed on the scale. By the scale for circles, a regular polygon may be inscribed in a circle, provided the sides do not exceed the numbers on the scale. Sometimes are added a scale for superficies and a scale for solids.

To divide a given line into a proposed number (11) of equal parts. 1. Shut the compasses. 2. Unscrew the milled nut, and move the slider until the line across it coincides with the 11th division on the scale. 3. Tighten the screw, that the slider may be immove able. 4. Open the compasses, so that the longer points may take in exactly the given line, and the shorter will give youth of that line.

To inscribe in a circle a regular polygon of 12 sides. 1. Shut the compasses. 2. Unscrew the milled nut, and set the division on the slider to coincide with the 12th division on the scale of circles. 3. Fasten the milled nut. 4. Open the compasses, so that the longer legs may take the radius, and the distance between the shorter legs will be the side of the required polygon.

To use the proportional compasses with an adjusting screw. The application being exactly the same as the simple one, we have nothing more to describe than the use and advantage of the adjusting screw. 1. Shut the legs close, slacken the screws of the nuts g and f; move the nuts and slider k to the division wanted, as near as can be readily done by the hand,

2.

and screw fast the nut f: then, by turning the adjuster h, the mark on the slider k may be brought exactly to the division: screw fast the nut g. Open the compasses; gently lift the end e of the screw of the nut fout of the hole in the bottom of the nut g; move the beam round its pillar a, and slip the point e into the hole in the pin n, which is fixed to the under leg; slacken the screw of the nut f; take the given line between the longer points of the compasses, and screw fast the nut f: then may the shorter points of the compasses be used, without any danger of the legs changing their position; this being one of the inconveniences that attended the proportional compasses, before this ingenious contrivance.

Fig. 10, plate3, represents a pair of beam compasses; they are used for taking off and transferring divisions from a diagonal or nonius scale, describing large arches, and bisecting lines or arches. It is the instrument upon which Mr. Bird principally depended, in dividing those instruments, whose accuracy has so much contributed to the progress of astronomy. These compasses consist of a long beam made of brass or wood, furnished with two brass boxes, the one fixed at the end, the other sliding along the beam, to any part of which it may be firmly fixed by the screw P. An adjusting screw and micrometer are adapted to the box A at the end of the beam; by these, the point connected therewith may be moved with extreme regularity and exactness, even less than the thousandth part of an inch.

Fig. 13, plate 3, is a small pair of beam compasses, with a micrometer and adjusting screw, for accurately ascertaining and laying down small distances, to the 1000th part of an inch or less.

Fig. 11, plate 3, represents a scale of equal parts, constructed by Mr. Sisson; that figured here contains two scales, one of three chains, the other of four chains in an inch, being those most frequently used; each of these is divided into 10 links, which

are again subdivided by a nonius into single links; the index carries the protracting pin for setting off the lengths of the several station lines on the plan. By means of an instrument of this kind, the length of a station line may be laid down on paper with as much exactness as it can be measured on land.

OF PARALLEL RULES.

Parallel lines occur so continually in every species of mathematical drawing, that it is no wonder so many instruments have been contrived to delineate them with more expedition than could be effected by the general geometrical methods: of the various contrivances for this purpose, the following are those most approved.

1. The common parallel rule, fig. A, plate 2. This consists of two straight rules, which are connected together, and always maintained in a parallel position by the two equal and parallel bars, which move very freely on their centres, or rivets, by which they are fastened to the straight rules.

2. The double parallel rule, fig. B, plate 2. This instrument is constructed exactly upon. the same principles as the foregoing, but with this advantage, that in using it, the moveable rule may always be so placed, that its ends may be exactly over, or even with, the ends of the fixed rule, whereas in the former kind, they are always shifting away from the ends of the fixed rule.

This instrument consists of two equal fiat rules, and a middle piece; they are connected-together by four brass bars, the ends of two bars are rivetted on the middle line of one of the straight rules; the ends of the other two bars are rivetted on the middle line of the other straight rule; the other ends of the brass bars are taken two and two, and rivetted on the middle piece, as is evident from the figure; it would be

needless to observe, that the brass bars move freely on their rivets, as so many centres.

3. Of the improved double parallel rule, fig. C, plate 2. The motions of this rule are more regular than those of the preceding one, but with somewhat more friction; its construction is evident from the figure; it was contrived by the ingenious mechanic, Mr. Haywood.

4. The cross barred parallel rule, fig. D, plate 2. In this, two straight rules are joined by two brass bars, which cross each other, and turn on their intersection as on a centre; one end of each bar moves on a centre, the other slides in a groove, as the rules recede from each other.

As the four parallel rules above described are all used in the same way, one problem will serve for them all; ex. gr. a right line being given, to draw a line parallel thereto by either of the foregoing instru

ments:

Set the edge of the uppermost rule to the given line; press the edge of the lower rule tight to the paper with one hand, and, with the other, move the upper rule, till its edge coincides with the given point; and a line drawn along the edge through the point is the line required.

5. Of the rolling parallel rule. This instrument was contrived by Mr. Eckhardt, and the simplicity of the construction does credit to the inventor; it must, however, be owned, that it requires some prac▾ tice and attention to use it with success.

Fig. E, plate 2, represents this rule; it is a rectangular parallelogram of black ebony, with slips of ivory laid on the edges of the rule, and divided into inches and tenths. The rule is supported by two small wheels, which are connected together by a long axis, the wheels being exactly of the same size, and their rolling surfaces being parallel to the axis; when they are rolled backwards or forwards, the axis and

rule will move in a direction parallel to themselves. The wheels are somewhat indented, to prevent their sliding on the paper; small ivory cylinders are sometimes affixed to the rollers, as in this figure; they are called rolling scales. The circumferences of these are so adjusted, that they indicate, with exactness, the parts of an inch moved through by the rule.

In rolling these rules, one hand only must be used, and the fingers should be placed nearly in the middle of the rule, that one end may not have a tendency to move faster than the other. The wheels only should touch the paper when the rule is moving, and the surface of the paper should be smooth and flat.

In using the rule with the rolling scales, to draw a line parallel to a given line at any determined distance, adjust the edge of the rule to the given line, and pressing the edge down, raise the wheels a little from the paper, and you may turn the cylinders round, to bring the first division to the index: then, if you move the rule towards you, look at the ivory cylinder on the left hand, and the numbers will shew in tenths of an inch, how much the rule moves. If you move the rule from you, then it will be shewn by the numbers on the right hand cylinder.

To raise a perpendicular from a given point on a given line. Adjust the edge of the rule to the line, placing any one of the divisions on the edge of the rule to the given point; then roll the rule to any distance, and make a dot or point on the paper, at the same division on the edge of the rule; through this point draw the perpendicular.

To let fall a perpendicular from any given point to a given line. Adjust the rule to the given line, and roll it to the given point; then, observing what division, or point, on the edge of the rule the given point comes to, roll the rule back again to the given line, and the division, or point, on the edge of the rule will shew the point on the given line, to which the perpendicular is to be drawn.