an instrument designed for drawing figures in what proportion you please, without any skill in the art.

The instrument is otherwise called a parallelogram. The common pentagraph (Plate CCCLXXXIII. fig. 14.) consists of four bars or wooden rulers, two of them from 15 to 18 inches long, the other two half that length. At the ends, and in the middle, of the longer rulers, as also at the ends of the shorter, are holes, upon the exact fixing whereof the perfection of the instrument chiefly depends. Those in the middle of the long rulers are to be at the same distance from those at the end of the long ones and those of the short ones; so that when put together they may always make a parallelogram.

The instrument is fitted together for use by several little pieces, particularly a little pillar, No. 1, having at one end a screw and nut, whereby the two long rulers are joined; and at the other a little knot for the instrument to slide on. The piece, No. 2, is a rivet with a screw and nut, wherewith each short ruler is fastened to the middle of each long one. The piece, No. 3, is a pillar, one end whereof, being hollowed into a screw, has a nut fitted to it. At the other end is a worm to screw into the table; when the instrument is to be used, it joins the ends of the two short rulers. The piece, No. 4, is a pen, porterpen, or pencil, screwed into a little pillar. Lastly, the piece, No. 5, is a bar's point, moderately blunt, screwed likewise into a little pillar.

Use of the Pentagraph, or Parallelogram. 1. To copy a design in the same scale or bigness as the original; screw the worm No. 3, into the table; lay a paper under the pencil No. 4, and the design under the point No. 5. This done, conducting the point over the several lines and parts of the design, the pencil will draw or repeat the same on the paper.

2. If the design be to be reduced—e.g., into half the space, the worm must be placed at the end of the long ruler, No. 4, and the paper and pencil in the middle. In this situation conduct the bar's point over the several lines of the design, as before; and the pencil at the same time will draw its copy in the proportion required; the pencil here only moving half the lengths that the point moves.

Hence, on the contrary, if the design be to be enlarged by one half, the bar's point, with the design, must be placed in the middle, at No. 3, the pencil and paper at the end of the long ruler, and the worm at the other.

3. To enlarge or reduce in other proportions, there are holes drilled at equal distances on each ruler, viz., all along the short ones, and half way of the long ones, in order for placing the bar's point, pencil, and worm, in a right line therein; i.e., if the piece carrying the point be put in the third hole, the two other pieces must be put in its third hole.

If, then, the point and design be placed at any hole of the great rulers, and the pencil with the paper at any hole of the short ruler, which forms the angle therein, the copy will be less than half the original. On the contrary, if it be placed at one of the holes of that short ruler, which is parallel to the long ruler, the copy will be greater than half the original.

The construction of this instrument requires a degree of accuracy which most of our instrument-makers are strangers to; for which reason there are very few of the instruments that succeed. Few will do anything tolerably but straight lines; and many of them not even those.

In order to prove that the figure described by a pentagraph is similar to the given figure, let C (fig. 15.) be the fixed centre of motion; P the pencil for tracing the given figure PP, and p the pencil which traces the other figure pp; p, &c., must be so adjusted, that p, C, and P, may lie in one straight line; then, since BP : AP :: BP : AC, whatever be the situation of the pentagraph, the angles PCP and pCp are vertical; and therefore PCp will in every position of the instrument be a right line: but PC : pC :: BA : Ap, in each of the two positions in the figure, and consequently the triangles PCP, pCp, are similar; and PP : pp (: PC : pC) :: BA : Ap, or in a given ratio. Hence it appears, that, by moving the pencil p, Ap may be equal to BA, or less in any proportion; and consequently pp may be equal to PP, or less, in the same proportion.