A TREATISE ON MATHEMATICAL INSTRUMENTS. 104 104 105 106 112 113 115 117 35 35 PART I.-ON MATHEMATICAL DRAWING INSTRUMENTS. In this branch of the subject the limits of our little work will not permit us to enter upon all the beautiful contrivances which have been invented for facilitating the operations of the draughtsman; but we shall endeavour to describe the constructions and applications of such as are in most general use, and, as far as our space will allow, to exhibit the principles upon which they are founded, so that the student may readily extend his views, after having completely mastered the matter here presented to him, to the principles of any other instruments, which may be useful to him in whatever particular professional branch of practical mathematics he may wish to employ himself. With this view we shall describe the instruments in the ordinary case of drawing instruments, as sold by any mathematical instrument maker; viz., Compasses with moveable point, ink point, and pencil point. Hair compasses. Bow compasses. Drawing pen and pricking Plain scale. Sector. And we shall also give some account of the following; viz., Whole and halves. Proportional compasses. Triangular compasses. Marquois's scales. Beam compasses. Plotting scales. The pantagraph. ON DRAWING COMPASSES. This instrument consists of two legs moveable about a joint, so that the points at the extremities of the legs may be set at any required distance from one another; it is used to transfer and measure distances, and to describe arcs and circles. The points of the compasses should be formed of well-tempered steel, that cannot be easily bent or blunted, the upper part being formed of brass or silver. The joint is framed of two substances; one side being of the same material as B the upper part of the compasses, either brass or silver, and the other of steel. This arrangement diminishes the wear of the parts, and promotes uniformity in their motion. If this uniformity be wanting, it is extremely difficult to set the compasses at any desired distance, for, being opened or closed by the pressure of the finger, if the joint be not good, they will move by fits and starts, and either stop short of, or go beyond the distance required; but, when they move evenly, the pressure may be regulated so as to open the legs to the desired extent, and the joint should be stiff enough to hold them in this position, and not to permit them to deviate from it in consequence of the small amount of pressure which is inseparable from their use. When greater accuracy in the set of the compasses is required than can be effected by the joint alone, we have recourse to the Hair Compasses, in which the upper part of one of the steel points is formed into a bent spring, which, being fastened at one extremity to the leg of the compasses almost close up to the joint, is held at the other end by a screw. A groove is formed in the shank, which receives the spring when screwed up tight; and, by turning the screw backwards, the steel point may be gradually allowed to be pulled backwards by the spring, and may again be gradually pulled forwards by the screw being turned forwards. Fig. 1 represents these compasses when shut; fig. 2 represents them open, with the screw turned backwards, Fig. 1 Fig. Fig. 3 and the steel point p, in consequence moved backwards by its spring s, from the position represented by the dotted lines, which it would have when screwed tight up. Fig. 3 represents a key, of which the two points fit into the two holes seen in the nut, n, of the joint; and by turning this ut the joint is made stiffer or easier at pleasure. To take a Distance with the Hair Compasses.-Open them as nearly as you can to the required distance, set the fixed leg on the point from which the distance is to be taken, and make the extremity of the other leg coincide accurately with the end of the required distance, by turning the screw. COMPASSES WITH MOVEABLE POINTS. If an arc or circle is to be described faintly, merely as a guide for the terminating points of other lines, the steel points are generally sufficient for the purpose, and are susceptible of adjustment with greater accuracy than a pencil point; but, in order to draw arcs or circles with ink or black lead, compasses with a moveable point are used. In the best description of these compasses the end of the shank is formed into a strong spring, which holds firmly the moveable point, or a pencil or ink point, as may be required. A lengthening bar may also be attached between the shank and the moveable point, so as to strike larger circles, and measure greater distances. The moveable point to be attached to the lengthening bar, as also the pen point and pencil point, are furnished with a joint, that they may be set nearly perpendicular to the paper. A, the compasses, with a moveable point at B. c and D, the joints to set each point perpendicular to the paper. E, the pencil point. F, the pen point. (This is represented with a dotting wheel, the pen point and the dotting point being similar in shape to each other.) G, the lengthening bar. To describe small arcs or circles a small pair of compasses, called bow compasses. with a permanent ink or pencil point, are used. They are formed with a round head, which rolls with ease be tween the fingers. The adjoining figures represent two constructions of pen bows, fig. 1 being well adapted to describe arcs of not more than one inch radius, and fig. 2 to describe arcs of small radii with exactness by means of the adjusting screw C. For copying and reducing drawings, compasses of a peculiar construction are used; the simplest form of which is that called wholes and halves, because the longer legs being twice the length of the shorter, when the former are opened to any given line, the shorter ones will be opened to the half of that line. By their means, then, all the lines of a drawing may be reduced to one-half, or enlarged to double their length. These compasses are also useful for dividing lines by continual bisections. PROPORTIONAL COMPASSES. By means of this ingenious instrument drawings may be reduced or enlarged, so that all the lines of the copy, or the areas or solids represented by its several parts, shall bear any required proportion to the lines, areas, or solids of the original drawing. They will also serve to inscribe regular polygons in circles, and to take the square roots and cube roots of numbers. In the annexed figure the scale of lines is placed on the leg A E, on the left-hand side of the groove, and the scale of circles, on the same leg, on the right-hand side of the groove. The scales of plans and solids are on the other face of the instrument. To set the instrument it must first be accurately closed, so that the two legs appear but as one; the nut c being then unscrewed, the slider may be moved, until the line across it coincides with any required division upon any one of the scales. Now tighten the screw, and the compasses are set. To reduce or enlarge the Lines of a Drawing.The line across the slider being set to one of the divisions, 2, 3, 4, &c., on the scale of lines, the points A, B will open to double, triple, four times, &c., the distances of the points D, E (Euc. vi. prop. 4). If, then, the points A and B be opened. to the lengths of the lines upon a drawing, the points D and E will prick off a copy with the lines reduced in the proportions of to 1, 1 to 1,4 to 1, &c.; but, if the points D and E be opened to the lengths of the lines upon a drawing, the points A and B will prick off a copy with the lines enlarged in the proportions of 2 to 1, 3 to 1, 4 to 1, &c. B C To inscribe in a Circle a regular Polygon of any number of Sides from 6 to 20.-The line across the slider being set to any number on the scale of circles, and the points A and B being opened to the length of any radius, the points D and E will prick off a polygon of that number of sides, in the circle described with this radius; thus, if the line across the slider be set to the division marked 12 on the scale of circles, and a circle be described with the radius A B, D E will be the chord of ath part of the circumference, and will prick off a regular polygon of 12 sides in it. To reduce or enlarge the Area of a Drawing.-The numbers upon the scale of plans are the squares of the ratios of the lengths of the opposite ends of the compasses, when the line across the slider is set to those numbers; and, the distances between the points being in the same ratio as the lengths of the corresponding ends (Euc. vi. prop. 4), the areas of the drawings, and of the several parts of the drawings, pricked off by these points, will have to one another the ratio of 1 to the number upon the scale of plans to which the instrument is set (Euc. vi. props. 19, 20; and xii. prop. 2). Thus, if the line across the slider be set to 4 on the scale of plans, the distance between the points A and B will be twice as great as the distance between D and E; and, if A and B be opened out to the lengths of the several lines of a drawing, D and E will prick off a copy occupying 4th the area; if the line across the |