In this branch of the subject the limits of our work will not permit us to enter apon all the beautiful contrivances that have been invented for facilitating the operations of the draughtsman; but we shall endeavor to describe the constructions and applications of such as are in the 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 movable point, ink point, and pencil point. Hair compasses. Bow compasses. Drawing pen and pricking point. And we shall also give some account of the following; viz., Beam Compasses. Plotting scales. The pantagraph. Sliding Rule. joint be beyond regulate stiff en from it their us can be Hair into a passes groove and, b to be wards Fig. the scr wards it woul Fig. the nu at plea To to N DRAWING COMPASSES. - This instrument consists of two legs movable about a t, so that the points at the extremities of the legs may be set at any required ance from one another; it is used to transfer and measure distances, and to cribe arcs and circles. The points of the compasses should be formed of well-tempered steel, that cannot ly be bent or blunted, the upper part being formed of brass or silver. The at is framed of two substances; one side being of the same material as the upper t of the compasses, either brass or silver, and the other of steel. This arrangeat diminishes the wear of the parts, and promotes uniformity in their motion. If s uniformity be wanting, it is extremely difficult to set the compasses at any ired distance, for, being opened or closed by the pressure of the finger, if the nt be not good, they will move by fits and starts, and either stop short of, or go yond the distance required; but, when they move evenly, the pressure may be gulated so as to open the legs to the desired extent, and the joint should be f enough to hold them in this position, and not to permit them to deviate m it in consequence of the small amount of pressure which is inseparable from eir use. When greater accuracy in the set of the compasses is required than n 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 to a bent spring, which, being fastened at one extremity to the leg of the comsses almost close up to the joint, is held at the other end by a screw. oove is formed in the shank, which receives the spring when screwed up tight; d, by turning the screw backwards, the steel point may be gradually allowed be pulled backwards by the spring, and may again be gradually pulled forards by the screw being turned forwards. A Fig. 2 represents these compasses when shut; Fig. 3 represents them open, with ne screw turned backwards, and the steel point p, in consequence moved backards by its spring, from the position represented by the dotted lines, which would have when screwed tight up. Fig. 4 represents a key, of which the two points fit into the two holes seen in se nut n, of the joint; and by turning this nut the joint is made stiffer or easier pleasure. To take a Distance with the Hair Compasses. - Open them as nearly as you can to MATHEMATICAL INSTRUMENTS. The required distance, set the fixed leg on the point from which the distance is o 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 MOVABLE 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 ccuracy than a pencil point; but, in order to draw arcs or circles with ink or black ead, compasses with a movable 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 movable point, or a pencil or ink point, as may be required. A lengthening par may also be attached between the shank and the movable point, so as to trike larger circles, and measure greater distances. The movable 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. Fig. 5, the compasses, with a movable point at B. c and D, the joints to set each point perpendicular to the paper. To describe small arcs or circles, a small pair of compasses, called bow compasses, vith a permanent ink or pencil point, are used. They are formed with a round head, which rolls with ease between the fingers. Figs. 9 and 10 represent two contructions of pen bows. Fig. 9 to describe arcs of small radii with exactness, by neans of the adjusting screw, c. For copying and reducing drawings, compasses of a peculiar construction are used; he simplest form of which is that called wholes and halves, (Fig. 11, Plate XLII,) Decause the longer legs being twice the length of the shorter, when the former are pened 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 nlarged 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 Fig. 1 groov appear set to will c the po oft of the enlarg To The 1 points prick thus, circles plans passes betwe ends, off by the S › reduced or enlarged, so that all the lines of the copy, or the areas of solids presented by its several parts, shall bear any required proportion of the lines, eas, or solids of the original drawing. They will also serve to inscribe regular olygons in circles, and to take the square roots and cube roots of numbers. In ig. 12 the scale of lines is placed on the leg, AE, on the left-hand side of the coove, and the scale of circles, on the same leg, on the right-hand side of the roove. 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 ppear but as one; the nut c, being then unscrewed, the slider may be moved, ntil the line across it coincide with any required division upon any one of the ales. Now tighten the screw, and the compasses are set. To reduce or enlarge the Lines of a Drawing. -The line across the slider being et to one of the divisions, 2, 3, 4, &c., on the scale of lines, the points A B, ill open to double, triple, four times, &c., the distances of the points DE. If, nen, the points A and B be opened to the lengths of the lines upon a drawing, he points D and E will prick off a copy with the lines reduced in the proportions f to 1, to 1,4 to 1, &c.; but, if the points D and E be opened to the lengths f the lines upon a drawing, the points A and B will prick off a copy with the lines nlarged in the proportions of 2 to 1, 3 to 1, 4 to 1, &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 rick off a polygon on 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 ircles, and a circle be described with the radius AB, DE will be the cord of a th part of the circumference, and will prick off a regular polygon of 12 sides in it. 1 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 comDasses, when the line across the slider is set to those numbers; and, the distances Detween the points being in the same ratio as the lengths of the corresponding ends, 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. Thus, if the line across the MATHEMATICAL INSTRUMENTS. be set to 4 on the scale of plans, the distance between the points a and • take the Square Root of a Number. The line across the slider being set to The numbers of the scale of solids are the cubes of the ratios of the lengths of The Cube Root of a given number may be found by setting the line across the lider to the number upon the scale of solids, and, opening the points AB, to take off the number upon any scale of equal parts, the points DE, will then take off the required cube root from the same scale. THE sented three in any points coincide transfer upon t with a the two the com "Anot THE sists of that a steel po more O required by taki effective mon pe the lin paper; before and in For represe adjustin called forceps division prick t the eye |