DIAGRAM, a figure so drawn that its geometrical relations may illustrate the rela tions between other quantities. The area of a rectangle is the product of its length and breadth; the diagram of the rectangle becomes the visible symbol, corresponding to the equation a = bl ; by analogy, the rectangle may be used to symbolize any quantity which is the product of two factors. Similarly, a parallelopiped may symbolize any quantity which is the product of 3 factors; e.g., interest, which is the product of prin cipal, rate, and time, i = p.t.r., may be symbolized by a diagram in which principal is figured by length, time by breadth, and rate by height, the total volume representing the interest.
The purpose of ordinary mathematical diagrams is simply illustration, and it is therefore necessary only that the ideas be clearly presented, accuracy of drawing being unimportant. Other diagrams, as some drawn by engineers and architects, are intended to furnish magnitudes or distances by actual measurement, and their execution cannot be too careful. Other diagrams, like those showing electric connections, require only a proper showing of the parts and the methods of uniting them. A. profile diagram shows such an outline as would be formed, for example, if a hill were cut through by a verti cal plane, and the material on one side of the plane were removed. Evidently a succes sion of such profiles might be laid upon the same sheet of paper, the lines being dis tinctively drawn, and the-whole would serve to compare several vertical profiles of the same mass. It is•not necessary that vertical and horizontal measurements should don form to the same scale, provided that each series of measurements is consistent with itself. Geographical profiles, which include upon a single sheet the outlines of entire continents or ocean-beds, have usually the vertical measurements on a scale several times greater than that used for horizontal distances; otherwise the diagram would be made inconveniently long, or the heights would be inconspicuously small. Yet the impression left by such a diagram is often mischievous, especially upon the illiterate. A topographer's contour map exhibits a series of curves, such as would he formed if a series of horizontal sections were made, and the outlines carefully laid down upon paper. The drawing may be understood to show the horizontal projections of the con tour lines upon a surface parallel to the system. In the representations of parts of machinery, particularly those designed to guide workmen in construction, several con nected views of the same object are required, each view giving some information which the others cannot furnish. Suppose three planes perpendicular to each other, like the bottom, one side, and one end of a rectangular box, and let an object, as a hexagonal nut, be placed within the triedral angle thus formed. Looking from the front, we see one image of the nut projected against the back of the box; from the side, a different image is seen against the end of the box; from above, a third form appears against the bottom of the box; while from one or other of these figures all the measurements of the nut may be obtained. If now the end of the box were swung outward into the plane
of the back of the box, and then both together were laid back into the plane of the bot tom, we should have the three co-existent drawings in one plane, and they may Le trans ferred to, or be constructed on, one sheet of paper. In many cases the :tune points will find representation upon each of the three diagrams, and the fact may be indicated by using the same letter for a point wherever it occurs; while the eye may be led from one position of it to another by lines, conventionally drawn, as fine, or dotted, or broken, to show that they are merely guides and not parts of the diagram.
Many devices have been invented by which diagrams illustrating natural phenomena may be automatically described. Let us suppose that a spring dynamometer is placed where it may receive the draft of a horse when moving a carriage. Let the movement of the spring be shown by an index whose motion is back and forth along a line in the direction of the draft. Fix a pencil to the index, and let its point rest upon a sheet of paper on a plane or a cylinder which moves at a uniform velocity in a direction perpen dicular to the motion of the index. The combined movements of the pencil and the paper beneath it will trace a line more or less irregular. If the force of draft were unvaried, the pencil would remain at a constant distance from the edge of the paper, and the trace would be parallel to the edge. If the paper does not move and the pencil varies, the line will be perpendicular to the edge. If both move, and the pencil be obedient to a diminishing force, the trace will be oblique, approaching one edge; while if the force increase, the oblique trace will diverge from the same edge. Such mechan ism is often arranged for instruments which indicate meteorological changes, as the force and direction of winds; or the pulsation of the arteries, as in the sphygmograph; or the movements of a clock, combined with the observations of an astronomer, as in the chronograph; applicable also to many physical problems. An application of the same principle has great importance in the indicator diagram, by which the pressure of the steam in the steam-engine and the work done by each stroke of the piston, becomes a matter of record for investigation. The paper moves with the movement of the piston, both in its excursion and return; the pencil moves at right angles to the direc tion of the motion of the paper, under the influence of the steam-pressure; and the dia gram drawn shows for each instant of the stroke the vol me and pressure of the steam, while the total area of the diagram indicates the amount f work done.