DIAGRAM, a figure drawn so that geometrical relations among its parts illustrate relations among the objects represented by the figure, supplemented sometimes by numerical or other entries on the figure itself to show relations not represented graphically by the figure. The purpose of a diagram is to present vividly to the eye the principal relations on which one desires to fix attention and even sometimes to show, by measurements on the figure, the exact numerical values of certain important quantities associated with the object which it represents. Owing to the generality of the concept, diagrams are useful for a great variety of specific purposes.
Mathematical Diagrams.—In mathematical treatises dia grams are used principally to help the reader follow the reasoning. Figures are drawn to represent to the eye the relations among the parts involved in a proposition to be proved and in the auxiliary propositions employed in the demonstration. In the proof itself attention is fixed upon the relations which are relevant to the matter in hand so that the demonstration is made general and is quite independent of the extraneous properties involved in the form of the particular figure employed. The construction of the figure is usually so well defined in words that the reader could easily draw one for himself if the author did not supply it. Such a diagram is a good one if it sets forth clearly those features which are involved in the proposition to be demonstrated.
Diagrams in Chemistry and Crystallography.—John Dal ton (q.v.) published many diagrams setting forth his conception of the configurations of the atoms in a large number of common chemical compounds; and the method has since been widely used by chemists. With the advance of chemistry it has been found that there are many pairs of substances such that the two sub stances in a given pair have the same molecular formula while they differ widely in some of their important properties. This difference of properties, where there is identity of molecular formulae and where we are forced to admit the same atomic linking, can be explained only by ascribing the difference to a different space distribution of the atoms in the molecule. This has given rise to the development of a branch of chemistry dealing with stereo-isomerism, a subject in which the space diagrams of the positions of atoms play an important role in the explana tion of structure. Similarly, in crystallography (q.v.) diagrams are likewise employed freely in the explanation of crystal struc ture. In recent years we have had a like analysis of atomic struc ture itself by means of diagrams.
Diagrams Showing Measurements.—In a quite different way, diagrams may be used for purposes of measurement ; they are then called metrical diagrams. The plans and designs of architects and engineers, necessarily drawn to scale and made as accurate as possible, are employed in determining various dimen sions of the objects represented, by measuring the lengths of cor responding lines in the diagrams. Such diagrams serve a purpose far beyond that of mere illustration; they afford a means of actual measurements relating to the objects themselves. They are strongly contrasted with diagrams of illustration, which are sug gestive merely and need not show the forms of :he parts provided only their relevant connections are properly exhibited. Of the latter kind are many diagrams in the mathematical subject of analysis situs (q.v.) and also those employed to show electrical connections, as, for instance, in the descriptions of radio receivers.
Geographical maps afford examples of diagrams of still another sort. In these the distances and relative positions among places on the earth are exhibited by their positions on the maps. By means of colours important features of various areas are often indicated, as, for instance, their political connections, or their geological character, or the distribution of rainfall or other cli matic features, or the distribution of terrestrial magnetism, or the variation of elevation above sea level. The heights of places above sea level are of ten indicated also by the insertion of num bers on the map to indicate the number of feet above sea level of each place so designated. Another (and more effective) method of serving the same purpose is that in which a line called a con tour line is drawn through all places on the map having the same height above sea level. When such contour lines are drawn suffi ciently close together and when each of them is marked in one or more places with a number to denote the height above sea level of the places through which it passes, we may obtain from them precise information concerning the character of the surface of the country. In this method the diagram is partly graphical and partly symbolic, and some things are presented by the contour lines and accompanying numbers which are not shown by the relations among the parts of the diagram itself.
Diagrams for Objects Having Three Dimensions.—It is possible to use a system of diagrams for the graphical representa tion of the relations among any set of magnitudes involving more than two variables. In particular, to represent the relations among the parts of an object having a distribution in three dimensions we may employ two or more diagrams, each of them showing the relations of parts in a single plane or plane section of the object. Thus construction engineers employ plans and elevations and sections in different planes. In such a system of diagrams a def inite indication must be given of the way in which the diagrams are severally related to the structure as a whole and to each other. Examples of this type are afforded by the plans for buildings or for bridge construction. (See BRIDGES and ARCHITECTURE.) But it is also possible to represent solids and other figures in three dimensions by means of a single diagram drawn in a plane. One of the objects of descriptive geometry (q.v.) is to develop methods for attaining this end.
The stereoscope (q.v.) furnishes a means for the use of two diagrams for the representation of three-dimensional objects in such a way that their forms are readily recognized. The two diagrams are two plane projections of the bodies taken from separated points of view which are yet near to each other. These two plane figures are nearly alike, their difference being due to the difference in point of view. When these two figures are placed in the stereoscope one of them is seen with one eye and the other with the other eye, in such a way that we intuitively identify the corresponding parts of the two figures. In pure geometry the method of projections, which underlies the diagrams used in the stereoscope, has led to many extensions of the science. In fact, on it are founded the principles and results of projective geometry (q.v.).
Diagrams in Mechanics.—It is probably in mechanics (q.v.), both theoretical and applied, that diagrams have been used for the greatest variety of specific purposes. Their application to statics is particularly convenient, owing to the fact that there is no motion of parts in a statical system. Consequently there has arisen an important branch of knowledge under the name of Graphic Statics. In the diagram of configuration it is convenient to represent the objects by points and to denote their relative positions by means of the relative positions of these points. This method is also applicable to the case of bodies in motion, the diagram of configuration then representing the relative positions of the objects at a given instant. If several diagrams of configura tion are constructed, one for each of several given instants of time, then, by a comparison of these diagrams, it is possible to see the relative displacements which have taken place in the various intervals of time involved; but the system of con figurations will not give the details of the motion during the interval between two consecutive instants for which diagrams of configuration have been constructed.
As an example of a different kind, let us consider the diagram in fig. I. This is a parallelogram formed from two directed lines AB and AC, issuing from a common point A, by drawing the related lines CD and BD. Attention is also placed on the directed diagonal AD issuing from A. We may use AB to denote a force applied to an object at A, the direction of AB representing the direction in which the force acts and the length of AB denoting the magnitude of the force. Similarly, AC will denote, in magnitude and direction, another force operating upon the same object at A. In mechanics it is shown that these two forces operating upon the body at A are equivalent, so far as their effect upon A is concerned, to a single force operating in the direction of the diagonal AD and of a magnitude represented by the length of AD. Thus a simple diagram enables us readily to find the (so-called) resultant force AD which is equivalent to the two given forces AB and AC.
Now BD is equivalent, in direction and length, to AC. Hence we might think of the resultant force AD as the third side of the triangle determined by the given forces AB and BD, where it is understood that the force represented by BD operates at A. Then we may call ABD the triangle of forces, whereas ABDC would be called the parallelogram of forces. Now the triangle of forces is capable of a ready generalization, which we shall describe by aid of fig. 2. Let several forces all operate upon an object at A; and let the magnitudes and directions of these forces be respectively those which are indicated by the directed lines AB, BC, CD, DE, placed end to end as in the figure. (These lines may or may not be in one plane.) Then AE will represent, both in magnitude and direction, a single force which is equivalent, in its effect upon the object at A, to the combined effects of the several forces denoted by AB, BC, CD and DE, respectively. This figure ABCDE is called a polygon of forces. If there is added to the system of given forces already described a single force EA acting upon A but having the direction and magnitude denoted by EA, then the new system of forces will be in equilibrium in the sense that their combined effect upon the object A will be to leave that object undisturbed in position. The diagram afforded by the poly gon of forces furnishes one of the most important means in mechanics for the analysis of the relations of forces.
Now the meaning of a diagram depends upon the point of view from which one considers it. This is well illustrated by the dia grams afforded by figs. I and 2, as we shall now show by giving interpretations of them in terms of velocities and accelerations.
If AB and AC in fig. I denote velocities, both in magnitude and in direction, then AD denotes the resultant velocity. Thus if AB denotes the velocity of a ship rela tive to the earth and if AC denotes the velocity with which one is walking relative to the deck of the ship, then AD will represent, both in magnitude and in direction, the velocity of the walker relative to the earth. The combination, or composition, of several velocities is rep resented in a similar way by the polygon in fig. 2. Again, if AB and AC in fig. I denote accelerations, then AD denotes the re sultant acceleration which is equivalent to a combination of the two given accelerations; and this may of course be extended to the case of the polygon of accelerations. These diagrams afford one of the most important means for the investigation of veloci ties and accelerations.
If the piston of an engine is moving back and forth along the line AB in fig. 4 and if the area ABDEC represents the work done on the piston in moving from A to B while the area ABDFC rep resents the work done by the piston against retarding forces on its return stroke, then the area CFDEC will represent the effec tive work which may be accomplished by the piston thrust in a single back and forth motion. The figure by means of which this effective work done by the piston is shown is known as the indi cator diagram of the motion of the piston. It is of fundamental importance in analysing the effective working capacity of the engine.
Other Diagrams.—Brief definitions of various other diagrams will now be given. In an Argand diagram the complex numbers x+yi are represented by corresponding points (x, y) with ref erence to a system of rectangular co-ordinates in a plane. An automatic diagram is one which is constructed automatically by a machine to show the related variations of two variables, as, for instance, the change of temperature with time during the day ; in these cases the completed diagram often consists of a graph drawn automatically upon specially prepared co-ordinate paper.
In many cases of this sort a piece of paper is made to move (uniformly or otherwise) in a given direction, let us say horizon tally, while a tracing pencil point is made to move vertically across it, the height of the tracing point varying proportionately to the magnitude of the quantity whose variation is to be registered. Machines of this sort are employed for the automatic registration of phenomena of many kinds, from those in meteorology and the theory of magnetism and electricity to those connected with the movements of plants and animals.
A strain diagram is a figure which shows the relation between the amount of stress applied by pressure or otherwise to a test piece of material and the strains which it undergoes on account of this stress. It is usually drawn automati cally by means of an instrument attached to the machine and the piece being tested, the deformations being amplified by aid of a suitable mechanism. A stress diagram is a figure in which each joint of a framework is represented by a funicular polygon (such a figure as is formed by a string supported at the ends and acted on by several pres sures), while each link in the frame is represented by a line belong ing to one or more of the funicular polygons ; it is also called a funicular diagram. A variation diagram sets forth the changes in the indicator diagram of an engine for successive strokes of the piston; it is used to determine whether the governor is acting properly. The word diagram is also used in numerous other com binations many of which are self-explanatory.
Diagrams appear in literature mainly as incidental to the sub jects in connection with which they are employed, as has been indicated in the course of the article. Consequently the bibliog raphies are to be found by consulting the articles dealing with these subjects. (R. D. CA.)