Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/167

Rh He could not have been, as is usually stated, a pupil of Democritus, as he was older than this philosopher, or cer tainly not younger. The circumstances of his death may have been confused with those of Protagoras. The writing in which he disclosed the Mysteries bore the name &amp;lt;pvyioi Aoyot, or a7ro7ru/DyiovTs. These are all the facts which are known of him, and none of his actual opinions are pre served. See Zeller, Geschichte der Griechischen Philosophic.  DIAGRAMS. A diagram is a figure drawn in such a manner that the geometrical relations between the parts of the figure help us to understand relations between other objects. A few have been selected for description in this article on account of their greater geometrical significance. Diagrams may be classed according to the manner in which they are intended to be used, and also according to the kind of analogy which we recognize between the diagram and the thing represented.

Diagrams of Illustration.

The diagrams in mathematical treatises are intended to help the reader to follow the mathematical reasoning. The construction of the figure is defined in words so that even if no figure were drawn the reader could draw one for himself. The diagram is a good one if those features which form the subject of the proposition are clearly represented. The accuracy of the drawing is therefore of smaller import ance than its distinctness.

Metrical Diagrams.

Diagrams are also employed in an entirely different way—namely, for purposes of measurement. The plans and designs drawn by architects and engineers are used to determine the value of certain real magnitudes by measur ing certain distances on the diagram. For such purposes it is essential that the drawing be as accurate as possible. We therefore class diagrams as diagrams of illustration, which merely suggest certain relations to the mind of the spectator, and diagrams drawn to scale, from which measure ments are intended to be made. Methods in which diagrams are used for purposes of measurement are called Graphical methods. Diagrams of illustration, if sufficiently accurate, may be used for purposes of measurement; and diagrams for measurement, if sufficiently clear, may be used for purposes of demonstration. There are some diagrams or schemes, however, in which the form of the parts is of no importance, provided their connections are properly shown. Of this kind are the diagrams of electrical connections, and those belonging to that department of geometry which treats of the degrees of cyclosis, periphraxy, linkedness, and knottedness.

Diagrams purely Graphic and mixed Symbolic and Graphic.

Diagrams may also be classed either as purely graphical diagrams, in which no symbols are employed except letters or other marks to distinguish particular points of the diagrams, and mixed diagrams, in which certain magnitudes are represented, not by the magnitudes of parts of the diagram, but by symbols, such as numbers written on the diagram. Thus in a map the height of places above the level of the sea is often indicated by marking the number of feet above the sea at the corresponding places on the map. There is another method in which a line called a contour line is drawn through all the places in the map whose height above the sea is a certain number of feet, and the number of feet is written at some point or points of this line. By the use of a series of contour lines, the height of a great number of places can be indicated on a map by means of a small number of written symbols. Still this method is not a purely graphical method, but a partly symbolical method of expressing the third dimension of objects on a diagram in two dimensions.

Diagrams in Pairs.

In order to express completely by a purely graphical method the relations of magnitudes involving more than two variables, we must use more than one diagram. Thus in the arts of construction we use plans and elevations and sections through different planes, to specify the form of objects having three dimensions. In such systems of diagrams we have to indicate that a point in one diagram corresponds to a point in another diagram. This is generally done by marking the corre sponding points in the different diagrams with the same letter. If the diagrams are drawn on the same piece of paper we may indicate corresponding points by drawing a line from one to the other, taking care that this line of correspondence is so drawn that it cannot be mistaken for a real line in either diagram. In the stereoscope the two diagrams, by the combined use of which the form of bodies in three dimensions is recognized, are projections of the bodies taken from two points so near each other that, by viewing the two diagrams simultaneously, one with each eye, we identify the corresponding points intuitively. The method in which we simultaneously contemplate two figures, and recognize a correspondence between certain points in the one figure and certain points in the other, is one of the most powerful and fertile methods hitherto known in science. Thus in pure geometry the theories of similar, reciprocal, and inverse figures have led to many extensions of the science. It is sometimes spoken of as the method or principle of Duality.

The study of the motion of a material system is much assisted by the use of a series of diagrams representing the configuration, displacement, and acceleration of the parts of the system.

Diagram of Configuration.

In considering a material system it is often convenient to suppose that we have a record of its position at any given instant in the form of a diagram of configuration. The position of any particle of the system is defined by drawing a straight line or vector from the origin, or point of reference, to the given particle. The position of the particle with respect to the origin is determined by the magnitude and direction of this vector. If in the diagram we draw from the origin (which need not be the same point of space as the origin for the material system) a vector equal and parallel to the vector which de termines the position of the particle, the end of this vector will indicate the position of the particle in the diagram of configuration. If this is done for all the particles, we shall have a system of points in the diagram of configuration, each of which corresponds to a particle of the material system, and the relative positions of any pair of these points will be the same as the relative positions of the material particles which correspond to them. We have hitherto spoken of two origins or points from which the vectors are supposed to be drawn one for the material system, the other for the diagram. These points, however, and the vectors drawn from them, may now be omitted, so that we have on the one hand the material 