586 VIII. ON THE STRUCTURE OF MATHEMATICS 

Let us write down in two columns the successive values for the two types of equations. Let us take the equationwith the graph shown in the
preceding diagram (Fig. 4) and the equation y = 2x as shown in Fig. 5. 







The equation y=2x involves the variables in the first degree and we see that the ratio of changes in the ordinates to corresponding changes in the abscissas remains constant (proportional). The triangles in Fig. 5, are either equal or similar, which necessitates the equality of angles and so the line OABCD is of necessity a straight line. In this case asgave usthe line passes
through the origin of coordinates.
The picture is entirely different in the case of the higher degree equation,
illustrated in Fig. 4. From the table of values of the function we see
that the value of the function increases increasingly more rapidly than the values of the independent variable and so the ordinates are not proportional to the abscissas. If in Fig. 4 we connect 0 with A, 0 with B, 0 with C, O with D, respectively, we see that the lines OA, OB, OC, and OD have different angles with the axis X'X; the respective triangles are not similar, and so there is no proportionality. The lines OA, OB, OC, OD . , do not represent a straight line as they have all different angles with the axis XX' and so the points A, B, C, D . , cannot lie on a straight line but'represent a broken line which, in the limit, when the points plotted become sufficiently near together, becomes a smooth and continuous curve.
The fact that equations in which the variables are only of the first degree, represent straight lines, and that equations of higher degrees represent curved lines is very important, as will appear later on. We must notice also that the problem of linearity is connected with proportionality.
These few simple notions concerning the use of coordinates will allow us to explain the geometrical meaning of the derivative and the differential. 
