SEMANTICS OF THE DIFFERENTIAL CALCULUS 585 

As we plot larger and larger numbers of points closer and closer together, in the limit, if we take indefinitely many such points, we approach a smooth line. It can be proved that an equation of the type given in this example; namely, where both variables are of the first order, always represents a straight line. Such equations are called therefore linear equations, as they represent straight lines.
The problem of linearity and nonlinearity is of extreme importance, and we will return to it later on. Here we are interested only in the definition and meaning of linearity of equations.
Let us consider next a simple equation of second degree, In assign
ing arbitrary values to x, we note that x^{2} is always positive (by the rule of signs) whether * is positive or negative. Hence, we may tabulate values of x *vith the double signmeaning either 





We see for each value of y we have two values for x which differ only in sign This means that we have points on two sides of the Y axis with numerically equal abscissas and, since for a =0, y =0, the beginning of our curve is at the origin y
of coordinates and the curve is symmetrical with respect to the Y axis.
If we connect the points D', C, B', A', 0, A, B, C, D, with straight lines we have a broken line. But if we choose smaller and smaller differences between the successive values of x, the broken line becomes smoother and smoother, and, in the limit, as we take increasingly smaller steps, or, in other words, plot indefinitely larger numbers of points in one interval, we approach a smooth, or continuous curve.
It must be noticed that in equations of higher orders the ratio of changes in
the function y to corresponding changes in the variable x vary from point to point, and so we have a curve instead of a straight line. It is necessary to become quite clear on this point so we may better compare the two different types of equations as to the law of their growth.

