[Illustration:  FIG. 9.]

In the next diagram (Fig. 10), reduced from one given by Mr. C.H.  Denyer in an article on the Price of Tea (Economic Journal, No. 9), the condition measured horizontally is Time; and, vertically, three variants are measured simultaneously, so that their relations to one another from time to time may be seen at a glance.  From this it is evident that, as the duty on tea falls, the price of tea falls, whilst the consumption of tea rises; and, in spite of some irregularity of correspondence in the courses of the three phenomena, their general causal connection can hardly be mistaken.  However, the causal connection may also be inferred by general reasoning; the statistical Induction can be confirmed by a Deduction; thus illustrating the combined method of proof to be discussed in the next chapter.  Without such confirmation the proof by Concomitant Variations would not be complete; because, from the complexity of the circumstances, social statistics can only yield evidence according to the method of Agreement in Variations.  For, besides the agents that are measured, there may always be some other important influence at work.  During the last fifty years, for example, crime has decreased whilst education has increased:  true, but at the same time wages have risen and many other things have happened.

[Illustration:  FIG. 10.]

It will be noticed that in the diagram the three lines, especially those of Price and Consumption (which may be considered natural resultants, in contrast with the arbitrary fixation of a Tax), do not depart widely from regular curves; and accordingly, assuming the causes at work to vary continuously during the intervals between points of measurement, curves may be substituted.  In fact, a curve often represents the course of a phenomenon more truthfully than can be done by a line that zigzags along the exact measurements; because it is less influenced by temporary and extraordinary causes that may obscure the operation of those that are being investigated.  On the other hand, the abrupt deviations of a punctilious zigzag may have their own logical value, as will appear in the next section.

In working with the Method of Variations one must allow for the occurrence in a series of ‘critical points,’ at which sudden and sometimes heterogeneous changes may take place.  Every substance exists at different temperatures in three states, gaseous, liquid, solid; and when the change takes place, from one state to another, the series of variations is broken.  Water, e.g., follows the general law that cooling is accompanied by decrease of volume between 212 deg. and 39 deg.  F.:  but above 212 deg., undergoes a sudden expansion in becoming a gas; and below 39 deg. begins to expand, until at 32 deg. the expansion is considerable on its becoming solid.  This illustrates a common experience that concomitant variations are most regular in the ‘median range,’ and are apt to become irregular at the extremities of the series, where new conditions begin to operate.