equivalent to the work performed; and whenever mechanical work is spent in generating heat” [as in rubbing two sticks together], “the heat generated is equivalent to the work thus spent.”  And an experiment of Joule’s is described, which consisted in fixing a rod with paddles in a vessel of water, and making it revolve and agitate the water by means of a string wound round the rod, passed over a pulley and attached to a weight that was allowed to fall.  The descent of the weight was measured by a graduated rule, and the rise of the water’s temperature by a thermometer.  “It was found that the heat communicated to the water by the agitation amounted to one pound-degree Fahrenheit for every 772 foot-pounds of work” expended by the falling weight.  As no other material change seems to take place during such an experiment, it shows that the progressive expenditure of mechanical energy is the cause of the progressive heating of the water.

The thermometer itself illustrates this method.  It has been found that the application of heat to mercury expands it according to a law; and hence the volume of the mercury, measured by a graduated index, is used to indicate the temperature of the air, water, animal body, etc., in which the thermometer is immersed, or with which it is brought into contact.  In such cases, if no other change has taken place, the heat of the air, water, or body is the cause of the rise of the mercury in its tube.  If some other substance (say spirit) be substituted for mercury in constructing a thermometer, it serves the same purpose, provided the index be graduated according to the law of the expansion of that substance by heat, as experimentally determined.

Instances of phenomena that do not vary together indicate the exclusion of a supposed cause (by Prop.  III (c)).  The stature of the human race has been supposed to depend on temperature; but there is no correspondence.  The “not varying together,” however, must not be confused with “varying inversely,” which when regular indicates a true concomitance.  It is often a matter of convenience whether we regard concomitant phenomena as varying directly or inversely.  It is usual to say—­’the greater the friction the less the speed’; but it is really more intelligible to say—­’the greater the friction the more rapidly molar is converted into molecular motion.’

The Graphic Method exhibits Concomitant Variations to the eye, and is extensively used in physical and statistical inquiries.  Along a horizontal line (the abscissa) is measured one of the conditions (or agents) with which the inquiry is concerned, called the Variable; and along perpendiculars (ordinates) is measured some phenomenon to be compared with it, called the Variant.

Thus, the expansion of a liquid by heat may be represented by measuring degrees of temperature along the horizontal, and the expansion of a column of the liquids in units of length along the perpendicular.