If we know the equation _pv_ = _k_(_1_ + _at_), we can find how much the volume changes when the pressure changes, that is, the rate of variation of _v_ with respect to _p_. But even if we don’t know that this equation applies, we can still find the rate of variation from our experiments. We see from the graph that, when the pressure increases from _p_↓{1} to _p_↓{2}, the volume decreases from _v_↓{1} to _v_↓{2} but that if the pressure is again increased to _p_↓{3}, that is, by a similar amount to the increase of pressure from _p_↓{1} to _p_↓{2}, the volume decreases from _v_↓{2} to _v_↓{3}. Now we find, by measurements made on the graph, that the decrease _v_↓{1} to _v_↓{2} is greater than the decrease _v_↓{2} to _v_↓{3}, and the latter decrease is greater again than the decrease from _v_↓{3} to _v_↓{4}. Evidently the rate of variation of volume is not like the rate of variation of pressure, that is, the same throughout, and when we look at the graph we see that the rate of variation is greatest where the slope of the curve is steepest. The latter is steepest near the point _a_, less steep near the point _b_, and still less steep near the point _c_. Now any _small_ part of the curve is indistinguishable from a straight line. Let us draw a straight line _ee_↓{1}, which appears to coincide with a small part of the curve near _a_, and similar straight lines _ff_↓{1}, and _gg_↓{1}, which also appear to coincide with small parts of the curve near _b_ and _c_. Then the steepness of the curve will be proportional to the angles which these straight lines make with the axis _op_, and these angles are measured by their tangents, that is, by the ratio _oe_↓{1}/_oe_, which is the tangent that _e_↓{1}_e_ makes with _op_, the ratio _of_↓{1}/_of_, and the ratio _og_↓{1}/_og_. [Illustration: FIG. 28.] The point _a_ on the curve corresponds with a pressure _a_↓{1} and a volume _a_↓{11}. The point _b_ corresponds with a pressure _b_↓{1} and a volume _b_↓{11}, and _c_ with a pressure _c_↓{1} and a volume _c_↓{11}. The _average_ rate of variation of the volume of the gas, as the pressure changes from _a_ to _c_, is therefore proportional to the sum of the tangents _oe_↓{1}/_oe_ and _og_↓{1}/_og_, divided by 2. THE NOTION OF THE LIMIT Suppose that we wish to find the rate of variation of volume for a pressure change in the immediate vicinity of the value _b_↓{1}, that is, the rate of variation as the pressure changes from a little less than _b_↓{1} to a little more than _b_↓{1}. If we find the point _b_ on the curve corresponding to _b_↓{1}, and if we then draw a line _ff_↓{1}, _touching_ the curve at the point _b_, we shall obtain the angle _off_↓{1}. It might appear now that the tangent of this angle, that is, the ratio _of_↓{1}/_of_, would give us a measure of the rate of variation of volume.