The philosophy of biology . when we look at thegraph we see that the rate of variation is greatest wherethe slope of the curve is steepest. The latter is steepestnear the point a, less steep near the point b, and stillless steep near the point c. Now any small part ofthe curve is indistinguishable from a straight line. Letus drawastraightlineeCi, which appears tocoincide with a smallpart of the curve neara, and similarstraightlines^i,andggi, whichalso appear to coin-cide with small partsof the curve near band c. Then thesteepness of the curvewill be proportionalto the angles whichthese straigh
The philosophy of biology . when we look at thegraph we see that the rate of variation is greatest wherethe slope of the curve is steepest. The latter is steepestnear the point a, less steep near the point b, and stillless steep near the point c. Now any small part ofthe curve is indistinguishable from a straight line. Letus drawastraightlineeCi, which appears tocoincide with a smallpart of the curve neara, and similarstraightlines^i,andggi, whichalso appear to coin-cide with small partsof the curve near band c. Then thesteepness of the curvewill be proportionalto the angles whichthese straight linesmake with the axis op, and these angles are measured by the ratio — which ?^ oe e^e makes with op, the ratio ^, and the ratio ^\0/ og The point a on the curve corresponds with apressure Ui and a volume an. The point b corre-sponds with a pressure 61 and a volume b^, andc with a pressure <h. and a volume Cu. The averagerate of variation of the volume of the gas, asthe pressure changes from a to c, is therefore. Fig. 28. by their tangents, that is,is the tangent that 346 THE PHILOSOPHY OF BIOLOGY proportional to the sum of the tangents —^ and ^ oe og divided by 2. THE NOTION OF THE LIMIT Suppose that we wish to find the rate of variationof volume for a pressure change in the immediatevicinity of the value bi, that is, the rate of variation asthe pressure changes from a little less than bi to a littlemore than bi. If we find the point b on the curvecorresponding to b}, and if we then draw a line J\,touching the curve at the point b, we shall obtain theangle oJ\. It might appear now that the tangent of this angle, that is, the ratio ^, would give us a measure of the rate of variation of volume. But the reasoning would be faulty. The line J^ionly touches the curve, it does not coincide with anelement of the curve. Also at the point bi the pressurehas a certain definite value, and there is no change. Atthe corresponding point bn the volume also has acertain definite val
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