Vii. The differential calculus

No elementary school child gets a chance of learning the differential calculus, and very few secondary school children do so. Yet I know from my own experience that children of twelve can learn it. As it is a mathematical tool used in most branches of science, this form a bar between the workers and many kinds of scientific knowledge. I have no intention of teaching the calcu­lus, but it is quite easy to explain what it is about, particularly to skilled work­ers. For a very large number of skilled workers use it in practice without knowing that they are doing so.

The differential calculus is concerned with rates of change. In practical life we constantly come across pairs of quantities which are related, so that af­ter both have been measured, when we know one, we know the other. Thus if we know the distance along the road from a fixed point we can find the height above sea level from a map with contour.If we know a time of day we can determine the air temperature on any particular day from a record of a ther­mometer made on that day. In such cases we often want to know the rate of change of one relative to the other.

If x and y are the two quantities then the rate of change of y relative to x is written dy/dx. For example if x is the distance of a point on a railway from London, measured in feet, and y the height above sea level, dy/dx is the gradi­ent of the railway. If the height increases by 1 foot while the distance x in­creases by 172 feet, the average value of dy/dx is 1/172. We say that the gradi­ent is 1 to 172. If x is the time measured in hours and fractions of an hour, and y the number of miles gone, then dy/dx is the speed in miles per hour. Of course, the rate of change may be zero, as on level road, and negative when the height is diminishing as the distance x increases. To take two more examples, if x the temperature, and y the length of a metal bar, dy/dx—:—y is the coeffi­cient of expansion, that is to say the proportionate increase in length per de­gree. And if x is the price of commodity, and y the amount bought per day, then -dyldx is called the elasticity of demand.

For example people must buy bread, but cut down on jam, so the de­mand for jam is more elastic than that for bread. This notion of elasticity is very important in the academic economics taught in our universities. Professors say that Marxism is out of date because Marx did not calculate such things. This would be a serious criticism if the economic "laws" of 1900 were eternal truths. Of course Marx saw that they were nothing of the kind and "elasticity of demand" is out of date in England today for the very good reason that most commodities are controlled or rationed.

The mathematical part of the calculus is the art of calculating dy/dx if y has some mathematical relations to x, for example is equal to its square or loga­rithm. The rules have to be learned like those for the area and volume of geo­metrical figures and have the same sort of value. No area is absolutely square, and no volume is absolutely cylindrical. But there are things in real life like enough to squares and cylinders to make the rules about them worth learning.So with the calculus. It is not exactly true that the speed of a falling body is proportional to the time it has been falling. But there is close enough to the truth for many purposes.

The differential calculus goes a lot further.Think of a bus going up a hill which gradually gets steeper. If x is the horizontal distance, and y the height, this means that the slope dy/dx is increasing. The rate of change of dy/dx with regard to y is written d^2y/dx². In this case it gives a measure of the curvature of the road surface. In the same way if x is time and distance, d^2y /dx² is the rate of change of speed with time, or acceleration. This is a quantity which good drivers can estimate pretty well, though they do not know they are using the basic ideas of the differential calculus.

If one quantity depends on several others, the differential calculus shows us how to measure this dependence. Thus the pressure of a gas varies with the temperature and the volume. Both temperature and volume vary dur­ing the stroke of a cylinder of a steam or petrol engine, and the calculus is needed for accurate theory of their action.

Finally, the calculus is a fascinating study for its own sake. In February 1917 I was one of a row wounded officers lying on stretchers on a steamer going down the river Tigris in Mesopotamia. I was reading a mathematical book on vectors, the man next to me was reading one on the calculus. As anti­dotes to pain we preferred them to novels. Some parts of mathematics are beautiful, like good verse or painting. The calculus is beautiful, but not because it is a product of "pure thought". It was invented as a tool to help men to cal­culate the movement of stars and cannon balls. It has the beauty of really efficient machine.