above five minutes when “bang went saxpence.” If he were to spend money at that rate all day long, say for hours, he would be spending shillings an hour, or £. . per day, or £. . a week, not counting the Sawbbath.
Now try to put some of these ideas into differential notation.
Let in this case stand for money, and let stand for time.
If you are spending money, and the amount you spend in a short time be called , the rate of spending it will be , or rather, should be written with a minus sign, as , because is a decrement, not an increment. But money is not a good example for the calculus, because it generally comes and goes by jumps, not by a continuous flow–you may earn £ a year, but it does not keep running in all day long in a thin stream; it comes in only weekly, or monthly, or quarterly, in lumps: and your expenditure also goes out in sudden payments.
A more apt illustration of the idea of a rate is furnished by the speed of a moving body. From London (Euston station) to Liverpool is miles. If a train leaves London at o’clock, and reaches Liverpool at o’clock, you know that, since it has travelled miles in hours, its average rate must have been miles per hour; because . Here you are really making a mental comparison between
