Order of magnitude calculations help us get a sense of scale of various measures of the environment. However, they are generally a poor way to look at how the environment changes through time. To improve our understanding of change, we need to delve into a few relationships from physics. First, though, let's talk about time series.

Time series: A plot that shows how a variable changes through time. The y-axis is the variable, whereas the x-axis is time. These types of plots are the backbone for any disciplines that look at the past history and/or future prediction of a variable. So, you can find these types of plots in geology, economics, ecology, history, archaeology, and political science, just to name a few.

Plots of rates through time: Any time you have a plot in which the y-axis variable is a rate (e.g., kg/yr, m/s, etc.) and the x-axis is time, you can calculate the total amount of your variable (e.g., kg, m, etc.) by measuring the area under the plotted curve. Formally, this is called integration. For example, the area under the curves in figure 4 of Hooke (2000) gives the total mass of earth moved by human activities since 5000 years ago.

Plots of amounts through time: Any time you have a plot in which the y-axis variable is an amount (e.g., kg, m, etc.) and the x-axis is time, you can calculate the rate at which your variable changed (e.g., kg/yr, m/s, etc.) by measuring the slope of the plotted curve. Formally, this is called differentiation. For example, the slope of the curve showing the amount of daylight through the year gives the change in daylight through time. Note that this is not a single value through the year, but instead changes from positive slopes (an increase in daylight from the winter to summer solstice) to negative slopes (an decrease in daylight from the summer to winter solstice).

Important note: Pay attention to the units that are plotted on the plot axes. You may need to do some unit conversions to get an appropriate answer. Fundamentally, you are multiplying the units of the two axes when you integrate. When you differentiate, you are dividing the units of the y-axis by the x-axis units.