Three linearity patters in the United States between 1609 and 1819 might be industry, population, or slavery.
It is helpful to think of linearity in terms of a graph. A linear relationship between two variables appears on a graph as a straight line. In historical change, one of the variables is time. Time might be represented by the "x," or horizontal, axis. Meanwhile, the "y," or vertical, axis can be any other variable. If the relationship is linear, the graph will show a straight line either ascending or descending. In order to produce a straight line, a particular process would have to demonstrate steady growth or decline.
What phenomena in United States history exhibited steady growth or decline between 1608 and 1819? In looking for areas of growth during this period, one might consider industry, population, slavery, infrastructure, and territory. In looking for areas of decline, one might consider the native population, forests, habitats, and certain animal species.
In order to show that these were linear, one would have to have some reasonable data points and statistics to demonstrate the relationship. To a large degree, the resulting graph will depend on the number and frequency of data points. The goal of the historian is to make it as reflective of reality as possible.