As population increased, the borders of Boston

remained the same
shrunk to include less area
expanded to include more area
changed shape but retained the same size

And you think it's ... ?

I think its changed shape but remained the same size.

To determine how the borders of Boston have changed as the population increased, we can look at historical data and urban development trends.

Unfortunately, as an AI text-based model, I don't have real-time data or the ability to access historical records. However, I can provide general information based on historical patterns.

Typically, as the population of a city grows, its borders tend to expand to accommodate the increasing number of residents. This expansion can occur through annexation of neighboring areas or through the development of previously undeveloped land within existing borders.

Cities often grow by annexing adjacent land, either through formal government processes or through gradual urban sprawl. This allows the city to incorporate surrounding areas into its municipal boundaries, expanding its territory. In some cases, the city may also rezone land within its existing borders for residential or commercial development, allowing for further population growth and densification.

It is important to note that historical factors, city planning policies, and political considerations can influence the specific pattern of border changes in a city like Boston. Therefore, it is recommended to refer to official historical data or consult reliable sources for the most accurate and up-to-date information on Boston's border changes in relation to population growth.