I

Multidimensional time is easier to see if we look at transportation. Consider the time table of a railway or subway system. Directions are typically shown by the station at the end of each line. The time table lists arrivals and departures – events in space and time. A railway station some distance away is in a certain direction both in space and in time within a system. For example, a famous distance-time graph by Etienne-Jules Marey shows the *Paris-Lyons line in the 1885*.

If we know something about the geography of the area, that is likely to be in our minds when reading a railway time table. But if we don’t know much about the geography, travel times provide a way to map the railway system. The “time scale” may even be more useful than the distance scale. Here are two examples of time scale maps for the Boston-area MBTA: *Time-Scale Commuter Rail Map* and *Time-Scale Subway Map*.

Another way to see time directions is simply to take a time-space or isochron map and remove indications of spatial directions and distances. See for example *Travel Times on Commuter Rail*. The point is that time directionality is real. To see it requires separating time directions from geographic directions.

II

A degree of space is an angular distance equal to 1/360th of a circle. An arc minute of space is an angular distance equal to 1/60th of one degree of space. An arc second of space is an angular distance equal to 1/60th of one minute of space.

A minute of time is an angular duration equal to 1/60th of a full rotation or cycle at a rate of one rotation per hour. A second of time is an angular duration equal to 1/60th of a full rotation or cycle at a rate of one rotation per minute.

Angular time is measured by a duration of angular movement. For example, if a motor turns clockwise at a rate of one rotation per hour for five minutes (like a minute hand), then turns at a rate of one rotation per minute for ten seconds (like a second hand), the angular duration of motion will be 5:10 minutes but the angular distance of motion will be 90:00 degrees.

III

A direction is in the context of a geometry. A moving object such as a vehicle has a travel distance and a travel time that are both scalars. The odometer is increasing no matter which direction the vehicle is moving. It is only relative to a space and time beyond the vehicle and its movement that one can speak of its direction.

This direction may be conceived spatially and/or temporally. Directions in space and time will be the same if space and time are proportional, that is, distance and duration are proportional. In that case, we might say either there is no time or there is no space, though either statement would not be not strictly correct. There is always both space and time but they may be equivalent, and one may be hidden behind the other so to speak.